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Kjš gÂ¥ò - 2011

Ïªj ò¤jf« gŸë¡ fšé¤ Jiw

têfh£Ljè‹go jahç¡f¥g£lJ.

bt¥ M¥br£ Kiwæš m¢Á£nlh® :

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fâå j£l¢R, tiufiy k‰W« m£ilgl« totik¥ò: î› Mdª

FG¤jiyt®

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Kidt® g. kâ fšÿç¢ rhiy,

gŸë¡ fšé Ïa¡Fe® br‹id-600 006.

thœ¤J¢ brŒÂ

muR bghJ¤ nj®éš fyªJ bfhŸS« 10M« tF¥ò khzt®fS¡F« k‰W«

f‰Ã¡F« MÁça®fS¡F« cjéL« tifæš, édh t§» cŸë£l ò¤jf§fis

cUth¡» btëæL« e‰gâia jäœehL bg‰nwh® MÁça® fHf« Áw¥ghf¢ brŒJ

tU»wJ.

10M« tF¥ò fâj ghlüyhÁça® FGédiu¡ bfh©nl, ‘SCORE BOOK’ v‹w

üš jahç¡f¥g£LŸsJ. bghJ¤ nj®éš fyªJ bfhŸS« 10M« tF¥ò khzt®fŸ

j§fŸ f‰wš miléid jh§fshfnt kÂ¥Ãl cjéL« tifæš ò¤jf¤ÂYŸs

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khzt®fŸ Ï¥ò¤jf¤Âid¡ T®ªJ go¤J, fâj m¿éid¥ bgU¡»¡

bfhŸtnjhL, bghJ¤ nj®éid e«Ã¡ifÍl‹ vÂ®bfhŸs KoÍ« vd e«ò»nw‹.

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bt‰¿ bgw ešthœ¤JfŸ.

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Namma Kalvi - The No.1 Educational Websitefor 9th, 10th, 11th, 12th,

TRB TET & TNPSC Materials

(v)

10M« tF¥ò fz¡F¥ ghlüèYŸs gæ‰Á édh¡fS¡F¤ Ô®ÎfŸ,

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eh§fŸ Ka‰Á brŒJŸnsh«.

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Ô®ÎfŸ

1. fz§fS« rh®òfS« ...... 1

2. bkŒba©fë‹ bjhl®tçirfS« bjhl®fS« ...... 23

3. Ïa‰fâj« ...... 57

4. mâfŸ ...... 121

5. Ma¤bjhiy toéaš ...... 141

6. toéaš ...... 171

7. K¡nfhzéaš ...... 190

8. mséaš ...... 211

9. brŒKiw toéaš ...... 234

10. tiugl§fŸ ...... 255

11. òŸëæaš ...... 273

12. ãfœjfÎ ...... 285

tif¥gL¤j¥g£l édh¡fŸ ...... 302

jahç¡f¥g£l édh¡fŸ ...... 379

khÂç édh¤jhŸfŸ ...... 404

muR khÂç édh¤jhŸ - kÂ¥ÕL Kiw ...... 429


Ô®Î - fz§fS« rh®òfS« 1

1. SETS AND FUNCTIONS

gæ‰Á 1.1 1. A B1 våš, A B B, = vd¡ fh£Lf (bt‹gl¤ij¥ ga‹gL¤jÎ«).

Ô®Î: A B B, =

2. A B1 våš, A B+ k‰W« \A B M»at‰iw¡ fh©f. (bt‹gl¤ij¥ ga‹gL¤Jf).Ô®Î: A B A+ = \A B z=

3. { , , }, { , , , } { , , , }k‰W«P a b c Q g h x y R a e f s= = = . våš, Ã‹tUtdt‰iw¡

fh©f:

(i) \P R (ii) Q R+ (iii) \R P Q+^ h

Ô®Î: (i) P = { , , }, { , , , }a b c Q g h x y= k‰W« { , , , }R a e f s= vd¡ bfhL¡f¥g£LŸsJ.

Ï¥bghGJ, \P R = { ; }t P t Rg! , P -š cŸs, R-š Ïšyhj cW¥òfŸ.

= { , }b c . (ii) Q R+ .

Q R+ v‹gJ Q k‰w« R M»a Ïu©L fz§fëY« cŸs cW¥òfŸ.vdnt, ,Q R+ Q= bt‰Wfz«.

(iii) \R P Q+^ h. Ï§F, P Q+ v‹gJ bt‰Wfz«. vdnt, \( ) { , , , } .R P Q R a e f s+ = =

4. { , , , , }, { , , } { , , , , , }k‰W«A B C4 6 7 8 9 2 4 6 1 2 3 4 5 6= = = ,

våš, Ã‹tUtdt‰iw¡ fh©f.

(i) A B C, +^ h, (ii) A B C+ ,^ h, (iii) \ \A C B^ h.Ô®Î: (i) B C+ = {2, 4, 6} + {1, 2, 3, 4, 5, 6} = {2, 4, 6}. vdnt, ( )A B C, + = {4, 6, 7, 8, 9} , {2, 4, 6} = {2, 4, 6, 7, 8, 9}.(ii) B C, = {2, 4, 6} , {1, 2, 3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}. vdnt, ( )A B C+ , = {4, 6, 7, 8, 9} + {1, 2, 3, 4, 5, 6} = {4, 6}(iii) \C B = {1, 2, 3, 4, 5, 6} \ {2, 4, 6} = {1, 3, 5} vdnt, \ ( \ )A C B = {4, 6, 7, 8, 9} \ {1, 3, 5} = {4, 6, 7, 8, 9}.

fz§fS« rh®òfS« 1www.nammakalvi.weebly.com

10-M« tF¥ò fz¡F - SCORE ò¤jf«2

5. { , , , , }, { , , , , }A a x y r s B 1 3 5 7 10= = - , vd¡ bfhL¡f¥g£LŸs fz§fS¡F,

fz§fë‹ nr®¥ò brayhdJ, gçkh‰W¥ g©ò cilaJ v‹gij rçgh®¡fÎ«.

Ô®Î: { , , , , }, {1,3,5,7, 10}A a x y r s B= = - vd bfhL¡f¥g£LŸsJ.

A B B A, ,= v‹gij rçgh®¥ngh«.Ï¥bghGJ, A B, = { , , , , } {1,3,5,7, 10}.a x y r s , -

= { , , , , , , , , , }a x y r s 1 3 5 7 10- g (1) B A, = {1,3,5,7, 10} { , , , , }a x y r s,-

= { , , , , , , , , , }a x y r s 1 3 5 7 10- g (2)(1) k‰W« (2) èUªJ, A B B A, ,= MF«. vdnt, fz§fë‹ nr®¥ò gçkh‰W¥g©ò cilaJ.

6. { , , , , , , , } { , , , , , , , }k‰W«A l m n o B m n o p2 3 4 7 2 5 3 2= = - M » a t ‰ ¿ ‰ F

fz§fë‹ bt£L, gçkh‰W¥ g©ò cilaJ v‹gij rçgh®¡fÎ«.

Ô®Î: A B B A+ += v‹gij rçgh®¥ngh«.Ï¥bghGJ, A B+ = { , , , , , , , } { , , , , , , , }l m n o m n o p2 3 4 7 2 5 3 2+ -

= { , , , , }m n o2 3 g (1) B A+ = { , , , , , , , } { , , , , , , , }m n o p l m n o2 5 3 2 2 3 4 7+-

= { , , , , }m n o2 3 g (2)(1) k‰W« (2) èUªJ, A B B A+ += MF«. vdnt, fz§fë‹ bt£L, gçkh‰W¥g©ò cilaJ.

7. A={ 42v‹gJx x; -‹ gfh¡ fhuâ}, { , }B x x x5 12 N1; # != k‰W«

{ , , , }C 1 4 5 6= ,våš, A B C A B C, , , ,=^ ^h h v‹gij rçgh®¡fÎ«.

Ô®Î: {2,3,7}, 6,7,8,9,10,11,12 {1,4,5,6}A B C= = =" ,

( ) ( )A B C A B C, , , ,= v‹gij rçgh®¥ngh«. Ï¥bghGJ, B C, = {6, 7, 8, 9, 10, 11, 12} , {1, 4, 5, 6} = {1, 4, 5, 6, 7, 8, 9, 10, 11, 12} vdnt, ( )A B C, , = {2, 3, 7} , {1, 4, 5, 6, 7, 8, 9, 10, 11, 12} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} g (1) Ï¥bghGJ, A B, = {2, 3, 7} , {6, 7, 8, 9, 10, 11, 12} = {2, 3, 6, 7, 8, 9, 10, 11, 12} vdnt, ( )A B C, , = {2, 3, 6, 7, 8, 9, 10, 11, 12} , {1, 4, 5, 6} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} g (2)(1) k‰W« (2) èUªJ, ( ) ( )A B C A B C, , , ,= .(F¿¥ò: fz§fë‹ nr®¥ò nr®¥ò¥g©ò cilaJ)

8. { , , , , }, { , , , , } { , , , }k‰W«P a b c d e Q a e i o u R a c e g= = = .M»a fz§fë‹ bt£L,

nr®¥ò¥ g©ò cilaJ v‹gij rçgh®¡fÎ«.Ô®Î: { , , , , }, { , , , , }P a b c d e Q a e i o u= = k‰W« { , , , }R a c e g= vd bfhL¡f¥

g£LŸsJ.

( ) ( )P Q R P Q R+ + + += v‹w bt£L¡fhd nr®¥ò¥ g©ig rçgh®¥ngh«.

P Q+ = { , , , , } { , , , , } { , }a b c d e a e i o u a e+ =


vdnt, ( )P Q R+ + = { , } { , , , } { , }a e a c e g a e+ = g (1) Q R+ = { , , , , } { , , , } { , }a e i o u a c e g a e+ =

vdnt, ( )P Q R+ + = { , , , , } { , } { , }a b c d e a e a e+ = g (2)(1) k‰W« (2) èUªJ ( ) ( )P Q R P Q R+ + + += .vdnt, fz§fë‹ bt£L nr®¥ò¥ g©ò cilaJ.

9. { , , , }; { , , , , } { , , , , , }k‰W«A B C5 10 15 20 6 10 12 18 24 7 10 12 14 21 28= = = , M»a fz§fS¡F \ \ \ \A B C A B C=^ ^h h v‹gJ bkŒahFkh vd MuhŒf. c‹

éil¡F j¡f fhuz« TWf.

Ô®Î:{ , , , }, { , , , , }A B5 10 15 20 6 10 12 18 24= = k‰W« { , , , , , }C 7 10 12 14 21 28= vd

bfhL¡f¥g£LŸsJ.

\B C = { , , , , }\{ , , , , , }6 10 12 18 24 7 10 12 14 21 28

= {6, 18, 24}vdnt, \ ( \ )A B C = {5, 10, 15, 20} \ {6, 18,24} = {5, 10, 15, 20} g (1) \A B = { , , , }\{ , , , , } { , , }5 10 15 20 6 10 12 18 24 5 15 20=

vdnt, ( \ ) \A B C = {5, 15, 20}\{7,10,12,14,21,28} = {5, 15, 20} g (2)(1) k‰W« (2) èUªJ, \ ( \ ) ( \ ) \A B C A B C! .

10. { , , , }, { , , }, { , , }k‰W«A B C5 3 2 1 2 1 0 6 4 2= - - - - = - - = - - - v ‹ f . \ \ ( \ ) \k‰W«A B C A B C^ h . M»at‰iw¡ fh©f. ÏÂèUªJ »il¡F« fz

é¤Âahr¢ brašgh£o‹ g©Ãid¡ TWf?Ô®Î: { 5, 3, 2, 1}, { 2, 1,0}A B= - - - - = - - k‰W« { , , }C 6 4 2= - - - vd

bfhL¡f¥g£LŸsJ. \B C = { , , }\{ , } { , }2 1 0 6 4 2 1 0- - - - - = -

vdnt, \( \ )A B C = { , , , }\{ , } { , , }5 3 2 1 1 0 5 3 2- - - - - = - - - g (1) \A B = { , , , }\{ , , } { , }5 3 2 1 2 1 0 5 3- - - - - - = - -

vdnt, ( \ )\A B C = { , }\{ , , } { , }5 3 6 4 2 5 3- - - - - = - - g (2)(1) k‰W« (2) èUªJ, \( \ ) ( \ )\A B C A B C! . vdnt, fz§fë‹ é¤Âahr« nr®¥ò¥ g©ò cilajšy.

11. { , , , , , , }, { , , , , , }A B3 1 0 4 6 8 10 1 2 3 4 5 6= - - = - - k‰W«

{ , , , , , },C 1 2 3 4 5 7= - M»at‰¿‰F Ã‹tUtdt‰iw rçgh®¡fÎ«. (i) A B C, +^ h= A B A C, + ,^ ^h h (ii) A B C+ ,^ h= A B A C+ , +^ ^h hbt‹gl§fis¥ ga‹gL¤Â rçgh®¡fÎ«

(iii) A B C, +^ h= A B A C, + ,^ ^h h (iv) A B C+ ,^ h= A B A C+ , +^ ^h h

Ô®Î: { , , , , , , }A 3 1 0 4 6 8 10= - - , { , , , , , }B 1 2 3 4 5 6= - - k‰W«

{ , , , , , }C 1 2 3 4 5 7= - vd bfhL¡f¥g£LŸsJ.

(i) B C+ = { 1, 2,3,4,5,6} { 1,2,3,4,5,7}+- - -

= { 1,3,4,5}- g(1)


vdnt, ( )A B C, + = { , , , , , , } { , , , }3 1 0 4 6 8 10 1 3 4 5+- - -

= { , , , , , , , , }3 1 0 3 4 5 6 8 10- -

A B, = { , , , , , , } { , , , , , }3 1 0 4 6 8 10 1 2 3 4 5 6,- - - -

= { , , , , , , , , , }3 2 1 0 3 4 5 6 8 10- - -

A C, = { , , , , , , } { , , , , , }3 1 0 4 6 8 10 1 2 3 4 5 7,- - -

= { , , , , , , , , , , }3 1 0 2 3 4 5 6 7 8 10- -

( ) ( )A B A C, + , ={ 3, 2, 1,0,3,4,5,6,8, } { 3, 1,0,2,3,4,5,6,7,8,10}10 +- - - - -

= { , , , , , , , , }3 1 0 3 4 5 6 8 10- - g (2)(1) k‰W« (2) èUªJ , ( ) ( ) ( )A B C A B A C, + , + ,= .

(ii) B C, = { , , , , , } { , , , , , }1 2 3 4 5 6 1 2 3 4 5 7,- - -

= { 2, 1,2,3,4,5,6,7}- -

vdnt, ( )A B C+ , = { , , , , , , } { , , , , , , , }3 1 0 4 6 8 10 2 1 2 3 4 5 6 7+- - - -

= { , , }1 4 6- g(1) A B+ = { , , , , , , } { , , , , , }3 1 0 4 6 8 10 1 2 3 4 5 6+- - - -

= { , , }1 4 6-

A C+ = { 3, 1,0,4,6,8,10} { 1,2,3,4,5, }7+- - -

= { , }1 4-

vdnt, ( ) ( )A B A C+ , + = { , , } { , } { , , }1 4 6 1 4 1 4 6,- - = - g(2)(1) k‰W« (2) èUªJ , ( ) ( ) ( )A B C A B A C+ , + , += .

(iii) ( ) ( ) ( )A B C A B A C, + , + ,= -‹ bt‹gl«.

(2) k‰W« (5) èUªJ , ( ) ( ) ( )A B C A B A C, + , + ,= .


(iv) ( ) ( ) ( )A B C A B A C+ , + , += -‹ bt‹gl«.

(2) k‰W« (5) èUªJ , ( ) ( ) ( )A B C A B A C+ , + , += .

gæ‰Á 1.2

1. ÑnH bfhL¡f¥g£LŸs fz§fis, bt‹gl§fë‹ _y« fh£Lf.

(i) ,{ , , , , }, { , , , } { , , , , }k‰W«U A B5 6 7 8 13 5 8 10 11 5 6 7 9 10g= = =

(ii) { , , , , , , , }, { , , , } { , , , , }k‰W«U a b c d e f g h M b d f g N a b d e g= = =

Ô®Î: (i) (ii)

2. ãHè£L¡ (nfho£L) fh£l¥g£LŸs x›bthU gFÂiaÍ« F¿p£L Kiwæš vGJf.

U, , , , ,A B C , + , l k‰W« \ M»a F¿pLfis njitahd Ïl§fëš ga‹gL¤Jf.


3. , k‰W«A B C M»a _‹W fz§fS¡F Ã‹ tUtdt‰iw és¡F« bt‹gl§fŸ

tiuf. (i) A B C+ + (ii) k‰W«A B v‹gd C-æ‹ c£fz§fŸ. nkY« mitfŸ bt£lh¡

fz§fŸ. (iii) \A B C+^ h(iv) \B C A,^ h (v) A B C, +^ h (vi) \C B A+^ h

(vii) C B A+ ,^ h.Ô®Î:

4. bt‹gl§fis¥ ga‹gL¤Â rçgh®¡fÎ« \A B A B A+ , =^ ^h h .Ô®Î:


(1) k‰W« (5) èUªJ , ( ) ( \ )A B A B A+ , = .

5. {4,8,12,16,20,24,28}U = , {8,16,24}A = k‰W« {4,16,20,28}B = . våš, k‰W«A B A B, +l l^ ^h h M»at‰iw¡ fh©f.Ô®Î: { , , , , , , }, { , , }U A4 8 12 16 20 24 28 8 16 24= = k‰W« { , , , }B 4 16 20 28= vd

bfhL¡f¥g£LŸsJ. A B, = { , , } { , , , } { , , , , , }8 16 24 4 16 20 28 4 8 16 20 24 28, =

vdnt, ( )A B, l = \ ( ) {4,8,12,16,20,24,28} \ {4,8,16,20,24,28} {12}U A B, = =

A B+ = {8,16,24} {4,16,20,28} { }16+ =

vdnt, ( )A B+ l = \( ) {4,8,12,16,20,24,28} \ {16} {4,8,12,20,24,28}U A B+ = =

6. { , , , , , , , }U a b c d e f g h= , { , , , } { , , },k‰W«A a b f g B a b c= =

våš, o kh®få‹ fz ãu¥Ã éÂfis¢ rçgh®¡fÎ«.Ô®Î: fz ãu¥Ã¡fhd o kh®få‹ éÂfŸ

( )A B A B(i) , +=l l l ( )A B A B(ii) + ,=l l l

{ , , , , , , , }, { , , , ,}U a b c d e f g h A a b f g= = k‰W« { , , }B a b c= .(i) A B, = { , , , } { , , } { , , , , }a b f g a b c a b c f g, =

( )A B, l = \ { , , , , , , , }\{ , , , , } { , , }U A B a b c d e f g h a b c f g d e h, = = g (1) Al = \ { , , , , , , , }\{ , , , } { , , , }U A a b c d e f g h a b f g c d e h= =

Bl = \ { , , , , , , , }\{ , , } { , , , , }U B a b c d e f g h a b c d e f g h= =

A B+l l = { , , , } { , , , , } { , , }c d e h d e f g h d e h+ = g (2)(1) k‰W« (2) èUªJ , ( )A B A B, +=l l l.(ii) A B+ = { , , , } { , , } { , }a b f g a b c a b+ =

( )A B+ l = \ { , , , , , , , } \ { , } { , , , , , }U A B a b c d e f g h a b c d e f g h+ = = g (3) Al = \ { , , , }; \ { , , , , }U A c d e h B U B d e f g h= = =l

A B,l l = { , , , } { , , , , } { , , , , , }c d e h d e f g h c d e f g h, = g (4)(1) k‰W« (2) èUªJ , ( )A B A B+ ,=l l l.

7. Ã‹tU« fz§fis¡ bfh©L o kh®få‹ fz é¤Âahr éÂfis¢ rçgh®¡fÎ«.

{ , , , , , , , }, { , , , } { , , , , }k‰W«A B C1 3 5 7 9 11 13 15 1 2 5 7 3 9 10 12 13= = = .Ô®Î: fz ãu¥Ã¡fhd o kh®få‹ éÂfŸ

\( ) ( \ ) ( \ )( ) A B C A B A Ci , += , \( ) ( \ ) ( \ )( ) A B C A B A Cii + ,=

A = 1,3,5,7,9,11,13,15 , {1,2,5,7}B =" , k‰W« C = { , , , , }3 9 10 12 13

(i) B C, = { , , , } { , , , , } { , , , , , , , , }1 2 5 7 3 9 10 12 13 1 2 3 5 7 9 10 12 13, =

vdnt, \( )A B C, = {1,3,5,7,9,11,13,15}\{1,2,3,5,7,9,10,12,13} { , }11 15= g (1) \A B = {1,3,5,6,7,9,11,13,15}\{1,2,5,7} {3, ,9,11,13,15}6=

\A C = { , , , , , , , }\{ , , , , } { , , , , }1 3 5 7 9 11 13 15 3 9 10 12 13 1 5 7 11 15=


( \ ) ( \ )A B A C+ = { , , , , } { , , , , } { , }3 9 11 13 15 1 5 7 11 15 11 15+ = g (2)(1) k‰W« (2) èUªJ , \( ) ( \ ) ( \ )A B C A B A C, += .(ii) B C+ = {1,2,5,7} {3,9,10,12,13}+ Q=

\( )A B C+ = {1,3,5,7,9,11,13,15}\ {1,3,5,7,9,11,13,15}Q = g (3) ( \ ) ( \ )A B A C, = { , , , , } { , , , , } { , , , , , , , }3 9 11 13 15 1 5 7 11 15 1 3 5 7 9 11 13 15, = g (4)

(3) k‰W« (4) èUªJ , \( ) ( \ ) ( \ )A B C A B A C+ ,= .

8. { , , , , , , , , }U 10 15 20 25 30 35 40 45 50= , {1, 5, 10, 15, 20, 30}B = k‰W« {7, 8, 15,20,35,45, 48}C = M»a fz§fS¡F \ \ \A B C A B A C+ ,=^ ^ ^h h h

v‹gij¢ rçgh®¡fÎ«.Ô®Î: { , , , , , , , , }, { , , , , , }A B10 15 20 25 30 35 40 45 50 1 5 10 15 20 30= = k‰W« { , , , , , , }C 7 8 15 20 35 45 48= vd bfhL¡f¥g£LŸsJ. B C+ = { , , , , , } { , , , , , , } { , }1 5 10 15 20 30 7 8 15 20 35 45 48 15 20+ =

\( )A B C+ = { , , , , , , , , }\{ , }10 15 20 25 30 35 40 45 50 15 20

= { , , , , , , }10 25 30 35 40 45 50 g (1) \A B = { , , , , , , , , }\{ , , , , , }10 15 20 25 30 35 40 45 50 1 5 10 15 20 30

= { , , , , }25 35 40 45 50

\A C = { , , , , , , , , }\{ , , , , , , }10 15 20 25 30 35 40 45 50 7 8 15 20 35 45 48

= { , , , , }10 25 30 40 50

( \ ) ( \ )A B A C, = {25,35,40,45,50} {10,25,30,40,50},

= { , , , , , , }10 25 30 35 40 45 50 g (2)(1) k‰W« (2) èUªJ , \ ( ) ( \ ) ( \ )A B C A B A C+ ,= .

9. bt‹gl§fis¥ ga‹gL¤Â Ã‹tUtdt‰iw¢ rçah vd¢ nrhÂ¤J¥ gh®¡fÎ«.

(i) A B C A B A C, + , + ,=^ ^ ^h h h (ii) A B C A B A C+ , + , +=^ ^ ^h h h

(iii) A B A B, += l ll^ h (iv) \ \ \A B C A B A C, +=^ ^ ^h h h

Ô®Î: (i)

(2) k‰W« (5) èUªJ , ( ) ( ) ( )A B C A B A C, + , + ,= .


(ii)

(2) k‰W« (5) èUªJ , ( ) ( ) ( )A B C A B A C+ , + , += .(iii)

(2) k‰W« (5) èUªJ , ( )A B A B, +=l l l.(iv)

(2) k‰W« (5) èUªJ , \( ) ( \ ) ( \ )A B C A B A C, += .


gæ‰Á 1.3 1. A, B v‹gd ÏU fz§fŸ k‰W« U v‹gJ mid¤J¡ fz« v‹f. nkY« 700n U =^ h ,

, k‰W« våš,n A n B n A B n A B200 300 100 ,+ += = = l l^ ^ ^ ^h h h h I¡ fh©f.

Ô®Î: ( )n A B, = ( ) ( ) ( )n A n B n A B 200 300 100++ - = + -

= 500 100 400- =

( )n A B+l l = ( ) ( ) ( )n A B n U n A B 700 400 300, ,= - = - =l

kh‰WKiw: ( )n A B+l l = ( ) ( ) ( ) ( ) ( ) ( )n A n B n A B n A n B n A B, ++ - = + -l l l l l l l

= 500 400 600 003+ - =

2. , , , ,våšn A n B n U n A B n A B285 195 500 410, ,= = = = l l^ ^ ^ ^ ^h h h h hI fh©f.

Ô®Î: ( )n A B+ = ( ) ( ) ( ) 2n A n B n A B 85 195 410 70,+ - = + - =

( )n A B,l l = [( ) ] ( ) ( )n A B n U n A B 500 70 430+ += - = - =l

3. A, B k‰W« C VnjD« _‹W fz§fŸ v‹f. nkY«, 17n A =^ h , ,n B 17=^ h ,n C n A B17 7+= =^ ^h h , ( ) ,n B C 6+ = n A C 5+ =^ h k‰W« n A B C 2+ + =^ h

våš, n A B C, ,^ hI¡ fh©f.Ô®Î: ( )n A B C, , = ( ) ( ) ( ) ( )n A n B n C n A B++ + - -

( ) ( ) ( )n B C n A C n A B C+ + + +- + . = 17 17 17 7 6 5 2 53 18+ + - - - + = - = 35.

4. Ã‹tU« fz§fS¡F n A B C n A n B n C n A B, , += + + - -^ ^ ^ ^ ^h h h h h n B C n A C n A B C+ + + +- +^ ^ ^h h h

v‹gij rçgh®¡fÎ«.

(i) { , , }, { , , , } { , , , }k‰W«A B C4 5 6 5 6 7 8 6 7 8 9= = =

(ii) { , , , , }, { , , } { , , }k‰W«A a b c d e B x y z C a e x= = = .Ô®Î: (i) {4,5,6}, ( ) 3, {5,6,7,8}, ( ) 4A n A B n B= = = = k‰W« { , , , }, ( )C n C6 7 8 9 4= =

A B C, , = { , , , , , }, ( )n A B C4 5 6 7 8 9 6, , = g (1) A B+ = { , , } { , , , } { , }, ( )n A B4 5 6 5 6 7 8 5 6 2+ += =

B C+ = { , , , } { , , , } { , , }, ( )n B C5 6 7 8 6 7 8 9 6 7 8 3+ += =

A C+ = { , , } { , , , } { }, ( )n A C4 5 6 6 7 8 9 6 1+ += =

A B C+ + = {4,5,6} {5,6,7,8} {6,7,8,9} {6}, ( ) 1n A B C+ + + += =

Ï¥bghGJ, ( ) ( ) ( ) ( ) ( ) ( ) ( )n A n B n C n A B n B C n A C n A B C+ + + + ++ + - - - +

= 3 4 4 2 3 1 1 6+ + - - - + = g (2)(1) k‰W« (2) èUªJ ,

( )n A B C, , = ( ) ( ) ( ) ( ) ( ) ( ) ( )n A n B n C n A B n B C n A C n A B C+ + + + ++ + - - - +

(ii) { , , , , }, { , , }A a b c d e B x y z= = k‰W« { , , }C a e x= ; ( ) , ( ) , ( )n A n B n C5 3 3= = =

A B+ = { , , , , } { , , } { }, ( ) 0a b c d e x y z n A B+ += =

B C+ = { , , } { , , } { }, ( ) 1x y z a e x x n B C+ += =

A C+ = { , , , , } { , , } { , }, ( ) 2a b c d e a e x a e n A C+ += =

A B C, , = { , , , , } { , , } { , , } { , , , , , , , }a b c d e x y z a e x a b c d e x y z, , =

vdnt, ( )n A B C, , = 8 g (1)


( ) ( ) ( ) ( ) ( ) ( ) ( )n A n B n C n A B n B C n A C n A B C+ + + + ++ + - - - +

= 5 3 3 0 1 2 0 8+ + - - - + = g (2)(1) k‰W« (2) èUªJ , ( )n A B C, ,

( ) ( ) ( ) ( ) ( ) ( ) ( )n A n B n C n A B n B C n A C n A B C+ + + + += + + - - - +

5. xU fšÿçæš nrUtj‰F 60 khzt®fŸ ntÂæaèY«, 40 ng® Ïa‰ÃaèY«,

30 ng® cæçaèY« gÂÎ brŒJŸsd®. 15 ng® ntÂæaèY« Ïa‰ÃaèY«,10

ng® Ïa‰ÃaèY« cæçaèY« k‰W« 5 ng® cæçaèY« ntÂæaèY« gÂÎ

brŒJŸsd®. _‹W ghl§fëY« xUtUnk gÂÎ brŒaéšiy våš, VnjD« xU

ghl¤Â‰fhtJ gÂÎ brŒJŸst®fë‹ v©â¡if ahJ? Ô®Î: C, P k‰W« B M»ait Kiwna ntÂæaš, Ïa‰Ãaš k‰W« cæçaš

ghl§fëš gÂÎ brŒJŸst®fë‹ fz§fŸ v‹f.

vdnt, ( ) 60, ( ) 40, ( ) 30n C n P n B= = = , ( ) , ( ) , ( ) , ( )n C P n P B n C B n C P B15 10 5 0+ + + + += = = =( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )n C P B n C n P n B n C P n P B n B C n C P B, , + + + + += + + - - - +

= 60 40 30 15 10 5 0 100+ + - - - + =

vdnt, VnjD« xU ghl¤Âš gÂÎ brŒJŸst®fŸ = 100.

6. xU efu¤Âš 85% ng® jäœ bkhê, 40% ng® M§»y bkhê k‰W« 20% ng® ÏªÂ

bkhê ngR»wh®fŸ. 32% ng® jäG« M§»yK«, 13% ng® jäG« ÏªÂÍ« k‰W«

10% ng® M§»yK« ÏªÂÍ« ngR»wh®fŸ våš, _‹W bkhêfisÍ« ngr¤

bjçªjt®fë‹ rjÅj¤Âid¡ fh©f.

Ô®Î: E, T k‰W« H M»ait Kiwna M§»y«, jäœ k‰W« ÏªÂ ngRgt®fë‹

fz§fŸ v‹f.

vdnt, ( ) 85, ( ) 40, ( ) 20n T n E n H= = =

nkY«, ( ) , ( ) , ( ) , ( )n E T n T H n E H n E T H32 13 10 100+ + + , ,= = = = .( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )n E T H n E n T n H n E T n T H n H E n E T H, , + + + + += + + - - - +

100 = ( )n E T H40 85 20 32 13 10 90 + ++ + - - - = +

vdnt, ( )n E T H+ + = 100 90 10- =

Mfnt, _‹W bkhêfisÍ« ngr bjçªjt®fŸ = 10%.

7. 170 tho¡ifahs®fëš 115 ng® bjhiy¡fh£ÁiaÍ«, 110 ng® thbdhèiaÍ«

k‰W« 130 ng® g¤Âç¡iffisÍ« ga‹gL¤Â»wh®fŸ v‹gij xU és«gu

ãWtd« f©l¿ªjJ. nkY«, 85ng® bjhiy¡fh£Á k‰W« g¤Âç¡ifiaÍ«, 75 ng®

bjhiy¡fh£Á k‰W« thbdhèiaÍ«, 95 ng® thbdhè k‰W« g¤Âç¡ifiaÍ«,

70 ng® _‹¿idÍ« ga‹gL¤J»wh®fŸ vdÎ« f©l¿ªjJ. bt‹gl¤Âš

étu§fis¢ F¿¤J, Ã‹tUtdt‰iw¡ fh©f.

(i) thbdhèia k£L« ga‹gL¤Jgt®fë‹ v©â¡if.

(ii) bjhiy¡fh£Áia k£L« ga‹gL¤Jgt®fë‹ v©â¡if.

(iii) bjhiy¡fh£Á k‰W« g¤Âç¡iffis¥ ga‹gL¤Â thbdhèia¥

ga‹gL¤jhjt®fë‹ v©â¡if.

Ô®Î: T, R k‰W« M M»ait Kiwna bjhiy¡fh£Á, thbdhè k‰W« g¤Âç¡if

M»at‰iw ga‹gL¤J« tho¡ifahs®fë‹ fz§fŸ v‹f,


vdnt, ( )n U = 170, ( ) 115n T =

( )n R = 110, ( ) 130n M =

( )n T M+ = 85, ( ) 75n T R+ =

( )n R M+ = 95, ( ) 70n T R M+ + =

bt‹gl¤ÂèUªJ

(i) thbdhèia ga‹gL¤Jgt®fë‹

v©â¡if = 10

(ii) bjhiy¡fh£Áia ga‹gL¤Jgt®fë‹

v©â¡if = 25

(iii) bjhiy¡fh£Á k‰W« g¤Âç¡ifia ga‹gL¤Â

thbdhèia ga‹gL¤jht®fë‹ v©â¡if = 15

8. 4000 khzt®fŸ gæY« xU gŸëæš , 2000 ngU¡F ÃbuŠR, 3000 ngU¡F¤ jäœ k‰W«

500 ngU¡F ÏªÂ bjçÍ«. nkY«, 1500 ngU¡F ÃbuŠR k‰W« jäœ, 300 ngU¡F

ÃbuŠR k‰W« ÏªÂ, 200 ngU¡F jäœ k‰W« ÏªÂ, 50 ngU¡F Ï«_‹W bkhêfS«

bjçÍ« våš, Ã‹tUtdt‰iw¡ fh©f.

(i) _‹W bkhêfS« bjçahjt®fë‹ v©â¡if.

(ii) VnjD« xU bkhêahtJ bjçªjt®fë‹ v©â¡if.

(iii) ÏU bkhêfŸ k£Lnk bjçªjt®fë‹ v©â¡if.

Ô®Î:

F, T k‰W« H M»ait Kiwna ÃbuŠR, jäœ k‰W« ÏªÂ bjçªjt®fë‹

fz§fŸ v‹f.

Ï¥bghGJ, ( )n U = , ( )n F4000 2000=

( )n T = , ( )n H3000 500=

( )n F T+ = 5 , ( )n F H1 00 300+ =

( )n T H+ = , ( )n F T H200 50+ + =

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )n F T H n F n T n H n F T n T H n F H n F T H, , + + + + += + + - - - +

= 2000 3000 500 1500 300 200 50+ + - - - + =3550.vdnt, ( )n F T H, , l = ( ) ( )n U n F T H 4000 3550 450, ,- = - = .

(i) _‹W bkhêfS« bjçahjt®fë‹ v©â¡if = 450

(ii) VnjD« xU bkhêiaahtJ bjçªjt®fë‹ v©â¡if = 3550

(iii) ÏU bkhêfŸ k£Lnk bjçªjt®fë‹ v©â¡if = 1850

9. 120 FL«g§fŸ cŸs xU »uhk¤Âš 93 FL«g§fŸ rikaš brŒtj‰F éwif¥

ga‹gL¤J»‹wd®. 63 FL«g§fŸ k©bz©bzæid¥ ga‹gL¤J»wh®fŸ.


45 FL«g§fŸ rikaš vçthÍit¥ ga‹gL¤J»wh®fŸ. 45 FL«g§fŸ éwF k‰W«

k©bz©bzŒ, 24 FL«g§fŸ k©bz©bzŒ k‰W« vçthÍ, 27 FL«g§fŸ

vçthÍ k‰W« éwF M»at‰iw¥ ga‹gL¤J»‹wd®. éwF, k©bz©bzŒ

k‰W« rikaš vçthÍ Ï«_‹iwÍ« ga‹gL¤J« FL«g§fë‹ v©â¡ifia¡

fh©f.

Ô®Î: F, K k‰W« G M»ait Kiwna éwF,

k©bz©bzŒ k‰W« rikaš vçthÍ

ga‹gL¤Jgt®fë‹ fz§fŸ v‹f.

( )n F` = 93, ( ) 63, ( )n K n G 45= =

( )n F K+ = 5, ( )n K G4 24+ =

( )n T H+ = 27, ( ) 120n F K G, , =

( )n F K G, ,

( ) ( ) ( ) ( ) ( ) ( ) ( )n F n K n G n F K n K G n F G n F K G+ + + + += + + - - - +

93 63 45 45 24 27 ( )n F K G& + ++ + - - - + = 120.

( )n F K G+ + = 120 105 15- = .

vdnt, _‹iwÍ« ga‹gL¤J« FL«g§fë‹ v©â¡if 15.

gæ‰Á 1.4

1. Ã‹tU« m«ò¡F¿¥ gl§fŸ rh®ig¡ F¿¡»‹wdth vd¡ TWf. c‹ éil¡F¤

jFªj fhuz« TWf.

Ô®Î: (i) (ii)

P š cŸs c v‹w cW¥ò¡F Q š

ãHšcU Ïšiy.

vdnt ÏJ xU rh®gšy.

kÂ¥gf« Lš cŸs x›bthU cW¥ò¡F«

M š xnu xU ãHš cU cŸsJ.

vdnt ÏJ xU rh®ò MF«.

2. bfhL¡f¥g£LŸs F = { (1, 3), (2, 5), (4, 7), (5, 9), (3, 1) } vD« rh®Ã‰F, kÂ¥gf«

k‰W« Å¢rf« M»at‰iw¡ fh©f.

Ô®Î: kÂ¥gf« = { , , , , }1 2 3 4 5 , Å¢rf« { , , , , }1 3 5 7 9= .

3. A = { 10, 11, 12, 13, 14 }; B = { 0, 1, 2, 3, 5 } k‰W« :f A Bi

" , i = 1,2,3. v‹f. ÑnH bfhL¡f¥g£LŸsit v›tif¢ rh®Ãid¡ F¿¡F«? éil¡fhd

jFªj fhuz« jUf.

(i) f1 = {(10, 1), (11, 2), (12, 3), (13, 5), (14, 3)},


(ii) f2 = {(10, 1), (11, 1), (12, 1), (13, 1), (14, 1)},

(iii) f3 = {(10, 0), (11, 1), (12, 2), (13, 3), (14, 5)}.

Ô®Î:{(10,1), (11,2),(12,3), (13,5), (14,3)}f(i)

1=

A š cŸs 12 k‰W« 14 v‹w cW¥òfŸ B-š xnu ãHš cU 3-I¥ bg‰WŸsJ.

vdnt, ÏJ x‹W¡F x‹W rh®ò.

B0 ! ¡F A-š K‹DU Ïšiy.

vdnt, ÏJ nkš rh®ò mšy. Mfnt, ÏJ x‹W¡F x‹W

rh®ò« mšy, nkš rh®ò« mšy.

{(10,1), (11,1), (12,1), (13,1), (14,1)}f(ii)2=

x›bthU x A! ¡F«, ( ) 1f x2

= .

vdnt, f2

MdJ xU kh¿è¢rh®ò MF«.

{(10,0),(11,1), (12,2),(13,3), (14,5)}f(iii)3=

f3‹ Ñœ A-‹ bt›ntW cW¥òfS¡F B-š bt›ntW ãHš

cU¡fŸ cŸsd. vdnt, f3 x‹W¡F x‹W rh®ò MF«.

nkY« ( )f A B3

= . vdnt, f3xU nkš rh®ò MF«.

Mfnt, f3 MdJ xU x‹W¡F x‹W k‰W« nkš rh®ò.

( f3 MdJ ÏU òw¢ rh®ò)

4. X = { 1, 2, 3, 4, 5 }, Y = { 1, 3, 5, 7, 9 } v‹f. X-èUªJ Y-¡fhd cwÎfŸ ÑnH

bfhL¡f¥g£LŸsd. Ït‰¿š vit rh®ghF«? c‹ éil¡fhd jFªj fhuz«

jUf. nkY«, mit rh®ò våš, v›tif¢ rh®ghF«? (i) R

1 = { ,x y^ h| 2y x= + , x X! , y Y! }

(ii) R2 = { (1, 1), (2, 1), (3, 3), (4, 3), (5, 5) }

(iii) R3 = { (1, 1), (1, 3), (3, 5), (3, 7), (5, 7) }

(iv) R4 = { (1, 3), (2, 5), (4, 7), (5, 9), (3, 1) }

Ô®Î: {( , ) | 2, , }R x y y x x X y Y(i)1

! != = +

y = x 2+ vd bfhL¡f¥g£LŸsJ.

2x = våš, y = 4 Yg .vdnt, R

1 xU rh®ò mšy.

nkY«, X-š cŸs 4-¡F Y-š ãHšcU Ïšiy. {(1,1), (2,1), (3,3), (4,3), (5,5)}R(ii)

2=

x›bthU cW¥ò¡F« xnu xU ãHš cU cŸsJ. vdnt, R2

xU rh®ò MF«.

1 k‰W« 2 M»a cW¥òfŸ 1 v‹w xnu ãHš cU bg‰WŸsJ.


vdnt, R2

x‹W¡F x‹W rh®ò mšy.

nkY«, 7 v‹w cW¥ò¡F K‹cU Ïšiy.

vdnt, R2

nkš rh®ò mšy.

Mfnt, R2

x‹W¡F x‹W rh®ò« mšy, nkš rh®ò« mšy.

{(1,1), (1,3), (3,5), (3,7), (5,7)}R(iii)3=

X1 ! ¡F Y-š 1, 3 M»a Ïu©L ãHš cU¡fŸ cŸsd.

vdnt, R3 xU rh®ò mšy.

{(1,3), (2,5), (4,7), (5,9), (3,1)}R(iv)4=

x›bthU cW¥ò¡F« xnu xU ãHš cU cŸsJ. vdnt, R4

xU

rh®ò MF«. X-‹ bt›ntW cW¥òfS¡F Y š bt›ntW ãHš

cU¡fŸ cŸsd. vdnt, ÏJ x‹W¡F x‹W rh®ò MF«.

Y-‹ x›bthU cW¥ò¡F« X-š K‹ cU cŸsJ.

vdnt, R4

xU nkš rh®ò MF«. Mfnt, R4

ÏU òw¢rh®ò MF«.

5. {( , 2), ( 5, ), (8, ), ( , 1)}R a b c d= - - - v‹gJ rkå¢ rh®ig¡ F¿¡F« våš,

, ,a b c k‰W« d M»at‰¿‹ kÂ¥òfis¡ fh©f.

Ô®Î: R MdJ rkå¢ rh®ò vd¡bfhL¡f¥g£LŸsJ.

vdnt, ( ) ,R x x x6= .Mfnt, 2, 5, 8a b c=- =- = k‰W« d 1=- .

6. A = { –2, –1, 1, 2 } k‰W« , :f xx

x A1 != ` j$ . våš, f -‹ Å¢rf¤ij¡ fh©f.

nkY«, f v‹gJ A-æèUªJ A-¡F xU rh®ghFkh?

Ô®Î: f = , 1xx

` j vd¡bfhL¡f¥g£LŸsJ. vdnt, ( ) .f xx1=

Mfnt, ( 2)f - = ; (1) 1.f21

21

11

-=- = =

( )f 1- = 1; (2) .f11

21

-=- =

vdnt, f ‹ Å¢rf« = , , ,21 1 1

21- -$ ..

nkY«, , A21

21 g- . vdnt, A-èUªJ A-¡F f xU rh®ò mšy.

7. f = { (2, 7), (3, 4), (7, 9), (–1, 6), (0, 2), (5, 3) } v‹gJ A = { –1, 0, 2, 3, 5, 7 }-æèUªJ B = { 2, 3, 4, 6, 7, 9 }-¡F xU rh®ò v‹f. f v‹w rh®ò

(i) x‹W¡F x‹whd rh®ghFkh? (ii) nkš rh®ghFkh? (iii) x‹W¡F x‹whd k‰W« nkš rh®ghFkh?Ô®Î: (i) A-‹ bt›ntW cW¥òfS¡F B-æš bt›ntW ãHš

cU¡fŸ cŸsd.


vdnt, f MdJ x‹W¡F x‹W rh®ò MF«.

(ii) B-‹ x›bthU cW¥ò¡F« A-æš xU K‹ cU cŸsJ.

vdnt, f MdJ xU nkš rh®ò MF«.

(iii) (i) k‰W« (ii) èUªJ, f xU ÏUòw¢ rh®ò MF«.

8. f = { (12, 2), (13, 3), (15, 3), (14, 2), (17, 17) } v‹w rh®Ãš 2 k‰W« 3 M»at‰¿‹

K‹cU¡fis¡ fh©f.

Ô®Î: 2-‹ K‹ cU¡fŸ 12 k‰W« 14. 3-‹ K‹ cU¡fŸ 13 k‰W« 15.

9. ÑnH bfhL¡f¥g£LŸs m£ltiz MdJ, {5,6,8,10}A = -æèUªJ

{ , , , }B 19 15 9 11= -¡F f x^ h = 2 1x - v‹wthW mikªj xU rh®ò våš, a k‰W« b M»adt‰¿‹ kÂ¥òfis¡ fh©f.

x 5 6 8 10f(x) a 11 b 19

Ô®Î: ( )f x = ,x x A2 1 6 !- vd¡ bfhL¡f¥g£LŸsJ

( )f 5 = 2 5 1 10 1 9# - = - =

( )f 8 = 2 8 1 16 1 15# - = - =

vdnt, a k‰W« b -‹ kÂ¥òfŸ Kiwna 9 k‰W« 15.F¿¥ò: ( ) 1,f x px x R!= + v‹w toéš cŸs rh®ò xU go¤jhd rh®ò MF«.

m¤jifa rh®òfŸ, x‹W¡F x‹W rh®òfŸ MF«.

10. A= { 5, 6, 7, 8 }; B = { –11, 4, 7, –10,–7, –9,–13 } v‹f.

f = {( ,x y ) : y = 3 2x- , x A! , y B! } vd tiuaW¡f¥g£LŸsJ.

(i) f -‹ cW¥òfis vGJf

(ii) mj‹ Jiz kÂ¥gf« ahJ?

(iii) Å¢rf« fh©f

(iv) v›tif¢ rh®ò vd¡ fh©f.

Ô®Î:{5,6,7,8}, { 11,4,7, 10, 7, 9, 13}A B= = - - - - - vd bfhL¡f¥g£LŸsJ.

Ï§F, y ( )f x= = 3 2 ,x x A6 !-

vdnt, ( )f 5 = 7, (6) 9f- =-

( )f 7 = 11, (8) 13f- =-

(i) {( , ), ( , ), ( , ), ( , )}f 5 7 6 9 7 11 8 13= - - - -

(ii) Jiz kÂ¥gf« = { , , , , , , }11 4 7 10 7 9 13- - - - -

(iii) Å¢rf« = { , , , }7 9 11 13- - - -


(iv) f -‹ Ñœ bt›ntW cW¥òfS¡F bt›ntW ãHš cU¡fŸ cŸsd.

vdnt, f MdJ x‹W¡F x‹W rh®ghF«.

( Ï§F f MdJ nkš rh®ò mšy )

11. ÑnH bfhL¡f¥g£LŸs tiugl§fëš vit rh®Ãid¡ F¿¡»‹wd? éil¡fhd

jFªj fhuz« jUf.

(i) bfhL¡f¥g£l tiugl« F¤J¡nfhL nrhjidia ãiwÎ

brŒ»wJ.

vdnt, ÏJ xU rh®ghF«.

(ii) bfhL¡f¥g£l tiugl« F¤J¡nfhL nrhjidia ãiwÎ

brŒ»wJ.

vdnt , ÏJ xU rh®ghF«.

(iii) tiugl¤Âš l v‹w F¤J¡nfhL tiugl¤ij A k‰W« B

v‹w ÏU òŸëfëš bt£L»wJ. vdnt, bfhL¡f¥g£l

tiugl« F¤J¡nfhL nrhjidia ãiwÎ brŒaéšiy.

Mfnt , ÏJ xU rh®gšy.

(iv) F¤J¡nfhL l MdJ tiugl¤ij ,A B k‰W« C v‹w

_‹W òŸëfëš bt£L»wJ. vdnt, bfhL¡f¥g£l

tiugl« F¤J¡nfhL nrhjidia ãiwÎ brŒaéšiy.

Mfnt, ÏJ xU rh®gšy.

(v) bfhL¡f¥g£l tiugl« F¤J¡nfhL nrhjidia ãiwÎ

brŒ»wJ.

vdnt, ÏJ xU rh®ghF«.

12. bfhL¡f¥g£LŸs rh®ò f = { (–1, 2), (– 3, 1), (–5, 6), (– 4, 3) } I (i) m£ltiz (ii) m«ò¡F¿ gl« M»at‰¿‹ _y« F¿¡fÎ«.

Ô®Î: {( , ), ( , ), ( , ), ( , )}f 1 2 3 1 5 6 4 3= - - - -

(i) m£ltiz

x 1- 3- 5- 4-

( )f x 2 1 6 3

(ii) m«ò¡F¿¥ gl«


13. A = { 6, 9, 15, 18, 21 } ; B = { 1, 2, 4, 5, 6 } k‰W« :f A B" v‹gJ

f x^ h = x33- vd tiuaW¡f¥g£oU¥Ã‹ rh®ò f -I

(i) m«ò¡F¿ gl« (ii) tçir¢ nrhofë‹ fz«

(iii) m£ltiz (iv) tiugl« M»at‰¿‹ _y« F¿¡fÎ«.

Ô®Î: { , , , , }, { , , , , }A B6 9 15 18 21 1 2 4 5 6= = vd bfhL¡f¥g£LŸsJ.f ‹ ãHš cU¡fis¡ fh©ngh«.

Ï¥bghGJ, ( )f x = ,x x A33 !-

vdnt, ( )f 6 = 1,3

6 3- = (9) 2f3

9 3= - =

( )f 15 = 4, (1 ) 5, (21) 6f f3

15 3 83

18 33

21 3- = = - = = - =

(i) m«ò¡F¿ gl«

(ii) tçir¡nrhofë‹ fz« {(6,1), (9,2), (15,4), (18,5), (21,6)}f =

(iii) m£ltiz

x 6 9 15 18 21( )f x 1 2 4 5 6

(iv) tiugl«

(6,1), (9,2), (15,4),(18,5), (21,6) M»a mid¤J¥ òŸëfS« nr®ªJ rh®ÃDila

tiugl¤ij xy-js¤Âš F¿¡»wJ.

14. A = {4, 6, 8, 10 } k‰W« B = { 3, 4, 5, 6, 7 } v‹f. :f A B" v‹gJ 1f x x21= +^ h

vd tiuaW¡f¥g£LŸsJ. rh®ò f I (i) m«ò¡F¿ gl« (ii) tçir¢ nrhofë‹

fz« (iii) m£ltiz M»at‰¿‹ _y« F¿¡fÎ«.

Ô®Î: ( )f x = ,x x A2

1 !+

vdnt, ( )f 4 = 1 3, (6) 1 4f24

26+ = = + =

( )f 8 = 1 5, (10) 1 6f28

210+ = = + = .


(i) m«ò¡F¿¥gl«:

(ii) tçir¢ nrhofë‹ fz«: {( , ), ( , ), ( , ), ( , )}f 4 3 6 4 8 5 10 6= .

(iii) m£ltiz:

x 4 6 8 10( )f x 3 4 5 6

15. rh®ò f : ,3 7- h6 " R Ñœ¡f©lthW tiuaW¡f¥g£LŸsJ.

f x^ h = ;

;

;

x x

x x

x x

4 1 3 2

3 2 2 4

2 3 4 7

2 1

1 1

#

# #

- -

-

-

* Ã‹tUtdt‰iw¡ fh©f.

(i) f f5 6+^ ^h h (ii) f f1 3- -^ ^h h (iii) f f2 4- -^ ^h h (iv) ( ) ( )

( ) ( )

f ff f2 6 1

3 1

-

+ - .

Ô®Î: njitahd Ïilbtëæš rh®Ã‹ kÂ¥òfis¡ fh©ngh«.

3, 2, 1x =- - - k‰W« 1 vD« nghJ ( ) 4 1f x x2= - .vdnt, ( 3) 35, ( 2) 15, ( 1) 3, (1) 3.f f f f- = - = - = =

x = 3, 4 vD« nghJ ( )f x = 3 2x - .vdnt, ( )f 3 = 7 k‰W« (4) 10f = .vdnt, 5x = k‰W« 6 våš, ( )f x = 2 3x - . ( )f 5 = 7, (6) 9f =

(i) ( ) ( )f f5 6+ = 7 9 16+ =

(ii) ( ) ( )f f1 3- - = 3 35 32- =-

(iii) ( ) ( )f f2 4- - = 15 10 5- =

(iv) ( ) ( )

( ) ( )f f

f f2 6 13 1

-+ - =

( )2 9 37 3

1510

32

-+ = = .

16. rh®ò f : ,7 6- h6 " R Ñœ¡f©lthW tiuaW¡f¥g£LŸsJ.

( )f x = ;

;

; .

x x x

x x

x x

2 1 7 5

5 5 2

1 2 6

2 1

1 1

#

# #

+ + - -

+ -

-

* Ã‹tUtdt‰iw¡ fh©f.

(i) 2 ( 4) 3 (2)f f- + (ii) ( 7) ( 3)f f- - - (iii) ( ) ( )

( ) ( )

f ff f

6 3 1

4 3 2 4

- -

- + .

Ô®Î: ,x 7 6=- - våš, ( ) 2 1f x x x2= + +


vdnt, ( )f 7- = ( ) ( )7 2 7 1 49 14 1 362- + - + = - + = k‰W«

( )f 6- = ( ) ( )6 2 6 1 36 12 1 252- + - + = - + = .

4, 3, 2x =- - k‰W« 1 våš, ( ) 5f x x= +

vdnt, ( )f 4- = 4 5 1,- + = ( )f 3- 2 , (1) 6f= = k‰W« ( )f 2 = 7.

x 4= våš, ( )f x x 1= - . vdnt, (4) 3.f =

(i) 2 ( 4) 3 (2)f f- + = 2 1 3 7 23# #+ = .

(ii) ( ) ( ) .f f7 3 36 2 34- - - = - =

(iii) ( ) ( )( ) ( )

f ff f

6 3 14 3 2 4

- -- + =

25 3 64 2 2 3

25 188 6

714 2

## #

-+ =

-+ = =

gæ‰Á 1.5

1. A k‰W« B , v‹gd Ïu©L fz§fŸ v‹f. A B, = A v‹gj‰F¤ njitahd k‰W«

nghJkhd f£L¥ghL.

(A) B A3 (B) A B3 (C) A B! (D) A B+ z=

Ô®Î:

( éil: (A) ) 2. A B1 våš, A B+ =

(A) B (B) \A B (C) A (D) \B A

Ô®Î:

( éil: (C) ) 3. P k‰W« Q v‹gd VnjD« Ïu©L fz§fŸ våš, P Q+ =

(A) : mšyJx x P x Q! !" , (B) : k‰W«x x P x Qb!" ,

(C) : k‰W«x x P x Q! !" , (D) : k‰W«x x P x Qb !" ,

Ô®Î: tiuaiuæ‹go { :P Q x x P+ != k‰W« }x Q! ( éil: (C) )

4. A= { p, q, r, s }, B = { r, s, t, u } våš, \A B =

(A) { , }p q (B) { , }t u (C) { , }r s (D) { , , , }p q r s

Ô®Î: \A B MdJ Bš Ïšyhj A‹ cW¥òfŸ MF«. ( éil: (A) )

5. ( )n p A6 @ = 64 våš, n A^ h =

(A) 6 (B) 8 (C) 4 (D) 5

Ô®Î: [ ( )] ( )n P A n A2 64 2 6( )n A 6 `= = = = . ( éil: (A) )


6. A, B k‰W« C M»a VnjD« _‹W fz§fS¡F, A B C+ ,^ h =

(A) A B B C, , +^ ^h h (B) A B A C+ , +^ ^h h

(C) ( )A B C, + (D) A B B C, + ,^ ^h h

Ô®Î: ( ) ( ) ( )A B C A B A C+ , + , += . ( éil: (B) ) 7. A, B M»a Ïu©L fz§fS¡F, {( \ ) ( \ )} ( )A B B A A B, + + =

(A) Q (B) A B, (C) A B+ (D) A B+l l

Ô®Î:

( éil: (A) )

8. ÑnH bfhL¡f¥g£LŸsitfëš jtwhd T‰W vJ?

(A) \A B = A B+ l (B) \A B A B+=

(C) \ ( )A B A B B, += l (D) \ ( ) \A B A B B,=

Ô®Î: \A B A B+= l vd m¿nth«. vdnt, \A B A B+! ( éil: (B) )

9. ,A B k‰W« C M»a _‹W fz§fS¡F \B A C,^ h =

(A) \ \A B A C+^ ^h h (B) \ \B A B C+^ ^h h

(C) \ \B A A C+^ ^h h (D) \ \A B B C+^ ^h h

Ô®Î: o kh®få‹ éÂ: \( ) ( \ ) ( \ )B A C B A B C, += ( éil: (B) )

10. n(A) = 20 , n(B) = 30 k‰W« ( )n A B, = 40 våš, ( )n A B+ =

(A) 50 (B) 10 (C) 40 (D) 70.Ô®Î: ( ) ( ) ( ) ( )n A B n A n B n A B 20 30 40 10+ ,= + - = + - = ( éil: (B) )

11. { ( x , 2), (4, y ) } xU rkå¢ rh®ig¡ F¿¡»wJ våš, ( , )x y =

(A) (2, 4) (B) (4, 2) (C) (2, 2) (D) (4, 4)Ô®Î: rkå¢rh®Ã‹ x›bthU cW¥ò« j‹ndhnl bjhl®ò bg‰WŸsJ. ( éil: (A) )

12. { (7, 11), (5, a ) } xU kh¿è¢rh®ig¡ F¿¡»wJ våš, ‘a ’-‹ kÂ¥ò

(A) 7 (B) 11 (C) 5 (D) 9Ô®Î: kh¿è¢ rh®Ãš mid¤J ãHš cU¡fS« rk«. ( éil: (B) )

13. ( )f x = 1 x-^ h v‹gJ N -èUªJ Z - ¡F tiuaW¡f¥ g£LŸsJ. f -‹ Å¢rf«

(A) { 1} (B) N (C) { 1, – 1 } (D) Z

Ô®Î: , ( )x f xN! = ( 1) .1x !- = ( éil: (C) )


14. f = { (6, 3), (8, 9), (5, 3), (–1, 6) } våš, 3-‹ K‹ cU¡fŸ

(A) 5 k‰W« –1 (B) 6 k‰W« 8 (C) 8 k‰W« –1 (D) 6 k‰W« 5

Ô®Î: 3 ‹ K‹ cU¡fŸ 5 k‰W« 6. ( éil: (D) )

15. A = { 1, 3, 4, 7, 11 } k‰W« B = {–1, 1, 2, 5, 7, 9 } v‹f. f = { (1, –1), (3, 2), (4, 1), (7, 5), (11, 9) } v‹wthW mikªj rh®ò :f A B" v‹gJ

(A) x‹W¡F x‹whd rh®ò (B) nkš rh®ò

(C) ÏUòw¢ rh®ò (D) rh®ò mšy ( éil: (A) )

Ô®Î:

16. bfhL¡f¥g£LŸs gl« F¿¡F« rh®ò, xU

(A) nkš rh®ò (B) kh¿è¢ rh®ò

(C) x‹W¡F x‹whd rh®ò (D) rh®ò mšy

Ô®Î: C2 ! - ¡F MdJ Ïu©L ãHš cU¡fŸ cŸsd.

vdnt ÏJ xU rh®ò mšy. ( éil: (D) )

17. A = { 5, 6, 7 }, B = { 1, 2, 3, 4, 5 } v‹f. ( )f x x 2= - v‹wthW tiuaiw brŒa¥g£l

rh®ò :f A B" Ï‹ Å¢rf«,

(A) { 1, 4, 5 } (B) { 1, 2, 3, 4, 5 } (C) { 2, 3, 4 } (D) { 3, 4, 5 }

Ô®Î: (5)f = , ( ) , ( )f f3 6 4 7 5= = . ( éil: (D) )

18. ( ) 5f x x2= + våš, ( )f 4- =

(A) 26 (B) 21 (C) 20 (D) – 20

Ô®Î: ( )f x = 5x2 +

& ( 4)f - = ( 4) 5 21.2- + = ( éil: (B) )

19. xU rh®Ã‹ Å¢rf« XUW¥ò¡ fzkhdhš, mJ xU

(A) kh¿è¢ rh®ò (B) rkå¢ rh®ò

(C) ÏUòw¢ rh®ò (D) x‹W¡F x‹whd rh®ò

Ô®Î: kh¿è¢ rh®ò. ( éil: (A) )

20. :f A B" xU ÏUòw¢ rh®ò k‰W« n(A) = 5 våš, n(B) =

(A) 10 (B) 4 (C) 5 (D) 25Ô®Î: A k‰W« B M»ait KoÎW fz§fŸ k‰W« f MdJ ÏUòw¢ rh®ò,vdnt ( ) ( )n A n B= ( éil: (C) )

Ô®Î - bkŒba©fë‹ bjhl®tçirfS« bjhl®fS« 23

gæ‰Á 2.1 1. n-MtJ cW¥ò bfhL¡f¥g£l Ã‹tU« bjhl®tçir x›bth‹¿Y« Kjš _‹W

cW¥òfis¡ fh©f.

(i) a n n

3

2n=

-^ h (ii) c 1 3n

n n 2= - +^ h (iii) z n n

4

1 2n

n

=- +^ ^h h .

Ô®Î: (i) Ï§F a n n

3

2n=

-^ h , n= 1, 2, 3,g .

n = 1, 2 k‰W« 3 våš

( ) ( )a

31 1 2

31 1

31

1=

-=

-= -

( ) ( )0a

32 2 2

32 0

2=

-= =

( ) ( )1a

33 3 2

33 1

3=

-= =

vdnt, bjhl® tçiræ‹ Kjš _‹W cW¥òfŸ Kiwna , 031- k‰W« 1.

(ii) Ï§F ( 1) 3cn

n n 2= -

+ , n = 1, 2 k‰W« 3 våš

( 1) (3) (3) 27c1

1 1 2 3= - =- =-

+

( 1) (3) (3) 81c2

2 2 2 4= - = =

+

( 1) (3) (3) 243c3

3 3 2 5= - =- =-

+

vdnt, bjhl® tçiræ‹ Kjš _‹W cW¥òfŸ Kiwna –27, 81 k‰W« –243.

(iii) Ï§F ( ) ( )z

n n4

1 2n

n

=- + , n = 1, 2 k‰W« 3 våš

( ) ( )( ) ( )( )z

41 1 1 2

41 3

43

1

1

=- +

=-

= -

( ) ( )( ) ( )( )2z

41 2 2 2

42 4

2

2

=- +

= =

( ) ( )( ) ( )( )z

41 3 3 2

4

1 3 5

415

3

3

=- +

=-

= -^ h .

vdnt, bjhl® tçiræ‹ Kjš _‹W cW¥òfŸ Kiwna , 243- k‰W«

415- .

2. x›bthU bjhl®tçiræ‹ n-MtJ cW¥ò ÑnH bfhL¡f¥g£LŸsJ. mit

x›bth‹¿Y« F¿¥Ãl¥g£LŸs cW¥òfis¡ fh©f.

(i) ; ,ann a a2 3

2n 7 9=

++ (ii) ; ,a n a a1 2 1

n

n n 3

5 8= - ++^ ^h h

(iii) ; ,a n n a a2 3 1.n

2

5 7= - + (iv) ( ) ( ); ,a n n a a1 1

n

n 2

5 8= - - +

bkŒba©fë‹ bjhl®tçirfS«

bjhl®fS« 2


Ô®Î: (i) Ï§F, 1, 2, 3, .ann n2 3

2 forn

g=++ =

n = 7 våš, ( )

a2 7 37 2

179

7=

++ =

n = 9 våš, ( )

a2 9 39 2

2111

9=

++ = .

(ii) Ï§F, ( 1) 2 ( 1) 1, 2, 3,,a n nn

n n 3g= - + =

+

n = 5 våš, ( 1) (2) (5 1) ( 1)(256)(6) 1536a5

5 8= - + = - =-

n = 8 våš, ( 1) (2) (8 1) (2) (9) (2048)(9) 18432a8

8 8 3 11= - + = = =

+ .

(iii) Ï§F, 2 3 1 1, 2, 3,a n n nforn

2g= - + =

n = 5 våš 2(5) 3(5) 1 2(25) 15 1 50 15 1 36a5

2= - + = - + = - + =

n = 7 våš 2(7) 3(7) 1 2(49) 21 1 98 20 78a7

2= - + = - + = - = .

(iv) Ï§F, ( 1) (1 ) 1, 2, 3, .a n n nforn

n 2g= - - + =

n = 5 våš ( 1) (1 ) ( )( ) ( )( )a 5 5 1 4 25 1 21 215

5 2= - - + = - - + = - =-

n = 8 våš ( 1) (1 8 8 ) ( 7 64) 57a8

8 2= - - + = - + = .

3. ( ),

,

k‰W« Ïu£il v©

k‰W« x‰iw v©

¥gil vD«nghJ

¥gil vD«nghJan n n n

n

n n n

3

1

2N

Nn 2

!

!=+

+*

vd tiuaW¡f¥g£l bjhl®tçiræ‹ 18-tJ k‰W« 25-tJ cW¥òfis¡ fh©f.

Ô®Î: n xU Ïu£ilgil v© våš, ( 3)a n nn= + .

vdnt, a18 = 18(18 3) 18(21) 378+ = = .

n xU x‰iw¥gil v© våš, an

n

1

2n 2=

+.

vdnt, a25

= ( )

25 1

2 25625 1

5062650

31325

2+

=+

= = .

4. ,

( ),

k‰W« Ïu£il v©k‰W« x‰iw v©

¥gil vD«nghJ¥gil vD«nghJ

bn n n

n n n n2

N

Nn

2!

!=

+)

vd tiuaW¡f¥g£l bjhl®tçiræ‹ 13 MtJ k‰W« 16 MtJ cW¥òfis¡ fh©f.

Ô®Î: n xU x‰iw¥gil v© våš, ( )b n n 2n= + .

vdnt, b13

= ( ) 13(13 2) 195n n 2+ = + =

n xU Ïu£il¥gil v© våš, 16 256.b n16

2 2= = =

5. 2, 3a a a1 2 1= = + k‰W« 2 5, 2,a a n

n n 12= +

- vd¡ bfh©l¤ bjhl®tçiræ‹

Kjš 5 cW¥òfis¡ fh©f.

Ô®Î: Ï§F 2, 3a a a1 2 1= = + k‰W« 2 5 2,a a n

n n 12= +

-.


a2

= 2 53 + =

a3 = 2 5 2 5 15a 5

2+ = + =^ h

a4

= 2 5 2 5 35a 153+ = + =^ h

a5

= 2 5 7535 + =^ h

vdnt, bjhl®tçiræ‹ Kjš IªJ cW¥òfŸ Kiwna 2, 5, 15, 35 k‰W« 75.

6. 1a a a1 2 3= = = k‰W« a a a

n n n1 2= +

- -, 3n 2 , vd¡ bfh©l¤ bjhl®tçiræ‹


Ô®Î: Ï§F, 1a a a1 2 3= = = k‰W« a a a

n n n1 2= +

- - , n 32 .

a4

= 1 1 2a a3 2+ = + =

a5

= 2 1 3a a4 3+ = + =

a6 = 3 2 5a a

5 4+ = + =

vdnt, bjhl®tçiræ‹ Kjš MW cW¥òfŸ Kiwna , , , ,1 1 1 2 3 k‰W« 5.

gæ‰Á 2.2

1. xU T£L¤ bjhl®tçiræ‹ Kjš cW¥ò 6 k‰W« bghJ é¤Âahr« 5 våš,

m¤bjhl®tçirÍ«, mj‹ bghJ cW¥igÍ« fh©f.

Ô®Î: T£L¤bjhl®tçiræ‹bghJtot« , , 2 , 3 ,a a d a d a d g+ + + .

Ï§F a 6= k‰W« .d 5=

vdnt, T£L¤bjhl®tçir 6, , 6 2 , 6 3 ,6 5 5 5 g+ + +^ ^ ^h h h

Mfnt, njitahdT£L¤bjhl®tçir 6, 11, 16, 21, gT£L¤bjhl®tçiræ‹bghJtot«

tn = ( )a n d1+ -

= n6 1 5+ -^ ^h h

= 6 5 5 5 1, 1 ,2 ,3 ,n n n g+ - = + = .

2. 125, 120, 115, 110, g v‹w T£L¤ bjhl®tçiræ‹ bghJ é¤Âahr¤ijÍ«

15 MtJ cW¥igÍ« fh©f.

Ô®Î: bfhL¡f¥g£l bjhl® tçir 125, 120, 115, 110, g xUT£L¤bjhl®tçir.

våš, a = 125, d = t t2 1- = 120 125- = 5-

T£L¤bjhl®tçiræ‹bghJtot«

tn = a n d1+ -^ h .

vdnt, t15

= 125 125 14 125 70 5515 1 5 5+ - - = + - = - =^ ^ ^h h h .

3. 24, 23 , 22 , 21 ,41

21

43 g v‹w T£L¤ bjhl®tçiræš 3 v‹gJ v¤jidahtJ

cW¥ò MF«?


Ô®Î: bfhL¡f¥g£l bjhl® tçir , , , ,24 2341 22

21 21

23 g

Ï§F, 24, 23 24a d41

43= = - = - k‰W« filÁ cW¥ò l 3= .

n v‹gJT£L¤bjhl®tçiræ‹cW¥òfë‹v©â¡if

n = 1d

l a- +

n& = 1

43

3 24-- + =

321 4 1 28 1 29#

-- + = + =` j

vdnt, Ï¤bjhl®tçiræ‹ 29tJ cW¥ò 3.

4. , , ,2 3 2 5 2 g v‹w T£L¤ bjhl®tçiræ‹ 12 MtJ cW¥ò ahJ?

Ô®Î: bfhL¡f¥g£l bjhl® tçir , 3 , 5 ,2 2 2 g .Ï§F, , 3 2a d2 2 2 2= = - =

xUT£L¤bjhl®tçiræ‹n tJ cW¥ò, ( )t a n d1n= + -

vdnt, t12

= 2 12 1 2 2+ -^ h

= 2 11 2 2+ ^ h = 2 22 2+

t12

= 23 2 .

5. 4, 9, 14, g v‹w T£L¤ bjhl®tçiræ‹ 17 MtJ cW¥ig¡ fh©f. Ô®Î: , , ,4 9 14 g ÏJxUT£L¤bjhl®tçir. Ï§F, 4, 9 4 5a d= = - =

T£L¤bjhl®tçiræ‹bghJtot«tn

tn = a n d1+ -^ h

vdnt, t17

= 4 17 1 5 4 16 5 84+ - = + =^ ^ ^h h h . 6. Ã‹tU« T£L¤ bjhl®tçiræš cŸs bkh¤j cW¥òfis¡ fh©f.

(i) 1, , , , .65

32

310g- - - (ii) 7, 13, 19, g , 205.

Ô®Î: (i) bfhL¡f¥g£lT£L¤bjhl®tçir 1, , , , .65

32

310g- - -

Ï§F, 1, 1a d65

61=- = - + = k‰W« l

310= .

vdnt, n = 1d

l a- +

Mfnt, n = 310 1

61

1+

+ = ( ) 1 26 1 27313 6 + = + = .

vdnt, T£L¤bjhl®tçir 27 cW¥òfis¡bfh©lJ.

(ii) bfhL¡f¥g£lT£L¤bjhl®tçir 7,13,19, ,205g .Ï§F, 7, 13 7 6a d= = - = k‰W« l = 205. T£L¤bjhl®tçiræ‹cW¥òfë‹v©â¡if,n =

dl a 1- +

= 6

205 7 1- + = 6

198 1+ = 34

vdnt, Ï¤bjhl®tçiræšbkh¤j«34 cW¥òfŸcŸsd.


7. xU T£L¤ bjhl®tçiræ‹ 9 MtJ cW¥ò ó¢Áa« våš, 19 MtJ cW¥Ã‹

ÏUkl§F 29 MtJ cW¥ò vd ã%Ã.

Ô®Î: Ï§F, t 09= & 8 0 8a d a dor+ = =- .

vdnt, t29

= a d d d d28 8 28 20+ =- + =

= d2 106 @ = d d2 8 18- +6 @ = a d2 18+6 @ = 2t

19

Mfnt, t29

= 2 t19

.

8. xU T£L¤ bjhl®tçiræš 10 k‰W« 18 MtJ cW¥òfŸ Kiwna 41 k‰W« 73

våš, 27 MtJ cW¥ig¡ fh©f.

Ô®Î: Ï§F, t 4110

=

& a d10 1 41+ - =^ h & 9a d+ = 41 g (1) t 73

18= & a d18 1 73+ - =^ h

& 17a d+ = 73 g (2)Ï¥bghGJ, (2) (1) 8d&- = 32 4d& = .nkY«, (1) 9(4)a& + = 41

& 41 36a = - = 5

vdnt, t27

= a d27 1 5 26 4 5 104 109+ - = + = + =^ ^h h .

Mfnt, 27 tJ cW¥ò 109.

9. 1, 7, 13, 19,g k‰W« 100, 95, 90, g M»a T£L¤ bjhl®tçirfë‹ n MtJ cW¥ò

rkbkåš, n-‹ kÂ¥ig¡ fh©f.

Ô®Î: Ï§FKjšT£L¤bjhl®tçiræš 1, 7, 13, 19, g .

,a d1 7 1 6= = - = vdntT£L¤bjhlç‹n tJ cW¥ò 1t n 1 6n= + -^ ^h h.

vdnt, Ïu©lhtJT£L¤bjhl®tçiræš100, 95, 90, g .

,a d100 95 100 5= = - =- .

vdnt, T£L¤bjhlç‹ n tJ cW¥ò 100s n 1 5n= + - -^ ^h h

bfhL¡f¥g£lãgªjidæ‹go t sn n

=

& n1 1 6+ -^ ^h h = n100 1 5+ - -^ ^h h

& 1 6 6n+ - = n100 5 5- +

& 11n = 110 10.n& =

10. 13Mš tFgL« <çy¡f äif KG v©fë‹ v©â¡ifia¡ fh©f.

Ô®Î: 13 Mš tFgL« Ïu©L Ïy¡f v©fŸ 13,26,39, 91.g

ÏJxUT£L¤bjhl®tçir. Ï§F, ,a d13 13= = k‰W« l 91= .


vdnt, n = d

l a 1- + .

= 13

91 13 11378 1 6 1 7- + = + = + = .

Mfnt, 13 MštFgL«<çy¡fäifKGv©fŸv©â¡if7. 11. xU bjhiy¡fh£Á¥ bg£o¤ jahç¥ghs® VHhtJ M©oš 1000 bjhiy¡fh£Á¥

bg£ofisÍ«, g¤jhtJ M©oš 1450 bjhiy¡fh£Á¥ bg£ofisÍ« jahç¤jh®.

x›bthU M©L« jahç¡F« bjhiy¡fh£Á¥ bg£ofë‹ v©â¡if ÓuhfÎ«

xU kh¿è v© msÎ« mÂfç¤jhš, Kjyh« M©oY«, 15 MtJ M©oY«

jahç¡f¥g£l bjhiy¡fh£Á¥ bg£ofë‹ v©â¡ifia¡ fh©f.

Ô®Î: x›bthUM©L« jahç¡F« bjhiy¡fh£Á¥ bg£ofë‹v©â¡if

ÓuhfÎ«,xUkh¿èv©msÎ«mÂfç¥gjhšÏJxUT£L¤bjhl®tçiria

cUth¡F«.

vdnt, t 10007= k‰W« t 1450

10=

ÏÂèUªJ, t7 = 6 1000a d+ = g (1)

t10

= 9 1450a d+ = g (2)(2) (1) &- 3d = 450 & d = 150.d 150= æ‹ kÂ¥ig (1) š ÃuÂæl,

( )a 6 150+ = 1000

a = 1000 900- = 100.

Kjyh«M©Ljahç¡f¥g£lbjhiy¡fh£Ábg£ofë‹v©â¡if 100.

vdnt, 14 100 14 100 2100 2200.t a d 15015

= + = + = + =^ h

vdnt, 15 tJM©ošjahç¡f¥g£lbjhiy¡fh£Á¥bg£ofë‹v©â¡if

2200 MF«.

12. xUt®, Kjš khj« `640, 2M« khj« `720, 3M« khj« `800-I nrä¡»wh®. mt®

j‹Dila nrä¥ig Ïnj bjhl®tçiræš bjhl®ªjhš, 25MtJ khj« mt®

nrä¡F« bjhifia¡ fh©f.

Ô®Î: x›bthUKjš,Ïu©lh«,_‹wh«khj«nrä¤jbjhif`640, 720, 800, ... .

ÏJxUT£L¤bjhl®tçirvåš a = 640, d = 720 – 640 = 80.

vdnt, t25

= 640 25 1 80+ -^ ^h h

= 640 24 80+ ^ h = 640 1920+

= 2560 Mfnt, 25 tJ khj« mtç‹ nrä¥ò bjhif ` 2560.

13. xU T£L¤ bjhl®tçiræš mL¤jL¤j _‹W cW¥òfë‹ TLjš 6 k‰W«

mt‰¿‹ bgU¡F¤ bjhif –120 våš, m«_‹W v©fis¡ fh©f.


Ô®Î: xUT£L¤bjhl®tçiræ‹_‹WcW¥òfŸKiwna , ,a d a a d- + .

vdnt, 6 3 6 2a d a a d a a& &- + + + = = =^ ^h h

nkY«, a d a a d- +^ ^ ^h h h = 120-

a a d2 2& -^ h = 120-

2 d4 2& -^ h = 120 4 60d2&- - =-

& d2 = 64 & d 8!=

8d = våš, mªj _‹W v©fŸ , ,6 2 10- .

8d =- våš, mªj _‹W v©fŸ , ,10 2 6- .


m›ÎW¥òfë‹ t®¡f§fë‹ TLjš 140 våš, m«_‹W v©fis¡ fh©f.

Ô®Î: xU T£L¤ bjhl®tçiræ‹ mL¤jL¤j _‹W v©fŸ , ,a d a a d- + v‹f.vdnt,TLjš

a d a a d- + + +^ ^ ^h h h = 18 & 3a = 18 6a& =

nkY«, a d a a d2 2 2- + + +^ ^h h = 401

& 2 2a ad d a a ad d2 2 2 2 2- + + + + + = 401

& 3 2a d2 2+ = 401

& 3 2d36 2+^ h = 401

& 2 140 108 2d d2 2

&= - = 32 4d& !=

d 4= , vD«nghJ mªj _‹W v©fŸ 2, 6, 10. d 4=- , vD«nghJ mªj _‹W v©fŸ 10, 6, 2.

15. xU T£L¤ bjhl®tçiræ‹ m-MtJ cW¥Ã‹ m kl§F mj‹ n-MtJ cW¥Ã‹ n kl§F¡F¢ rkbkåš, m¡T£L¤ bjhl®tçiræ‹ (m + n)-MtJ cW¥ò ó¢Áa«

vd¡fh£Lf.

Ô®Î: bfhL¡f¥g£lit m t n tm n

=

& m a m d n a n d1 1+ - = + -^ ^h h6 6@ @& am m m d na n n d1 1+ - = + -^ ^h h

& 0m n a m m n n d1 1- + - - - =^ ^ ^h h h6 @& m n a m m n n d 02 2

- + - - + =^ h 6 @

& m n a m n m n d 02 2- + - - - =^ ^ ^h h h6 @

& m n a m n m n d1 0- + - + - =^ ^h h6 @& a m n d1 0+ + - =^ h

& 0tm n

=+

.


16. xUt® tUl¤Â‰F jåt£o 14% jU« KjÄ£oš 25,000-I KjÄL brŒjh®. x›bthU

tUl KoéY« »il¡F« mrš k‰W« jåt£o nr®ªj bkh¤j¤ bjhif xU T£L¤

bjhl®tçiria mik¡Fkh? m›thbwåš, 20 M©LfS¡F¥ ÃwF KjÄ£oš

cŸs bjhifia¡ fh©f.

Ô®Î: jåt£o I = P R T100# # =

10025000 14 1# #

= 250 14# = 3500

bjhif = mrš+t£o

= 25,000 5003+ = 28,500

Kjyh« M©L ÏWÂæš cŸs bjhif = ` 28,500

Ïu©lh« M©L ÏWÂæš cŸs bjhif = 28,500 + 3500 = ` 32,000

Kjyh«, Ïu©lh«, _‹wh« M©L ÏWÂæš cŸs bjhif `28,500, `32,000,

`35,500, g . Ï§FmL¤jL¤jÏu©Lv©fS¡fhdé¤Âahr« 3500.

vdntÏJxUT£L¤bjhl®tçir.

Ï§F, , ,d a3500 28 500= =

20 M©LfS¡FÃwF,KjÄ£ošcŸsbjhif t20

.

t20

= a d19+

= 28,500 + 19 (3500) = 28500 + 66500 = ` 95,000.

17. a, b, c M»ad T£L¤ bjhl®tçiræš ÏU¥Ã‹ ( ) ( )a c b ac42 2- = - vd ãWÎf.

Ô®Î: Ï§F , ,a b c xUT£L¤bjhl®tçir.

vdnt, 2b = a c+ ..... (1)

& 4b2 = a c 2+^ h

& 4b ac4 2- = 4a c ac2+ -^ h

& 4 b ac2-^ h = a c 2-^ h

kh‰WKiw: L.H.S. = b ac4 2-^ h

= b ac4 42-

= 4a c ac2+ -^ h ((1)iaga‹L¤Â)

= a c 2-^ h = R.H.S.

18. a, b, c M»ad T£L¤ bjhl®tçiræš ÏU¥Ã‹ , ,bc ca ab1 1 1 M»ad xU T£L¤

bjhl®tçiræš ÏU¡F« vd ãWÎf.

Ô®Î: Ï§F , ,a b c xUT£L¤bjhl®tçir.

vdnt, , ,abca

abcb

abcc M»adnkY«,T£L¤bjhl®tçir

& , ,bc ca ab1 1 1 M»adxUT£L¤bjhl®tçir.


19. , ,a b c2 2 2 M»ad T£L¤ bjhl®tçiræš ÏU¥Ã‹ , ,

b c c a a b1 1 1+ + +

M»adÎ«

T£L¤ bjhl®tçiræš ÏU¡F« vd¡fh£Lf.

Ô®Î: bfhL¡f¥g£l , ,a b c2 2 2 M»adxUT£L¤bjhl®tçir

våš, b a2 2- = c b2 2

- (bghJé¤Âahr«)

& b a b a+ -^ ^h h = c b c b+ -^ ^h h

& b c c a

b a+ +

-^ ^h h

= a b c a

c b+ +

-^ ^h h

[ a b b c c a+ + +^ ^ ^h h h Mš tF¡f]

& b c c ab c c a+ ++ - -

^ ^h h =

c a a bc a a b+ ++ - -

^ ^h h

& b c c a

b c c a

+ +

+ - +

^ ^^ ^

h hh h =

c a a b

c a a b

+ +

+ - +

^ ^^ ^

h hh h

& c a b c1 1+

-+

= a b c a1 1+

-+

& c a2+

= a b b c1 1+

++

. vdnt, , ,b c c a a b1 1 1+ + +

xUT£L¤bjhl®tçir.

20. , , ,a b c x y z0 0 0x y z ! ! != = k‰W« b ac2= , våš, , ,

x y z1 1 1 M»ad xU T£L¤

bjhl®tçiræš ÏU¡F« vd¡ fh£Lf.

Ô®Î: Ï§F , 0, 0, 0a b c x y zx y z ! ! != = . våš a b c kx y z= = =

& , ,a k b k c kx y z1 1 1

= = =

nkY«, b ac2=

vdnt, y

1 2

kc m = k kx z1 1

& k y2

= kx z1 1+

& y2 =

x z1 1+

Mfnt, , ,x y z1 1 1 xUT£L¤bjhl®tçir.

gæ‰Á 2.3

1. Ã‹tU« bjhl®tçirfëš vJ bgU¡F¤ bjhl®tçir vd¡ fh©f. bgU¡F¤

bjhl®tçirfshf cŸsdt‰¿‹ bghJ é»j« fh©f.

(i) 0.12, 0.24, 0.48,g . (ii) 0.004, 0.02, 0.1,g . (iii) , , , ,21

31

92

274 g .

(iv) 12, 1, ,121 g . (v) , , ,2

2

1

2 2

1 g . (vi) 4, 2, 1, ,21 g- - - .

Ô®Î: (i) Ï§F ..

.

. 20 120 24

0 240 48 g= = =

vdnt, bghJé»j«2. MjyhšÏ¤bjhl®tçirxUbgU¡F¤bjhl®tçir.


(ii) . , . , .0 004 0 02 0 1g

Ï§F, ..

..

0 0040 02

0 020 1 5g= = =

vdnt, bghJé»j«5. Mjyhš,Ï¤bjhl®tçirxUbgU¡F¤bjhl®tçir.

(iii) Ï§F

2131

3192

92274

32g= = = = vdnt, bghJé»j« .

32 .

MjyhšÏ¤bjhl®tçirxUbgU¡F¤bjhl®tçir.

(iv) Ï§F 121

1121

121g= = = . vdntbghJé»j« .

121

Mfnt, Ï¤bjhl®tçirxUbgU¡F¤bjhl®tçir.

(v) Ï§F 2

2

1

2

12 2

1

21g= = = vdnt, bghJé»j«

21 .

Mjhš, Ï¤bjhl®tçirxUbgU¡F¤bjhl®tçir.

(vi) Ï§F 42

21!-

-- , Ïªjbjhl®tçiræšbghJé»j«rkäšiy.

vdnt, Ï¤bjhl®tçirxUbgU¡F¤bjhl®tçirašy.

2. , ,1, 2,41

21 g- - v‹w bgU¡F¤ bjhl®tçiræš 10 MtJ cW¥igÍ«, bghJ

é»j¤ijÍ« fh©f.

Ô®Î: bjhl®tçiræ‹bgU¡Fé»j«

r

4121

211

12 2g=

-

=-

= - = =-

bjhl® tçiræ‹ Kjš cW¥ò 41 .

bjhl®tçiræ‹bghJtot« , , ,t a r n 1 2 3n

n 1 g= =-

vdnt, t41 2

1010 1= - -` ^j h = 2

2

1 22

9 7- =-^ h .

3. xU bgU¡F¤ bjhl®tçiræš 4MtJ k‰W« 7 MtJ cW¥òfŸ Kiwna 54 k‰W«

1458 våš, m¤bjhl®tçiria¡ fh©f.

Ô®Î: Ï§F t 544= k‰W« t 1458

7= .

, , ,t a r n 1 2 3n

n 1 g= =- v‹wN¤Âu¤ijga‹gL¤Â

a r 543= k‰W« a r 14586

=

& a r

a r54

14583

6

= & r r27 33&= =


våš, a r3 = 54

& a 33^ h = 54

& a2754= = 2

vdnt, bgU¡F¤bjhl®tçir , , ,2 2 3 2 3 2 32 3g^ ^ ^ ^ ^ ^h h h h h h

2, 6, 18, 54,& g

4. xU bgU¡F¤ bjhl®tçiræš Kjš k‰W« MwhtJ cW¥òfŸ Kiwna 31 ,

7291

våš, m¥bgU¡F¤ bjhl®tçiria¡ fh©f.

Ô®Î: Kjš cW¥ò 31 k‰W« Mwh« cW¥ò

7291 . våš, a

31= k‰W« a r

72915

=

a r5 = 7291 & r

31

72915

=

r5 = 2431 & r

315 5

= ` j & r31=

vdnt, njitahdbgU¡F¤bjhl®tçir , , ,31

31

31

31

31 2

g` ` ` ` `j j j j j

, , ,31

91

271& g

5. Ã‹tU« bgU¡F¤ bjhl®tçiræš bfhL¡f¥g£l cW¥ò v¤jidahtJ cW¥ò

vd¡ fh©f.

(i) 5, 2, , ,54

258 g , is

15625128 ? (ii) 1, 2, 4, 8,g , is 1024 ?

Ô®Î: (i) 5, 2, , ,54

258 g

15625128 xUbgU¡F¤bjhl®tçiræš

5,a r

52

254

52g= = = = =

, , , ,t a r n 1 2 3n


( )( )

a r15625128

625 252n 17

= =-

& 552

5

2n 1

6

7

=-

` j

& 52

52n 1 7

=-

` `j j & n 1 7- = & n 8=

(ii) bfhL¡f¥g£lit 1, 2, 4, 8,g1024 xUbgU¡F¤bjhl®tçir

Kjš cW¥ò a 1= k‰W«bghJé»j« r12

24 2g= = = =

, , ,t a r n 1 2 3n


a rn 1- = 1024 = 210

& 2 n 1-^ h = 210

& n 1- = 10 & n 11= . vdnt, 11 tJ cW¥ò 1024.


6. 162, 54, 18,g k‰W« , , ,812

272

92 g M»a bgU¡F¤ bjhl®tçirfë‹ n MtJ

cW¥ò rkbkåš, n-‹ kÂ¥ò fh©f.

Ô®Î: 162, 54, 18, g xUbgU¡F¤bjhl®tçiræš

,a r16216254

5418

31g= = = = = .

vdnt, 162t31

n

n 1=

-` j ..... (1)

, , ,812

272

92 g v‹wbgU¡F¤bjhl®tçiræš

,a r812

812272

27292

3g= = = = =

t812 3

nn 1

= -^ h ..... (2)

n tJ cW¥òfŸ rkbkåš.

16231 n 1-

` j = 812 3 1n-^ h

& 3

162n 1-

= 3812 n 1-

& 3n 1 2-^ h = 2

162 81#

& 3n 1 2-^ h = 812

& 3 1n- = 81 34=

& n 1- = 4 & n 5=

vdnt, bgU¡F¤bjhl®tçirfëš 5 tJcW¥òrk«.

7. xU bgU¡F¤ bjhl®tçiræ‹ Kjš cW¥ò 3 k‰W« IªjhtJ cW¥ò 1875 våš,

mj‹ bghJ é»j« fh©f.

Ô®Î: xU bgU¡F¤ bjhl® tçiræ‹Kjš k‰W« IªjhtJcW¥òKiwna 3 k‰W« 1875 våš,

3, 1875a t a r5

4& = = =

& r3 4 = 1875

& r4 = 625 = 54 & r 5=

vdnt, bghJé»j«5.

8. xU bgU¡F¤ bjhl®tçiræ‹ mL¤jL¤j _‹W cW¥òfë‹ TLjš 1039 k‰W«

mt‰¿‹ bgU¡f‰gy‹ 1 våš, m¤bjhl®tçiræ‹ bghJ é»j¤ijÍ«,

m«_‹W cW¥ò¡fisÍ« fh©f.

Ô®Î: xUbgU¡F¤bjhl®tçiræ‹Kjš_‹WcW¥òfŸ , ,ra a ar .


våš, ra a ar+ + =

1039

ar

r1 1+ +` j = 1039

& r

r r 12+ +c m =

1039 g (1)

vdnt, ra a ar 1=` ^ ^j h h

& a 13= mšyJ a 1=

a 1= våš (1) &

r

r r 12+ +^ h =

1039

& r r10 10 102+ + = 39r

& r r10 29 102- + = 0

& r r2 5 5 2- -^ ^h h = 0

& r25= mšyJ

52

r25= våš _‹W cW¥òfŸ , ,

52 1

25 ; r

52= våš _‹W cW¥òfŸ , ,

25 1

52 .

9. xU bgU¡F¤ bjhl®tçiræš mL¤jL¤j 3 cW¥òfë‹ bgU¡F¤ bjhif 216 k‰W« mitfëš Ïu©ou©L cW¥ò¡fë‹ bgU¡f‰gy‹fë‹ TLjš 156

våš, mªj cW¥òfis¡ fh©f.


våš 216ra a ar =` ^ ^j h h

& a 2163= & a 6=

nkY«, ra a a ar ar

ra 156+ + =` ^ ^ ^ ^ `j h h h h j

& ra a r a2

2 2+ + = 156

& ar

r1 12+ +` j = 156

& 36 1r

r r2+ +c m = 156

& r

r r 12+ + =

313

& r r3 3 32+ + = 13r

& r r3 10 32- + = 0

& r r3 1 3- -^ ^h h = 0vdnt, r 3= mšyJ

31

3r = våš _‹W cW¥òfŸ 2, 6, 18

r31= våš _‹W cW¥òfŸ 18, 6, 2



mt‰¿‹ jiyÑêfë‹ TLjš 47 våš, m›ÎW¥òfis¡ fh©f.


våš, ra a ar+ + = 7

& ar

r1 1+ +` j = 7

& ar

r r 12+ +c m = 7 g (1)

nkY«, ar

a ar1 1+ + =

47

& a

rr

1 1 1+ +8 B = 47

& a r

r r1 12+ +; E =

47 g (2)

(1) ' (2) »il¥gJ, a 7742

#=

& a 42= & a 2!=

a 2= xU äif v©,

( )1 & 2r

r r 12+ +c m = 7

& 2 2 2r r2+ + = r7

& r r2 5 22- + = 0

& r r2 1 2- -^ ^h h = 0

& r 2= mšyJ 21

,a r2 2= = våš _‹W cW¥òfŸ 1, 2, 4

,a r221= = , våš _‹W cW¥òfŸ 4, 2, 1.

11. xU bgU¡F¤ bjhl®tçiræš Kjš _‹W cW¥òfë‹ TLjš 13 k‰W« mt‰¿‹

t®¡f§fë‹ TLjš 91 våš, m¤bjhl®tçiria¡ fh©f.

Ô®Î: xUbgU¡F¤bjhl®tçiræ‹_‹WcW¥ò¡s , ,a ar ar2 våš,

a r r1 2+ +^ h = 13 (1)

a r r12 2 4+ +^ h = 91 (2)

(2) ' (1) &

a r r

a r r

1

12 2 2

2 2 4

+ +

+ +

^

^

h

h = 16991

137=

& r r

r r r r

1

1 12 2

2 2

+ +

+ + - +

^

^ ^

h

h h = 137

& r r

r r

1

12

2

+ +

- + = 137


& 3 10 3r r2- + = 0

& r r3 3 1- -^ ^h h = 0

& r 3= mšyJ 31

r 3= våš _‹W cW¥òfŸ 1, 3, 9; r31= våš _‹W cW¥òfŸ 9, 3, 1.

12. xUt® M©o‰F 5% T£L t£o jU« xU t§»æš 1000-I it¥ò ãÂahf it¤jhš, 12 M« tUlKoéš »il¡F« bkh¤j¤ bjhifia¡ fh©f.

Ô®Î: mrš ` 1000. xUtUl¤Â‰fhdt£o = 1000 1005` j

KjštUlKoéšcŸsbjhif

100 100001005+ ` j = 1000 1

1005+` j

Ïu©lhtJtUl¤Â‰fhdt£o = 1000 15005

1005+` `j j

Ïu©lhtJtUlKoéšcŸsbjhif

= 1000 11005 1000 1

1005

1005+ + +` ` `j j j

= 1000 11005 1

1005+ +` `j j

= 1000 11005 2

+` j

Ï›thW bjhl®ªjhš 12M«M©LKoéš»il¡F«bjhif

= ` 1000 11005 12

+` j = ` 1051000100

12` j

F¿¥ò: nknycŸsKiwiaga‹gL¤jhkšT£Lt£o¡fz¡Ffë‹bkh¤j

bjhif¡fhQ«N¤Âu«A P i1 n= +^ h , A v‹gJbkh¤jbjhif, P v‹gJ mrš

,i r r100

= v‹gJM©Lt£oé»j«n M©Lfë‹v©â¡if.

A = 1000 11005 12

+` j = ` 1000100105 12` j

13. xU ãWtd« 50,000-¡F xU m¢R¥ÃuÂ ÏaªÂu¤ij th§F»wJ. m›éaªÂu«

x›bthU M©L« j‹ kÂ¥Ãš 15% ÏH¡»wJ vd kÂ¥Ãl¥gL»wJ. 15

M©LfS¡F¥ ÃwF mªj m¢R¥ÃuÂ ÏaªÂu¤Â‹ kÂ¥ò v‹d?

Ô®Î: KjštUlÏWÂæšm¢R¥ÃuÂÏaªÂu¤Â‹kÂ¥ò 50,000t10085

1#= ` j

Ïu©lh«tUlÏWÂæšm¢R¥ÃuÂÏaªÂu¤Â‹kÂ¥ò

t2

= 5000010085

10085

# #` j

= 5000010085 2

# ` j

vdnt, 15 M©LfS¡F¥ÃwFm¢R¥ÃuÂÏaªÂu¤Â‹kÂ¥ò

t15

= ` 5000010085 15

` j .


14. , , ,a b c d M»ad xU bgU¡F¤ bjhl®tçiræš mikªjhš,

.a b c b c d ab bc cd- + + + = + +^ ^h h

Ô®Î: , , ,a b c d v‹gdxUbgU¡F¤bjhl®tçiræšmikªJŸsd.

Ïj‹bghJé»j«r v‹f. vdnt, , ,b ar c ar d ar2 3= = = .

vdnt, a b c b c d- + + +^ ^h h = a ar ar ar ar ar2 2 3- + + +^ ^h h

= a r r r r r1 12 2 2- + + +^ ^h h

= a r a r a r2 2 3 2 4+ +

= ab bc cd+ +

kh‰WKiw: , , ,a b c d xUbgU¡F¤bjhl®tçiræšmikªJŸsd.

våš, b ac2= , ad bc= k‰W« c bd2

= . j‰nghJ, a b c b c d- + + +^ ^h h

= ab ac ad b bc bd bc c cd2 2+ + - - - + + +

= ab ac b ad bc bd c bc cd2 2+ - + - + - + + +^ ^ ^h h h

= ab bc cd+ + .

15. , , ,a b c d M»ad xU bgU¡F¤ bjhl®tçiræš mikªjhš , , ,a b b c c d+ + + v‹gitÍ« bgU¡F¤ bjhl®tçiræš mikÍ« vd ãWÎf.

Ô®Î: , , ,a b c d v‹gdxUbgU¡F¤bjhl®tçiræšmikªJŸsd.

Ïj‹bghJé»j«r våš, , ,b ar c ar d ar2 3= = =

Ï¥bghGJ, a bb c

++ =

a arar ar2

++

= a r

ar r

1

1

+

+

^^

hh = r

nkY«, b cc d

++ =

ar ar

ar ar2

2 3

+

+ = 1

ar r

ar rr

1

2

+

+=

^^

hh

vdnt, , ,a b b c c d+ + + xUbgU¡F¤bjhl®tçir.

gæ‰Á 2.4

1. Ã‹tUtdt‰¿‹ TLjš fh©f. (i) Kjš 75 äif KG¡fŸ (ii) Kjš 125 Ïaš

v©fŸ,

Ô®Î: (i) 1 2 3 75g+ + + + xUT£L¤bjhl®tçirvåš,

,a d1 2 1 1= = - = k‰W« n 75=

Sn = n a n d

22 1+ -^ h6 @ v‹wN¤Âu¤ij¥ga‹gL¤Â

s75

= ( ) ( )275 2 1 75 1 1+ -^ h6 @


= 275 2 74+6 @ =

275 766 @ = 75 38#6 @

vdnt, S75

= 2850

F¿¥òiu: nkYŸs fz¡»id S n a l2n

= +6 @ v‹w N¤Âu¤ij ga‹gL¤ÂÍ«

Ô®Î fhzyh«. Ï§F, 75, 1n a= = .

` S75

= 275 1 75+6 @ =

275 76 2850=^ h

(ii) 1 2 3 125g+ + + + xUT£L¤bjhl®tçirvåš,

, ,a d n l1 1 125= = = =

` S125

= 2

125 1 125+6 @ S n a l2n

` = +` j6 @

= 2

125 126# = 125 63#

S125

= 7875

2. n MtJ cW¥ò 3 2n+ v‹wthW mikªj xU T£L¤ bjhl®tçiræ‹

Kjš 30 cW¥òfë‹ T£l‰gyid¡ fh©f.

Ô®Î: xUT£L¤bjhl®tçiræ‹n MtJ cW¥ò n3 2+ .

tn = 3 2 ( )( )n n5 1 2+ = + - ( ( 1)a n d+ - v‹wtoéšcŸsJ )

S30

= 230 2 5 30 1 2+ -^ ^ ^h h h6 @ = 15 10 58+6 @ = 15 68#

S30

= 1020

3. Ã‹tU« T£L¤bjhl®fë‹ T£l‰gyid¡ fh©f.

(i) 38 35 32 2g+ + + + . (ii) 6 5 4 2541

21 g+ + + cW¥òfŸ tiu.

Ô®Î: (i) 38 35 32 2g+ + + + v‹wT£L¤bjhl®tçiræš

, ,a d l38 35 38 3 2= = - =- =

( 1)l a n d &= + - nd

l a 13

2 38 1= - + =-- + =13

vdnt, S13

= 213 2 38+6 @ ( )S n a l

2n= +6 @

= 213 406 @ = 260=

(ii) 6 5 4 2541

21 g+ + + cW¥òfŸcŸsT£L¤bjhl®tçiræš

6, 5 6a d41

43= = - = - k‰W« n 25= .

Sn = n a n d

22 1+ -^ h6 @

vdnt, S25

= 225 2 6 25 1

43+ - -^ ^ `h h j8 B

= 3225 12 24

4+ -` j8 B =

225 12 18-6 @ 75=- .


4. Ã‹tU« étu§fis¡ bfh©l T£L¤ bjhl®fë‹ TLjš Sn fh©f.

(i) 5,a = 30,n = 121l = (ii) 50,a = 25,n = 4d =-

Ô®Î: (i) Ï§F , ,a n l5 30 121= = =

Sn = n a l

2+6 @

S30

= 230 5 121+6 @ = 15 1266 @ = 1890.

(ii) Ï§F ,a n50 25= = k‰W« d 4=-

Sn = n a n d

22 1+ -^ h6 @

S25

= 225 2 50 25 1 4+ - -^ ^ ^h h h6 @

= 225 100 24 4+ -^ h6 @ =

225 100 96-6 @ =

225 4 50=6 @

5. 1 2 3 42 2 2 2

g- + - + v‹w bjhlç‹ Kjš 40 cW¥òfë‹ T£l‰gyid¡ fh©f.

Ô®Î: bfhL¡f¥g£l bjhl® tçiria Ñœ¡f©lthW vGj

1 4 9 16 25 36- + - + - +^ ^ ^h h h . . . 20 cW¥òfŸ

= 3 7 11- + - + - +^ ^ ^h h h . . . 20 cW¥òfŸ

Ï¡T£L¤bjhl®tçiræš, 3,a d 4=- =- k‰W« n 20= .

Sn = n a n d

22 1+ -^ h6 @

S20

= 220 6 19 4- + -^ ^h h6 @ = 10 82-^ h = – 820

kh‰WKiw: 1 2 3 4 39 402 2 2 2 2 2g- + - + + -

= 1 2 3 4 39 402 2 2 2 2 2g+ + + + + +

2 2 4 6 402 2 2 2g- + + + +^ h

= 1 2 3 40 2 2 1 2 202 2 2 2 2 2 2 2g g+ + + + - + + +^ ^h h

= 40

6

41 818

6

20 21 41-

^ ^ ^ ^ ^ ^h h h h h h

= 20 41 27 28 820- =-^ ^h h6 @

6. xU T£L¤ bjhlçš Kjš 11 cW¥òfë‹ TLjš 44 k‰W« mj‹ mL¤j

11 cW¥òfë‹ TLjš 55 våš, m¤bjhliu¡ fh©f.

Ô®Î: Sn = ( )n a n d

22 1+ -6 @

vdnt, S 4411

= 44a d211 2 11 1& + - =^ ^h h6 @ & a d5 4+ = (1)

nkY«, S S 5522 11

= + = 44 55 99+ =


& a d222 2 22 1 99+ - =^ h6 @ & a d2 21 9+ = (2)

(1) k‰W« (2) I Ô®¡f, a1139= k‰W« d

111= .

vdnt,njitahdT£L¤bjhl®tçir ( ) ( )a a d a d2 g+ + + + +

= 1139

1139

111

1139

112 g+ + + + +` `j j =

1139

1140

1141

1142 g+ + + + .

7. 60, 56, 52, 48,g v‹w T£L¤ bjhl®tçiræ‹ Kjš cW¥ÃèUªJ bjhl®¢Áahf

v¤jid cW¥òfis¡ T£odhš TLjš 368 »il¡F«?

Ô®Î: 60, 56, 52, 48,g xUT£L¤bjhl®tçirvåš,

,a d60 56 60 52 56 4g= = - = - = =-

nkY«, S 368n= . våš, ' 'n ‹ kÂ¥ò

Sn = n a n d

22 1+ -^ h6 @

368 = n n2

2 60 1 4+ - -^ ^ ^h h h6 @

& n n2120 4 4- +6 @ = 368

& n n2124 4-6 @ = 368 & n n62 2-^ h = 368

& n n2 62 3682- + = 0 & n n31 1842

- + = 0& n n8 23- -^ ^h h = 0 & n 8= mšyJ 23vdnt, 8 cW¥òfŸ mšyJ 23 cW¥òfë‹ TLjš 368 »il¡F«.

8. 9 Mš tFgL« mid¤J _‹¿y¡f Ïaš v©fë‹ TLjš fh©f.

Ô®Î: 9 Mš tF¥gL« 3 Ïy¡f Ïaš v©fŸ , , , ,108 117 126 999g .

ÏJxUT£L¤bjhl®tçirvåš, ,a d108 9= = k‰W« l 999= .

nkY«, ( )l a n d1 &= + - n = d

l a 1- +

& = 9

999 108 1- + = 9

891 1+ = 99 1 100+ =

vdnt, ( )S n a l2n

&= + S100

= 2

100 999 108+6 @

= 50 1107 55350=^ h .

9. xU T£L¤ bjhlç‹ 3 MtJ cW¥ò 7 k‰W« mj‹ 7 MtJ cW¥ghdJ 3 MtJ

cW¥Ã‹ _‹W kl§if él 2 mÂf«. m¤bjhlç‹ Kjš 20 cW¥òfë‹

T£l‰gyid¡ fh©f.

Ô®Î: Ï§F t 73= k‰W« t t2 3 23

7 3= + =

t a n d1n= + -^ h

a d2+ = 7 g (1) a d6+ = 23 g (2)


( ) ( )2 1- & d4 = 16 & d = 4

d 4= våš (1) 2a 4& + ^ h = 7 & a = 1-

Ï¥bghGJ, { 2 ( 1) }S n a n d2n

&= + - S20

= 220 2 1 19 4- +^ ^h h6 @

= 10 740.2 76- + =6 @

10. 300-¡F« 500-¡F« ÏilnaÍŸs 11 Mš tFgL« mid¤J Ïaš v©fë‹

T£l‰gy‹ fh©f.

Ô®Î: xUT£L¤bjhl®tçiræ‹a = 308, l = 495 k‰W«

bghJé¤Âahr«, d = 11.

nkY«, ( )l a n d1 &= + - n = d

l a 1- + = 111

495 308 18- + =

S n a l2n

= +6 @ v‹wN¤Âu¤ij¥ga‹gL¤Â

S18

= 218 308 495+6 @ = 9 8036 @ = 7227.

11. 1 6 11 16 148xg+ + + + + = våš, x-‹ kÂ¥Ãid¡ fh©f.

Ô®Î: bfhL¡f¥g£lT£L¤bjhl®tçiræš 1a = k‰W« 5d =

bfhL¡f¥llit 1 6 11 16 148xg+ + + + + =

j‰nghJ, Sn = 148 & n a n d

22 1+ -^ h6 @ = 148

& 2 5n n2

1 1+ -^ ^h h6 @ = 148

& n n2

5 3-^ h = 148 & n n5 3 2962- - = 0

& n n5 37 8+ -^ ^h h = 0 & n 8= mšyJ 537-

vdnt, n 8= [Ï§F, n537= - V‰W¡bfhŸsj¡fjšy]

Mfnt, x = t8 = a +7d = 1+7(5) = 36.

12. 100-¡F« 200-¡F« ÏilnaÍŸs 5 Mš tFglhj mid¤J Ïaš v©fë‹


Ô®Î: T v‹gJ 100¡F« 200¡F« Ïilna cŸs Ïaš v©fë‹ TLjš.vdnt, 101 102 103 199T g= + + + + , xUT£L¤bjhl®tçirvåš,

101 , 199a l= = k‰W« 99n =

Mfnt, 99 150 14850T299 101 199 #= + = =^ h .

S v‹gJ 100¡F« 200¡F« Ïilna 5 Mš tFgL« Ïaš v©fë‹ TLjš.vdnt, 105 110 115 195S g= + + + + . Ï§FÏJxUT£L¤bjhl®tçir

, ,a d l105 5 195= = = .

vdnt, ( )l a n d1 &= + - n = d

l a 1- + = 1955105 1- + = 18 1 19+ = .

S n a l2n

= +6 @ v‹wN¤Âu¤ijga‹gL¤Â


S = 219 195 105+6 @ = 19 150 2850# = .

vdnt, 100¡F« 200¡F« Ïilna 5 Mš tFglhj Ïaš v©fë‹ TLjš

T – S = 14850 2850 12000.- =

13. xU f£Lkhd FGk«, xU ghy¤ij f£o Ko¡f¤ jhkjkhF« nghJ, jhkjkhF«

x›bthU ehS¡F« mguhj¤bjhif¡ f£lnt©L«. Kjš ehŸ jhkj¤Â‰F

mguhj« `4000 nkY« mL¤JtU« x›bthU ehS¡F« Kªija ehis él `1000

mÂf« brY¤j nt©oæU¡F«. tuÎ bryÎ¤Â£l¤Â‹go m¡FGk« bkh¤j

mguhj¤bjhifahf `1,65,000 brY¤j ÏaY« våš, v¤jid eh£fS¡F ghy«

Ko¡F« gâia jhkj¥gL¤jyh«?

Ô®Î: Ï§F ,a d4000 1000= = .

ghy«Ko¡F«gâiajhkj¥gL¤J«n eh£fë‹mguhj¤bjhif.

vdnt, Sn = 1,65,000 ( bfhL¡f¥g£lJ )

& n n2

2 4000 1 1000+ -^ ^ ^h h h6 @ = 1,65,000

& n n8000 1000 1000+ -6 @ = 3,30,000

& 7 330n n2+ - = 0

& 15n n22+ -^ ^h h = 0 & n 15= mšyJ 22-

mÂf g£rkhf 15 eh£fŸghy«f£L«gâiajhkj¥gL¤jyh«.

14. 8% Åj« jåt£o jU« ãWtd¤Âš x›bthU M©L« 1000 it¥ò¤ bjhifahf

brY¤j¥gL»wJ. x›bthU M©o‹ ÏWÂæš bgW« t£oia¡ fz¡»Lf. bgW«

t£o¤bjhiffŸ xU T£L¤ bjhl®tçiria mik¡Fkh? m›thW mik¡f

KoÍkhdhš, 30 M©Lfë‹ Koéš »il¡F« bkh¤j t£oia¡ fh©f.

Ô®Î: 8% jåt£o¡F`1000 x›bthUM©L«it¥ò¤bjhifahfbrY¤j¥

gL»wJ.

KjštUlt£o, t 10001008 80

1#= =

Ïu©lh«tUlt£o, t 20001008 160

2#= =

vdnt,t£o¤bjhif 80 , 160 , 240, g ÏJxUT£L¤bjhl®tçirvåš, a 80= k‰W« d 80= .

Mfntbkh¤jt£o { 2 ( 1) }S n a n d2n

&= + - S230 160 29 80

30= + ^ h6 @ = ` 7200

15. xU bjhlç‹ Kjš n cW¥òfë‹ TLjš 3 2n n2- våš, m¤bjhluhdJ xU

T£L¤ bjhl® vd ãWÎf.

Ô®Î: Ï§F, Sn = 3 2n n2

-

nkY«, Sn 1-

= n n3 1 2 12- - -^ ^h h

= n n n3 2 1 2 22- + - +6 @ = n n3 8 52

- +


Ï¥bghGJ, tn = S S

n n 1-

-

= n n n n3 2 3 8 52 2- - - +6 @

= n n n6 5 6 6 1 1 1 6- = - + = + -^ h .

vdnt, tn

= a n d1+ -^ h .

Mifahš, ÏJxUT£L¤bjhl®tçirvåš, ,a d1 6= = .

16. xU fofhu« xU kâ¡F xU Kiw, 2 kâ¡F ÏU Kiw, 3 kâ¡F _‹W Kiw

v‹wthW, bjhl®ªJ rçahf x›bthU kâ¡F« xè vG¥ò« våš, xU ehëš

m¡fofhu« v¤jid Kiw xè vG¥ò«?

Ô®Î: xUfofhu«x›bthUkâ¡F«vG¥ò«xèxUT£L¤bjhl®.

1 2 3 12g+ + + +

vdnt,Sn= n a l2

+6 @& S12

= 212 1 12 6 13 78+ = =^ h6 @ .

xUehëšfofhu«vG¥ò«xèmsé‹v©â¡if = 2 78# = 156 Kiw. 17. Kjš cW¥ò a, Ïu©lh« cW¥ò b k‰W« filÁ cW¥ò c vd¡ bfh©l xU T£L¤

bjhlç‹ T£l‰gy‹ b a

a c b c a

2

2

-

+ + -

^^ ^

hh h vd¡fh£Lf.

Ô®Î: bfhL¡f¥g£LŸsgo ,t a t b1 2= = k‰W« filÁ cW¥ò t l c

n= =

bghJé¤Âahr«, d = t t2 1- = b a-

vdnt, tn = a n d c1+ - =^ h & a n b a c1+ - - =^ ^h h

& n 1- = b ac a

-- & n =

b ab c a2

-+ -

Mfnt, Sn = n a l

2+6 @

= b a

b c aa c

2

2

-

+ -+

^^

^hh

h

18. xU T£L¤ bjhlçš n2 1+^ h cW¥òfŸ ÏU¥Ã‹ x‰iw¥gil cW¥òfë‹

T£l‰gyD¡F«, Ïu£il¥gil cW¥òfë‹ T£l‰gyD¡F« ÏilnaÍŸs é»j« :n n1+^ h vd ãWÎf.

Ô®Î: xUT£L¤bjhl®tçiræš n2 1+^ hcW¥òfŸcŸsd.

T k‰W« S M»ad x‰iwgil cW¥òfŸ k‰W« Ïu£il¥ gil cW¥òfë‹ T£L¥

gyid¡F¿¡»wJ.

våš, T = t t t tn1 3 5 2 1

g+ + + ++

= 21n t t

n1 2 1+ +

+` j 6 @ (n 1+ cW¥òfŸ)

= 21 { ( ) }n a a n d2+ + +` j 6 @

= na nd

2

12

++

^^

hh6 @ = n a nd1+ +^ ^h h.


våš, S = t t t tn2 4 6 2

g+ + + +

= n t t2 n2 2

+6 @ (n cW¥òfŸ)

= { } { }n a d a n d2

2 1+ + + -^ h6 @

= n a nd2

2 2+6 @ = n a nd+^ h

Mfnt, ST =

n a nd

n a nd

nn1 1

+

+ += +

^^ ^

hh h .

19. xU T£L¤ bjhlçš Kjš m cW¥òfë‹ T£l‰gyD¡F«, Kjš n cW¥òfë‹

T£l‰gyD¡F« ÏilnaÍŸs é»j« :m n2 2 våš, m MtJ cW¥ò k‰W« n

MtJ cW¥ò M»aitfŸ :m n2 1 2 1- -^ ^h h v‹w é»j¤Âš mikÍ« vd¡

fh£Lf.

Ô®Î: bfhL¡f¥g£LŸsgo,

S

S

n

m

n

m2

2

= & 2n a n d

m a m d

n

m

22 1

2 1

2

2

+ -

+ -=

^

^

h

h

6

6

@

@

& a n d

a m d

nm

2 1

2 1

+ -

+ -=

^^

hh

& an mnd nd am mnd md2 2+ - = + -

& an nd am md2 2- = -

& a n m n m d2 - = -^ ^h h & a d2 =

vdnt, t

t

n

m = a n d

a m d

1

1

+ -

+ -

^^

hh =

a n a

a m a

1 2

1 2

+ -

+ -

^ ^^ ^

h hh h

= a n

a m

1 2 2

1 2 2

+ -

+ -

66

@@ =

nm2 12 1

-- .

20. xU njh£l¡fhu® rçtf toéš Rt® x‹¿id mik¡f Â£läL»wh®. rçtf¤Â‹

Ú©l Kjš tçir¡F 97 br§f‰fŸ njit¥gL»wJ. Ã‹ò x›bthU tçiræ‹

ÏUòwK« Ïu©ou©L br§f‰fŸ Fiwthf it¡f nt©L«. m›totik¥Ãš 25

tçirfëU¥Ã‹, mt® th§f nt©oa br§f‰fë‹ v©â¡if v¤jid?Ô®Î: x›bthUtçiræY«ga‹gL¤Âabr§f‰fë‹v©â¡ifxUT£L¤

bjhl®tçiræšmik»wJ.

vdntbr§f‰fë‹v©â¡if = 97 93 89 g+ + + 25 cW¥òfŸ

,a d97 4= =- k‰W« n 25= .

nkY«, Sn = n a n d

22 1+ -^ h6 @

S25

= 225 2 97 25 1 4+ - -^ ^ ^h h h6 @

= 225 194 96-6 @ =

225 986 @

njitahdbr§f‰fë‹v©â¡if = 1225.


gæ‰Á 2.5

1. 25

65

185 g+ + + v‹w bgU¡F¤ bjhlç‹ Kjš 20 cW¥òfë‹ TLjiy¡ fh©f.

Ô®Î: bfhL¡f¥g£lbgU¡F¤bjhl®tçir25

65

185 g+ + + våš,

,a r25

2565

65185

31g= = = = = k‰W« n = 20.

Sn =

ra r11 n

--^ h

vdnt, S20

= 1

31

25 1

31 20

-

- ` j; E = 2

5

32

131 20

- ` j; E =

415 1

31 20

- ` j; E

2. 91

271


Ô®Î: bfhL¡f¥g£lbgU¡F¤bjhl®tçir 91

271

811 g+ + +

a = , r91

91271

31= = k‰W« n = 27.

vdnt, Sn=

ra r11 n

--^ h & S

27=

131

91 1

31 27

-

- ` j; E =

32

91 1

31 27

- ` j; E =

61 1

31 27

- ` j; E.

3. Ã‹tU« étu§fis¡ bfh©l bgU¡F¤ bjhlç‹ TLjš Sn fh©f.

(i) 3,a = 384,t8= 8n = . (ii) 5,a = 3r = , 12n = .

Ô®Î: (i) bfhL¡f¥g£lit ,a 3= 384,t8= n 8= .

t8 = a r8 1$ =- a r7

& a r7 = 384 & r3 7^ h = 384

& r7 = 128 & r 27 7= & r = 2

vdnt, Sn=

ra r11 n

--^ h & S

8 =

1 23 1 28

--6 @ = 3 256 1 765- =6 @ .

(ii) bfhL¡f¥g£lit ,a 5= r 3= , n 12=

vdnt, Sn=

ra r

11n

--^ h & S

12 =

3 15 3 112

--6 @ =

25 3 112

-6 @.

4. Ã‹tU« KoÎW bjhl®fë‹ TLjš fh©f.

(i) 1 0.1 0.01 0.001 .0 1 9g+ + + + +^ h (ii) 1 11 111 g+ + + 20 cW¥òfŸ tiu.

Ô®Î: (i) bfhL¡f¥g£lbgU¡F¤bjhl®tçir1 0.1 0.01 0.001 .0 1 9g+ + + + + ^ h

, .a r1 0 1= = k‰W« .t 0 1n

9= ^ h


Mfnt, tn = a rn 1

#- & ( . )0 1 9 = .1 0 1 n 1-^ ^h h

& 1n - = 9 & n 10= .

vdnt, Sn =

ra r11 n

--^ h & S

10 =

.

.

1 0 1

1 1 0 1 10

-

-^

^ ^h

h h6 @ =

.

.

0 9

1 0 1 10- ^ h .

(ii) bfhL¡f¥g£lbgU¡F¤bjhl®tçir1 11 111 g+ + + 20 cW¥òfŸ tiu

S20

= 1 11 111 g+ + + 20 cW¥òfŸ tiu

9 Mš bgU¡» tF¡f,

S20

= [91 9 99 999 g+ + + 20 cW¥òfŸ tiu]

= [91 10 1 100 1 1000 1 g- + - + - +^ ^ ^h h h 20 cW¥òfŸ tiu]

= 91 10 10 102 3 g+ + +^6 20 cW¥òfŸ tiu) 20- @

= 91

10 110 10 1

2020

#-

--

^ h; E' 1.

( Ï§F, n cW¥òfë‹T£L¤bjhif r

a r11n

--^ h )

= 10 18110

92020

- -^ h6 @ .

5. Ã‹tU« bjhl®fëš, v¤jid cW¥òfis¡ T£odhš

(i) 3 9 27 g+ + + TLjš 1092 »il¡F«?

(ii) 2 6 18 g+ + + TLjš 728 »il¡F«?

Ô®Î: (i) bfhL¡f¥g£l n cW¥òfë‹ TLjš 3 9 27 t 1092n

g+ + + + = .

ÏJxUbgU¡F¤bjhl®tçir 3, 3.a r39

927 g= = = = = våš,

S 1092n=

& r

a r11n

--^ h =1092 &

3 13 3 1n

--6 @ = 1092

& 3 1 728n- = & 3 729n

= = 36

vdnt, bfhL¡f¥g£lcW¥òfë‹v©â¡if, n 6=

(ii) bfhL¡f¥g£lbgU¡F¤bjhl®tçir2 6 18 g+ + +

, ,a r226 3= = = k‰W« S 728

n= .

vdnt, Sr

a r11

n

n

=--^ h &

3 12 3 1n

--6 @ = 728 & 3 1n

- = 728

& 3n = 729 3 6n6&= =

vdnt, bfhL¡f¥g£lcW¥òfë‹v©â¡if, n 6= .

6. xU bgU¡F¤ bjhlçš Ïu©lhtJ cW¥ò 3 k‰W« mj‹ bghJ é»j« .54 våš,

T£L¤ bjhlçYŸs Kjš cW¥ÃèUªJ bjhl®¢Áahf 23 cW¥òfë‹ TLjš

fh©f.


Ô®Î: bfhL¡f¥g£lit ,t r354

2= = k‰W« n 23=

3t2

&= ar = 3 .a

543

415& = =

vdnt, Sr

a r11

n

n

=--^ h & S

23 =

154

415

154

23

-

- ` j; E =

1415

5

154 23

- ` j; E

= 475 1

54 23

- ` j; E.

7. bghJ é»j« äif v©zhf cŸs xU bgU¡F¤ bjhlçš 4 cW¥òfŸ cŸsd. mj‹

Kjš Ïu©L cW¥òfë‹ TLjš 9 k‰W« filÁ Ïu©L cW¥òfë‹ TLjš 36

våš, m¤bjhliu¡ fh©f.

Ô®Î: bgU¡F¤bjhl®tçiræ‹eh‹FcW¥òfŸa ,ar , ar2 , ar3 .

bfhL¡f¥g£lit 9 ( ) ;a ar 1g+ = ( )ar ar 36 22 3 g+ =

vdnt, (2) & r a ar2+^ h = 36

& r 92^ h = 36 & r 42= & r 2!=

0, 2r r> = . vdnt, a + ar = 9 & a = 3.

Mfnt, njitahd eh‹F v©fŸ 3 + 3(2) + 3(22 ) + 3(23 ) = 3 + 6 + 12 + 24.

8. Ã‹tU« bjhl®fë‹ Kjš n cW¥òfë‹ TLjš fh©f.

(i) 7 77 777 g+ + + . (ii) 0.4 0.94 0.994 g+ + + .Ô®Î: (i) bfhL¡f¥g£lbgU¡F¤bjhl®tçir 7 77 777 g+ + + n cW¥òfŸ

7 IbghJ¡fhuâahfvL¤J9 Mš bgU¡» tF¡f

Sn =

97 9 99 999 g+ + +6 n cW¥òfŸ tiu]

= 97 10 1 100 1 1000 1 g- + - + - +^ ^ ^h h h6 n cW¥òfŸ]

= (97 10 100 1000 g+ + +6 n cW¥òfŸ) (1 1 1 ng- + + cW¥òfŸ)

= n97 10 10 102 3 g+ + +6 cW¥òfŸ n- @

= 10 110 1 n

97 10

n

-- -c m; E

( bgU¡F bjhlç‹ TLjš r

a r11n

--^ h )

= n8170 10 1

97n

- -^ h

(ii) bfhL¡f¥g£lbgU¡F¤bjhl®tçir 0.4 0.94 0.994 g+ + + .

Sn = . . .0 4 0 94 0 994 g+ + + n cW¥òfŸ


Sn = . . .1 0 6 1 0 06 1 0 006 g- + - + - +^ ^ ^h h h n cW¥òfŸ

= (1 1 1 g+ + + n cW¥òfŸ) ( . . .0 6 0 06 0 006 g- + + + n cW¥òfŸ)

= n 6101

10

1

10

12 3

g- + + +; n cW¥òfŸ]

= 101

n 61

101

1101 n

--

- ` j8>

BH = n 6

91 1

101 n

- - `` j j8 B = n32 1

101 n

- - ` j8 B

( bgU¡F bjhlç‹ TLjš r

a r11n

--^ h )

9. bjh‰WnehŒ guÎ« fhy¤Âš, Kjš thu¤Âš 5 ngU¡F clšey¡ FiwÎ V‰g£lJ.

clšey¡FiwÎ‰w x›bthUtU« Ïu©lhtJ thu ÏWÂæš 4 ngU¡F m¤bjh‰W

nehia¥ gu¥òt®. Ï›tifæš bjh‰WnehŒ guédhš 15 MtJ thu ÏWÂæš

v¤jid ng®, m¤bjh‰Wnehædhš ghÂ¡f¥ggLt®?

Ô®Î: bjh‰W nehahš ghÂ¡f¥gL« k¡fë‹ v©â¡if thu« thçahf xU

bgU¡F¤bjhl®tçiriamik¡»wJ. våš, 15 M« thu ÏWÂæš bjh‰W

nehahšghÂ¡f¥g£lk¡fë‹v©â¡if

S15

= 5 4 5 4 20 4 80# # # g+ + + +^ ^ ^h h h 15 cW¥òfŸ

= 5 20 80 g+ + + 15 cW¥òfŸ

ÏJxUbgU¡F¤bjhl®tçirvåš, 5, 4a r= = , n 15=

vdnt, Sr

a r11

n

n

=--^ h & S

15 =

4 15 4 115

--6 @ =

35 4 115

-6 @. 10. e‰gâ brŒj xU ÁWtD¡F¥ gçrë¡f éU«Ãa njh£l¡fhu® Áy kh«gH§fis

gçrhf më¡f K‹tªjh®. m¢ÁWt‹ cldoahf 1000 kh«gH§fis¥ bg‰W¡

bfhŸsyh« mšyJ Kjš ehëš 1 kh«gH«, Ïu©lh« ehëš 2 kh«gH§fŸ,

_‹wh« ehëš 4 kh«gH§fŸ, eh‹fh« ehëš 8 kh«gH§fŸ g vDkhW 10

eh£fS¡F¥ bg‰W¡ bfhŸsyh« vd ÏU thŒ¥òfŸ më¤jh®. m¢ÁWt‹ mÂf

v©â¡ifÍŸs kh«gH§fis¥ bgw vªj thŒ¥Ãid nj®ªbjL¡f nt©L«?

Ô®Î: xU khzt‹ 10 eh£fS¡F bg‰W¡ bfh©l kh«gH§fŸ

S10

= 1 2 4 8 g+ + + + to 10 cW¥òfŸ

ÏJxUbgU¡F¤bjhl®tçirvåš, ,a r1 2= = k‰W« n 10= .

vdnt, Sr

a r11

n

n

=--^ h & S

10 = ( )

2 11 2 110

--6 @ = 2 110

- = 1023.

Mfnt, mªj khzt‹ 10 eh£fS¡F kh«gH« bg‰W¡bfhŸS« thŒ¥ig

nj®ªbjL¤jh‹. 11. Ïu£il¥gil v©â¡ifæš cW¥òfis¡ bfh©l xU bgU¡F¤ bjhlç‹

x‰iw¥ gil v©fshš F¿¡f¥gL« cW¥òfë‹ TLjè‹ _‹W kl§F,

m¥bgU¡F¤ bjhlçYŸs mid¤J cW¥òfë‹ TLjY¡F¢ rkbkåš, mj‹

bghJ é»j¤ij¡ fh©f.


Ô®Î:

bfhL¡f¥g£lit Sn2 = 3 (x‰iw¥ gil v©fë‹ TLjš)

& Sn2 = 3 t t t t

n1 3 5 2 1g+ + + +

-6 @

& 3arr a ar ar ar

11 n

n2

2 4 2 2g-- = + + + + -c m 6 @

& = a r r r3 1n2 2 2 2 1

g+ + +-^ ^h h6 @

& = 3ar

r

1

1n

2

2

-

- ^ h= G & 2r

r13 1 &+

= =

vdnt, bghJé»j« 2.

12. xU bgU¡F¤ bjhlç‹ Kjš n, 2n k‰W« 3n M»a cW¥òfë‹ TLjšfŸ Kiwna

, k‰W«S S S1 2 3

våš, S S S S S1 3 2 2 1

2- = -^ ^h h vd ãWÎf.

Ô®Î: bfhL¡f¥g£LŸsgo ,Sr

a rS

ra r

11

11n n

1 2

2

=--

=--^ ^h h k‰W« S

ra r11 n

3

3

=--^ h

S S S1 3 2

-^ h = r

a rr

a rr

a r11

11

11n n n3 2

--

--

---^ ^ ^h h h; ;E E

= r

a rr r

1

11 1

nn n

2

23 2

-

-- - +

^

^

h

h= 6G @ = r

a rr r

1

1 nn n

2

22 3

-

--

^

^

h

h= 6G @

= r

a r r r

1

1 1n n n

2

2 2

-

- -

^

^ ^

h

h h = r

a r r

1

1n n

2

2 2 2

-

-

^

^

h

h (1)

nkY« S S2 1- = a

rr a

rr

11

11n n2

-- -

--c cm m

= r

a r r1

1 1n n2

-- - +6 @ =

ra r r1

1n

n

--6 @

& S S2 1

2-^ h = r

a r r

1

1n n

2

2 2 2

-

-

^

^

h

h = S S S1 3 2

-^ h . ((1)Iga‹gL¤Â)

gæ‰Á 2.6

1. Ã‹tU« bjhl®fë‹ TLjiy¡ fh©f.

(i) 1 + 2 + 3 + g + 45 (ii) 16 17 18 252 2 2 2

g+ + + +

(iii) 2 + 4 + 6 + g + 100 (iv) 7 + 14 +21 g + 490

(v) 5 7 9 392 2 2 2

g+ + + + (vi) 16 17 353 3 3

g+ + +

Ô®Î: (i) Ï§F 1 + 2 + 3 + g nn n

2

1+ =

+^ h .

vdnt, 1 + 2 + 3 + g + 45 = 2

45 45 1+^ h = 45 23 1035# = .

(ii) 16 17 18 252 2 2 2

g+ + + +

= (1 2 3 25 ) 1 2 3 152 2 2 2 2 2 2 2g g+ + + + - + + + +^ h


= k kkk

2 2

1

15

1

25-

==

// . { ( )( ).k

n n n6

1 2 1n2

1=

+ +/ }

= 6

25 25 1 50 1

6

15 15 1 30 1+ +-

+ +^ ^ ^ ^h h h h

= 6

25 26 51

6

15 16 31-

^ ^ ^ ^ ^ ^h h h h h h = 5525 1240 4285- = .

(iii) 2 + 4 + 6 + g + 100

= 2 1 2 3 50g+ + + +^ h = 2 k1

50/ . ( k

n n

2

1

k

n

1=

+

=

^ h/ )

= 25502

2 50 50 1+=

^ ^h h

(iv) 7 + 14 +21 g + 490

= 7 1 2 3 70g+ + + +^ h = 7 7k2

70 70 1

1

70=

+^ h; E/ ( kn n

2

1

k

n

1=

+

=

^ h/ )

= 7 35 71 17395# # = .

(v) 5 7 9 392 2 2 2

g+ + + +

= 1 2 3 39 2 4 6 38 1 32 2 2 2 2 2 2 2 2 2g g+ + + + - + + + + - +^ ^ ^h h h

= 2 10k 1 2 3 192

1

392 2 2 2 2g- + + + + -6 @/ = 4 10k k2

1

392

1

19- -/ /

{ ( )( ).k

n n n6

1 2 1n2

1=

+ +/ }

= 106

39 39 1 78 14

6

19 19 1 38 1+ +-

+ +-

^ ^ ^ ^h h h h; E

= 20540 9880 10 10650- - =

(vi) 16 17 353 3 3

g+ + +

= 1 2 3 35 1 2 153 3 3 3 3 3 3g g+ + + + - + + +^ ^h h

= k k3

1

353

1

15-/ / . k

n n

2

1

k

n3

1

2

=+

=

^c

hm) 3/

= ( )2

35 35 12

15 15 12 2+

-+^ h; ;E E =

235 36

215 162 2# #-8 8B B

= 35 18 15 82 2# #-^ ^h h = 630 1202 2-^ ^h h

= ( )( ) 382500.630 120 630 120+ - =

2. Ã‹tUtdt‰¿‰F k-‹ kÂ¥ò¡ fh©f.

(i) 1 2 3 6084k3 3 3 3

g+ + + + = (ii) 1 2 3 2025k3 3 3 3

g+ + + + =

Ô®Î: (i) bfhL¡f¥g£LŸsgo 1 2 3 6084k3 3 3 3

g+ + + + =

& nk

3

1/ = 6084

& k k

2

1 2+^ h; E = 6084 = 782

& ( )k k 1+ = 156 = 1312 # & k = 12 .


(ii) bfhL¡f¥g£LŸsgo 1 2 3 2025k3 3 3 3

g+ + + + =

& nk

3

1/ = 2025

& k k

2

1 2+^ h; E = 2025 = 452 & k k 1+^ h= 109 #

vdnt, k = 9

3. 1 2 3 171pg+ + + + = våš, 1 2 3 p3 3 3 3

g+ + + + -æ‹ kÂ¥ig¡ fh©f.

Ô®Î: bfhL¡f¥g£LŸsgo 1 2 3 171pg+ + + + =

& np

1/ = 171

p p

2

1+^ h = 171

& p p

2

1 2+^ h; E = 1712

& 1 2 3 p3 3 3 3g+ + + + = 1712 = 29241

4. 1 2 3 8281k3 3 3 3

g+ + + + = våš, 1 2 3 kg+ + + + -æ‹ kÂ¥ig¡ fh©f.

Ô®Î: bfhL¡f¥g£LŸsgo 1 2 3 8281k3 3 3 3

g+ + + + =

& k k

2

1 2+^ h; E = 8281 = 912

& k k

2

1+^ h = 91

& 1 2 3 kg+ + + = 91 5. 12 br.Û, 13br.Û, g , 23 br.Û M»adt‰iw Kiwna g¡f msÎfshf¡ bfh©l

12 rJu§fë‹ bkh¤j¥ gu¥gsÎ¡ fh©f.

Ô®Î: bfhL¡f¥g£l 12 rJu§fë‹ g¡f§fŸ

12br.Û, 13 br.Û, 14 br.Û, g , 23 br.Û,våš.

12 rJu§fë‹ gu¥ò = 12 13 14 232 2 2 2g+ + + +

= 1 2 3 23 1 2 3 1122 2 2 2 2 2 2g g+ + + + - + + + +^ ^h h

= k k2

1

232

1

11/-/ / { ( )( )

kn n n

61 2 1n

2

1=

+ +/ }

= 6

23 23 1 46 1

6

11 11 1 22 1+ +-

+ +^ ^ ^ ^h h h h

= 6

23 24 476

11 12 23# # # #-

= 4324 506 3818- = r.br.Û.

6. 16 br.Û, 17 br.Û, 18 br.Û, g , 30 br.Û M»adt‰iw Kiwna g¡f msÎfshf¡

bfh©l 15 fd¢rJu§fë‹ fdmsÎfë‹ TLjš fh©f.

Ô®Î: bfhL¡f¥g£l 15 fdrJu§fë‹ g¡f§fë‹


16 br.Û, 17 br.Û, 18 br.Û, g , 30 br.Û våš,

15 fd rJu§fë‹ fd msÎ = 16 17 18 303 3 3 3g+ + + +

= 1 2 3 30 1 2 3 153 3 3 3 3 3 3 3g g+ + + + - + + + +^ ^h h

= k k3

1

303

1

15-/ / k

n n

2

1

k

n3

1

2

=+

=

^c

hm) 3/

= 2

30 30 1

2

15 15 12 2+-

+^ ^h h; ;E E

= 15 31 15 82 2# #-^ ^h h = 465 1202 2-^ ^h h

= 216225 14400 201825- = f.br.Û.

gæ‰Á 2.7

rçahd éilia¤ nj®ªbjL¡fÎ«.

1. Ã‹tUtdt‰WŸ vJ bkŒahd¡ T‰wšy? (A) Ïaš v©fë‹ fz« N -š tiuaiw brŒa¥g£l bkŒba© kÂ¥òila¢ rh®ò

xUbjhl®tçirahF«.

(B) x›bthU rh®ò« xU bjhl® tçiræid¡ F¿¡F«. (C) xUbjhl®tçir,Koéèv©â¡ifæšcW¥òfis¡bfh©oU¡fyh«. (D) xUbjhl®tçir,KoÎWv©â¡ifæšcW¥òfis¡bfh©oU¡fyh«.

Ô®Î: x›bthU rh®ò« xU bjhl® tçiræid¡ F¿¡F«. ( éil: (B) )

2. 1, 1, 2, 3, 5, 8, g v‹w bjhl®tçiræ‹ 8 MtJ cW¥ò

(A) 25 (B) 24 (C) 23 (D) 21

Ô®Î: Ãnghdh» bjhl®tçir, , .F F F n 2>n n n1 2= +

- - ( éil: (D) )

3. , , , ,21

61

121

201 g v‹w bjhl®tçiræš, cW¥ò

201 -¡FmL¤jcW¥ò

(A) 241 (B)

221 (C)

301 (D)

181

Ô®Î: bghJ cW¥ò ( )

tn n 1

1n=

+ ( éil: (C) )

4. a, b, c, l, m v‹gdT£L¤bjhl®tçiræšÏU¥Ã‹ 4 6 4a b c l m- + - + =

(A) 1 (B) 2 (C) 3 (D) 0

Ô®Î: 4 6 4a b c l m- + - + ( ) ( ) ( ) 0a m b l c c c c4 6 2 4 2 6= + - + + = - + = ( éil: (D) )

5. a, b, c v‹gdxUT£L¤bjhl®tçiræšcŸsdvåš, b ca b

-- =

(A) ba (B)

cb (C)

ca (D) 1 ( éil: (D) )


6. 100 n +10 v‹gJ xU bjhl®tçiræ‹ n MtJ cW¥ò våš, mJ

(A) xUT£L¤bjhl®tçir (B) xUbgU¡F¤bjhl®tçir

(C) xUkh¿è¤bjhl®tçir (D) xUT£L¤bjhl®tçirÍ«mšybgU¡F¤bjhl®tçirÍ«mšy

Ô®Î: 100n + 10 = 110 + 100(n–1) v‹gJ a + (n–1)d v‹wtoéšcŸsJ.

vdnt, , 1,2,t an b nn

g= + = ,Ï§F a k‰W« b v‹gJ kh¿èfŸ ( éil: (A) )

7. , , ,a a a1 2 3

gv‹gdxUT£L¤bjhl®tçiræYŸsd.nkY« a

a

23

7

4 = våš, 13tJ

cW¥ò

(A) 23 (B) 0 (C) a12 1 (D) a14 1

Ô®Î: 2( a + 3d ) = 3( a + 6d ) & 3a + 18d – 2a – 6d = 0

& a + 12d = 0 ( éil: (B) )

8. , , ,a a a1 2 3

g v‹gJ xU T£L¤ bjhl®tçir våš, , , ,a a a5 10 15

g v‹w bjhl®

tçirahdJ

(A) xUbgU¡F¤bjhl®tçir

(B) xUT£L¤bjhl®tçir

(C) xUT£L¤bjhl®tçirÍ«mšybgU¡F¤bjhl®tçirÍ«mšy

(D) xUkh¿è¤bjhl®tçir

Ô®Î: , , ,a a a5 10 15

g = a + 4d, a+ 9d, a+ 14d,g .

ÏJxUT£L¤bjhl®tçir,bghJé»j« = 5d . ( éil: (B) )

(Terms of an A.P. selected at equal intervals consecutively, again form an A.P. )

9. xUT£L¤bjhl®tçiræ‹mL¤jL¤j_‹WcW¥òfŸk + 2, 4k – 6, 3k – 2 våš, k -‹ kÂ¥ò

(A) 2 (B) 3 (C) 4 (D) 5

Ô®Î: (k+2) + (3k–2) = 2(4k–6) & 4k = 8k – 12 & k = 3. ( éil: (B) )

10. a, b, c, l, m. n v‹gdT£L¤bjhl®tçiræšmikªJŸsdvåš, 3a + 7, 3b + 7, 3c + 7, 3l + 7, 3m + 7, 3n + 7 v‹w bjhl®tçir

(A) xUbgU¡F¤bjhl®tçir (B) xUT£L¤bjhl®tçir

(C) xUkh¿è¤bjhl®tçir

(D) xUT£L¤bjhl®tçirÍ«mšybgU¡F¤bjhl®tçirÍ«mšy

Ô®Î: xUT£L¤bjhl®tçiræšx›bthUcW¥igÍ«xnukh¿èahšT£odhY«

fê¤jhY«m¤bjhl®tçirT£L¤bjhl®tçirahfÏU¡F«. ( éil: (B) )


11. xUbgU¡F¤bjhl®tçiræš3 MtJ cW¥ò 2 våš, mj‹ Kjš 5 cW¥òfë‹

bgU¡f‰gy‹

(A) 52 (B) 25 (C) 10 (D) 15

Ô®Î: bgU¡F¤bjhl®tçir , , , ,r

ara a ar ar

2

2 . _‹wh« cW¥ò a = 2.bgU¡f‰gy‹ a5= 25

(Ï§F, (2n + 1) ) cW¥òfë‹ bgU¡f‰gy‹ 2( )n2 1+ ) ( éil: (B) )

12. a, b, c v‹gdxUbgU¡F¤bjhl®tçiræšcŸsdvåš, b ca b

-- =

(A) ba (B)

ab (C)

ca (D)

bc

Ô®Î: b ca b

-- =

1

1

bbc

aab

-

-

`

`

j

j =

( )( )

b ra r11-- =

ba . F¿¥ò: ,b ar c ar2= = ( éil: (A) )

13. ,x x2 2+ , 3 3x + v‹gd xU bgU¡F¤ bjhl®tçiræèU¥Ã‹ ,x5 x10 10+ , 15 15x + v‹w bjhl®tçirahdJ




Ô®Î: bfhL¡f¥g£l bjhl®tçir ,x5 x10 10+ , 15 15x + , 5 Mš bgU¡»dhš

»il¡F«. ( éil: (B) ) 14. –3, –3, –3,g v‹w bjhl®tçirahdJ

(A) xUT£L¤bjhl®tçirk£L«

(B) xUbgU¡F¤bjhl®tçirk£L«


(D) xUT£L¤bjhl®tçirk‰W«bgU¡F¤bjhl®tçir

Ô®Î: xU kh¿è bjhl®tçir T£L¤ bjhl® tçiræY« bgU¡F¤ bjhl®

tçiræY«ÏU¡F«. ( éil: (D) )

15. xUbgU¡F¤bjhl®tçiræ‹Kjšeh‹FcW¥òfë‹bgU¡f‰gy‹256, mj‹

bghJ é»j« 4 k‰W« mj‹ Kjš cW¥ò äif v© våš, mªj¥ bgU¡F¤

bjhl®tçiræ‹ 3 tJ cW¥ò

(A) 8 (B) 161 (C)

321 (D) 16

Ô®Î: eh‹F cW¥òfŸ , , ,r

ara ar ar

3

3 ; a4 = 256 & a = 4. ar = 16. ( éil: (D) )

16. xUbgU¡F¤bjhl®tçiræš t53

2= k‰W« t

51

3= våš,mj‹bghJé»j«

(A) 51 (B)

31 (C) 1 (D) 5 ( éil: (B) )


17. x 0! våš, 1 sec sec sec sec secx x x x x2 3 4 5

+ + + + + =

(A) (1 )( )sec sec sec secx x x x2 3 4

+ + + (B) (1 )( )sec sec secx x x12 4

+ + +

(C) (1 )( )sec sec sec secx x x x3 5

- + + (D) (1 )( )sec sec secx x x13 4

+ + +

Ô®Î: nfhit ( )( )secsec

secsec sec sec

xx

xx x x

11

11 16 2 2 4

=-- =

-- + + ( éil: (B) )

18. 3 5t nn= - v‹gJxUT£L¤bjhl®tçiræ‹ n MtJcW¥òvåš,m¡T£L¤

bjhl®tçiræ‹ Kjš n cW¥ò¡fë‹ TLjš

(A) n n21 5-6 @ (B) n n1 5-^ h (C) n n

21 5+^ h (D) n n

21 +^ h

Ô®Î: ;a t 21

= =- Sn = ( ) ( )n a l n n2 2

2 3 5+ = - + - = ( )n n2

1 5- ( éil: (A) )

19. am n- , am , am n+ v‹wbgU¡F¤bjhl®tçiræ‹bghJé»j«

(A) am (B) a m- (C) an (D) a n-

( éil: (C) )

20. 1 + 2 + 3 +. . . + n = k våš, 13 n23 3

g+ + + v‹gJ

(A) k2 (B) k3 (C) k k

2

1+^ h (D) k 1 3+^ h

( éil: (A) )

Ô®Î - Ïa‰fâj« 57

3. ALGEBRA

gæ‰Á 3.1

Ú¡fš Kiwia¥ ga‹gL¤Â Ã‹tU« rk‹gh£L¤ bjhF¥òfis¤ Ô®:

1. 2 7x y+ = , 2 1x y- = .

Ô®Î: x y2+ = 7 g (1)

x y2- = 1 g (2)

Ï¥bghGJ, (1) + (2) & 2x = 8 & x = 4.

x = 4 våš, (1) & y4 2+ = 7 & y = 23 .

vdnt, Ô®Î ,423` j.

2. 3 8x y+ = , 5 10x y+ = .

Ô®Î: x y3 + = 8 g (1)

x y5 + = 10 g (2)

Ï¥bghGJ (1) – (2) & x2- = – 2 & x = 1

x = 1 våš, (1) & y3 + = 8 & y = 5

vdnt, Ô®Î (1, 5).

3. 4xy2

+ = , 2 5x y3+ = .

Ô®Î: bfhL¡f¥g£l rk‹ghLfë‹ bjhF¥ò

xy2

+ = 4 g (1)

x y3

2+ = 5 g (2)

( )1 2 &# x y2 + = 8 g (3)

( )2 3 &# x y6+ = 15 g (4)

( ) ( )3 4 2 &#- y11- = 22- & y = 2

y 2= våš, (4) & ( )x 6 2+ = 15 & x = 3

vdnt, Ô®Î (3, 2).

4. 11 7x y xy- = , 9 4 6x y xy- = .

Ô®Î: 0x = våš 0y = k‰W« 0y = våš 0x = MF«.

vdnt, (0, 0) v‹gJ Ïªj bjhF¥Ã‹ xU Ô®Î.

Mfnt, k‰bwhU Ô®Î ÏU¥Ã‹ mJ 0, 0x y^ ^ vd¡ bfhŸnth«.

x›bthU rk‹gh£o‹ ÏUòw§fisÍ« xy Mš tF¡f

y x11 7- = 1 k‰W«

y x9 4- = 6

Ï§F ax1= k‰W« b

y1= v‹f.

Ïa‰fâj« 3


j‰nghJ nk‰f©l rk‹ghLfŸ Ã‹tUkhW mikÍ«

a b7 11- + = 1 g (1)

a b4 9- + = 6 g (2)

( ) ( )1 4 2 7 &# #- b19- = – 38 & b = 2

2b = våš, (1) & ( )a7 11 2- + = 1

& – 7a = 1 – 22 & a = 3

Ï§F, 3a = våš ; 2x b31= = våš y

21= .

vdnt, bjhF¥Ã‹ Ïu©L Ô®ÎfŸ (0, 0) , ,31

21` j.

5. x y xy3 5 20+ = ,

x y xy2 5 15+ = , 0, 0x y! ! .

Ô®Î: 0, 0x y^ ^ v‹gjhš, bfhL¡f¥g£l rk‹ghLfë‹ bjhF¥ò

x y3 5+ =

xy20 g (1)

x y2 5+ =

xy15 g (2)

0, 0,x y^ ^ våš x›bthU rk‹gh£o‹ ÏUòw§fisÍ« xy Mš bgU¡f. x y5 3+ = 20 g (3) ` x y5 2+ = 15 g (4) ( ) ( )3 4 &- y = 5

5y = våš (3) & ( )x5 3 5+ = 20 & x5 = 5 1.x& =

vdnt, Ô®Î (1, 5).

6. 8 3 5x y xy- = , 6 5 2x y xy- =- .

Ô®Î: 0x = våš 0y = k‰W« 0y = våš 0x = MF«.

vdnt, (0, 0) v‹gJ Ïªj bjhF¥Ã‹ xU Ô®Î.

Mfnt, k‰bwhU Ô®Î ÏU¥Ã‹ mJ 0, 0x y^ ^ vd¡ bfhŸnth«.

x y8 3- = xy5 g (1) x y6 5- = xy2- g (2)x›bthU rk‹gh£o‹ ÏUòw§fisÍ« xy Mš tF¡f

y x8 3- = 5 5

x y3 8& - =- g (3)

k‰W« y x6 5- = 2

x y2 5 6&- - = g (4)

Ï§F ax1= k‰W« b

y1= , v‹f.

(3) & a b3 8- = – 5 g (5)

( )4 & a b5 6- = 2 g (6)


( ) ( )5 5 6 3 &# #- b22- = – 31 & b = 2231

b2231= våš (5) & a3 8

2231- ` j = – 5 & a =

1123 .

a1123= våš

xx1

1123

2311&= = .

b2231= våš .

yy1

2231

3122&= =

vdnt, bjhF¥Ã‹ Ïu©L Ô®ÎfŸ (0, 0) , ,2311

3122` j.

7. 13 11 70x y+ = , 11 13 74x y+ = .


x y13 11+ = 70 g (1)

x y11 13+ = 74 g (2)

(1)-cl‹ (2)-I¡ T£l, x y 6+ = g (3)

(1) èUªJ (2)-I¡ fê¡f, x y 2- =- g (4)

(3) (4) 2x&+ =

2x = våš (4) & y2 - = – 2 & y = 4.

vdnt, Ô®Î (2, 4).

8. 65 33 97x y- = , 33 65 1x y- = .


x y65 33- = 97 g (1)

x y33 65- = 1 g (2)

( ) ( )1 2 &+ x y- = 1 g (3)

( ) ( )1 2 &- x y+ = 3 g (4)

(3)-cl‹ (4)-I¡ T£l, x 2= . 2x = våš (4) 1.y& =

vdnt, Ô®Î (2, 1).

9. 17x y15 2+ = , , 0, 0

x yx y1 1

536 ! !+ = .


x y15 2+ = 17 g (1)

x y1 1+ =

536 g (2)

ax1= k‰W« b

y1= v‹f.

( )1 & 15a b2+ = 17 g (3)


( )2 & a b+ = 536 g (4)

( ) ( )3 4 15 &#- b13- = – 91 b& = 7

b = 7 våš, ( )4 & ( )a5 5 7+ = 36 & a = 51

a51= våš, 5.

xx1

51 &= = 7b = våš, 7 .

yy1

71&= =

vdnt, Ô®Î ,571` j.

10. x y2

32

61+ = , 0, 0, 0

x yx y3 2 ! !+ = .


x y2

32+ =

61 g (1)

x y3 2+ = 0 g (2)

ax1= k‰W« b

y1= , våš (1) & a b2

32+ =

61 g (3)

(2) 3 2 0a b& + = g (4)

( ) 3 (4) 23 &# #- 2b- = 21 .b

41& =-

b41=- våš, (4) & a3 2

41+ -` j = 0 3 .a a

21

61& &= =

a61= våš, 6x = k‰W« b

41=- våš, 4.y =-

vdnt, Ô®Î (6, – 4).

Exercise 3.2

1. FW¡F¥ bgU¡fš Kiwia¥ ga‹gL¤Â Ã‹tU« rk‹ghLfë‹

bjhF¥òfis¤ Ô®¡f.

(i) 3 4 24x y+ = , 20 11 47x y- = (ii) 0.5 0.8 0.44x y+ = , 0.8 0.6 0.5x y+ =

(iii) 2,x y x y23

3

5

3 2 613- =- + = (iv) 2, 13

x y x y5 4 2 3- =- + =

Ô®Î: (i) bfhL¡f¥g£l rk‹ghLfë‹ bjhF¥ò

x y3 4 24+ - = 0

x y20 11 47- - = 0. FW¡F¥ bgU¡fš Kiwia¥ ga‹gL¤Â

x y 1

4 24 3 4

11 47 20 11

& -

- - -

& ( ) ( )( )

x4 47 11 24- - - -

= ( )( ) ( )( )

y24 20 47 3- - -

= ( ) ( )( )3 11 20 4

1- -


& x452-

= y339 113

1-

=-

& x = 4,y113452

113339 3

-- = =

-- =

vdnt, Ô®Î (4, 3).

(ii) bfhL¡f¥g£l rk‹ghLfë‹ bjhF¥ò

. . .x y0 5 0 8 0 44+ - = 0

. . .x y0 8 0 6 0 5+ - = 0

x›bthU rk‹gh£o‹ ÏUòwK« 100 Mš bgU¡f.

x y50 80 44+ - = 0

x y80 60 50+ - = 0.

& x y 1

80 44 50 80

60 50 80 60

-

-

& 80( 50) 60( 44)

x- - -

= ( ) ( ) ( ) ( )

y80 44 50 50 50 60 80 80

1- - -

=-

& x1360-

= y1020 3400

1-

=-

& x = .34001360 0 4

-- = ; y = .

34001020 0 3

-- =

vdnt, Ô®Î (0.4, 0.3).

(iii) 9 10x y6- = – 2, x y

62 3

613+

= .

bfhL¡f¥g£l rk‹ghLfë‹ bjhF¥ig Ã‹tUkhU vGjyh«.

x y9 10 12- + = 0

x y2 3 13+ - = 0

& x y 1

10 12 9 10

3 13 2 3

- -

-

& x130 36-

= y24 117 27 20

1+

=+

& x = 2 ; 3.y4794

47141= = =

vdnt, Ô®Î (2, 3).

(iv) ax1= k‰W« b

y1= v‹f.

bfhL¡f¥g£l rk‹ghLfë‹ bjhF¥ig Ã‹tUkhU vGjyh«.

a b5 4 2- + = 0 a b2 3 13+ - = 0


& a b 1

4 2 5 4

3 13 2 3

- -

-

& a52 6-

= b4 65 15 8

1+

=+

& a46

= b69 23

1=

& 2 ; 3.a b2346

2369= = = =

2a = våš, ;x21= 3b = våš, .y

31=

vdnt, Ô®Î ,21

31` j.

2. Ã‹tU« fz¡FfS¡fhd¢ rk‹ghLfis mik¤J, mt‰¿‹ Ô®Îfis¡ fh©f: (i) xU v© k‰bwhU v©â‹ _‹W kl§ifél 2 mÂf«. Á¿a v©â‹

4 kl§fhdJ bgça v©izél 5 mÂf« våš, m›bt©fis¡ fh©f.

Ô®Î: bgça v© k‰W« Á¿a v© Kiwna x, y vd¡ bfhŸf.

bfhL¡f¥g£l ãgªjidfë‹go,

x y3- = 2 3 2x y& - - = 0 g (1)

y x4 - = 5 4 5x y& - + - = 0 g (2)

(1) cl‹ (2)I T£l y& = 7

y 7= våš, (1) & x = 23

vdnt, njitahd v©fŸ 23 k‰W« 7.

(ii) ÏU eg®fë‹ tUkhd§fë‹ é»j« 9 : 7. mt®fë‹ bryÎfë‹ é»j« 4 : 3. x›bthUtU« khjbkh‹W¡F ` 2000 nrä¡f Koªjhš, mt®fSila

khjhªÂu tUkhd¤ij¡ fh©f.

Ô®Î: ÏUtç‹ khj tUkhd§fŸ Kiwna x k‰W« y v‹f.


:x y = :9 7

x7& = y x y9 7 9 0& - = g (1)nkY«

( ) : ( )x y2000 2000- - = 4 : 3

& x3 6000- = y4 8000-

& 3 4x y 2000- + = 0 g (2)

(1) IÍ« (2) IÍ« FW¡F¥ bgU¡fš Kiwia¥ ga‹gL¤Â¤ Ô®¡f,

x y 1

9 0 7 9

4 2000 3 4

- -

- -


& x18000 0- +

= y0 14000 28 27

1-

=- +

& x = 18,000 ; 14,000.y1

180001

14000-

- = =-

- =

vdnt, ÏUtç‹ khj tUkhd§fŸ Kiwna ` 18,000 k‰W« ` 14,000.

(iii) xU <çy¡f v©â‹ kÂ¥ò mj‹ Ïy¡f§fë‹ TLjš nghš 7 kl§F cŸsJ.

Ïy¡f§fis ÏlkhWjš brŒa »il¡F« v© bfhL¡f¥g£l v©izél 18

FiwÎ våš, m›bt©iz¡ fh©f.

Ô®Î: v©â‹ g¤jh« ÏlÏy¡f« k‰W« x‹wh« ÏlÏy¡f« Kiwna x k‰W« y v‹f. njitahd v© x y10 + .


x y10 + = ( )x y7 +

x y2& - = 0 g (1)

nkY«, y x10 + = x y10 18+ -

x y 2& - + + = 0 g (2)

(1) cl‹ (2) I T£l y 2= k‰W« .x 4=

vdnt, bfhL¡f¥g£l v© 42.

(iv) 3 eh‰fhèfŸ k‰W« 2 nkirfë‹ bkh¤j éiy ` 700. nkY« 5 eh‰fhèfŸ k‰W«

3 nkirfë‹ bkh¤jéiy ` 1100. våš, 2 eh‰fhèfŸ k‰W« 3 nkirfë‹

bkh¤jéiyia¡ fh©f.

Ô®Î: eh‰fhèæ‹ éiy ` x k‰W« nkiræ‹ éiy ` y v‹f.


x y3 2+ = 700 3 2 700 0x y& + - = g (1)

nkY« x y5 3+ = 1100 5 3 1100 0x y& + - = g (2)

(1) IÍ« (2) IÍ« FW¡F¥ bgU¡fš Kiwia¥ ga‹gL¤Â¤ Ô®¡f,

x y 1

2 700 3 2

3 1100 5 3

-

-

x2200 2100

&- +

= y3500 3300 9 10

1- +

=-

vdnt, x = 100 ; 200.y1

1001

200-

- = =-

- =

2 eh‰fhèfŸ k‰W« 3 nkirfë‹ éiy

= 2(100) 3(200) 200 600+ = + = ` 800.


(v) xU br›tf¤Â‹ Ús¤ij 2 br. Û mÂfç¤J mfy¤ij 2 br.Û Fiw¤jhš, mj‹

gu¥ò 28 r. br.Û Fiw»wJ. Ús¤ij 1 br.Û Fiw¤J mfy¤ij 2 br.Û mÂfç¤jhš,

br›tf¤Â‹ gu¥ò 33 r. br. Û mÂfç¡F« våš, br›tf¤Â‹ gu¥ig¡ fh©f.

Ô®Î: br›tf¤Â‹ Ús« k‰W« mfy« Kiwna x br.Û., y br.Û. v‹f.

vdnt,gu¥ò = xy

bfhL¡f¥g£l ãgªjidfë‹go, ( )( )x y2 2+ - = xy 28-

& xy x y2 2 4- + - = xy 28-

& x y 12- + + = 0 g (1)nkY« ( )( )x y1 2- + = xy 33+

& xy x y2 2+ - - = xy 33+

& x y2 35- - = 0 g (2)

(1) cl‹ (2) I T£l, x = 23

x = 23 våš (1) & y = 11

vdnt, br›tf¤Â‹ Ús« = 23 br.Û, br›tf¤Â‹ mfy« = 11 br.Û.

Mfnt, br›tf¤Â‹ gu¥gsÎ = 253 r.br.Û.

(vi) Óuhd ntf¤Âš xU F¿¥Ã£l öu¤ij xU bjhl® t©o F¿¥Ã£l neu¤Âš

flªjJ. bjhl® t©oæ‹ ntf« kâ¡F 6 ».Û vd mÂfç¡f¥g£oUªjhš

m¤öu¤ij¡ fl¡f, F¿¥Ãl¥g£oUªj neu¤ij él 4 kâ neu« Fiwthf

m¤bjhl® t©o vL¤J¡ bfh©oU¡F«. bjhl® t©oæ‹ ntf« kâ¡F 6 ».Û

vd Fiw¡f¥g£oUªjhš, mnj öu¤ij¡ fl¡f F¿¥Ãl¥g£oUªj neu¤ijél 6

kâneu« mÂfç¤ÂU¡F« våš, gaz öu¤ij¡ f©LÃo.

Ô®Î: Ïuæè‹ ntf« x ».Û./kâ k‰W« mj‹ fhy« y kâfŸ v‹f. öu« = ntf« × fhy« = xybfhL¡f¥g£l ãgªjidfë‹go, ( )( )x y6 4+ - = xy

& xy x y4 6 24- + - = xy

& x y4 6 24- + - = 0

& x y2 3 12- + - = 0 g (1)

nkY« ( )( )x y6 6- + = xy

& xy x y6 6 36+ - - = xy

& x y 6- - = 0 g (2)

(1) (2) 2 &#+ y = 24

y 24= våš , (2) & x 30= .

vdnt, Ïuæš gaz¤Â‹ öu«, 24 30 720xy #= = ».Û.


gæ‰Á 3.3

1. Ã‹tU« ÏUgo gšYW¥ò¡nfhitfë‹ ó¢Áa§fis¡ fh©f. ó¢Áa§fS¡F«

bfG¡fS¡F« Ïilna cŸs bjhl®òfis¢ rçgh®¡f.

(i) x x2 82- - .

Ô®Î: ( )p x = 2 8 ( 4)( 2)x x x x2- - = - + v‹f.

Ï§F, (4)p = 0 k‰W« ( )p 2 0- = MF«.

vdnt, ( )p x = x x2 82- - -‹ ó¢Áa§fŸ Kiwna 4 k‰W« – 2.

ó¢Áa§fë‹ TLjš = 4 – 2 = 2 g (1)

ó¢Áa§fë‹ bgU¡f‰gy‹ = (4) (– 2) = – 8 g (2)

mo¥gil¤ bjhl®òfëèUªJ:

ó¢Áa§fë‹ TLjš = ( )2

‹bfG

‹bfG

x

x

12

2 -

-- =

- -= g (3)

ó¢Áa§fë‹ bgU¡f‰gy‹ = 8‹bfG

kh¿è

x 18

2 -= - =- g (4)

rk‹ghL (1) k‰W« (3) nkY« (2) k‰W« (4)-èUªJ mo¥gil¤ bjhl®òfŸ

rçgh®¡f¥g£ld.

(ii) 4 4 1x x2- + .

Ô®Î: ( )p x = ( )( )x x x x4 4 1 2 1 2 12- + = - - v‹f.

Ï§F, p x^ h = 0 våš ,x21

21= (Ïu©L ó¢Áa§fŸ).

vdnt, ( )p x x x4 4 12= - + -‹ ó¢Áa§fŸ Kiwna

21 k‰W«

21 .

ó¢Áa§fë‹ TLjš = 21

21 1+ = g (1)

ó¢Áa§fë‹ bgU¡f‰gy‹ = 21

21

41

# = g (2)


ó¢Áa§fë‹ TLjš = ( )

‹bfG

‹bfG

x

x

44

12 -

-- =

- -= g (3)

ó¢Áa§fë‹ bgU¡f‰gy‹ = ‹bfG

kh¿è

x 41

2 -= g (4)


rçgh®¡f¥g£ld.


(iii) 6 3 7x x2- - .

Ô®Î: ( )p x = ( )( )x x x x6 7 3 2 3 3 12- - = - + v‹f.

Ï§F, p23` j = 0 k‰W« p

31 0- =` j

vdnt, ( )p x = 6 7 3x x2- - -‹ ó¢Áa§fŸ Kiwna

23 k‰W«

31- .


31

67- = g (1)


31

21

#- =- g (2)



‹bfG

‹bfG

x

x

67

67

2 -

-- =

- -= g (3)


kh¿è

x 63

21

2 -= - = - g (4)


rçgh®¡f¥g£ld.

(iv) 4 8x x2+ .

Ô®Î: ( )p x = 4 8 ( )x x x x4 22+ = + v‹f.

Ï§F, (0)p = 0 k‰W« ( ) 0p 2- = .

vdnt, ( )p x = 4 8x x2+ -‹ ó¢Áa§fŸ Kiwna 0 k‰W« – 2.

ó¢Áa§fë‹ TLjš = 0 – 2 = – 2 g (1)

ó¢Áa§fë‹ bgU¡f‰gy‹ = (0) (– 2) = 0 g (2)


ó¢Áa§fë‹ TLjš = ‹bfG

‹bfG

x

x

48 2

2 -

-- = - =- g (3)


kh¿è

x 40 0

2 -= = g (4)


rçgh®¡f¥g£ld.

(v) 15x2- .

Ô®Î: ( )p x = ( )( )x x x15 15 152- = + - v‹f.

Ï§F, ( )p 15- = 0 k‰W« ( ) 0p 15 = .

vdnt, ( )p x = 15x2 - -‹ ó¢Áa§fŸ Kiwna 15- k‰W« 15 .


ó¢Áa§fë‹ TLjš = 15 15 0- + = g (1)

ó¢Áa§fë‹ bgU¡f‰gy‹ = ( )( )15 15 15- =- . g (2)



‹bfG

x

x

10 0

2 -

-- = = g (3)


kh¿è

x 115 15

2 -= - =- g (4)


rçgh®¡f¥g£ld.

(vi) 3 5 2x x2- + .

Ô®Î: ( )p x = ( )( )x x x x3 5 2 3 2 12- + = - - v‹f.

Ï§F, p32` j = 0 k‰W« ( ) 0p 1 = .

vdnt, ( )p x x x3 5 22= - + -‹ ó¢Áa§fŸ Kiwna

32 k‰W« 1.

ó¢Áa§fë‹ TLjš = 1 .32

35+ = g (1)

ó¢Áa§fë‹ bgU¡f‰gy‹ = (1)32

32=` j g (2)



‹bfG

‹bfG

x

x

35

35

2 -

-- =

- -= g (3)


kh¿è

x 32

2 -= g (4)


rçgh®¡f¥g£ld.

(vii) x x2 2 2 12- + .

Ô®Î: ( )p x = 2 2 1 ( )( )x x x x2 2 1 2 12- + = - - v‹f.

Ï§F, p2

1c m = 0 (ÏUKiw)

vdnt, ( )p x x x2 2 2 12= - + -‹ ó¢Áa§fŸ Kiwna

2

1 k‰W« 2

1 .


1

2

1

2

2 2+ = = g (1)


1

2

121=c cm m g (2)




‹bfG

‹bfG

x

x

35

35

2 -

-- =

- -= g (3)


kh¿è

x 32

2 -= g (4)


rçgh®¡f¥g£ld.

(viii) 2 143x x2+ - .

Ô®Î: ( )p x = ( )( )x x x x2 143 13 112+ - = + - v‹f.

Ï§F, ( 13)p - = 0 k‰W« ( ) 0p 11 = .

vdnt, ( )p x x x2 1432= + - -‹ ó¢Áa§fŸ Kiwna – 13 k‰W« 11.

ó¢Áa§fë‹ TLjš = – 13 + 11 = – 2 g (1)

ó¢Áa§fë‹ bgU¡f‰gy‹ = (– 13) (11) = – 143 g (2)



‹bfG

x

x

12 2

2 -

-- = - =- g (3)


kh¿è

x 1143 143

2 -= - =- g (4)


rçgh®¡f¥g£ld.

2. Ã‹tU« x›bthU ÃçéY« bfhL¡f¥g£LŸs nrho v©fis Kiwna

ó¢Áa§fë‹ TLjyhfÎ« k‰W« mitfë‹ bgU¡f‰gydhfÎ« bfh©l

gšYW¥ò¡nfhitia¡ fh©f.

(i) 3, 1.

Ô®Î: ÏUgo gšYW¥ò¡ nfhit ( )p x -‹ ó¢Áa§fŸ a k‰W« b v‹f.

Ï§F, a b+ = 3 k‰W« 1ab = vd bfhL¡f¥g£LŸsJ

vdnt, ( )p x = ( )x x2 a b ab- + + = x x3 12- + .

(ii) 2, 4.

Ô®Î: ÏUgo gšYW¥ò¡ nfhit ( )p x -‹ ó¢Áa§fŸ a k‰W« b v‹f. Ï§F, a b+ = 2 k‰W« 4ab = vd bfhL¡f¥g£LŸsJ

vdnt, ( )p x = ( )x x2 a b ab- + + = x x2 42- + .


(iii) 0, 4.



vdnt, ( )p x = ( )x x2 a b ab- + + = (0) 4 4x x x2 2- + = + .

(iv) ,251 .



vdnt, ( )p x = ( )x x2 a b ab- + + = x x2512

- + .

(v) ,131 .



vdnt, ( )p x = ( )x x2 a b ab- + + = x x31 12

- + .

(vi) , 421 - .


Ï§F, a b+ = 21 k‰W« 4ab =- vd bfhL¡f¥g£LŸsJ

vdnt, ( )p x = ( )x x2 a b ab- + + = x x21 42

- - .

(vii) ,31

31- .


Ï§F, a b+ = 31 k‰W«

31ab =- vd bfhL¡f¥g£LŸsJ

vdnt, ( )p x = ( )x x2 a b ab- + + = .x x31

312

- -

(viii) , 23 .



vdnt, ( )p x = ( )x x2 a b ab- + + = x x3 22- + .

Exercise 3.4 1. bjhFKiw tF¤jiy¥ ga‹gL¤Â Ã‹tUtdt‰¿‰F <Î k‰W« ÛÂ fh©f.

(i) ( 3 5x x x3 2+ - + ) ' ( 1x - ).

Ô®Î: ( )p x = x x x3 53 2+ - + v‹f.

tF¡F« nfhit( )x 1- . vdnt tF¡F« nfhitæ‹ ó¢Áa« 1.1 1 1 –3 5

0 1 2 –11 2 –1 4 " ÛÂ

vdnt, <Î x x2 12= + - , ÛÂ = 4.


(ii) (3 2 7 5x x x3 2- + - ) ' ( 3x + ).

Ô®Î: ( )p x = x x x3 2 7 53 2- + - v‹f.

tF¡F« nfhit( )x 3+ . vdnt tF¡F« nfhitæ‹ ó¢Áa« – 3.– 3 3 – 2 7 – 5

0 – 9 33 – 1203 – 11 40 – 125 " ÛÂ

vdnt, <Î x x3 11 402= - + , ÛÂ = – 125.

(iii) (3 4 10 6x x x3 2+ - + )' ( 3 2x - ).

Ô®Î: ( )p x = x x x3 4 10 63 2+ - + v‹f.

tF¡F« nfhit ( )x3 2- . vdnt tF¡F« nfhitæ‹ ó¢Áa« 32 .

32 3 4 – 10 6

0 2 4 – 43 6 – 6 2 " ÛÂ

Mfnt, x x x3 4 10 63 2+ - + = 3 6 6 2.x x x

32 2

- + - +` ^j h

= ( ) ( )x x x3 231 3 6 6 22

- + - + .

vdnt, <Î = ( )x x x x31 3 6 6 2 22 2

+ - = + - .

ÛÂ = 2.

(iv) (3 4 5x x3 2- - ) ' (3 1x + ).

Ô®Î: ( )p x = x x3 4 53 2- - v‹f.

tF¡F« nfhit ( )x3 1+ . vdnt tF¡F« nfhitæ‹ ó¢Áa« 31- .

31- 3 – 4 0 – 5

0 – 135

95-

3 – 535

950- " ÛÂ

Mfnt, ( )x x3 4 53 2- - = x x x

31 3 5

35

9502

+ - + -` `j j

= ( )x x x3 131 3 5

35

9502

+ - + -` j

vdnt, <Î = x x x x31 3 5

35

35

952 2

- + = - +` j .

ÛÂ = 950- .

(v) (8 2 6 5x x x4 2- + - )' (4 1x + ).

Ô®Î: ( )p x = x x x8 2 6 54 2- + - v‹f.

tF¡F« nfhit ( 1)x4 + . vdnt tF¡F« nfhitæ‹ ó¢Áa« 41- .


41- 8 0 – 2 6 – 5

0 – 221

83

3251-

8 – 223-

851

32211- " ÛÂ

Mfnt, x x8 2 64 54 2- + - = x x x x

41 8 2

23

851

322113 2

+ - - + -` `j j

= ( )x x x x4 141 8 2

23

851

322113 2

+ - - + -` j

vdnt, <Î = x x x x x x41 8 2

23

851 2

21

83

32513 2 3 2

- - + = - - +` j

ÛÂ = 32211- .

(vi) (2 7 13 63 48x x x x4 3 2- - + - )' (2 1x - ).

Ô®Î: ( )p x = x x x x2 7 13 63 484 3 2- - + - v‹f.

tF¡F« nfhit ( 1)x2 - . vdnt tF¡F« nfhitæ‹ ó¢Áa« 21 .

21 2 – 7 – 13 63 – 48

0 1 – 3 – 8255

2 – 6 – 16 55241- " ÛÂ

x x x x2 7 13 63 484 3 2- - + - = ( )x x x x

21 2 6 16 55

2413 2

- - - + -` j

= ( ) ( )x x x x2 121 2 6 16 55

2413 2

- - - + -

vdnt, <Î = (2 6 16 55) 3 8x x x x x x21

2553 2 3 2

- - + = - - + .

ÛÂ = 241- .

2. 10 35 50 29x x x x4 3 2+ + + + vD« gšYW¥ò¡nfhitia 4x + Mš tF¡f¡

»il¡F« <Î 6x ax bx3 2- + + våš a, b M»at‰¿‹ kÂ¥òfisÍ« k‰W«

ÛÂiaÍ« fh©f.

Ô®Î: ( )p x = 10 35 50 29x x x x4 3 2+ + + + v‹f.

tF¡F« nfhit ( 4)x + . vdnt tF¡F« nfhitæ‹ ó¢Áa« – 4.– 4 1 10 35 50 29

0 – 4 – 24 – 44 – 241 6 11 6 5 " ÛÂ

Mfnt, ( )( )x x x x x x x x10 35 50 29 4 6 11 6 54 3 2 3 2+ + + + = + + + + +

vdnt, <Î x x x6 11 63 2+ + + . Mdhš, x x x6 11 63 2

+ + + = x ax bx 63 2- + + .

bfG¡fis x¥ÃLifæš, a 6=- k‰W« b 11= . ÛÂ 5.


3. 8 2 6 7x x x4 2- + - v‹gij 2 1x + Mš tF¡f¡ »il¡F« <Î 4 3x px qx

3 2+ - +

våš, p, q M»at‰¿‹ kÂ¥òfisÍ«, ÛÂiaÍ« fh©f.

Ô®Î: ( )p x = x x x8 2 6 74 2- + - v‹f.

tF¡F« nfhit (2 1)x + . vdnt tF¡F« nfhitæ‹ ó¢Áa« 21- .

21- 8 0 – 2 6 – 7

0 – 4 2 0 – 38 – 4 0 6 – 10 " ÛÂ

Mfnt, x x x8 2 6 74 2- + - = ( )x x x

21 8 4 6 103 2

+ - + -` j

= (2 1) ( 4 6) 10x x x21 8 3 2

+ - + -

vdnt, <Î ( )x x x x21 8 4 6 4 2 33 2 3 2

- + = - + . ÛÂ –10.

Mdhš, 4 2 3x x3 2- + = 4 3x px qx3 2

+ - + . bfG¡fis x¥ÃLifæš, p 2=- k‰W« q 0= . ÛÂ – 10.

gæ‰Á 3.5 1. Ã‹tU« gšYW¥ò¡nfhitfis fhuâ¥gL¤Jf.

(i) 2 5 6x x x3 2- - + .

Ô®Î: ( )p x = x x x2 5 63 2- - + v‹f.

( )p x -‹ bfG¡fë‹ TLjš, 1 2 5 6 0- - + = .

Mfnt, ( )p x -¡F ( 1)x - xU fhuâahF«.1 1 – 2 – 5 6

0 1 – 1 – 61 – 1 – 6 0 " ÛÂ

k‰bwhU fhuâ x x 62- - = 3 2 6 ( )( )x x x x x2 32

- + - = + - .vdnt, 2 5 6x x x3 2

- - + = ( )( )( )x x x1 2 3- + - .

(ii) 4 7 3x x3- + .

Ô®Î: ( )p x = x x4 7 33- + v‹f.

( )p x -‹ bfG¡fë‹ TLjš, 4 7 3 0- + = .

Mfnt, ( )p x -¡F ( 1)x - xU fhuâahF«.1 4 0 – 7 3

0 4 4 – 34 4 – 3 0 " ÛÂ

k‰bwhU fhuâ x x4 4 32+ - = ( )( )x x x x x4 6 2 3 2 3 2 12

+ - - = + - .vdnt, 4 7 3x x3

- + = ( )( )( )x x x1 2 3 2 1- + - .


(iii) 23 142 120x x x3 2- + - .

Ô®Î: ( )p x = x x x23 142 1203 2- + - v‹f.

( )p x -‹ bfG¡fë‹ TLjš, 1 23 142 120 0- + - = .

Mfnt, ( )p x -¡F ( 1)x - xU fhuâahF«.1 1 – 23 142 – 120

0 1 – 22 1201 – 22 120 0 " ÛÂ

k‰bwhU fhuâ x x22 1202- + = ( )( )x x x x x12 10 120 12 102

- - + = - - .

vdnt, 23 142 120x x x3 2- + - = ( )( )( )x x x1 12 10- - - .

(iv) 4 5 7 6x x x3 2- + - .

Ô®Î: ( )p x = x x x4 5 7 63 2- + - v‹f.

( )p x -‹ bfG¡fë‹ TLjš, 4 5 7 6 0- + - = .

Mfnt, ( )p x -¡F ( 1)x - xU fhuâahF«.1 4 – 5 7 – 6

0 4 – 1 64 – 1 6 0 " ÛÂ

k‰bwhU fhuâ x x4 62- + .

vdnt, x x x4 5 7 63 2- + - = ( )( )x x x1 4 62

- - + .

(v) 7 6x x3- + .

Ô®Î: ( )p x = x x7 63- + v‹f.

( )p x -‹ bfG¡fë‹ TLjš, 1 7 6 0- + = .


0 1 1 – 61 1 – 6 0 " ÛÂ

k‰bwhU fhuâ x x 62+ - = ( )( )x x x x x3 2 6 3 22

+ - - = + - .vdnt, 7 6x x3

- + = ( )( )( )x x x1 2 3- - + . (vi) 13 32 20x x x

3 2+ + + .

Ô®Î: ( )p x = x x x13 32 203 2+ + + v‹f.

( )p x -‹ x‰iwgil mL¡Ffë‹, bfG¡fë‹ TLjš 1 32 33+ = .( )p x -‹ Ïu£ilgil mL¡Ffë‹, bfG¡fë‹ TLjš 13 20 33+ = .

Mfnt, ( )p x -¡F ( 1)x + xU fhuâahF«.– 1 1 13 32 20

0 – 1 – 12 – 201 12 20 0 " ÛÂ


k‰bwhU fhuâ x x12 202+ + = x x x10 2 202

+ + + .

= ( ) ( ) ( )( )x x x x x10 2 10 10 2+ + + = + + .

vdnt, 13 32 20x x x3 2+ + + = ( )( )( )x x x1 10 2+ + + .

(vii) 2 9 7 6x x x3 2- + + .

Ô®Î: ( )p x = x x x2 9 7 63 2- + + v‹f.

(1) 0, ( 1) 0p p! !- Mfnt ( 1)x - k‰W« ( 1)x + M»ad ( )p x -¡F fhuâfsšy.

Mfnt, x-¡F ntW kÂ¥òfis¥ ÃuÂæ£L ( )p x -‹ ó¢Áa§fis fhzyh«.

2x = Mf ÏU¡F«nghJ (2) 0.p = vdnt, ( )p x -¡F ( )x 2- xU fhuâ.2 2 – 9 7 6

0 4 – 10 – 62 – 5 – 3 0 " ÛÂ

k‰bwhU fhuâ x x2 5 32- - = ( )( )x x3 2 1- + .

vdnt, 2 9 7 6x x x3 2- + - = ( )( )( )x x x2 3 2 1- - + .

(viii) 5 4x x3- + .

Ô®Î: ( )p x = x x5 43- + v‹f.

( )p x -‹ bfG¡fë‹ TLjš, 1 5 4 0- + = .


0 1 1 – 41 1 – 4 0 " ÛÂ

k‰bwhU fhuâ x x 42+ - .

vdnt, 5 4x x3- + = ( )( )x x x1 42

- + - .

(ix) 10 10x x x3 2- - + .

Ô®Î: ( )p x = x x x10 103 2- - + v‹f.

( )p x -‹ bfG¡fë‹ TLjš, 1 10 1 10 0- - + = .

Mfnt, ( )p x -¡F ( 1)x - xU fhuâahF«.1 1 – 10 – 1 10

0 1 – 9 – 101 – 9 – 10 0 " ÛÂ

k‰bwhU fhuâ 9 10 ( )( )x x x x10 12- - = - +

vdnt, 10 10x x x3 2- - + = ( )( )( )x x x1 10 1- - + .

(x) 2 11 7 6x x x3 2+ - - .

Ô®Î: ( )p x = x x x2 11 7 63 2+ - - v‹f.

( )p x -‹ bfG¡fë‹ TLjš, 2 11 7 6 0+ - - = .


Mfnt, ( )p x -¡F ( 1)x - xU fhuâahF«.1 2 11 – 7 – 6

0 2 13 62 13 6 0 " ÛÂ

k‰bwhU fhuâ ( )( )x x x x2 13 6 6 2 12+ + = + + .

vdnt, 2 11 7 6x x x3 2+ - - = ( 1)( )( )x x x6 2 1- + + .

(xi) 14x x x3 2+ + - .

Ô®Î: ( )p x = x x x 143 2+ + - v‹f.

(1) 0p ! våš ( )p x ¡F ( 1)x - xU fhuâašy.

nkY«, ( )p 1 0!- . våš, ( )p x ¡F ( )x 1+ ¡F xU fhuâašy.

( )p 2 0= . våš, ( )p x ¡F ( )x 2- is a factor of ( )p x .2 1 1 1 – 14

0 2 6 141 3 7 0 " ÛÂ

k‰bwhU fhuâ x x3 72+ + .

vdnt, 14x x x3 2+ + - = ( )( )x x x2 3 72

- + + .

(xii) 5 2 24x x x3 2- - + .

Ô®Î: ( )p x = x x x5 2 243 2- - + v‹f.

( )p x -‹ bfG¡fë‹ TLjš, 1 5 2 24 0!- - + .

Mfnt, ( )p x -¡F ( 1)x - xU fhuâašy.

nkY«, ( )p 1- = 1 5 2 24 0!- - + + . vdnt, ( )p x -¡F ( 1)x + xU fhuâašy.

2x =- Mf ÏU¡F«nghJ ( ) .p 2 0- = vdnt, ( )p x -¡F ( )x 2+ xU fhuâahF«.

– 2 1 – 5 – 2 240 – 2 14 – 241 – 7 12 0 " ÛÂ

k‰bwhU fhuâ ( )( )x x x x7 12 3 42- + = - - .

vdnt, 5 2 24x x x3 2- - + = ( )( )( )x x x2 3 4+ - - .

gæ‰Á 3.6

1. Ã‹tUtdt‰¿‹ Û. bgh. t fh©f.

(i) 7x yz2 4 , 21x y z

2 5 3 .

Ô®Î: x yz7 2 4 = x yz7 2 4#

x y z21 2 5 3 = 7 3 x y z2 5 3# #

Û.bgh.t. = x yz7 2 3 .


(ii) x y2 , x y

3 , x y2 2 .

Ô®Î: Û.bgh.t. = x y2 .

(iii) 25bc d4 3 , 35b c

2 5 , 45c d3 .

Ô®Î: bc d25 4 3 = bc d52 4 3#

b c35 2 5 = 5 7b c2 5#

c d45 3 = c d5 32 3#

Û.bgh.t. = c5 3 .

(iv) 35x y z5 3 4 , 49x yz

2 3 , 14xy z2 2 .

Ô®Î: x y z35 5 3 4 = x y z7 5 5 3 4# #

x yz49 2 3 = 7 x yz7 2 3# #

xy z14 2 2 = xy z7 2 2 2# #

Û.bgh.t. = xyz7 2 .

2. Ã‹tUtdt‰¿‹ Û. bgh. t fh©f.

(i) c d2 2- , c c d-^ h.

Ô®Î: c d2 2- = ( )( )c d c d+ -

( )c c d- = ( )c c d-

Û.bgh.t. = ( )c d- .

(ii) 27x a x4 3- , x a3 2-^ h .

Ô®Î: x a x274 3- = ( ) ( )( )x x a x x a x ax a3 3 3 93 3 3 2 2

- = - + +

( )x a3 2- = ( )( )x a x a3 3- -

Û.bgh.t. = ( )x a3- .

(iii) 3 18m m2- - , 5 6m m

2+ + .

Ô®Î: m m3 182- - = ( )( )m m6 3- +

m m5 62+ + = ( )( )m m2 3+ +

Û.bgh.t. = ( )m 3+ .

(iv) 14 33x x2+ + , 10 11x x x

3 2+ - .

Ô®Î: x x14 332+ + = ( )( )x x11 3+ +

10 11x x x3 2+ - = ( )( )x x x11 1+ -

Û.bgh.t. = ( )x 11+ .

(v) 3 2x xy y2 2+ + , 5 6x xy y

2 2+ + .

Ô®Î: x xy y3 22 2+ + = ( )( )x y x y2+ +

x xy y5 62 2+ + = ( )( )x y x y3 2+ +

Û.bgh.t. = ( )x y2+ .


(vi) 2 1x x2- - , 4 8 3x x

2+ + .

Ô®Î: x x2 12- - = ( )( )x x2 1 1+ -

x x4 8 32+ + = ( )( )x x2 1 2 3+ +

Û.bgh.t. = ( )x2 1+ .

(vii) 2x x2- - , 6x x

2+ - , 3 13 14x x

2- + .

Ô®Î: x x 22- - = ( )( )x x2 1- +

x x 62+ - = ( )( )x x2 3- +

x x3 13 142- + = ( )( )x x2 3 7- -

Û.bgh.t. = ( )x 2- .

(viii) 1x x x3 2- + - , 1x

4- .

Ô®Î: x x x 13 2- + - = ( )( )x x1 12

+ -

x 14- = ( )( )( )x x x1 1 12

+ - +

Û.bgh.t. = ( )( )x x1 12+ - .

(ix) 24 x x x6 24 3 2- -^ h, 20 x x x2 3

6 5 4+ +^ h.

Ô®Î: ( )x x x24 6 24 3 2- - = 4 6 ( 1)(3 2) .x x x22

# # + -

( )x x x20 2 36 5 4+ + = ( )( )x x x4 5 2 1 14

# # + +

bghJ fhuâfŸ4, , (2 1)x x2+

Û.bgh.t. = ( )x x4 2 12+ .

(x) a a1 35 2- +^ ^h h , a a a2 1 32 3 4- - +^ ^ ^h h h .

Ô®Î: ( ) ( )a a1 35 2- + , ( ) ( ) ( )a a a1 3 23 4 2

- + -

Û.bgh.t. = ( ) ( )a a1 33 2- +

3. ÑnH bfhL¡f¥g£LŸs gšYW¥ò¡nfhitfë‹ nrhofS¡F tF¤jš go Kiwæš

Û. bgh. t fh©f.

(i) 9 23 15x x x3 2- + - , 4 16 12x x

2- + .

Ô®Î: ( )f x = x x x9 23 153 2- + -

k‰W« ( )g x = ( )x x x x4 16 12 4 4 32 2- + = - + , v‹f.

vdnt, tF¤Â 4 3x x2- + MF«.

x 5-

x x4 32- + x x x9 23 153 2

- + -

x x x4 33 2- +

x x5 20 152- + -

x x5 20 152- + -

0

ÛÂ = 0. vdnt, Û.bgh.t. ( ( ), ( ))f x g x = x x4 32- + .


(ii) 3 18 33 18x x x3 2+ + + , 3 13 10x x

2+ + .

Ô®Î: ( )f x = x x x3 18 33 183 2+ + + .

k‰W« ( )g x = x x3 13 102+ + v‹f.

vdnt, tF¤Â x x3 13 102+ + MF«.

x35+

x x3 13 102+ + x x x3 18 33 183 2

+ + +

x x x3 13 103 2+ +

(–) (–) (–)

5 23 18x x2+ +

5x x365

3502

+ +

x34

34+

( 1)x34& + ! 0. (

34 v‹gJ ( )g x -‹ tF¤Âašy)

x3 10+

x 1+ x x3 13 102+ +

x x3 32+

(–) (–)

x10 10+

x10 10+ 0

ÛÂ = 0. vdnt, Û.bgh.t. ( ( ), ( ))f x g x = x 1+ .

(iii) 2 2 2 2x x x3 2+ + + , 6 12 6 12x x x

3 2+ + + .

Ô®Î: ( )f x = ( )x x x2 13 2+ + +

k‰W« ( )g x = ( )x x x6 2 23 2+ + + v‹f.

(2 v‹gJ ( )f x k‰W« ( )g x -‹ bghJ¡fhuâ)

vdnt, tF¤Â x x x 13 2+ + + MF«.

1

x x x 13 2+ + + x x x2 23 2

+ + +

x x x 13 2+ + +

1 0x2 !+


x 1+

1x2 + x x x 13 2+ + +

x3 x+

x 12+

x 12+

0

ÛÂ = 0. vdnt, Û.bgh.t. ( ( ), ( ))f x g x = ( 1)x2 2+ .

(iv) 3 4 12x x x3 2- + - , 4 4x x x x

4 3 2+ + + .

Ô®Î: ( )f x = x x x3 4 123 2- + -

k‰W« ( )g x = ( )x x x x4 43 2+ + + v‹f.

vdnt, tF¤Â x x x4 43 2+ + + MF«.

1

x x x4 43 2+ + + x x x3 4 123 2

- + -

x x x4 43 2+ + +

x4 2- 16-

( )x4 42& - + ! 0. ( 4- v‹gJ ( )f x -‹ fhuâašy)

x 1+

x 42+ x x x4 43 2

+ + +

x3 x4+(–) (–)

x2 + 4

x2 + 4 (–) (–) 0

ÛÂ = 0. vdnt, Û.bgh.t. ( ( ), ( ) )f x g x = x 42+ .

gæ‰Á 3.7

Ã‹tUtdt‰¿‰F Û.bgh.k fh©f.

1. x y3 2 , xyz .

Ô®Î: ; .x y xyz3 2 vdnt, Û.bgh.k. x y z3 2= .

2. 3x yz2 , 4x y

3 3 .

Ô®Î: 3 ; 4 .x yz x y2 3 3 vdnt, Û.bgh.k. = 3 4 12x y z x y z3 3 3 3# = .


3. a bc2 , b ca

2 , c ab2 .

Ô®Î: a bc2 = a bc2 ; b ca ab c2 2= ; c ab2 = abc2

vdnt, Û.bgh.k. = a b c2 2 2 .

4. 66a b c4 2 3 , 44a b c

3 4 2 , 24a b c2 3 4 .

Ô®Î: a b c66 4 2 3 = a b c11 2 3 4 2 3# # #

a b c44 3 4 2 = a b c11 2 2 3 4 2# # #

a b c24 2 3 4 = a b c2 3 2 2 2 3 4# # # #

vdnt, Û.bgh.k. = a b c11 2 2 3 2 4 4 4# # # # # = a b c264 4 4 4 .

5. am 1+ , am 2+ , am 3+ .

Ô®Î: am 1+ = a am# ; a a am m2 2

#=+

am 3+ = a am 3#

vdnt, Û.bgh.k. = a a am m3 3# = + .

6. x y xy2 2

+ , x xy2+ .

Ô®Î: x y xy2 2+ = ( )xy x y+

x xy2+ = ( )x x y+

vdnt, Û.bgh.k. = ( )xy x y+ .

7. 3 a 1-^ h, 2 a 1 2-^ h , a 12-^ h.

Ô®Î: ( )a3 1- = ( )a3 1- ; ( ) ( )a a2 1 2 12 2- = -

a 12- = ( )( )a a1 1- + .

vdnt, Û.bgh.k. = ( 1) ( 1) .a a6 2- +

8. 2 18x y2 2- , 5 15x y xy

2 2+ , 27x y

3 3+ .

Ô®Î: x y2 182 2- = ( )( )x y x y2 3 3+ -

x y xy5 152 2+ = ( )xy x y5 3+

x y273 3+ = ( )( )x y x xy y3 3 92 2

+ - +

Û.bgh.k. = 2 5 ( 3 )( 3 )( 3 9 )xy x y x y x xy y2 2# # + - - +

vdnt, Û.bgh.k. = ( )( )( )xy x y x y x xy y10 3 3 3 92 2+ - - + .

9. x x4 32 3+ -^ ^h h , x x x1 4 3 2- + -^ ^ ^h h h .Ô®Î: ( ) ( )x x4 32 3

+ - ; ( )( )( )x x x1 4 3 2- + -

vdnt, Û.bgh.k. = ( ) ( ) ( )x x x4 3 12 3+ - - .

10. 10 x xy y9 62 2+ +^ h, 12 x xy y3 5 2

2 2- -^ h, 14 x x6 2

4 3+^ h.

Ô®Î: ( )x xy y10 9 62 2+ + = ( )x y2 5 3 2

# +

( )x xy y12 3 5 22 2- - = ( )( )x y x y2 3 3 22

# + -


( )x x14 6 24 3+ = 2 7 (3 1)x x2 3

# # +

Û.bgh.k. = 2 7 5 3 (3 ) (3 1)( 2 )x x y x x y2 3 2# # # + + -

vdnt, Û.bgh.k. = ( ) ( )( )x x y x y x420 3 2 3 13 2+ - + .

gæ‰Á 3.8

1. Ã‹tU« x›bthU nrho gšYW¥ò¡nfhitfë‹ Û. bgh. k fh©f.

(i) 5 6x x2- + , 4 12x x

2+ - Ït‰¿‹ Û. bgh. t 2x - .

Ô®Î: ( )f x = 5 6, ( ) 4 12k‰W«x x g x x x2 2- + = + - v‹f.

Û.bgh.t. = x 2- vd¡ bfhL¡f¥g£LŸsJ.

Ï§F,

Û.bgh.k. # Û.bgh.t. = ( ) ( )f x g x#

vdnt, Û.bgh.k. = ( ) ( ) ( )( )Û.bgh.t.f x g x

xx x x x

25 6 4 122 2#

=-

- + + -

= ( )( )( )( )x

x x x x2

3 2 6 2-

- - + -

Mfnt, Û.bgh.k. = ( )( )( )x x x3 2 6- - + .

(ii) 3 6 5 3x x x x4 3 2+ + + + , 2 2x x x

4 2+ + + Ït‰¿‹ Û. bgh. t 1x x

2+ + .

Ô®Î: ( )f x = x x x x3 6 5 34 3 2+ + + + k‰W« ( )g x = x x x2 24 2

+ + + v ‹ f . Û.bgh.t.= x x 12

+ + vd¡ bfhL¡f¥g£LŸsJ.

vdnt, Û.bgh.k. = ( ) ( )Û.bgh.t.f x g x#

mjhtJ, ( )f x k‰W« ( )g x M»a Ïu©ilÍ« Û.bgh.t-Mš tF¡f ÏaY«.

vdnt, ( )f x -I Û.bgh.t-Mš tF¡f

x x2 32+ +

x x 12+ + x x x x3 6 5 34 3 2

+ + + +

x x x4 3 2+ +

x x x2 5 53 2+ +

x x x2 2 23 2+ +

x x3 3 32+ +

x x3 3 32+ +

0

Û.bgh.k. = ( )

( )( )( )

x x

x x x x x x x

1

1 2 3 2 22

2 2 4 2

+ +

+ + + + + + +

vdnt, Û.bgh.k. = ( )( )x x x x x2 3 2 22 4 2+ + + + + .


(iii) 2 15 2 35x x x3 2+ + - , 8 4 21x x x

3 2+ + - Ït‰¿‹ Û. bgh. t 7x + .

Ô®Î: ( )f x = x x x2 15 2 353 2+ + - k‰W« ( )g x = x x x8 4 213 2

+ + - v‹f. Û.bgh.t.= x 7+ vd¡ bfhL¡f¥g£LŸsJ.



vdnt, ( )f x -I Û.bgh.t-Mš tF¡f

x x2 52+ -

x 7+ x x x2 15 2 353 2+ + -

x x2 143 2+

x x22+

x x72+

x5 35- -

x5 35- -

0

Û.bgh.k. = ( )( )( )x

x x x x x x7

7 2 5 8 4 212 3 2

++ + - + + -

vdnt, Û.bgh.k. = ( )( )x x x x x2 5 8 4 212 3 2+ - + + - .

(iv) 2 3 9 5x x x3 2- - + , 2 10 11 8x x x x

4 3 2- - - + Ït‰¿‹ Û. bgh. t 2 1x - .

Ô®Î: ( )f x = x x x2 3 9 53 2- - + k‰W« ( )g x = x x x x2 10 11 84 3 2

- - - + v ‹ f . Û.bgh.t. = x2 1- vd¡ bfhL¡f¥g£LŸsJ.



vdnt, ( )g x -I Û.bgh.t-Mš tF¡f

x x5 83- -

x2 1- x x x x2 10 11 84 3 2- - - +

x x2 4 3-

x x10 112- -

x x10 52- +

x16 8- +

x16 8- +

0

Û.bgh.k. = ( )

( )( )( )x

x x x x x x2 1

2 1 5 8 2 3 9 53 3 2

-- - - - - +

vdnt, Û.bgh.k. = ( )( )x x x x x5 8 2 3 9 53 3 2- - - - + .


2. Ã‹tUtdt‰¿š Kiwna p x^ h k‰W« q x^ h M»at‰¿‹ Û.bgh.k, k‰W«

Û.bgh.t nkY« p x^ h M»ad bfhL¡f¥g£LŸsd. q x^ h v‹w k‰bwhU

gšYW¥ò¡nfhitia¡ fh©f.

(i) x x1 22 2+ +^ ^h h , x x1 2+ +^ ^h h, x x1 22+ +^ ^h h.

Ô®Î: Û.bgh.k. = ( 1) ( 2)x x2 2+ + ; Û.bgh.t. = ( )( )x x1 2+ +

( )p x = ( ) ( )x x1 22+ + v‹f

mjhtJ, Û.bgh.k. # Û.bgh.t. = ( ) ( )p x q x# .

( )q x& = ( ) ( ) ( )

( ) ( ) ( )( )Û.bgh.k. Û.bgh.t.p x x x

x x x x

1 2

1 2 1 22

2 2# =

+ +

+ + + +

vdnt, ( )q x = ( )( )x x1 2 2+ + .

(ii) x x4 5 3 73 3+ -^ ^h h , x x4 5 3 7 2+ -^ ^h h , x x4 5 3 73 2+ -^ ^h h .

Ô®Î: Û.bgh.k. = ( ) ( )x x4 5 3 73 3+ - ; Û.bgh.t. = ( )( )x x4 5 3 7 2

+ -

( )p x = ( ) ( )x x4 5 3 73 2+ - v‹f


( )q x& = ( ) ( ) ( )

( ) ( ) ( )( )Û.bgh.k. Û.bgh.t.p x x x

x x x x

4 5 3 7

4 5 3 7 4 5 3 73 2

3 3 2# =

+ -

+ - + -

vdnt, ( )q x = ( ) ( )x x3 7 4 53- + .

(iii) x y x x y y4 4 4 2 2 4- + +^ ^h h, x y

2 2- , x y

4 4- .

Ô®Î: Û.bgh.k. = ( )( )x y x x y y4 4 4 2 2 4- + + ; Û.bgh.t. = ( )x y2 2

-

( )p x = x y4 4- v‹f.


( )q x& = ( )

( )( )( )Û.bgh.k. Û.bgh.t.p x x y

x y x x y y x y4 4

4 4 4 2 2 4 2 2# =

-

- + + -

vdnt, ( )q x = ( )( )x x y y x y4 2 2 4 2 2+ + - .

(iv) x x x4 5 13- +^ ^h h, x x5

2+^ h, x x x5 9 2

3 2- -^ h.

Ô®Î: Û.bgh.k. = ( 4 )(5 1) ( )( )( )x x x x x x x2 2 5 13- + = + - +

Û.bgh.t. = ( ) ( )x x x x5 5 12+ = +

( )p x = ( )( )x x x x x x5 9 2 5 1 23 2- - = + - v‹f


( )q x& = ( ) ( )( )

( )( )( )( )( )Û.bgh.k. Û.bgh.t.p x x x x

x x x x x x5 1 2

2 2 5 1 5 1# =+ -

+ - + +

vdnt, ( )q x = ( )( )x x x2 5 1+ + .

(v) x x x x1 2 3 32

- - - +^ ^ ^h h h, x 1-^ h, x x x4 6 33 2- + -^ h.

Ô®Î: Û.bgh.k. = ( )( )( )x x x x1 2 3 32- - - + ; Û.bgh.t. = x 1-

( )p x = 4 6 3 ( )( )x x x x x x1 3 33 2 2- + - = - - + v‹f



& ( )q x = ( ) ( )( )

( )( )( )( )Û.bgh.k. Û.bgh.t.p x x x x

x x x x x

1 3 3

1 2 3 3 12

2# =

- - +

- - - + -

vdnt, ( )q x = ( )( )x x1 2- - .

(vi) 2 x x1 42

+ -^ ^h h, x 1+^ h, x x1 2+ -^ ^h h.

Ô®Î: Û.bgh.k. = ( )( ) ( )( )( )x x x x x2 1 4 2 1 2 22+ - = + + -

Û.bgh.t. = x 1+ k‰W« ( )p x = ( )( )x x1 2+ - v‹f


& ( )q x = ( ) ( )( )

( )( )( )( )Û.bgh.k. Û.bgh.t.p x x x

x x x x1 2

2 1 2 2 1# =+ -

+ + - +

vdnt, ( )q x = ( )( )x x2 1 2+ + .

gæ‰Á 3.9

Ã‹tUtdt‰iw vëa toé‰F¢ RU¡Ff.

(i) x x

x x

3 12

6 92

2

-

+ .

Ô®Î: x x

x x

3 12

6 92

2

-

+ = ( )( )

x xx x

xx

3 43 2 3

42 3

-+

=-+ .

(ii) x

x

1

14

2

-

+ .

Ô®Î: x

x

1

14

2

-

+ = ( )( )x x

x

x1 1

1

1

12 2

2

2+ -

+ =-

.

(iii) x x

x

1

12

3

+ +

- .

Ô®Î: x x

x

1

12

3

+ +

- = ( )( )( )

x x

x x xx

1

1 11

2

2

+ +

- + += - .

(iv) x

x

9

272

3

-

- .

Ô®Î: x

x

9

272

3

-

- = ( )( )

( )( )x x

x x xx

x x3 3

3 3 93

3 92 2

+ -- + +

=+

+ + .

(v) x x

x x

1

12

4 2

+ +

+ + . (F¿¥ò: 1x x4 2+ + = x x1

2 2 2+ -^ h )

Ô®Î: x x

x x

1

12

4 2

+ +

+ + = ( )( )

x x

x x x xx x

1

1 11

2

2 22

+ +

+ + - += - + .

(vi) x x

x

4 16

84 2

3

+ +

+ .

Ô®Î: x x

x

4 16

84 2

3

+ +

+ = ( ) ( ) ( )( )

( )( )

x x

x

x x x x

x x x

4 2

2

2 4 2 4

2 2 42 2 2

3 3

2 2

2

+ -

+ =+ + - +

+ - +

= x x

x

2 4

22+ +

+ .


(vii) x x

x x

2 5 3

2 32

2

+ +

+ - .

Ô®Î: x x

x x

2 5 3

2 32

2

+ +

+ - = ( )( )( )( )

x xx x

xx

2 3 12 3 1

11

+ ++ -

=+- .

(viii) x x

x

9 2 6

2 1622

4

+ -

-^ ^h h

.

Ô®Î: ( )( )x x

x

9 2 6

2 1622

4

+ -

- = ( ) ( )

(( ) )

x x

x

9 2 3

2 92

2 2 2

+ -

-

= ( )( )

( )( )( )( )

x x

x x xx

2 9 3

2 9 3 33

2

2

+ -

+ + -= + .

(ix) x x x

x x x

4 2 3

3 5 42

2

- - -

- - +

^ ^

^ ^

h h

h h.

Ô®Î: ( )( )

( )( )

x x x

x x x

4 2 3

3 5 42

2

- - -

- - + = ( )( )( )( )( )( )x x xx x x

xx

4 3 13 4 1

11

- - +- - -

=+- .

(x) x x x

x x x

10 13 40

8 5 502

2

+ - +

- + -

^ ^

^ ^

h h

h h.

Ô®Î: ( )( )

( )( )

x x x

x x x

10 13 40

8 5 502

2

+ - +

- + - = ( )( )( )( )( )( )x x xx x x

10 8 58 10 5

1+ - -- + -

= .

(xi) x x

x x

8 6 5

4 9 52

2

+ -

+ + .

Ô®Î: x x

x x

8 6 5

4 9 52

2

+ -

+ + = ( )( )( )( )

x xx x

xx

4 5 2 14 5 1

2 11

+ -+ +

=-+ .

(xii) x x x

x x x x

7 3 2

1 2 9 142

2

- - +

- - - +

^ ^

^ ^ ^

h h

h h h.

Ô®Î: ( )( )

( )( )( )

x x x

x x x x

7 3 2

1 2 9 142

2

- - +

- - - + = ( )( )( )

( )( )( )( )( )

x x xx x x x

x7 2 1

1 2 7 22

- - -- - - -

= - .

gæ‰Á 3.101. Ã‹tU« é»jKW nfhitfis¥ bgU¡», éilia¢ RU¡»a toéš vGJf.

(i) x

x xxx

22

23 6

2

#+-

-+ .

Ô®Î: x

x xxx

22

23 6

2

#+-

-+ = ( ) ( )

3x

x xxx

x22

23 2

#+-

-+

= .

(ii) x

x

x x

x x

4

81

5 36

6 82

2

2

2

#-

-

- -

+ +

Ô®Î: x

x

x x

x x

4

81

5 36

6 82

2

2

2

#-

-

- -

+ + = ( )( )( )( )

( )( )( )( )

x xx x

x xx x

2 29 9

9 44 2

#+ -+ -

- ++ +

= xx

29

-+ .


(iii) x x

x x

x

x x

20

3 10

8

2 42

2

3

2

#- -

- -

+

- +

Ô®Î: x x

x x

x

x x

20

3 10

8

2 42

2

3

2

#- -

- -

+

- + = ( )( )( )( )

( )( )x xx x

x x x

x x5 45 2

2 2 4

2 42

2

#- +- +

+ - +

- +

= x 41+

.

(iv) x x

xxx

x xx x

3 216

644

2 84 16

2

2

3

2

2

2

# #- +-

+-

- -- +

Ô®Î: 4 16x x

xxx

x xx x

3 216

644

2 8

2

2

2

3

2

2# #- +-

+-

- -- +

= ( )( )( )( )

( )( )

( )( )( )( )x x

x x

x x x

x xx xx x

2 14 4

4 4 16

2 24 24 16

2

2

# #- -+ -

+ - +

+ -- +- +

= x 11-

.

(v) x x

x x

x x

x x

2

3 2 1

3 5 2

2 3 22

2

2

2

#- -

+ -

+ -

- -

Ô®Î: x x

x x

x x

x x

2

3 2 1

3 5 2

2 3 22

2

2

2

#- -

+ -

+ -

- - = ( )( )( )( )

( )( )( )( )

x xx x

x xx x

2 13 1 1

3 1 22 1 2

#- +- +

- ++ -

= xx

22 1++ .

(vi) x x

x

x x

x x

x x

x

2 4

2 1

2 5 3

8

2

32 2

4

2# #+ +

-

+ -

-

-

+

Ô®Î: x x

x

x x

x x

x x

x

2 4

2 1

2 5 3

8

2

32 2

4

2# #+ +

-

+ -

-

-

+

= ( )( )

( )( )( )x x

xx x

x x x xx x

x

2 4

2 12 1 32 2 4

23

2

2

# #+ +

-- +

- + +-+ = 1.

2. Ã‹tUtd‰iw vëa toéš RU¡Ff.

(i) x

x

x

x1 1

2

2

'+ -

.

Ô®Î: x

x

x

x1 1

2

2

'+ -

= ( ).

xx

x

x x

xx

1

1 1 12

#+

+ -= -^ h

(ii) x

xxx

49

3676

2

2

'-

-++

Ô®Î: ( )( )( )( )( )( )

.x

xxx

x x xx x x

xx

49

3667

7 7 66 6 7

76

2

2

#-

-++ =

+ - ++ - +

=--

(iii) x

x x

x x

x x

25

4 5

7 10

3 102

2

2

2

'-

- -

+ +

- -

Ô®Î:

( )

( )

( )

( )( 5)( )( )( )

( 5)( )( )( )

( )( )

.x

x x

x x

x xx xx x

x xx x

xx

25

4 5

3 10

7 105

5 12

5 251

2

2

2

2

# #-

- -

- -

+ +=

+ -- +

- ++ +

=-+


(iv) x x

x x

x x

x x

4 77

11 28

2 15

7 122

2

2

2

'- -

+ +

- -

+ +

Ô®Î: x x

x x

x x

x x

4 77

11 28

2 15

7 122

2

2

2

'- -

+ +

- -

+ +

= ( )( )( )( )

( )( )( )( )

x xx x

x xx x

xx

11 77 4

3 45 3

115

#- ++ +

+ +- +

=-- .

(v) x x

x x

x x

x x

3 10

2 13 15

4 4

2 62

2

2

2

'+ -

+ +

- +

- -

Ô®Î: x x

x x

x x

x x

3 10

2 13 15

4 4

2 62

2

2

2

'+ -

+ +

- +

- -

= ( )( )( )( )

( )( )( )( )

x xx x

x xx x

5 22 3 5

2 3 22 2

1#+ -+ +

+ -- -

= .

(vi) x

x x

x x

x

9 16

3 4

3 2 1

4 42

2

2

2

'-

- -

- -

-

Ô®Î: x

x x

x x

x

9 16

3 4

3 2 1

4 42

2

2

2

'-

- -

- -

-

= ( )( )( )( )

( )( )( )( )

( )x xx x

x xx x

xx

3 4 3 43 4 1

4 1 13 1 1

4 3 43 1

#+ -- +

+ -+ -

=++ .

(vii) x x

x x

x x

x x

2 9 9

2 5 3

2 3

2 12

2

2

2

'+ +

+ -

+ -

+ -

Ô®Î: x x

x x

x x

x x

2 9 9

2 5 3

2 3

2 12

2

2

2

'+ +

+ -

+ -

+ - = ( )( )( )( )

( )( )( )( )

x xx x

x xx x

2 3 32 1 3

2 1 12 3 1

#+ +- +

- ++ -

= xx

11

+- .

gæ‰Á 3.11

1. Ã‹tUtdt‰iw ÏU gšYW¥ò¡nfhitfë‹ xU Ã‹dkhf (é»jKW

nfhitahf) vëa toéš RU¡Ff.

(i) x

xx2 2

83

-+

-.

Ô®Î: x

xx2 2

83

-+

- =

xx

x xx

2 28

223 3 3

--

-=

--

= ( )( )x

x x xx x

22 2 4

2 42

2

-- + +

= + + .

(ii) x x

x

x x

x

3 2

2

2 3

32 2+ +

+ +- -

- .

Ô®Î: x x

x

x x

x

3 2

2

2 3

32 2+ +

+ +- -

- = ( )( ) ( )( )x x

xx x

x2 1

23 1

3+ +

+ +- +

-

= x x x11

11

12

++

+=

+.


(iii) x

x x

x x

x x

9

6

12

2 242

2

2

2

-

- - +- -

+ - .

Ô®Î: x

x x

x x

x x

9

6

12

2 242

2

2

2

-

- - +- -

+ -

= ( )( )( )( )

( )( )( )( )

x xx x

x xx x

xx

xx

3 33 2

4 36 4

32

36

+ -- +

+- ++ -

=++ +

++

= ( )x

x xxx

xx

32 6

32 8

32 4

++ + + =

++ =

++ .

(iv) x x

x

x x

x

7 10

2

2 15

32 2- +

- +- -

+ .

Ô®Î: x x

x

x x

x

7 10

2

2 15

32 2- +

- +- -

+ = ( )( ) ( )( )x x

xx x

x5 2

25 3

3- -

- +- +

+

= x x x51

51

52

-+

-=

-.

(v) x x

x x

x x

x x

3 2

2 5 3

2 3 2

2 7 42

2

2

2

- +

- + -- -

- - .

Ô®Î: x x

x x

x x

x x

3 2

2 5 3

2 3 2

2 7 42

2

2

2

- +

- + -- -

- -

= ( )( )( )( )

( )( )( )( )

x xx x

x xx x

xx

xx

2 12 3 1

2 1 22 1 4

22 3

24

- -- -

-+ -+ -

=-- -

--

= x

x xxx

22 3 4

21

-- - + =

-+ .

(vi) x x

x

x x

x x

6 8

4

20

11 302

2

2

2

+ +

- -- -

- + .

Ô®Î: x x

x

x x

x x

6 8

4

20

11 302

2

2

2

+ +

- -- -

- +

= ( )( )( )( )

( )( )( )( )

x xx x

x xx x

xx

xx

2 42 2

5 46 5

42

46

+ ++ -

-- +- -

=+- -

+-

= x

x xx4

2 64

4+

- - + =+

.

(vii) xx

x

xxx

12 5

1

11

3 22

2

++ +

-

+ ---` j> H .

Ô®Î: xx

x

xxx

12 5

1

11

3 22

2

++ +

-

+ ---` j= G

= ( )( )

( )xx

x xx

xx

12 5

1 11

13 22

++ +

+ -+ -

--

= ( )( )

( )( ) ( )( )x x

x x x x x1 1

2 5 1 1 3 2 12

+ -+ - + + - - +


= ( )( )x x

x x x x x x x1 1

2 2 5 5 1 3 3 2 22 2 2

+ -- + - + + - - + +

= ( )( ) ( )( )

( )x x

xx x

xx1 1

2 21 1

2 11

2+ -

- =+ -

-=

+.

(viii) x x x x x x3 2

15 61

4 32

2 2 2+ +

++ +

-+ +

.

Ô®Î: x x x x x x3 2

15 61

4 32

2 2 2+ +

++ +

-+ +

= ( )( ) ( )( ) ( )( )x x x x x x1 2

12 31

3 12

+ ++

+ +-

+ +

= ( )( )( )

( )x x x

x x x1 2 3

3 1 2 2+ + +

+ + + - +

= ( )( )( )x x x

x x1 2 3

2 4 2 4 0+ + +

+ - - = .

2. x

x

2

12

3

+

- cl‹ vªj é»jKW¡ nfhitia¡ T£l x

x x

2

3 2 42

3 2

+

+ + »il¡F«?

Ô®Î: ( )p x v‹gJ njitahd é»jKW nfhit v‹f.

Mfnt, 1 ( )x

x p x2

3

2+

- + = x

x x

2

3 2 42

3 2

+

+ +

& ( )p x = x

x x

x

x

2

3 2 4

2

12

3 2

2

3

+

+ + -+

-

= x

x x x

2

3 2 4 12

3 2 3

+

+ + - +

vdnt, ( )p x = x

x x

2

2 2 52

3 2

+

+ + .

3. vªj é»jKW nfhitia x

x x2 1

4 7 53 2

-- + -èUªJ fê¡f 2 5 1x x

2- + »il¡F«?

Ô®Î: ( )p x v‹gJ njitahd é»jKW nfhit v‹f.

Mfnt, ( )x

x x p x2 1

4 7 53 2

-- + - = x x2 5 12

- +

& ( )p x = ( )

( )x

x x x x2 1

4 7 5 2 5 13 2

2

-- + - - +

= ( )x

x x x x x2 1

4 7 5 4 12 7 13 2 3 2

-- + - + - +

vdnt, ( )p x = x

x x2 1

5 7 62

-- + .

4. P = x yx+

, Q = x yy

+ våš,

P Q P Q

Q1 22 2-

--

I¡ fh©f.

Ô®Î: P Q P Q

Q1 22 2-

--

= ( )( )P Q P Q P Q

Q1 2

--

+ -


= ( )( ) ( )( )P Q P QP Q Q

P Q P QP Q2

+ -+ -

=+ -

-

= P Q

x yx

x yy

x yx y

1 1 1+

=

++

+

=

++

vdnt, P Q P Q

Q1 22 2-

--

= 1.

gæ‰Á 3.12

1. Ã‹tUtdt‰¿‰F t®¡f_y« fh©f.

(i) 196a b c6 8 10 .

Ô®Î: a b c196 6 8 10 = a b c a b c14 142 6 8 10 3 4 5=

(ii) 289 a b b c4 6- -^ ^h h .

Ô®Î: 289 a b b c4 6- -^ ^h h = ( ) ( )a b b c172 4 6- -

= 17 ( ) ( )a b b c2 3- - .

(iii) 44x x11 2+ -^ h .

Ô®Î: 11 44x x2+ -^ h = x x x x x22 121 44 22 1212 2+ + - = - +

= ( ) ( )x x11 112- = - .

(iv) 4x y xy2- +^ h .

Ô®Î: 4x y xy2- +^ h = x xy y xy x xy y2 4 22 2 2 2- + + = + +

= ( ) ( )x y x y2+ = + .

(v) 121x y8 6 ' 81x y

4 8 .

Ô®Î: 121 81x y x y8 6 4 8

' = x y

x y

y

xyx

81

121

9

11911

4 8

8 6

2 2

2 4 2

= = .

(vi) x y a b b c

a b x y b c

25

644 6 10

4 8 6

+ - +

+ - -

^ ^ ^

^ ^ ^

h h h

h h h .

Ô®Î: 25

64

x y a b b c

a b x y b c4 6 10

4 8 6

+ - +

+ - -

^ ^ ^

^ ^ ^

h h h

h h h

= ( ) ( ) ( )

( ) ( ) ( )

x y a b b c

a b x y b c

5

82 4 6 10

2 4 8 6

+ - +

+ - - = ( ) ( ) ( )

( ) ( ) ( )

x y a b b c

a b x y b c58

2 3 5

2 4 3

+ - +

+ - - .

2. Ã‹tUtdt‰¿‰F t®¡f_y« fh©f.

(i) 16 24 9x x2- + .

Ô®Î: 16 24 9x x2- + = ( ) ( )x x4 3 4 32

- = - .


(ii) x x x x x25 8 15 2 152 2 2- + + - -^ ^ ^h h h.

Ô®Î: 25 8 15 2 15x x x x x2 2 2- + + - -^ ^ ^h h h

= ( )( )( )( )( )( )x x x x x x5 5 3 5 5 3+ - + + - +

= ( ) ( ) ( ) ( )( )( )x x x x x x5 5 3 5 5 32 2 2+ - + = + - + .

(iii) 4 9 25 12 30 20x y z xy yz zx2 2 2+ + - + - .

Ô®Î: 4 9 25 12 30 20x y z xy yz zx2 2 2+ + - + -

= ( ) ( ) ( ) ( )( ) ( )( ) ( )( )x y z x y y z z x2 3 5 2 2 3 2 3 5 2 5 22 2 2+ - + - + - + - - + -

= ( )x y z x y z2 3 5 2 3 52- - = - - .

(iv) 2xx

14

4+ + .

Ô®Î: 1 2xx

4

4+ + = ( ) ( )xx

xx

1 2 12 2

2

22

2+ +d dn n

= xx

xx

1 122

2 22

+ = +c cm m .

(v) x x x x x x6 5 6 6 2 4 8 32 2 2+ - - - + +^ ^ ^h h h.

Ô®Î: 6 5 6 6 2 4 8 3x x x x x x2 2 2+ - - - + +^ ^ ^h h h

= ( )( )( )( )( )( )x x x x x x2 3 3 2 3 2 2 1 2 1 2 3+ - - + + +

= ( ) ( ) ( ) ( )( )( )x x x x x x2 3 3 2 2 1 2 3 3 2 2 12 2 2+ - + = + - + .

(vi) x x x x x x2 5 2 3 5 2 6 12 2 2- + - - - -^ ^ ^h h h.

Ô®Î: x x x x x x2 5 2 3 5 2 6 12 2 2- + - - - -^ ^ ^h h h

= ( )( )( )( )( )( )x x x x x x2 1 2 3 1 2 2 1 3 1- - + - - +

= ( ) ( ) ( ) ( )( )( )x x x x x x2 1 2 3 1 2 1 2 3 12 2 2- - + = - - + .

gæ‰Á 3.13 1. Ã‹tU« gšYW¥ò¡ nfhitfë‹ t®¡f_y¤ij tF¤jš Kiw _y« fh©f.

(i) 4 10 12 9x x x x4 3 2- + - + .

Ô®Î: x x2 32- +

x2 x x x x4 10 12 94 3 2- + - +

x4

x x2 22- x x4 103 2

- +

x x4 43 2- +

x x2 4 32- + x x6 12 92

- +

6 12 9x x2- +

0

vdnt, x x x x x x4 10 12 9 2 34 3 2 2- + - + = - + .


(ii) 4 8 8 4 1x x x x4 3 2+ + + + .

Ô®Î: x x2 2 12+ +

x2 2 x x x x4 8 8 4 14 3 2+ + + +

x4 4

4 2x x2+ x x8 83 2

+

x x8 43 2+

4 1x x42+ + x x4 4 12

+ +

x x4 4 12+ +

0

vdnt, x x x x x x4 8 8 4 1 2 2 14 3 2 2+ + + + = + + .

(iii) 9 6 7 2 1x x x x4 3 2- + - + .

Ô®Î: x x3 12- +

x3 2 x x x x9 6 7 2 14 3 2- + - +

x9 4

x x6 2- x x6 73 2

- +

x x6 3 2- +

6 2 1x x2- + x x6 2 12

- +

x x6 2 12- +

0

vdnt, x x x x x x9 6 7 2 1 3 14 3 2 2- + - + = - + .

(iv) 4 25 12 24 16x x x x2 3 4

+ - - + .

Ô®Î: x x4 3 22- +

x4 2 x x x x16 24 25 12 44 3 2- + - +

x16 4

x x8 32- x x24 253 2

- +

x x24 93 2- +

x x8 6 22- + x x16 12 42

- +

x x16 12 42- +

0

vdnt, x x x x x x16 24 25 12 4 4 3 24 3 2 2- + - + = - + .


2. Ã‹tU« gšYW¥ò¡nfhitfŸ KGt®¡f§fŸ våš, a, b M»at‰¿‹ kÂ¥òfis¡

fh©f.

(i) 4 12 37x x x ax b4 3 2- + + + . (ii) 4 10x x x ax b

4 3 2- + - + .

(iii) 109 60 36ax bx x x4 3 2+ + - + . (iv) 40 24 36ax bx x x

4 3 2- + + + .

Ô®Î: (i) x x2 3 72- +

x2 2 x x x ax b4 12 374 3 2- + + +

x4 4

x x4 32- 12 37x x3 2

- +

x x12 93 2- +

x x4 6 72- + 2 x ax b8 2

+ +

x x28 42 492- +

0bfhL¡f¥g£l gšYW¥ò¡nfhit xU KG t®¡fkhjyhš a 42=- k‰W« b 49= .

(ii)

x x2 32- +

x2 x x x ax b4 104 3 2- + - +

x4

x x2 22- x x4 103 2

- +

x x4 43 2- +

x x2 4 32- + x ax b6 2

- +

x x6 12 92- +

0bfhL¡f¥g£l gšYW¥ò¡nfhit xU KG t®¡fkhjyhš 2a 1= k‰W« 9b = .

x x6 5 7 2- +

(iii) 6 36 60 109x x bx ax2 3 4- + + +

36

x12 5- x x60 109 2- +

x x60 25 2- +

x x12 10 7 2- + x bx ax84 2 3 4

+ +

x x x84 70 492 3 4- +

0bfhL¡f¥g£l gšYW¥ò¡nfhit xU KG t®¡fkhjyhš a 49= k‰W« b 70=- .


(iv)

x x6 2 3 2+ +

6 x x bx ax36 24 40 2 3 4+ + - +

36

12 2x+ x x24 40 2+

x x24 4 2+

12 4 3x x2+ + x bx ax36 2 3 4- +

x x x36 12 92 3 4+ +

0bfhL¡f¥g£l gšYW¥ò¡nfhit xU KG t®¡fkhjyhš a 9= k‰W« b 12=- .

gæ‰Á 3.14

fhuâ¥gL¤J« Kiwæš ÑnH bfhL¡f¥g£l ÏUgo¢ rk‹ghLfis¤ Ô®¡f.

(i) 81x2 3 2+ -^ h = 0.

Ô®Î: ( )x2 3 812+ - = 0 kh‰WKiw:

& x x4 12 9 812+ + - = 0 ( )x2 3 92 2

+ - = 0

& x x4 24 12 722+ - - = 0 (2 3 9)(2 3 9)x x& + + + - = 0

& ( )( )x x6 4 12+ - = 0 (2 12)(2 6)x x& + - = 0

& x 6 0+ = or x4 12 0- = 6x& =- or x = 3

6x& =- or x = 3

vdnt, Ô®Î fz« {– 6, 3}.

(ii) 3 5 12x x2- - = 0.

Ô®Î: x x3 5 122- - = 0

& ( )( )x x3 3 4- + = 0 3x& = or x34=-

vdnt, Ô®Î fz« ,34 3-$ ..

(iii) 2 3x x5 52+ - = 0.

Ô®Î: x x5 2 3 52+ - = 0

& ( )( )x x5 5 3+ - = 0 x 5& =- or x5

3=

vdnt, Ô®Î fz« ,55

3-' 1.


(iv) 3 x 62-^ h = 3x x 7+ -^ h .

Ô®Î: ( )x3 62- = ( )x x 7 3+ -

& x x x3 7 18 32 2- - - + = 0

& x x2 7 152- - = 0

& ( )( )x x5 2 3- + = 0

5x& = mšyJ x23=-

vdnt, Ô®Î fz« ,23 5-$ ..

(v) 3xx8- = 2.

Ô®Î: xx

3 8- = 2

& 3 2 8x x2- - = 0

( )( )x x2 3 4- + = 0

2x& = mšyJ x34=-

vdnt, Ô®Î fz« ,34 2-$ ..

(vi) xx1+ =

526 .

Ô®Î: xx1+ =

526

& x x5 26 52- + = 0

& ( )( )x x5 5 1- - = 0

5x& = mšyJ x51=

vdnt, Ô®Î fz« ,51 5$ ..

(vii) x

xx

x1

1+

+ + = 1534 .

Ô®Î: x

xx

x1

1+

+ + = 1534

& x x

x x x2 12

2 2

+

+ + + = 1534

& x x30 30 152+ + = x x34 342

+

& ( )( )x x2 5 2 3+ - = 0

x25& =- mšyJ x

23=


23-$ ..

(viii) 1a b x a b x2 2 2 2 2

- + +^ h = 0.

Ô®Î: ( )a b x a b x 12 2 2 2 2- + + = 0 & ( 1)( 1)a x b x2 2

- - = 0.

xa

12

& = mšyJ xb

12

= . vdnt, Ô®Î fz« ,a b

1 12 2' 1.

(ix) 2 5x x1 12+ - +^ ^h h = 12.

Ô®Î: ( ) ( )x x2 1 5 12+ - + = 12 kh‰WKiw: Let y x 1= + . Then,

& ( )x x x2 2 1 5 5 122+ + - - - = 0 y y2 5 122

- - = 0

& x x2 152- - = 0 ( )( )y y2 3 4& + - = 0

& ( )( )x x3 2 5- + = 0 ( )( )x x2 5 3& + - = 0

Mfnt, 3x = mšyJ x25=- Mfnt, 3x = mšyJ x

25=- .

vdnt, Ô®Î fz« ,325-$ ..

(x) 3 5x x4 42- - -^ ^h h = 12.

Ô®Î: ( ) ( )x x3 4 5 42- - - = 12 kh‰WKiw: Let y x 4= - . Then,

& x x x3 24 48 5 20 122- + - + - = 0 3 5 12y y2

- - = 0

& x x3 29 562- + = 0 ( )( )y y3 4 3& + - = 0

& ( )( )x x3 8 7- - = 0 ( )( )x x3 8 7& - - = 0

Mfnt, x38= mšyJ x 7= Mfnt, x

38= mšyJ x 7=

vdnt, Ô®Î fz« ,38 7$ ..


gæ‰Á 3.15

1. t®¡f¥ ó®¤Â Kiwæš Ã‹tU« rk‹ghLfis¤ Ô®¡f.

(i) 6 7x x2+ - = 0.

Ô®Î: x x6 72+ - = 0

& ( )x x2 32+ = 7 (

26 3= )

& ( )x x2 3 92+ + = 7 + 9 ( ( )3 92

= )

& ( )x 3 2+ = 16

& x 3+ = 4! 1x& = mšyJ x 7=- .

vdnt, Ô®Î fz« {– 7, 1}.

(ii) 3 1x x2+ + = 0.

Ô®Î: x x3 12+ + = 0

& x x223

492

+ +` j = 149- +

& x23 2

+` j = 45

& x23+ =

25!

& x = 23

25!- & x =

23 5- - mšyJ x

23 5= - + .


3 52

3 5- - - +' 1.

(iii) 2 5 3x x2+ - = 0.

Ô®Î: x x2 5 32+ - = 0

& x x25

232

+ - = 0 (ÏUòwK« 2-Mš tF¡f)

& x x25

16252

+ + = 1625

23+ F¿¥ò:

21

25

16252

=` j8 B .

& x45 2

+` j = 1649

& x45+ =

47! & x =

45

47!-

x21& = mšyJ x 3=- . vdnt, Ô®Î fz« ,3

21-$ ..

(iv) 4 4x bx a b2 2 2+ - -^ h = 0.

Ô®Î: ( )x bx a b4 42 2 2+ - - = 0 (ÏUòwK« 4-Mš tF¡f)

& x bx2+ = a b

4

2 2-

& x bx b4

22

+ + = a b b4 4

2 2 2- +


& x b2

2+` j = a

4

2

& x b2

+ = a2

! & x = b a2 2!-

Mfnt, x a b2

= - mšyJ ( )x

a b2

=-+

vdnt, Ô®Î fz« ( ),

a b a b2 2

- + -' 1.

(v) x x3 1 32- + +^ h = 0.

Ô®Î: ( )x x3 1 32- + + = 0 ÏUòwK«

23 1

2+c m T£l

& ( )x x3 12

3 122

- + + +c m = 2

3 1 32

+ -c m

& x2

3 12

- +c m; E = 4

3 2 3 1 4 3+ + -

& x2

3 12

- +c m; E = 2

3 12

-c m

& x2

3 1- +c m = 2

3 1! -c m

& x = 2

3 12

3 1!+ -c cm m

Mfnt, x 3= mšyJ x 1= . vdnt, Ô®Î fz« { , }1 3 .

(vi) xx

15 7

-+ = 3 2x + .

Ô®Î: xx

15 7-+ = x3 2+

& x5 7+ = ( )( )x x3 2 1+ -

& x x3 6 92- - = 0

& x x2 32- - = 0 ( F¿¥ò

22 1

2- =` j )

& x x2 12- + = 1 + 3

& ( )x 1 2- = 4

& x 1- = 2! & x = 1 2!

Mfnt, x 3= mšyJ x 1=- .

vdnt, Ô®Î fz« { , }1 3- .

2. ÏUgo¢ N¤Âu¤ij¥ ga‹gL¤Â Ã‹tU« rk‹ghLfis¤ Ô®¡f.

(i) 7 12x x2- + = 0.

Ô®Î: x x7 122- + = 0. ax bx c 02

+ + = v‹w ÏUgo rk‹gh£o‹go.

Ï§F, , ,a b c1 7 12= =- =

x = a

b b ac2

42!- -


= 2

7 49 482

7 1! !- =

& x = 28 mšyJ x

26= & x = 4 mšyJ x = 3.

vdnt, Ô®Î fz« , .4 3" ,

(ii) 15 11 2x x2- + = 0.

Ô®Î: x x15 11 22- + = 0. ax bx c 02


Ï§F, 1 , ,a b c5 11 2= =- =

Mfnt, x = a

b b ac2

42!- -

= ( )2 15

11 121 12030

11 121 12030

11 1! ! !- = - =

& x = 3012 mšyJ x

3010= & x =

52 mšyJ x

31= .


31$ ..

(iii) xx1+ = 2

21 .

Ô®Î: xx1+ = 2

21

& 1x

x2 + = 25

& x x2 5 22- + = 0 , ax bx c 02


Ï§F, , , 2a b c2 5= =- =

Mfnt, x = a

b b ac2

42!- -

= 4

5 25 164

5 94

5 3! ! !- = =

& x = 2 mšyJ x21=

vdnt, Ô®Î fz« ,21 2$ ..

(iv) 3 2a x abx b2 2 2

- - = 0.

Ô®Î: a x abx b3 22 2 2- - = 0. 0Ax Bx C2


Ï§F, , ,A a B ab C b3 22 2= =- =- .

vdnt, x = A

B B AC2

42!- -

= ( )

( )( )

a

ab a b a b

2 3

4 3 22

2 2 2 2! - -

= a

ab a b a b

a

ab ab

6

24

6

52

2 2 2 2

2! !+ =


& x = a

ab abab

6

52

+ = mšyJ x = .a

ab abab

6

532

2- = -

vdnt, Ô®Î fz« ,ab

ab

32-$ ..

(v) a x 12+^ h = x a 1

2+^ h.

Ô®Î: ( )a x 12+ = ( )x a 12

+

& ax a2+ = ( )x a 12

+

& ( )ax x a a12 2- + + = 0. 0Ax Bx C2


Ï§F, , ( ),A a B a C a12= =- + =

vdnt, x = A

B B AC2

42!- -

= ( ) ( )a

a a a2

1 1 42 2 2 2!+ + -

= ( )a

a a a a2

1 2 1 42 4 2 2!+ + + -

= ( ) ( ) ( )a

a a aa

a a2

1 2 12

1 12 4 2 2 2 2! !+ - +=

+ -

= ( ) ( )a

a a2

1 12 2!+ -

Mfnt, x = aa a22 2

= mšyJ x = a a22 1=

vdnt, Ô®Î fz« ,a

a1$ ..

(vi) 36 12x ax a b2 2 2- + -^ h = 0.

Ô®Î: 3 12 ( )x ax a b6 2 2 2- + - = 0. 0Ax Bx C2


Ï§F, , , ( )A B a C a b36 12 2 2= =- = -

vdnt, x = A

B B AC2

42!- -

= ( )

( )( )a a a b2 36

12 144 4 362 2 2! - -

= a a a b72

12 144 144 1442 2 2! - +

= a b a b72

12 14472

12 122! !=

Mfnt, x = ( ) ( )a b a b72

126

+=

+ (m) x = ( ) ( )a b a b72

126

-=

-

vdnt, Ô®Î fz« ,a b a b6 6- +$ ..


(vii) xx

xx

11

43

+- +

-- =

310 .

Ô®Î: xx

xx

11

43

+- +

-- =

310

& ( )( )

( )( ) ( )( )x x

x x x x1 4

1 4 3 1+ -

- - + - + = 310

& x x

x x x x

3 4

5 4 2 32

2 2

- -

- + + - - = 310

& x x

x x

3 4

2 7 12

2

- -

- + = 310

& x x6 21 32- + = x x10 30 402

- -

& x x4 9 432- - = 0 ( 0ax bx c2

+ + = -‹ go)

Ï§F, , ,a b c4 9 43= =- =-

vdnt, x = a

b b ac2

42!- -

= ( )( )( )

2 49 81 4 4 43

89 769! !- -

=

Mfnt, x = 8

9 769+ mšyJ 8

9 769-


9 7698

9 769- +' 1.

(viii) a x a b x b2 2 2 2 2

+ - -^ h = 0.

Ô®Î: ( )a x a b x b2 2 2 2 2+ - - = 0. 0Ax Bx C2


Ï§F, , ,A a B a b C b2 2 2 2= = - =-

Mfnt, x = A

B B AC2

42!- -

= ( )

( ) ( ) ( )( )

a

a b a b a b

2

42

2 2 2 2 2 2 2!- - - - -

= ( ) ( ) ( )

a

b a a a b b

a

b a a b

2

2

22

2 2 4 2 2 4

2

2 2 2 2! !- + +=

- +

Mjyhš, x = a

b a a b

a

b

2 2

2 2 2 2

2

2- + + =

mšyJ x = a

b a a b

21

2

2 2 2 2- - - =-

vdnt, Ô®Î fz« ,a

b12

2

-) 3.


gæ‰Á 3.16

1. xU v© k‰W« mj‹ jiyÑê M»at‰¿‹ TLjš 865 våš, mªj v©iz¡

fh©f.

Ô®Î: njitahd v© x k‰W« mj‹ jiyÑê x1 vd bfhŸf. ( )x 0!

bfhL¡f¥g£l ãgªjidæ‹go

xx1+ =

865

& x

x 12+ =

865

& x x8 65 82- + = 0

& ( )( )x x8 1 8- - = 0

& x8 1- = 0 mšyJ x 8- = 0 & x = 81 mšyJ x = 8

vdnt, njitahd v© 8.

2. Ïu©L äif v©fë‹ t®¡f§fë‹ é¤Âahr« 45. Á¿a v©â‹ t®¡f« MdJ,

bgça v©â‹ eh‹F kl§»‰F¢ rk« våš, mªj v©fis¡ fh©f.

Ô®Î: x k‰W« y Ïu©L äif v©fŸ v‹f.nkY«, x y< .

bfhL¡f¥g£l ãgªjidæ‹go

x2 = y4 g (1)

k‰W« y x2 2- = 45 g (2)

& y y4 452- - = 0 [ (1)-‹ go ]

& ( )( )y y9 5- + = 0

& y 9- = 0 mšyJ y 5+ = 0 & y = 9 mšyJ y = – 5

bfhL¡f¥g£l v© äif v© våš y = 9.

y = 9I (1)-š ÃuÂæl, x2 = 4 9, 6x&# =

vdnt, njitahd v©fŸ 6 k‰W« 9.

3. xU étrhæ 100r.Û gu¥gséš xU br›tf tot¡ fhŒf¿¤ njh£l¤ij mik¡f

éU«Ãdh®. mtçl« 30 Û ÚsnkÍŸs KŸf«Ã ÏUªjjhš Å£o‹ kÂšRtiu¤

njh£l¤Â‹ eh‹fhtJ g¡f ntèahf it¤J¡ bfh©L m¡f«Ãahš _‹W

g¡fK« ntèia mik¤jh®. njh£l¤Â‹ g¡f msÎfis¡ fh©f.

Ô®Î: AB = x Û k‰W« BC = y Û, br›tf tot fhŒf¿ njh£l¤Â‹ g¡f§fŸ v‹f.

CD v‹gJ kÂšRt®.

KŸf«Ãæ‹ Ús« = 30 Û

& y x y+ + = 30

& x y2+ = 30


& y = x2

30 -` j g (1)

fhŒf¿ njh£l¤Â‹ gu¥gsÎ = 100 r.Û.

& xy = 100

& x x2

30 -` j = 100 ( (1)-‹ go )

& x x30 2- = 200

& 30 200x x2- + = 0

& ( )( )x x20 10- - = 0

& x 20- = 0 mšyJ x 10- = 0 & x = 20 mšyJ x = 10

x 10= I (1)-š ÃuÂæl y = 2

30 10 10- =

nkY«, x 20= våš, (1) & y = 2

30 20 5- = .

vdnt, br›tf taè‹ Ús, mfy§fŸ Kiwna 10 Û, 10 Û (mšyJ) 20 Û, 5 Û.

4. xU br›tf tot ãy« 20 Û Ús« k‰W« 14 Û mfy« bfh©lJ. mij¢ R‰¿

btë¥òw¤Âš mikªJŸs Óuhd mfyKŸs ghijæ‹ gu¥ò 111 r.Û våš,

ghijæ‹ mfy« v‹d?Ô®Î: AB = 20 Û k‰W« BC = 14 Û v‹gd ABCD v‹w br›tf ãy¤Â‹ Ús,

mfy§fŸ v‹f. x Û v‹gJ ghijæ‹ mfykhF«.

PQ = x20 2+ k‰W« QR = x14 2+ v‹gd PQRS v‹w br›tf¤Â‹ Ús, mfy§fŸ

.ghijæ‹ gu¥ò = 111 r.Û.

& PQRS-v‹w br›tf¤Â‹ gu¥ò – ABCD -v‹w br›tf¤Â‹ gu¥ò = 111 r.Û.

& (20 2 )(14 2 ) (20 14)x x #+ + - = 111

& x x x20 14 40 28 4 20 142# #+ + + - = 111

& 4 68 111x x2+ - = 0

& ( )( )x x2 37 2 3+ - = 0

& x = 237- mšyJ x

23=

Ús« äif v© våš, x23= .

vdnt, btë¥òw ghijæ‹ mfy« 1.5 Û.

5. Óuhd ntf¤Âš xU bjhl® t©oahdJ (train) 90 ».Û öu¤ij¡ flªjJ.

mjDila ntf« kâ¡F 15 ».Û mÂfç¡f¥g£oUªjhš, gaz« brŒÍ« neu« 30 ãäl§fŸ FiwªÂU¡F« våš, bjhl® t©oæ‹ Óuhd ntf« fh©f.

Ô®Î: Ïuæè‹ tH¡fkhd ntf« x ».Û./kâ.

T1 v‹gJ x ».Û./kâ ntf¤Âš 90 ».Û. öu« fl¡f MF« neu«.

T2

v‹gJ x + 15 ».Û./kâ¡F ntf¤Âš 90 ».Û. öu« fl¡f MF« neu«.

fhy« = ntf«öu« , våš T

1 =

x90 k‰W« T

2 =

x 1590+

.


bfhL¡f¥g£l ãgªjidfë‹go

T T1 2- =

6030

& x x90

1590-+

= 21

& ( )

( )x xx x

1590 15 90

++ - =

21

& x x

x x

15

90 1350 902+

+ - = 21

& x x15 27002+ - = 0

& ( )( )x x60 45+ - = 0

& x = – 60 mšyJ x = 45

ntf« äif v© våš, x = 45.

vdnt, Ïuæè‹ tH¡fkhd ntf« 45 ».Û./kâ.

6. mirt‰w Úçš xU ÏaªÂu¥gl»‹ ntf« kâ¡F 15 ».Û. v‹f. m¥glF Únuh£l¤Â‹

Âiræš 30 ».Û öu« br‹W, ÃwF vÂ®¤ Âiræš ÂU«Ã 4 kâ 30 ãäl§fëš

Û©L« òw¥g£l Ïl¤Â‰F ÂU«Ã tªjhš Úç‹ ntf¤ij¡ fh©f.

Ô®Î: Úç‹ ntf« x ».Û./kâ.

ÏaªÂugl»‹ ntf« 15 ».Û./kâ.

vdnt, Únuh£l Âiræ‹ ntf« k‰W« vÂ®Âiræ‹ ntf« Kiwna

( )x15 + ».Û./kâ¡F k‰W« ( )x15 - ».Û./kâ

T1 v‹gJ 30 ».Û. öu¤ij Únuh£l¤ Âiræš fl¡F« neu«.

T2

v‹gJ 30 ».Û. öu¤ij Únuh£l¤Â‹ vÂ®Âiræš fl¡F« neu«.

fhy« = ntf«öu« , våš T

1 =

x1530+

k‰W« T2

= x15

30-

.

mjhtJ, T1 + T

2 = 4 kâ. 30 ãäl« = 4

21 kâ.

& x x15

301530

-+

+ =

29

& ( )( )

( ) ( )x x

x x15 15

30 15 30 15- +

+ + - = 29

& ( )x9 225 2- = 1800

& x225 2- = 200

& x = 5!

ntf« xU äif v© våš .x 5= vdnt, Úç‹ ntf« 5 ».Û./kâ.

7. xU tUl¤Â‰F K‹ò, xUtç‹ taJ mtUila kfå‹ taij¥nghš 8 kl§F.

j‰nghJ mtUila taJ, kfå‹ taÂ‹ t®¡f¤Â‰F¢ rk« våš, mt®fSila

j‰nghija taij¡ fh©f.


Ô®Î:kf‹ jªij

j‰nghija taJ x M©LfŸ y M©LfŸ

Xuh©L¡F K‹ taJ ( )x 1- M©LfŸ ( )y 1- M©LfŸ

bfhL¡f¥g£lÂ‹ go y = x2 g (1)

k‰W« y 1- = ( )x8 1-

y& = x8 7- g (2)

(2)-I (1)-š ÃuÂæl x8 7- = x2

& x x8 72- + = 0 & ( )( )x x7 1- - = 0

& x 7= mšyJ x 1=

x 1= vd ÏU¡f ÏayhJ,vdnt x 7= .

vdnt, jªijæ‹ taJ kfå‹ taijél mÂf« våš kfå‹ taJ 7 M©LfŸ

k‰W« jªijæ‹ taJ = 7 492= M©LfŸ.

8. xU rJu§f¥ gyifæš 64 rk rJu§fŸ cŸsd. x›bthU rJu¤Â‹ gu¥ò

6.25 r.brÛ. v‹f. rJu§f¥ gyifæš eh‹F¥ g¡f§fëY« btë¥òw rJu§fis

x£o 2 br.Û mfy¤Âš g£ilahd Xu« cŸsJ våš, rJu§f¥ gyifæ‹ g¡f¤Â‹

Ús¤Âid¡ fh©f.

Ô®Î: rJu§f gyifæ‹ g¡f« = x br.Û. v‹f

rJu§f gyifæš xU rJu¤Â‹ gu¥ò = 6.25 r.br.Û.

vdnt, 64 rJu¤Â‹ gu¥ò = .64 6 25#

& ( )x 4 2- = 400

& x 4- = 20! & x = 24 mšyJ – 16

vdnt, rJu¤Â‹ g¡f« äif v© våš x = 24 br.Û.

9. xU ntiyia¢ brŒa A-¡F B-ia él 6 eh£fŸ Fiwthf¤ njit¥gL»wJ.

ÏUtU« nr®ªJ m›ntiyia¢ brŒjhš mij 4 eh£fëš Ko¡f ÏaY« våš, B

jåna m›ntiyia v¤jid eh£fëš Ko¡f ÏaY«?

Ô®Î: B xU ntiyia brŒa vL¤J¡ bfhŸS« eh£fŸ x v‹f.

vdnt, A xU ntiyia brŒa vL¤J¡ bfhŸS« eh£fŸ ( 6)x - v‹f.

A xU ehëš brŒÍ« ntiy = x 61-

B xU ehëš brŒÍ« ntiy = x1

A k‰W« B ÏUtU« xU ehëš brŒÍ« ntiy = 41

vdnt, x x61 1-

+ = 41

& ( )x x

x x66

-+ - =

41


& x x14 242- + = 0

& ( )( )x x12 2- - = 0

& x 12 0- = mšyJ x 2 0- = & x 12= mšyJ x 2=

x 2= mDkÂ¡f¥glhjjhš, x 12= .vdnt, B xU ntiy Ko¡f vL¤J¡bfhŸs eh£fŸ 12.

10. xU Ïuæš ãiya¤ÂèUªJ Ïu©L bjhl® t©ofŸ xnu neu¤Âš òw¥gL»‹wd.

Kjš t©o nk‰F Âiria neh¡»Í«, Ïu©lh« t©o tl¡F Âiria neh¡»Í«

gaz« brŒ»‹wd. Kjš t©oahdJ Ïu©lhtJ t©oia él kâ¡F 5 ». Û

mÂf ntf¤Âš brš»wJ. Ïu©L kâ neu¤Â‰F¥ ÃwF mt‰¿‰F ÏilnaÍŸss

bjhiyÎ 50 ».Û. våš, x›bthU t©oæ‹ ruhrç ntf¤Âid¡ fh©f.

Ô®Î: Ïu©lh« Ïuæè‹ ntf« x ».Û./kâ¡F våš

Kjyh« Ïuæè‹ ntf« = ( )x 5+ ».Û./kâ¡F

O v‹gJ Ïuæš ãiya«.

Kjš Ïuæš 2 kâ neu¤Âš flªj öu« = 2 (x + 5) = OA

Ïu©lh« Ïuæš 2 kâ neu¤Âš flªj öu« = 2x = OB

Ãjhfu° nj‰w¤Â‹go, OA OB2 2+ = AB2

& [ ( )] ( )x x2 5 22 2+ + = 502

& x x8 40 24002+ - = 0

& x x5 3002+ - = 0

& ( )( )x x20 15+ - = 0

& x 20 0+ = mšyJ x 15 0- = & x 20=- mšyJ x 15=

ntf« xU äif vdnt, 15x =

Mfnt, Ïu©lh« Ïuæè‹ ntf« 15 ».Û./kâ¡F

k‰W« Kjš Ïuæè‹ ntf« 20 ».Û./kâ¡F.

gæ‰Á 3.17

1. rk‹ghLfë‹ _y§fë‹ j‹ikia MuhŒf.

(i) 8 12 0x x2- + = .

Ô®Î: x x8 122- + = 0

ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš , ,a b c1 8 12= =- =

Ï¥bghGJ, D = b ac42- = ( ) ( )( )8 4 1 12 64 48 0>2

- - = -

vdnt, _y§fŸ bkŒba©fŸ k‰W« rkk‰wit.

(ii) 2 3 4 0x x2- + = .

Ô®Î: x x2 3 42- + = 0


ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš , ,a b c2 3 4= =- =

Ï¥bghGJ, D = b ac42- = ( ) ( )( )3 4 2 4 0<2

- -

vdnt, _y§fŸ bkŒba©fŸ mšy.

(iii) 9 12 4 0x x2+ + = .

Ô®Î: x x9 12 42+ + = 0

ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš , ,a b c9 12 4= = =

Ï¥bghGJ, D = b ac42- = ( ) ( )( )12 4 9 4 02

- =

vdnt, _y§fŸ bkŒba©fshfÎ« rkkhfÎ« ÏU¡F«.

(iv) 3 2 2 0x x62- + = .

Ô®Î: x x3 2 6 22- + = 0

ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš , ,a b c3 2 6 2= =- =

Ï¥bghGJ, D = b ac42- = ( ) ( )( ) ( )2 6 4 3 2 4 6 24 02

- - = - =

vdnt, _y§fŸ bkŒba©fshfÎ« rkkhfÎ« ÏU¡F«.

(v) 1 0x x53

322

- + = .

Ô®Î: x x53

32 12

- + = 0

ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš , , 1a b c

53

32= =- =

Ï¥bghGJ, D = b ac42- = 4 (1)

32

53

94

512

4588 0<

2- - = - =-` `j j

vdnt, _y§fŸ bkŒba©fŸ mšy.

(vi) 4x a x b ab2 2- - =^ ^h h .

Ô®Î: ( )( )x a x b2 2- - = ab4

( )x x a b ab2 2 42& - + + = ab4 ( )x x a b22

& - + = 0

ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš 1, ( ),a b a b c2 0= =- + =

Ï¥bghGJ, D = b ac42- = [ ( )] ( )( ) ( )a b a b2 4 1 0 4 0>2 2

- + - = +

vdnt, _y§fŸ bkŒba©fŸ k‰W« rkk‰wit.

2. Ã‹tU« rk‹ghLfë‹ _y§fŸ bkŒba©fŸ k‰W« rkkhdit våš, k Ï‹

kÂ¥òfis¡f©LÃo.

(i) 2 10 0x x k2- + = .

Ô®Î: x x k2 102- + = 0

ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš , ,a b c k2 10= =- =

_y§fŸ rk« våš,

D = b ac42- = 0 & ( ) ( )( )k10 4 22

- - = 0

& k8 = 100 k225& =


(ii) 12 4 3 0x kx2+ + = .

Ô®Î: x kx12 4 32+ + = 0

ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš 2, ,a b k c1 4 3= = =

_y§fŸ rk« våš,

D = b ac42- = 0 & ( ) ( )( )k4 4 12 32

- = 0

& k2 = 16144 9=

vdnt, k = 3! .

(iii) 2 5 0x k x 22+ - + =^ h .

Ô®Î: ( )x k x2 2 52+ - + = 0

& 2 4 5x kx k2+ - + = 0

& ( )x kx k2 5 42+ + - = 0

ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš 1, ,a b k c k2 5 4= = = -

_y§fŸ rk« våš,

D = b ac42- = 0 & ( ) ( )( )k k2 4 1 5 42

- - = 0

& k k4 52+ - = 0

& ( )( )k k5 1+ - = 0 & 5k =- mšyJ k 1= .

(iv) 2 1 0k x k x1 12

+ - - + =^ ^h h .

Ô®Î: ( ) ( )k x k x1 2 1 12+ - - + = 0

ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš

, 2( ),a k b k c1 1 1= + =- - =

_y§fŸ rk« våš,

D = b ac42- = 0

& [ ( )] ( )( )k k2 1 4 1 12- - - + = 0

& 4( 1) 4( 1)k k2- - + = 0

& ( ) ( )k k1 12- - + = 0 & k k32

- = 0

vdnt, k = 0 , 3.

3. 2 2 0x a b x a b2 2 2+ + + + =^ ^h h v‹w rk‹gh£o‹ _y§fŸ bkŒba©fŸ mšy

vd¡ fh£Lf.

Ô®Î: ( ) ( )x a b x a b2 22 2 2+ + + + = 0

0Ax Bx C2+ + = v‹w rk‹gh£il x¥ÃLifæš

1, 2( ), 2( )A B a b C a b2 2= = + = +

Mfnt, D = 4B AC2-


= [2( )] 4(1)( )( )a b a b22 2 2+ - +

= ( )a ab b a b4 2 8 82 2 2 2+ + - -

= a ab b a b4 8 4 8 82 2 2 2+ + - -

= 4 8 4 4 ( 2 )a ab b a ab b2 2 2 2- + - =- - +

= 4( )a b 0<2- - , 6 a, b R! .

vdnt, _y§fŸ bk©ba©fŸ mšy.

4. 3 2 0p x pqx q2 2 2

- + = v‹w rk‹gh£o‹ _y§fŸ bkŒba©fŸ mšy vd¡

fh£Lf.

Ô®Î: bfhL¡f¥g£l rk‹ghL 3 2p x pqx q2 2 2- + = 0

ax bx c 02+ + = v‹w rk‹gh£il x¥ÃLifæš , ,a p b pq c q3 22 2

= =- =

mjhtJ, D = b ac42-

= ( ) ( )( )pq p q2 4 32 2 2- -

= 4 12 8 0p q p q p q <2 2 2 2 2 2- =-

vdnt, _y§fŸ bk©ba©fŸ mšy.

5. 0ad bc !- vd mikªj 2 0a b x ac bd x c d2 2 2 2 2+ - + + + =^ ^h h

v‹w rk‹gh£o‹ _y§fŸ rkbkåš, ba

dc= vd ãWÎf.

Ô®Î: ( ) ( )a b x ac bd x c d22 2 2 2 2+ - + + + = 0


, 2( ),A a b B ac bd C c d2 2 2 2= + =- + = +

mjhtJ, D = 4B AC 02- = ( _y§fŸ rk« )

& [ ( )] ( )( )ac bd a b c d2 42 2 2 2 2- + - + + = 0

& ( ) ( )ac bd a c a d b c b d4 42 2 2 2 2 2 2 2 2+ - + + + = 0

& a c abcd b d a c a d b c b d22 2 2 2 2 2 2 2 2 2 2 2+ + - - - - = 0

& a d b c abcd22 2 2 2- - + = 0

& 2a d abcd b c2 2 2 2- + = 0

& ( )ad bc 2- = 0 & ad bc- = 0

& ad = bc

& ba =

dc . ( , , ,a b c d v‹gd ó¢Áak‰w v©fŸ)

6. 0x a x b x b x c x c x a- - + - - + - - =^ ^ ^ ^ ^ ^h h h h h h -‹ _y§fŸ v¥bghGJ«

bkŒba©fŸ v‹W«, a b c= = vd Ïšyhéoš k£Lnk m«_y§fŸ rkk‰wit

v‹W« ãWÎf.

Ô®Î: ( )( ) ( )( ) ( )( )x a x b x b x c x c x a- - + - - + - - = 0


& x ax bx ab x bx cx bc x cx ax ca2 2 2- - + + - - + + - - + = 0

& ( )x a b c x ab bc ca3 22- + + + + + = 0


, 2( ),A B a b c C ab bc ca3= =- + + = + + .

mjhtJ, D = 4B AC2-

= [ ( )] ( )( )a b c ab bc ca2 4 32- + + - + +

= ( ) ( )a b c ab bc ca4 122+ + - + +

= 4[( ) 3( )]a b c ab bc ca2+ + - + +

= [ ]a b c ab bc ca4 2 2 2+ + - - -

= [ ]a b c ab bc ca2 2 2 2 2 2 22 2 2+ + - - -

= [( ) ( ) ( ) ]a b b c c a2 2 2 2- + - + - > 0

vdnt, _y§fŸ v¥bghGJ« bkŒba©fŸ.

nkY«, a b c= = våš 0D = . vdnt, _y§fŸ rkkhdit.

7. rk‹ghL 2 0m x mcx c a12 2 2 2

+ + + - =^ h -‹ _y§fŸ rk« våš,c a m12 2 2= +^ h

vd ãWÎf.

Ô®Î: ( )m x mcx c a1 22 2 2 2+ + + - = 0


( ), 2 ,A m B mc C c a1 2 2 2= + = = -

bfhL¡f¥g£l _y§fŸ rkbkåš,

D = 4B AC2- = 0

& ( ) ( )mc m c a2 4 12 2 2 2- + -^ h = 0

& 4 ( )m c c a m c a m42 2 2 2 2 2 2 2- - + - = 0

& m c c a m c a m2 2 2 2 2 2 2 2- + - + = 0

& c a a m2 2 2 2- + + = 0

vdnt, c2 = ( )a m12 2+ .

gæ‰Á 3.18

1. ÑnH bfhL¡f¥g£LŸs rk‹ghLfë‹ _y§fë‹ TLjš k‰W« bgU¡f‰gy‹

M»at‰iw¡ fh©f.

(i) 6 5 0x x2- + = .

Ô®Î: x x6 52- + = 0.

ax bx c 02+ + = ax bx c 02

+ + = v‹w rk‹gh£o‹ go , ,a b c1 6 5= =- = .


a k‰W« b v‹gd bfhL¡f¥g£l rk‹gh£o‹ _y§fŸ v‹f.

vdnt, _y§fë‹ TLjš, a b+ = ( )6

ab

16

- =--

=

_y§fë‹ bgU¡f‰gy‹, ab = ac

15 5= =

vdnt, _y§fë‹ TLjš k‰W« bgU¡f‰gy‹ Kiwna 6 k‰W« 5.

(ii) 0kx rx pk2+ + = .

Ô®Î: kx rx pk2+ + = 0.

ax bx c 02+ + = v‹w rk‹gh£o‹ go , ,a k b r c pk= = = .


vdnt, _y§fë‹ TLjš, a b+ = ab

kr- =-


kpk

p= =

vdnt, _y§fë‹ TLjš k‰W« bgU¡f‰gy‹ Kiwna kr- k‰W« p.

(iii) 3 5 0x x2- = .

Ô®Î: x x3 52- = 0.

ax bx c 02+ + = v‹w rk‹gh£o‹ go , ,a b c3 5 0= =- = .


vdnt, _y§fë‹ TLjš, a b+ = ( )ab

35

35- =-

-=


30 0= =

vdnt, _y§fë‹ TLjš k‰W« bgU¡f‰gy‹ Kiwna 35 k‰W« 0.

(iv) 8 25 0x2- = .

Ô®Î: x8 252- = 0.

ax bx c 02+ + = v‹w rk‹gh£o‹ go , ,a b c8 0 25= = = .


vdnt, _y§fë‹ TLjš, a b+ = ab

80 0- = = F¿¥ò: x

8

252= ,


825=- vdnt, ,x

8

25

8

25= -

vdnt, _y§fë‹ TLjš k‰W« bgU¡f‰gy‹ Kiwna 0 k‰W« 825- .

2. bfhL¡f¥g£LŸs _y§fis¡ bfh©l ÏUgo¢ rk‹ghLfis mik¡fÎ«.

(i) 3 , 4 (ii) 3 7+ , 3 7- (iii) ,2

4 72

4 7+ -

Ô®Î: (i) bfhL¡f¥g£l _y§fŸ 3, 4.a k‰W« b v‹gd bfhL¡f¥g£l rk‹gh£o‹ _y§fŸ v‹f.

_y§fë‹ TLjš, a b+ = 3 + 4 = 7

_y§fë‹ bgU¡f‰gy‹, ab = 3 (4) = 12


njitahd rk‹ghL, x2 - (_y§fë‹ TLjš) x + _y§fë‹ bgU¡f‰gy‹ = 0.

vdnt, njitahd rk‹ghL x x7 12 02- + = .

(ii) bfhL¡f¥g£l _y§fŸ ,3 7 3 7+ - .a k‰W« b v‹gd bfhL¡f¥g£l rk‹gh£o‹ _y§fŸ v‹f.

_y§fë‹ TLjš, a b+ = 3 7 3 7 6+ + - =

_y§fë‹ bgU¡f‰gy‹, ab = ( )( )3 7 3 7 9 7 2+ - = - =



(iii) bfhL¡f¥g£l _y§fŸ ,2

4 72

4 7a b= + = - .a k‰W« b v‹gd bfhL¡f¥g£l rk‹gh£o‹ _y§fŸ v‹f.

_y§fë‹ TLjš, a b+ = 2

4 72

4 728 4+ + - = =

_y§fë‹ bgU¡f‰gy‹, ab = 2

4 72

4 74

16 749+ - = - =c cm m


vdnt, njitahd rk‹ghL x x449 02

- + = & 4 16 0x x 92- + = .

3. 3 5 2x x2- + = 0 v‹w rk‹gh£o‹ _y§fŸ a , b våš, Ã‹tUtdt‰¿‹

kÂ¥òfis¡ fh©f.

(i) ba

ab

+ (ii) a b- (iii) 2 2

ba

ab

+

Ô®Î: (i) 3 5 2x x 02- + = . ax bx c 02

+ + = v‹w rk‹gh£o‹go

, ,a b c3 5 2= =- = .

vdnt, a b+ = ( )ab

35

35- =-

-= k‰W« ab =

ac

32= .

Mfnt, ba

ab

+ = ( ) 22 2 2

aba b

aba b ab+

=+ -

= 32

32

35 2

925 12

23

613

2-

= - =` `

`j j

j .

(ii) vdnt, ( )2a b- = ( ) 42a b ab+ -

= 35 4

32

925 24

912

- = - =` `j j

Mfnt, a b- = 31!


(iii) 2

2 2

b

aab

+ = ( ) ( )33 3 3

aba b

aba b ab a b+

=+ - +

=

32

27125

930

27125 90

23

1835

#-

= - = .

4. a , b v‹gd 3 6 4x x2- + = 0, v‹D« rk‹gh£o‹ _y§fŸ våš,

2 2a b+ -‹

kÂ¥ò¡ fh©f.

Ô®Î: x x3 6 4 02- + = . ax bx c 02


3, ,a b c6 4= =- = .

vdnt, a b+ = ( )ab

36

2- =--

=

ab = ac

34=

Mfnt, 2 2a b+ = ( ) 2 2 234

342 2a b ab+ - = - =` j

2 2a b+ = 34 .

5. a , b v‹gd 2 3 5x x2- - = 0-‹ _y§fŸ våš,

2a k‰W« 2

b M»at‰iw

_y§fshf¡ bfh©l ÏUgo¢ rk‹ghL x‹¿id mik¡f.

Ô®Î: x x2 3 5 02- - = . ax bx c 02


, ,a b c2 3 5= =- =- .

vdnt, a b+ = ( )ab

23

23- =-

-=

ab = ac

25= -

njitahd _y§fë‹ TLjš, 2 2a b+ = ( ) 22a b ab+ -

= 223

25

49

210

4292

- - = + =` `j j

nkY«, _y§fë‹ bgU¡f‰gy‹, ( )( ) ( )25

4252 2 2 2

a b ab= = - =` j


& x x429

4252

- +` j = 0

vdnt, njitahd rk‹ghL x x4 29 25 02- + = .

6. 3 2x x2- + = 0-‹ _y§fŸ a , b våš, a- k‰W« b- M»at‰iw _y§fshf¡

bfh©l ÏUgo¢ rk‹ghL x‹¿id mik¡f.

Ô®Î: 3 0x x 22- + = . ax bx c 02


, 3,a b c1 2= =- = .

a b+ = ( )ab

13

3- =--

=

ab = ac

12 2= =

njitahd rk‹gh£o‹ _y§fŸ a- k‰W« b- .


_y§fë‹ TLjš = ( ) 3a b a b- - =- + =-

_y§fë‹ bgU¡f‰gy‹ = ( )( ) 2a b ab- - = =


& 3 2x x2+ + = 0.

F¿¥ò: 3 2x x2+ + = 0 v‹w rk‹gh£il mila x-¡F gÂè –xI x x3 2

2- + = 0

v‹w rk‹gh£oš ÃuÂæl »il¡F«.

7. a , b v‹gd 3 1x x2- - = 0-‹ _y§fŸ våš, 1

2a

k‰W« 12b

M»at‰iw


Ô®Î: x x3 1 02- - = . ax bx c 02


1, 3,a b c 1= =- =-

mjhtJ, a b+ = ( )3

ab

13- =-

-= k‰W« ab =

ac

11 1= - =-

njitahd rk‹gh£o‹ _y§fŸ 12a

k‰W« 12b

.

_y§fë‹ TLjš, ( )

1 12 2 2

2 2

a b ab

a b+ =

+

= ( )

( )

( )

( )2

1

3 2 11

9 2 112

2

2

2

ab

a b ab+ -=

-

- -= + =

_y§fë‹ bgU¡f‰gy‹, 11 1 122

2

a b ab= =c c cm m m


& (11) 1x x2- + = 0


8. 3 6 1x x2- + = 0 v‹w rk‹gh£o‹ _y§fŸ a , b våš, Ñœ¡fhQ« _y§fis¡

bfh©l rk‹ghLfismik¡f.

(i) ,1 1a b

(ii) ,2 2a b b a (iii) 2 , 2a b b a+ +

Ô®Î: 3 0x x6 12- + = . ax bx c 02


3, ,a b c6 1= =- =

vdnt, a b+ = ( )ab

36

2- =--

= k‰W« ab = ac

31=

(i) njitahd rk‹gh£o‹ _y§fŸ 1a

k‰W« 1b

.

_y§fë‹ TLjš, 1 1

312 6

a b aba b

+ =+

= =


_y§fë‹ bgU¡f‰gy‹, 1 1 1

311 3

a b ab= = =` cj m

njitahd rk‹ghL, x2 - (_y§fë‹ TLjš) x + _y§fë‹ bgU¡f‰gy‹ = 0 x x6 32

& - + = 0


(ii) njitahd rk‹gh£o‹ _y§fŸ 2a b k‰W« 2b a .

_y§fë‹ TLjš, ( ) ( )31 2

322 2a b b a ab a b+ = + = =

_y§fë‹ bgU¡f‰gy‹, ( )( ) ( )31

2712 2 3 3 3 3

a b b a a b ab= = = =` j

njitahd rk‹ghL, x2 - (_y§fë‹ TLjš) x + _y§fë‹ bgU¡f‰gy‹ = 0

& x x32

2712

- +` j = 0


(iii) njitahd rk‹gh£o‹ _y§fŸ 2a b+ k‰W« 2b a+ .

_y§fë‹ TLjš, ( ) ( ) ( ) ( )2 2 3 3 2 6a b b a a b+ + + = + = =

_y§fë‹ bgU¡f‰gy‹, (2 )(2 ) 4 2 22 2a b b a ab a b ab+ + = + + +

= [( ) ] (2) 2322 2 5 2 5

3122a b ab ab+ - + = - +` `j j8 B

= 3

12 2235

325- + =8 B .


& (6)x x3252

- + = 0


9. 4 3 1x x2- - = 0 v‹w rk‹gh£o‹ _y§fë‹ jiyÑêfis _y§fshf¡ bfh©l

rk‹ghL x‹¿id mik¡f.

Ô®Î: x x4 3 1 02- - = . ax bx c 02


, ,a b c4 3 1= =- =- .

vdnt, a b+ = ( )ab

43

43- =-

-= k‰W« ab =

ac

41= -

njitahd rk‹gh£o‹ _y§fŸ 1a

k‰W« 1b

.

_y§fë‹ TLjš, 31 1

4143

a b aba b

+ =+

=-

=-


_y§fë‹ bgU¡f‰gy‹, 1 1 1

411 4

a b ab= =

-=-` cj m


& ( 3) ( 4)x x2- - + - = 0

vdnt, njitahd rk‹ghL 3 4x x 02+ - = .

F¿¥ò: x ¡F gÂyhf x1 I x x4 3 1 02

- - = v‹w rk‹gh£oš ÃuÂæ£lhš

»il¡F« òÂa rk‹ghL 3 4x x 02+ - = .

10. 3 81x kx2+ - = 0 v‹w rk‹gh£o‹ xU _y« k‰bwhU _y¤Â‹ t®¡fbkåš, k-‹

kÂ¥ig¡ fh©f.

Ô®Î: a k‰W« 2b a= v‹gd x kx3 81 02+ - = v‹w rk‹gh£o‹ _y§fŸ.

ax bx c 02+ + = v‹w rk‹gh£o‹go , ,a b k c3 81= = =- .

_y§fë‹ TLjš a b+ = ab k

32

& a a- + = - g (1)

_y§fë‹ bgU¡f‰gy‹ ab = ( ) 27 ( 3)ac

3812 3 3

& &a a a= - =- = -

vdnt, a = – 3

3a =- våš (1) ( ) ( ) 18.k k3

3 3 2& &- = - + - =-

11. 2 64 0x ax2- + = v‹w rk‹gh£o‹ xU _y« k‰bwhU _y¤Â‹ ÏUkl§F våš,

a-‹ kÂ¥ig¡ fh©f.

Ô®Î: a k‰W« 2b a= v‹gd x ax2 64 02- + = v‹w rk‹gh£o‹ _y§fŸ.

0Ax Bx C2+ + = v‹w rk‹gh£o‹go , ,A B a C2 64= =- = .

_y§fë‹ TLjš, a b+ = 2( )

AB a a

2 2& a a- + =-

-= .a

6& a = g (1)

_y§fë‹ bgU¡f‰gy‹, ab = (2 ) 2 32.AC

264 2

& &a a a= =

& a6

2` j = 16 ((1)‹ go)

vdnt, 24.a !=

12. 5 1x px2- + = 0 v‹w rk‹gh£od _y§fŸ a k‰W« b v‹f. nkY« a b- = 1

våš, p-‹ kÂ¥ig¡ fh©f.

Ô®Î: bfhL¡f¥g£l rk‹ghL x px5 1 02- + =

ax bx c 02+ + = v‹w rk‹gh£o‹go , ,a b p c5 1= =- = .

vdnt, a b+ = ab p

5- =` j g (1)

ab = 51 g (2)

bfhL¡f¥g£LŸsgo, a b- = 1 g (3)


vdnt, ( ) 42a b ab+ - = ( )2a b-

& p25 5

42

- = 1 ( (1), (2) & (3)I ga‹gL¤Â )

& p 202- = 25 & p2 = 45

vdnt, p = 3 5! .

gæ‰Á 3.19


1. 6x – 2y = 3, kx – y = 2 v‹w bjhF¥Ã‰F xnubahU Ô®Î c©blåš,

(A) k = 3 (B) k 3! (C) k = 4 (D) k 4!

Ô®Î: ,x y kx y6 2 3 2- = - = . våš, 6, 2, , 1a b a k b1 1 2 2= =- = =- .

xnuxU Ô®Î¡F, a

a

2

1 ! b

b

k6

12

2

1 & !-- k& ! 3 (éil. (B) )

2. ÏU kh¿fëš cŸs neçaš rk‹ghLfë‹ bjhF¥ò xU§fikahjJ våš, mt‰¿‹

tiugl§fŸ

(A) x‹¿‹ ÛJ x‹W bghUªJ« (B) xU òŸëæš bt£o¡ bfhŸS«

(C) vªj¥ òŸëæY« bt£o¡ bfhŸshJ (D) x-m¢ir bt£L«

Ô®Î: rk‹ghLfë‹ bjhF¥ò xU§fikahjJ våš Ô®Î VJäšiy. (éil. (C) )

3. x – 4y = 8 , 3x – 12y =24 v‹D« rk‹ghLfë‹ bjhF¥Ã‰F

(A) Koéè v©â¡ifæš Ô®ÎfŸ cŸsd

(B) Ô®Î Ïšiy (C) xnubahU Ô®Î k£L« c©L (D) xU Ô®Î ÏU¡fyh« mšyJ ÏšyhkY« ÏU¡fyh«.

Ô®Î: , , , , ,a b c a b c1 4 8 3 12 241 1 1 2 2 2= =- =- = =- =- .

a

a

b

b

2

1

2

1= =c

c

31

2

1 = . (våš, Koéè v©â¡ifæš Ô®ÎfŸ cŸsd) (éil. (A) )

4. p x^ h = (k +4)x2+13x+3k v‹D« gšYW¥ò¡nfhitæ‹ xU ó¢Áa« k‰bwh‹¿‹

jiyÑêahdhš, k-‹ kÂ¥ò

(A) 2 (B) 3 (C) 4 (D) 5

Ô®Î: ( )k x x k4 13 32+ + + . våš, 4, 3, 3a k b c k1= + = = .

a k‰W« 1a

v‹gd gšYW¥ò¡ nfhitæ‹ ó¢Áa§fŸ våš,

( )( ) .k

k k14

3 2&aa

=+

= (éil. (A) )


5. 2 ( 3) 5f x x p x2

= + + +^ h v‹D« gšYW¥ò¡nfhitæ‹ ÏU ó¢Áa§fë‹ TLjš

ó¢Áabkåš p-‹ kÂ¥ò. (A) 3 (B) 4 (C) –3 (D) –4

Ô®Î: ( 3) 5x p x2 2+ + + . våš , ,a b p c2 3 5= = + = .

ó¢Áa§fë‹ TLjš = ( )0 3 0.

ab p

p2

3& &-

- += + = (éil. (C) )

6. x x2 72- + v‹gij x+4 Mš tF¡F« nghJ »il¡F« ÛÂ

(A) 28 (B) 29 (C) 30 (D) 31Ô®Î: ÛÂ nj‰w¤Â‹go

( )f 4- = ( ) ( )4 2 4 7 16 8 7 312- - - + = + + = . (éil. (D) )

7. 5 7 4x x x3 2- + - v‹gij x–1Mš tF¡F« nghJ »il¡F« <Î

(A) 4 3x x2+ + (B) 4 3x x

2- + (C) 4 3x x

2- - (D) 4 3x x

2+ -

Ô®Î: bjhFKiw gF¤jè‹go

1 1 – 5 7 – 40 1 – 4 3

<Î " 1 – 4 3 – 1 " ÛÂ

(éil. (B) )

8. x 13+^ h k‰W« 1x

4- M»adt‰¿‹ Û. bgh.t

(A) x 13- (B) 1x

3+ (C) x +1 (D) x 1-

Ô®Î: ( )f x = ( ) ( )( )x x x x1 1 13 2+ = + - +

( )g x = ( )( )( )x x x x1 1 1 14 2- = + - +

Û.bgh.t. = x +1 (éil. (C) )

9. 2x xy y2 2- + k‰W« x y

4 4- M»adt‰¿‹ Û. bgh.t

(A) 1 (B) x+y (C) x–y (D) x y2 2-

Ô®Î: ( )f x = ( )x xy y x y22 2 2- + = -

( )g x = ( )( )( )x y x y x y x y4 4 2 2- = - + + (éil. (C) )

10. x a3 3- k‰W« (x – a)2M»adt‰¿‹ Û. bgh.k

(A) ( )x a x a3 3- +^ h (B) ( )x a x a

3 3 2- -^ h

(C) x a x ax a2 2 2- + +^ ^h h (D) x a x ax a2 2 2

+ + +^ ^h h

Ô®Î: ( )f x = ( )( )x a x a x ax a3 3 2 2- = - + +

( )g x = ( ) ( )x a x a2 2- = -

vdnt, Û.bgh.k. = x a x ax a( ) ( )2 22- + + (éil. (C) )

11. k Ne vD«nghJ , ,a a ak k k3 5+ +

M»at‰¿‹ Û. bgh.k

(A) ak 9+

(B) ak

(C) ak 6+

(D) ak 5+


Ô®Î: ( )f x = ; ( ) .a g x a a ak k k3 3#= =+

( )h x = a a ak k5 5#=+

vdnt, Û.bgh.k. a 5k+ . (éil. (D) )

12. 6x x

x x5 62

2

- -

+ + v‹D« é»jKW nfhitæ‹ äf¢ RU¡»a tot«

(A) xx

33

+- (B)

xx

33

-+ (C)

xx

32

-+ (D)

xx

23

+-

Ô®Î: x x

x x

6

5 62

2

- -

+ + = ( )( )( )( )x xx x

3 23 2

- ++ + (éil. (B) )

13. a ba b-+ k‰W«

a b

a b3 3

3 3

+

- M»ad ÏU é»jKW nfhitfŸ våš, mt‰¿‹ bgU¡f‰gy‹

(A) a ab b

a ab b2 2

2 2

- +

+ + (B) a ab b

a ab b2 2

2 2

+ +

- + (C) a ab b

a ab b2 2

2 2

+ +

- - (D) a ab b

a ab b2 2

2 2

- -

+ +

Ô®Î: a ba b

a b

a b3 3

3 3

#-+

+

- = ( )( )

( )( )

( )( )a ba b

a b a ab b

a b a ab b2 2

2 2

#-+

+ - +

- + + (éil. (A) )

14. x

x325

2

+- v‹gij

9x

x 52-

+ Mš tF¡F« nghJ »il¡F« <Î

(A) (x –5)(x–3) (B) (x –5)(x+3) (C) (x +5)(x–3) (D) (x +5)(x+3)

Ô®Î: x

x

x

x325

9

52

2'

+-

-

+ = ( )( ) ( )( )x

x xx

x x3

5 55

3 3#

++ -

++ - (éil. (A) )

15. a ba3

- cl‹

b ab3

- I¡ T£l, »il¡F« òÂa nfhit

(A) a ab b2 2+ + (B) a ab b

2 2- + (C) a b

3 3+ (D) a b

3 3-

Ô®Î: a ba

b ab3 3

-+

-= ( )( )

a ba

a bb

a ba b a ab b3 3 2 2

--

-=

-- + + (éil. (A) )

16. 49 ( 2 )x xy y2 2 2- + -‹ t®¡f_y«

(A) 7 x y- (B) 7 x y x y+ -^ ^h h (C) 7( )x y2

+ (D) 7( )x y2

-

Ô®Î: ( )x xy y49 22 2 2- + = ( ) ( )x y x y7 72 4 2

- = - ( éil. (D))

17. 2 2 2x y z xy yz zx2 2 2+ + - + - -‹ t®¡f_y«

(A) x y z+ - (B) x y z- + (C) x y z+ + (D) x y z- -


Ô®Î: x y z xy yz zx2 2 22 2 2+ + - + -

= ( ) ( ) ( )( ) ( )( ) ( )( )x y z x y x2 5 2 2 2 22 2 2+ - + - + - + - - + -

= ( ) | |x y z x y z2- - = - - ( éil. (D))

18. 121 ( )x y z l m4 8 6 2

- -‹ t®¡f_y«

(A) 11x y z l m2 4 4

- (B) 11 ( )x y z l m34 4

-

(C) 11x y z l m2 4 6

- (D) 11 ( )x y z l m32 4

-

Ô®Î: ( )x y z l m121 4 8 6 2- = ( ) | |x y z l m x y z l m11 112 4 8 6 2 2 4 3

- = - ( éil. (D))

19. ax bx c 02+ + = v‹w rk‹gh£o‹ _y§fŸrk« våš, c-‹ kÂ¥ò

(A) ab2

2

(B) ab4

2

(C) ab2

2

- (D) ab4

2

-

Ô®Î: rk‹gh£o‹ _y§fŸ rkbkåš b ac42& - = 0

b2 = 4ac cab4

2& = ( éil. (B))

20. 5 16 0x kx2+ + = v‹w rk‹gh£o‰F bkŒba© _y§fŸ Ïšiybaåš,

(A) k582 (B) k

582-

(C) k58

581 1- (D) k0

581 1

Ô®Î: x kx5 16 02+ + = . våš, , ,a b k c1 5 16= = = .

rk‹gh£o‹ _y§fŸ bkŒa‰wit b ac4 0<2& -

( ) ( )( )k5 4 1 162- < 0 25 64

58k k< <2 2 2

& & ` j k58

58< <& - . ( éil. (C))

21. 3 -I xU _ykhf¡ bfh©l ÏUgo¢ rk‹ghL

(A) 6 0x x 52- - = (B) 0x x6 5

2+ - =

(C) 0x x5 62- - = (D) 0x x5 6

2- + =

Ô®Î: (A) k‰W« (B) M»at‰iw fhuâgL¤j KoahJ.

(C) 5 6x x2- - = ( )( )x x6 1- + (D) 5 6x x2

- + = ( )( )x x3 2- - . ( éil. (D))

22. 0x bx c2- + = k‰W« x bx a 0

2+ - = M»a rk‹ghLfë‹ bghJthd _y«

(A) b

c a2+ (B)

bc a2- (C)

ac b2+ (D)

ca b2+

Ô®Î: b c2a a- + = b a2a a+ -

vdnt, a = b

c a2+ ( éil. (A))


23. ,a 0=Y vd mikªj rk‹ghL ax bx c 02

+ + = -‹ _y§fŸ k‰W«a b våš,

Ã‹tUtdt‰WŸ vJ bkŒašy?

(A) a

b ac22

2

22

a b+ = - (B) acab =

(C) aba b+ = (D)

cb1 1

a b+ =-

Ô®Î: a b+ = ;ab

acab- = ( éil. (C))

24. ax bx c 02+ + = v‹w ÏUgo¢ rk‹gh£o‹ _y§fŸ a k‰W« b våš,

1a

k‰W« 1b

M»adt‰iw _y§fshf¡ bfh©l ÏUgo¢rk‹ghL

(A) ax bx c 02+ + = (B) 0bx ax c2

+ + =

(C) 0cx bx a2+ + = (D) 0cx ax b2

+ + =

Ô®Î: x-¡F gÂyhf x1 I ax bx c 02

+ + = v‹w rk‹gh£oš ÃuÂæl »il¡F«

rk‹ghL cx bx a 02+ + = ( éil. (C))

25. b = a + c v‹f. 0ax bx c2+ + = v‹w rk‹gh£o‹ _y§fŸ rk« våš,

(A) a c= (B) a c=-

(C) a c2= (D) a c2=-

Ô®Î: 4 4 0 .b ac a c ac a c a c2 2 2& & &= + = - = =^ ^h h ( éil. (A))

Ô®Î - mâfŸ 121

gæ‰Á 4.1

1. xU bghGJ ngh¡F Ú® éisah£L¥ ó§fhé‹ f£lz é»j« Ã‹tUkhW:

thu eh£fëš f£lz« (`) éLKiw eh£fëš f£lz« (`)taJ tªnjh® 400 500ÁWt® 200 250_¤j¡ Fokf‹ 300 400

taJ tªnjh®, ÁWt® k‰W« _¤j¡FokfD¡fhd f£lz é»j§fS¡fhd

mâfis vGJf. nkY« mt‰¿‹ tçirfis vGJf.

Ô®Î: taJ tªnjh®, ÁWt® k‰W« _¤j FokfD¡fhd f£lz é»j§fis mâ

toéš ÏUtêfëš Ã‹tUkhW vGjyh«

(i) A = 400

200

300

500

250

400

f p. mâ A-‹ gçkhd« (tçir) 3 # 2 MF«

(ii) B = 400500

200

250

300

400c m. mâ B-‹ gçkhd« (tçir) 2 # 3 MF«

2. xU efu¤Âš 6 nkšãiy¥ gŸëfŸ, 8 ca®ãiy¥ gŸëfŸ k‰W« 13 bjhl¡f¥

gŸëfŸ cŸsd. Ïªj étu§fis 3 1# k‰W« 1 3# tçirfis¡ bfh©l

mâfshf F¿¡fÎ«.

Ô®Î: bfhL¡f¥g£l étu§fis 3 # 1 tçirÍŸs mâahf A = 6

8

13

f p vd vGjyh«.

nkY«, bfhL¡f¥g£l étu§fis 1 # 3 tçirÍŸs mâahf B = 6 8 13^ h vd

vGjyh«

3. Ã‹tU« mâfë‹ tçirfis¡ fh©f.

(i) 1

2

1

3

5

4-

-e o (ii)

7

8

9

f p (iii) 3

6

2

2

1

4

6

1

5

-

-f p (iv) 3 4 5^ h (v)

1

2

9

6

2

3

7

4

-

J

L

KKKKK

N

P

OOOOO

Ô®Î: (i) 1

2

1

3

5

4-

-e o v‹w mâæš 2 ãiufS« 3 ãušfS« cŸsd. vdnt,

Ï›tâæ‹ tçir 2 # 3 MF«.

(ii) 7

8

9

f p v‹w mâæš 3 ãiufS« 1 ãuY« cŸsJ. vdnt, Ï›tâæ‹ tçir

3 1# MF«.

mâfŸ 4


(iii) 3

6

2

2

1

4

6

1

5

-

-f p v‹w mâæš 3 ãiufS« 3 ãušfS« cŸsd. vdnt,

Ï›tâæ‹ tçir 3 # 3 MF«.

(iv) 3 4 5^ h v‹w mâæš 1 ãiuÍ« 3 ãušfS« cŸsjhš, Ï›tâæ‹ tçir 1 # 3 MF«.

(v)

1

2

9

6

2

3

7

4

-

J

L

KKKKK

N

P

OOOOO v‹w mâæš 4 ãiufS« 2 ãušfS« cŸsjhš, Ï›tâæ‹

tçir 4 # 2 MF«.

4. 8 cW¥òfŸ bfh©l xU mâ¡F v›tif tçirfŸ ÏU¡f ÏaY«?

Ô®Î: 8 cW¥òfŸ bfh©l xU mâæ‹ tçir 1 # 8, 2 # 4, 4 # 2 k‰W« 8 # 1 vd ÏU¡f ÏaY«.

5. 30 cW¥òfŸ bfh©l mâ¡F v›tif tçirfŸ ÏU¡f ÏaY«?

Ô®Î: 30 cW¥òfŸ bfh©l xU mâæ‹ tçir

1 # 30, 2 # 15, 3 # 10, 5 # 6, 6 # 5, 10 # 3, 15 # 2 k‰W« 30 # 1 vd ÏU¡f

ÏaY«.

6. Ã‹tUtdt‰iw¡ bfh©L 2 2# tçirÍila mâ A aij

= 6 @-ia¡ fh©f (i) a ij

ij= (ii) 2a i j

ij= - (iii) a

i ji j

ij=

+-

Ô®Î: bghJthf 2 # 2 tçirÍŸs mâ A = a

a

a

a11

21

12

22

e o v‹w toéš ÏU¡F«.

(i) a ijij

= , Ï§F i = 1, 2 k‰W« j = 1, 2

(1)(1) 1, (1)(2) 2, (2)(1) 2, (2)(2) 4a a a a11 12 21 22= = = = = = = =

Mfnt, njitahd mâ A = 1

2

2

4c m

(ii) 2 –a i jij= , Ï§F i = 1, 2 k‰W« j = 1, 2

2(1) 1 1, 2(1) 2 0, 2(2) 1 3, 2(2) 2 2a a a a11 12 21 22= - = = - = = - = = - =


0

2c m

(iii) ai ji j

ij=

+- , Ï§F i = 1, 2 k‰W« j = 1, 2

0, , , 0a a a a1 11 1

20

1 21 2

31

2 12 1

31

2 22 2

11 12 21 22=

+- = = =

+- =- =

+- = =

+- =

Ô®Î - mâfŸ 123


31

31

0

-J

L

KKK

N

P

OOO MF«.

7. Ã‹tUtdt‰iw¡ bfh©L 3 2# tçiria¡ bfh©l mâ A aij

= 6 @-æid¡ fh©f.

(i) aji

ij= (ii) ( )

ai j

22

ij

2

=- (iii) a i j

2

2 3ij=

-

Ô®Î: bghJthf 3 × 2 tçirÍŸs mâ A= a

a

a

a

a

a

11

21

31

12

22

32

J

L

KKK

N

P

OOO v‹w toéš ÏU¡F«

(i) aji

i j= , Ï§F i = 1, 2, 3 k‰W« j = 1, 2

1,a a11

21

11 12= = = , 2, 1a a

12

22

21 22= = = = , 3,a a

13

23

31 32= = =

Mfnt, njitahd mâ A =

1

2

3

21

1

23

J

L

KKKKKK

N

P

OOOOOO

MF«.

(ii) a i j

2

2ij

2

=-^ h , Ï§F i = 1, 2, 3 k‰W« j = 1, 2

,a2

1 2

21

11

2

=-

=^ h a

2

1 4

29

12

2

=-

=^ h , 0,a

2

2 221

2

=-

=^ h

2a2

2 4

24

22

2

=-

= =^ h , ,a

2

3 2

21

31

2

=-

=^ h a

2

3 4

21

32

2

=-

=^ h

Mfnt, njitahd mâ A = 0 2

21

21

29

21

J

L

KKKKKK

N

P

OOOOOO

MF«.

(iii) 2 3a

i j

2ij=

- , Ï§F i = 1, 2, 3 k‰W« j = 1, 2

| | | | ,a2

2 321

21

11= - = - = | | | | 2a

22 6

24

12- = - = , | | ,a

24 3

21

21= - =

| | 1a2

4 622

22= - = = , | | ,a

26 3

23

31= - = | | 0a

26 6

32= - =

Mfnt, njitahd mâ A =

2

1

0

21

21

23

J

L

KKKKKK

N

P

OOOOOO

MF«.

8. A

1

5

6

1

4

0

3

7

9

2

4

8

=

-

-f p våš, (i) mâæ‹ tçiria¡ fh©f. (ii) a24

k‰W« a32

M»a

cW¥òfis vGJf (iii) cW¥ò 7 mikªJŸs ãiu k‰W« ãuiy¡ fh©f.


Ô®Î: A = 1

5

6

1

4

0

3

7

9

2

4

8

-

-f p

(i) mâ A-š 3 ãiufS« 4 ãušfS« cŸsd. vdnt, mâ A-‹ tçir 3 × 4 (ii) a

24 v‹w cW¥ò Ïu©lh« ãiuæš eh‹fh« ãuèš ÏU¡F«. vdnt, a

24= 4

MF«.

a32

v‹w cW¥ò _‹wh« ãiuæš Ïu©lh« ãuèš ÏU¡F«. vdnt, a32

= 0 MF«.

(iii) cW¥ò 7 v‹gJ Ïu©lhtJ ãiuæš _‹whtJ ãuèš cŸsJ. mjhtJ

7a23

= MF«.

9. A

2

4

5

3

1

0

= f p våš, A-æ‹ ãiu ãuš kh‰W mâia¡ fh©f.

Ô®Î: mâ A-æ‹ ãiu ãuš kh‰W mâ AT v‹gJ A-æ‹ ãiufis ãušfshfÎ«,

ãušfis ãiufshfÎ« kh‰¿ vGj »il¥gjhF«.

Mfnt, AT = 2

3

4

1

5

0c m

10. A

1

2

3

2

4

5

3

5

6

=

-

-f p våš, ( )A AT T= v‹gjid¢ rçgh®¡f.

Ô®Î: A = 1

2

3

2

4

5

3

5

6-

-f p. g (1)

mâ A-‹ ãiufis ãušfshfÎ«, ãušfis ãiufshfÎ« kh‰¿ vGj, eh«

bgWtJ

AT = 1

2

3

2

4

5

3

5

6-

-f p

mâ AT -‹ kh‰W mâ

AT T^ h = 1

2

3

2

4

5

3

5

6-

-f p g (2)

(1) k‰W« (2) M»at‰¿èUªJ, AT T^ h = A

gæ‰Á 4.2 1. Ã‹tU« mâ¢rk‹gh£oèUªJ x, y k‰W« z-fë‹ kÂ¥òfis¡ fh©f.

x y

z

5 2

0

4

4 6

12

0

8

2

+ -

+=

-e co m

Ô®Î: bfhL¡f¥g£l mâfŸ rk mâfŸ v‹gjhš, x¤j cW¥òfŸ rkkhF«. x¤j

cW¥òfis x¥Ãl, eh« bgWtJ

Ô®Î - mâfŸ 125

5 2 12x + = & 5 10 2x x&= =

4 8y - =- & 4y =-

4 6 2z + = & 4 4 1z z&=- =-

Mfnt, 2, 4, 1x y z= =- =-

2. x y

x y

2

3

5

13

+

-=e co m våš, x k‰W« y-fë‹ Ô®Îfis¡ fh©f.

Ô®Î: bfhL¡f¥g£l mâfŸ rk mâfŸ v‹gjhš, x¤j cW¥òfŸ rkkhF«. x¤j

cW¥òfis x¥Ãl

2 5 2 5 0x y x y&+ = + - = k‰W« 3 13 3 13 0x y x y&- = - - =

FW¡F¥ bgU¡fš Kiwæš rk‹ghLfis¤ Ô®¥ngh«.

x y13 15 5 26 6 1

1- -

=- +

=- -

x y28 21 7

1&-

= =-

Mfnt, 4, 3x y728

721=

-- = =

-=-

3. A2

9

3

5

1

7

5

1=

--

-e eo ovåš, A-‹ T£lš ne®khW mâia¡ fh©f.

Ô®Î: A = 2

9

3

5

1

7

5

1--

-e eo o 2

9

3

5

1

7

5

1=

-+

-

-

-e eo o

= 2 1

9 7

3 5

5 1

1

16

2

6

-

- -

-

+=

-

-e eo o

–A v‹gJ mâ A-‹ T£lY¡fhd ne®khW MF«. Mfnt, T£lš ne®khW

A- = 1

16

2

6

1

16

2

6-

-

-=

-

-e eo o

4. A3

5

2

1= c m k‰W« B 8

4

1

3=

-c mvåš, 2C A B= + v‹w mâia¡ fh©f.

Ô®Î: A3

5

2

1= c m k‰W« B

8

4

1

3=

-c m vd bfhL¡f¥g£LŸsJ.

vdnt, C = 2A + B

= 23

5

2

1

8

4

1

3

6

10

4

2

8

4

1

3+

-= +

-c c c cm m m m

= 6 8

10 4

4 1

2 3

14

14

3

5

+

+

-

+=e co m

5. k‰W«A B4

5

2

9

8

1

2

3=

-

-=

- -e eo o våš, 6 3A B- v‹w mâia¡ fh©f.

Ô®Î: A4

5

2

9=

-

-e o k‰W«B

8

1

2

3=

- -e o vd bfhL¡f¥g£LŸsJ.


A B6 3- = 6 34

5

2

9

8

1

2

3

-

--

- -e eo o

= 24

30

12

54

24

3

6

9

24 24

30 3

12 6

54 9

0

33

18

45

-

-+

- -=

-

+

- -

- +=

-

-e c e eo m o o

6. a b2

3

1

1

10

5+

-=c c cm m m våš, a k‰W« b M»adt‰¿‹ kÂ¥òfis¡ fh©f.

Ô®Î: a b2

3

1

1

10

5+

-=c c cm m m vd bfhL¡f¥g£LŸsJ.

a

a

b

b

a b

a b

2

3

10

5

2

3

10

5& &+

-=

-

+=c c c e cm m m o m

x¤j cW¥òfis x¥Ãl, eh« bgWtJ

2a – b = 10 g (1)

3a + b = 5 g (2)

(1) k‰W« (2) M»at‰iw¡ T£l¡ »il¥gJ 5a = 15 & a = 3.

a = 3 vd (2)-š ÃuÂæl, eh« bgWtJ 9 + b = 5 & b = – 4

Mfnt, 3, 4a b= =-

7. 2 3X Y2

4

3

0+ = c m k‰W« 3 2X Y

2

1

2

5+ =

-

-e o våš, X k‰W« Y M»a

mâfis¡ fh©f

Ô®Î: X Y2 3+ = 2

4

3

0c m g (1)

X Y3 2+ = 2

1

2

5-

-e o g (2)

Kjèš Y-I Ú¡Fnth«,

X Y1 2 4 6&# +^ h = 22

4

3

0

4

8

6

0=c cm m g (3)

3 9 6X Y2 &# +^ h = 32

1

2

5

6

3

6

15-

-=

-

-e eo o g (4)

(4)-èUªJ (3)-I fê¡f »il¥gJ

5X = 6

3

6

15

4

8

6

0

6

3

6

15

4

8

6

0-

-- =

-

-+

-

-

-e c e eo m o o

= X2

11

12

15

52

511

512

3&

-

-=

-

-J

L

KKKK

f

N

P

OOOO

p

X-I (1)-š ÃuÂæl, eh« bgWtJ

Ô®Î - mâfŸ 127

Y3 = 2 2X2

4

3

0

2

4

3

0 3

2

4

3

0 6

52

511

512

54

522

524

- = - = +--

- -J

L

KKK

J

L

KKK

f f f

N

P

OOO

N

P

OOO

p p p

= 2

4

3

0 6

54

522

524-

+

+

-

J

L

KKK

N

P

OOO =

6

56

542

539

-

J

L

KKK

N

P

OOO

Mfnt, Y = –3

1 56

542

539

6

J

L

KKKK

N

P

OOOO =

–

52

514

513

2

J

L

KKKK

N

P

OOOO

8. Ô®¡f : x

y

x

y3

2 9

4

2

2+

-=

-f e ep o o.

Ô®Î: 3–

x

y

x

y

2 9

4

2

2 + =-

e e co o m vd bfhL¡f¥g£LŸsJ

& –x

y

x

y

6

3

9

4

2

2 +-

=e e co o m x x

y y

6

3

9

4

2

2&+

-=

-e co m

x¤j cW¥òfis x¥Ãl, eh« bgWtJ

6 9x x2+ =- 3 4y y2

- =

& 6 9 0x x2+ + = 3 4 0y y2

& - - =

& 0x 3 2+ =^ h 0y y1 4& + - =^ ^h h

& 3, 3x =- - ,–y 1 4& =

9. , k‰W«A B O3

5

2

1

1

2

2

3

0

0

0

0= =

-=c c cm m m våš, Ã‹tUtdt‰iw

rçgh®¡f : (i) A B B A+ = + (ii) ( ) ( )A A O A A+ - = = - + .

Ô®Î: , k‰W«A B O3

5

2

1

1

2

2

3

0

0

0

0= =

-=c c cm m m vd bfhL¡f¥g£LŸsJ.

(i) A + B = 3

5

2

1

1

2

2

3+

-c cm m = 3 1

5 2

2 2

1 3

4

7

0

4

+

+

-

+=e co m g (1)

B + A = 1

2

2

3

3

5

2

1

-+c cm m = 1 3

2 5

2 2

3 1

4

7

0

4

+

+

- +

+=e co m g (2)

(1) k‰W« (2)-èUªJ, eh« bgWtJ A B B A+ = +

(ii) A + (–A) = 3

5

2

1

3

5

2

1

3 3

5 5

2 2

1 1+

-

-

-

-=

-

-

-

-c e em o o = O

0

0

0

0=c m g (3)

(–A) + A = 3

5

2

1

3

5

2

1

3 3

5 5

2 2

1 1

-

-

-

-+ =

- +

- +

- +

- +e c eo m o = O

0

0

0

0=c m g (4)

(3) k‰W« (4)-èUªJ, eh« bgWtJ A + (–A) = (–A) + A = O


10. ,A B

4

1

0

1

2

3

2

3

2

2

6

2

0

2

4

4

8

6

= - =f fp p k‰W« C

1

5

1

2

0

1

3

2

1

=

-

-

f p våš,

( ) ( )A B C A B C+ + = + + v‹gjid¢ rçgh®¡f..Ô®Î:

B + C = 2

6

2

0

2

4

4

8

6

1

5

1

2

0

1

3

2

1

+

-

-

f fp p =2 1

6 5

2 1

0 2

2 0

4 1

4 3

8 2

6 1

3

11

3

2

2

3

1

10

7

+

+

+

+

+

-

-

+

+

=f fp p

Mfnt, ( )A B C+ + = 4

1

0

1

2

3

2

3

2

3

11

3

2

2

3

1

10

7

- +f fp p

= 4 3

1 11

0 3

1 2

2 2

3 3

2 1

3 10

2 7

+

+

+

+

- +

+

+

+

+

f p = 7

12

3

3

0

6

3

13

9

f p g (1)

nkY«, A + B = 4

1

0

1

2

3

2

3

2

2

6

2

0

2

4

4

8

6

- +f fp p= 4 2

1 6

0 2

1 0

2 2

3 4

2 4

3 8

2 6

6

7

2

1

0

7

6

11

8

+

+

+

+

- +

+

+

+

+

=f fp p

vdnt, (A + B) + C = 6

7

2

1

0

7

6

11

8

1

5

1

2

0

1

3

2

1

+

-

-

f fp p

= 6 1

7 5

2 1

1 2

0 0

7 1

6 3

11 2

8 1

7

12

3

3

0

6

3

13

9

+

+

+

+

+

-

-

+

+

=f fp p g (2)

(1) k‰W« (2) M»at‰¿èUªJ, eh« bgWtJ ( ) ( )A B C A B C+ + = + + .

11. xU ä‹dQ FGk« jdJ é‰gid¡fhf th§F« ä‹dQ¥ bghU£fis

f©fhâ¡F« bghU£L jdJ _‹W é‰gid¡ Tl§fëš é‰gid brŒa¥gL«

bghGJngh¡F¢ rhjd§fŸ g‰¿a étu§fis¥ gÂÎ brŒjJ. Ïu©L thu§fëš

eilbg‰w é‰gid étu§fŸ Ã‹tU« m£ltidæš F¿¥Ãl¥g£LŸsd.

T.V. DVD Videogames CD Players

thu« Ifil I 30 15 12 10fil II 40 20 15 15fil III 25 18 10 12

thu« IIfil I 25 12 8 6fil II 32 10 10 12fil III 22 15 8 10

mâfë‹ T£liy¥ ga‹gL¤Â Ïu©L thu§fëš é‰gid brŒa¥g£l

rhjd§fë‹ TLjiy¡ fh©f.

Ô®Î - mâfŸ 129

Ô®Î: Kjš thu¤Âš, _‹W é‰gid¡ Tl§fëš é‰gid brŒa¥gL« rhjd§fŸ

g‰¿a étu§fë‹ mâ mik¥ò

TV DVD Video CD

A = 30

40

25

15

20

18

12

15

10

10

15

12

f p filfilfil

I

II

III

Ïnjngh‹W, Ïu©lh« thu¤Âš _‹W é‰gid¡ Tl§fëš é‰gid brŒa¥gL«

rhjd§fŸ g‰¿a étu§fë‹ mâ mik¥ò

TV DVD Video CD

B = 25

32

22

12

10

15

8

10

8

6

12

10

f p filfilfil

I

II

III

Mfnt, Kjš k‰W« Ïu©lh« thu§fëš _‹W é‰gid¡ Tl§fëY« é‰gid

brŒa¥g£l rhjd§fë‹ TLjè‹ mâ mik¥ò

A B+ = 30

40

25

15

20

18

12

15

10

10

15

12

25

32

22

12

10

15

8

10

8

6

12

10

+f fp p

= 30 25

40 32

25 22

15 12

20 10

18 15

12 8

15 10

10 8

10 6

15 12

12 10

+

+

+

+

+

+

+

+

+

+

+

+

f p

TV DVD Video CD

= 55

72

47

27

30

33

20

25

18

16

27

22

f p filfilfil

I

II

III

12. xU Ú¢rš Fs¤Â‰F xU ehS¡fhd EiHÎ¡ f£lz« Ã‹tUkhW

xU ehS¡fhd EiHÎ¡ f£lz« ( ` ) cW¥Ãd® ÁWt® bgçat®

Ã‰gfš 2 kâ¡F K‹ò 20 30Ã‰gfš 2 kâ¡F Ã‹ò 30 40

cW¥Ãd® mšyhjt®


cW¥Ãd® mšyhjt®fS¡F V‰gL« TLjš f£lz¤ij¡ F¿¡F« mâia¡ fh©f.

Ô®Î: Ú¢rš Fs¤Âš xU ehS¡F cW¥Ãd®fS¡fhd EiHÎ¡ f£lz¤ij

F¿¡F« mâ A v‹f.

ÁWt® bgçat®

A = 2030

30

40c m ™

™

2

2

Ã‰gf kâ¡F K‹ò

Ã‰gf kâ¡F Ã‹ò

Ú¢rš Fs¤Âš xU ehS¡F cW¥Ãd® mšyhjt®fS¡fhd EiHÎ¡ f£lz¤ij

F¿¡F« mâ B v‹f.


ÁWt® bgçat®

B = 25

40

35

50c m

™

™

2

2



Mfnt, cW¥Ãd® mšyhjt®fS¡F V‰gL« TLjš f£lz¤ij F¿¡F« mâ

B – A = –25

40

35

50

20

30

30

40c cm m = 25 20

40 30

35 30

50 40

-

-

-

-e o

ÁWt® bgçat®

= 5

10

5

10c m ™

™

2

2



gæ‰Á 4.3 1. Ã‹tUtdt‰WŸ x›bth‹¿Y« mâfë‹ bgU¡f‰ gy‹ tiuaW¡f¥g£LŸsjh

vd¤ Ô®khå¡f. m›th¿U¥Ã‹, bgU¡f‰gyå‹ tçiria vGJf.

(i) AB, Ï§F ,A a B bx xij ij4 3 3 2

= =6 8@ B (ii) PQ, Ï§F ,P p Q qx xij ij4 3 4 3

= =6 6@ @ (iii) MN, Ï§F ,M m N n

x xij ij3 1 1 5= =6 6@ @ (iv) RS, Ï§F ,R r S s

x xij ij2 2 2 2= =6 6@ @

Ô®Î: (i) A-‹ ãušfë‹ v©â¡ifÍ«, B-‹ ãiufë‹ v©â¡ifÍ« rk«.

Mfnt, bgU¡f‰gy‹ mâ AB tiuaW¡f¥gL»‹wJ k‰W« AB-æ‹

tçir 4 × 2 MF«.

(ii) P-‹ ãušfë‹ v©â¡ifÍ«, Q-‹ ãiufë‹ v©â¡ifÍ« rkkhf

Ïšiy.

Mfnt, Ï›éU mâfë‹ bgU¡fš PQ-I tiuaW¡f ÏayhJ.

(iii) M-‹ ãušfë‹ v©â¡ifÍ«, N-‹ ãiufë‹ v©â¡ifÍ« rk«.

Mfnt, Ï›éU mâfë‹ bgU¡fš MN tiuaW¡fgL»wJ k‰W«

MN-‹ tçir 3×5 MF«.

(iv) R-‹ ãušfë‹ v©â¡ifÍ«, S-‹ ãiufë‹ v©â¡ifÍ« rk«.

Mfnt, bgU¡f‰gy‹ mâ RS tiuaW¡fgL»wJ k‰W« RS-‹ tçir

2×2 MF«.

2. Ã‹tUtdt‰¿‰F mâfë‹ bgU¡fš tiuaW¡f¥gLkhdhš, mt‰iw¡ fh©f.

(i) 2 15

4-^ ch m (ii) 3

5

2

1

4

2

1

7

-c cm m

(iii) 2

4

9

1

3

0

4

6

2

2

7

1-

--

-

e fo p (iv) 6

32 7

--e ^o h

Ô®Î: (i) A = (2 –1) k‰W« B = 5

4c m v‹f.

mâ A-‹ tçir 1 × 2 k‰W« mâ B-‹ tçir 2 × 1.A-‹ ãušfë‹ v©â¡ifÍ«, B-‹ ãiufë‹ v©â¡ifÍ« rk«. Mfnt, AB tiuaW¡f¥g£LŸsJ.

nkY« AB = (2 –1) 5

4c m = ( (2)(5) + (–1)4 ) = ( 6 ).

Ô®Î - mâfŸ 131

(ii) A = 3

5

2

1

-c m k‰W« B = 4

2

1

7c m v‹f.

mâ A-‹ tçir 2 × 2 k‰W« mâ B-‹ tçir 2 × 2.A-‹ ãušfë‹ v©â¡ifÍ«, B-‹ ãiufë‹ v©â¡ifÍ« rk«.Mfnt, AB tiuaW¡f¥g£LŸsJ.

nkY« AB = 3

5

2

1

4

2

1

7

-c cm m

= ( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

3 4

5 4

2 2

1 2

3 1

5 1

2 7

1 7

+

+

- +

+

-e o = 12 4

20 2

3 14

5 7

-

+

-

+e o = 8

22

11

12

-c m

(iii) A = 2

4

9

1

3

0-

-e o k‰W« B =

4

6

2

2

7

1

-

-

f p v‹f.

mâ A-‹ tçir 2 × 3 k‰W« mâ B-‹ tçir 3 × 2. A-‹ ãušfë‹ v©â¡ifÍ«, B-‹ ãiufë‹ v©â¡ifÍ« rk«.Mfnt, AB tiuaW¡f¥g£LŸsJ.

nkY« AB = 2

4

9

1

3

0

4

6

2

2

7

1-

--

-

e fo p = 8 54 6

16 6 0

4 63 3

8 7 0

- +

+ -

+ -

- +e o = 40

22

64

1

-c m

(iv) A = 6

3-e o k‰W« B = (2 – 7) v‹f.

mâ A-‹ tçir2 × 1 k‰W« mâ B-‹ tçir 1 × 2.A-‹ ãušfë‹ v©â¡ifÍ«, B-‹ ãiufë‹ v©â¡ifÍ« rk«.Mfnt, AB tiuaW¡f¥g£LŸsJ.

nkY« AB = 6

32 7

6 2 6 7

3 2 3 7

12 42

6 21

# #

# #-- =

-

- - -=

-

-c ^ ^

^ ^ ^e cm h h

h h ho m

3. xU gH éahghç j‹Dila filæš gH§fis é‰gid brŒ»wh®. M¥ÃŸ, kh«gH«

k‰W« MuŠR M»at‰¿‹ x›bth‹¿‹ é‰gid éiy Kiwna ` 20, ` 10 k‰W«

` 5 MF«. 3 eh£fëš é‰gidahF« gH§fë‹ v©â¡iffë‹ étu§fŸ ÑnH

ju¥g£LŸsd.

ehŸ M¥ÃŸ kh«gH« MuŠR

1 50 60 302 40 70 203 60 40 10

x›bthU ehëY« »il¤j bkh¤j é‰gid¤ bjhifia¡ F¿¥ÃL« xU mâia

vGJf. ÏÂèUªJ gH§fë‹ é‰gidæš »il¤j bkh¤j¤ bjhifia¡ fz¡»Lf.

Ô®Î: M¥ÃŸ, kh«gH« k‰W« MuŠR M»a x›bth‹¿‹ é‰gid éiyia¡

F¿¡F« mâ A v‹f. éiy

A = 20

10

5

f p M¥ÃŸ

kh«gH«

MuŠR


_‹W eh£fëš é‰gidahF« gH§fë‹ v©â¡ifia F¿¡F« mâ B v‹f.

M¥ÃŸ kh«gH« MuŠR

B = 50

40

60

60

70

40

30

20

10

f p ehŸ

ehŸ

ehŸ

1

2

3

Mfnt, x›bthU ehS« »il¤j bkh¤j é‰gid¤ bjhifia¡ F¿¥ÃL« mâ

T = BA.

& T = 50

40

60

60

70

40

30

20

10

20

10

5

f fp p

= 1000

800

1200

600

700

400

150

100

50

+

+

+

+

+

+

f p = 1750

1600

1650

f p ehŸ

ehŸ

ehŸ

1

2

3

Mfnt, »il¤j bkh¤j bjhif = ` (1750+1600+1650) = ` 5000kh‰WKiw: _‹W gH§fë‹ bkh¤j é‰gidia F¿¡F« mâ C. C = ( 50+40+60 60+70+40 30+20+10 ) = ( 150 170 60 ) Mfnt, mid¤J gH§fisÍ« é‰wÂš »il¤j bkh¤j bjhif

T = CA

= (150 170 60) 20

10

5

f p = (3000 + 1700 + 300) = (5000)

Mfnt, T = ` 5000

4. x

y

x1

3

2

3 0

0

9

0

0=c c cm m m våš, x k‰W« y-æ‹ kÂ¥òfis¡ fh©f. .

Ô®Î: x

y

x1

3

2

3 0

0

9

0

0=c c cm m m x

x

y

y

0

3 0

0 2

0 3&

+

+

+

+e o = x

9

0

0c m x

x

y

y3

2

3& c m = x

9

0

0c m

x¤j cW¥òfis x¥Ãl, ek¡F »il¥gJ

3x = 9 x 3& = k‰W« 2y = 0 y 0& = .

5. ,,A Xx

yC

5

7

3

5

5

11= = =

-

-c c em m o k‰W« AX C= våš, x k‰W« y-fë‹

kÂ¥òfis¡ fh©f.

Ô®Î: A = 5

7

3

5c m, X = x

yc m k‰W« C = 5

11

-

-e o vd bfhL¡f¥g£LŸsJ.

AX = C x

y

5

7

3

5

5

11& =

-

-c c em m o x

x

y

y

5

7

3

5&

+

+e o = 5

11

-

-e o

x¤j cW¥òfis x¥Ãl, ek¡F »il¥gJ

Ô®Î - mâfŸ 133

5x + 3y = – 5 & 5x + 3y + 5 = 0 7x + 5y = –11 & 7x + 5y + 11 =0FW¡F¥ bgU¡fš Kiwæš rk‹ghLfis¤ Ô®¥ngh«.

x y 1

3

5

5

11

5

7

3

5

& x y33 25 35 55 25 21

1-

=-

=-

& x y8 20 4

1=-

= 2, 5x y48

420& = = = - =-

6. A1

2

1

3=

-c m våš, A A I O4 52

2- + = vd ãWÎf.

Ô®Î: A2 = A.A

= 1

2

1

3

1

2

1

3

- -c cm m = 1 2

2 6

1 3

2 9

-

+

- -

- +e o = 1

8

4

7

- -c m

A A I4 52

2- + = 4 5

1

8

4

7

1

2

1

3

1

0

0

1

- --

-+c c cm m m

= 1

8

4

7

4

8

4

12

5

0

0

5

- -+

-

- -+c e cm o m

= 1 4 5

8 8 0

4 4 0

7 12 5

- - +

- +

- + +

- +e o = 0

0

0

0c m = O

7. k‰W«A B3

4

2

0

3

3

0

2= =c cm m våš, AB k‰W« BA M»at‰iw¡ fh©f. mit

rkkhf ÏU¡Fkh?

Ô®Î: A k‰W« B v‹gd 2 × 2 tçir bfh©l rJu mâfŸ. Mfnt, AB k‰W« BA M»at‰¿‹ bgU¡f‰gy‹ tiuaW¡f¥gL»wJ.

AB = 3

4

2

0

3

3

0

2c cm m = 9 6

12 0

0 4

0 0

+

+

+

+e o = 15

12

4

0c m g (1)

BA = 3

3

0

2

3

4

2

0c cm m = 9 0

9 8

6 0

6 0

+

+

+

+e o = 9

17

6

6c m g (2)

(1) k‰W« (2)-èUªJ, AB ! BA.

8. , k‰W«A B C1

1

2

2

1

3

0

1

2

2 1=-

= =c f ^m p h våš, ( ) ( )AB C A BC= v‹gij

rçgh®¡fÎ«.

Ô®Î: A-‹ tçir 2 × 3, B-‹ tçir 3 × 1 k‰W« C-‹ tçir 1 × 2 MF«.


Mfnt, AB-‹ tçir 2 × 1 k‰W« BC-‹ tçir 3 × 2 MF«.

AB = 1

1

2

2

1

3

0

1

2

-c fm p = 0 2 2

0 2 6

4

8

+ +

+ +=e co m

vdnt, (AB)C = 4

8c m (2 1) = 8

16

4

8c m g (1)

BC = 0

1

2

f p (2 1) = 0

2

4

0

1

2

f p

vdnt, A(BC) = 1

1

2

2

1

3

0

2

4

0

1

2

-c fm p = 0 4 4

0 4 12

0 2 2

0 2 6

+ +

+ +

+ +

+ +e o

= 8

16

4

8c m g (2)

(1) k‰W« (2)- èUªJ, ( ) ( )AB C A BC= .

Mfnt, mâfë‹ bgU¡fš nr®¥ò¥ g©òilaJ.

9. k‰W«A B5

7

2

3

2

1

1

1= =

-

-c em o våš, ( )AB B AT T T

= v‹gij rç¥gh®¡fÎ«.

Ô®Î: A = 5

7

2

3c m, B = 2

1

1

1-

-e o

A, B v‹gd 2 × 2 tçirÍila rJu mâfŸ. Mfnt, bgU¡fš mâ AB tiuaW¡f¥g£LŸsJ.

AB = 5

7

2

3

2

1

1

1-

-c em o = 10 2

14 3

5 2

7 3

-

-

- +

- +e o = 8

11

3

4

-

-e o

Mfnt, ( )AB T = 8

3

11

4- -e o g (1)

BT = 2

1

1

1-

-e o ; AT = 5

2

7

3c m

Mfnt, B AT T = 2

1

1

1

5

2

7

3-

-e co m = 10 2

5 2

14 3

7 3

-

- +

-

- +e o

= 8

3

11

4- -e o g (2)

(1) k‰W« (2)-èUªJ, ( )AB B AT T T= .

10. k‰W«A B5

7

2

3

3

7

2

5= =

-

-c em o v‹w mâfŸ x‹W¡bfh‹W bgU¡fš ne®khW

mâ vd ãWÎf.

Ô®Î - mâfŸ 135

Ô®Î: A = 57

2

3c m k‰W« B = 3

7

2

5-

-e o vd bfhL¡f¥g£LŸsJ.

A k‰W« B M»ad 2 × 2 tçirÍila rJu mâfŸ. Mfnt AB k‰W« BA M»at‰¿‹ bgU¡fš mâfŸ tiuaW¡f¥g£LŸsJ.

AB = 57

2

3

3

7

2

5-

-c em o = 15 14

21 21

10 10

14 15

-

-

- +

- +e o = 1

0

0

1c m = I g (1)

BA = I3 2

7 5

5 2

7 3

15 14 6 6

35 35 14 15

1 0

0 1

-

-=

- -

- + - += =c c c cm m m m g (2)

(1) k‰W« (2)-èUªJ, ek¡F »il¥gJ AB BA I= = .

vdnt, bfhL¡f¥g£l mâfŸ mâ¥bgU¡fè‹ Ñœ x‹W¡bfh‹W ne®khW

mâfshF«.

11. Ô®¡f: xx

11

2

0

3 50

- -=^ e c ^h o m h.

Ô®Î: (x 1) ( )x1

2

0

3 50

- -=e co m & (x 1) x

x

0

2 15

+

- -e o = (0)

& (x 1) x

x2 15- -e o = (0) & ((x)(x) + (1)(–2x–15)) = (0)

& (x2 – 2x – 15) = ( 0 )Mfnt, x x2 15 0

2- - = & ( )( )x x3 5 0+ - = & ,x 3 5=- .

12. k‰W«A B1

2

4

3

1

3

6

2=

-

-=

-

-e eo o våš, ( )A B A AB B22 2 2!+ + + vd

ãWÎf.

Ô®Î: A + B = 1

2

4

3

1

3

6

2-

-+

-

-e eo o

= 1 1

2 3

4 6

3 2

0

1

2

1

-

- +

- +

-=e co m

( )A B 2+ = (A + B) (A + B)

= 0

1

2

1

0

1

2

1c cm m = 0 2

0 1

0 2

2 1

+

+

+

+e o = 2

1

2

3c m g (1)

A2 = AA = 1

2

4

3

1

2

4

3-

-

-

-e eo o

= 1 8

2 6

4 12

8 9

9

8

16

17

+

- -

- -

+=

-

-e eo o

AB = 1

2

4

3

1

3

6

2-

- -

-e eo o


= 1 12

2 9

6 8

12 6

- -

+

+

- -e o = 13

11

14

18

-

-e o

2AB = 2 13

11

14

18

26

22

28

36

-

-=

-

-e eo o

B2 = BB = 1

3

6

2

1

3

6

2

-

-

-

-e eo o

= 1 18

3 6

6 12

18 4

+

- -

- -

+e o = 19

9

18

22-

-e o

Mfnt, 2A AB B2 2+ + = 9

8

16

17

26

22

28

36

19

9

18

22-

-+

-

-+

-

-e e eo o o

= 9 26 19

8 22 9

16 28 18

17 36 22

- +

- + -

- + -

- +e o

= 2

5

6

3

-c m g (2)

(1) k‰W« (2) M»at‰¿èUªJ, ( ) 2A B A AB B2 2 2!+ + +

13. , k‰W«A B C3

7

3

6

8

0

7

9

2

4

3

6= = =

-c c cm m m våš, ( ) k‰W«A B C AC BC+ + v‹w

mâfis¡ fh©f. nkY«, ( )A B C AC BC+ = + v‹gJ bkŒahFkh?

Ô®Î: A + B = 3 8

7 0

3 7

6 9

11

7

10

15

+

+

+

+=e co m

(A + B) C = 11

7

10

15

2

4

3

6

-c cm m = 22 40

14 60

33 60

21 90

+

+

- +

- +e o

= 62

74

27

69c m g (1)

AC = 3

7

3

6

2

4

3

6

-c cm m

= 6 12

14 24

9 18

21 36

+

+

- +

- +e o = 18

38

9

15c m

BC = 8

0

7

9

2

4

3

6

-c cm m

= 16 28

0 36

24 42

0 54

+

+

- +

+e o = 44

36

18

54c m

AC + BC = 18

38

9

15c m+ 44

36

18

54c m = 62

74

27

69c m g (2)

(1) k‰W« (2) M»at‰¿èUªJ ( )A B C AC BC+ = + .

Ô®Î - mâfŸ 137

gæ‰Á 4.4


1. Ã‹tUtdt‰WŸ vªj T‰W bkŒahdjšy?

(A) Âiræè mâahdJ xU rJu mâahF«.

(B) _iy é£l mâahdJ xU rJu mâahF«.

(C) Âiræè mâahdJ xU _iy é£l mâahF«.

(D) _iy é£l mâahdJ xU Âiræè mâahF«.

Ô®Î: xU Âiræè mâ, _iyé£l mâahF«. Mdhš xU _iyé£l mâ

Âiræè mâahf ÏU¡f nt©oa mtÁa« Ïšiy. ( éil. (D) ) 2. A a

ij m n=

#6 @ v‹gJ xU rJu mâ våš,

(A) m n1 (B) m n2 (C) m 1= (D) m = nÔ®Î: xU rJu mâæš ãiufë‹ v©â¡ifÍ«, ãušfë‹ v©â¡ifÍ«

rk«.

Mfnt, A aij m n

=#

6 @ v‹gJ xU rJumâ. Mfnt, m = n MF«. ( éil. (D) )

3. x

y x

y3 7

1

5

2 3

1

8

2

8

+

+ -=

-e eo o våš, x k‰W« y-fë‹ kÂ¥òfŸ Kiwna

(A) –2 , 7 (B) 31- , 7 (C)

31- ,

32- (D) 2 , –7

Ô®Î: mâfŸ rk« v‹gjhš, x¤j cW¥òfŸ rk«. Mfnt,

3 7 1 2 ; 1 8 7x x y y& &+ = =- + = = ( éil. (A) )

4. k‰W«A B1 2 3

1

2

3

= - =

-

-

^ fh p våš, A + B =

(A) 0 0 0^ h (B) 0

0

0

f p (C) 14-^ h (D) tiuaW¡f¥gléšiy

Ô®Î: bt›ntW tçirfŸ bfh©l mâfë‹ T£liy tiuaW¡f ÏayhJ. A-‹ tçir 1 × 3 k‰W« B-‹ tçir 3 × 1. vdnt A + B tiuaW¡fglhjJ. ( éil. (D) )

5. xU mâæ‹ tçir 2 3# våš, m›tâæš cŸs cW¥òfë‹ v©â¡if

(A) 5 (B) 6 (C) 2 (D) 3

Ô®Î: m × n tçir bfh©l mâ mn cW¥òfis¥ bg‰¿U¡F«. ( éil. (B) )

6. 4x

8 4

8

2

1

1

2=c cm m våš, x-‹ kÂ¥ò

(A) 1 (B) 2 (C) 41 (D) 4

Ô®Î: x

8 4

8

8

4

4

2=c cm m. x¤j cW¥òfis x¥ÃL« nghJ, eh« bgWtJ 4x =

( éil. (D) )


7. A-‹ tçir 3 4# k‰W« B-‹ tçir 4 3# våš, BA-‹ tçir

(A) 3 3# (B) 4 4# (C) 4 3# (D) tiuaW¡f¥gléšiy

Ô®Î:B-‹ tçir 4 3# , A-‹ tçir 3 4# . Mfnt, BA-‹ tçir 4 4# . ( éil. (B) )

8. A1

0

1

21 2# =c ^m h våš, A-‹ tçir

(A) 2 1# (B) 2 2# (C) 1 2# (D) 3 2#

Ô®Î: (1 2)A1

0

1

2m n2 2

1 2=

##

#c m

2n& = k‰W« 1m = . Mfnt, A-‹ tçir 1 2# . ( éil. (C) )

9. A k‰W« B v‹gd rJu mâfŸ. nkY« AB = I k‰W« BA = I våš, B v‹gJ

(A) myF mâ (B) ó¢Áa mâ

(C) A-‹ bgU¡fš ne®khW mâ (D) A-

Ô®Î: tiuaiwæ‹go AB BA I= = våš, B v‹gJ A-‹ bgU¡fš ne®khW

mâahF«. ( éil. (C) )

10. x

y

1

2

2

1

2

4=c c cm m m våš, x k‰W« y-fë‹ kÂ¥òfŸ Kiwna

(A) 2 , 0 (B) 0 , 2 (C) 0 , 2- (D) 1 , 1

Ô®Î: x

y

1

2

2

1

2

4=c c cm m m 2

2

x y

x y

2

4&

+

+=e co m

Mfnt, 2 2 (1) ; 2 4 (2)x y x yg g+ = + =

(1) k‰W« (2) M»at‰¿èUªJ, ; .x y2 0= = ( éil. (A) )

F¿¥ò: neuoahf ÃuÂæLtj‹ _y« 2 , 0x y= = v‹git Ïu©L ne®¡nfhL

fisÍ« ãiwÎ brŒtij fhzyh«.

11. A1

3

2

4=

-

-e o k‰W« A B O+ = våš, B =

(A) 1

3

2

4-

-e o (B) 1

3

2

4

-

-e o (C) 1

3

2

4

-

-

-

-e o (D) 1

0

0

1c m

Ô®Î: B A1

3

2

4

1

3

2

4=- =-

-

-=

-

-e eo o. ( éil. (B) )

12. A4

6

2

3=

-

-e o, våš, A2 =

(A) 16

36

4

9c m (B) 8

12

4

6

-

-e o (C) 4

6

2

3

-

-e o (D) 4

6

2

3

-

-e o

Ô®Î - mâfŸ 139

Ô®Î: A AA4

6

2

3

4

6

2

3

16 12

24 18

8 6

12 9

4

6

2

32= =

-

-

-

-=

-

-

- +

- +=

-

-e e e eo o o o

( éil. (D) ) 13. A-‹ tçir m n# k‰W« B-‹ tçir p q# v‹f. nkY«, A k‰W« B M»adt‰¿‹

TLjš fhz ÏaYbkåš,

(A) m p= (B) n = q (C) n = p (D) m = p, n = qÔ®Î: xnu tçir bfh©l mâfS¡F k£Lnk T£lš tiuaW¡f¥g£LŸsJ.

Ï§F, m n# = p q# v‹whš k£Lnk A k‰W« B mâfë‹ TLjš fhz ÏaY«.

Mfnt, m p= k‰W« n = q MF«. ( éil. (D) )

14. a

1

3

2

2

1

5

0-=c e cm o m våš, a-‹ kÂ¥ò

(A) 8 (B) 4 (C) 2 (D) 11

Ô®Î: 2 3

0

5

0

a a

1

3

2

2

1

5

0&

-=

-=c e c c cm o m m m

& a a2 3 5 4&- = = ( éil. (D) )

15. Aa

c

b

a=

-e o k‰W« A I2

= våš,

(A) 1 02a bc+ + = (B) 1 0

2a bc- + =

(C) 1 02a bc- - = (D) 1 0

2a bc+ - =

Ô®Î: A I0

0 1

0

0

12

2

2&a bc

bc a=

+

+=e co m

1 1 02 2& &a bc a bc+ = - - = ( éil. (C) )

16. A aij 2 2

=#

6 @ k‰W« a i jij= + våš, A =

(A) 1

3

2

4c m (B) 2

3

3

4c m (C) 2

4

3

5c m (D) 4

6

5

7c m

Ô®Î: Aa

a

a

a2

3

3

411

21

12

22

= =e co m ( éil. (B) )

17. a

c

b

d

1

0

0

1

1

0

0

1

-=

-c c em m o våš, a, b, c k‰W« d M»adt‰¿‹ kÂ¥òfŸ Kiwna

(A) , , ,1 0 0 1- - (B) 1, 0, 0, 1 (C) , , ,1 0 1 0- (D) 1, 0, 0, 0

Ô®Î: a

c

b

d

1

0

0

1

- -=

-c em o. x¤j cW¥òfis x¥Ãl

1, 0, 0a b c=- = = k‰W« 1d =- ( éil. (A) )


18. A7

1

2

3= c m k‰W« A B

1

2

0

4+ =

-

-e o våš, mâ B =

(A) 1

0

0

1c m (B) 6

3

2

1-e o (C) 8

1

2

7

- -

-e o (D) 8

1

2

7-e o

Ô®Î: B 1

2

0

4

7

1

2

3=

-

--e co m =

8

1

2

7

- -

-e o. ( éil. (C) )

19. x5 1

2

1

3

20- =^ f ^h p h våš, x-‹ kÂ¥ò

(A) 7 (B) 7- (C) 71 (D) 0

Ô®Î: 20x5 1

2

1

3

- =^ f ^h p h

(10 3) (20) 13 20 7x x x& &- + = - = =- ( éil. (B) )

20. A k‰W« B v‹gd xnu tçirÍila rJu mâfŸ våš, Ñœ¡f©litfëš vJ

bkŒahF«?

(A) ( )AB A BT T T= (B) ( )A B A BT T T T

=

(C) ( )AB BAT= (D) ( )AB B AT T T

=

Ô®Î: ãiu ãuš kh‰W mâ¡fhd Ã‹-ÂU¥òif éÂæ‹ go, ( )AB B AT T T= .

( éil. (D) )

Ô®Î - Ma¤bjhiy toéaš 141

gæ‰Á 5.1 1. Ã‹tU« òŸëfis Ïiz¡F« nfh£L¤ J©Lfë‹ eL¥òŸëfis¡ fh©f.

(i) ,1 1-^ h k‰W« ,5 3-^ h (ii) ,0 0^ h k‰W« ,0 4^ h.

Ô®Î: (i) ( , ) (1, 1), ( , ) ( 5,3)x y x y1 1 2 2

= - = - v‹f.

(1, –1), (–5, 3) v‹w òŸëfis Ïiz¡F« nfh£L¤J©o‹ ika¥òŸë,

, , , ( 2,1)x x y y

2 2 21 5

21 3

24

221 2 1 2

+ += - - + = - = -c ` `m j j

(ii) ( , ) (0,0), ( , ) (0,4)x y x y1 1 2 2

= = v‹f.

(0, 0), (0, 4) v‹w òŸëfis Ïiz¡F« nfh£L¤J©o‹ ika¥òŸë,

, (0, 2)2

0 02

0 4+ + =` j

2. Ã‹tU« òŸëfis Kidfshf¡ bfh©l K¡nfhz§fë‹ eL¡nfh£L

ika§fis¡ fh©f.

(i) , , , ,k‰W«1 3 2 7 12 16-^ ^ ^h h h (ii) , , , ,k‰W«3 5 7 4 10 2- - -^ ^ ^h h h

Ô®Î: (i) ( , ) (1,3), ( , ) (2,7)x y x y1 1 2 2

= = k‰W« ( , ) (12, 16)x y3 3

= - v‹f.

eL¡nfh£L ika«, ,x x x y y y

3 31 2 3 1 2 3+ + + +

c m = ,3

1 2 123

3 7 16+ + + -` j

(5, 2)= - .

(ii) ( , ) (3, 5), ( , ) ( 7,4)x y x y1 1 2 2

= - = - k‰W« ( , ) (10, 2)x y3 3

= - v‹f.

eL¡nfh£L ika«, ,3

3 7 103

5 4 2- + - + -` j = , (2, 1)36

33- = -` j .

3. xU t£l¤Â‹ ika« (-6, 4). m›t£l¤Â‹ xU é£l¤Â‹ xU Kid, MÂ¥òŸë

våš, k‰bwhU Kidia¡ fh©f.

Ô®Î: é£l¤Â‹ xU Kid MÂ¥òŸë (0, 0). k‰bwhU Kid ( , )x y v‹f.

t£l¤Â‹ ika« é£l¤Â‹ ika¥òŸë MF«.

Mfnt, , ( 6, 4)x y2

02

0+ += -c m .

x, y m¢R¤bjhiyÎfis ÏUòwK« rk¥gL¤j, eh« bgWtJ

6 12x x2

0 &+ =- =- , 4 8.y

y2

0&

+= =

vdnt, é£l¤Â‹ k‰bwhU Kid ( , )12 8- .

Ma¤bjhiy toéaš 5


4. òŸë (1, 3)-I eL¡nfh£L ikakhf¡ bfh©l K¡nfhz¤Â‹ ÏU KidfŸ

(-7, 6) k‰W« (8, 5) våš, K¡nfhz¤Â‹ _‹whtJ Kidia¡ fh©f.

Ô®Î: K¡nfhz¤Â‹ c¢ÁfŸ (-7, 6), (8, 5) k‰W« eL¡nfh£L ika« ( , )1 3 vd

bfhL¡f¥g£LŸsJ. _‹whtJ c¢Á ( , )x y v‹f.

Mfnt, , (1, 3)x y3

7 83

6 5- + + + +=c m

& , (1, 3)x y3

13

11+ +=c m

x, y m¢R¤bjhiyÎfis ÏUòwK« rk¥gL¤j, eh« bgWtJ

1 2x x3

1 &+ = = k‰W« 3 2.y

y3

11&

+= =-

vdnt, K¡nfhz¤Â‹ _‹whtJ c¢Á (2, )2- .

5. ÃçÎ¢ N¤Âu¤ij¥ ga‹gL¤Â, A(1,0), B(5,3), C(2,7) k‰W« D (–2, 4) v‹w tçiræš

vL¤J¡bfhŸs¥g£l òŸëfŸ xU Ïizfu¤Â‹ c¢ÁfshF« vd ãWÎf.

Ô®Î: Ïizfu¤Â‹ _iyé£l§fŸ x‹iwbah‹W ÏUrk¡ T¿L« vd eh«

m¿nth«. Ï¥nghJ,

AC-‹ ika¥òŸë , , .2

1 22

0 723

27+ + =` `j j

BD-‹ ika¥òŸë , ,2

5 22

3 423

27- + =` `j j.

AC k‰W« BD-‹ ika¥òŸëfŸ x‹nwahF«. Mfnt ABCD X® Ïizfu«.

6. (3, 4) k‰W« (– 6, 2) M»a òŸëfis Ïiz¡F« nfh£L¤ J©oid btë¥òwkhf

3 : 2 v‹w é»j¤Âš Ãç¡F« òŸëæ‹ m¢R¤ bjhiyÎfis¡ fh©f.

Ô®Î: ( , )A 3 4 k‰W« ( , )B 6 2- v‹gd bfhL¡f¥g£l òŸëfŸ, ( , )P x y v‹gJ

AB-ia 3 : 2 v‹w é»j¤Âš btë¥òwkhf Ãç¡F« òŸë v‹f.

Mfnt, P(x, y) = ,Pl m

lx mx

l m

ly my2 1 2 1

-

-

-

-c m.

( , )( ) ( )

,( ) ( )

( 24, 2) .P x y P P3 2

3 6 2 33 2

3 2 2 4=

-- -

--

= - -c m

7. (– 3, 5) k‰W« (4, – 9) M»a òŸëfis Ïiz¡F« nfh£L¤ J©oid c£òwkhf


Ô®Î: ( 3, )A 5- k‰W« ( , )B 4 9- v‹gd bfhL¡f¥g£l òŸëfŸ, ( , )P x y v‹gJ

AB-I 1 : 6 v‹w é»j¤Âš c£òwkhf Ãç¡F« òŸë v‹f.

Mfnt, P(x, y) = ,Pl m

lx mx

l m

ly my2 1 2 1

+

+

+

+c m

( , )( ) ( )

,( ) ( )

( 2, 3) .P x y P P1 6

1 4 6 31 6

1 9 6 5=

++ -

+- +

= -c m


8. A(–6, –5), B(–6, 4) v‹gd ÏU òŸëfŸ v‹f. nfh£L¤J©L AB-æ‹ nkš

AP = 92 AB v‹wthW mikªJŸs òŸë P-ia¡ fh©f.

Ô®Î: P(x, y) v‹gJ nfh£L¤J©L AB-‹ nkš AP AB92= v‹wthW mikªjŸs

òŸë v‹f. vdnt

AP AB92= & 9AP = 2AB

= 2( )AP PB+

7AP = 2PB . & PBAP

72= .

` : 2 : 7.AP PB =

Mfnt, P v‹gJ AB-I 2 : 7 v‹w é»j¤Âš c£òwkhf Ãç¡F«.

ÃçÎ N¤Âu¤Â‹ go,

( , )( ) ( )

,( ) ( )

, ( 6, 3)P x y2 7

2 6 7 62 7

2 4 7 59

12 429

8 35=+

- + -++ -

= - - - = - -c `m j

Mfnt, P(x, y) = P(–6, –3).

9. A(2, –2) k‰W« B(–7, 4) v‹w òŸëfis Ïiz¡F« nfh£L¤ J©il _‹W rk

ghf§fshf¥ Ãç¡F« òŸëfis¡ fh©f.

Ô®Î: P, Q v‹gd nfh£L¤J©L AB-I _‹W rkghf§fshf AP PQ QB= =

v‹wthW Ãç¡F« òŸëfŸ v‹f.

Mfnt, P v‹gJ AB-ia 1 : 2 v‹w é»j¤ÂY«, Q v‹gJ AB-ia 2 : 1 v‹w

é»j¤ÂY« c£òwkhf Ãç¡»‹wd. ÃçÎ N¤Âu¤Â‹ go

( ) ( ),

( ) ( ), ( 1, 0) .P

1 21 7 2 2

1 21 4 2 2

37 4

34 4=

+- +

++ -

= - + - = -c `m j

Mfnt, òŸë P v‹gJ (–1, 0) MF«.

nkY«, ( ) ( )

,( ) ( )

( , ) .Q2 1

2 7 1 22 1

2 4 1 24 2=

+- +

++ -

= -c m

Mfnt, òŸë Q v‹gJ (–4, 2) MF«.

F¿¥ò: Q v‹gJ PB-‹ ika¥òŸë. Mfnt òŸë , ( 4, 2) .Q21 7

20 4= - - + = -` j

Ïnjngh‹W P v‹gJ AQ-‹ ika¥òŸë MF«.

10. A(–4, 0) k‰W« B(0, 6) v‹w òŸëfis Ïiz¡F« nfh£L¤ J©il eh‹F

rkghf§fshf¥ Ãç¡F« òŸëfis¡ fh©f.

Ô®Î: P, Q, R v‹gd AB v‹w nfh£L¤J©il eh‹F rkghf§fshf Ãç¡F«

òŸëfŸ v‹f.

Q v‹gJ AB-‹ ika¥òŸë.

vdnt, Q = 24 0 ,

20 6 ( , )2 3- + + = -` j .

P v‹gJ AQ-‹ ika¥òŸë


P = , , ,24 2

20 3

26

23 3

23- - + = - = -` ` `j j j.

R v‹gJ QB-‹ ika¥òŸë R , ,22 0

23 6 1

29- + + = -` `j j.

Mfnt, njitahd òŸëfŸ ( 3, ), ( 2,3), ( 1, ) .P Q R23

29- - -

F¿¥ò: P v‹gJ AB-I 1 : 3 v‹w é»j¤ÂY«, R v‹gJ AB-I 3 : 1 v‹w

é»j¤ÂY« c£òwkhf Ãç¡F«. ÃçÎ N¤Âu¤ij ga‹gL¤ÂÍ« P k‰W« R òŸëfis fhzyh«.

11. (6, 4) k‰W« (1, –7) v‹w òŸëfis Ïiz¡F« nfh£L¤ J©oid x-m¢R Ãç¡F«

é»j¤ij¡ fh©f.

Ô®Î: ( , )A 6 4 k‰W« ( , )B 1 7- v‹gd bfhL¡f¥g£l òŸëfŸ.

P(x, 0) v‹gJ AB-I l : m v‹w é»j¤Âš c£òwkhf Ãç¡»wJ

v‹f.

ÃçÎ thŒ¥gh£o‹ go, ( , 0)( ) ( )

,( ) ( )

P x Pl m

l ml m

l m1 6 7 4=

++

+- +

c m

y m¢R¤ bjhiyÎfis rk¥gL¤j, 0 7 4 .l ml m l m

ml7 4

74& &

+- + = - =- =

Mfnt, x m¢rhdJ nfh£L¤J©L AB-I 4 : 7 v‹w é»j¤Âš c£òwkhf Ãç¡»wJ

12. (–5, 1) k‰W« (2, 3) v‹w òŸëfis Ïiz¡F« nfh£L¤ J©oid

y-m¢R Ãç¡F« é»j¤ijÍ« k‰W« Ãç¡F« òŸëiaÍ« fh©f.

Ô®Î: ( , )A 5 1- k‰W« ( , )B 2 3 v‹gd bfhL¡f¥g£l òŸëfŸ.

P(0, y) v‹gJ AB-ia l : m v‹w é»j¤Âš c£òwkhf Ãç¡»wJ v‹f.

ÃçÎ thŒ¥gh£o‹ go, (0, )( ) ( )

,( ) ( )

P y Pl m

l ml m

l m2 5 3 1=

++ -

++

c m g (1)

x m¢R¤ bjhiyÎfis rk¥gL¤j, 0 2 5 0 .l ml m l m

ml2 5

25& &

+- = - = =

vdnt, njitahd é»j« : 5 : 2l m = .

nkY« (1)-èUªJ, eh« bgWtJ (0, ) ,( ) ( )

( , ) .P y P P05 2

5 3 2 10

717=

++

=c m

Mfnt, y m¢R Ãç¡F« òŸë ,0717` j MF«.

13. xU K¡nfhz¤Â‹ KidfŸ (1, –1), (0, 4) k‰W« (–5, 3) våš, m«K¡nfhz¤Â‹

eL¡nfhLfë‹ (medians) Ús§fis¡ fz¡»lÎ«.

Ô®Î: ( , 1)A 1 - , (0,4)B k‰W« ( 5,3)C - v‹gd K¡nfhz¤Â‹ c¢ÁfŸ. D, E, F v‹gd Kiwna BC, CA k‰W« AB-‹ ika¥òŸëfŸ v‹f.

vdnt, BC-‹ ika¥òŸë , ( , ) .D D2

0 52

4 325

27- + = -` j


AC-‹ ika¥òŸë , ( 2,1) .E E2

1 521 3- - + = -` j

AB-‹ ika¥òŸë , , .F F2

1 021 4

21

23+ - + =` `j j

Mfnt, eL¡nfhL AD-‹ Ús«, 1 1AD2 2

25

27= + + - -` `j j

2 2

27

29

449

481

4130

2130= + = + = =-` `j j .

eL¡nfhL BE-‹ Ús«, BE 2 0 1 42 2= - - + -^ ^h h 4 9 13= + = .

eL¡nfhL CF-‹ Ús«, CF 5 32 2

21

23= + + -` `j j

.2 2

211

23

4121

49

2130= + = + =-` `j j

Mfnt, ABCT -‹ eL¡nfhLfë‹ Ús§fŸ 2130 , 13 ,

2130 MF«.

gæ‰Á 5.2

1. Ñœ¡ f©l òŸëfis Kidfshf¡ bfh©l K¡nfhz§fë‹ gu¥òfis¡ fh©f.

(i) (0, 0), (3, 0) k‰W« (0, 2) (ii) (5, 2), (3, -5) k‰W« (-5, -1) (iii) (–4, –5), (4, 5) k‰W« (–1, –6)

Ô®Î: (i) A(0, 0), B(3, 0) k‰W« C(0, 2) v‹gd K¡nfhz¤Â‹ c¢ÁfŸ.

ABCT gu¥gsÎ = ( )x y x y x y x y x y x y21

1 2 2 3 3 1 2 1 3 2 1 3+ + - + +^ h6 @

= [(0 6 0) (0 0 0)] 321 + + - + + = r.myFfŸ

(ii) bfhL¡f¥g£l òŸëfis gl¤Âš tçir¥go F¿¡fÎ«. A(-5, -1), B(3, -5) k‰W« C(5, 2) v‹gd K¡nfhz¤Â‹ c¢ÁfŸ v‹f.

ABCT -‹ gu¥gsÎ = 21 5

1

3

5

5

2

5

1

-

- -

-

-) 3

[( ) ( )] ( )21 25 6 5 3 25 10

21 26 38 32= + - - - - - = + = .

Mfnt, K¡nfhz¤Â‹ gu¥ò 32 r. myFfŸ.

(iii) bfhL¡f¥g£l òŸëfis gl¤Âš tçir¥go F¿¡fÎ«. A(– 4, –5), B(–1, –6) k‰W« C(4, 5) v‹gd K¡nfhz¤Â‹ c¢ÁfŸ v‹f..


5

1

6

4

5

4

5

-

-

-

-

-

-) 3

[( ) ( )] ( )21 24 5 20 5 24 20

21 38 19= - - - - - = = .

Mfnt, K¡nfhz¤Â‹ gu¥ò 19 r. myFfŸ.


2. tçiræš mikªj K¡nfhz¤Â‹ KidfS« mit mik¡F« K¡nfhz¤Â‹

gu¥gsÎfS« ÑnH¡ bfhL¡f¥g£LŸsd. x›bth‹¿Y« a-‹ kÂ¥ig¡ fh©f.

c¢ÁfŸ gu¥ò (rJu myFfŸ)

(i) (0, 0), (4, a), (6, 4) 17 (ii) (a, a), (4, 5), (6,–1) 9

(iii) (a, –3), (3, a), (–1,5) 12

Ô®Î: (i) A( , )0 0 , B(4, a) k‰W« C(6, 4) v‹gd K¡nfhz¤Â‹ c¢ÁfŸ v‹f.

nkY« ABCT -‹ gu¥gsÎ 17 r. myFfŸ vd bfhL¡f¥g£LŸsJ.

Mfnt, a2

1 0

0

4 6

4

0

017=' 1

& (16 6 ) 17a21 - = 16 6a 34& - = .a 3& =-

Mfnt, a-‹ kÂ¥ò –3.

(ii) A(a, a), B(4, 5) k‰W« C(6,–1) v‹gd K¡nfhz¤Â‹ c¢ÁfŸ v‹f. nkY«

ABCT -‹ gu¥gsÎ 9 r. myFfŸ vd bfhL¡f¥g£LŸsJ.

Mfnt, a

a

a

a21 4

5

6

19

-=) 3

[(5 4 6 ) (4 30 )] 9a a a a21& - + - + - = 8 34 18a& &- = a

213= .

Mfnt, a-‹ kÂ¥ò 213 .

(iii) A(a, –3), B(3, a) k‰W« C(–1 , 5) v‹gd K¡nfhz¤Â‹ c¢ÁfŸ v‹f. nkY«

ABCT -‹ gu¥gsÎ 12 r. myFfŸ vd bfhL¡f¥g£LŸsJ.

vdnt, 12a

a

a

21

3

3 1

5 3-

-

-=) 3

[( 15 3) ( 9 5 )] 12a a a21 2

& + + - - - + =

a a4 3 02& - + = ( )( ) 0a a3 1& - - = 3a& = k‰W« 1a =

Mfnt, a-‹ kÂ¥òfŸ 1, 3.

3. Ã‹tU« òŸëfŸ xnu ne®¡nfh£oš mikÍ« òŸëfsh vd MuhŒf.

(i) (4, 3), (1, 2) k‰W« (–2, 1) (ii) (–2, –2), (–6, –2) k‰W« (–2, 2)

(iii) ,23 3-` j, (6, –2) k‰W« (–3, 4)

Ô®Î: (i) A(4, 3), B(1, 2) k‰W« C(–2, 1) v‹gd bfhL¡f¥g£l òŸëfŸ v‹f.


3

1

2

2

1

4

3

-' 1

= (8 1 6) (3 4 4)21 0+ - - - + =

Mfnt, bfhL¡f¥g£l òŸëfŸ xnu ne®¡nfh£oš mikÍ«.


(ii) A(-2, -2), B(-6, -2) k‰W« C(-2, 2) v‹gd bfhL¡f¥g£l òŸëfŸ v‹f.

ABCT -‹ gu¥gsÎ

= [(4 12 4) (12 4 4)]21 2

2

6

2

2

2

2

2 21-

-

-

-

- -

-= - + - + -) 3

[(8 12) 12]21 8 0!= - - =-

Mfnt, bfhL¡f¥g£l òŸëfŸ xnu ne®¡nfh£oš mikahJ.

(iii) A23 ,3-` j, B(6, -2) k‰W« C(-3, 4) v‹gd bfhL¡f¥g£l òŸëfŸ v‹f.

ABCT -‹ gu¥gsÎ

= [(3 24 9) (18 6 6)]21

3

6

2

3

4 3212

323

-

-= + - - + -

- -Z

[

\

]]

]

_

`

a

bb

b= 0.

Mfnt, bfhL¡f¥g£l òŸëfŸ xnu ne®¡nfh£oš mikÍ«.

4. bfhL¡f¥g£oU¡F« òŸëfŸ xU nfhliktd våš, x›bth‹¿Y« k-‹ kÂ¥ig¡

fh©f.

(i) (k, –1), (2, 1) k‰W« (4, 5) (ii) , , , ,k‰W« k2 5 3 4 9- -^ ^ ^h h h

(iii) , , , ,k‰W«k k 2 3 4 1-^ ^ ^h h h

Ô®Î: (i) A(k, -1), B(2, 1) k‰W« C(4, 5) bfhL¡f¥g£l òŸëfŸ v‹f.

_‹W òŸëfS« xnu ne®¡nfh£oš miktjhš ABCT -‹ gu¥ò ó{ía« MF«.

Mfnt, 0k k

21

1

2

1

4

5 1- -=) 3

[( 6) (2 5 )] 0k k& + - + = 1k& = .

Mfnt, k-‹ kÂ¥ò 1.(ii) , , ,A B2 5 3 4- -^ ^h h k‰W« ,C k9^ h v‹gd bfhL¡f¥g£l òŸëfŸ v‹f._‹W òŸëfS« xnu ne®¡nfh£oš miktjhš ABCT -‹ gu¥ò ó{ía« MF«.

Mfnt

& 0k2

1 2

5

3

4

9 2

5- - -=) 3

& [( 8 3 45) ( 15 36 2 )] 0k k- + - - - - + = 2.k& =

Mfnt, k-‹ kÂ¥ò 2.(iii) , , ,A k k B 2 3^ ^h h k‰W« ,C 4 1-^ h v‹gd bfhL¡f¥g£l òŸëfŸ v‹f._‹W òŸëfS« xnu ne®¡nfh£oš miktjhš, ABCT -‹ gu¥ò ó{ía« MF«.

Mfnt,

& 0k

k

k

k21 2

3

4

1-=) 3

& (3 2 4 ) (2 12 ) 0k k k k- + - + - = 6 14 0 .k k37& &- = =

Mfnt, k-‹ kÂ¥ò 37 .


5. Ã‹tUtdt‰iw Kidfshf¡ bfh©l eh‰fu§fë‹ gu¥gsÎfis¡ fh©f.

(i) , , , , , ,k‰W«6 9 7 4 4 2 3 7^ ^ ^ ^h h h h (ii) , , , , , ,k‰W«3 4 5 6 4 1 1 2- - - -^ ^ ^ ^h h h h

(iii) , , , , , ,k‰W«4 5 0 7 5 5 4 2- - - -^ ^ ^ ^h h h h

Ô®Î: (i) bfhL¡f¥g£l òŸëfis fofhu KŸnsh£l Âir¡F vÂ®Âiræš

mikÍkhW tçirahf gl¤Âš F¿¡fÎ«.

bfhL¡f¥g£l òŸëfis , , , , ,A B C4 2 7 4 6 9^ ^ ^h h h k‰W«

,D 3 7^ h v‹f.

eh‰fu« ABCD-‹ gu¥ò = 21 4

2

7

4

6

9

3

7

4

2' 1

= [( ) ( )]21 16 63 42 6 14 24 27 28+ + + - + + +

= [(127 93) (34) 17.21

21- = =

Mfnt, eh‰fu« ABCD-‹ gu¥ò 17 r. myFfŸ.

(ii) bfhL¡f¥g£l òŸëfis fofhu KŸnsh£l Âir¡F vÂ®Âiræš mikÍkhW

tçirahf gl¤Âš F¿¡fÎ«.

bfhL¡f¥g£l òŸëfis , , , , ,A B C5 6 4 1 1 2- - -^ ^ ^h h h k‰W« ,D 3 4-^ h v‹f.

eh‰fu« ABCD-‹ gu¥ò

21 5

6

4

1

1

2

3

4

5

6=

-

- -

- -

-) 3

[(5 8 4 18) ( 24 1 6 20)]21= + + + - - - - -

= [(35 51) (86) 43.21

21+ = =

Mfnt, eh‰fu« ABCD-‹ gu¥ò 43 r. myFfŸ.(iii) bfhL¡f¥g£l òŸëfis fofhu KŸnsh£l Âir¡F vÂ®Âiræš mikÍkhW

tçirahf gl¤Âš F¿¡fÎ«.

bfhL¡f¥g£l òŸëfis , , , , ,A B C4 2 5 5 0 7- - -^ ^ ^h h h k‰W« ,D 4 5-^ h v‹f. eh‰fu« ABCD-‹ gu¥ò

21 4

2

5

5

0

7

4

5

4

2=

-

- -

- -

-) 3

[(20 35 0 8) ( 10 0 28 20)]21= + + + - - + - -

= [( 5 ) ( ) .21 63 8

21 121 60 5+ = =

Mfnt, eh‰fu« ABCD-‹ gu¥ò 60.5 r. myFfŸ.

6. , , ( , ) ,k‰W«h a b k0 0^ ^h h v‹gd xU ne®¡nfh£oš mikÍ« òŸëfŸ våš,

K¡nfhz¤Â‹ gu¥Ã‰fhd N¤Âu¤ij¥ ga‹gL¤Â 1ha

kb+ = vd ãWÎf.

Ï§F 0k‰W«h k ! .


Ô®Î: , , ( , )h a b0^ h k‰W« ,k0^ h M»a _‹W òŸëfS« xnu ne®¡nfh£oš

mikªJŸsd.

vdnt, h a

b k

h

21

0

0

0' 1 = 0

& ( 0) (0 0 )hb ak kh+ + - + + = 0& hb ak+ = kh

, 0h k ! v‹gjhš, ÏUòwK« hk Mš tF¡f, eh« bgWtJ ha

kb 1+ = .

7. xU K¡nfhz¤Â‹ KidfŸ , , , ,k‰W«0 1 2 1 0 3-^ ^ ^h h h våš, Ïj‹ g¡f§fë‹

eL¥òŸëfis Ïiz¤J cUth¡F« K¡nfhz¤Â‹ gu¥gsit¡ fh©f.

nkY«, Ï¢Á¿a K¡nfhz¤Â‹ gu¥gsé‰F«, bfhL¡f¥g£l K¡nfhz¤Â‹

gu¥gsé‰F« cŸs é»j¤ij¡ fh©f.

Ô®Î: , , , ,k‰W«A B C0 1 2 1 0 3-^ ^ ^h h h v‹gd K¡nfhz¤Â‹ c¢ÁfŸ. nkY« D, E, F v‹gd Kiwna BC, CA k‰W« AB-‹ eL¥òŸëfŸ v‹f.

BC-‹ eL¥òŸë D2

2 0 ,2

1 3 ( , )D 1 2+ + =` j .

AC-‹ eL¥òŸë E , ( , )E2

0 02

3 1 0 1+ - =` j .

AB-‹ eL¥òŸë F 0 , ( , )F22

21 1 1 0+ - + =` j .

vdnt, TDEF-‹ gu¥ò = 21 1

2

0

1

1

0

0

2' 1

= [( ) ( )]21 1 0 2 0 1 0+ + - + + =1 r. myFfŸ.

ABCT -‹ gu¥ò = 21 2

1

0

3

0

1

2

1-) 3

= [( 0) ( 2)]21 6 0 0 0+ + - + - = 4 r. myFfŸ.

Mfnt, DEFT k‰W« ABCT -‹ gu¥òfë‹ é»j« 1 : 4 MF«.

gæ‰Á 5.3 1. Ã‹tU« rhŒÎfis¡ bfh©l ne®¡nfhLfë‹ rhŒÎ¡ nfhz§fis¡ fh©f.

(i) 1 (ii) 3 (iii) 0Ô®Î: (i) ne®¡nfh£o‹ rhŒÎ, 1m = Mfnt, ne®¡nfh£o‹ rhŒÎ¡ nfhz« 1 45tan &i i= = c

(ii) ne®¡nfh£o‹ rhŒÎ, m 3=

Mfnt, ne®¡nfh£o‹ rhŒÎ¡ nfhz« 60tan 3 &i i= = c

(iii) ne®¡nfh£o‹ rhŒÎ, 0m =

Mfnt, ne®¡nfh£o‹ rhŒÎ¡ nfhz« 0 0tan &i i= = c

2. Ã‹tU« rhŒÎ¡ nfhz§fis¡ bfh©l ne®¡nfhLfë‹ rhŒÎfis¡ fh©f.

(i) 30c (ii) 60c (iii) 90c


Ô®Î: (i) rhŒÎ¡nfhz« 30i = c vd bfhL¡f¥g£LŸsJ.

Mfnt, ne®¡nfh£o‹ rhŒÎ, 30tan tanm3

1i= = =c .(ii) rhŒÎ¡nfhz« 06i = c. Mfnt, ne®¡nfh£o‹ rhŒÎ, 0tan tanm 6 3i= = =c .(iii) rhŒÎ¡nfhz« 09i = c. ne®¡nfh£o‹ rhŒÎ, 0tan tanm 9i= = =c tiuaW¡f¥glhjJ. Mfnt, ne®¡nfh£o‹ rhŒÎ tiuaW¡f¥gléšiy.

3. bfhL¡f¥g£l òŸëfŸ têna bršY« ne®¡nfh£o‹ rhŒÎfis¡ fh©f.

(i) (3 , -2), (7 , 2) (ii) (2 , -4) k‰W« MÂ¥òŸë (iii) ,1 3 2+^ h, ,3 3 4+^ h

Ô®Î: (i) ( , )x y1 1

k‰W« ( , )x y2 2

M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ

mx x

y y

2 1

2 1=-

-

Mfnt, (3, –2) k‰W« (7, 2) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ ( )

1.m7 3

2 2=

-- -

=

(ii) (2, –4) k‰W« (0, 0) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ

mx x

y y

2 1

2 1=-

- =

( )( )

2.0 20 4- +- -

=-

(iii) ,1 3 2+^ h, ,3 3 4+^ h M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ

mx x

y y

2 1

2 1=-

- =

( )1

3 3 1 3

4 2

3 3 1 3

222

+ - +

- =+ - -

= = .

4. bfhL¡f¥g£l òŸëfŸ tê¢ bršY« ne®¡nfhLfë‹ rhŒÎ¡ nfhz§fis¡

fh©f.

(i) ,1 2^ h, ,2 3^ h (ii) ,3 3^ h, ,0 0^ h (iii) (a , b), (-a , -b) Ô®Î: (i) ,1 2^ h, ,2 3^ h M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ

mx x

y y

2 1

2 1=-

- = 1

2 13 2-- =

Mfnt, rhŒÎ¡nfhz« 1tani = 45 .& i = c (ii) ,3 3^ h k‰W« ,0 0^ h M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ

m0 30 3

33

3

1=-- = =

Mfnt, rhŒÎ¡nfhz« tan3

1i = & 30 .i = c

(iii) (a , b) k‰W« (-a , -b) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ

ma ab b

ab

ab

22=

- -- - =

-- =

Mfnt, rhŒÎ¡nfhz«, tanabi = v‹w bjhl®ÃèUªJ »il¡»wJ.


5. MÂ¥ òŸë têahfÎ«, ,0 4-^ h k‰W« (8, 0) M»a òŸëfis Ïiz¡F«

nfh£L¤J©o‹ eL¥òŸë têahfÎ« bršY« nfh£o‹ rhŒit¡ fh©f.

Ô®Î: ,0 4-^ h k‰W« (8 , 0) M»a òŸëfis Ïiz¡F«

ne®¡nfh£L¤J©o‹ eL¥òŸë ,2

0 824 0+ - +` j = (4, 2)-

(4, –2) k‰W« (0, 0) M»a òŸë tê¢bršY«

ne®¡nfh£o‹ rhŒÎ ( ).m

0 40 2

21=

-- -

= -

6. rJu« ABCD-‹ g¡f« AB MdJ x-m¢R¡F Ïizahf cŸsJ våš,

Ã‹tUtdt‰iw¡ fh©f.

(i) AB-‹ rhŒÎ (ii) BC-‹ rhŒÎ (iii) _iyé£l« AC-‹ rhŒÎ

Ô®Î: (i) g¡f« AB MdJ x m¢R¡F Ïiz v‹gjhš, AB-‹ rhŒÎ, m 0= .

(ii) BC AB= v‹gjhš BC, x-m¢Rl‹ V‰gL¤J« nfhz« 90i = c.

rhŒÎ tanm 90= c, tiuaW¡f¥glhjJ

Mfnt, BC-‹ rhŒÎ tiuaW¡f¥gléšiy.(iii) _iyé£l« AC MdJ DAB+ -I ÏUrk¡ T¿L«.

vdnt, 45 . . ., 45BAC i e+ i= =c c

Mfnt, _iyé£l« AC-‹ rhŒÎ 45tan tanm 1i= = =c .

7. rkg¡f K¡nfhz« ABC-‹ g¡f« BC MdJ x-m¢Á‰F Ïiz våš, AB k‰W« BC M»at‰¿‹ rhŒÎfis¡ fh©f

Ô®Î: rkg¡f TABC-š g¡f« BC MdJ x-m¢R¡F Ïiz. nkY« .ABC 60+ = c vdnt, g¡f« AB-‹ rhŒÎ¡nfhz« 60c. AB-‹ rhŒÎ, 60tanm 3= =c .nkY«, BC MdJ x-m¢R¡F Ïiz v‹gjhš

BC-‹ rhŒÎ, 0 0tanm = =c . 8. rhŒéid¥ ga‹gL¤Â, Ñœ¡f©l òŸëfŸ xnu ne®¡nfh£oš mikÍ« vd

ãWÎf.

(i) (2 , 3), (3 , -1) k‰W« (4 , -5) (ii) (4 , 1), (-2 , -3) k‰W« (-5 , -5) (iii) (4 , 4), (-2 , 6) k‰W« (1 , 5)

Ô®Î: (i) A(2 , 3), B(3, -1) k‰W« C(4, -5) v‹gd bfhL¡f¥g£l òŸëfŸ v‹f.

AB-‹ rhŒÎ 4m3 21 3

1=

-- - =-

BC-‹ rhŒÎ 4m4 35 1

2=

-- + =-

AB-‹ rhŒÎ = BC-‹ rhŒÎ. nkY« B v‹gJ bghJ¥òŸë.

Mfnt, A, B k‰W« C M»ait xnu ne®¡nfh£oš mikÍ« òŸëfshF«.


(ii) A(4, 1), B(-2, -3) k‰W« C(-5, -5) v‹gd bfhL¡f¥g£l òŸëfŸ v‹f.

AB-‹ rhŒÎ m2 43 1

64

32

1=

- -- - =

-- = .

BC-‹ rhŒÎ m5 25 3

32

32

2=

- +- + =

-- = .



(iii) A(4, 4), B(-2, 6) k‰W« C(1, 5) v‹gd bfhL¡f¥g£l òŸëfŸ v‹f

AB-‹ rhŒÎ .m2 46 4

62

31

1=

- -- =

-=-


31

2=

+- =- .



9. (a, 1), (1, 2) k‰W« (0, b + 1) v‹gd xnu ne®¡nfh£oš mikÍ« òŸëfŸ våš,

a b1 1+ = 1 vd ãWÎf.

Ô®Î: ( , 1), (1, 2)A a B k‰W« (0, 1)C b+ v‹gd bfhL¡f¥g£l òŸëfŸ v‹f

AB-‹ rhŒÎ .ma a1

2 111

1=

-- =

-

BC-‹ rhŒÎ m b b0 11 2

11

2=

-+ - =

--

_‹W òŸëfS« xnu ne®¡nfh£oš mikÍ« v‹gjhš m m1 2= .

a

b11

11&

-=

-- a b ab& + =

ÏUòwK« ab Mš tF¡f, eh« bgWtJ 1a b1 1+ = .

10. A(–2, 3), B(a, 5) M»a òŸëfis Ïiz¡F« ne®¡nfhL k‰W« C(0, 5), D(–2, 1) M»a òŸëfis Ïiz¡F« ne®¡nfhL M»ad Ïiz nfhLfŸ våš, a-‹

kÂ¥ig¡ fh©f.

Ô®Î: ne®¡nfhLfŸ AB k‰W« CD Ïiz v‹gjhš, mj‹ rhŒÎfŸ rk«

Mfnt, AB-‹ rhŒÎ = CD-‹ rhŒÎ.

a 25 3

2 01 5&

+- =

- -- 2 1a& + = 1a& =-

Mfnt, a-‹ kÂ¥ò –1.

11. A(0, 5), B(4, 2) v‹w òŸëfis Ïiz¡F« ne®¡nfhlhdJ, C(-1, -2), D (5, b) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‰F¢ br§F¤J våš, b-‹ kÂ¥ig¡

fh©f.

Ô®Î: AB-‹ rhŒÎ m1 =

4 02 5

43

-- = -


CD-‹ rhŒÎ m2 = b b

5 12

62

++ = +

ne®¡nfhLfŸ AB k‰W« CD v‹gd br§F¤J v‹gjhš m m1 2

= 1-

& 1b43

62- + =-` `j j 2 8 6.b b& &+ = =

Mfnt, b-‹ kÂ¥ò 6.

12. TABC-‹ KidfŸ A(1, 8), B(-2, 4), C(8, -5). nkY«, M, N v‹gd Kiwna AB, AC Ït‰¿‹ eL¥òŸëfŸ våš, MN-‹ rhŒit¡ fh©f. Ïij¡ bfh©L MN k‰W« BC M»a ne®¡nfhLfŸ Ïiz vd¡ fh£Lf.

Ô®Î: A(1, 8), B(-2, 4), C(8, -5) v‹gd K¡nfhz¤Â‹ KidfŸ.

AB-‹ eL¥òŸë ,M2

1 22

8 4- +` j , 6M21= -` j

AC-‹ eL¥òŸë , ,N N2

8 125 8

29

23+ - + =` `j j

MN-‹rhŒÎ m6

109

1

29

21

23

2102

3 12

=+

-= = -

- g (1)


109

2=

+- - = - g (2)

(1) k‰W« (2)-èUªJ, m m1 2= .

Mfnt, ne®¡nfhLfŸ BC k‰W« MN M»ait ÏizahF«.

13. (6 , 7), (2 , -9) k‰W« (-4 , 1) M»ad xU K¡nfhz¤Â‹ KidfŸ våš,

K¡nfhz¤Â‹ eL¡nfhLfë‹ rhŒÎfis¡ fh©f.

Ô®Î: A(6 , 7), B(2 , -9) k‰W« C(-4 , 1) M»ait K¡nfhz¤Â‹ KidfŸ

v‹f. nkY« D, E, F v‹gd Kinwa BC, CA k‰W« AB-‹ eL¥òŸëfŸ v‹f.

Mfnt, AD, BE k‰W« CF M»ait ABCT eL¡nfhLfshF«.

BC-‹ eL¥òŸë D , ( 1, 4) .D2

2 429 1- - + = - -` j

CA-‹ eL¥òŸë E , (1, 4) .E24 6

21 7- + + =` j

AB-‹ eL¥òŸë , (4, 1) .F F2

6 22

7 9+ - = -` j

vdnt, AD-‹ rhŒÎ = 1 64 7

711

711

- -- - =

-- = ,

BE-‹rhŒÎ = 1 24 9

113 13

-+ =

-=-

CF-‹ rhŒÎ = 4 41 1

82

41

+- - = - = - .

Mfnt, eL¡nfhLfë‹ rhŒÎfŸ , 13711 - k‰W«

41-


14. A(-5, 7), B(-4, -5) k‰W« C(4, 5) M»ad TABC-‹ KidfŸ våš,

K¡nfhz¤Â‹ F¤Jau§fë‹ rhŒÎfis¡ fh©f.

Ô®Î: A(-5, 7), B(-4, -5) k‰W« C(4, 5) M»ad TABC-‹ KidfshF«.

AD, BE k‰W« CF M»ad ABCT -‹ F¤J¡nfhLfŸ v‹f.

AB-‹ rhŒÎ = 124 55 7

- +- - =-

F¤J¡nfhL CF MdJ AB-¡F br§F¤J v‹gjhš,

CF-‹ rhŒÎ = 121 .

BC-‹ rhŒÎ = 4 45 5

810

45

++ = = .

F¤J¡nfhL AD MdJ BC-¡F br§F¤J v‹gjhš, AD-‹ rhŒÎ = .

54-

AC-‹ rhŒÎ 4 55 7

92

+- = - .

F¤J¡nfhL BE MdJ CA-¡F br§F¤J v‹gjhš, BE-‹ rhŒÎ = .29

Mfnt, F¤J¡nfhLfë‹ rhŒÎfŸ , , .121

54

29-

15. rhŒéid¥ ga‹gL¤Â (1 , 2), (-2 , 2), (-4 , -3) k‰W« (-1, -3) v‹gd mnj

tçiræš X® Ïizfu¤ij mik¡F« vd ãWÎf.

Ô®Î: bfhL¡f¥g£l òŸëfis tiugl¤Âš fofhu KŸnsh£l Âir¡F

vÂ®Âiræš tçirahf mikÍkhW F¿¡fÎ«.

A(-4 , -3), B(-1, -3), C(1 , 2) k‰W« D(-2 , 2) M»ait Ïizfu¤Â‹

KidfŸ v‹f.

AB-‹ rhŒÎ = 01 43 3

- +- + = ; CD-‹ rhŒÎ = 0

2 12 2- -

- = .

AB k‰W« CD M»at‰¿‹ rhŒÎfŸ rk« v‹gjhš,

AB MdJ CD-¡F ÏizahF«. g (1)

BC-‹ rhŒÎ = 1 12 3

25

++ = ; AD-‹ rhŒÎ =

2 42 3

25

- ++ = .

BC k‰W« AD M»at‰¿‹ rhŒÎfŸ rk« v‹gjhš,

BC MdJ AD-¡F ÏizahF«. g (2)

(1) k‰W« (2) M»at‰¿èUªJ, eh‰fu« ABCD-æ‹ vÂ® g¡f§fŸ Ïiz

v‹gjhš ABCD X® Ïizfu«.

16. ABCD v‹w eh‰fu¤Â‹ KidfŸ Kiwna A(-2 ,-4), B(5 , -1), C(6 , 4) k‰W«

D(-1, 1) våš, Ïj‹ vÂ®¥ g¡f§fŸ Ïiz vd¡ fh£Lf.


Ô®Î: A(-2,-4), B(5 , -1), C(6 , 4) k‰W« D(-1, 1) M»ait eh‰fu¤Â‹

KidfŸ vd bfhL¡f¥g£LŸsJ.

AB-‹ rhŒÎ = 5 21 4

73

+- + = ; CD-‹ rhŒÎ =

1 61 4

73

73

- -- =

-- = .

AB k‰W« CD-‹ rhŒÎfŸ rk« v‹gjhš, AB MdJ CD-¡F Ïiz. g (1)

AD-‹ rhŒÎ = 51 21 4

- ++ = ; BC-‹ rhŒÎ = 5.

6 54 1-+ =

BC k‰W« AD-‹ rhŒÎfŸ rk« v‹gjhš, BC MdJ AD-¡F Ïiz. g (2)

(1) k‰W« (2) M»at‰¿èUªJ, eh‰fu« ABCD-æ‹ vÂ® g¡f§fŸ Ïiz

v‹gjhš ABCD X® Ïizfu«.

gæ‰Á 5.4

1. x-m¢ÁèUªJ 5 myFfŸ bjhiyéš cŸsJ« x-m¢R¡F ÏizahdJkhd

ne®¡nfhLfë‹ rk‹ghLfis¡ fh©f.

Ô®Î: x-m¢R¡F Ïizahd ne®¡nfh£o‹ rk‹ghL y k= .

x-m¢R¡F ÏizahfÎ« x-m¢ÁèUªJ 5 myF öu¤Âš mikªJŸsJkhd

ne®¡nfh£o‹ rk‹ghLfŸ , .y y5 5= =-

2. (-5, -2) v‹w òŸë tê¢ brštJ« Mam¢RfS¡F ÏizahdJkhd


Ô®Î: x-m¢R¡F ÏizahfÎ« (–5, –2) v‹w òŸë tê¢brštJkhd ne®¡nfh£o‹

rk‹ghL 2y =- .

y-m¢R¡F ÏizahfÎ« (–5, –2) v‹w òŸë tê¢brštJkhd ne®¡nfh£o‹

rk‹ghL x 5=- .

3. ÑnH¡ bfhL¡f¥g£LŸs étu§fS¡F ne®¡nfhLfë‹ rk‹ghLfis¡ fh©f.

(i) rhŒÎ -3; y-bt£L¤J©L 4. (ii) rhŒÎ¡nfhz« 60c, y-bt£L¤J©L 3.

Ô®Î:

(i) rhŒÎ 3m =- k‰W« y-bt£L¤J©L 4c = vd bfhL¡f¥g£LŸsJ. rhŒÎ-bt£L¤J©L mik¥Ãš ne®¡nfh£o‹ rk‹ghL .y mx c= +

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL

3 4y x=- + mšyJ x y3 4 0+ - =

(ii) rhŒÎ¡nfhz« i = 60c k‰W« y-bt£L¤J©L = 3 vd bfhL¡f¥g£LŸsJ. Mfnt, rhŒÎ 60tanm 3= =c rhŒÎ-bt£L¤J©L mik¥Ãš ne®¡nfh£o‹ rk‹ghL .y mx c= +

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL 3y x3= + mšyJ 3 0x y 3- + =


4. xU ne®¡nfhL y-m¢ir MÂ¥òŸë¡F nkyhf 3 myFfŸ öu¤Âš bt£L»wJ

k‰W« tan21i = (i v‹gJ ne®¡nfh£o‹ rhŒÎ¡nfhz«) våš, mªj

ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

Ô®Î: rhŒÎ tanm21i= = , y-bt£L¤J©L c 3= vd bfhL¡f¥g£LŸsJ.

rhŒÎ - bt£L¤J©L mik¥Ãš ne®¡nfh£o‹ rk‹ghL .y mx c= +

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL 3y x21= + & x y2 6 0- + = .

5. Ã‹tU« ne®¡nfhLfë‹ rhŒÎ k‰W« y bt£L¤J©L M»adt‰iw¡ fh©f.

(i) 1y x= + , (ii) 5 3x y= (iii) 4 2 1 0x y- + = , (iv) 10 15 6 0x y+ + = .

Ô®Î: (i) y x 1= + I y mx c= + v‹w ne®¡nfh£L rk‹gh£Ll‹ x¥Ãl, eh«

bgWtJ rhŒÎ m 1= k‰W« y-bt£L¤J©L 1c = MF«.

(ii) x y5 3= v‹w rk‹gh£il y x35= v‹wthW kh‰¿aik¤J y mx c= + v‹w

ne®¡nfh£L rk‹gh£Ll‹ x¥Ãl,

rhŒÎ m35= k‰W« y-bt£L¤J©L 0c = MF«.

(iii) x y4 2 1 0- + = 2 4 1 2 .y x y x21& &= + = +

2y x21= + I y mx c= + v‹w rk‹gh£Ll‹ x¥Ãl, eh« bgWtJ,

rhŒÎ m 2= k‰W« y-bt£L¤J©L c21= MF«.

(iv) x y10 15 6 0+ + = .y x32

52& = - -

y x32

52= - - I y mx c= + v‹w rk‹gh£Ll‹ x¥Ãl, eh« bgWtJ

rhŒÎ m32= - k‰W« y-bt£L¤J©L c

52= - MF«.

6. Ã‹tU« étu§fS¡F, ne®¡nfhLfë‹ rk‹ghLfis¡ fh©f.

(i) rhŒÎ -4 ; (1, 2) v‹w òŸë tê¢ brš»wJ.

(ii) rhŒÎ 32 ; (5, -4) v‹w òŸë tê¢ brš»wJ.

Ô®Î: (i) rhŒÎ m 4=- k‰W« òŸë ( , ) (1, 2)x y1 1

= vd bfhL¡f¥g£LŸsJ.

rhŒÎ - òŸë mik¥Ãš ne®¡nfh£o‹ rk‹ghL ( )y y m x x1 1

- = -

2 4( 1) 4 6 0.y x x y& &- =- - + - =

(ii) rhŒÎ m32= k‰W« òŸë ( , ) (5, 4)x y

1 1= - vd bfhL¡f¥g£LŸsJ.


- = - . 4 ( 5) 3( 4) 2( 5) 2 3 22 0y x y x x y

32& & &+ = - + = - - - = .

7. rhŒÎ¡ nfhz« 30c bfh©l k‰W« (4, 2), (3, 1) M»a òŸëfis Ïiz¡F«

ne®¡nfh£L¤ J©o‹ eL¥òŸë tê¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡

fh©f.

Ô®Î: rhŒÎ¡nfhz« 30i = c vd bfhL¡f¥g£LŸsJ.


Mfnt, rhŒÎ 3

tanm 30 1= =c .

(4, 2) k‰W« (3, 1) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ ika¥òŸë

( , ) , , .x y2

4 32

2 127

23

1 1= + + =` `j j


- = - .

& ( )y x y x23

3

127 3 2 3 2 7&- = - - = -^ h

2 2 (3 7) 0.x y3 3& - + - =

8. Ã‹tU« òŸëfë‹ tê¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

(i) (– 2, 5) k‰W« (3, 6) (ii) (0, – 6) k‰W« (– 8, 2).

Ô®Î: (i) bfhL¡f¥g£l òŸëfis ( , ) ( 2,5), ( , ) (3,6)x y x y1 1 2 2

= - = v‹f.

Ïu©L òŸëfŸ tê¢bršY« ne®¡nfh£o‹ rk‹ghL y y

y y

x x

x x

2 1

1

2 1

1

-

-=

-

-.

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL y x6 5

53 2

2--

=++

y x15

52&

-= + x y5 27 0& - + = .

(ii) bfhL¡f¥g£l òŸëfis ( , ) (0, 6), ( , ) ( 8,2)x y x y1 1 2 2

= - = - v‹f.

Ïu©L òŸëfŸ tê¢bršY« ne®¡nfh£o‹ rk‹ghL y y

y y

x x

x x

2 1

1

2 1

1

-

-=

-

-.

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL y x2 6

68 0

0++

=- -

- 6 0.x y& + + =

9. P(1, -3), Q(-2, 5) k‰W« R(-3, 4) M»a Kidfis¡ bfh©l TPQR-š Kid

R-èUªJ tiua¥gL« eL¡nfh£o‹ rk‹gh£il¡ fh©f.

Ô®Î : P(1, -3), Q(-2, 5) k‰W« R(-3, 4) M»ait TPQR-‹ KidfŸ. M v‹gJ

PQ-‹ eL¥òŸë v‹f.

vdnt, , ( ,1) .M M2

1 223 5

21- - + = -` j

R(-3, 4) k‰W« ,M21 1-` j M»a òŸëfis

Ïiz¡F« eL¡nfhL RM-‹ rk‹ghL

( )y x1 4

4

3

3

21-

-=

- -

- -- ^ h

( )y x34

52 3

&--

=+ 6 5 2 0.x y& + - =

10. ne®¡nfh£o‹ rk‹ghL fhQ« Kiwia¥ ga‹gL¤Â, Ã‹tU« òŸëfŸ

xnune®¡nfh£oš mikÍ« vd¡ fh£Lf.

(i) (4, 2), (7, 5) k‰W« (9, 7) (ii) (1, 4), (3, -2) k‰W« (-3, 16).Ô®Î: (i) (4, 2), (7, 5) k‰W« (9, 7) M»ad bfhL¡f¥g£l òŸëfŸ.

(4, 2), (7, 5) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rk‹ghL


y x5 2

27 4

4--

=-- 2 0.x y& - - = g (1)

_‹whtJ òŸë (9, 7)I (1)š ÃuÂæl, 9 7 2 0- - = MF«.

Mfnt, bfhL¡f¥g£l _‹W òŸëfS« xnu ne®¡nfh£oš mikÍ«.

(ii) (1, 4), (3, -2) k‰W« (-3,16) M»ad bfhL¡f¥g£l òŸëfŸ.

(1, 4), (3, -2) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rk‹ghL

y x2 4

43 1

1 &- -

-=

-- 6 2 14 0.x y+ - = 3 7 0.x y& + - = g (1)

_‹whtJ òŸë (-3, 16) I (1)š ÃuÂæl, ( ) .3 3 16 7 0- + - =

vdnt, (-3, 16) v‹w òŸëahdJ (1, 4), (3, –2) v‹w òŸëfis Ïiz¡F«

ne®nfh£o‹ ÛJ mikÍ«.

Mfnt, bfhL¡f¥g£l _‹W òŸëfS« xnu ne®¡nfh£oš mikÍ«.

11. ÑnH bfhL¡f¥g£LŸs x, y-bt£L¤J©Lfis¡ bfh©l ne®¡nfhLfë‹

rk‹ghLfis¡ fh©f.

(i) 2 k‰W« 3 (ii) 31- k‰W«

23 (iii)

52 k‰W«

43- .

Ô®Î: (i) x-bt£L¤J©L, a 2= k‰W« y-bt£L¤J©L, b 3= vd bfhL¡f¥

g£LŸsJ

bt£L¤J©L mik¥Ãš ne®¡nfh£o‹ rk‹ghL 1.ax

by

+ =

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL 1x y2 3+ = 3 2 6 0.x y& + - =

(ii) x-bt£L¤J©L, a31=- k‰W« y-bt£L¤J©L, b

23= vd bfhL¡f¥

g£LŸsJ

njitahd ne®¡nfh£o‹ rk‹ghL 1.ax

by

+ =

1x y

31

23-

+ = 9 2 3 0x y& - + =

(iii) x-bt£L¤J©L, a52= k‰W« y-bt£L¤J©L, b

43=- vd bfhL¡f¥

g£LŸsJ

bt£L¤J©L mik¥Ãš ne®¡nfh£o‹ rk‹ghL 1.ax

by

+ =

1x y

52

43

& +-

= 15 8 6 0.x y& - - =

12. ÑnH¡ bfhL¡f¥g£LŸs ne®¡nfhLfë‹ rk‹ghLfëèUªJ x, y-bt£L¤

J©Lfis¡ fh©f.

(i) 5 3 15 0x y+ - = (ii) 2 16 0x y- + = (iii) 3 10 4 0x y+ + =

Ô®Î: (i) bfhL¡f¥g£l rk‹ghL x y5 3 15 0+ - = 5 3 15.x y& + =


ÏUòwK« 15 Mš tF¡f, 1x y3 5+ = g (1)

(1) I bt£L¤J©L mik¥Ãyhd ne®¡nfh£L rk‹ghLl‹ 1ax

by

+ = cl‹

x¥Ãl eh« bgWtJ,

x-bt£L¤J©L, a 3= k‰W« y-bt£L¤J©L, b 5= .

(ii) bfhL¡f¥g£l rk‹ghL 2 1 0x y 6- + = 2 16x y& - =-

ÏUòwK« –16 Mš tF¡f, 1x y8 16-+ = g (1)


by

+ = cl‹

x¥Ãl eh« bgWtJ,

x-bt£L¤J©L, a 8=- k‰W« y-bt£L¤J©L, b 16= .

(iii) bfhL¡f¥g£l rk‹ghL x y3 10 4 0+ + = 3 10 4x y& + =-

ÏUòwK« –4 Mš tF¡f, 1 1x y x y43

410

34

52

&-

+-

=-

+-

=^ ^h h

g (1)


by

+ = cl‹

x¥Ãl eh« bgWtJ,

x-bt£L¤J©L, a34=- k‰W« y-bt£L¤J©L,

5b 2=- .

F¿¥ò: ne®¡nfh£L rk‹gh£oš y 0= vd ÃuÂæl x-bt£L¤J©o‹ kÂ¥igÍ«,

0x = vd ÃuÂæl y-bt£L¤J©o‹ kÂ¥igÍ« bgwyh«.

13. (3, 4) v‹w òŸë tê¢ brštJ«, bt£L¤J©Lfë‹ é»j« 3 : 2 vd cŸsJkhd


Ô®Î: a, b v‹gd Kiwna ne®¡nfh£o‹ x k‰W« y-‹ bt£L¤J©LfŸ v‹f.

Mfnt, a : b = 3 : 2. nkY« a = 3k k‰W« b = 2k v‹f. (k v‹gJ ó{íak‰w kh¿è)

Mfnt, bt£L¤J©L mik¥Ãyhd ne®¡nfh£L rk‹ghL

1kx

ky

3 2+ = g (1)

Ï¡nfhL (3, 4) v‹w òŸë tê¢brštjhš, eh« bgWtJ

1 1 3.k k k k

k33

24 1 2& &+ = + = =

k 3= vd (1)-š ÃuÂæl, 1x y9 6+ = 2 3 18 0.x y& + - =

14. (2, 2) v‹w òŸë tê¢ brštJ«, bt£L¤J©Lfë‹ TLjš 9 MfÎ« bfh©l


Ô®Î: a, b v‹gd Kiwna ne®¡nfh£o‹ x k‰W« y-‹ bt£L¤J©LfŸ v‹f

Mfnt, a+b = 9 mšyJ .b a9= -

Mfnt, bt£L¤J©L mik¥Ãyhd ne®¡nfh£o‹ rk‹gh£o‹ go


1ax

ay

9+

-= g (1)

Ï¡nfhL (2, 2) v‹w òŸë tê¢brštjhš, eh« bgWtJ

1a a2

92+-

= 9 18 0a a2& - + =

( 3)( 6) 0a a& - - = Mfnt, 3a = mšyJ 6a =

a 3= vD« nghJ (1)-èUªJ, 1 2 6 0.x yx y

3 6&+ = + - =

a 6= vD« nghJ (1)-èUªJ, 1 2 6 0.x yx y

6 3&+ = + - =

15. (5, -3) v‹w òŸë têahf¢ bršY«, mséš rkkhfÎ«, Mdhš F¿ bt›ntwhfÎ«

cŸs bt£L¤ J©Lfis¡ bfh©l ne®¡nfh£o‹ rk‹gh£oid¡ fh©f.

Ô®Î: a v‹gJ ne®¡nfh£o‹ x-bt£L¤J©L v‹f. Mfnt, y-bt£L¤J©L a-

vdnt, bt£L¤J©L ne®¡nfh£o‹ rk‹ghL

1ax

ay

x y a&+-

= - = (1)g

Ï¡nfhL (5, –3) v‹w òŸë tê¢brštjhš, 5 3 8.a = + =

8a = vd (1)-š ÃuÂæl, njitahd ne®¡nfh£o‹ rk‹ghL 8 0.x y- - =

16. (9, -1) v‹w òŸë tê¢ brštJ« x-bt£L¤J©lhdJ, y-bt£L¤J©o‹ msit¥

nghš K«kl§F bfh©lJkhd ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

Ô®Î: bt£L¤J©L mik¥Ãš ne®¡nfh£o‹ rk‹ghL 1ax

by

+ = .

x-bt£L¤J©lhdJ, y-bt£L¤J©oid¥ nghš K«kl§F v‹gjhš, a b3= .

Mfnt, ne®¡nfh£o‹ rk‹ghL, 1 3 3 .bx

by

x y b3

&+ = + = g (1)

Ï¡nfhL (9, –1) v‹w òŸë tê¢ brštjhš, 3 2.b b9 3 &= - =

b = 2 vd (1)-š ÃuÂæl, njitahd ne®¡nfh£o‹ rk‹ghL 3 6 0.x y+ - =

17. xU ne®¡nfhL Mam¢Rfis A k‰W« B M»a òŸëfëš bt£L»‹wJ. AB-‹

eL¥òŸë (3, 2) våš, AB-‹ rk‹gh£il¡ fh©f.

Ô®Î: bt£L¤J©L mik¥Ãš ne®¡nfh£o‹ rk‹ghL 1ax

by

+ = g (1)

x-m¢R (1)I A vD« òŸëæl¤J bt£L« nghJ, y = 0.

vdnt, (1) 1 .ax x a& &= = Mfnt, A v‹gJ (a, 0).

nkY« y-m¢R (1)I B vD« òŸëæl¤J bt£L« nghJ, x = 0.

vdnt, (1)-èUªJ 1 .by

y b&= = Mfnt B v‹gJ (0, b).

AB -‹ ika¥òŸë , (3,2)a b20

20+ + =` j


3 6a a20 &+ = = k‰W« 2 4b b

20 &+ = =

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL xy

6 41+ =

& 2 3 12x y+ = mšyJ 2 3 12 0x y+ - = .

18. x-bt£L¤J©lhdJ y-bt£L¤J©o‹ msit él 5 myFfŸ mÂfkhf¡

bfh©l xU ne®¡nfhlhdJ (22, -6) v‹w òŸë tê¢ brš»‹wJ våš,

m¡nfh£o‹ rk‹gh£il¡ fh©f.

Ô®Î: y-bt£L¤J©L a v‹f. vdnt x-bt£L¤J©L a 5+ .

Mfnt, bt£L¤J©L mik¥Ãš ne®¡nfh£o‹ rk‹ghL 1ax

ay

5++ = g (1)

Ï¡nfhL (22, –6) v‹w òŸë tê¢ brštjhš a a522 6 1+

- =

( )( )

1a a

a a5

22 6 5&

+- +

= 11 30a a 02& - + = 5a& = k‰W« 6a =

5a = vD« nghJ, (1)-èUªJ 1 2 10 0x yx y

10 5&+ = + - = .

a 6= vD« nghJ, (1)-èUªJ 1 6 11 66 0x yx y

11 6&+ = + - = .

Mfnt, njitahd ne®¡nfhLfë‹ rk‹ghLfŸ

2 10 0, 6 11 66 0.x y x y+ - = + - =

19. ABCD v‹w rhŒrJu¤Â‹ ÏU KidfŸ A(3, 6) k‰W« C(-1, 2) våš, mj‹ _iy

é£l« BD têahf¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

Ô®Î: rhŒrJu¤Â‹ _iyé£l§fŸ AC k‰W« BD x‹iwbah‹W ÏUrk¡ T¿L«

k‰W« br§F¤jhf bt£o¡bfhŸS«.

AC-‹ rhŒÎ = 1 32 6 1

- -- = . Mfnt, BD-‹ rhŒÎ 1=- .

AC-‹ ika¥òŸë = , (1, 4) .2

3 12

6 2- + =` j

Mfnt, rhŒÎ –1MfÎ« (1, 4) v‹w òŸë têahfÎ« bršY« ne®nfhL BD-‹

rk‹ghL

4 1( 1) 4 1y x y x x y 5 0& &- =- - - =- + + - = .

20. A(-2, 6), B(3, -4) M»a òŸëfis Ïiz¡F« ne®¡nfh£L¤ J©il P v‹w

òŸë c£òwkhf 2 : 3 v‹w é»j¤Âš Ãç¡»wJ. òŸë P têahf¢ bršY«

rhŒÎ 23 cila, ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

Ô®Î: AB-I c£òwkhf 2 : 3 v‹w é»j¤Âš Ãç¡F« òŸë P v‹f.

Mfnt òŸë ,Pl m

lx mx

l m

ly my2 1 2 1

+

+

+

+c m = ( ) ( )

,( ) ( )

2 32 3 3 2

2 32 4 3 6

++ -

+- +

c m (0, 2)= .

rhŒÎ 23 MfÎ« (0, 2) v‹w òŸë têahfÎ« bršY« ne®nfh£o‹ rk‹ghL

2 ( 0) 2 4 3 3 2 4 0y x y x x y23 & &- = - - = - + = .


gæ‰Á 5.5

1. Ã‹tU« ne®¡nfhLfë‹ rhŒÎfis¡ fh©f.

(i) 3 4 6 0x y+ - = , (ii) 7 6y x= + , (iii) 4 5 3x y= + .

Ô®Î: (i) 0ax by c+ + = v‹w ne®¡nfh£o‹ rhŒÎ ba-

Mfnt, x y3 4 6 0+ - = v‹w ne®¡nfh£o‹ rhŒÎ = ba

43- =- .

(ii) 0ax by c+ + = v‹w ne®¡nfh£o‹ rhŒÎ ba-

Mfnt, 7 6 0x y- + = v‹w ne®¡nfh£o‹ rhŒÎ = ba

17 7- =

-- = .

(iii) 0ax by c+ + = v‹w ne®¡nfh£o‹ rhŒÎ ba-

Mfnt, 4 5 3 0x y- - = v‹w ne®¡nfh£o‹ rhŒÎ = ba

54

54- =

-- = .

2. 2 1 0x y+ + = k‰W« 3 6 2 0x y+ + = M»a ne®¡nfhLfŸ Ïiz vd ãWÎf.

Ô®Î: x y2 1 0+ + = v‹w ne®¡nfh£o‹ rhŒÎ .m21

1=-

x y3 6 2 0+ + = v‹w ne®¡nfh£o‹ rhŒÎ .m21

2=-

rhŒÎfŸ m m1 2= v‹gjhš Ïu©L ne®nfhLfS« ÏizahF«.

kh‰WKiw: ;a

a

b

b

a

a

b

b

31

62

31

2

1

2 2

1

2

1 1&= = = = .

vdnt, ne®nfhLfŸ ÏizahF«.

3. 3 5 7 0x y- + = , 15 9 4 0x y+ + = M»a ne®¡nfhLfŸ x‹W¡F x‹W br§F¤J

vd ãWÎf.

Ô®Î: x y3 5 7 0- + = v‹w ne®¡nfh£o‹ rhŒÎ .m53

53

1=

-- =

x y15 9 4 0+ + = v‹w ne®¡nfh£o‹ rhŒÎ .m915

35

2= - = -

1.m m53

35

1 2# = - =-` `j j Mfnt, ne®nfhLfŸ x‹W¡bfh‹W br§F¤J.

F¿¥ò: 3(15) ( 5)(9) 0a a b b1 2 1 2

+ = + - = . Mfnt, ne®nfhLfŸ br§F¤jhF«.

4. 5 3k‰W«yx p ax y

2= - + = v‹gd Ïiz våš, a-‹ kÂ¥ig¡ fh©f.

Ô®Î: y x p2= - k‰W« 5 3ax y+ = M»ait bfhL¡f¥g£l ne®¡nfhLfŸ.

0xy

p2

& - - = k‰W« 3 5 0ax y- + =

ne®¡nfhLfŸ Ïiz v‹gjhš, 6.a

a

b

b

aa1

32

1

2

21

1 & &= =-

-=

5. 5 2 9 0x y- - = , 2 11 0ay x+ - = M»a ne®¡nfhLfŸ x‹W¡F x‹W br§F¤J

våš, a-‹ kÂ¥ig¡ fh©f.

Ô®Î: x y5 2 9 0- - = v‹w ne®¡nfh£o‹ rhŒÎ .m25

25

1=

-- =


11 0ay x2+ - = v‹w ne®¡nfh£o‹ rhŒÎ .ma2

2= -

ne®¡nfhLfŸ x‹W¡bfh‹W br§F¤J v‹gjhš,

1 1 5.m ma

a25 2

1 2 & &=- - =- =` `j j

Mfnt, a-‹ kÂ¥ò 5.

6. 8 1 0px p y2 3+ - + =^ h , 8 7 0px y+ - = M»ad br§F¤J ne®¡nfhLfŸ våš,

p-‹ kÂ¥òfis¡ fh©f.

Ô®Î: px p y8 2 3 1 0+ - + =^ h v‹w ne®¡nfh£o‹ rhŒÎ .mpp

2 38

1=

--

px y8 7 0+ - = v‹w ne®¡nfh£o‹ rhŒÎ m p82

=- .

ne®¡nfhLfŸ x‹W¡bfh‹W br§F¤J v‹gjhš,

1 1m mpp p

2 38

81 2&=-

-- -

=-c `m j

3 2 0 ( 1)( 2) 0 1, 2.p p p p p2& & &- + = - - = =

7. ,h 3^ h, (4, 1) M»a òŸëfis Ïiz¡F« ne®¡nfhL«, 7 9 19 0x y- - = v‹w

ne®¡nfhL« br§F¤jhf bt£o¡ bfhŸ»‹wd våš, h-‹ kÂ¥ig¡ fh©f.

Ô®Î: ,h 3^ h k‰W« (4, 1) M»a òŸëfis Ïiz¡F«

ne®¡nfh£o‹ rhŒÎ

.mh h4

1 34

21=

-- =

--

x y7 9 19 0- - = v‹w ne®¡nfh£o‹ rhŒÎ .m97

2=

ne®¡nfhLfŸ x‹W¡bfh‹W br§F¤J v‹gjhš

1 1 1m mh h42

97

36 914

1 2& & &=-

-- =-

-- =-` `j j h

922= .

8. 3 7 0x y- + = v‹w ne®¡nfh£o‰F ÏizahdJ« (1, -2) v‹w òŸë tê¢

brštJkhd ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

Ô®Î: x y3 7 0- + = v‹w ne®¡nfh£o‰F Ïizahf bršY« ne®¡nfh£o‹

rk‹ghL 3 0.x y k- + =

Ï¡nfhL (1, –2) v‹w òŸë tê¢brštjhš, eh« bgWtJ 3(1) ( 2) 0 5.k k&- - + = =-

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL 3 0x y 5- - =

9. 2 3 0x y- + = v‹w ne®¡nfh£o‰F¢ br§F¤jhdJ« (1, -2) v‹w òŸë tê¢


Ô®Î: x y2 3 0- + = v‹w ne®¡nfh£o‰F br§F¤jhd

ne®¡nfh£o‹ rk‹ghL 0x y k2 + + = .Ï¡nfhL, (1, –2) v‹w òŸë tê¢brštjhš

2(1) (2) 0 0.k k&- + = =

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL 0x y2 + = .


10. (3, 4), (-1, 2) v‹w òŸëfis Ïiz¡F« ne®¡nfh£L¤J©o‹ ika¡

F¤J¡nfh£o‹ (perpendicular bisector) rk‹gh£il¡ fh©f.

Ô®Î: A(3, 4) k‰W« B(-1, 2) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ eL¥òŸë

, , .2

3 12

4 2 1 3- + =` ^j h

(3, 4) k‰W« (–1, 2) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rk‹ghL

4 16 2 6y x y x2 4

41 3

3 & &--

=- -

- - + =- + 2 5 0.x y- + =

Mfnt, 2 0x y 5- + = v‹w ne®¡nfh£o‰F br§F¤jhd ne®¡nfh£o‹ rk‹ghL

0x y k2 + + = .Ï¡nfhL (1, 3) v‹w òŸë tê¢brštjhš,

2(1) 3 0 5.k k&+ + = =-

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL 0x y2 5+ - = .F¿¥ò: rhŒÎ-òŸë mik¥ò thŒ¥gh£oid ga‹gL¤ÂÍ« njitahd

ne®¡nfh£o‹ rk‹gh£oid bgwyh«.

11. 2 3 0x y+ - = , 5 6 0x y+ - = M»a ne®¡nfhLfŸ rªÂ¡F« òŸë têahfÎ«,

(1, 2), (2, 1) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‰F ÏizahfÎ« cŸs


Ô®Î: A(1, 2), B(2, 1) v‹gd bfhL¡f¥g£l òŸëfŸ.

nkY« bfhL¡f¥g£l rk‹ghLfŸ

2 3 (1) ; 5 6 (2) .x y x yg g+ = + =

(2)-èUªJ (1)-I fê¡f, eh« bgWtJ 3 3 1x x&= =

Mfnt, .y 1=

vdnt, bt£L« òŸë (1, 1).(1, 2) k‰W« (2, 1) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ

1.m1 22 1=-- =-

Mfnt, njitahd ne®¡nfh£o‹ (ÏiznfhL) rhŒÎ –1rhŒÎ –1 MfÎ« (1, 1) v‹w òŸë tê¢brštJkhd ne®¡nfh£o‹ rk‹ghL 1 1( 1) 2 0y x x y&- =- - + - = .

12. 5 6 1x y- = , 3 2 5 0x y+ + = M»a ne®¡nfhLfŸ rªÂ¡F« òŸë têahfÎ«,

3 5 11 0x y- + = v‹w ne®¡nfh£o‰F br§F¤jhfÎ« mikÍ« ne®¡nfh£o‹

rk‹gh£il¡ fh©f.

Ô®Î: bfhL¡f¥g£l rk‹ghLfŸ

5 6 1 (1) ; 3 2 5 (2)x y x yg g- = + =-

(1) k‰W« (2) I Ô®¡f, eh« bt£L« òŸëia¥ bgwyh«.

bt£L« òŸë (–1, –1).


x y3 5 11 0- + = v‹w ne®¡nfh£o‹ rhŒÎ .m53

53=

-- =

Mfnt, njitahd (br§F¤J) ne®¡nfh£o‹ rhŒÎ 35- .

rhŒÎ 35- MfÎ« (–1, –1) v‹w òŸë tê¢brštJkhd ne®¡nfh£o‹ rk‹ghL

1 ( 1) 5 3 8 0.y x x y35 &+ = - + + + =

13. 3 9 0x y- + = , 2 4x y+ = M»a ne®¡nfhLfŸ bt£L« òŸëÍl‹, 2 4 0x y+ - = ,

2 3 0x y- + = M»a ne®¡nfhLfŸ bt£L« òŸëia Ïiz¡F« ne®¡nfh£o‹


Ô®Î: bfhL¡f¥g£l rk‹ghLfëèUªJ

3 9 (1)x y g- =- ; 2 4 (2) .x y g+ =

2 4 (3)x y g+ = ; 2 3 ( )x y 4g- =- .

(1) k‰W« (2) I Ô®¡f, »il¡F« bt£L« òŸë (–2, 3).

(3) k‰W« (4) I Ô®¡f, »il¡F« bt£L« òŸë (1, 2).

(–2, 3) k‰W« (1, 2) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rk‹ghL

y y

y y

x x

x x

2 1

1

2 1

1

-

-=

-

- & y x y x

2 33

1 22

13

32&

--

=++

--

= +

3 7 0x y& + - =

Mfnt, njitahd ne®¡nfh£o‹ rk‹ghL 3 7 0x y+ - = .

14. TABC-‹ KidfŸ A(2, -4), B(3, 3) k‰W« C(-1, 5) våš, B-èUªJ tiua¥gL«

F¤J¡nfh£L tê¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

Ô®Î: A(2, -4), B(3, 3) k‰W« C(-1, 5) v‹gd K¡nfhz¤Â‹ c¢ÁfŸ.

BD v‹gJ K¡nfhz¤Â‹ c¢Á B-æèUªJ tiua¥gL«

F¤J¡nfhL v‹f.

AC-‹ rhŒÎ 3.1 25 4

39

- -+ =

-=-

Mfnt, F¤J¡nfhL BD-‹ rhŒÎ 31 . (AC BD= )

rhŒÎ 31 MfÎ« (3, 3) v‹w òŸë tê¢brštJkhd ne®¡nfh£o‹ rk‹ghL

3 ( 3) 3 9 3 3y x y x x y31 6 0& &- = - - = - - + = .

15. TABC-‹ KidfŸ A(-4,4 ), B(8 ,4), C(8,10) våš, A-èUªJ tiua¥gL«

eL¡nfh£L tê¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

Ô®Î: Kid A-æèUªJ tiua¥gL« eL¡nfhL AD v‹f.

BC-‹ ika¥òŸë, , (8, 7) .D D2

8 82

4 10+ + =` j

A(-4,4 ), D(8, 7) M»a òŸëfis Ïiz¡F« eL¡nfhL AD-‹ rk‹ghL

.y x7 4

48 4

4--

=++ 4 16 4 4 20 0.y x x y& &- = + - + =


16. MÂ¥òŸëæèUªJ 3 2 13x y+ = v‹w ne®¡nfh£o‰F tiua¥gL« br§F¤J¡

nfh£o‹ mo¥òŸëia¡ (foot of the perpendicular) fh©f.

Ô®Î: x y3 2 13+ = v‹w ne®¡nfh£o‰F tiua¥gL« br§F¤J¡nfhL OP. nkY«

mj‹ br§F¤J mo P v‹f.

Mfnt, ne®¡nfhL OP-‹ rk‹ghL x y k2 3 0- + = .

Ï¡nfhL MÂ¥òŸë O(0, 0) tê¢brštjhš, .k 0=

Mfnt, ne®¡nfhL OP-‹ rk‹ghL 2 3 0x y- = .

3 2 13x y+ = (1)g k‰W«

2 3 0x y- = (2)g M»a ne®¡nfhLfŸ bt£L« òŸë P MF«.

(1) k‰W« (2) I Ô®¡f, eh« bgWtJ , .x y3 2= =

Mfnt, br§F¤J¡ nfh£o‹ mo P(3, 2) MF«.

17. xU t£l¤Â‹ ÏU é£l§fë‹ rk‹ghLfŸ 2 7x y+ = , 2 8x y+ = k‰W«

t£l¤Â‹ ÛJ mikªJŸs xU òŸë (0, -2) våš, Ï›t£l¤Â‹ Mu¤ij fh©f.

Ô®Î: bfhL¡f¥g£l ÏU é£l§fë‹ rk‹ghLfŸ

2 7 (1) ; 2 8 (2)x y x yg g+ = + =

mt‰¿‹ bt£L« òŸëfŸ rªÂ¡F« òŸë t£lika« C.

(1) k‰W« (2) I Ô®¡f ek¡F »il¥gJ t£l¤Â‹ ika« C(3, 2).

t£l¤Â‹ ÛJ mikªj òŸë P(0, –2) v‹f.

Mfnt, t£l¤Â‹ Mu« ( ) ( ) 5r CP 3 0 2 2 252 2= = - + + = = myFfŸ

18. 2 3 4 0x y- + = , 2 3 0x y- + = M»a ne®¡nfhLfŸ rªÂ¡F« òŸëiaÍ«,

(3, -2), (-5, 8) M»a òŸëfis Ïiz¡F« ne®¡nfh£L¤J©o‹

eL¥òŸëiaÍ«, Ïiz¡F« nfh£L¤J©o‹ rk‹gh£il¡ fh©f.

Ô®Î: 2 3 4 0 ( )x y 1g- + = k‰W«

2 3 0 (2)x y g- + = M»a ne®¡nfhLfŸ bt£L« òŸë P v‹f.

(1) k‰W« (2) I Ô®¡f ek¡F »il¡F« bt£L« òŸë P(1, 2). (3, –2) k‰W« (–5, 8) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ ika¥òŸë

, ( 1, 3) .M M2

3 522 8- - + = -` j

Mfnt, njitahd ne®¡nfhL MP-‹ rk‹ghL

y x3 2

21 1

1--

=- -

-

1 2 4 2 5 0.x y x y& &- =- + + - =

19. ÏUrkg¡f K¡nfhz« TPQR-š PQ = PR k‰W« mo¥g¡f« QR v‹gJ x-m¢Á‹

ÛJ mik»wJ v‹f. nkY«, Kid P MdJ y-m¢Á‹ ÛJ mik»wJ. PQ-‹

rk‹ghL 2 3 9 0x y- + = våš, PR têahf bršY« ne®¡nfh£o‹ rk‹gh£il¡

fh©f.


Ô®Î: PQ-‹ rk‹ghL 2 3 9 0 ( )x y 1g- + =

òŸë P, y-m¢Á‹ ÛJ mikªJŸsjhš x 0=

vdnt, (1) 2(0) 3 9 0 3.y y& &- + = =

Mfnt, òŸë P v‹gJ (0, 3) MF«.

òŸë Q, x-m¢Á‹ ÛJ mikªJŸsjhš 0y = .

vdnt, (1) 2 0 9 0 .x x29& &- + = =-

Mfnt, òŸë Q v‹gJ 29 , 0-` j MF«.

PQ PR= k‰W« QR MdJ x-m¢Á‹ ÛJ mikªJŸsjhš, òŸë R v‹gJ ,29 0` j

MF«.

Mfnt, ne®¡nfhL PR-‹ rk‹ghL

y x0 3

3

0

0

29-

-=

- -

- 9 27 6 2 3 9 0.y x x y& &- =- + - =

gæ‰Á 5.6

rçahd éilia¤ nj®ªbjL.

1. ,a b-^ h, ,a b3 5^ h M»a òŸëfis Ïiz¡F« ne®¡nfh£L¤ J©o‹ eL¥òŸë

(A) ,a b2-^ h (B) ,a b2 4^ h

(C) ,a b2 2^ h (D) ,a b3- -^ h

Ô®Î: eL¥òŸë = , ( , 2 )a a b b a b23

25 2+ - + =` j . ( éil. (C) )

2. ,A 1 3-^ h, ,B 3 9-^ h M»a òŸëfis Ïiz¡F« ne®¡nfh£L¤ J©il 1 : 3 v‹w

é»j¤Âš Ãç¡F« òŸë P

(A) ,2 1^ h (B) ,0 0^ h

(C) ,35 2` j (D) ,1 2-^ h

Ô®Î: òŸë P = ( ) ( ),

( ) ( ), (0, 0)

1 31 3 3 1

1 31 9 3 3

43 3

49 9

+- +

++ -

= - + - =c `m j .

( éil. (B) )

3. ,A 3 4^ h, ,B 14 3-^ h M»at‰iw Ïiz¡F« ne®¡nfh£L¤J©L x-m¢ir P Ïš

rªÂ¡»‹wJ våš, m¡nfh£L¤J©il P Ãç¡F« é»j«

(A) 4 : 3 (B) 3 : 4 (C) 2 : 3 (D) 4 : 1

Ô®Î: ne®¡nfhL x-m¢ir bt£Läl¤Âš y 0= .

( ) ( )0 3 4 0 3 4

l ml m

l m l mml3 4

34& & & &

+- +

= - + = = = ( éil. (A) )


4. (–2, –5), (–2, 12), (10, –1) M»a òŸëfis Kidfshf¡ bfh©l K¡nfhz¤Â‹

eL¡nfh£L ika« (centroid)

(A) ,6 6^ h (B) ,4 4^ h (C) ,3 3^ h (D) ,2 2^ h

Ô®Î: eL¡nfh£L ika« = 3

2 2 10 ,3

5 12 1 ( , )2 2- - + - + - =` j ( éil. (D) )

5. ,1 2^ h, ,4 6^ h, ,x 6^ h, ,3 2^ h v‹gd Ï›tçiræš X® Ïizfu¤Â‹ KidfŸ våš,

x-‹ kÂ¥ò

(A) 6 (B) 2 (C) 1 (D) 3

Ô®Î: Ïizfu¤Â‹ _iyé£l§fŸ x‹iwbah‹W ÏUrk¡ T¿L«

AC-‹ ika¥òŸë = BD-‹ ika¥òŸë

, , 6x x x2

12

2 62

4 32

6 22

12

4 3& &+ + = + + + = + =` `j j ( éil. (A) )

6. (0,0), ,2 0^ h, ,0 2^ h M»a òŸëfshš mikÍ« K¡nfhz¤Â‹ gu¥ò

(A) 1 r.myFfŸ (B) 2 r.myFfŸ (C) 4 r.myFfŸ (D) 8 r.myFfŸ

Ô®Î : K¡nfhz¤Â‹ gu¥ò = (4) 221 0

0

2

0

0

2

0

0 21= =' 1 r.myFfŸ

(mšyJ) (2)(2) 2ab21

21= = r.myFfŸ ( éil. (B) )

7. ,1 1^ h, ,0 1^ h, ,0 0^ h, ,1 0^ h M»a òŸëfshš mikÍ« eh‰fu¤Â‹ gu¥ò

(A) 3 r.myFfŸ (B) 2 r.myFfŸ (C) 4 r.myFfŸ (D) 1 r.myFfŸ

Ô®Î: gu¥ò = (2) 121 1

1

0

1

0

0

1

0

1

1 21= =' 1 r.myFfŸ

(mšyJ) rJu¤Â‹ gu¥ò, (1) 1a2 2= = r.myFfŸ. ( éil. (D) )

8. x-m¢R¡F Ïizahd ne®¡nfh£o‹ rhŒÎ¡ nfhz«

(A) 0c (B) 60c (C) 45c (D) 90c

Ô®Î: x-m¢R¡F Ïizahd nfh£o‹ rhŒÎ¡ nfhz« 0c. ( éil. (A) ) 9. ,3 2-^ h, , a1-^ h M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ

23- våš,

a-‹ kÂ¥ò

(A) 1 (B) 2 (C) 3 (D) 4

Ô®Î: rhŒÎ = 2 4 12 4.a a a1 3

223 & &

- -+ = - + = = ( éil. (D) )

10. ,2 6-^ h, ,4 8^ h M»a òŸëfis Ïiz¡F« ne®¡nfh£o‰F¢ br§F¤jhd

ne®¡nfh£o‹ rhŒÎ

(A) 31 (B) 3 (C) -3 (D)

31-

Ô®Î: ,2 6-^ h k‰W« ,4 8^ h M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ

.4 28 6

31

+- = Mfnt, br§F¤J¡nfh£o‹ rhŒÎ –3. ( éil. (C) )


11. 9 2 0x y- - = , 2 9 0x y+ - = M»a ne®¡nfhLfŸ rªÂ¡F« òŸë

(A) ,1 7-^ h (B) ,7 1^ h (C) ,1 7^ h (D) ,1 7- -^ h

Ô®Î: ( ) ( )x y x y9 2 0 1 2 9 0 2- - = + - =

rk‹ghLfis Ô®¡f¡ »il¡F« òŸë (1, 7). ( éil. (C) ) 12. 4 3 12 0x y+ - = v‹w ne®¡nfhL y-m¢ir bt£L« òŸë

(A) ,3 0^ h (B) ,0 4^ h (C) ,3 4^ h (D) ,0 4-^ h

Ô®Î: y-m¢ir bt£L« òŸëæl¤J, 0x = . vdnt, m¥òŸë (0, 4) ( éil. (B) )

13. 7 2 11y x- = v‹w ne®¡nfh£o‹ rhŒÎ

(A) 27- (B)

27 (C)

72 (D)

72-

Ô®Î: rhŒÎ .mba

72

72= - =- - =` j ( éil. (C) )

14. (2, 7) v‹w òŸë tê¢ brštJ«, x-m¢Á‰F ÏizahdJkhd ne®¡nfh£o‹

rk‹ghL

(A) x 2= (B) x 7=- (C) y 7=- (D) y 2=

Ô®Î: x-m¢R¡F Ïizahd ne®¡nfh£o‹ rk‹ghL y k= .

Ï¡nfhL (2, –7) v‹w òŸë tê¢brštjhš, »il¡F« ne®¡nfhL y 7=- ( éil. (C) )

15. 2 3 6 0x y- + = v‹w ne®¡nfh£o‹ x, y-bt£L¤J©LfŸ Kiwna

(A) 2, 3 (B) 3, 2 (C) -3, 2 (D) 3, -2

Ô®Î: x-bt£L¤J©il fhz, rk‹gh£oš y 0= vd ÃuÂæl 3.x =-

y-bt£L¤J©il fhz, rk‹gh£oš 0x = vd ÃuÂæl 2.y = ( éil. (C) )

16. xU t£l¤Â‹ ika« (-6, 4). xU é£l¤Â‹ xU Kid (-12, 8) våš, mj‹ kW

Kid

(A) (-18, 12) (B) (-9, 6) (C) (-3, 2) (D) (0, 0)

Ô®Î: (-12, 8) k‰W« (x, y) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ ika¥òŸë

, ( 6, 4) , .x yx y

212

28

0 0&- + += - = =c m ( éil. (D) )

17. MÂ¥òŸë tê¢ brštJ« 2 3 7 0x y+ - = v‹w nfh£o‰F¢ br§F¤Jkhd

ne®¡nfh£o‹ rk‹ghL

(A) x y2 3 0+ = (B) x y3 2 0- =

(C) y 5 0+ = (D) y 5 0- =

Ô®Î: njitahd ne®¡nfhL 3 0.x y k2- + = Ï¡nfhL MÂ¥òŸë tê¢brštjhš k 0= ( éil. (B) )


18. y-m¢Á‰F ÏizahdJ« ,2 5-^ h v‹w òŸë tê¢ brštJkhd ne®¡nfh£o‹

rk‹ghL

(A) x 2 0- = (B) x 2 0+ = (C) y 5 0+ = (D) y 5 0- =

Ô®Î: y-m¢R¡F Ïizahd ne®¡nfh£o‹ rk‹ghL x k=

Ï¡nfhL (–2, 5) tê¢brštjhš, 2 2 0.x x&=- + = ( éil. (B) )

19. (2, 5), (4, 6), ,a a^ h M»a òŸëfŸ xnu ne®¡nfh£oš mik»‹wd våš, a-‹ kÂ¥ò

(A) -8 (B) 4 (C) -4 (D) 8

Ô®Î: rhŒÎfŸ rk« vdnt, .aa a

4 26 5

25 8&

-- =

-- = ( éil. (D) )

20. 2y x k= + v‹w ne®¡nfhL (1, 2) v‹w òŸë tê¢ brš»‹wJ våš, k-‹

kÂ¥ò

(A) 0 (B) 4 (C) 5 (D) -3

Ô®Î: 2y x k= + v‹w nfhL (1, 2) v‹w òŸë tê¢brštjhš.

2(1) 0k k2&+ = = ( éil. (A) ) 21. rhŒÎ 3 MfÎ«, y bt£L¤J©L -4 MfÎ« cŸs ne®¡nfh£o‹ rk‹ghL

(A) x y3 4 0- - = (B) x y3 4 0+ - =

(C) x y3 4 0- + = (D) x y3 4 0+ + =

Ô®Î: 3, 4m c= =- Mfnt 3 4 3 4 0.y mx c y x x y& &= + = - - - =

( éil. (A) )

22. 0y = k‰W« 4x =- M»a ne®¡nfhLfŸ bt£L« òŸë

(A) ,0 4-^ h (B) ,4 0-^ h (C) ,0 4^ h (D) ,4 0^ h

Ô®Î: ne®¡nfhLfŸ bt£L« òŸë ,4 0-^ h ( éil. (B) )

23. 3x + 6y + 7 = 0 k‰W« 2x + ky = 5 M»a ne®¡nfhLfŸ br§F¤jhdit våš,

k-‹ kÂ¥ò

(A) 1 (B) –1 (C) 2 (D) 21

Ô®Î: 1m m1 2 &=- 1 1.k

k63 2 &- - =- =-` `j j

(mšyJ) a a b b k k0 3 2 6 0 11 2 1 2

& &+ = + = =-^ h ( éil. (B))

Ô®Î - toéaš 171

gæ‰Á 6.1

1. D k‰W« E M»a òŸëfŸ Kiwna ABCT -‹ g¡f§fŸ AB k‰W« AC-fëš DE BC<

v‹¿U¡FkhW mikªJŸsd. (i) AD = 6 br. Û, DB = 9 br. Û k‰W« AE = 8 br. Û, våš AC-I fh©f.

(ii) AD = 8 br. Û, AB = 12 br. Û k‰W« AE = 12 br. Û, våš CE-I fh©f.

(iii) AD = 4x – 3, BD = 3x – 1 , AE = 8x – 7 k‰W« EC = 5x – 3, våš x ‹ kÂ¥ò

fh©f. Ô®Î: (i) ABCT -š DE BC< . vdnt, njš° nj‰w¤Â‹go, eh«

bgWtJ

DBAD

ECAE= 9 12EC

ADAE DB EC

68& &# #= = = br.Û

AC AE EC` = + = 8 + 12 = 20 br.Û (ii) AD = 8 br.Û, AB = 12 br.Û k‰W« AE = 12 br.Û vd bfhL¡f¥g£LŸsJ

Mfnt, 12 8 4BD AB AD= - = - = br.Û

ABCT -š DE BC< . vdnt, njš° nj‰w¤Â‹go, eh« bgWtJ

DBAD

ECAE= 4 6EC

ADAE DB EC

812& &# #= = = br.Û

` 6CE = br.Û

(iii) AD = 4x – 3, BD = 3x – 1 , AE = 8x – 7 k‰W« EC = 5x – 3. ABCT -š DE BC< . vdnt, njš° nj‰w¤Â‹go, eh« bgWtJ

DBAD

ECAE= &

xx

xx

3 14 3

5 38 7

-- =

--

& (4 3)(5 3)x x- - = (8 7)(3 1)x x- -

& 4 2 2x x2- - = 0

& 2 1x x2- - = 0

& ( 1)(2 1)x x- + = 0

Mfnt, , 1x x21=- = . bjhiyÎ x

21!- v‹gjhš, 1x = MF«.

2. gl¤Âš AP = 3 br.Û, AR = 4.5 br.Û, AQ = 6 br.Û, AB = 5 br.Û k‰W«

AC = 10 br. Û våš, AD-‹ Ús« fh©f.

Ô®Î: bfhL¡f¥g£l étu§fëèUªJ APAB

35= k‰W«

AQAC

610

35= = vd

bgW»nwh«.

vdnt, ABCT -š APAB

AQAC=

Mjyhš, njš° nj‰w¤Â‹ kWjiyæ‹go, PQ BC< MF«.

RD = x v‹f. ABDT -š PR BD< . APAB

ARAD=

.. x

35

4 54 5& = +

toéaš 6


& 13.5 3 .x 22 5+ = & 3x = 9 & x = 3.Mfnt, 4.5 3 7.5AD AR RD= + = + = br.Û.

kh‰WKiw: PB AB AP 5 3 2= - = - = br.Û, QC AC AQ 10 6 4= - = - = br.Û

PBAP

23= k‰W«

QCAQ

46

23= = . vdnt, ABCT -š

PBAP

QCAQ

=

Mjyhš, njš° nj‰w¤Â‹ kWjiyæ‹go, PQ BC<

ABDT -š, PBAP

RDAR= . .

RD23 4 5& =

` RD = . 33

4 5 2# =

Mfnt, 4.5 3 7.5AD AR RD= + = + = br.Û

3. E k‰W« F v‹w òŸëfŸ Kiwna PQRT -‹ g¡f§fŸ PQ k‰W« PR M»at‰¿‹

ÛJ mikªJŸsd. Ã‹tUtdt‰¿‰F EF QR< v‹gjid¢ rçgh®¡f.

(i) PE = 3.9 br. Û, EQ = 3 br.Û, PF = 3.6 br.Û k‰W« FR = 2.4 br. Û.

(ii) PE = 4 br. Û, QE = 4.5 br.Û, PF = 8 br.Û k‰W« RF = 9 br. Û.

Ô®Î: (i) PE = 3.9 br.Û, EQ = 3 br.Û, PF = 3.6 br.Û k‰W« FR = 2.4 br.Û vd

bfhL¡f¥g£LŸsJ.

PQRT -š . 1.3EQPE

33 9= = k‰W«

.

. 1.FRPF

2 43 6 5= =

EQPE

FRPF` !

Mfnt, EF MdJ QR-¡F ÏizahdJ mšy.

(ii) PE = 4 br.Û, QE = 4.5 br.Û, PF = 8 br.Û k‰W« RF = 9 br.Û vd bfhL¡f¥g£LŸsJ.

.EQPE

4 54

98= = k‰W«

FRPF

98=

Mfnt, PQRT -š EQPE

FRPF=

Mfnt, é»jrk nj‰w¤Â‹ kWjiyæ‹go, EF QR<

4. gl¤Âš AC BD< k‰W« CE DF< . OA =12 br. Û, AB = 9 br. Û, OC = 8 br. Û.k‰W« EF = 4.5 br. Û våš, FO-it¡ fh©f.

Ô®Î: OBDT -š AC BD< .

Mfnt, njš° nj‰w¤Â‹go, eh« bgWtJ

ABOA =

CDOC 6

CDCD

912 8

128 9& & #= = = br.Û

ODFT -š CE DF< . Mfnt, njš° nj‰w¤Â‹go

CDOC =

EFOE

.. 6OE OE

68

4 5 68 4 5& & #= = = br.Û

vdnt, 6 4.5OF OE EF= + = + = 10.5 br.Û.


5. ABCD v‹w eh‰fu¤Âš, AB-¡F Ïiz CD v‹f. AB-¡F Ïizahf tiua¥g£l

xU ne®¡nfhL AD-I P-æY« BC-I Q-æY« rªÂ¡»wJ våš, PDAP

QCBQ

= vd

ãWÎf.

Ô®Î: mik¥ò: PQ it E-š bt£LkhW BD I Ïiz¡fÎ«.

DABT -š PE AB< . Mfnt, njš° nj‰w¤Â‹ (BPT) go

PDAP

EDBE= g(1)

BCDT -š EQ DC< . Mfnt, njš° nj‰w¤Â‹ (BPT) go

EDBE

QCBQ

= g(2)

(1) k‰W« (2)-èUªJ, eh« bgWtJ PDAP

QCBQ

= .

6. gl¤Âš PC QK< k‰W« BC HK< . AQ = 6 br. Û, QH = 4 br. Û, HP = 5 br. Û, KC = 18 br. Û våš, AK k‰W« PB-¡fis¡ fh©f.

Ô®Î: APCT -š PC QK< .

Mfnt, njš° nj‰w¤Â‹ go, eh« bgWtJ QPAQ

KCAK= .

AKQPAQ

KC& #= [aQP QH HP 4 5 9= + = + = ]

AK& = 18 1296# = br.Û

nkY« ABCT -š BC HK< . Mfnt, njš° nj‰w¤Â‹ go, eh« bgWtJ

HBAH

KCAK=

HB10& =

1812 [a AH = AQ + QH = 6+ 4 = 10]

& 15HB12

10 18#= = br.Û

Mfnt, PB = 15 5 10HB HP- = - = br.Û

7. gl¤Âš DE AQ< k‰W« DF AR< våš, EF QR< vd ãWÎf.

Ô®Î: PQAT -š, DE AQ< . Mfnt, njš° nj‰w¤Â‹ go, eh« bgWtJ

EQPE =

DAPD g(1)

nkY«, PART -š DF AR< . Mfnt, njš° nj‰w¤Â‹ go

DAPD =

FRPF g(2)

(1) k‰W« (2)-èUªJ, eh« bgWtJ EQPE

FRPF=

Mfnt, njš° nj‰w¤Â‹ kWjiyæ‹go, eh« bgWtJ EF QR< . 8. gl¤Âš DE AB< k‰W« DF AC<

våš, EF BC< vd ãWÎf.

Ô®Î: ABPT -š DE AB< . Mfnt, njš° nj‰w¤Â‹go

DAPD =

EBPE g (1)


nkY« CAPT -š DF AC< . Mfnt, njš° nj‰w¤Â‹go

DAPD =

FCPF g (2)

(1) k‰W« (2) M»at‰¿èUªJ, eh« bgWtJ EBPE

FCPF=

Mfnt, njš° nj‰w¤Â‹ kWjiyæ‹go, EF BC< .

9. AD v‹gJ ABCT -š A+ -‹ c£òw nfhz ÏUrkbt£o. mJ BC-I D-š rªÂ¡»wJ.

(i) BD = 2 br. Û, AB = 5 br. Û, DC = 3 br. Û våš, AC fh©f.

(ii) AB = 5.6 br. Û, AC = 6 br. Û k‰W« DC = 3 br. Û våš, BC fh©f.

(iii) AB = x, AC = x – 2, BD = x + 2 k‰W« DC = x – 1 våš, x-‹ kÂ¥ig¡ fh©f.

Ô®Î: (i) AD v‹gJ A+ -‹ c£òw ÏUrkbt£o vd bfhL¡f¥g£LŸsJ.

ABCT -š, nfhz ÏUrkbt£o nj‰w¤Â‹go

ACAB =

DCBD

& AC5 =

32

& AC = 7.52

5 3# = br.Û

(ii) AD v‹gJ A+ -‹ c£òw ÏUrkbt£o vd bfhL¡f¥g£LŸsJ.

ABCT -š, nfhz ÏUrkbt£o nj‰w¤Â‹go, eh« bgWtJ

ACAB =

DCBD

& .65 6 = . 2.8BD BD

3 65 6 3& #= = br.Û

Mfnt, BC = 2.8 3 5.8BD DC+ = + = br.Û

(iii) AD v‹gJ A+ -‹ c£òw ÏUrkbt£o vd bfhL¡f¥g£LŸsJ.

ABCT -š, nfhz ÏUrkbt£o nj‰w¤Â‹go, eh« bgWtJ

ACAB =

DCBD

& x

x2-

= xx

12

-+

& ( 1)x x - = ( 2)( 2)x x+ -

& x x2- = x 42

- x` = 4 10. Ã‹tUtdt‰WŸ AD v‹gJ ABCT -š A+ -‹ nfhz ÏUrkbt£o MFkh vd¢

nrhÂ¡f.

(i) AB = 4 br. Û, AC = 6 br. Û, BD = 1.6 br. Û, k‰W« CD = 2.4 br. Û, (ii) AB = 6 br. Û, AC = 8 br. Û, BD = 1.5 br. Û. k‰W« CD = 3 br. Û.

Ô®Î: (i) DCBD =

.

.2 41 6

32= g (1)

ACAB =

64

32= g (2)


(1) k‰W« (2) M»at‰¿èUªJ, eh« bgWtJ DCBD

ACAB=

Mfnt, nfhz ÏUrkbt£o nj‰w¤Â‹go, AD v‹gJ A+ -‹ nfhz ÏUrkbt£o

MF«.

(ii) DCBD = .

31 5

21= g (1)

ACAB =

86

43= g (2)

(1) k‰W« (2) M»at‰¿èUªJ, DCBD

ACAB!

Mfnt, AD v‹gJ A+ -‹ ÏUrkbt£o mšy.

11. MP v‹gJ MNOT -š M+ -‹ btë¥òw ÏUrkbt£o. nkY«, ÏJ NO-‹

Ú£Áæid P-æš rªÂ¡»wJ. MN = 10 br.Û, MO = 6 br.Û, NO = 12 br.Û våš, OP-I fh©f.

Ô®Î: MP-v‹gJ M+ -‹ btë¥òw ÏUrkbt£o vd bfhL¡f¥g£LŸsJ.

MNOT -š nfhz ÏUrkbt£o nj‰w¤Â‹go

OPNP =

MOMN

& OP

OP12 + = MOMN [ 12NP NO OP OPa = + = + ]

& 12OP

OP+ = 610

& 72 6 OP#+ = 10 4OP OP 72&# # =

` OP = 18 br.Û

12. ABCD v‹w eh‰fu¤Âš B+ k‰W« D+ M»at‰¿‹ ÏUrkbt£ofŸ AC-I E-š

bt£L»‹wd våš, BCAB

DCAD= vd ãWÎf.

Ô®Î: DE-v‹gJ D+ -‹ c£òw ÏUrkbt£o vd bfhL¡f¥g£LŸsJ.

ADCT -š, nfhz ÏUrkbt£o nj‰w¤Â‹go, ECAE =

DCAD g(1)

BE v‹gJ B+ -‹ c£òw ÏUrkbt£o.

ABCT -š nfhz ÏUrkbt£o nj‰w¤Â‹go ECAE =

BCAB g(2)

(1) k‰W« (2) M»at‰¿èUªJ, eh« bgWtJ BCAB

DCAD= .

13. ABCT -š A+ -‹ c£òw ÏUrkbt£o BC-I D-æY«, A+ -‹ btë¥òw ÏUrkbt£o

BC-‹ Ú£Áæid E-æY« rªÂ¡»‹wd våš, BEBD

CECD= vd ãWÎf.

Ô®Î: ABCT -š, AD v‹gJ A+ -‹ c£òw ÏUrkbt£o.

Mfnt, ABCT -š nfhz ÏUrkbt£o nj‰w¤Â‹go

DCBD =

ACAB g (1)


ABCT -š, AE v‹gJ A+ -‹ btë¥òw ÏUrkbt£o.

Mfnt, ABCT -š CEBE =

ACAB g (2)

(1) k‰W« (2) M»at‰¿èUªJ, eh« bgWtJ

DCBD

CEBE= &

BEBD

CEDC= (mšyJ)

BEBD

CECD=

14. ABCD v‹w eh‰fu¤Âš AB = AD. AE k‰W« AF v‹gd Kiwna BAC+ k‰W«

DAC+ M»at‰¿‹ c£òw ÏUrkbt£ofŸ våš, EF BD< vd ãWÎf.

Ô®Î: ABCT -š, AE v‹gJ BAC+ -‹ c£òw ÏUrkbt£o.

Mfnt, nfhz ÏUrkbt£o nj‰w¤Â‹go

ACAB =

ECBE g (1)

ADCT -š AF v‹gJ DAC+ -‹ c£òw ÏUrkbt£o.

Mfnt, nfhz ÏUrkbt£o nj‰w¤Â‹go

ACAD =

FCDF mšyJ

ACAB

FCDF= g (2) [ AB = AD ]

(1) k‰W« (2) M»at‰¿èUªJ, eh« bgWtJ ECBE

FCDF=

Mfnt, CDBT -š njš° nj‰w¤Â‹ kWjiyæ‹go, EF BD< .

kh‰WKiw: ADCT -š AF-v‹gJ DAC+ -‹ c£òw ÏUrkbt£o.

Mfnt, nfhz ÏUrkbt£o nj‰w¤Â‹go,

ADAC =

FDCF g (1)

ABCT -š, AE v‹gJ BAC+ -‹ c£òw ÏUrkbt£o.

Mfnt, nfhz ÏUrkbt£o nj‰w¤Â‹go, ABAC =

EBCE .

AB = AD v‹gjhš, eh« bgWtJ ADAC =

EBCE g (2)

(1) k‰W« (2) M»at‰¿èUªJ, FDCF

EBCE=

Mfnt, CDBT -š mo¥gil é»jrk nj‰w¤Â‹ kWjiyæ‹go EF BD< MF«.

gæ‰Á 6.2 1. Ã‹tU« gl§fŸ x›bth‹¿Y« bjçahjdt‰¿‹ kÂ¥òfis¡ fh©f. všyh

Ús§fS« br‹o Û£lçš bfhL¡f¥g£LŸsd. (msÎfŸ msÎ¤Â£l¥go Ïšiy)

(i) (ii) (iii) a


Ô®Î: (i) ABCT k‰W« ADET M»at‰¿èUªJ,

ABC+ = ADE+ (x¤j nfhz§fŸ)k‰W« A+ = A+ (bghJnfhz«)

éÂKiwæ‹go, ABC ADET T+ . vdnt, AEAC

DEBC= .

& x

x8+

= 248 &

xx8 3

1+

= & 4x = br.Û.

nkY«, EAGT k‰W« ECFT M»ait tobth¤j K¡nfhz§fshF«.

Mfnt, EAEC

AGCF= & AG

ECCF EA#= [a EA = EC + CA = 8 + 4 = 12]

y& = 12 9.86# = Mfnt, 9y = br.Û.

(ii) HBCT -š, FG BC< .HFG HBC

HBHF

BCFG&` T T+ = 3.6x x

104

9 104 9& & #= = = br.Û.

FBDT k‰W« FHGT M»at‰iw vL¤J¡bfhŸnth«. Ï§F BD GH<

FBD FHG`+ += [x‹Wé£l nfhz§fŸ] BFD HFG+ += [F¤bjÂ® nfhz§fŸ]Mfn,t AA éÂKiwæ‹go, eh« bgWtJ FBD FHGT T+

& FDFG

FBFH=

yx3 6

4&+

= .y 33 6

32&

+=

& 2 6 .y 10 8 &+ = 2.4y = br.Û.

AEGT k‰W« ABCT M»at‰iw vL¤J¡bfhŸnth«. Ï§F EG BC< Mfnt, tobth¤j K¡nfhz§fS¡fhd AA éÂKiwæ‹go, eh« bgWtJ AEG ABCT T+

& ABAE =

BCEG

& z

z5+

= x y9+

zz5 9

6&+

=

& 3z = 2 10z + & 10z = br.Û.

(iii) gl¤ÂèUªJ, EFCD xU ÏizfukhF«. nkY«, 7EF DC= = br.Û, 6DE CF= = br.Û. AEFT k‰W« ABCT M»at‰iw fUJf.

gl¤ÂèUªJ, AEF ABCT T+ v‹gJ bjëth»wJ

ACAF =

BCEF

x

x6+

= 127 & x x12 7 42= +

& x = 8.4 br.Û BDGT k‰W« BCFT M»at‰iw¡ fUJf.

gl¤ÂèUªJ, BDG BCFT T+ v‹gJ bjëth»wJ

BCBD` =

CFDG

DG = BCBD CF# & 6 2.5y

125

#= = br.Û

A+ bghJ¡nfhz«.

AEG+ = ABC+

x¤j nfhz§fŸ

AEF ABC+ +=

x¤j nfhz§fŸ

A+ bghJ¡nfhz«

` AA tiuaiw¥go

AEF ABCT T+


2. xU ãH‰gl¡ fUéæ‹ gl¢RUëš, 1.8 Û cauKŸs xU kåjå‹ Ã«g¤Â‹ Ús«

1.5 br.Û. v‹f. fUéæ‹ by‹ìèUªJ gl¢RUŸ 3 br.Û öu¤Âš ÏUªjhš, mt®

ãH‰gl¡ fUéæèUªJ v›tsÎ öu¤Âš ÏU¥gh®?

Ô®Î: AB v‹gJ kåjå‹ cau« v‹f.

CD v‹gJ kåjå‹ Ã«g¤Â‹ Ús«.

L v‹gJ ãH‰gl fUéæš cŸs by‹Á‹ ãiy.

LM v‹gJ kåjD¡F« by‹R¡F« Ïil¥g£l bjhiyÎ.

LN v‹gJ by‹R¡F« gl¢RUS¡F« Ïil¥g£l bjhiyÎ.

nkY«, AB || CD, AB = 1.8 Û, CD = 1.5 br.Û, LN = 3 br. Û,

LABT k‰W« LCDT M»at‰¿èUªJ, eh« bgWtJ

LAB+ = LCD+ [x‹Wé£l nfhz§fŸ]

BLA+ = DLC+ [F¤bjÂ® nfhz§fŸ]

tobth¤j K¡nfhz§fS¡fhd AA éÂKiwæ‹go LAB LCDT T+

& CDAB =

LNLM &

.1 5180 = LM

3

& LM = .

3 360CDAB LN

1 5180

# #= = br.Û. = 3.6 Û

Mfnt, kåjD¡F« ãH‰gl fUé¡F« Ïilna cŸs bjhiyÎ 3.6 Û.

3. 120 br.Û. cauKŸs xU ÁWä xU és¡F¡ f«g¤Â‹ moæèUªJ éy»,

mj‰F nebuÂuhf 0.6 Û./é ntf¤Âš elªJ¡ bfh©oU¡»whŸ. és¡F, jiu

k£l¤ÂèUªJ 3.6 Û. cau¤Âš cŸsJ våš, ÁWäæ‹ ãHè‹ Ús¤ij 4

édhofS¡F ÃwF fh©f.

Ô®Î: AB v‹gJ és¡F f«g¤Â‹ cau«, CD v‹gJ ÁWäæ‹ cau« k‰W« CE v‹gJ ÁWäæ‹

ãHè‹ Ús« v‹f.

Ã‹d®, AB = 3.6 Û, CD = 120 br. Û = 1.2 ÛÁWä el¡F« ntf« 0.6 Û/é vd bfhL¡f¥g£LŸsJ.Mfnt, 4 édhofëš ÁWä flªj bjhiyÎ AC 4 0.6 2.4#= = Û

ECDT k‰W« EABT -æèUªJ, CD AB< v‹gJ bjëth»wJ. ECD+ = EAB+ [x¤j nfhz§fŸ] E+ = E+ [bghJ¡nfhz«]` tobth¤j K¡nfhz§fS¡fhd AA éÂKiwæ‹go, ECD EABT T+

Mfnt, EAEC =

ABCD

. ECEC

2 4 + =

.

.3 61 2

31= & .EC EC3 2 4= +

& EC = 1.2 Û

Mfnt, 4 édhofS¡F ÃwF ÁWäæ‹ ãHè‹ Ús« 1.2 Û MF«.


4. xU ÁWä fl‰fiuæš mtŸ jªijÍl‹ ÏU¡»whŸ. flèš ÚªJ« xUt‹

bjhl®ªJ Úªj Koahkš Úçš j¤jë¤J¡ bfh©oU¥gij f©lhŸ. mtŸ clnd

nk‰»š 50 Û bjhiyéš ÏU¡F« j‹ jªijia cjé brŒÍkhW T¡Fuè£lhŸ.

Ïtis él ÏtŸ jªij xU glF¡F 10 Û mU»èUªjh®. Ït® m¥glif¥

ga‹gL¤Â _œ»¡ bfh©oU¥gtid mila nt©Lbkåš, 126 Û m¥gl»š bršy

nt©L«. mnj rka¤Âš m¢ÁWä Ú® C®Â x‹¿š gl»èUªJ 98 Û öu¤Âš

bršY« xUtid¡ fh©»whŸ. Ú® C®Âæš ÏU¥gt® _œ»¡ bfh©oU¥gtU¡F

»H¡»š cŸsh®. mtiu¡ fh¥gh‰w C®Âæš cŸst® v›tsÎ bjhiyÎ bršy

nt©L«? (gl¤ij¥ gh®¡f) (Ï›édh k£L« nj®Î¡Fçaj‹W)

Ô®Î: A v‹gJ jªij ã‰Fäl«

C v‹gJ ÁWä ã‰Fäl«

B v‹gJ glF cŸs Ïl«

D v‹gJ Ú® C®Â cŸs Ïl« k‰W«

E v‹gJ Úçš _œ» bfh©oU¥gt® cŸs Ïl« v‹f.

BC = x Û v‹f. Ã‹d®, AB = ( 10)x - Û

ABCT k‰W« DBET M»at‰iw vL¤J¡bfhŸnth«.

ABC+ = DBE+ [F¤bjÂ® nfhz§fŸ]

BAC+ = BDE+ [x‹Wé£l nfhz§fŸ]

Mfnt, tobth¤j K¡nfhz§fS¡fhd AA éÂKiwæ‹go, ABC DBET T+

DBAB =

BEBC

DEAC=

DBAB =

BEBC x x

9810

126& - =

& x = 28

1260 = 45 ` BC = 45 Û

nkY«, BEBC

DEAC= DE& =

BCAC BE#

& DE = 45

50 126# = 140

DE = 140 ÛMfnt, Úçš _œ» bfh©oU¥gtiu fh¥gh‰w, Ú® C®Âæš cŸst® 140 Û gaz«

brŒant©L«.

5. ABCT -š g¡f§fŸ AB k‰W« AC-æš Kiwna P k‰W« Q v‹w òŸëfŸ cŸsd.

AP = 3 br.Û, PB = 6 br.Û, AQ = 5 br.Û k‰W« QC = 10 br.Û våš, BC = 3 PQ vd

ãWÎf.

Ô®Î: bfhL¡f¥g£litfëèUªJ, ABAP

93

31= = ,

ACAQ

155

31= =

APQT k‰W« ABCT M»at‰¿èUªJ, eh« bgWtJ

ABAP =

ACAQ

nkY« A+ = A+ [bghJ¡nfhz«]


Mfnt, tobth¤j K¡nfhz§fS¡fhd SAS éÂKiwæ‹go, APQ ABCT T+

& ABAP =

ACAQ

BCPQ

=

ABAP =

BCPQ

BCPQ

93& = & BC = 3PQ

6. ABCT -š, AB = AC k‰W« BC = 6 br.Û v‹f. nkY«, AC-š D v‹gJ AD = 5 br.Û

k‰W« CD = 4 br.Û v‹W ÏU¡FkhW xU òŸë våš, BCD ACBT T+ vd ãWÎf.

Ïj‹ _y« DB-ia¡ fh©f.

Ô®Î: ABCT -š, AB=AC vd bfhL¡f¥g£LŸsJ.

ACBC =

96

32= ;

CBCD

64

32= =

BCDT k‰W« ACBT M»at‰¿èUªJ, eh« bgWtJ

ACBC =

CBCD

k‰W« C+ = C+ [bghJ¡nfhz«]

Mfnt, tobth¤j K¡nfhz§fS¡fhd SAS éÂKiwæ‹go, eh« bgWtJ

BCD ACBT T+

& ABBD =

ACBC

ACBD

96& = [ ]AB ACa =

& BD9

= 96

Mfnt, BD = 6 br. Û

7. ABCT -‹ g¡f§fŸ AB k‰W« AC-fë‹ nkš mikªj òŸëfŸ Kiwna D k‰W«

E v‹f. nkY«, DE BC< , AB = 3 AD k‰W« ABCT -‹ gu¥gsÎ 72 br.Û 2 våš,

eh‰fu« DBCE-‹ gu¥gsit¡ fh©f.

Ô®Î: gl¤ÂèUªJ, DE BC< k‰W« AB = 3AD vd bfhL¡f¥g£LŸsJ.

ABAD& =

31

ADET k‰W« ABCT M»at‰iw vL¤J¡bfhŸnth«.

ADE+ = ABC+ [x¤j nfhz§fŸ]

k‰W« A+ = A+ [bghJ¡nfhz«]

Mfnt, tobth¤j K¡nfhz§fS¡fhd AA éÂKiwæ‹go, eh« bgWtJ

ADE ABCT T+

‹ gu¥ò‹ gu¥ò

ABCADE

TT =

AB

AD2

2

‹ gu¥òADE72 9

1&T

=

Mfnt, ADET -‹ gu¥ò = 8 r.br.Û

eh‰fu« DBCE-‹ gu¥ò = ABCT -‹ gu¥ò – ADET -‹ gu¥ò

= 72 8 64- = br.Û2

8. ABCT -‹ g¡f Ús§fŸ 6 br.Û, 4 br.Û, 9 br.Û k‰W« PQR ABC3 3+ . PQRT -‹xU g¡f« 35 br.Û våš, PQRT -‹ R‰wsÎ äf mÂfkhf v›tsÎ ÏU¡f¡ TL«?


Ô®Î: PQR ABCT T+ vd bfhL¡f¥g£LŸsJ

& ABPQ =

Ï‹ R‰wsÎÏ‹ R‰wsÎ

BCQR

ACPR

ABC

PQR

T

T= = g(1)

QR = 35 v‹f.

PQRT -‹ R‰wsÎ mÂfg£rkhf ÏU¡f

nt©Lkhdhš QR-‹ x¤j g¡f« BC Mf

ÏU¡f nt©L«.

Ï‹ R‰wsÎÏ‹ R‰wsÎ

ABC

PQR

T

T = BCQR

435=

Mfnt, PQRT -‹ äf mÂfkhd R‰wsÎ = 19435

# = 166.25 br.Û.

kh‰WKiw: PQR ABCT T+ v‹gjhš, eh« bgWtJ

PQ6

= PR435

9= (QR v‹gJ g¡f« BC-‹ x¤j g¡f«)

PQ6

= 6 52.5PQ435

435

2105& #= = =

PR9

= 9 78.75PR435

435

4315& #= = =

Mfnt, PQRT -‹ äf mÂfkhd R‰wsÎ

= PQ QR PR+ + = 52.5 35 78.75 166.25+ + = br.Û

9. gl¤Âš DE BC< . nkY«, BDAD

53= våš,

(i) ‹ gu¥gsÎ‹ gu¥gsÎ

ABCADE

--

TT ,

(ii) ‹ gu¥gsÎ

rçtf« ‹ gu¥gsÎABC

BCED-

-T

M»adt‰¿‹ kÂ¥òfis¡ fh©f.

Ô®Î: (i) ABC-š, DE || BC. ` ADE ABCT T+

BDAD = 3 , 5AD k BD k

53 & = =

Ï‹ gu¥òÏ‹ gu¥ò

ABC

ADE

T

T = AB

AD2

2

( )

( )

k

k

8

3649

2

2

& =

(ii) TADE-‹ gu¥ò = 9k k‰W« ABCT -‹ gu¥ò 64k=

rçtf« BCED-‹ gu¥ò

= ABCT -‹ gu¥ò – TADE-‹ gu¥ò

= 64 9 55k k k- =

‹ gu¥ò

rçtf« ‹ gu¥òABC

BCEDT

= kk

6455

= 6455 .

(mšyJ) ‹gu¥ò

rçtf« ‹gu¥ò

ABC

BCED

T

= ‹ gu¥ò

‹ gu¥ò ‹ gu¥ò

ABC

ABC ADE

T

T T-

= ‹gu¥ò

‹gu¥ò

ABC

ADE1

T

T- = 1

64

9

64

55- =


10. muR, xU khefçš ga‹gL¤j¥glhj ãy¥gFÂ x‹¿š òÂa bjhê‰ng£ilæid

ãWt¤ Â£läL»wJ. ãHè£l¥ gFÂ òÂajhf mik¡f¥gL« bjhê‰ng£il

gFÂæ‹ gu¥gsit¡ F¿¡»wJ. Ï¥gFÂæ‹ gu¥gsit¡ fh©f.

Ô®Î: ,EAB EDCT T M»at‰¿èUªJ AB CD< v‹gJ bjëÎ

nkY«, AEB+ = DEC+ [F¤bjÂ® nfhz§fŸ]

EAB+ = EDC+ [x‹Wé£l nfhz§fŸ]

tobth¤j K¡nfhz§fë‹ AA tiuaiwæ‹go, eh«

bgWtJ EAB EDCT T+

DCAB =

EGEF

& EF = ( . )DCAB EG

13 1 4# #= = 4.2km .

òÂajhf mik¡f¥gL« bjhê‰ng£il gFÂæ‹ gu¥gsÎ

= EABT -‹ gu¥ò

= AB EF21# #

= 3 4.2 6.321# # = r.Ñ.Û

11. xU ÁWt‹ itu¤Â‹ FW¡F bt£L¤ njh‰w toéš, gl¤Âš fh£oathW xU g£l«

brŒjh‹. Ï§F AE = 16 br.Û, EC = 81 br.Û. mt‹ BD v‹w FW¡F¡ F¢Áæid¥

ga‹gL¤j éU«ò»wh‹. m¡F¢Áæ‹ Ús« v›tsÎ ÏU¡fnt©L«?

Ô®Î: bfhL¡f¥g£l gl¤Âš ADCT xU br§nfhz K¡nfhz«. k‰W« DE AC=

[xU br§nfhz K¡nfhz¤Âš KidæèUªJ f®z¤Â‰F br§F¤J¡nfhL

tiuªjhš br§F¤J¡nfh£o‹ ÏUòwK« mikªj K¡nfhz§fŸ

x‹W¡bfh‹W tobth¤jit. nkY« jå¤jåna KGikahd K¡nfhz¤Â‰F

tobth¤jitahF« vd ek¡F bjçÍ«]

EAD EDCT T+ vdnt, EDEA

ECED=

& ED2 = 16 81EA EC# #=

& ED = 4 9 3616 81# #= = .

ABDT xU ÏUrkg¡f br§nfhz K¡nfhz« k‰W« AE BD=

BE = ED

Mfnt, BD = ED2 = 2 × 36 = 72 cm.

12. xU khzt‹ bfho¡f«g¤Â‹ cau¤Âid¡ fz¡»l éU«ò»wh‹.

bfho¡f«g¤Â‹ c¢Áæ‹ vÂbuhë¥ig¡ f©zhoæš fhQ« tifæš, xU

ÁW f©zhoia¤ jiuæš it¡»wh‹. m¡f©zho mtåläUªJ 0.5Û

bjhiyéš cŸsJ. f©zho¡F« bfhof«g¤Â‰F« Ïilna cŸs bjhiyÎ

3Û k‰W« mtDila »ilk£l¥ gh®it¡ nfhL jiuæèUªJ 1.5 Û cau¤Âš

cŸsJ våš, bfho¡f«g¤Â‹ cau¤ij¡ fh©f. (khzt‹, f©zho k‰W«

bfho¡ f«g« M»ad xnu ne®¡nfh£oš cŸsd.)


Ô®Î: AB k‰W« ED v‹gd Kiwna khzt‹ k‰W« bfho¡f«g¤Â‹ cau§fŸ

v‹f.

C v‹gJ f©zhoæš bfhof«g¤Â‹ gLòŸë v‹f.

k‰W«ABC EDCT T M»at‰¿š 90 ,ABC EDC BCA DCE+ + + += = =c

Mfnt, tobth¤j K¡nfhz§fë‹ AA tiuaiwæ‹go, eh« bgWtJ ABC EDCT T+ .

EDAB =

DCBC . .

ED1 5

30 5& =

& . ED0 5 = 4.5 & 9ED = Û. vdnt, bfhof«g¤Â‹ cau« 9 Û.

13. xU nk‰Tiu gl¤Âš fh£oathW FW¡F bt£L¤ njh‰w¤ij¡ bfh©LŸsJ.

ÏÂš

(i) tobth¤j K¡nfhz§fis¤ bjçªbjL¡fÎ«.

(ii) Tiuæ‹ cau« h-I¡ fh©f.

Ô®Î: (i) xU br§nfhz K¡nfhz¤Â‹ KidæèUªJ

f®z¤Â‰F br§F¤J¡ nfhL tiuªjhš br§F¤J¡nfh£o‹

ÏUòwK« mikªj K¡nfhz§fŸ x‹W¡ bfh‹W tobth¤

jit. nkY« jå¤jåna KGikahd K¡nfhz¤Â‰F

tobth¤ jitahF« vd ek¡F bjçÍ«

bfhL¡f¥g£l gl¤ÂèUªJ »il¡F« tobth¤j K¡nfhz§fŸ

(i) WZY YZXT T+ , (ii) WYX YZXT T+ k‰W« (iii) WZY WYXT T+ (mšyJ)(i) XWY YWZT T+ , (ii) YWZ XYZT T+ k‰W« (iii) XWY XYZT T+

(ii) XWY XYZT T+ -š YWZ XYZT T+ -š

YZWY

XZXY= & h

8 106= (mšyJ)

XYYW

XZYZ= & h

6 108=

4.8h` = Û 4.8h` = Û

gæ‰Á 6.3 1. gl¤Âš TP xU bjhLnfhL. A, B v‹gd t£l¤Â‹ ÛJŸs òŸëfŸ. BTP 72+ = c

k‰W« +ATB = 43c våš +ABT-I¡ fh©f.

Ô®Î: BAT+ = 72PTB+ = c [kh‰W t£l¤J©Lfëš mikªj nfhz§fŸ]ABTT -æèUªJ

ATB BAT ABT+ + ++ + = 180c

43 72 ABT++ +c c = 180c

ABT` + = 65c

2. xU t£l¤Âš AB, CD v‹D« ÏU eh©fŸ x‹iwbah‹W c£òwkhf P-æš bt£o¡

bfhŸ»‹wd.

(i) CP = 4 br.Û, AP = 8 br.Û., PB = 2 br.Û våš, PD-I¡ fh©f.

(ii) AP = 12 br.Û, AB = 15 br.Û, CP = PD våš, CD-I¡ fh©f.


Ô®Î: (i) xU t£l¤Âš AB k‰W« CD v‹w ÏU eh©fŸ x‹iwbah‹W

c£òwkhf P v‹w òŸëæš bt£o¡bfhŸtjhš, PA PB# = PC PD#

& PD = PC

PA PB# = 44

8 2# = br.Û.

(ii) CP = PD, AP = 12 br. Û k‰W« AB = 15 br. Û. vd bfhL¡f¥g£LŸsJ.

vdnt, 15 15 12 3AP PB PB&+ = = - = br.Û

Mfnt, PA PB# = PC PD#

& PD2 = PA PB# PC PDa =6 @& PD2 = 36 & 6PD = br.Û

CD = 2 12PD = br.Û.

3. AB k‰W« CD v‹w ÏU eh©fŸ t£l¤Â‰F btëna P vD« òŸëæš bt£o¡

bfhŸ»‹wd.

(i) AB = 4 br.Û, BP = 5 br.Û k‰W« PD = 3 br.Û våš, CD-I¡ fh©f.

(ii) BP = 3 br.Û, CP = 6 br.Û k‰W« CD = 2 br.Û våš, AB-I¡ fh©f.

Ô®Î: (i) AB k‰W« CD v‹w ÏU eh©fŸ x‹iwbah‹W btë¥òwkhf P v‹w

òŸëæš bt£o¡bfhŸtjhš,

PA PB# = PC PD#

& 9 5# = [3 ] (3)CD+

& 3 CD+ = 153

9 5# = br.Û

Mfnt, CD = 12 br.Û

(ii) CP = 6 br.Û vd bfhL¡f¥g£LŸsJ. vdnt,

& CD PD+ = 6 4PD` = br.Û.

PA PB# = PC PD#

vdnt, ( )AB PB PB#+ = PC PD#

(3 ) 3AB& #+ = 6 4#

& 3 AB+ = 83

6 4# =

Mfnt, AB = 5 br.Û.

4. xU t£l«, TABC-š g¡f« BC-I P-š bjhL»wJ. m›t£l« AB k‰W« AC-fë‹

Ú£Áfis Kiwna Q k‰W« R-š bjhL»wJ våš, AQ = AR = 21 (TABC-‹ R‰wsÎ)

vd ãWÎf.

Ô®Î: btëna cŸs xU òŸëæèUªJ t£l¤Â‰F tiua¥gL« ÏU bjhLnfh£L

J©Lfë‹ Ús§fŸ rk« vd eh« m¿nth«. vdnt,


BQ = BP g(1) [òŸë B-æèUªJ tiuag¥g£l bjhLnfhLfŸ] CP = CR g(2) [òŸë C-æèUªJ tiuag¥g£l bjhLnfhLfŸ] AQ = AR g(3) [òŸë A-æèUªJ tiuag¥g£l bjhLnfhLfŸ] ABCT -‹ R‰wsÎ = AB BC CA+ +

= AB BP PC CA+ + +

= ( ) ( )AB BP PC CA+ + +

= ( ) ( )AB BQ CR CA+ + + (1) k‰W« (2)I ga‹gL¤j

= 2AQ AR AR AR AR+ = + = , (3)I ga‹gL¤j

AR = 21 ( ABCT -‹ R‰wsÎ )

Mfnt, (3)èUªJ eh« bgWtJ, AR = AQ = 21 ( ABCT -‹ R‰wsÎ )

5. xU Ïizfu¤Â‹ všyh¥ g¡f§fS« xU t£l¤Âid bjhLkhdhš

m›éizfu« xU rhŒrJukhF« vd ãWÎf.

Ô®Î: ABCD xU Ïizfu« v‹f. AB, BC, CD k‰W« DA v‹w g¡f§fŸ t£l¤ij

bjhL« òŸëfŸ Kiwna P,Q,R k‰W« S v‹f.

btëna cŸs òŸëæèUªJ t£l¤Â‰F tiua¥gL« bjhLnfhLfë‹ Ús§fŸ

rk« vd eh« m¿nth«. vdnt,

(1) ; (2) ; (3) ; (4)AP AS BP BQ CR CQ DR DSg g g g= = = =

(1), (2), (3) k‰W« (4) I T£l, eh« bgWtJ

AP BP CR DR+ + + = AS BQ CQ DS+ + +

( ) ( )AP BP CR DR+ + + = ( ) ( )AS DS BQ CQ+ + +

AB CD+ = AD BC+

& 2 2AB AD= mšyJ AB AD= [a ABCD xU Ïizfu« ,AB CD BC AD= = ]

Mfnt, AB BC CD AD= = = ABCD` X® rhŒrJukhF«.

6. xU jhkiu¥ óthdJ j©Ù® k£l¤Â‰F nkš 20 br.Û cau¤Âš cŸsJ. j©o‹

ÛÂ¥ gFÂ j©Ù® k£l¤Â‰F ÑnH cŸsJ. fh‰W ÅR«nghJ j©L jŸs¥g£L,

jhkiu¥ óthdJ j©o‹ Mu«g ãiyæèUªJ 40 br.Û öu¤Âš j©Ùiu¤

bjhL»wJ. Mu«g ãiyæš j©Ù® k£l¤Â‰F¡ ÑnH cŸs j©o‹ Ús« fh©f?

Ô®Î: j©ÙU¡FŸ _œ»ÍŸs j©o‹ mo¥gFÂ O v‹f. Ã‹d® B v‹gJ jhkiu¥ó v‹f. AB v‹gJ Ú®k£l¤Â‰F nkš cŸs k‰W«

OA v‹gJ Ú®k£l¤Â‰F Ñœ cŸs j©o‹ ÚskhF«.

OA = x br.Û v‹f.

fh‰W ÅR« nghJ jhkiu¥óé‹ j©L j©Ùç‹ nk‰gu¥ig

bjhL«òŸë C v‹f.


vdnt, OC = OA + AB = x + 20 br.Û

Ãjhfu° nj‰w¤Â‹go, eh« bgWtJ OC OA AC2 2 2= +

( 20)x 2+ = 40x2 2

+

40 400x x2+ + = 1600x2 +

x40 = 1200 x` = 30 br.Û

Mfnt, j©Ùç‹ k£l¤Â‰F ÑnH cŸs j©o‹ Ús« 30 br.Û MF«.

7. br›tf« ABCD-‹ c£òw òŸë O-éèUªJ br›tf¤Â‹ KidfŸ A, B, C, D

Ïiz¡f¥g£LŸsd våš, OA OC OB OD2 2 2 2+ = + vd ãWÎf.

Ô®Î: O têahf, EOF AB< vd tiuf.

Ã‹d®, ABFE k‰W« EFCD v‹gd br›tf§fshF«.

br§nfhz OEAT -š, Ãjhfu° nj‰w¤Â‹go, OA OE EA2 2 2= + g (1)

br§nfhz OFCT -š, Ãjhfu° nj‰w¤Â‹go, OC OF FC2 2 2= + g (2)

br§nfhz OFBT -š, Ãjhfu° nj‰w¤Â‹go, OB OF FB2 2 2= + g (3)

br§nfhz OEDT -š, Ãjhfu° nj‰w¤Â‹go, OD OE ED2 2 2= + g (4)

(3) k‰W« (4) I T£l,

OB OD2 2+ = OF FB OE ED2 2 2 2

+ + +

= ( ) ( )OE FB OF ED2 2 2 2+ + +

= ( ) ( )OE EA OF FC2 2 2 2+ + +

[ ABCDa k‰W« EFCD v‹gd br›tf§fŸ, FB = EA k‰W« ED = FC

= OA OC2 2+ (1) k‰W« (2) I ga‹gL¤j

gæ‰Á 6.4

1. TABC-‹ g¡f§fŸ AB k‰W« AC M»at‰iw xU ne®¡nfhL Kiwna D k‰W« E-òŸëfëš bt£L»wJ. nkY«, m¡nfhL BC-¡F Ïiz våš

ACAE =

(A) DBAD (B)

ABAD (C)

BCDE (D)

ECAD

Ô®Î: njš° nj‰w¤Â‹go, ACAE

ABAD= ( éil. (B) )

2. TABC-š AB k‰W« AC-fëYŸs òŸëfŸ D k‰W« E v‹gd DE < BC v‹wthW

cŸsd. nkY«, AD = 3 br.Û, DB = 2 br.Û k‰W« AE = 2.7 br.Û våš, AC =

(A) 6.5 br.Û (B) 4.5 br.Û (C) 3.5 br.Û (D) 5.5 br.Û

Ô®Î: njš° nj‰w¤Â‹go

BDAD =

ECAE & EC =

ADAE BD# = . 1.8

32 7 2# = br.Û

AC = 2.7 1.8 4.5AE EC+ = + = br.Û ( éil. (B) )


3. TPQR-š RS v‹gJ R+ -‹ nfhz c£òw ÏUrkbt£o. PQ = 6 br.Û, QR = 8 br.Û,

RP = 4 br.Û våš, PS = (A) 2 br.Û (B) 4 br.Û (C) 3 br.Û (D) 6 br.Û

Ô®Î: PS x= br.Û. 6SQ x= - br.Û v‹f.

RS v‹gJ PRQ+ -‹ nfhz ÏUrkbt£o, vdnt

QRPR =

SQPS

84& = 2 6 2

xx x x x

6& &

-= - = ( éil. (A) )

4. gl¤Âš ACAB

DCBD= , 40B c+ = k‰W« 60C c+ = våš, BAD+ =

(A) 30c (B) 50c

(C) 80c (D) 40c

Ô®Î: ACAB

DCBD AD&= v‹gJ BAC+ -‹ nfhz ÏUrkbt£o

vdnt, ABC BCA CAB+ + ++ + = 180c 40 60 2 BAD++ + = 180c & 40BAD+ = c ( éil. (D) )

5. gl¤Âš x-‹ kÂ¥ghdJ

(A) 4 2$ (B) 3 2$ (C) 0 8$ (D) 0 4$

Ô®Î: njš° nj‰w¤Â‹go, .BDAD

ECAE x x

8 104 3 2& &= = = ( éil. (B) )

6. ABCT k‰W« DEFT -fëš k‰W«B E C F+ + + += = våš,

(A) DEAB

EFCA= (B)

EFBC

FDAB= (C)

DEAB

EFBC= (D)

FDCA

EFAB=

Ô®Î: tobth¤j K¡nfhz§fë‹ AA tiuaiwæ‹go, ~ABC DEFT T .

Mfnt, DEAB

EFBC= ( éil. (C) )

7. bfhL¡f¥g£l gl¤Â‰F¥, bghUªjhj T‰¿id¡ f©l¿f.

(A) ADBT + ABCT (B) ABDT + ABCT

(C) BDCT + ABCT (D) ADBT + BDCT

Ô®Î: ~ABD ABCT T v‹gJ jtwhd T‰whF« ( éil. (B) ) 8. 12 Û ÚsKŸs xU ne®¡F¤jhd F¢Á, 8 Û ÚsKŸs ãHiy¤ jiuæš V‰gL¤J»wJ.

mnj neu¤Âš xU nfhòu« 40 Û ÚsKŸs ãHiy¤ jiuæš V‰gL¤J»wJ våš,

nfhòu¤Â‹ cau«

(A) 40 Û (B) 50 Û (C) 75 Û (D) 60 Û

Ô®Î: nfhòu ãHè‹ Ús«nfhòu¤Â‹ cau« =

F¢Áæ‹ ãHè‹ Ús«F¢Áæ‹ cau«

nfhòu¤Â‹ cau« = 40812

# = 60 Û. ( éil. (D) )


9. ÏU tobth¤j K¡nfhz§fë‹ g¡f§fë‹ é»j« 2:3 våš, mt‰¿‹

gu¥gsÎfë‹ é»j«

(A) 9:4 (B) 4:9 (C) 2:3 (D) 3:2

Ô®Î: 2 : 3 4 : 92 2= . ( éil. (B) )

10. K¡nfhz§fŸ ABC k‰W« DEF tobth¤jit. mt‰¿‹ gu¥gsÎfŸ Kiwna 100 br.Û 2, 49 br.Û 2 k‰W« BC = 8.2 br.Û våš, EF =


Ô®Î: ‹ gu¥ò‹ gu¥ò

DEFABC

EF

BC

EF

BC49100

2

2

2

2&

TT

= =

Mfnt, . 5.74EFBC EF

710

107 8 2& #= = = br.Û ( éil. (B) )

11. ÏU tobth¤j K¡nfhz§fë‹ R‰wsÎfŸ Kiwna 24 br.Û, 18 br.Û v‹f. Kjš

K¡nfhz¤Â‹ xU g¡f« 8 br.Û våš, k‰bwhU K¡nfhz¤Â‹ mj‰F x¤j

g¡f«

(A) 4 br.Û (B) 3 br.Û (C) 9 br.Û (D) 6 br.Û

Ô®Î: ™

ˆˆ

Ïu©lh« K¡nfhz Â‹ R‰wsÎKj K¡nfhz Â‹ R‰wsÎ = ™

ˆ ˆˆ

Ïu©lh« K¡nfhz Â‹ mj‰F x j g¡f«Kj K¡nfhz Â‹ xU g¡f«

Ïu©lh« K¡nfhz¤Â‹ x¤j g¡f« = 624

8 18# = br.Û ( éil. (D) )

12. AB, CD v‹gd xU t£l¤Â‹ ÏU eh©fŸ. mit Ú£l¥gL«nghJ P-š rªÂ¡»‹wd k‰W« AB = 5 br.Û, AP = 8 br.Û, CD = 2 br.Û våš, PD =


Ô®Î: PD = x br.Û & PA × PB = PC × PD

8 × 3 = (x + 2) x & x x2 24 02+ - = & x = 4 ( éil. (D) )

13. gl¤Âš eh©fŸ AB k‰W« CD v‹gd P-š bt£L»‹wd AB = 16 br.Û, PD = 8 br.Û, PC = 6 k‰W« AP >PB våš, AP =

(A) 8 br.Û (B) 4 br.Û

(C) 12 br.Û (D) 6 br.Û

Ô®Î: PA = x br.Û & PA × PB = PC × PD

x (16 – x) = 6 × 8 & x x16 48 02- + = & (x – 4) (x – 12) = 0

x = 4 mšyJ x = 12. Mdhš, AP > PB ` AP = 12 br.Û ( éil. (C) )

14. P v‹D« òŸë, t£l ika« O-éèUªJ 26 br.Û bjhiyéš cŸsJ. P-æèUªJ

t£l¤Â‰F tiua¥g£l PT v‹w bjhLnfh£o‹ Ús« 10 br.Û våš, OT =


Ô®Î: 26 10 24OP OT TP OT2 2 2 2 2 2&= + = - = ( éil. (D) )


15. gl¤Âš, PAB 120+ = c våš, BPT+ =

(A) 120o (B) 30o

(C) 40o (D) 60o

Ô®Î: 120BCP+ + c = 180° (` ABCP xU t£l eh‰fu« ) ` BCP+ = 60c mjdhš, BPT BCP+ += = 60c ( éil. (D) )

16. O-it ikakhf cila t£l¤Â‰F PA, PB v‹gd btë¥òŸë P-æèUªJ

tiua¥g£l¤ bjhLnfhLfŸ. Ï¤bjhLnfhLfS¡F Ïilæš cŸs nfhz« 40o

våš, POA+ = (A) 70o (B) 80o (C) 50o (D) 60o

Ô®Î: , 20OAP OBP APOT T +, = c

OAPT -š 90 20 180POA+ + + =c c c ` POA 70+ = c ( éil. (A) ) 17. gl¤Âš, PA, PB v‹gd t£l¤Â‰F btënaÍŸs òŸë

P-æèUªJ tiua¥g£l¤ bjhLnfhLfŸ. nkY« CD v‹gJ Q v‹w òŸëæš t£l¤Â‰F bjhLnfhL. PA = 8 br.Û, CQ = 3 br.Û

våš, PC =

(A) 11 br.Û (B) 5 br.Û

(C) 24 br.Û (D) 38 br.Û

Ô®Î: PB = PA & PC + BC = 8 & PC + PQ = 8 & PC = 5 br.Û ( éil. (B) )

18. br§nfhz ABCD -š B 90+ = c k‰W« BD AC= . BD = 8 br.Û, AD = 4 br.Û våš, CD =


Ô®Î: DBDC

DADB= & DCB DBA

DBDC

DADB&T T+ =

DC DA# = DB2

4DC = 82 & DC = 16 br.Û ( éil. (B) ) 19. Ïu©L tobth¤j K¡nfhz§fë‹ gu¥gsÎfŸ Kiwna 16 br.Û 2, 36 br.Û 2.

Kjš K¡nfhz¤Â‹ F¤Jau« 3 br.Û våš, k‰bwhU K¡nfhz¤Âš mjid

x¤j F¤Jau«

(A) 6.5 br.Û (B) 6 br.Û (C) 4 br.Û (D) 4.5 br.Û

Ô®Î: ™ ™

ˆˆ

ˆ ˆ

ˆ ˆÏu©lh« K¡nfhz Â‹ gu¥ò

Kj K¡nfhz Â‹ gu¥ò

Ïu©lh« K¡nfhz Â‹ F Jau«

Kj K¡nfhz Â‹ F Jau«2

2=^

^

h

h

Ïu©lh« K¡nfhz¤Â‹ F¤Jau« = 16

36 9# = 4.5 br.Û ( éil. (D) )

20. ÏU tobth¤j K¡nfhz§fŸ ABCD k‰W« DEFD M»at‰¿‹ R‰wsÎfŸ

Kiwna 36 br.Û, 24 br.Û. nkY«, DE = 10 br.Û våš, AB =


Ô®Î: Ï‹ R‰wsÎÏ‹ R‰wsÎ

DEF

ABC

TT =

DCAB , AB =

2436 10# = 15 br.Û ( éil. (C) )


gæ‰Á 7.1

1. Ã‹tUtd x›bth‹W« K‰bwhUik MFkh vd¡ fh©f.

(i) cos sec sin22 2i i i+ = + , (ii) cot cos sin2 2i i i+ = .

Ô®Î: (i) 45oi= våš, 2 2.5cos sec2

1 2212 2 2 2

i i+ = + = + =c ^m h

k‰W« 2+sini = 2 + 2

1

45oi= våš, 2cos sec sin2 2 !i i i+ +

vdnt , 2cos sec sin2 2i i i+ = + MdJ xU K‰bwhUik mšy.

(ii) 30oi= våš, 3cot cos 323

232 2

i i+ = + = +^ h k‰W« sin21

412 2

i = =` j

30oi= våš, cot cos sin2 2!i i i+ .

vdnt , cot cos sin2 2i i i+ = MdJ xU K‰bwhUik mšy.

2. Ã‹tU« K‰bwhUikfis ãWÎf.

(i) sec cosec sec cosec2 2 22

i i i i+ =

Ô®Î: (i) sec cosec22

i i+

= cos sin

1 12 2i i

+ = cos sin

sin cos2 2

2 2

i i

i i+ = cos sin

12 2i i

= sec cosec22

i i

F¿¥ò: Ï¡fz¡»š cŸs ÏU cW¥òfë‹ T£lš k‰W« bgU¡fš rkkhf cŸsJ.

(ii) cos

sin cosec cot1 i

i i i-

= + .

Ô®Î: cos

sin1 i

i-

= cos

sincoscos

1 11

#ii

ii

- ++c m = (1 )

cos

sin cos1 2

#i

i i-

+

= (1 )sin

sin cos2i

i i+ = 1sincosii+

= sin sin

cos1i i

i+ = cosec coti i+

(iii) sinsin sec tan

11

ii i i

+- = - .

Ô®Î: sinsin

sinsin

sinsin

11

11

11

#ii

ii

ii

+- =

+-

--

kh‰W Kiw:

( )( )cos cosec cot1 i i i- +

= cosec cot cotsincos2i i iii+ - -

= sin i

F¿¥ò: 90i = c vd¡bfh©L«

rçgh®¡fyh«

K¡nfhzéaš7

Ô®Î - K¡nfhzéaš 191

= sin

sin

1

12

2

i

i

-

-^ h

= cos

sin12

2

i

i-^ h = cossin1ii-

= sec tani i-

(iv) 1sec tan

cos sini ii i

-= + .

Ô®Î: sec tan

cosi ii

- =

sec tan sec tan

cos sec tan

i i i i

i i i

- +

+

^ ^^

h hh

= 1coscos cos

sinii i

i+c m sec tan 12 2a i i- =^ h

= sin1 i+ .

kh‰W Kiw: ( )(1 )sec tan sin sec tan tancossin2i i i i i iii- + = + - -

= cossin cos1 2

ii i- =

(v) sec cosec tan cot2 2i i i i+ = + .

Ô®Î: sec cosec2 2i i+ = tan cot1 12 2i i+ + +^ ^h h

= tan cot 22 2i i+ +

= tan cot tan cot22 2i i i i+ + tan cot 1a i i =^ h

= tan cot 2i i+^ h = tan coti i+

(vi) sin cos

cos sin cot1

1 2

i ii i i+

+ - =^ h

.

Ô®Î: sin cos

cos sin1

1 2

i ii i+

+ -^ h

= sin cos

sin cos1

1 2

i ii i+

- +^ h

= sin coscos cos

1

2

i ii i++

^ h

= ( )sin coscos cos

11

i ii i

++

^ h =

sincosii = coti

(vii) 1sec sin sec tan1i i i i- + =^ ^h h .

Ô®Î: sec sin sec tan1i i i i- +^ ^h h

= cos

sin sec tan1 1i

i i i- +^ ^h h

= 1cos cos

sin sec tani i

i i i- +c ^m h

= sec tan sec tani i i i- +^ ^h h

= sec tan2 2i i-^ h = 1

kh‰W Kiw: ( ) ( (1 ))cot sin cosi i i+

= ( )( )cos cos1i i+

= cos cos2i i+

= cos sin1 2i i+ -

kh‰W Kiw: [ ( )] ( )sec sin sec tan1i i i i- +

= ( )( )sec tan sec tani i i i- +

= sec tan 12 2i i- =


(viii) cosec cot

sini ii+

= 1 cosi- .

Ô®Î: cosec cot

sini ii+

= cosec cot

sincosec cotcosec cot

#i ii

i ii i

+ --c m

= cosec cot

sinsin sin

cos

2 2

1

i i

i

-

-i i

ic m = sin

sinsin

sincos1

#ii

iii- c m

cosec cot 12 2a i i- =^ h

= cos1 i-

kh‰W Kiw:I kh‰W Kiw:II

sin sincos

sin1i i

ii

+ =

cossin

1

2

ii

+ ( )( )cos cosec cot1 i i i- +

= coscos

11 2

ii

+- = cosec cot cot

sincos2i i iii+ - -

= 1

1 1

cos

cos cos

i

i i

+

- +^ ^h h = sin sin

cos sin1 2

i ii i- =

= cos1 i-

3. Ã‹tU« K‰bwhUikfis ãWÎf.

(i) sin

sin

coscos sec

1

90

1 902

i

i

ii i

+

-+

- -=

c

c^

^h

h.

Ô®Î: sin

sin

coscos

1

90

1 90i

i

ii

+

-+

- -

c

c^

^h

h

= 1sin

cossin

cos1 i

iii

++

-

= sin sin

cos sin cos sin

1 1

1 1

i i

i i i i

+ -

- + +

^ ^^ ^

h hh h

= sin

cos cos sin cos cos sin

1 2i

i i i i i i

-

- + +

= cos

cos22i

i = cos2i

= sec2 i sin cos1 2 2a i i- =^ h

(ii) 1cot

tantan

cot sec cosec1 1i

iii i i

-+

-= + .

Ô®Î: cot

tantan

cot1 1i

iii

-+

-

= tantan

tancot

1 1

2

ii

ii

--

-


= tan

tantan1

1 12

ii

i--

^`

hj

= ( )tan tan

tan1

13

i ii-

-

= tan tan

tan tan tan

1

1 12

i i

i i i

-

- + +

^^ ^

hh h

= tan

tan tan12

ii i+ +

= tansec

tantan2

ii

ii+ = 1

cos sincos1

2#i i

i +

= 1cos sin1 1

#i i

+ = sec cosec 1i i+

(iii) tan

sin

cot

coscos sin

1

90

1

90

i

i

i

ii i

-

-+

-

-= +

c c^ ^h h .

Ô®Î: tan

sin

cot

cos

1

90

1

90

i

i

i

i

-

-+

-

-c c^ ^h h

=

cossin

cos

sincos

sin

1 1iii

iii

-+

-

= cos sin

cossin cos

sin2 2

i ii

i ii

-+

-

= cos sin

cos sin1 2 2

i ii i

--^ h

= cos sin

cos sin cos sin

i i

i i i i

-

+ -^ ^h h

= cos sini i+

(iv) .cosec

tan

cotcosec sec

1

90 1 2i

i

ii i

+

-+ + =

c^ h

Ô®Î: cosec

tan

cotcosec

1

90 1i

i

ii

+

-+ +c^ h

= cosec

cotcot

cosec1

1ii

ii

++ +

= sin

coscossin

11

ii

ii

++ +

= cos sin

cos sin

1

12 2

i i

i i

+

+ +

^^

hh

kh‰W Kiw:

= tan

costansin tan

1 1ii

ii i

-+

-

= tan

cos sincossin

1 i

i iii

-

- c m

= cos sincos sin cos sin

2 2

i ii i i i-- = +

kh‰W Kiw:

= ( )( )

cot coseccot cosec

112 2

i ii i

++ +

= ( )

( ) ( )cot cosec

cosec cosec cosec1

1 1 22 2

i ii i i

+- + + +

= ( )( )

cot coseccosec cosec

sec1

2 12

i ii i

i++

=

kh‰W Kiw:

sin

cos

cos

sin

cos

sin

sin

cos

1 1i

i

i

i

i

i

i

i

-

+

-

= ( ) ( )cos sin cos

sin

sin cos sin

cos2 2

i i i

i

i i i

i

-+

-

= ( )sin cos sin cos

sin cos3 3

i i i i

i i

-

-

= ( )

( ) ( )

sin cos sin cos

sin cos sin cos sin cos2 2

i i i i

i i i i i

-

- + +

( ( )( ))a b a b a ab b3 3 2 2a - = - + +

= 1sin cos

sin coscosec sec

1

i i

i ii i

+= +


= cos sin

cos sin sin1

1 22 2

i ii i i

++ + +

^ h

= 1

1

cos sin

sin2

i i

i

+

+

^^

hh = sec2 i

(v) .cot coseccot cosec cosec cot

11

i ii i i i- ++ - = +

Ô®Î: cot coseccot cosec

11

i ii i- ++ -

= ( )cot cosec

cot cosec cosec cot1

2 2

i ii i i i

- ++ - - ( 1)cosec cot2 2a i i- =

= 1

( )( )cot cosec

cot cosec cosec cot cosec coti i

i i i i i i

- ++ - + -

= ( )( ( ))cot cosec


1i i

i i i i

- ++ - -

= ( )

( )( )cot cosec


1i i

i i i i

- ++ - +

= cot coseci i+

(vi) 2cot cosec tan sec1 1i i i i+ - + + =^ ^h h .

Ô®Î: cot cosec tan sec1 1i i i i+ - + +^ ^h h

= sincos

sin cossin

cos1 1 1 1

ii

i ii

i+ - + +c cm m

= (( ) )(( ) )sin cos

sin cos cos sin1 1i i

i i i i+ - + +

= ( )sin cos

sin cos 12

i ii i+ -

= sin cos

sin cos sin cos2 12 2

i ii i i i+ + -

= sin cossin cos1 2 1i ii i+ - ( 1)sin cos2 2a i i+ =

= sin cossin cos2 2i ii i =

(vii) sin cossin cos

sec tan11 1

i ii i

i i+ -- + =

-.

Ô®Î: sin cossin cos

11

i ii i+ -- +

kh‰W Kiw:( ) ( 1)cosec cot cot coseci i i i+ - +

= cosec cot cosec cosec cot2 2

i i i i i- + +

cot cosec coti i i- +

= cot cosec cosec cosec 12 2

i i i i+ - + -

= cot cosec 1i i+ -


gFÂ k‰W« bjhFÂia cos i Mš tF¡f

= cossin

coscos

cos

cossin

coscos

cos1

1

+ -

- +

ii

ii

i

ii

ii

i

= 1tan sec

tan sec1i ii i+ -- +

= tan sectan sec

11

i ii i+ -+ -

= ( )tan sec

tan sec sec tan1

2 2

i ii i i i

+ -+ - - ( 1)sec tan2 2a i i- =

= ( )( )tan sec

tan sec sec tan sec tan1i i

i i i i i i

+ -+ - + -

= ( )( ( ))tan sec

tan sec sec tan1

1i i

i i i i

+ -+ - -

= ( )( )tan sec

tan sec sec tan11

i ii i i i

+ -+ - +

= tan seci i+

= ( )sec tansec tansec tan

#i ii ii i+--c m

= sec tansec tan2 2

i ii i--c m

= sec tan

1i i-

( 1)sec tan2 2a i i- =

(viii) tan

tan

sin

sin sin

1 2 90 1

902 2i

i

i

i i

-=

- -

-

c

c

^

^

h

h .

Ô®Î: tan

tan

12i

i

-= 1

cossin

cossin

2

2-

iiii

=

coscos sin

cossin

2

2 2

ii iii

-

= cossin

cos sin

cos2 2

2

#ii

i i

i

-

= cos sin

sin cos2 2i i

i i

-

= ( )

cos cos

sin sin

1

902 2i i

i i

- -

-c

^ h ( 1 )sin cos2 2a i i= -

= ( )

cos

sin sin

2 1

902i

i i

-

-c

= ( )

( )

sin

sin sin

2 90 1

902 i

i i

- -

-

c

c

kh‰W Kiw:

( )( )sec tan sin cos 1i i i i- - +

= tan seccossin sin tan1

2i i

ii i i- + - + -

= sincossin1 1 2

iii- + -c m

= sin cos1i i- +

sin cossin cos

sec tan11 1

i ii i

i i- +- + =

-


(ix) cosec cot sin sin cosec cot

1 1 1 1i i i i i i-

- = -+

.

Ô®Î: Ï§F cosec cot 12 2i i- = ( )( ) 1cosec cot cosec cot& i i i i- + = g (1)

cosec cot sin

1 1i i i-

- = ( )cosec cot coseci i i+ - (1) èUªJ

= cosec cosec cot

1i

i i-

-^ h

= sin cosec cot1 1i i i-

+ (1) èUªJ

kh‰W Kiw:I kh‰W Kiw:II

cosec cot cosec cot

1 1i i i i-

++

cosec cot sin

1 1i i i-

-

= cosec cot

cosec cot cosec cot2 2i i

i i i i

-

+ + - = ( ) ( )cosec cot cosec cot

cosec cot

sin1

i i i i

i i

i- +

+-

^ h

= cosecsin sin sin

2 2 1 1ii i i

= = + = cosec cot

cosec cot cosec2 2i i

i i i-

+ -

= cosec cot coseci i i+ - = coti g (1)

sin cosec cot1 1i i i-

+

= coseccosec cot

cosec cot2 2

ii i

i i-

-

-^ h

= cosec cosec coti i i- + = coti g(2)

(1) k‰W« (2), èUªJ ãWt¥g£lJ

(x) ( )tan cosec

cot sec sin cos tan cot2 2

2 2

i i

i i i i i i+

+ = +^ h.

Ô®Î: tan cosec

cot sec2 2

2 2

i i

i i

+

+

= ( )

( )1

tan cot

cot tan

1

12 2

2 2

i i

i i

+ +

+ += g (1)

( , )sec tan cosec cot1 12 2 2 2i i i i= + = +

( )sin cos tan coti i i i+

= sin coscossin

sincosi i

ii

ii+c m

= sin cos 12 2i i+ = g (2)

(1) k‰W« (2) èUªJ ãWt¥g£lJ.

kh‰W Kiw:

[( ) ( )] ( )sin cos tan cot tan cosec2 2

i i i i i i+ +

= ( )( )sin cos tan cosec2 2 2 2i i i i+ +

= ( )( )sec cot1 1 12 2i i- + +

= sec cot2 2i i+


4. sec tanx a bi i= + k‰W« tan secy a bi i= + våš, x y a b2 2 2 2- = - vd

ãWÎf.

Ô®Î: x y2 2-

= ( ) ( )sec tan tan seca b a b2 2i i i i+ - +

= 2 ( 2 )sec tan sec tan tan sec tan seca b ab a b ab2 2 2 2 2 2 2 2i i i i i i i i+ + - + +

= sec tan tan seca a b b2 2 2 2 2 2 2 2i i i i- + -

= ( ) ( )sec tan tan seca b2 2 2 2 2 2i i i i- + -

= a b2 2- ( 1)sec tan2 2a i i- =

5. tan tanni a= k‰W« ,sin sinmi a= våš, cosn

m

1

12

22i =

-

- , n !+1 vd ãWÎf..

Ô®Î: bfhL¡f¥g£LŸs bjhl®òfëèUªJ a I Ú¡Fnth«.

tan tanni a= k‰W« sin sinmi a= vd¡ bfhL¡f¥g£LŸsJ.

cottan

n& ai

= k‰W« cosecsin

mai

=

cosec cot2 2a a- = 1

sin tan

m n2

2

2

2

i i- = 1

sin

cosm n2

2 2 2

i

i- = 1

cosm n2 2 2i- = sin2i

cosm n2 2 2i- = 1 cos2i-

1m2- = ( 1)cos n2 2i -

n

m

1

12

2

-

- = cos2i .

6. , k‰W«sin cos tani i i v‹gd bgU¡F¤ bjhlçš (G.P.) ÏU¥Ã‹ .cot cot 16 2i i- = vd ãWÎf.

Ô®Î: , k‰W«sin cos tani i i v‹gd bgU¡F¤ bjhlçš (G.P.) ÏU¡»‹wd.

vdnt, sincosii =

costanii

cos3& i = sin2i g (1)

cot cot6 2i i- = sin

cos

sin

cos6

6

2

2

i

i

i

i- = ( )

sin

cos

sin

cos6

3 2

2

2

i

i

i

i-

= ( )

sin

sin

sin

cos6

2 2

2

2

i

i

i

i- (1) I ga‹gL¤j

= sin

sin

sin

cos6

4

2

2

i

i

i

i- = sin sin

cos12 2

2

i i

i-

= sin

cos12

2

i

i- = sin

sin2

2

i

i = 1.


gæ‰Á 7.2 1. xU Rik C®ÂæèUªJ (truck) Rikia Ïw¡f VJthf 30c V‰w¡ nfhz¤Âš xU

rhŒÎ¤ js« (ramp) cŸsJ. rhŒÎ¤ js¤Â‹, c¢Á jiuæèUªJ 0.9 Û cau¤Âš

cŸsJ våš, rhŒÎ¤ js¤Â‹ Ús« ahJ?

Ô®Î: rhŒÎ js¤Â‹ c¢Áia C vdÎ«, rhŒÎ¤ js¤Â‹ Ús¤ij AC vdÎ«

bfhŸf.

0CAB 3+ = c k‰W« BC =0.9 Û vd¡ bfhL¡f¥g£LŸsJ.

br§nfhz TABC-š, 30sin c = ACBC

sin

AC BC30

=c

= 0.9 2 1.8 .Û# =

vdnt, rhŒÎ¤ js¤Â‹ Ús« 1.8 Û. 2. cau« 150 br.Û cŸs xU ÁWä xU és¡F¡ f«g¤Â‹ K‹ ã‹wthW 150 3 br.Û

ÚsKŸs ãHiy V‰gL¤J»whŸ våš, és¡F¡ f«g¤Â‹ c¢Áæ‹ V‰w¡

nfhz¤ij¡ fh©f.

Ô®Î: ÁWäæ‹ cau« BC vdÎ«, ÁWäæ‹ ãHš AB vdÎ« bfhŸf.

és¡F¡ f«g¤Â‹ c¢Áæ‹ V‰w¡nfhz« i v‹f.

150 , 150br.Û br.ÛAB BC3= = vd¡ bfhL¡f¥g£LŸsJ.

br§nfhz TABC-š, tani = ABBC

150 3

150

3

1= = = 30tan c

& i = 30c

vdnt, és¡F¡ f«g¤Â‹ c¢Áæ‹ V‰w¡nfhz« 30c.

3. A k‰W« B v‹w ó¢ÁfS¡F Ïil¥g£l öu« 2 Û ÏU¡F« tiuæš, x‹W vG¥ò«

xèia k‰wJ nf£f ÏaY«. Rt‰¿èUªJ 1 Û öu¤Âš jiuæYŸs ó¢Á A MdJ

xU ÁyªÂahš c©z¥gL« ãiyæš cŸs ó¢Á B -ia Rt‰¿š fh©»wJ. A-æèUªJ B-¡F V‰w¡ nfhz« 30c Mf ÏU¡F«nghJ A MdJ B-¡F

v¢rç¡if xè éL¤jhš, ÁyªÂ¡F Ïiu »il¡Fkh mšyJ »il¡fhjh? (A-æ‹ v¢rç¡if xèia B nf£F«nghJ mJ j¥ÃéL« vd¡ bfhŸf.)Ô®Î: jiuæYŸs ó¢Á A-æ‹ ãiy A k‰W« Rt‰¿YŸs ó¢Á B æ‹ ãiy B v‹f.

1 30Û k‰W«OA BAO o+= = vd¡ bfhL¡f¥g£LŸsJ.

br§nfhz TAOB-š, cos 30c = ABAO

& AB = cosAO30c

& AB = 3

2 = 3

2

3

33

2 3# =


= . 2 0.5773

2 1 732# #= = 1.154 Û

ó¢ÁfS¡F Ïil¥g£l öu« 1.154 Û . Ï¤bjhiyÎ 2 Û ¡F Fiwthf cŸsjhš A-æ‹

v¢rç¡if xèia B nf£f ÏaY«. vdnt, ÁyªÂ¡F Ïiu »il¡fhJ.

4. ÏuÎ neu¤Âš xUt® ml® nkf_£l¤ js¤Âid¡ (cloud ceiling) fhz Ãufhrkhd

xU és¡»‹ xëæid nkf¤ij neh¡» br§F¤jhf¢ brY¤J»wh®.

m›és¡»èUªJ 100 Û öu¤Âš, jiuæèUªJ 1.5 Û cau¤Âš bghU¤j¥g£l

Ânahliy£ (Theodolite) _y« nkf_£l¤ij¥ gh®¡F«nghJ V‰w¡nfhz« 60c våš, ml® nkf_£l¤ js« v›tsÎ cau¤Âš cŸsJ? (Ïªjédh nj®Î¡F

cçajšy)

Ô®Î: és¡»‹ ãiy B v‹f. ml® nkf_£l js¤Â‹ cau« AC v‹f.

Ânahliy£o‹ ãiy E v‹f. 100 , 1.5 60k‰W«Û ÛBE BC AEB+= = = c vd¡

bfhL¡f¥g£LŸsJ.

br§nfhz ABET -š, 60tanBEABo

= 60tanAB BE& #= c

& AB = 3 100# = 1.732 100#

& AB = 173.2 Û

AC = AB + BC = 173.2 Û + 1.5 Û = 174.7 Û

ml® nkf_£l js« 174.7 Û cau¤Âš cŸsJ.

kh‰W Kiw: jiu k£l¤ÂèUªJ ml® nkf_£l js¤Â‰F cŸs cau« h våš,

tanh x y i= + . x- v‹gJ Ânahliy£o‰F« jiu k£l¤Â‰F« cŸs öu«, y-v‹gJ

Ânahliy£o‰F« és¡»‰F« Ïilna cŸs öu« k‰W« i v‹gJ V‰w¡nfhz«.

h = 1.5 100 60tan#+ c

= 1.5 100 3#+ 1.5 100 1.732#= +

= 1.5 173.2+ = 174.7 Û.

ml®nkf_£l js« 174.7 Û cau¤Âš cŸsJ.

5. 40 br.Û ÚsKŸs xU jåCryhdJ (Simple pendulum), xU KG miyé‹ nghJ,

mj‹ c¢Áæš 60c nfhz¤ij V‰gL¤J»wJ. mªj miyéš, Crš F©o‹

Jt¡f ãiy¡F«, ÏWÂ ãiy¡F« Ïilna cŸs äf¡ Fiwªj öu¤ij¡ fh©f?

Ô®Î: jå CryhdJ O v‹w òŸëia bghW¤J miyÎ V‰gL¤J»‹wJ v‹f.

A , B v‹gd Crš F©o‹ Jt¡f ãiy k‰W« ÏWÂ ãiy

v‹f.

OA = OB = 40 br.Û k‰W« 0AOB 6+ = c. AB -¡F

br§F¤jhf OC-I tiuf. OA = OB Mjyhš, OC MdJ AB-¡F ika¡F¤J¡nfhL k‰W« AOB+ -‹ nfhz

ÏUrkbt£o MF«. 30AOC` + = c.


br§nfhz TOCA-š, 30sin c = OAAC

AC = 30 40 20 br.ÛsinOA21

#= =c

AB-‹ ika¥òŸë C Mjyhš,

AB = 2AC = 2 20 40 br.Û# =

Crš F©o‹ Jt¡f ãiy¡F« , ÏWÂ ãiy¡F« Ïilna cŸs äf¡Fiwªj öu« 40 br.Û.

6. x‹W¡bfh‹W nebuÂuhf cŸs ÏU ku§fë‹ ÛJ A, B v‹w ÏU fh¡iffŸ 15 Û k‰W«

10 Û cau§fëš mk®ªJ¡ bfh©oUªjd. mit jiuæèU¡F« xU tilæid

Kiwna 45c k‰W« 60c Ïw¡f¡ nfhz¤Âš gh®¡»‹wd. mit xnu neu¤Âš »s«Ã¡

Fiwthd ÚsKŸs¥ ghijæš rkntf¤Âš gwªJ, m›tilia vL¡f Ka‰Á¤jhš

vªj gwit bt‰¿ bgW«? (ÏU ku§fë‹ mo, til M»ad xnu ne®¡nfh£oš cŸsd)

Ô®Î: A, B v‹w ÏU fhf§fë‹ ãiyfŸ Kiwna A, B v‹f.

ku§fë‹ mofis C k‰W« E v‹f. D v‹gJ tilæ‹ ãiy v‹f.

AC = 15 Û, BE = 10 Û, 45ADC+ = c k‰W« BDE 60+ = c vd¡ bfhL¡f¥g£LŸsJ.

br§nfhz TACD-š, 45sin c =ADAC &

2

1 = AD15

AD = 15 15 1.414 21.21Û2 #= =

tilia vL¡f, fhf« A gwªj öu« 21.21 Û.

br§nfhzTBED-š, 60sin c = BDBE

BD = sinBE60c

BD3

20( =

= 3

20

3

33

20 3# #=

= .3

20 1 732# = .20 0 574#

BD = 11.48 Û tilia vL¡f fhf« B gwªjöu« 11.48 Û. tilia vL¡f fhf« B gwªj öu«,

fhf« A gwªj öu¤ij él äf¡FiwÎ. vdnt fhf« B bt‰¿ bgW«.

7. t£l toéš cŸs xU ó§fhé‹ ika¤Âš xU és¡F¡ f«g« ã‰»wJ. t£l¥ gçÂæš mikªj P, Q v‹D« ÏU òŸëfŸ és¡F¡ f«g¤Âdoæš 90c nfhz¤ij V‰gL¤J»‹wd. nkY«, P-æèUªJ és¡F¡ f«g¤Â‹ c¢Áæ‹

V‰w¡ nfhz« 30c MF«. PQ = 30 Û våš, és¡F¡ f«g¤Â‹ cau¤ij¡ fh©f.Ô®Î: ó§fhé‹ ika¤ij O vdÎ«, és¡F¡ f«g¤ij OR vdÎ« bfhŸf. PQ = 30 Û, POQ 90o+ = vd¡ bfhL¡f¥g£LŸsJ.

br§nfhz TOPQ-š, 90POQ+ = c, OP = OQ = Mu« . vdnt , OPQ OQP 45+ += = c

OP = 45cosPQ # c


OP = 2

30 = 2

30 2 15 2=

br§nfhz TROP-š, 30tan c = OPOR

OR = 30tanOP # c

OR = 15 23

1# =

3

15 2

3

3#

= 5 Û3

15 6 6=

vdnt, és¡F¡ f«g¤Â‹ cau« 5 Û6 .kh‰W Kiw : ó§fhé‹ ika¤ij O vdÎ«, és¡F¡ f«g¤ij OR vdÎ«

bfhŸf. PQ = 30 Û, POQ 90o+ = , OP = OQ (Mu«) vd¡ bfhL¡f¥g£LŸsJ.

br§nfhzTOPQ-š, OP OQ PQ2 2 2+ =

2 30OP2 2= & OP

230 302 #=

OP2

30=

br§nfhzTRPO-š, 0tanOPOR3 o

= &OPOR

3

1=

OR OP

3 2 3

30

#= = ( )OP

2

30a =

= 5 Û6

30

6

6630 6 6# = =

8. 700 Û cau¤Âš gwªJ¡ bfh©oU¡F« xU bAèfh¥lçèUªJ xUt® X® M‰¿‹

ÏU fiufëš nebuÂuhf cŸs ÏU bghU£fis 30 , 45c c Ïw¡f¡ nfhz§fëš

fh©»wh® våš, M‰¿‹ mfy¤ij¡ fh©f. ( 1.7323 = )

Ô®Î: gh®it¥ òŸë C v‹f . M‰¿‹ ÏU fiufëš nebuÂuhf cŸs ÏU

bghU£fis A, B v‹f. CD AB= vd tiuf . CD v‹gJ bAèfh¥lU¡F«

M‰W¡F« Ïilna cŸs öu« v‹f.

CD = 700 Û, 30 , 45CAD CBD+ += =c c vd¡ bfhL¡f¥g£LŸsJ.

br§nfhz CDBT -š, 45 700tanDBCD DB CD&= = =c Û

br§nfhz TCAD-š, 30tan c = ADCD &

tanAD CD

30=

c

AD = 700 Û3

M‰¿‹ mfy«, AB = AD DB+

= 700 700 700( 1) 700( .732 )3 3 1 1+ = + = +

= 700(2.732) 1912.400 1912.4 Û= =

vdnt , M‰¿‹ mfy« 1912.4 Û.


9. rkjs¤Âš ã‹W bfh©oU¡F« X v‹gt®, mtçläUªJ 100 Û öu¤Âš gwªJ

bfh©oU¡F« xU gwitæid 30c V‰w¡ nfhz¤Âš gh®¡»wh®. mnj neu¤Âš,

Y v‹gt® 20 Û cauKŸs xU f£ll¤Â‹ c¢Áæš ã‹W bfh©L mnj gwitia

45c V‰w¡ nfhz¤Âš gh®¡»wh®. ÏUtU« ã‹W bfh©L mt®fS¡»ilnaÍŸs

gwitia vÂbuÂuhf¥ gh®¡»‹wd® våš, Y-æèUªJ gwit cŸs öu¤ij¡

fh©f.

Ô®Î: gwitæ‹ ãiyia B v‹f.BD XA= k‰W« CY BD= tiuf. CD = 20 Û. BX = 100 Û, 30 , 45BXD BYCo o+ += = vd¡ bfhL¡f¥g£LŸsJ. gwit¡F« Y v‹gtU¡F« cŸs öu« BY v‹f.

br§nfhz TBDX-š, 30sin c = BXBD

& BD = 30sinBX o# & 50 Û

2100 =

BC BD CD= - = 50 20 30Û Û Û- =

br§nfhz TBCY-š, sin45c = BYBC &

BY2

1 30=

& BY = 30 Û2

gwit¡F« Y v‹gtU¡F« cŸs öu« 30 Û2 . 10. tF¥giwæš mk®ªJ¡ bfh©oU¡F« xU khzt‹ fU«gyifæš »ilãiy¥

gh®it¡ nfh£oèUªJ 1.5 Û cau¤Âš cŸs Xéa¤ij 30c V‰w¡ nfhz¤Âš

fh©»wh‹. Xéa« mtD¡F¤ bjëthf¤ bjçahjjhš neuhf¡ fU«gyifia

neh¡» ef®ªJ Û©L« mªj Xéa¤ij 45c V‰w¡ nfhz¤Âš bjëthf¡

fh©»wh‹ våš, mt‹ ef®ªj öu¤ij¡ fh©f.Ô®Î: khzt‹ Xéa¤ij Kiwna 30c k‰W« 45c V‰w¡nfhz§fëš gh®¡F«

ãiyfis A k‰W« B v‹f. Mfnt, khzt‹ ef®ªj öu« AB MF«.

30 , 45DAC DBC+ += =c c k‰W« CD = 1.5Û vd¡ bfhL¡f¥g£LŸsJ.

br§nfhz TDCB-š, tan 45c = BCCD = .

BC1 5 & 1.5 ÛBC=

br§nfhz TDCA-š, tan 30c =ACCD

& 3

1 = .AC1 5 & 1.5 ÛAC 3=

AB = AC BC- 1.5 .3 1 5= - 1.5( 1)3= -

= 1.5(1.732 1) 1.5(0.732)- = = 1.098 Û vdnt, khzt‹ ef®ªj öu« 1.098 Û.

11. xU ÁWt‹ 30 Û cauKŸs f£ll¤Â‰F vÂnu F¿¥Ã£l öu¤Âš ã‰»wh‹.

mtDila¡ »ilãiy¥ gh®it¡nfhL jiuk£l¤ÂèUªJ 1.5 Û cau¤Âš

cŸsJ. mt‹ f£ll¤ij neh¡» elªJ bršY« nghJ, m¡f£ll¤Â‹ c¢Áæ‹

V‰w¡ nfhz« 30c-èUªJ 60cMf ca®»wJ. mt‹ f£ll¤ij neh¡» elªJ

br‹w¤ öu¤ij¡ fh©f.


Ô®Î: bjhl¡f¤Âš khzt‹ ã‰F« ãiyia Al vdÎ«, f£ol« C Dl neh¡» khzt‹ ef®ªj ãiy Bl vdÎ« bfhŸf. gl¤Âš ABC v‹gJ khztQila

»ilãiy¥ gh®it¡nfhL. 1.5 , 30Û ÛAA BB CC C D= = = =l l l l vd¡

bfhL¡f¥g£LŸsJ. 30 60k‰W«DAC DBCo o+ += =

30 1.5 28.5ÛCD C D CC= - = - =l l .

khzt‹ ef®ªj öu« AB.

br§nfhzTDCB-š , 60tan c = BCCD &

tanBC CD

60=

c

& BC = .

3

28 5 = .

3

28 5

3

3# = (28.5)

33 = 9.5 3^ h

br§nfhzTDCA-š , 30tan c = ACCD &

tanAC CD

30=

c

& AC = 28.5 3

& AB BC+ = 28.5 3

& AB = . .28 5 3 9 5 3-^ ^h h 19 Û3=

khzt‹ ef®ªj öu« 19 Û3 . 12. 200 mo cauKŸs fy§fiu és¡f¤Â‹ c¢ÁæèUªJ, mj‹ fh¥ghs® njhâ

k‰W« glF M»at‰iw gh®¡»wh®. fy§fiu és¡f¤Â‹ mo, njhâ k‰W«

glF M»ad xnu Âiræš xnu ne®¡nfh£oš mik»‹wd. njhâ, glF

M»at‰¿‹ Ïw¡f¡ nfhz§fŸ Kiwna k‰W«45 30c c v‹f. Ï›éu©L«

ghJfh¥ghf ÏU¡f nt©Lbkåš, mitfS¡F Ïil¥g£l öu« FiwªjJ 300 moahf ÏU¡f nt©L«. Ïilbtë Fiwªjhš fh¥ghs® v¢rç¡if xè vG¥g

nt©L«. mt® v¢rç¡if xè vG¥g nt©Lkh?Ô®Î: njhâ k‰W« glF M»at‰¿‹ ãiyfŸ Kiwna A , B v‹f.

D v‹gJ gh®it¥ òŸë v‹f. CD v‹gJ fy§fiu és¡f« v‹f.

CD = 200 mo, 30 45k‰W«DAC DBCo o+ += = vd¡ bfhL¡f¥g£LŸsJ. AB v‹gJ njhâ k‰W« glF M»at‰¿‰F Ïil¥g£löu«.

br§nfhz TDBC-š, DBC CDB 45+ += = c. vdnt, BC = CD = 200 mo

br§nfhzTDCA-š, 30tan c= ACCD

& AC = tan30200

c

& AC = 200 3

AB = AC – BC 200 2003= -

= 200( 1)3 - 200(1.732 1)= -

= 200(0.732) = 146.4 mo


njhâ k‰W« glF M»at‰¿‰F Ïil¥g£löu« 146.4 mo.

ÏJ 300 moia él Fiwthf cŸsjhš fh¥ghs® v¢rç¡if xè vG¥g nt©L«.

13. jiuæš ã‹Wbfh©oU¡F« xU ÁWt‹ fh‰¿š, »ilãiy¡ nfh£oš khwhj

cau¤Âš ef®ªJ¡ bfh©oU¡F« xU gÿid¡ fh©»wh‹. xU F¿¥Ã£l

ãiyæš ÁWt‹ 60c V‰w¡ nfhz¤Âš gÿid¡ fh©»wh‹. 2 ãäl§fŸ fêªj

Ã‹d® mnj ãiyæèUªJ ÁWt‹ Û©L« gÿid¥ gh®¡F«nghJ V‰w¡nfhz«

30c Mf¡ Fiw»wJ. fh‰¿‹ ntf« 29 3 Û/ãäl« våš, jiuæèUªJ gÿå‹

cau« fh©f.

Ô®Î: gh®it¥òŸë A v‹f. E , D v‹gd Kiwna

V‰w¡nfhz§fŸ 60c k‰W« 30c vd mikªj gÿå‹

ãiyfŸ MF«. B, C v‹w òŸëfŸ BC = ED k‰W« BE = CD v‹wthW jiuæš mikªJŸsd.

60 30k‰W«EAB DACo o+ += =

vd¡ bfhL¡f¥g£LŸsJ.

fh‰¿‹ ntf« = 29 3 Û / ãäl«

ÏU ãäl§fëš gÿ‹ ef®ªj öu«

BC = ED = 29 2 58 Û3 3# = (öu«= ntf«# neu«)

br§nfhz TABE-š, tan 60c = ABBE

BE = 60tanAB AB 3o= g (1)

br§nfhz ADCT - š, 30tan c = ACCD

ACBE= CD BEa =^ h

& BE = 30tanAC CA AB BC

3 3= = +c = AB BC

3 3+

& BE = AB BE33 58

358+ = + ((1)k‰W« BC 58 3= èUªJ)

& BE 131-` j = 58 58 87 .ÛBE

23& #= =

jiuæèUªJ gÿ‹ gw¡F« cau« 87 Û.

14. xU neuhd beLŠrhiy xU nfhòu¤ij neh¡»¢ brš»wJ. nfhòu¤Â‹ c¢Áæš

ã‹W¡ bfh©oU¡F« xUt® Óuhd ntf¤Âš tªJ¡bfh©oU¡F« xU C®Âia

30c Ïw¡f¡ nfhz¤Âš fh©»wh®. 6 ãäl§fŸ fêªj Ã‹d® mªj C®Âæ‹

Ïw¡f¡ nfhz« 60cvåš, nfhòu¤ij mila C®Â nkY« v¤jid ãäl§fŸ

vL¤J¡ bfhŸS«?

Ô®Î: gh®it¥òŸë A æèUªJ B, C v‹gd Kiwna thfd¤Â‹

Ïw¡f¡ nfhz§fŸ 60c k‰W« 30c vd mikªj òŸëfŸ MF«.


30 60 .k‰W«ABD ACDo o+ += =

br§nfhz TBDA-š, 30tanBDADo

=

& BD = tanAD BD AD30

3& =c

g (1)

br§nfhzTCDA-š, 0tanCDAD6 o

= & 0tanAD CD6 o= CD3= g (2)

BC = BD – CD = AD CD3 -

= CD CD3 3 -^ ^h h = CD CD CD3 2- = ( (2) èUªJ) BC I mila vL¤J¡bfh©l neu« 6 ãäl§fŸ,

CD I mila neu« = BC2

= 26 = 3 ãäl§fŸ.

nfhòu¤ij mila thfd« nkY« 3 ãäl§fŸ vL¤J¡bfhŸS«.

15. xU bra‰if¡ nfhS¡F xnu Âiræš óäæš mikªJŸs ÏU f£L¥gh£L

ãiya§fëèUªJ m¢bra‰if¡ nfhë‹ V‰w¡ nfhz§fŸ Kiwna 30c k‰W« 60c vd cŸsd. m›éU ãiya§fŸ, bra‰if¡nfhŸ M»a Ïit _‹W«

xnu br§F¤J¤ js¤Âš mik»‹wd. ÏU ãiya§fS¡F« Ïilna cŸs

öu« 4000 ».Û våš, bra‰if¡nfhS¡F«, óä¡F« Ïilna cŸs öu¤ij¡

fh©f. ( 1.7323 = )

Ô®Î: A , B v‹gd óäæYŸs ÏU f£L¥gh£L ãiya§fŸ v‹f .

D v‹gJ bra‰if¡nfhë‹ ãiy v‹f .

bra‰if¡nfhS¡F« , óä¡F« Ïilna cŸs öu« CD v‹f.

AB = 4000 ».Û, 30 60k‰W«DAC DBC+ += =c c vd¡ bfhL¡f¥g£LŸsJ.

br§nfhzTBCD-š, 6tan 0c = BCDC

DC = 60tanBC o

& DC = BC3 g (1)

br§nfhzTACD-š, tan30c = ACDC

& DC = 30tanAC o

& DC = BC

3

4000 + g (2)

(1) k‰W« (2) èUªJ BC3 = BC

3

4000 +

BC3 = 4000 BC+ & BC2 4000= & BC = 2000 ».Û. BC = 2000 vd (1) š ÃuÂæl, DC = 20003 #

= 1.732 2000# = 3464 ».Û.

bra‰if¡nfhS¡F« , óä¡F« Ïilna cŸs öu« 3464 ».Û.


16. 60 Û cauKŸs xU nfhòu¤ÂèUªJ xU f£ll¤Â‹ c¢Á k‰W« mo M»at‰¿‹

Ïw¡f¡ nfhz§fŸ Kiwna 30 60k‰W«c c våš, f£ll¤Â‹ cau¤ij¡ fh©f.

Ô®Î: AE v‹gJ f£ll« k‰W« BD v‹gJ nfhòu« v‹f.

AE = BC v‹wthW EC || AB tiuf. AE = h Û v‹f.

vdnt, BC = h Û.

BD = 60 Û, 30 60k‰W«DEC DABo o+ += = vd¡

bfhL¡f¥g£LŸsJ.

CD = BD BC- h60= -

br§nfhz TDAB š, 60tan c = ABBD

& AB = tanBD60c

= 3

60 g (1)

br§nfhz TDEC š, tan30c = tanEC

CD EC CD30

& =c

& AB = (60 )h 3- ( )EC AB= g (2)

(1) k‰W« (2) & (60 )h 3- = 3

60

& 60 h- = 360

& 60 h- = 20 h& = 40 Û

f£ll¤Â‹ cau« 40 Û.

17. 40 Û cauKŸs xU nfhòu¤Â‹ c¢Á k‰W« mo M»at‰¿èUªJ xU fy§fiu

és¡f¤Â‹ c¢Áæ‹ V‰w¡ nfhz§fŸ Kiwna 30ck‰W« 60c våš, fy§fiu

és¡f¤Â‹ cau¤ij¡ fh©f. fy§fiu és¡f¤Â‹ c¢ÁæèUªJ nfhòu¤Â‹

mo¡F cŸs öu¤ijÍ« fh©f.

Ô®Î: nfhòu¤ij AE vdÎ«,fy§fiu és¡f¤ij BD vdÎ« bfhŸf.

AE = BC v‹WŸsthW EC || AB tiuf.vdnt AB = EC MF«.

AE = 40 Û, 60DAB+ = c k‰W« 30DEC+ = cvd¡

bfhL¡f¥g£LŸsJ.

CD = x Û v‹f. BD = BC + CD = 40 + x.

br§nfhzTABD-š, 6tan 0c = ABBD

AB = tan

x AB x60

40

3

40&+ = +c

g (1)


br§nfhzTECD-š, 30tan c = ECCD =

ABCD AB EC=^ h

& AB = tan

x AB x30

3& =c

g (2)

(1) k‰W« (2) èUªJ x3 = x

3

40 + & x x3 40= +

& x2 = 40

fy§fiu és¡f¤Â‹ cau«, BD = 40 + x = 40 + 20 = 60 Û

br§nfhz TABD-š, 60sin c = ADBD

& AD = sinBD60c

40AD3

120 3& = =

nfhòu¤Â‹ moæèUªJ fy§fiu és¡f¤Â‹ c¢Á¡F cŸs öu«40 3 Û.

18. xU Vçæ‹ nk‰gu¥Ãš xU òŸëæèUªJ fhQ«nghJ, 45 Û cau¤Âš gwªJ

bfh©oU¡F« xU bAèfh¥lç‹ V‰w¡ nfhz« 30c Mf cŸsJ.

m¥òŸëæèUªJ mnj neu¤Âš j©Ùçš bAèfh¥lç‹ ãHè‹ Ïw¡f¡

nfhz« 60c våš, bAèfh¥lU¡F« Vçæ‹ nk‰gu¥Ã‰F« Ïil¥g£l¤

öu¤ij¡ fh©f.

Ô®Î:Vçæ‹ nk‰gu¥ÃèUªJ 45Û cau¤Âš

gh®it¥òŸë A cŸsJ v‹f.Vçæ‹ nk‰gu¥ò BD v‹f. vdnt, AB = 45 Û. bAèfh¥lç‹ ãiyia

F vdÎ« k‰W« j©Ùçš mj‹ vÂbuhë¥ig

C vdÎ« bfhŸf.

FD = h Û v‹f. DC = h Û . AE || BD tiuf.

ED = 45Û, 30 60 .k‰W«FAE CAEo o+ += =

br§nfhz TAEF-š, 30tan c = AEFE

& 3

1 = AE

h 45- ( FE = FD – ED )

& AE = 45h 3-^ h g (1)

br§nfhz TCEA-š, 6tan 0c =AEEC =

AEED DC+

& AE 3 = 45 + h

& h = 45 45AE h3 45 3- = - -^ ^h h ((1) èUªJ )

= 3h –180& 2h = 180 & h = 90Vçæ‹ nk‰gu¥ÃèUªJ bAèfh¥lU¡F cŸs bjhiyÎ 90 Û.


gæ‰Á 7.3


1. sin sec12 2i i-^ h =

(A) 0 (B) 1 (C) tan2i (D) cos2i

Ô®Î: 1sin sec cos sec12 22 2

i i i i- = =^ h ( éil: (B) )

2. tan sin12 2i i+^ h =

(A) sin2i (B) cos2i (C) tan2i (D) cot2i

Ô®Î: tan sin sec sincos

sin tan12 2

2

222 2

i i i ii

i i+ = = =^ h ( éil: (C) )

3. cos cot1 12 2i i- +^ ^h h =

(A) sin2i (B) 0 (C) 1 (D) tan2i

Ô®Î: cos cot sin cosec1 1 12 2 2 2i i i i- + = =^ ^h h ( éil: (C) )

4. sin cos cos sin90 90i i i i- + -c c^ ^h h =

(A) 1 (B) 0 (C) 2 (D) –1

Ô®Î: sin cos cos sin cos cos sin sin90 90 1i i i i i i i i- + - = + =c c^ ^h h ( éil: (A) )

5. 1cos

sin1

2

ii-

+ =

(A) cosi (B) tani (C) coti (D) coseci

Ô®Î: 1 11

)(1 (1 )

cossin

cos

cos coscos cos

1

1 12

ii

i

i ii i-

+= -

+

+ -= - - =

^ h

( éil: (A) ) 6. cos sinx x

4 4- =

(A) 2 1sin x2

- (B) 2 1cos x2

- (C) 1 2sin x2

+ (D) 1 2 .cos x2

-

Ô®Î: ( )( )cos sin cos sin cos sinx x x x x x2 2 2 24 4

- = + -

(1 ) 2cos sin cos cos cosx x x x x 12 2 2 2 2= - = - - = - ( éil: (B) )

7. tani = xa våš,

a x

x2 2+

-‹ kÂ¥ò

= (A) cosi (B) sini (C) coseci (D) seci

Ô®Î: 1 tan sec

cosa x

x 1

1

1 1

x

a 22 2

2

2 i ii

+=

+=

+= = ( éil: (A) )

8. secx a i= , tany b i= våš, ax

b

y2

2

2

2

- -‹ kÂ¥ò=

(A) 1 (B) –1 (C) tan2i (D) cosec2i

Ô®Î: sec tan sec tanax

b

y

aa

bb 1

2

2 2

2

2 22 2

2

2

2

2i i i i- = - = - = . ( éil: (A) )


9. cot tan

seci ii

+ =

(A) coti (B) tani (C) sini (D) – coti

Ô®Î: cot tan

sec sin

sin coscos sin

cos1

2 2i ii i

+= =

i ii ii

+ ( éil: (C) )

10. tan

sin sin

cot

cos cos90 90

i

i i

i

i i-+

-c c^ ^h h =

(A) tani (B) 1 (C) –1 (D) sini

Ô®Î: tan

sin sin

cot

cos cos cos sin sin cos90 90

cossin

sincosi

i i

i

i i i i i i-+

-= +

ii

ii

c c^ ^h h

= cos sin 12 2i i+ = ( éil: (B) ) 11. gl¤Âš, AC =

(A) 25 Û (B) 25 3 Û

(C) 3

25 Û (D) 25 2 Û

Ô®Î: 60 25 60tan tanAC AC25

25 3o o&= = = Û ( éil: (B) )

12. gl¤Âš, ABC+ =

(A) 45c (B) 30c

(C) 60c (D) 05 c

Ô®Î: tan tanABC ABC ABC100

100 3 3 60o& &+ + += = = ( éil: (C) )

13. xU nfhòu¤ÂèUªJ 28.5 Û öu¤Âš ã‹W bfh©oU¡F« xUt® nfhòu¤Â‹

c¢Áia 45c V‰w¡ nfhz¤Âš fh©»wh®. mtUila »ilãiy¥ gh®it¡ nfhL

jiuæèUªJ 1.5 Û cau¤Âš cŸsJ våš, nfhòu¤Â‹ cau«

(A) 30 Û (B) 27.5 Û (C) 28.5 Û (D) 27 ÛÔ®Î: nfhòu¤Â‹ cau« = tanx y i+

= 1.5 28.5 45 1.5 28.5 30Ûtan o#+ = + = ( éil: (A) )

14. gl¤Âš, sini = 1715 . våš, BC =

(A) 85 Û (B) 65 Û

(C) 95 Û (D) 75 Û

Ô®Î: sini = 1715 . gl¤ÂèUªJ, sini =

ACBC

ACBC BC

1715

1715 85 75&` #= = = Û ( éil:(D) )


15. 1 tan sin sin1 12i i i+ - +^ ^ ^h h h =

(A) cos sin2 2i i- (B) sin cos2 2i i-

(C) sin cos2 2i i+ (D) 0

Ô®Î: 1tan sin sin sec cos sin cos1 1 12 2 2 2 2i i i i i i i+ - + = = = +^ ^ ^h h h

( éil: (C) ) 16. cot cos cos1 1 12i i i+ - +^ ^ ^h h h =

(A) tan sec2 2i i- (B) sin cos2 2i i-

(C) sec tan2 2i i- (D) cos sin2 2i i-

Ô®Î: 1cot cos cos cosec sin sec tan1 1 12 2 2 2 2i i i i i i i+ - + = = = -^ ^ ^h h h

( éil: (C) ) 17. 1 1 1cos cot2 2i i- + +^ ^h h =

(A) 1 (B) –1

(C) 2 (D) 0

Ô®Î: 1cos cot sin cosec1 1 1 1 1 02 2 2 2#i i i i- + + =- + =- + =^ ^h h

( éil: (D) ) 18.

11

cottan

2

2

i

i

++ =

(A) cos2i (B) tan2i

(C) sin2i (D) cot2i

Ô®Î: cottan

cosecsec

cossin tan

11

2

2

2

2

2

22

i

i

i

i

i

i i++ = = = ( éil: (B) )

19. 1

sintan12

2i

i+

+ =

(A) cosec cot2 2i i+ (B) cosec cot2 2i i-

(C) cot cosec2 2i i- (D) sin cos2 2i i-

Ô®Î:

1sintan

sinsec

sin cos cosec cot1

1 122

22

2 2 2 2ii

ii

i i i i++

= + = + = = -

( éil: (B) ) 20. 9 9tan sec2 2i i- =

(A) 1 (B) 0

(C) 9 (D) –9

Ô®Î: 9 9 9( ) 9tan sec sec tan2 2 2 2i i i i- =- - =- ( éil: (D) )

Ô®Î - mséaš 211

gæ‰Á 8.1

1. xU Â©k ne®t£l cUisæ‹ Mu« 14 br.Û k‰W« cau« 8 br.Û. våš, mj‹

tisgu¥ò k‰W« bkh¤j¥ òw¥gu¥ig¡ fh©f.

Ô®Î: xU Â©k ne®t£l cUisæ‹ Mu« r = 14 br.Û k‰W« cau« h = 8 br.Û.

vd¡ bfhL¡f¥g£LŸsJ. tisgu¥ò, CSA = 2 rhr

= 2 × × ×722 14 8= 704

vdnt, tisgu¥ò = 704 r. br.Û.

bkh¤j¥ òw¥gu¥ò, TSA = 2 ( )r h rr +

= 2 × 722 ×14(8+14) = 88 × 22 = 1936

vdnt, bkh¤j¥ òw¥gu¥ò = 1936 r.br.Û.

2. xU Â©k ne®t£l cUisæ‹ bkh¤j¥ òw¥gu¥ò 660 r. br.Û. mj‹ é£l« 14 br.Û.

våš, m›ÎUisæ‹ cau¤ijÍ«, tisgu¥igÍ« fh©f.

Ô®Î: r k‰W« h v‹gd Kiwna xU Â©k ne®t£l cUisæ‹ Mu« k‰W« cau«

v‹f.

bkh¤j òw¥gu¥ò = 660 r.br.Û., k‰W« é£l« 2r = 14 br.Û vd¡ bfhL¡f¥g£LŸsJ.

vdnt, 2r = 14 & r = 7 bkh¤j¥ òw¥gu¥ò, 2rr(h + r) = 660

2×722 ×7×(h + 7) = 660

& h = ×2 22660 – 7 = 8 br.Û.

vdnt, tisgu¥ò òw¥gu¥ò, 2 rhr = 2×722 ×7×8 = 352 r. br.Û..

3. xU Â©k ne®t£l cUisæ‹ tisgu¥ò k‰W« mo¢R‰wsÎ Kiwna

4400 r.br.Û k‰W« 110 br.Û. våš, m›ÎUisæ‹ cau¤ijÍ«, é£l¤ijÍ«

fh©f.

Ô®Î: xU Â©k ne®t£l cUisæ‹ tisgu¥ò k‰W« mo¢R‰wsÎ Kiwna

4400 r.br.Û k‰W« 110 br.Û. vd¡ bfhL¡f¥g£LŸsJ. cUisæ‹ moghf¤Â‹ R‰wsÎ, 2 rr = 110 br.Û

2×722 ×r = 110

Mfnt, é£l«, 2r = ×22

110 7 = 35 br.Û

cUisæ‹ tisgu¥ò, rh2r = 110×h = 4400

Mfnt, cUisæ‹ cau«. h = 1104400 = 40 br.Û

mséaš 8


4. xU khëifæš, x›bth‹W« 50 br.Û. MuK«, 3.5 Û cauK« bfh©l 12 ne® t£l

cUis tot¤ ö©fŸ cŸsd. m¤ö©fS¡F t®z« ór xU rJu Û£lU¡F

` 20 Åj« v‹d brythF«?

Ô®Î: r k‰W« h v‹gd Kiwna ne®t£l cUis toéyhd öâ‹ Mu« k‰W«

cau« v‹f. Mu«, r = 50 br.Û = 0.5 Û k‰W« cau«, h = 3.5 Û vd¡

bfhL¡f¥g£LŸsJ.

cUis toéyhd öâ‹ tisgu¥ò,

rh2r = 2×722 ×0.5×3.5 = 11 Û2

xU rJu Û£lU¡F« t®z« ór MF« bryÎ = ` 20 Mfnt, 12 ö©fS¡F t®z« ór MF« bryÎ = 12×20×11 = ` 2640

5. xU Â©k ne® t£l cUisæ‹ bkh¤j¥ òw¥gu¥ò 231 r. br.Û. mj‹ tisgu¥ò

bkh¤j òw¥gu¥Ãš _‹¿š Ïu©L g§F våš, mj‹ Mu« k‰W« cau¤ij¡

fh©f.

Ô®Î: Â©k ne®t£l cUisæ‹ bkh¤j òw¥gu¥ò = 231 r.br.Û vd¡ bfhL¡f¥

g£LŸsJ.

nkY«, tisgu¥ò = 32 × bkh¤j òw¥gu¥ò

rh2r = 32 ×231 = 154 r.br.Û

cUisæ‹ bkh¤j òw¥gu¥ò ( )r h r2r + = 231 rh2r +2 r2r = 231 2 r2r = 231 – 154 = 77

r2 = 277r

= ××

××

2 2277 7

2 27 7=

Mfnt, cUisæ‹ Mu« r = 27 br.Û

cUisæ‹ tisgu¥ò, rh2r = 154 & 2×722 ×

27 ×h = 154

& h = × ×× ×

2 22 7154 7 2 = 7

Mfnt, cUisæ‹ cau«, h = 7 br.Û. 6. xU Â©k ne® t£l cUisæ‹ bkh¤j òw¥gu¥ò 1540 br.Û2 . mj‹ caukhdJ,

mo¥g¡f Mu¤ij¥nghš eh‹F kl§F våš, cUisæ‹ cau¤ij¡ fh©f.

Ô®Î: cUisæ‹ bkh¤j òw¥gu¥ò, TSA = 1540 br.Û2 k‰W« cau« h = r4 vd

bfhL¡f¥g£LŸsJ. nkY«, h = r4 & r h4

= . bkh¤j òwgu¥ò, ( )r h r2r + = 1540

2×722 × h

4 h h

4+` j = 1540

h45 2

= 2 22

1540 7 4## #

& h2 = 140 × 7 × 54 = 28 × 7 × 4

& h = × ×28 7 4 = 28

Mfnt, cUisæ‹ cau« = 28 br.Û.


7. Ïu©L ne® t£l cUisfë‹ Mu§fë‹ é»j« 3 : 2 v‹f. nkY« mt‰¿‹

cau§fë‹ é»j« 5 : 3 våš, mt‰¿‹ tisgu¥òfë‹ é»j¤ij¡ fh©f.

Ô®Î: r1 k‰W« r2 M»ad Ïu©L ne®t£l cUisfë‹ Mu§fŸ v‹f.

nkY«, h1 k‰W« h2 mt‰¿‹ cau§fŸ v‹f.

r1 : r2 = 3 : 2 k‰W« h1 : h2 = 5 : 3 vd¡ bfhL¡f¥g£LŸsJ.cUisfë‹ tisgu¥òfë‹ é»j« 2 r h

1 1r : 2 r h

2 2r = 3 × 5 : 2 × 3 = 5 : 2.

8. xU cŸÇl‰w cUisæ‹ btë¥òw tisgu¥ò 540r r.br.Û. mj‹ cŸé£l«

16 br.Û k‰W« cau« 15 br.Û. våš, mj‹ bkh¤j òw¥gu¥ig¡ fh©f.

Ô®Î: R, r k‰W« h v‹gd Kiwna cŸÇl‰w cUisæ‹ btëMu« , cŸ Mu« k‰W«

cau« v‹f. nkY«, h = 15 br.Û, cŸé£l« 2r = 16 br.Û vd¡ bfhL¡f¥g£LŸsJ. btë¥òw tisgu¥ò = 540r br.Û2 & Rh2r = 540r

btë Mu« R = 2 15

540# # rr = 18 br.Û

bkh¤j òw¥gu¥ò = ( )( )R r R r h2r + - +

= 2 (18 8) 18 8 15r + - +^ h

= 2 26 25# # #r =1300r

Mfnt, bkh¤j òw¥gu¥ò = 1300r br.Û2 .

9. xU cUis tot ÏU«ò¡FHhæ‹ btë¥òw é£l« 25 br.Û, mj‹ Ús« 20 br.Û.

k‰W« mj‹ jok‹ 1 br.Û våš, m¡FHhæ‹ bkh¤j¥ òw¥gu¥ig¡ fh©f.

Ô®Î: R, r k‰W« h v‹gd Kiwna cUistoéyhd

ÏU«ò¡FHhæ‹ btë¥òw, c£òw Mu§fŸ k‰W« cau« v‹f.

2R = 25 br.Û & R = 12.5 br.Û k‰W« mj‹ jok‹ w = 1 br.Û


cŸ Mu«, r = R – w = 12.5 – 1 = 11.5 br.Û

bkh¤j òw¥gu¥ò = ( )( )R r R r h2r + - +

= 2 ( . . ) ( . . )722 12 5 11 5 20 12 5 11 5# # + + -

= 2722 24 21# # # = 3168

Mfnt, bkh¤j òw¥gu¥ò = 3168 br.Û2 .

10. xU Â©k ne®t£l¡ T«Ã‹ Mu« k‰W« cau« Kiwna 7 br.Û k‰W« 24 br.Û.

våš, mj‹ tisgu¥ò k‰W« bkh¤j¥ òw¥gu¥ig¡ fh©f.

Ô®Î: r, h k‰W« l v‹gd Kiwna xU Â©k ne®t£l¡ T«Ã‹ Mu«, cau« k‰W«

rhÍau« v‹f. r = 7 br.Û k‰W« cau« h = 24 br.Û vd bfhL¡f¥g£LŸsJ.

rhÍau«, l = h r2 2+ = 7 242 2

+ = 25 br.Û


Mfnt, T«Ã‹ tisgu¥ò = rlr

= 7722 25# # = 550 br.Û2

nkY«, bkh¤j òw¥gu¥ò = ( )r l rr +

= 7 (2 )722 5 7# # + = 704 br.Û2

11. xU Â©k ne® t£l¡ T«Ã‹ c¢Á¡nfhz« k‰W« Mu« Kiwna 60ck‰W« 15 br.Û

våš, mj‹ cau« k‰W« rhÍau¤ij¡ fh©f.

Ô®Î: gl¤Âš, OAB v‹gJ T«ò. OC CB= tiuf. AOB 60+ = c c¢Á¡nfhz« k‰W« AC = 15 br.Û vd bfhL¡f¥g£LŸsJ. Mfnt,

AOC+ = 30AOB2 2

60+ = =c c

br§nfhz OCAT -š 30tan c =

OCAC

OC3

1 15& =

& &OC = 15 3

Mfnt, T«Ã‹ cau« 15 3 br.Û.nkY«, sin30c =

AOAC

AO21 15& =

& AO = 30

Mfnt, T«Ã‹ rhÍau« 30 br.Û.kh‰WKiw : OAB v‹gJ rkg¡f K¡nfhz«. AB = 2AC = 30 br.Û vdnt,

rhÍau« OA = 30 br.Û. T«Ã‹ cau« = 30

23 15 3# =

(a v‹gJ rkg¡f K¡nfhz¤Â‹ g¡f« våš, mj‹ cau« a23 )

12. xU Â©k ne® t£l¡ T«Ã‹ mo¢R‰wsÎ 236 br.Û . k‰W« mj‹ rhÍau« 12 br.Û

våš, m¡T«Ã‹ tisgu¥ig¡ fh©f.

Ô®Î: Â©k ne® t£l¡ T«Ã‹ rhÍau«, l 12= br.Û k‰W« mo¢R‰wsÎ, r2r = 236 br.Û vd¡ bfhL¡f¥g£LŸsJ.

r2r = 236 & r 118r = br.Û

Mfnt, T«Ã‹ tisgu¥ò, rlr = 118 12 1416# = br.Û2

13. ne®t£l T«ò toéš Fé¡f¥g£l be‰Féaè‹ é£l« 4.2 Û k‰W« mj‹ cau«

2.8 Û. v‹f. Ïªbe‰Féaiy kiHæèUªJ ghJfh¡f »¤jh‹ Jâahš äf¢rçahf

_l¥gL»wJ våš, njitahd »¤jh‹ Jâæ‹ gu¥ig¡ fh©.

Ô®Î: r k‰W« h v‹gd Kiwna T«ò tot be‰Féaè‹ Mu« k‰W«

cau« v‹f.

h = 2.8 Û k‰W« é£l« 2r = 4.2 Û & r = 2.1 Û vd¡ bfhL¡f¥g£LŸsJ.

vdnt, rhÍau« l = h r2 2+ = . .2 8 2 12 2

+ = 3.5 Û


njitahd »¤jh‹ Jâæ‹ gu¥ò, rlr = 2.1 3.5722

# # = 23.1 r.Û

Mfnt,be‰F éaiy kiHæèUªJ ghJfh¡f¤

njitahd »¤jh‹ Jâæ‹ gu¥ò3 = 23.1 r.Û.

14. 180c ika¡ nfhzK« 21 br.Û. MuK« bfh©l t£lnfhz toéyhd ÏU«ò¤

jf£o‹ Mu§fis Ïiz¤J xU T«ò cUth¡f¥gL»wJ våš, m¡T«Ã‹

Mu¤ij¡ fh©f.

Ô®Î: ika¡nfhz« 180i = c k‰W« t£l¡nfhz¥ gFÂæ‹ Mu« r = 21 br.Û. vd¡ bfhL¡f¥g£LŸsJ.

t£l¡nfhz¥ gFÂæ‹ éë«òfis Ïiz¡f xU

cŸÇl‰w T«ò »il¡F«. Ï¡T«Ã‹ Mu« R v‹f.

T«Ã‹ mo¥gu¥Ã‹ R‰wsÎ = t£léšè‹ Ús«

R2r = r360

2#i r

Mfnt, T«Ã‹ Mu« R = 21360180

# = 10.5 br.Û

15. xU ne®t£l Â©k¡ T«Ã‹ MuK« rhÍauK« 3 : 5 v‹w é»j¤Âš cŸsd.

m¡T«Ã‹ tisgu¥ò 60r r.br.Û våš, mj‹ bkh¤j¥ òw¥gu¥ig¡ fh©f.

Ô®Î: r k‰W« l v‹gd Kiwna Â©k ne®t£l¡ T«Ã‹ Mu« k‰W« rhÍau« v‹f.

Mu« k‰W« rhÍau¤Â‹ é»j« = 3 : 5

r : l = 3 : 5 lr

53& = & r l

53=

T«Ã‹ tisgu¥ò, rlr = 60r

& l l53

# #r = 60r

& l2 = 6035

##

rr

= 100

Mfnt, T«Ã‹ rhÍau«, l = 10 br.Û. nkY«, r = l53 = 6 br.Û

nkY«, T«Ã‹ bkh¤j¥gu¥ò ( )r l rr + = 6 (6 10)722

# # +

= 7

2112 = 30175 br.Û2

16. 98.56 r.br.Û òw¥gu¥ò bfh©l xU Â©k¡ nfhs¤Â‹ Mu¤ij¡ fh©f.

Ô®Î: nfhs¤Â‹ òw¥gu¥ò = 98.56 br.Û2 vd bfhL¡f¥g£LŸsJ.

& 4 r2r = 98.56

& 4 r722 2

# # = 98.56

& r = 4 22

98.56 7#

# = . .7 84 2 8=

Mfnt, nfhs¤Â‹ Mu«, r = 2.8 br.Û. 17. xU Â©k miu¡nfhs¤Â‹ tisgu¥ò 2772 r.br.Û våš, mj‹ bkh¤j¥

òw¥gu¥ig¡ fh©f.

Ô®Î: Â©k miu¡nfhs¤Â‹ tisgu¥ò 2772 r.br.Û vd

bfhL¡f¥g£LŸsJ. mj‹ Mu« r v‹f.


2 r2r = 2772 br.Û2

& r2r = 2

2772 = 1386

Mfnt, miu¡nfhs¤Â‹ bkh¤j¥gu¥ò r3 2r = 3 × 1386 = 4158 br.Û2 .

18. Ïu©L Â©k miu¡nfhs§fë‹ Mu§fŸ 3 : 5 v‹w é»j¤Âš cŸsd.

m¡nfhs§fë‹ tisgu¥òfë‹ é»j« k‰W« bkh¤j¥ òw¥gu¥òfë‹ é»j«

M»at‰iw¡ fh©f.

Ô®Î: r1 k‰W« r2 v‹gd Ïu©L Â©k miu¡nfhs§fë‹ Mu§fŸ v‹f.

r1 : r2 = 3 : 5 vd bfhL¡f¥g£LŸsJ.

Mfnt, tisgu¥òfë‹ é»j« = 2 : 2r r1

2

2

2r r = 3 : 52 2 = 9 : 25

bkh¤j¥gu¥Ã‹ é»j« = 3 : 3r r1

2

2

2r r = 3 : 52 2= 9 : 25

19. xU cŸÇl‰w miu¡nfhs¤Â‹ btë Mu« k‰W« cŸ Mu« Kiwna 4.2 br.Û

k‰W« 2.1 br.Û våš mj‹ tisgu¥ò k‰W« bkh¤j òw¥gu¥ig¡ fh©f.

Ô®Î: R k‰W« r v‹gd cŸÇl‰w miu¡nfhs¤Â‹ btë

k‰W« cŸ Mu§fŸ v‹f. R = 4.2 br.Û k‰W« r = 2.1 br.Û vd

bfhL¡f¥g£LŸsJ

vdnt, tisgu¥ò 2 ( )R r2 2r + = 2 (4.2 2.1 )2 2r +

= 2 (17.64 4.41)r + = 44.1r br.Û2

bkh¤j òw¥gu¥ò = 2 ( ) ( )R r R r2 2 2 2r r+ + -

= .44 1r + (4.2 2.1 )2 2r -

= .44 1r+ (17.64 4.41)r -

vdnt, bkh¤j òw¥gu¥ò = 44.1 13.23 57.33r r r+ = br.Û2

20. miu¡nfhs tot nk‰Tiuæ‹ c£òw tisgu¥Ã‰F t®z« ór nt©oÍŸsJ.

mj‹ c£òw mo¢R‰wsÎ 17.6 Û våš, xU rJu Û£lU¡F ` 5 Åj«, t®z« ór

MF« bkh¤j bryit¡ fh©f.

Ô®Î: miu¡nfhs tot nk‰Tiuæ‹ c£òw Mu« r v‹f.

mj‹ c£òw¢R‰wsÎ, 2 rr = 17.6 Û vd bfhL¡f¥g£LŸsJ

& r = . 2.82 2217 6 7

## =

tisgu¥ò, r2 2r = 2 2.8 2.8722

# # # = 49.28 Û2

xU rJuÛ£lU¡F t®z« ór MF« bryÎ = 5

Mfnt, t®z« ór MF« bkh¤j bryÎ = 49.28 × 5 = ` 246.40.


gæ‰Á 8.2

1. xU Â©k cUisæ‹ Mu« 14 br.Û. mj‹ cau« 30 br.Û våš, m›ÎUisæ‹

fdmsit¡ fh©f.

Ô®Î: r k‰W« h v‹gd Kiwna Â©k cUisæ‹ Mu« k‰W« cau« v‹f.

r = 14 br.Û k‰W« h = 30 br.Û vd¡ bfhL¡f¥g£LŸsJ.

cUisæ‹ fdmsÎ = r h2r

= 14 14 30722

# # #

= 18480 br.Û3

2. xU kU¤JtkidæYŸs nehahë xUtU¡F ÂdK« 7 br.Û é£lKŸs cUis

tot »©z¤Âš to¢rhW (Soup) tH§f¥gL»wJ. m¥gh¤Âu¤Âš 4 br.Û

cau¤Â‰F to¢rhW xU nehahë¡F tH§f¥g£lhš, 250 nehahëfS¡F tH§f¤

njitahd to¢rh¿‹ fdmséid¡ fh©f.

Ô®Î: r k‰W« h v‹gd Kiwna cUis tot »©z¤Â‹ Mu« k‰W« cau« v‹f.

h = 4br.Û k‰W« é£l« 2r = 7 br.Û vd¡ bfhL¡f¥g£LŸsJ.

vdnt, r = 27 br.Û

»©z¤Âš cŸs to¢rh¿‹ bfhŸssÎ r h2r =

722

27

27 4 154# # # = br.Û3

250ÂdK« nehahëfS¡F tH§f

njitahd to¢rhW3 = 154 × 250 = 38500 br.Û3

Mfnt, njitahd to¢rhW 100038500 = 38.5 è£l®

3. xU Â©k cUisæ‹ Mu« k‰W« cau¤Â‹ TLjš 37 br.Û. v‹f. nkY«, mj‹

bkh¤j òw¥gu¥ò 1628 r.br.Û våš, m›ÎUisæ‹ fdmsit¡ fh©f.

Ô®Î: Â©k cUisæ‹ Mu« k‰W« cau« Kiwna r, h v‹f.

cUisæ‹ Mu« k‰W« cau¤Â‹ TLjš (r + h) = 37 br.Û,

bkh¤j¥ òw¥gu¥ò = 1628 r.br.Û vd bfhL¡f¥g£LŸsJ.

cUisæ‹ bkh¤j òw¥gu¥ò 2 r h rr +^ h = 1628 r.br.Û

& 2 rr = 37

1628

& r = 37

162821

227

# #

Mfnt, cUisæ‹ Mu«, r = 7 br.Û

Mu« k‰W« cau¤Â‹ TLjš, r + h = 37 & h = 30 br.Û

Mfnt, cUisæ‹ fdmsÎ r h2r = 7 30 4620722 2

# # = f.br.Û

4. 62.37 f.br.Û. fdmsÎ bfh©l xU Â©k ne®t£l cUisæ‹ cau« 4.5 br.Û våš, m›ÎUisæ‹ Mu¤ij¡ fh©f.

Ô®Î: r k‰W« h v‹gd Kiwna cUisæ‹ Mu« k‰W« cau« v‹f.


cau«, h = 4.5 br.Û, fdmsÎ 62.37 f.br.Û vd bfhL¡f¥g£LŸsJ.

cUisæ‹ fdmsÎ r h2r = 62.37 f.br.Û

r2 = .h

62 37r

= 62.37.

.227

4 51 4 41# # =

& r = . 2.14 41 =

Mfnt, cUisæ‹ Mu« r = 2.1 br.Û.

5. Ïu©L ne® t£l cUisfë‹ Mu§fë‹ é»j« 2 : 3. nkY« cau§fë‹ é»j«

5 : 3 våš, mt‰¿‹ fdmsÎfë‹ é»j¤ij¡ fh©f.

Ô®Î: r1 k‰W« r2 v‹gd Ïu©L cUisfë‹ Mu§fŸ k‰W« h1 , h2 v‹gd

mt‰¿‹ cau§fŸ v‹f.

r1 : r2 = 2 : 3 k‰W« h1 : h2 = 5 : 3 vd bfhL¡f¥g£LŸsJ.

r

r

2

1 = 32 & r

1 = r

32

2 k‰W«

h

h

2

1 = 35 & h

1= h

35

2

cUisfë‹ fdmsÎfë‹ é»j«

2 : 2r h r h1

2

1 2

2

2r r = 2 : 2r h r h

32

352

2

2

2 2

2

2#r r` j

= : 12720

vdnt, cUisfë‹ fdmsÎfë‹ é»j« = 20 : 27.

6. xU cUisæ‹ Mu« k‰W« cau¤Â‹ é»j« 5 : 7. nkY« mj‹ fdmsÎ 4400 f.br.Û våš, m›ÎUisæ‹ Mu¤ij¡ fh©f.

Ô®Î: r k‰W« h v‹gJ cUisæ‹ Mu« k‰W« cau« v‹f.

r : h = 5 : 7 &h = r57 vd bfhL¡f¥g£LŸsJ

cUisæ‹ fdmsÎ r h2r = 4400

r r722

572

# # = 4400

& r3 = 22 7

4400 7 5## # = 1000

Mfnt, cUisæ‹ Mu«, r = 10 br.Û.

7. 66 br.Û # 12 br.Û vD« msÎ¡ bfh©l xU cnyhf¤ jf£oid 12 br.Û cauKŸs

xU cUisahf kh‰¿dhš »il¡F« cUisæ‹ fdmsit¡ fh©f.

Ô®Î: r k‰W« h v‹gJ cUisæ‹ Mu« k‰W« cau« v‹f.

br›tftot jf£o‹ g¡f§fŸ 66 br.Û. × 12 br.Û. vd bfhL¡f¥g£LŸsJ.

cnyhf¤ jf£oid 12 br.Û cauKŸs xU cUisahf kh‰¿dhš »il¡F« cUisæ‹

mog¡f¢R‰wsÎ cnyhf¤ jf£o‹ Ús¤Â‰F rk«.

vdnt, & l = 66 br.Û. k‰W« b = 12 br.Û


mog¡f¢ R‰wsÎ r2r = l

& 2 r722

# # = 66 & r2 2266 7

221

##= =

cUisæ‹ cau« = jf£o‹ mfy« & h = b = 12 br.Û

cUisæ‹ fdmsÎ = 221r h

722 122 2

# #r = ` j = 4158 br.Û3

8. xU bg‹ÁyhdJ xU ne® t£l cUis toéš cŸsJ. bg‹Áè‹ Ús« 28 br.Û

k‰W« mj‹ Mu« 3 ä.Û. bg‹ÁèDŸ mikªj ikæ‹ (»uh~ig£)-‹ Mu« 1 ä.Û

våš, bg‹Áš jahç¡f ga‹gL¤j¥g£l ku¥gyifæ‹ fdmsit¡ fh©f.

Ô®Î: cUis toéyhd bg‹Áè‹ Mu« k‰W« cau« (Ús«) Kiwna

R k‰W« h v‹f. r v‹gJ bg‹Áè‹ cŸsikªj ikæ‹ Mu« v‹f.

R = 3 ä.Û, h = 28 br.Û = 280 ä.Û k‰W« r = 1 ä.Û


Mfnt, ikæ‹ fdmsÎ = h R r2 2r -^ h = 280722 3 12 2

# # -^ h

= 22 40 8 7040# # =

= 7040 ä.Û3 = 7.04 br.Û3 .

9. xU Â©k¡ T«Ã‹ Mu« k‰W« rhÍau« Kiwna 20 br.Û k‰W« 29 br.Û. våš

m¤Â©k¡ T«Ã‹ fdmsit¡ fh©f.

Ô®Î: r h k‰W« l v‹gd Kiwna T«Ã‹ Mu«, cau« k‰W« rhÍau« v‹f.

r = 20 br.Û k‰W« l = 29 br.Û vd¡ bfhL¡f¥g£LŸsJ.

vdnt, h = l r2 2- = 29 202 2

- = 21 br.Û

Mfnt, T«Ã‹ fdmsÎ = r h31 2r = 20 21

31

722 2

# # # = 8800 br.Û3

10. ku¤Âdhyhd xU Â©k¡ T«Ã‹ mo¢R‰wsÎ 44 br.Û. k‰W« mj‹ cau«

12 br.Û våš m¤Â©k¡ T«Ã‹ fdmsit¡ fh©f.

Ô®Î: r k‰W« h M»ad ku¤Âdhyhd Â©k T«Ã‹ Mu« k‰W« cau« v‹f.

h = 12 br.Û. k‰W« mo¢R‰wsÎ 44 br.Û vd¡ bfhL¡f¥g£LŸsJ.

Â©k T«Ã‹ mo¢R‰wsÎ, 2 rr = 44

r = 244r

= 2 2244 7## = 7 br.Û

Mfnt, T«Ã‹ fdmsÎ = r h31 2r

= 7 131

722 22

# # #

vdnt, T«Ã‹ fdmsÎ = 166 br.Û3 .


11. xU gh¤Âu« T«Ã‹ Ïil¡f©l toéš cŸsJ. mj‹ nk‰òw Mu« k‰W« cau«

Kiwna 8 br.Û k‰W« 14 br.Û v‹f. m¥gh¤Âu¤Â‹ fdmsÎ 3

5676 br.Û3 våš,

mo¥g¡f¤ÂYŸs t£l¤Â‹ Mu¤Âid¡ fh©f.

Ô®Î: R, r k‰W« h v‹gd T«Ã‹ Ïil¡f©l toéš mikªj gh¤Âu¤Â‹ nk‰òw

Mu«, mo¥òw Mu« k‰W« cau« v‹f. nk‰òw Mu« R = 8 br.Û, cau« h = 14 br.Û

k‰W« fdmsÎ = 3

5676 br.Û3vd¡ bfhL¡f¥g£LŸsJ.

fdmsÎ = ( )h R r Rr31 2 2r + + =

35676

& 14 (8 8 )r r31

722 2 2

# # # #+ + = 3

5676

& 64 8r r2+ + =

3 22 145676 3 7# #

# # = 129

& 8 64r r2+ + = 129

8r r2+ = 65

& r(r +8) = 5 × 13 & r = 5Mfnt, Mu« r = 5 br.Û.

12. xU ne®t£l¡ T«Ã‹ Ïil¡f©l¤Â‹ ÏUòwK« mikªj t£l éë«òfë‹

R‰wsÎfŸ Kiwna 44 br.Û k‰W« 8.4r br.Û v‹f. mj‹ cau« 14 br.Û våš,

m›éil¡f©l¤Â‹ fdmsit¡ fh©f.

Ô®Î: R, r k‰W« h v‹gd Kiwna T«ò tot Ïil¡f©l¤Â‹ nk‰òw Mu«,

mo¥òw Mu« k‰W« cau« v‹f. nk‰òw R‰wsÎ 2 44Rr = br.Û, mo¥òw R‰wsÎ

2 .r 8 4r r= br.Û k‰W« h = 14 br.Û vd¡ bfhL¡f¥g£LŸsJ. vdnt,

R = 244

227 7

## = k‰W« . .r

28 4 4 2rr= = br.Û

Ïil¡f©l¤Â‹ fdmsÎ = ( )h R r Rr31 2 2r + +

= 14( . . )31

722 7 4 2 7 4 22 2

# # #+ +

= ( . . )344 49 29 4 17 64+ +

vdnt, Ïil¡f©l¤Â‹ fdmsÎ = 1408.57 br.Û3 .

13. 5 br.Û, 12 br.Û. k‰W« 13 br.Û. g¡f msÎfŸ bfh©l xU br§nfhz ABCT MdJ 12 br.Û. ÚsKŸs mj‹ xU g¡f¤ij m¢rhf¡ bfh©L RH‰w¥gL«nghJ

cUthF« T«Ã‹ fdmsit¡ f©LÃo.

Ô®Î: br§nfhz ABCT -‹ g¡f§fŸ 5 br.Û, 12 br.Û k‰W« 13br.Û

vd¡ bfhL¡f¥g£LŸsJ.12 br.Û ÚsKŸs g¡f¤ij m¢rhf¡ bfh©L K¡nfhz¤ij

RH‰¿dhš »il¡F« T«Ã‹ Mu« k‰W« cau« Kiwna 5 br.Û k‰W« 12 br.Û.

T«Ã‹ fdmsÎ = r h31 2r 5 5 12

31

722

# # # #= = 7

2200 = 314 .br.Û72 3


14. xU Â©k ne® t£l¡ T«Ã‹ MuK« cauK« 2 : 3 v‹w é»j¤Âš cŸsJ. mj‹

fdmsÎ 100.48 f.br.Û våš, m¡T«Ã‹ rhÍau¤ij¡ f©LÃo. (r = 3.14 v‹f)

Ô®Î: r, h k‰W« l v‹gd Kiwna ne®t£l¡ T«Ã‹ Mu«, cau« k‰W« rhÍau«

v‹f.

r : h = 2 : 3 vd¡ bfhL¡f¥g£LŸsJ.

vdnt, hr r h

32

32&= =

T«Ã‹ fdmsÎ, r h31 2r = 100.48

& 3.14 h h31

32 2

# # #` j = 100.48

& h3 = .. 8 273 14 4

100 48 3 9## # #=

& h = 8 273# = 2 × 3 = 6 br.Û

nkY«, r = h32 & r = 2

36 4# = br.Û.

vdnt, l r h2 2= + = 6 42 2

+ = 2 3 22 2+ = 2 13 br.Û.

15. xU ne® t£l¡ T«Ã‹ fdmsÎ 216r f.br.Û k‰W« m¡T«Ã‹ Mu« 9 br.Û våš,

mj‹ cau¤ij¡ fh©f.

Ô®Î: r k‰W« h v‹gd T«Ã‹ Mu« k‰W« cau« v‹f.

r = 9 br.Û, fdmsÎ 216r vd bfhL¡f¥g£LŸsJ.

T«Ã‹ fdmsÎ, r h31 2r = 216r

9 h31 2# # #r = 216r

Mfnt, T«Ã‹ cau«, h = 9 9

216 3# #

#rr = 8 br.Û

16. nfhs toéyikªj 200 ÏU«ò F©LfŸ (ball bearings) x›bth‹W« 0.7 br.Û Mu«

bfh©lJ. ÏU«Ã‹ ml®¤Â 7.95 »uh«/br.Û3 våš ÏU«ò¡ F©Lfë‹

ãiwia¡ fh©f. (ãiw = fdmsÎ ×ml®¤Â)

Ô®Î: r v‹gJ nfhstot ÏU«ò F©o‹ Mu« v‹f.

r = 0.7 br.Û vd¡ bfhL¡f¥g£LŸsJ.

ÏU«ò F©o‹ fdmsÎ = r34 3r

= 0.7 0.7 0.734

722

# # # #

200 ÏU«ò F©Lfë‹ fdmsÎ = . .3

88 0 049 200 287 46# # = br.Û3 .

1 br.Û3ÏU«Ã‹ ml®¤Â = 7.95 »

vdnt, 200 ÏU«ò F©o‹ ãiw = 287.46 × 7.95 = 2285.316 »

Mfnt, 200 ÏU«ò F©o‹ ãiw = . .1000

2285 316 2 29= ».»


17. xU cŸÇl‰w nfhs¤Â‹ btë k‰W« cŸ Mu§fŸ Kiwna 12 br.Û k‰W« 10 br. Û

våš, m¡nfhs¤Â‹ fd msit¡ fh©f.

Ô®Î: R k‰W« r v‹gd cŸÇl‰w nfhs¤Â‹ btë¥òw k‰W«

c£òw Mu« v‹f.

R = 12 br.Û k‰W« r = 10 br.Û vd¡ bfhL¡f¥g£LŸsJ.

cŸÇl‰w nfhs¤Â‹ fdmsÎ = ( )R r34 3 3r -

= (12 10 )34

722 3 3

# - = (1728 1000)2188 -

= 7282188

# = 305032 br.Û3 .

18. X® miu¡nfhs¤Â‹ fd msÎ 1152r f.br. Û. våš, mj‹ tisgu¥ò fh©f.

Ô®Î: miu¡nfhs¤Â‹ Mu« r v‹f. mj‹ fdmsÎ 1152r br.Û3 vd¡

bfhL¡f¥g£LŸsJ.

miu¡nfhs¤Â‹ fdmsÎ, r32 3r = 1152r

& r3 = 115223

# = 1728& r = 17283 br.Û = 12 br.Û

Mfnt, miu¡nfhs¤Â‹ tisgu¥ò = 2 r2r & 2 288r2# #r r= br.Û2 .

19. 14 br.Û g¡f msÎfŸ bfh©l xU fd¢rJu¤Âš ÏUªJ bt£obaL¡f¥gL«

äf¥bgça T«Ã‹ fdmsit¡ fh©f.

Ô®Î: fdrJu¤Â‹ g¡f« 14 br.Û vd¡ bfhL¡f¥g£LŸsJ.

fdrJu¤ÂèUªJ äf¥bgça T«ò bt£obaL¡f¥g£lhš

T«Ã‹ Mu«, r = fdrJu¤Â‹ g¡f«2

= 214 7= br.Û.

nkY«, T«Ã‹ cau«, h = fd¢ br›tf¤Â‹ g¡f« = 14 br.Û

T«Ã‹ fdmsÎ = 7 7 14r h31

31

7222

# # # #r =

Mfnt, T«Ã‹ fdmsÎ = 718.673

2156 = br.Û3 .

20. 7 br.Û Mu« bfh©l nfhs tot gÿåš fh‰W brY¤j¥gL«nghJ mj‹ Mu«

14 br.Û Mf mÂfç¤jhš, m›éU ãiyfëš gÿå‹ fdmsÎfë‹ é»j¤ij¡

fh©f.

Ô®Î: r1 k‰W« r2 v‹gd Kiwna nfhstoéyhd gÿåš fh‰W brY¤j¥gLtj‰F

K‹ò«, Ã‹ò« mj‹ Mu§fŸ v‹f.

r1 = 7 br.Û k‰W« r2 =14 br.Û vd¡ bfhL¡f¥g£LŸsJ.

vdnt, : :r r 7 141 2

= .

Mfnt, Ï›éU ãiyfëY« nfhstot gÿå‹ fdmsÎfë‹ é»j«

4 : 4r r1

3

2

3r r = 7 :143 3

= 7 7 7 :14 14 14# # # # = 1 : 8


gæ‰Á 8.3

1. xU éisah£L g«gukhdJ (Top) T«Ã‹ ÛJ miu¡nfhs« Ïizªj toéš

cŸsJ. miu¡nfhs¤Â‹ é£l« 3.6 br.Û k‰W« g«gu¤Â‹ bkh¤j cau« 4 . 2 br.Û

våš, mj‹ bkh¤j¥ òw¥gu¥ig¡ fh©f.

Ô®Î: miu¡nfhs gFÂ: é£l« 2r = 3.6 br.Û & r = 1.8 br.Û

T«ò¥ gFÂ: Mu« r = 1.8 br.Û, cau« h = 4.2 – 1.8 = 2.4 br.Û

vdnt, rhÍau« l = . .h r 2 4 1 82 2 2 2+ = +

= . ( . ) ( )0 6 4 3 0 6 52 2+ = = 3 br.Û.

g«gu¤Â‹ bkh¤j òw¥gu¥ò = miu¡nfhs¤Â‹ tisgu¥ò + T«Ã‹ tisgu¥ò

= 2 r rl2r r+ = (2 )r r lr +

= 1.8(2 1.8 3)# #r +

= 1.8 6.6# #r

vdnt, g«gu¤Â‹ bkh¤j òw¥gu¥ò = 11.88r br.Û2 . 2. xU fd cUt«, miu¡nfhs¤Â‹ ÛJ cUis Ïizªj toéš cŸsJ.

m¡fdÎUt¤Â‹ é£l« k‰W« bkh¤j cau« Kiwna 21 br.Û k‰W« 25.5 br.Û

våš, mj‹ fd msit¡ fh©f.

Ô®Î: miu¡nfhs gFÂ: é£l«, 2r = 21 br.Û & r = 221 br.Û

T«ò¥ gFÂ: Mu«, r = 221 br.Û, cau«, h = 15 br.Û

fd cUt¤Â‹ fdmsÎ = miu¡nfhs¤Â‹

fdmsÎ

cUisæ‹

fdmsÎ+e eo o

= r r h32 3 2r r+ = ( )r r h

322r +

= 722

221

221

32

221 15# # # +` j

= 33 22 762221 3# # = br.Û3 .

3. xU kUªJ¡ F¥ÃahdJ xU cUisæ‹ ÏUòwK« miu¡nfhs§fŸ Ïizªj

toéš cŸsJ. kUªJ¡ F¥Ãæ‹ bkh¤j Ús« 14 ä.Û k‰W« é£l« 5 ä.Û våš

m«kUªJ¡ F¥Ãæ‹ òw¥gu¥ig¡ fh©f.

Ô®Î: miu¡nfhs tot gFÂæ‹ Mu«, r = 25 ä.Û.

cUis tot gFÂæ‹ Mu«, r = 25 ä.Û

cau«, h = bkh¤j cau« – 2(Mu«) = 14 – 5 = 9 ä.Û

kUªJ F¥Ãæ‹ bkh¤j òw¥gu¥ò = 2cUisæ‹

tisgu¥ò

miu¡nfhs¤Â‹

tisgu¥ò#+e eo o

= 2 4rh r2r r+ = 2 ( 2 )r h rr +

= 2722

25 9 2

25 220# # #+ =` j

Mfnt, kUªJ F¥Ãæ‹ bkh¤j òw¥gu¥ò = 220 ä.Û2 .


4. xU TlhukhdJ cUisæ‹ ÛJ T«ò Ïizªj toéš cŸsJ. Tlhu¤Â‹ bkh¤j

cau« 13.5 Û k‰W« é£l« 28 Û. nkY« cUis¥ ghf¤Â‹ cau« 3 Û våš,

Tlhu¤Â‹ bkh¤j òw¥gu¥ig¡ fh©f.

Ô®Î: cUis tot gFÂ: cau« h = 3 Û, é£l« 2r = 28 Û & r = 14T«ò tot gFÂ: cau« h1 = bkh¤j cau« – cUisæ‹ cau«

h1 = 13.5 – 3 = 10.5 Û

Mfnt, Mu«, r = 14 Û

rhÍau«, l = .h r 10 5 141

2 2 2 2+ = +

= 0.7 ( . ) ( )15 20 0 7 252 2+ =

= 17.5 Û

Tlhu¤Â‹ bkh¤j òw¥gu¥ò = cUisæ‹ tisgu¥ò + T«Ã‹ tisgu¥ò

= 2 (2 )rh rl r h lr r r+ = + = 14(2 3 17.5)722

# # +

= 44 (6 17.5)# + = 44 23.5 1034# =

Mfnt, Tlhu¤Â‹ bkh¤j òw¥gu¥ò = 1034 r.Û

5. fëk©iz¥ ga‹gL¤Â xU khzt‹ 48 br.Û cauK« 12 br.Û MuK« bfh©l

ne® t£lÂ©k¡ T«ig¢ brŒjh®. m¡T«ig k‰bwhU khzt® xU Â©k¡

nfhskhf kh‰¿dh®. m›thW kh‰w¥g£l òÂa nfhs¤Â‹ Mu¤ij¡ fh©f.

Ô®Î: r1 k‰W« h v‹gJ T«Ã‹ Mu« k‰W« cau« v‹f. nkY«, r2 v‹gJ

nfhs¤Â‹ Mu« v‹f.

r1 = 12 br.Û, h = 48 br.Û vd¡ bfhL¡f¥g£LŸsJ.T«ò tot¤ij nfhs totkhf kh‰¿a Ã‹ò,

nfhs¤Â‹ fdmsÎ = T«Ã‹ fdmsÎ

r34

2

3r = r h31

1

2r

& r2

3 = 31 12 48

432

# # # #r

r = 123

Mfnt, nfhs¤Â‹ Mu« = 12 br.Û.

6. 24 br.Û MuKŸs xU Â©k cnyhf nfhskhdJ cU¡f¥g£L 1.2 ä.Û MuKŸs

Óuhd cUis¡ f«Ãahf kh‰w¥g£lhš, m¡f«Ãæ‹ Ús¤ij¡ fh©f.

Ô®Î: r1 k‰W« r2 v‹gJ Kiwna nfhs« k‰W« cUisæ‹ Mu«

v‹f. nkY« h v‹gJ Óuhd cUis tot f«Ãæ‹ Ús« v‹f.

r1 = 24 br.Û = 240 ä.Û, r2 = 1.2 ä.Û = 1012 ä.Û vd¡ bfhL¡f¥g£LŸsJ.

Â©k¡nfhs« cUistot Óuhd f«Ãahf kh‰w¥g£l Ã‹ò,

Óuhd cUistot f«Ãæ‹ fdmsÎ = nfhs¤Â‹ fdmsÎ


r h2

2r = r34

1

3r

h1012

1012

# # #r = 240 240 24034# # # #r

& h = 34 240 240 240

1210

1210

# # # # # #r

r

= 12800000 ä.Û

Mfnt, cUis¡ f«Ãæ‹ Ús« = 12.8100000012800000 = ».Û

7. 5 br.Û cŸt£lMuK« 24 br.Û cauK« bfh©l T«ò tot gh¤Âu¤Âš KG

mséš j©Ù® cŸsJ. Ï¤j©ÙuhdJ 10 br.Û cŸMuKŸs cUis tot

fhè¥ gh¤Âu¤Â‰F¥ kh‰w¥gL«nghJ, cUis¥ gh¤Âu¤Âš cŸs Ú® k£l¤Â‹

cau¤ij¡ fh©f.

Ô®Î: T«òtot gh¤Âu«: Mu« r1 = 5 br.Û, cau« h1 = 24 br.Û

cUis tot gh¤Âu« : Mu« r2 = 10 br.Û, cau« = h2 v‹f.

cUistot gh¤Âu¤ÂYŸs

j©Ùç‹ fdmsÎe o = T«òtot gh¤Âu¤ÂYŸs

j©Ùç‹ fdmsÎe o

r h2

2

2r = r h

31

1

2

1r

10 h2

2# #r = 5 24

31 2# # #r

h2 = 31

10 105 5 24

## #

# # #rr

Mfnt, cUis tot gh¤Âu¤Âš cŸs j©Ù® k£l¤Â‹ cau« = 2 br.Û. 8. Á¿jsÎ j©Ù® ãu¥g¥g£l 12 br.Û é£lKŸs cUis tot¥ gh¤Âu¤Âš 6 br.Û

é£lKŸs xU Â©k¡ nfhs¤ij KGtJkhf _œf¢ brŒjhš, cUis tot¥

gh¤Âu¤Âš ca®ªj Ú® k£l¤Â‹ cau¤ij¡ fh©f.

Ô®Î: r1 k‰W« r2 v‹gd Kiwna cUis k‰W« nfhs¤Â‹ Mu§fŸ v‹f. nkY«

h v‹gJ cUistot¥ gh¤Âu¤Âš ca®ªj j©Ùç‹ cau« v‹f.

cUisæ‹ é£l« 2r1 = 12 br.Û k‰W« nfhs¤Â‹ é£l« 2r2 = 6 br.Û vd¡

bfhL¡f¥£LŸsJ. vdnt, r1 = 6 br.Û, r2 = 6 br.Û

Â©k¡nfhs¤ij j©Ùçš _œf¢brŒj Ã‹ò,

ca®ªj j©Ù® k£l¤Â‹ fdmsÎ = nfhs¤Â‹ fdmsÎ

r h1

2r = r34

2

3r

6 6 h# # #r = 3 3 334# # # #r

h = 34

6 63 3 3 1## #

# # #rr =

Mfnt, cUistot¥ gh¤Âu¤Âš ca®ªj j©Ù® k£l« = 1 br.Û. 9. 7 br.Û cŸ Mu« bfh©l cUis tot FHhæ‹ têna 5 br.Û / édho ntf¤Âš

j©Ù® ghŒ»wJ. miu kâ neu¤Âš m¡FHhŒ têna ghŒªj j©Ùç‹

fd msit¡ (è£lçš) fh©f.

Ô®Î: cUis tot¡FHhæ‹ Mu«, r = 7 br.Û


Úç‹ ntf« = 5 br.Û/édho. neu« = 30 ãäl§fŸ

= 30 × 60 = 1800 édhofŸ

xU édhoæš btëna‰w¥gL« Úç‹ fdmsÎ

r h2r = 7 7 5722

# # # = 770 f.br.Û

vdnt, miukâ neu¤Âš btëna‰w¥gL« j©Ùç‹ fdmsÎ 770 × 1800 = 1386000 f.br.Û

= 1000

1386000 = 1386 è£l®.

kh‰WKiw:

miukâ neu¤Âš btëna‰w¥gL« j©Ùç‹ fdmsÎ = FHhæ‹ FW¡F¥gu¥ò × neu« × ntf«

= ( r2r ) × 5 × 1800 = 722 7 7# #` j × 5 × 1800

= 1386000 f.br.Û

= 1000

1386000 = 1386 è£l®.

10. 4 Û é£lK« 10 Û cauKŸs cUis tot¤ bjh£oæYŸs j©ÙuhdJ 10 br.Û

é£lKŸs xU cUis tot FHhŒ têna kâ¡F 2.5 ».Û ntf¤Âš

btëna‰w¥gL»wJ. bjh£oæš ghÂasÎ j©Ù® btëna‰w¥gl MF« neu¤ij¡

fh©f. (Mu«g ãiyæš bjh£o KGtJ« j©Ù® ãu¥g¥g£LŸsJ vd¡ bfhŸf.)

Ô®Î : r1 k‰W« h1 v‹gJ cUistot bjh£oæ‹ Mu« k‰W« cau« v‹f. r2 v‹gJ cUistot FHhæ‹ Mu« v‹f. nkY« T v‹gJ ghÂasÎ j©Ùiu

bjh£oæèUªJ btëna‰w¥gl MF« neu« v‹f.

cUis tot bjh£o:

é£l« 2r1 = 4 Û & r1 = 2 Û k‰W« cau« h1 = 10 Û cUis tot FHhŒ:

é£l« 2r2 = 10 br.Û & r2= 1005 Û

ntf« = 2.5 ».Û/kâ = 2500 Û/kâ.

h2

= neu« × ntf« = T × 2500 v‹f.

cUis tot FHhŒ têahf btëna‰w¥g£l Úç‹ fdmsÎ

= 21 (cUis tot bjh£oæ‹ fdmsÎ)

r h2

2

2r =

21 r h

1

2

1r` j

T1005

1005 2500# # # #r = 2 2 10

21# # # #r

T = 2

2 2 105 5

100 10025001# # # #

# ## #

rr

= .2580 3 2= kâ

Mfnt, bjh£oæèUªJ ghÂasÎ j©Ù® btëna‰w¥gl 3 kâ 12 ãäl« MF«.


11. 18 br.Û MuKŸs Â©k cnyhf¡ nfhskhdJ cU¡f¥g£L _‹W Á¿a bt›ntW

msÎŸs nfhs§fshf th®¡f¥gL»wJ. m›thW th®¡f¥g£l Ïu©L Â©k¡

nfhs§fë‹ Mu§fŸ Kiwna 2 br.Û k‰W« 12 br.Û våš _‹whtJ nfhs¤Â‹

Mu¤ij¡ fh©f.

Ô®Î: R v‹gJ Â©k nfhs¤Â‹ Mu« v‹f. r1, r2 k‰W« r3 v‹gd òÂjhf

th®¡f¥gL« nfhs§fë‹ Mu§fŸ v‹f. R = 18 br.Û. r1 = 2 br.Û k‰W«

r2 = 12 br.Û vd¡ bfhL¡f¥g£LŸsJ.

_‹W òÂa nfhs§fë‹ fdmsÎ = Â©k nfhs¤Â‹ fdmsÎ

r r r34

34

34

1

3

2

3

3

3r r r+ + = R34 3r

( )r r r34

1

3

2

3

3

3r + + = R34 3# #r

2 12 r3 3

3

3+ + = 183

r3

3 = 5832 1736- = 4096 = 163

& r3 = 16Mfnt, _‹wthJ nfhs¤Â‹ Mu« = 16 br.Û.

12. xU cŸÇl‰w cUis tot¡ FHhæ‹ Ús« 40 br.Û. mj‹ cŸ k‰W« btë Mu§fŸ

Kiwna 4 br.Û k‰W« 12 br.Û. m›ÎŸÇl‰w cUis¡ FHhŒ cU¡f¥g£L 20 br.Û

ÚsKŸs Â©k ne® t£l cUisahf kh‰W«nghJ »il¡F« òÂa cUisæ‹

Mu¤ij¡ fh©f.

Ô®Î : R, r k‰W« h v‹gd Kiwna

cŸÇl‰w cUis tot FHhæ‹ btë,

cŸ Mu§fŸ k‰W« cau« v‹f. r1 k‰W« h1 v‹gJ Kiwna cŸÇl‰w cUisæ‹ Mu«

k‰W« cau« v‹f.

cŸÇl‰w cUis: R = 12 br.Û, r = 4 br.Û, h = 40 br.Û cŸÇl‰w cUis: h1 = 20 br.Û

Â©k cUisæ‹ fdmsÎ = cŸÇl‰w cUisæ‹ fdmsÎ

r h1

2

1r = ( )h R r2 2r -

20r1

2# #r = 40(12 4 )2 2

#r -

r1

2 = ( )20

40 144 16#

#r

r - = 2 × 128 = 256

r1

& = 16.

Mfnt, Â©k cUisæ‹ Mu« = 16 br.Û.

13. 8 br.Û é£lK« 12 br.Û cauK« bfh©l xU ne® t£l Â©k ÏU«ò¡ T«ghdJ

cU¡f¥g£L 4 ä.Û MuKŸs Â©k¡ nfhs tot F©Lfshf th®¡f¥g£lhš

»il¡F« nfhs tot F©Lfë‹ v©â¡ifia¡ fh©f.

Ô®Î: r k‰W« h v‹gd Kiwna Â©k¡ T«Ã‹ Mu« k‰W« cau« v‹f.

nkY«, nfhstot ÏU«ò F©o‹ Mu« r1 v‹f.


r = 4 br.Û = 40 ä.Û, h = 12 br.Û = 120 ä.Û,r1 = 4 ä.Û vd¡ bfhL¡f¥g£LŸsJ. nfhstot F©Lfë‹ v©â¡if n v‹f.

n × (nfhstot F©o‹ fdmsÎ) = T«Ã‹ fdmsÎ

n r34

1

3# # r = r h

31 2# # #r

4n34 3

# # #r = 40 12031 2# # #r

n = 75031

4 4 440 40 120

43

## # #

# # # #r

r =

Mfnt, »il¡F« nfhstot F©Lfë‹ v©â¡if 750.

14. 12 br.Û é£lK« 15 br.Û cauK« bfh©l ne®t£l cUis KGtJ« gå¡Têdhš

(ice cream) ãu¥g¥g£LŸsJ. Ï¥gå¡THhdJ 6 br.Û é£lK«, 12 br.Û cauK«

bfh©l T«Ã‹ ÛJ nk‰òw« miu¡nfhs« mikªjthW ãu¥g¥gL»wJ våš,

v¤jid T«òfëš gå¡Têid KGtJkhf ãu¥gyh« vd¡ fh©f.

Ô®Î : r1 k‰W« h1 v‹gd Kiwna cUisæ‹ Mu«

k‰W« cau« v‹f. nkY«, r2 k‰W« h2 v‹gJ T«Ã‹

Mu« k‰W« cau« v‹f.

ne®t£l cUis: é£l« 2r1 = 12 br.Û & r1 = 6 br.Û k‰W« h1 = 15 br.Û

T«ò: é£l«, 2r2 = 6 br.Û & r2 = 3 br.Û k‰W« h2 = 12 br.Û gå¡Tœ ãu¥g¥gL« T«òfë‹ v©â¡if

= T«Ã‹ fdmsÎ miu¡nfhs¤Â‹ fdmsÎ

cUisæ‹ fdmsÎ+

= r h r

r h

2

2

2 2

3

1

2

1

31

32r r

r

+ =

( )r h r2

6 6 15

2

2

2 231 # #

# # #

r

r

+

= ( )3 3 12 2 3

6 6 15

31 # # #

# #

+ =

36 6

1815 10

## # =

Mfnt, gå¡Tœ ãu¥g¥gL« T«òfë‹ v©âif 10.

15. 4.4 Û ÚsK« 2 Û mfyK« bfh©l xU fd¢br›tf tot¤ bjh£oæš

kiHÚ® nrfç¡f¥gL»wJ. Ï¤bjh£oæš 4 br.Û cau¤Â‰F nrfç¡f¥g£l

kiH ÚuhdJ 40br.Û MuKŸs cUis tot fhè¥ gh¤Âu¤Â‰F

kh‰w¥gL«nghJ m¥gh¤Âu¤Âš cŸs j©Ù® k£l¤Â‹ cau¤ij¡ fh©f.

Ô®Î: cUistot gh¤Âu«: Mu« r = 40 br.Û,j©Ùç‹ k£l¤Â‹ cau« = h v‹f.

fd¢br›tf¤ bjh£o: Ús« l = 4.4 Û = 440 br.Û,

mfy« b = 2 Û = 200 br.Û k‰W« cau« h1= 4 br.Û vd bfhL¡f¥g£LŸsJ.

kiH ÚuhdJ cUistot gh¤Âu¤Â‰F kh‰¿a Ã‹ò,


cUistot gh¤Âu¤ÂYŸs

j©Ùç‹ fdmdÎe o = fd¢br›tf bjh£oæYŸs

j©Ùç‹ fdmsÎe o

r h2r = lbh1

40 40 h722

# # # = 440 200 4# #

& h = 40 40 22

440 200 4 7# #

# # # = 70

Mfnt, cUis tot gh¤Âu¤Âš cŸs j©Ù® k£l¤Â‹ cau« 70 br.Û. 16. kzyhš ãu¥g¥g£l xU cUis tot thëæ‹ cau« 32 br.Û k‰W« Mu«

18 br.Û. m«kzš KGtJ« jiuæš xU ne®t£l¡ T«ò toéš bfh£l¥gL»wJ.

m›thW bfh£l¥g£l kz‰T«Ã‹ cau« 24 br.Û våš, m¡T«Ã‹ Mu« k‰W«

rhÍau¤ij¡ fh©f.

Ô®Î: r k‰W« h v‹gd Kiwna cUis tot thëæ‹ Mu«

k‰W« cau« v‹f. r1, h1 k‰W« l1 v‹gd Kiwna T«ò tot

kz‰Féaè‹ Mu«, cau« k‰W« rhÍau« v‹f.

cUis tot thë : Mu« r = 18 br.Û, cau« h = 32 br.Û

T«ò tot Féa : cau« h1 = 24 br.Û

T«ò tot kz‰Féaè‹ fd msÎ

1 = cUis tot thëæšcŸs kzè‹ fd msÎ'

r h31

1

2

1r = r h2r

24r31

1

2# # #r = 18 18 32# # #r

r1 = 18 18 4# # = 18×2 = 36

Mfnt, Mu«, r1 = 36 br.Û

nkY«, rhÍau«, l1 = r h 36 241

2

1

2 2 2+ = +

= 12 3 22 2+ = 12 13 br.Û

17. 14 Û é£l« k‰W« 20 Û MHKŸs xU »zW cUis toéš bt£l¥gL»wJ.

m›thW bt£L«nghJ njh©obaL¡f¥g£l k© Óuhf gu¥g¥g£L 20 Û # 14 Û

msÎfëš mo¥g¡fkhf¡ bfh©l xU nkilahf mik¡f¥g£lhš, m«nkilæ‹

cau« fh©f.

Ô®Î: r k‰W« h v‹gd cUis tot »z‰¿‹ Mu« k‰W« cau«

v‹f. nkY« l, b k‰W« h1 v‹gd Kiwna br›tf tot nkilæ‹

Ús«, mfy« k‰W« cau« v‹f.

cUis tot »zW: é£l« 2r = 14 Û & r = 7 Û, MH« (cau«) h = 20 Û

br›tf tot nkil: Ús« l = 20 Û, mfy« b = 14 Ûfd¢br›tf tot nkilæ‹ fdmsÎ = cUistot »z‰¿‹ fdmsÎ lbh

1 = r h2r

20 × 14 × h1 = 7 7 20

722

# # #

h1 = 7 7

722

20 1420 11#

## # =

Mfnt, nkilæ‹ cau« 11 Û.


gæ‰Á 8.4rçahd éilia¤ nj®ªbjL¡fÎ«.

1. 1 br.Û MuK« k‰W« 1 br.Û cauK« bfh©l xU ne® t£l cUisæ‹ tisgu¥ò

(A) r br.Û2 (B) 2r br.Û2 (C) 3r br.Û2 (D) 2 br.Û2

Ô®Î: cUisæ‹ tisgu¥ò = 2 2 1rh 1# # #r r= =2r br.Û2 ( éil. (B) )

2. xU ne®t£l cUisæ‹ MukhdJ mj‹ cau¤Âš ghÂ våš mj‹ bkh¤j¥

òw¥gu¥ò

(A) 23 rh r.m (B) h

32 2r r.m (C) h

23 2r r.m (D)

32 rh r.m

Ô®Î: Mu«, r = h2

bkh¤j òw¥gu¥ò = 2 ( ) 2r h r h h h h h h2 2 2

323 2r r r r+ = + = =` `j j r.myFfŸ.

( éil. (C) )

3. xU ne®t£l cUisæ‹ mo¥g¡f¥ gu¥ò 80 r. br.Û. mj‹ cau« 5 br.Û våš, T«Ã‹

fd msÎ

(A) 400 br.Û3 (B) 16 br.Û3 (C) 200 br.Û3 (D)3

400 br.Û3

Ô®Î: mog¡f gu¥ò = 80r2r = br.Û2 , fdmsÎ V = 0 5 400r h 82#r = = br.Û3.

( éil. (A) )

4. xU ne®t£l cUisæ‹ bkh¤j òw¥gu¥ò 200 r. br.Û.r k‰W« mj‹ Mu« 5 br.Û

våš mj‹ cau« k‰W« Mu¤Â‹ TLjš


Ô®Î: cUisæ‹ bkh¤j òw¥gu¥ò, TSA = 200r br.Û2 k‰W« Mu«, r = 5 br.Û

2 ( )r h rr + 200r= r.br.Û 2 5( ) 200 ( ) 20h r h r& &#r r+ = + = br.Û

( éil. (A) ) 5. a myFfŸ MuK«, b myFfŸ cauK« bfh©l xU ne®t£l cUisæ‹ tisgu¥ò

(A) a b2r r.br.Û (B) 2rab r.br.Û (C) 2r r.br.Û (D) 2 r.br.Û

Ô®Î: cUisæ‹ Mu« r = a, cau« h = b,

tisgu¥ò, CSA = rh ab2 2r r= r.br.Û ( éil. (B) )

6. xU ne®t£l¡ T«ò k‰W« ne®t£l cUisæ‹ MuK« cauK« Kiwna rk«

cUisæ‹ fd msÎ 120 br.Û3 våš, T«Ã‹ fd msÎ

(A) 1200 br.Û3 (B) 360 br.Û3 (C) 40 br.Û3 (D) 90 br.Û3

Ô®Î: T«Ã‹ fdmsÎ = 31 cUisæ‹ fdmsÎ = (120) 40

31 = br.Û3

( éil. (C) )


7. ne® t£l¡ T«Ã‹ é£l« k‰W« cau« Kiwna 12 br.Û k‰W« 8 br.Û våš mj‹

rhÍau«


Ô®Î: rhÍau«, l = 10r h 6 8 1002 2 2 2+ = + = = br.Û ( éil. (A) )

8. xU ne® t£l¡ T«Ã‹ mo¢R‰wsÎ k‰W« rhÍau« Kiwna 120r br.Û k‰W« 10 br.Û våš mj‹ tisgu¥ò

(A) 1200r br.Û2 (B) 600r br.Û2 (C) 300r br.Û2 (D) 600 br.Û2

Ô®Î: T«Ã‹ mo¥gu¥Ã‹ R‰wsÎ r2 120r r= & r = 60

tisgu¥ò, CSA = 60rl 10 600# #r r r= = br.Û2 ( éil. (B) )

9. xU ne® t£l¡ T«Ã‹ fd msÎ k‰W« mo¥g¡f¥ gu¥ò Kiwna 48r br.Û3 k‰W« 12r br.Û2våš, mj‹ cau«


Ô®Î: T«Ã‹ fdmsÎ = 48r h31 2r r= Û3

mog¡f gu¥ò = 12r2r r= br.Û2 .

48 12 48 12r h h h31

31

1248 32

& &# # #r r r rrr= = = = br.Û ( éil. (D) )

10. 5 br.Û cauK«, 48 r.br.Û mo¥g¡f¥ gu¥ò« bfh©l xU ne® t£l¡ T«Ã‹ fd msÎ


Ô®Î: T«Ã‹ cau« h = 5br.Û, mog¡f gu¥ò = 48r2r = br.Û2

fdmsÎ = 48 5 80r h31

312# #r = = br.Û3 ( éil. (C) )

11. Ïu©L cUisfë‹ cau§fŸ Kiwna 1:2 k‰W« mt‰¿‹ Mu§fŸ Kiwna 2:1 M»a é»j§fëèU¥Ã‹, mt‰¿‹ fdmsÎfë‹ é»j«

(A) 4 : 1 (B) 1 : 4 (C) 2 : 1 (D) 1 : 2

Ô®Î: h1 : h2 = 1 : 2, r1 : r2 = 2 : 1fd msÎfë‹ é»j«

: : 2 1 :1 2 4 : 2 2 :1V V r h r h31

31

1 2 1

2

1 2

2

2

2 2& # #r r= = = ( éil. (C) )

12. 2 br.Û Mu« cŸs xU nfhs¤Â‹ tisgu¥ò

(A) 8r br.Û2 (B) 16 br.Û2 (C) 12r br.Û2 (D) 16r br.Û2 .

Ô®Î: nfhs¤Â‹ tisgu¥ò, CSA = 4 4 2 16r2 2#r r r= = br.Û2 ( éil. (D) )


13. xU Â©k miu¡nfhs¤Â‹ é£l« 2 br.Û våš mj‹ bkh¤j òw¥gu¥ò

(A) 12 br.Û2 (B) 12r br.Û2 (C) 4r br.Û2 (D) 3r br.Û2 .

Ô®Î: miu¡nfhs¤Â‹ bkh¤j òw¥gu¥ò, TSA = 3 3 1 3r2 2#r r r= = br.Û2

(éil. (D) )

14. 169 r f.br.Û. fd msÎ bfh©l nfhs¤Â‹ Mu«

(A) 34 br.Û (B)

43 br.Û (C)

23 br.Û (D)

32 br.Û.

Ô®Î: nfhs¤Â‹ fdmsÎ V = 169 r f.br.Û.

r34 3r = r

169

109

43

64273

& #r r r= = r43& = ( éil. (B) )

15. Ïu©L nfhs§fë‹ tisgu¥òfë‹ é»j« 9 : 25. mt‰¿‹ fd msÎfë‹ é»j«

(A) 81 : 625 (B) 729 : 15625 (C) 27 : 75 (D) 27 : 125.

Ô®Î: S1 : S2 = 9 : 25

& r12

: r22

= 9 : 25

& r1 : r2 = 3 : 5

:r r1

3

2

3 = 3 : 53 3 = 27 :125

:V V1 2

= 27 :125 . ( éil. (D) ) 16. a myFfŸ Mu« bfh©l Â©k miu¡nfhs¤Â‹ bkh¤j¥ òw¥gu¥ò

(A) 2r a2 r.m (B) 3ra2 r.m (C) 3ra r.m (D) 3a2 r.m.

Ô®Î: miu¡nfhs¤Â‹ bkh¤j¥ òw¥gu¥ò, TSA=3 3r a2 2r r= r.myFfŸ (éil(B))

17. 100r r.br.Û tisgu¥ò bfh©l nfhs¤Â‹ Mu«

(A) 25 br.Û (B) 100 br.Û (C) 5 br.Û (D) 10 br.Û.

Ô®Î: nfhs¤Â‹ tisgu¥ò = 100r

4 r2& r = 100 25 5r r4

1002& &r

rr= = = ( éil. (C) )

18. xU nfhs¤Â‹ tisgu¥ò 36r r.br.Û våš, mj‹ fd msÎ

(A) 12r br.Û3 (B) 36r br.Û3 (C) 72r br.Û3 (D)108r br.Û3

Ô®Î: nfhs¤Â‹ tisgu¥ò = 4 36 9 3r r rcm2 2 2& &r r= = =

fdmsÎ = 3 36r34

343 3

#r r r= = br.Û3 ( éil. (B) )

19. 12r br.Û2 bkh¤j¥gu¥ò bfh©l Â©k miu¡nfhs¤Â‹ tisgu¥ò

(A) 6r br.Û2 (B) 24r br.Û2 (C) 36r br.Û2 (D) 8r br.Û2 .

Ô®Î: miu¡nfhs¤Â‹ bkh¤j¥gu¥ò, TSA 3 r2r = 12r br.Û2 .

tisgu¥ò, CSA, 2 r2r = 8r

r

3

12 22

2#

r

r r r= br.Û2 ( éil. (D) )


20. xU nfhs¤Â‹ MukhdJ k‰bwhU nfhs¤Â‹ Mu¤Âš ghÂ våš mt‰¿‹ fd

msÎfë‹ é»j«

(A) 1 : 8 (B) 2: 1 (C) 1 : 2 (D) 8 : 1

Ô®Î: rr

212= vd¡ bfhL¡f¥g£LŸsJ. r r2

1 2=

:V V1 2

= 4 : 4 : 2 : 1 : 8r r r r r r81

3

2

3

1

3

13

1

3

1

3r r = = =^ h . ( éil. (A) )

21. xU Â©k nfhs¤Â‹ tisgu¥ò 24 br.Û2 mªj nfhs¤ij Ïu©L

miu¡nfhs§fshf¥ Ãç¤jhš »il¡F« miu¡nfhs§fëš x‹¿‹ bkh¤j¥

òw¥gu¥ò


Ô®Î: nfhs¤Â‹ tisgu¥ò r4 2r = 24br.Û2

miu¡nfhs¤Â‹ bkh¤j¥ òw¥gu¥ò TSA, 3 r2r = 24 18r

r

4

32

2

#r

r = br.Û2

( éil. (D) )

22. Ïu©L T«òfŸ rk Mu§fŸ bfh©LŸsd. nkY« mt‰¿‹ rhÍau§fë‹ é»j« 4 : 3 våš, tisgu¥òfë‹ é»j«

(A) 16 : 9 (B) 8 : 6 (C) 4 : 3 (D) 3 : 4

Ô®Î: r1 = r

2 k‰W« :l l

1 2 = 4 : 3.

vdnt, :r l r l1 1 2 2r r = :l l

1 2 = 4 : 3 ( éil. (C) )

c§fS¡F bjçÍkh?Ã¤jhfu° v©fŸ

a b c2 2 2+ = våš ( , , )a b c xU Ã¤jhfu° v©fshF«.

AB BC AC2 2 2+ = våš ( ,AB BC , AC ) xU Ã¤jhfu° v©fshF«.

Áy Ã¤jhfu° v©fŸ ...

(3, 4, 5). mjhtJ ( , , )n n n3 4 5 , Ï§F n = 1, 2, 3, g(5, 12, 13). mjhtJ ( , , )n n n5 12 13 , Ï§F n = 1, 2, 3, g(7, 24, 25). mjhtJ ( , , )n n n7 24 25 , Ï§F n = 1, 2, 3, g(8, 15, 17). mjhtJ ( , , )n n n8 15 17 , Ï§F n = 1, 2, 3, g(12, 35, 37). mjhtJ ( , , )n n n12 35 37 , Ï§F n = 1, 2, 3, g(20, 21, 29). mjhtJ (2 , , )n n n0 21 29 , Ï§F n = 1, 2, 3, g

Ïnjngh‹W, eh« Ãw Ã¤jhfu° v©fisÍ« f©l¿ayh«


gæ‰Á 9.1

1. 4.2 br.Û MuKŸs xU t£l« tiuªJ m›t£l¤Â‹ nkš VnjD« xU òŸëia¡

F¿¡f. t£l¤Â‹ ika¤ij¥ ga‹gL¤Â m¥òŸë têna bjhLnfhL tiuf.

bfhL¡f¥g£lit: t£l¤Â‹ Mu« = 4.2 br.Û

tiuKiw:

(i) O-it ikakhf¡ bfh©L 4.2 br.Û. MuKŸs t£l« tiuf.

(ii) t£l¤Â‹ nkš P v‹w òŸëia¡ F¿¤J OP I Ïiz¡f. (iii) P-ia ikakhf¡ bfh©L OP-š L v‹w Ïl¤Âš bt£L«go xU t£léš

tiuf.

(iv) m›éšè‹ nkš LM! = MN! = PL v‹wthW M, N v‹w òŸëfis F¿.

(v) +MPN-‹ nfhz ÏUrkbt£o PT tiuf.

(vi) TP I T l tiu Ú£o njitahd bjhLnfhL T lPT tiuf.P v‹w òŸë têahf OP-¡F¢ br§F¤jhf ne®¡nfhL PT tiuªJ« bjhLnfhL

tiuayh«. Ï§F PT v‹gJ òŸë P-æš mikªj bjhLnfhL MF«.

brŒKiw toéaš9

Ô®Î - brŒKiw toéaš 235

2. 4.8 br.Û MuKŸs xU t£l« tiuf. t£l¤Â‹ nkš VnjD« xU òŸëia¡ F¿.

bjhLnfhL - eh© nj‰w¤ij¥ ga‹gL¤Â m¥òŸë têna bjhLnfhL tiuf.

bfhL¡f¥g£lit: t£l¤Â‹ Mu« = 4.8 br.Û

tiuKiw:

(i) O-it ikakhf¡bfh©L 4.8 br.Û. MuKŸs t£l« tiuf.

(ii) t£l¤Â‹nkš P v‹w òŸëia¡ F¿¡f.

(iii) òŸë Ptêna VnjD« xU eh© PQ tiuf.

(iv) P k‰W« Q-I jé®¤J R v‹w òŸëia òŸëfŸ P,Q k‰W« R v‹gd fofhu

KŸ efU« vÂ®Âiræš mikÍkhW F¿¡fÎ«.

(v) PR k‰W« QR-M»adt‰iw Ïiz¡f.

(vi) òŸë R-š AB!

tiuf. mJ RQ k‰W« RP-I Kiwna A k‰W« B-æš rªÂ¡F«.

(vii) P-I ikakhf RA(=RB)I MukhfÎ« xU t£l« tiuf. mJ PQ-I C-š

rªÂ¡F«.

(viii) C-I ikakhfÎ« AB-MukhfÎ« bfh©L k‰bwhU éš tiuf. mJ K‹ò

tiuªj éšiy D-š rªÂ¡F«.

(viii) PQ I T k‰W« T l tiuÚ£o T lPT ia tiuayh«.


3. 10 br.Û é£lKŸs xU t£l« tiuf. t£l¤Â‹ ika¤ÂèUªJ 13 br.Û.

bjhiyéš P v‹w òŸëia¡ F¿¤J m¥òŸëæèUªJ t£l¤Â‰F PA k‰W«

PB v‹w bjhLnfhLfŸ tiuªJ mj‹ Ús§fis fz¡»Lf.

bfhL¡f¥g£lit: t£l¤Â‹ é£l« = 10 br.Û

t£l¤Â‹ Mu« = 5 br.Û

tiuKiw:

(i) O-it ikakhf¡ bfh©L 5 br.Û. MuKŸs t£l« tiuf.

(ii) t£l¤Â‰F btëna, t£l ika« O-æèUªJ 13 br.Û. bjhiyéš P v‹w òŸëia F¿¤J OP-ia Ïiz¡f.

(iii) OP ‹ ika¡F¤J¡nfhL tiuf. mJ OP-ia M-š bt£l£L«.


(iv) M-I ikakhfÎ«, MO (= MP)I MukhfÎ« bfh©L k‰bwhU t£l« tiuf.

(v) Ïu©L t£l§fS« A k‰W« B-š rªÂ¡F«. (vi) PA k‰W« PB tiuf. Ïitna njitahd bjhLnfhLfŸ MF«.

bjhLnfh£o‹ Ús«, PA = 12 br.Û.

rçgh®¤jš: br§nfhz OPAT -š

PA = OP OA2 2- = 13 12

2 2-

= 169 25- = 144 ` PA = 12 br.Û.

4. 6 br.Û MuKŸs xU t£l« tiuªJ mj‹ ika¤ÂèUªJ 10 br.Û bjhiyé

YŸs xU òŸëia¡ F¿¡f. m¥òŸëæèUªJ t£l¤Â‰F bjhLnfhLfŸ

tiuªJ mj‹ Ús§fis fz¡»Lf.

bfhL¡f¥g£lit: t£l¤Â‹ Mu« = 6 br.Û


tiuKiw:



(iii) OP ‹ ika¡F¤J¡nfhL tiuf. mJ OP-ia M-š bt£l£L«. (iv) M-I ikakhfÎ«, MO (= MP)I MukhfÎ« bfh©L k‰bwhU t£l« tiuf.



rçgh®¤jš: br§nfhz OPAT -š PA = OP OA2 2

- = 10 62 2- = 100 36- = 64


5. 3 br.Û MuKŸs t£l¤Â‹ ika¤ÂèUªJ 9 br.Û bjhiyéš xU òŸëia¡

F¿¡f. m¥òŸëæèUªJ t£l¤Â‰F bjhLnfhLfŸ tiuªJ, mj‹ Ús§fis

fz¡»Lf.


tiuKiw:


(ii) t£l¤Â‰F btëna, t£l ika« O-æèUªJ 9 br.Û. bjhiyéš P v‹w

òŸëia F¿¤J OP-ia Ïiz¡f. (iii) OP ‹ ika¡F¤J¡nfhL tiuf. mJ OP-ia M-š bt£l£L«. (iv) M-I ikakhfÎ«, MO (= MP)I MukhfÎ« bfh©L k‰bwhU t£l« tiuf.


bjhLnfh£o‹ Ús«, PA = 8.5 br.Û.

rçgh®¤jš: br§nfhz OPAT -š PA = OP OA

2 2- = 9 3

2 2-

= 81 9- = 72 = 6 2 = .6 1 414# = 8.484 br.Û.

bjhLnfh£o‹ Ús«, PA = 8.5br.Û.

gæ‰Á 9.2

1. AB = 5.2 br.Û ÚsKŸs nfh£L¤J©o‹ ÛJ 48c nfhz« V‰gL¤J« t£l¥gFÂia

mik¡f.

bfhL¡f¥g£lit: AB = 5.2 br.Û. c¢Ánfhz« = 48°.


tiuKiw:

(i) nfh£L¤J©L AB = 5.2 br.Û tiuf. (ii) òŸë A-š, BAX+ =48c mik¡f.

(iii) AY AX= tiuf. (iv) AB-‹ ika¡F¤J¡nfhL tiuf. mJ AY -ia O-š rªÂ¡f£L«.

(v) O-it ikakhfÎ«, OA I MukhfÎ« bfh©L xU t£l« tiuf.

(vi) t£l¤Â‹ kh‰W t£l¤J©oš C v‹w òŸëia F¿¤J AC k‰W« BC ia

Ïiz¡f. (vii) ABCT v‹gJ njitahd K¡nfhz« MF«.

2. DPQR-š mo¥g¡f« PQ = 6 br.Û., 60R+ = c k‰W« c¢Á R-èUªJ PQ-¡F

tiua¥g£l F¤J¡nfh£o‹ Ús« 4 br.Û vd ÏU¡FkhW DPQR tiuf.

bfhL¡f¥g£lit: TPQR-š PQ = 6 br.Û., R 60+ = c,

R-æèUªJ PQ-¡F tiua¥g£l F¤J¡nfh£o‹ Ús« = 4 br.Û.

tiuKiw:

(i) nfh£L¤J©L PQ = 6 br.Û tiuf.

(ii) QPX+ = 06 c vd ÏU¡F«go PX tiuf.

(iii) PY PX= tiuf.


(iv) PQ-‹ ika¡F¤J¡ nfhL tiuf. mJ PY k‰W« PQ-M»adt‰iw Kiwna O k‰W« M òŸëfëš rªÂ¡»wJ.

(v) O-ia ikakhfÎ«, OA-it MukhfÎ« bfh©l t£l« tiuf.

(vi) t£l¥ gFÂ PKQ v‹gJ nfhz« 40c bfh©oU¡F«.

(vii) ika¡F¤J¡nfhL MO-š, MH = 4 br.Û ÏU¡F« go H v‹w òŸëia F¿¡f.

(viii) PQ-¡F Ïizahf 'RHR tiuf. mJ t£l¤ij R k‰W« 'R -fëš rªÂ¡F«.

(ix) PR k‰W« QR Ïiz¡f ÏJnt njitahd K¡nfhz« PQRD MF«.

Ï§F, 3 'PQR v‹gJ njitahd k‰bwhU K¡nfhz« MF«.

3. PQ = 4 br.Û., R+ = 25c k‰W« c¢Á R-èUªJ PQ -¡F tiua¥g£l F¤J¡nfh£o‹

Ús« 4.5 br. Û. v‹w msÎfŸ bfh©l PQRD tiuf.

bfhL¡f¥g£lit: DPQR-š PQ = 4 br.Û., R+ = 25c, R-æèUªJ PQ-¡F

tiua¥g£l F¤J¡nfh£o‹ Ús« = 4.5 br.Û.

tiuKiw:

(i) nfh£L¤J©L PQ = 4 br.Û tiuf.

(ii) QPX+ = 25c vd ÏU¡F«go PX tiuf.


(iii) PY PX= tiuf.

(iv) PQ-‹ ika¡F¤J¡ nfhL tiuf. mJ PY k‰W« PQ-M»adt‰iw Kiwna O k‰W« M òŸëfëš rªÂ¡»wJ.

(v) O-ia ikakhfÎ«, OA-it MukhfÎ« bfh©l t£l« tiuf.

(vi) t£l¥ gFÂ PKQ v‹gJ nfhz« 25c bfh©oU¡F«.

(vii) ika¡F¤J¡nfhL MO-š, MH = 4.5 br.Û ÏU¡F« go H v‹w òŸëia

F¿¡f.

(viii) PQ-¡F Ïizahf 'RHR tiuf. mJ t£l¤ij R k‰W« 'R -fëš rªÂ¡F«.

(ix) PR k‰W« QR Ïiz¡f ÏJnt njitahd K¡nfhz« PQRD MF«.

4. ABCD -š, BC = 5 br.Û., 45A+ = c k‰W« c¢Á A-èUªJ BC-¡F tiua¥g£l

eL¡nfh£o‹ Ús« 4 br.Û vd ÏU¡F« go ABCD tiuf.

bfhL¡f¥g£lit: ABCD -š, BC = 5 br.Û., 45A+ = c, eL¡nfhL AM = 4.5 br.Û.

tiuKiw:

(i) BC = 5 br.Û. msÎŸs xU nfh£L¤J©L tiuf.

(ii) òŸë B têna CBX 45+ = c vd ÏU¡F« go BX tiuf.


(iii) BY=BX tiuf.

(iv) BC-‹ ika¡F¤J¡nfhL tiuf. mJ BY k‰W« BC-fis O k‰W«

M -òŸëfëš rªÂ¡»wJ.

(v) O-it ikakhfÎ«, OB-ia MukhfÎ« bfh©L t£l« tiuf. t£l¤Âš

òŸë K-I¡ F¿¡f.

(vi) bgça éš BKC MdJ nfhz« 45c-I¡ bfh©oU¡F«.

(vii) M-I ikakhf¡ bfh©L 4 br.Û MuKŸs xU éš tiuf. mJ t£l¤ij A k‰W« Al òŸëfëš rªÂ¡F«.

(viii) AB, AC M»adt‰iw Ïiz¡f.

(ix) ABC3 mšyJ A BCT l v‹gJ njitahd K¡nfhz« MF«.

5. BC = 5 br.Û., 40BAC+ = c k‰W« c¢Á A-èUªJ BC-¡F tiua¥g£l eL¡nfh£o‹

Ús« 6 br.Û. v‹w msÎfŸ bfh©l ABCT tiuf. nkY« c¢Á A-èUªJ tiua¥g£l

F¤J¡nfh£o‹ Ús« fh©f.

bfhL¡f¥g£lit: ABCT -š, BC = 5 br.Û., 40BAC+ = c, eL¡nfhL AM = 4.5 br.Û.


tiuKiw:


(ii) òŸë B têna 4CBX 0+ = c vd ÏU¡F« go BX tiuf.

(iii) BY=BX tiuf.









(ix) CB I CZ tiu.

(x) AE CZ= . (xi) F¤J¡nfhL AE ‹ Ús« 3.8 br.Û.

gæ‰Á 9.3

1. PQ = 6.5 br.Û., QR = 5.5 br.Û., PR = 7 br.Û. k‰W« PS = 4.5 br.Û. v‹w msÎfŸ

bfh©l t£l eh‰fu« PQRS tiuf.

bfhL¡f¥g£lit: t£leh‰fu« PQRS-š, PQ = 6.5 br.Û., QR = 5.5 br.Û.,

PR = 7 br.Û. k‰W« PS = 7 br.Û.


tiuKiw:

(i) cjé¥gl« tiuªJ, mÂš bfhL¡f¥g£l msÎfis¡ F¿¡fÎ«.

nfh£L¤J©L PQ = 6.5 br.Û. tiuf.

(ii) òŸëfŸ P k‰W« Q-ia ikakh¡ bfh©L Kiwna 7 br.Û. k‰W« 5.5 br.Û.

MuKŸs t£l é‰fŸ tiuªJ, mit rªÂ¡F« òŸë R I¡ fh©f.

(iii) PR k‰W« QR-fis Ïiz¡f.

(iv) PQ k‰W« QR-‹ ika¡F¤J¡nfhLfŸ tiuªJ mit rªÂ¡F« òŸë

O-it¡ fh©f.

(v) O-it ikakhfÎ« k‰W« OP(=OQ=OR)-ia MukhfÎ« bfh©L PQRD -‹

R‰W t£l« tiuf.

(vi) P-it ikakhf¡ bfh©L 4.5 br.Û MuKŸs xU éš tiuf. mJ R‰W

t£l¤ij S -š rªÂ¡F«.

(vii) PS k‰W« RS-fis Ïiz¡f.

(viii) njitahd t£leh‰fu« PQRS MF«.

2. AB = 6 br.Û., AD = 4.8 br.Û., BD = 8 br.Û. k‰W« CD = 5.5 br.Û. v‹w msÎfŸ

bfh©l t£l eh‰fu« ABCD .bfhL¡f¥g£lit: t£l eh‰fu« ABCD-š, AB = 6 br.Û., AD = 4.8 br.Û.,

BD = 8 br.Û. k‰W« CD = 5.5 br.Û.


tiuKiw:

(i) cjé¥gl« tiuªJ mÂš bfhL¡f¥g£l msÎfis¡ F¿¡f. AB = 6 br.Û

msÎŸs xU nfh£L¤J©L tiuf.

(ii) òŸëfŸ A k‰W« B-ia ikakhf¡ bfh©L Kiwna 4 br.Û k‰W« 4.8 br.Û MuKŸs t£lé‰fŸ (arcs) tiuf. mitfŸ mJ D v‹w òŸëæš

rªÂ¡F«. AD k‰W« BD -fis Ïiz¡f.

(iii) nfh£L¤J©LfŸ AB k‰W« BD-fë‹ ika¡F¤J¡nfhLfŸ tiuªJ mit

rªÂ¡F« òŸë O-I fh©f.

(iv) O-I ikakhfÎ« k‰W« OA (= OB = OD) -I MukhfÎ« bfh©L ABDT -‹

R‰W t£l« tiuf.

(v) òŸë D-I ikakhf¡ bfh©L 5.5 br.Û MuKŸs t£léš tiuf mJ R‰W

t£l¤ij C-š rªÂ¡f£L«.

(vi) CD k‰W« BC M»adt‰iw Ïiz¡f. ÏJnt njitahd t£leh‰fu« ABCD MF«.

3. PQ = 5.5 br.Û., QR = 4.5 br.Û., QPR 45+ = c k‰W« PS = 3 br.Û. M»a msÎfŸ


bfhL¡f¥g£lit: t£l eh‰fu« PQRS-š, PQ = 5.5 br.Û., QR = 4.5 br.Û.,

45QPR+ = c k‰W« PS = 3 br.Û.


tiuKiw:

(i) cjé¥gl« tiuªJ mÂš bfhL¡f¥g£l msÎfis¡ F¿¡f.

PQ = 6 br.Û ÚsKŸs xU nfh£L¤J©L tiuf.

(ii) òŸë B têna 45QPX+ = c vD« go PX tiuf.

(iii) òŸë Q-ia ikakhf¡ bfh©L 4.5 br.Û. MuKŸs t£léš tiuf. mJ PX-ia rªÂ¡F« òŸë R v‹f. QR-ia Ïiz¡f.

(iv) PQ k‰W« QR-¡fë‹ ika¡F¤J¡nfhLfŸ tiuªJ mit rªÂ¡F«

òŸë O v‹f.

(v) O-it ikakhfÎ« k‰W« OP (= OQ = OR) MukhfÎ« bfh©L PQRT -‹

R‰W t£l« tiuf. (vi) òŸë P-ia ikakhf¡ bfh©L 3.5 br.Û MuKŸs t£léš tiuf.mJ

R‰Wt£l¤ij rªÂ¡F« òŸë S v‹f.

(vii) PS k‰W« RS M»adt‰iw Ïiz¡f.

(viii) njitahd t£leh‰fu« PQRS MF«.

4. AB = 7 br.Û., A 80+ = c, AD = 4.5 br.Û. k‰W« BC = 5 br.Û. v‹w msÎfŸ bfh©l

t£l eh‰fu« ABCD tiuf.

bfhL¡f¥g£lit: t£l eh‰fu« ABCD-š, AB = 7 br.Û., 80A+ = c,

AD = 4.5 br.Û. k‰W« BC = 5 br.Û.


tiuKiw:

(i) cjé¥gl« tiuªJ mÂš bfhL¡f¥g£l msÎfis¡ F¿¡f.

AB = 7 br.Û ÚsKŸs xU nfh£L¤J©L tiuf.

(ii) òŸë A têna 80BAX+ = c vD« go AX tiuf.

(iii) òŸë A-ia ikakhf¡ bfh©L 4.5 br.Û. MuKŸs t£léš tiuf. mJ AX-ia rªÂ¡F« òŸë D v‹f. BD-ia Ïiz¡f.

(iv) AB k‰W« AD-¡fë‹ ika¡F¤J¡nfhLfŸ tiuªJ mit rªÂ¡F«

òŸë O v‹f.

(v) O-it ikakhfÎ« k‰W« OA (= OB = OD) MukhfÎ« bfh©L ABDT -‹

R‰W t£l« tiuf. (vi) òŸë B -ia ikakhf¡ bfh©L 5 br.Û MuKŸs t£léš tiuf.

mJ R‰Wt£l¤ij rªÂ¡F« òŸë C v‹f.

(vii) BD k‰W« CD M»adt‰iw Ïiz¡f.

(viii) njitahd t£leh‰fu« ABCD MF«.

5. KL = 5.5 br.Û., KM = 5 br.Û., LM = 4.2 br.Û. k‰W« LN = 5.3 br.Û. M»a msÎfŸ

bfh©l t£l eh‰fu« KLMN tiuf.

bfhL¡f¥g£lit: t£leh‰fu«KLMN-š,KL = 5.5 br.Û., KM = 5 br.Û.,

LM = 4.2 br.Û. k‰W« LN = 5.3 br.Û.

tiuKiw:


nfh£L¤J©L KL = 5.5 br.Û. tiuf.


(ii) òŸëfŸ K k‰W« L-ia ikakh¡ bfh©L Kiwna5 br.Û. k‰W« 4.2 br.Û.

MuKŸs t£l é‰fŸ tiuªJ, mit rªÂ¡F« òŸë M I¡ fh©f.

(iii) KM k‰W« LM-fis Ïiz¡f.

(iv) KL k‰W« LM-‹ ika¡F¤J¡nfhLfŸ tiuªJ mit rªÂ¡F« òŸë

O-it¡ fh©f.

(v) O-it ikakhfÎ« k‰W« OK(=OL=OM)-ia MukhfÎ« bfh©L KLMT -‹

R‰W t£l« tiuf.

(vi) L-it ikakhf¡ bfh©L 5.3 br.Û MuKŸs xU éš tiuf. mJ R‰W

t£l¤ij N-š rªÂ¡F«.

(vii) KN k‰W« MN-fis Ïiz¡f.

(viii) njitahd t£leh‰fu« KLMN MF«.

6. EF = 7 br.Û., EH = 4.8 br.Û., FH = 6.5 br.Û. k‰W« EG = 6.6 br.Û. msÎfŸ bfh©l

t£l eh‰fu« EFGH tiuf.

bfhL¡f¥g£lit: t£leh‰fu«EFGH -š,EF = 7 br.Û., EH = 4.8 br.Û.,

FH = 6.5 br.Û. k‰W« EG = 6.6 br.Û.


tiuKiw:


nfh£L¤J©L EF = 5.5 br.Û. tiuf.

(ii) òŸëfŸ E k‰W« F-ia ikakh¡ bfh©L Kiwna 4.8 br.Û. k‰W« 6.5 br.Û.

MuKŸs t£l é‰fŸ tiuªJ, mit rªÂ¡F« òŸë H I¡ fh©f.

(iii) EH k‰W« FH-fis Ïiz¡f.

(iv) EF k‰W« FH-‹ ika¡F¤J¡nfhLfŸ tiuªJ mit rªÂ¡F« òŸë

O-it¡ fh©f.

(v) O-it ikakhfÎ« k‰W« OE(=OF=OH)-ia MukhfÎ« bfh©L EHFT -‹

R‰W t£l« tiuf.

(vi) E-it ikakhf¡ bfh©L 6.6 br.Û MuKŸs xU éš tiuf. mJ R‰W

t£l¤ij G-š rªÂ¡F«.

(vii) HG k‰W« FG-fis Ïiz¡f.

(viii) njitahd t£leh‰fu« EFGH MF«.

7. AB = 6 br.Û., 70ABC+ = c, BC = 5 br.Û. k‰W« 30ACD+ = c M»a msÎfŸ

bfh©l t£leh‰fu« ABCD tiuf.

bfhL¡f¥g£lit: t£leh‰fu« ABCD -š,AB = 6 br.Û., 70ABC+ = c,

BC = 5 br.Û. k‰W« 30ACD+ = c


tiuKiw:

(i) X® cjé¥gl« tiuªJ mÂš bfhL¡f¥g£l msÎfis F¿¡f.

AB = 6 br.Û.. ÚsKŸs nfh£L¤J©L tiuf.

(ii) òŸë B-š, 70ABX+ = c vD« go BX tiuf.

(iii) B-ia ikakhf¡ bfh©L 5 br.Û MuKŸs X® éš tiuªJ mJ BX -ia

rªÂ¡F« òŸë C v‹f.

(iv) AC-ia Ïiz¡f.

(v) AB k‰W« BC-¡fë‹ ika¡F¤J¡ nfhLfŸ tiuf. mitfŸ òŸë O-š

rªÂ¡f£L«.

(vi) O-it ikakhfÎ« k‰W« OA(= OB = OC) -I MukhfÎ« bfh©L

ABCT -‹ R‰W t£l« tiuf.

(vii) òŸë G š, 30ACY+ = c vD«go CY tiuf.

(viii) CY MdJ R‰Wt£l¤ij bt£L« òŸë D v‹f. AD-I Ïiz¡f. j‰nghJ

njitahd t£leh‰fu« ABCD MF«.

8. PQ = 5 br.Û., QR = 4 br.Û., 35QPR+ = c k‰W« 70PRS+ = c M»a msÎfŸ


bfhL¡f¥g£lit: t£leh‰fu« PQRS -š,PQ = 5 br.Û., QR = 4 br.Û.,

35QPR+ = c k‰W« 70PRS+ = c.


tiuKiw:

(i) X® cjé¥gl« tiuªJ mÂš bfhL¡f¥g£l msÎfis F¿¡f.

PQ = 5 br.Û. ÚsKŸs nfh£L¤J©L tiuf.

(ii) òŸë P-š, 35QPX+ = cvD« go PX tiuf.

(iii) Q-ia ikakhf¡ bfh©L 4 br.Û MuKŸs X® éš tiuªJ mJ PX-ia

rªÂ¡F« òŸë R v‹f.

(iv) QR-ia Ïiz¡f.

(v) PQ k‰W« QR-fë‹ ika¡F¤J¡ nfhLfŸ tiuf. mit òŸë O-š

rªÂ¡f£L«.

(vi) O-it ikakhfÎ« k‰W« OP (= OQ = OR) -I MukhfÎ« bfh©L

PQRT -‹ R‰W t£l« tiuf.

(vii) òŸë R-š, PRY 70+ = c vD«go RY tiuf.

(viii) RY MdJ R‰Wt£l¤ij bt£L« òŸë S v‹f. PS-I Ïiz¡f. j‰nghJ

njitahd t£leh‰fu« PQRS MF«.

9. AB = 5.5 br.Û., 50ABC+ = c, 60BAC+ = c k‰W« 30ACD+ = c M»a msÎfŸ

bfh©l t£l eh‰fu« ABCD tiuf.

bfhL¡f¥g£lit: t£leh‰fu« ABCD -š,AB = 5.5 br.Û., 50ABC+ = c,

60BAC+ = c k‰W« 30ACD+ = c

tiuKiw:

(i) cjé¥gl« tiuªJ, mÂš bfhL¡f¥g£l msÎfis¡F¿¡f.

AB = 5.5 br.Û. msÎŸs nfh£l¤J©L tiuf.

(ii) òŸë B-š 50ABX+ = cvD« go BX tiuf.


(iii) òŸë A-š 60BAY+ = cvD« go AY tiuf. AY MdJ BX I C-š

rªÂ¡f£L«.

(iv) AB k‰W« BC-fë‹ ika¡F¤J¡nfhLfŸ tiuªJ mit rªÂ¡F« òŸë O-ia¡ fh©f.

(v) òŸë O-it ikakhfÎ« OA( = OB = OC )ia MukhfÎ« bfh©L

ABC3 -‹ R‰W t£l« tiuf.

(vi) òŸë C-š, 30ACZ+ = c vD« go AD tiuf. mJ t£l¤ij òŸë D-š

rªÂ¡f£L«.

(vii) CD-I Ïiz¡f.

njitahd t£l eh‰fu« ABCD MF«.

10. AB = 6.5 br.Û., 110ABC+ = c, BC = 5.5 br.Û. k‰W« AB || CD v‹wthW mikÍ«

t£leh‰fu« ABCD tiuf.

bfhL¡f¥g£lit: t£leh‰fu« ABCD š, AB = 6.5 br.Û., 110ABC+ = c, BC = 5.5 br.Û. k‰W« AB CD<


tiuKiw:

(i) cjé¥gl« tiuªJ, mÂš bfhL¡f¥g£l msÎfis F¿¡f.

AB =6.5 br.Û. msÎŸs xU nfh£L¤J©L tiuf.

(ii) òŸë B-š, ABX+ = 110c vD«go BX tiuf.

(iii) B-ia ikakhf¡ bfh©L 5.5 br.Û MuKŸs X® t£léš tiuf

mJ BX -ia rªÂ¡F« òŸë C v‹f.

(iv) AB k‰W« BC-fë‹ ika¡F¤J¡nfhLfŸ tiuf mit rªÂ¡F« òŸë O-v‹f.

(v) O-ia ikakhfÎ«, OA (= OB = OC)-I MukhfÎ« bfh©L ABCD -‹

R‰W t£l« tiuf.

(vi) DY AB< vD«go DY tiuf mJ t£l¤ij rªÂ¡F« òŸë D v‹f.

BC-ia Ïiz¡f.

(vii) njitahd t£l eh‰fu« ABCD MF«.

Ô®Î - tiugl§fŸ 255

gæ‰Á 10.1

1. Ã‹tU« rh®òfë‹ tiugl« tiuf.

(i) y x3 2= .

Ô®Î:

x – 3 – 2 – 1 0 1 2 3

x2 9 4 1 0 1 4 9

y x3 2= 27 12 3 0 3 12 27

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )3 27 2 12 1 3 0 0 1 3 2 12 3 27- - -

tiugl§fŸ 10


(ii) y x4 2=-

Ô®Î:

x – 3 – 2 – 1 0 1 2 3

x2 9 4 1 0 1 4 9

4y x2=- – 36 – 16 – 4 0 – 4 – 16 – 36

òŸëfŸ: ( 3, ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )36 2 16 1 4 0 0 1 4 2 16 3 36- - - - - - - - -

(iii) 6 8y x x x x2 4 2= + + = + +^ ^h h

Ô®Î: y x x6 82

= + + v‹f.

x – 5 – 4 – 3 – 2 – 1 0 1 2

x2 25 16 9 4 1 0 1 4

x6 – 30 – 24 – 18 – 12 – 6 0 6 128 8 8 8 8 8 8 8 8y 3 0 – 1 0 3 8 15 24

òŸëfŸ: ( 5,3), ( 4,0), ( 3, 1), ( 2,0), ( 1,3), (0,8), (1,15), (2,24)- - - - - -


(iv) y x x2 32= - + .

Ô®Î:

x – 3 – 2 – 1 0 1 2 3

x2 2 18 8 2 0 2 8 18

x- 3 2 1 0 – 1 – 2 – 33 3 3 3 3 3 3 3y 24 13 6 3 4 9 18

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )3 24 2 13 1 6 0 3 1 4 2 9 3 18- - -


2. tiugl« _y« Ã‹tU« rk‹ghLfis¤ Ô®¡fÎ«.

(i) 4 0x2- = .

Ô®Î: y x 42= - v‹f.

x – 3 – 2 – 1 0 1 2 3

x2 9 4 1 0 1 4 9– 4 – 4 – 4 – 4 – 4 – 4 – 4 – 4y 5 0 – 3 – 4 – 3 0 5

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )3 5 2 0 1 3 0 4 1 3 2 0 3 5- - - - - -

Ô®¡f: y = x 42-

0 = x 42-

y = 0

Ïªj tistiu x- m¢ir ( , )2 0- k‰W« (2, 0) M»a òŸëfëš bt£L»wJ.

Mfnt, x-Ma¤bjhiyÎfŸ – 2 k‰W« 2 MF«. Mfnt, Ô®Î fz«{– 2, 2}.


(ii) x x3 10 02- - = .

Ô®Î: y x x3 102= - - v‹f.

x – 3 – 2 – 1 0 1 2 3 4 5 6

x2 9 4 1 0 1 4 9 16 25 36

x3- 9 6 3 0 – 3 – 6 – 9 – 12 – 15 – 18– 10 – 10 – 10 – 10 – 10 – 10 – 10 – 10 – 10 – 10 – 10

y 8 0 – 6 – 10 – 12 – 12 – 10 – 6 0 8

òŸëfŸ: ( 3,8), ( 2,0), ( 1, 6), (0, 10), (1, 12)- - - - - - (2, 12), (3, 10), (4, 6), (5,0), (6,8)- - -

Ô®¡f: y = x x3 102- -

0 = x x3 102- -

y = 0

Ïªj tistiu x- m¢ir ( , )2 0- k‰W« (5, 0) M»a òŸëfëš bt£L»wJ.

Mfnt, x-Ma¤bjhiyÎfŸ – 2 k‰W« 5 MF«. Mfnt, Ô®Î fz«{– 2, 5}.


(iii) 0x x5 1- - =^ ^h h .

Ô®Î: ( 5)( 1) 0x x &- - = x x6 5 02- + = . Mfnt, y x x6 52

= - + v‹f.

x – 1 0 1 2 3 4 5 6

x2 1 0 1 4 9 16 25 36

x6- 6 0 – 6 – 12 – 18 – 24 – 30 – 365 5 5 5 5 5 5 5 5y 12 5 0 – 3 – 4 – 3 0 5

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )1 12 0 5 1 0 2 3 3 4 4 3 5 0 6 5- - - -

Ô®¡f: y = x x6 52- +

0 = x x6 52- +

y = 0

Ïªj tistiu x- m¢ir (1, 0) k‰W« (5, 0) M»a òŸëfëš bt£L»wJ.

Mfnt, x-Ma¤bjhiyÎfŸ 1 k‰W« 5 MF«. Mfnt, Ô®Î fz«{1, 5}.

(iv) 0x x2 1 3+ - =^ ^h h .

Ô®Î: (2 1)( 3) 0 2 5 3 0x x x x2&+ - = - - = . vdnt, y x x2 5 32

= - - v‹f.

x – 1 0 1 2 3 4

x2 2 2 0 2 8 18 32

x5- 5 0 – 5 – 10 – 15 – 20 – 3 – 3 – 3 – 3 – 3 – 3 – 3 y 4 – 3 – 6 – 5 0 9

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , ), ( , )1 4 0 3 1 6 2 5 3 0 4 9- - - -


Ô®¡f: y = x x2 5 32- -

0 = x x2 5 32- -

y = 0

Ïªj tistiu x- m¢ir (– 0.5, 0) k‰W« (3, 0) M»a òŸëfëš bt£L»wJ.

Mfnt, x-Ma¤bjhiyÎfŸ – 0.5 k‰W« 3 MF«. Mfnt, Ô®Î fz«{–0.5, 3}.

3. y x2

= -‹ tiugl« tiuªJ, mjid¥ ga‹gL¤Â x x4 5 02- - = v‹w

rk‹gh£il¤ Ô®¡fÎ«.

Ô®Î: y x2=

x – 2 – 1 0 1 2 3 4 5 6

y x2= 4 1 0 1 4 9 16 25 36

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )2 4 1 1 0 0 1 1 2 4 3 9 4 16 5 25 6 36- - Ô®¡f: y = x x0 02

+ +

0 = x x4 52- -

y = x4 5+

y x4 5= + v‹w ne®¡nfh£o‰fhd m£ltizia mik¥ngh«.

x – 2 – 1 0 1 2y x4 5= + – 3 1 5 9 13

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , )2 3 1 1 0 5 1 9 2 13- - -


.

ne®¡nfhL k‰W« tistiu M»ait bt£o¡bfhŸS« òŸëfŸ

( , )1 1- k‰W« ( , )5 25 .x-Ma¤bjhiyÎfŸ – 1 k‰W« 5 MF«. Mfnt, Ô®Î fz«{–1, 5}.

4. 2 3y x x2

= + - -‹ tiugl« tiuªJ, mjid¥ ga‹gL¤Â 6 0x x2- - = v‹w


Ô®Î: y x x2 32= + -

x – 3 – 2 – 1 0 1 2 3

x2 9 4 1 0 1 4 9

x2 – 6 – 4 – 2 0 2 4 6– 3 – 3 – 3 – 3 – 3 – 3 – 3 – 3y 0 – 3 – 4 – 3 0 5 12

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )3 0 2 3 1 4 0 3 1 0 2 5 3 12- - - - - - Ô®¡f: y = x x2 32

+ -

0 = x x 62- -

y = x3 3+

y x3 3= + v‹w ne®¡nfh£o‰fhd m£ltizia mik¥ngh«.

x – 2 – 1 0 1 2y x3 3= + – 3 0 3 6 9

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , )2 3 1 0 0 3 1 6 2 9- - -


ne®¡nfhL k‰W« tistiu M»ait bt£o¡bfhŸS« òŸëfŸ ( , )2 3- - k‰W« ( , 2)3 1 .

x-Ma¤bjhiyÎfŸ – 2 k‰W« 3 MF«. Mfnt, Ô®Î fz«{– 2, 3}.

5. 2 6y x x2

= + - -‹ tiugl« tiuªJ, mjid¥ ga‹gL¤Â 2 10 0x x2+ - = v‹w

rk‹gh£il¤ Ô®¡fÎ«.Ô®Î: 6y x x2 2

= + -

x – 3 – 2 – 1 0 1 2 3

x2 2 18 8 2 0 2 8 18

x – 3 – 2 – 1 0 1 2 3– 6 – 6 – 6 – 6 – 6 – 6 – 6 – 6y 9 0 – 5 – 6 – 3 4 15

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )3 9 2 0 1 5 0 6 1 3 2 4 3 15- - - - - -

Ô®¡f: y = x x2 62+ -

0 = x x2 102+ -

y = 4

y 4= v‹gJ x-m¢R¡F Ïizahd ne®¡nfhL MF«.



(–2.5, 4) k‰W« (2, 4).

x-Ma¤bjhiyÎfŸ – 2.5 k‰W« 2 MF«. Mfnt, Ô®Î fz«{– 2.5, 2}.

6. 8y x x2

= - - -‹ tiugl« tiuªJ, mjid¥ ga‹gL¤Â 2 15 0x x2- - = v‹w


Ô®Î: y x x 82

= - -

x – 4 – 3 – 2 – 1 0 1 2 3 4 5

x2 16 9 4 1 0 1 4 9 16 25

x- 4 3 2 1 0 – 1 – 2 – 3 – 4 – 5– 8 – 8 – 8 – 8 – 8 – 8 – 8 – 8 – 8 – 8 – 8y 12 4 – 2 – 6 – 8 – 8 – 6 – 2 4 12

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , )4 12 3 4 2 2 1 6 0 8- - - - - - - ( , ), ( , ), ( , ), ( , ), ( , )1 8 2 6 3 2 4 4 5 12- - -


Ô®¡f: y = x x 82- -

0 = x x2 152- -

y = x 7+

y x 7= + v‹w ne®¡nfh£o‰fhd m£ltizia mik¥ngh«.

x – 3 – 2 – 1 0 1 2 3 4 5y x 7= + 4 5 6 7 8 9 10 11 12

òŸëfŸ: ( 3,4), ( 2,5), ( 1,6), (0,7), (1,8), (2,9), (3,10), (4,11), (5,12)- - -


(– 3, 4) k‰W« (5, 12).Mfnt, x-Ma¤bjhiyÎfŸ – 3 k‰W« 5 MF«. Mfnt, Ô®Î fz«{–3, 5}.

7. 12y x x2

= + - -‹ tiugl« tiuªJ, mjid¥ ga‹gL¤Â 2 2 0x x2+ + = v‹w



Ô®Î: y x x 122

= + -

x – 5 – 4 – 3 – 2 – 1 0 1 2 3 4

x2 25 16 9 4 1 0 1 4 9 16

x – 5 – 4 – 3 – 2 – 1 0 1 2 3 4– 12 – 12 – 12 – 12 – 12 – 12 – 12 – 12 – 12 – 12 – 12

y 8 0 – 6 – 10 – 12 – 12 – 10 – 6 0 8

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , )5 8 4 0 3 6 2 10 1 12- - - - - - - - ( , ), ( , ), ( , ), ( , ), ( , )0 12 1 10 2 6 3 0 4 8- - -

Ô®¡f: y = x x 122+ -

0 = 2x x 22+ +

y = x 14- -

y x 14=- - v‹w ne®¡nfh£o‰fhd m£ltizia mik¥ngh«.

x – 3 – 2 – 1 0 1 2 3y x 14=- - – 11 – 12 – 13 – 14 – 15 – 16 – 17

òŸëfŸ: ( 3, 11), ( 2, 12), ( 1, 13), (0, 14), (1, 15), (2, 16), (3, 17)- - - - - - - - - -


y x 14=- - v‹w ne®¡nfhL y x x 122= + - v‹w tistiuia v§F«

bt£léšiy.Mfnt, x x2 2 02

+ + = -¡F bkŒ _y§fŸ VJäšiy.

gæ‰Á 10.2 1. xU ngUªJ kâ¡F 40 ».Û. ntf¤Âš brš»wJ. Ïj‰Fça öu-fhy bjhl®Ã‰fhd

tiugl« tiuf. Ïij¥ ga‹gL¤Â 3 kâneu¤Âš Ï¥ngUªJ gaâ¤j¤

öu¤ij¡ f©LÃo.

Ô®Î:

x 1 2 3 4 5y 40 80 120 160 200

m£ltizæèUªJ, x-‹ kÂ¥ò mÂfç¡F« nghJ, y-‹ kÂ¥ò mÂfç¥gij

fhzyh«.

Mfnt, Ï«khWghL xU ne®khWghL MF«.

y x\ & y kx= xy

k& =

Ï§F, k -v‹gJ é»jrk kh¿è. bfhL¡f¥g£l kÂ¥òfS¡F, eh« bgWtJ

k = 40140

280

3120

4160= = = =

k` = 40; ,xy

y x40 40= =

F¿¥ò: Ïªj tiugl¤ÂèUªJ

ne®¡nfhlhdJ MÂ¥òŸë têahf¢

brštij¡ fhzyh«. vdnt, y x\ & y kx=

xy

k& = .


4y x0= v‹w cwÎ xU ne®¡nfh£L tiugl¤ij mik¡F«.

tiugl¤ÂèUªJ, 3 kâ neu¤Âš ngUªJ gaâ¤j öu« = 120 ».Û.

2. th§f¥g£l neh£L¥ ò¤jf§fë‹ v©â¡if k‰W« mj‰fhd éiy étu«

Ã‹tU« m£ltizæš ju¥g£LŸsJ.

neh£L¥ò¤jf§fë‹

v©â¡if (x)2 4 6 8 10 12

éiy ` (y) 30 60 90 120 150 180

Ïj‰fhd tiugl« tiuªJ mj‹ _y« (i) VG neh£L¥ ò¤jf§fë‹ éiyia¡

fh©f. (ii) 165-¡F th§f¥gL« neh£L¥ ò¤jf§fë‹ v©â¡ifia¡ fh©f. Ô®Î:

x 2 4 6 8 10 12y 30 60 90 120 150 180


fhzyh«.



y x\ , y kx= xy

k& =


k = 15230

460

690

8120

10150

12180= = = = = =

& y = 5x1 .

tiugl¤ÂèUªJ, (i) VG neh£L¥ò¤jf§fë‹ éiy ` 105.

(ii) ` 165-¡F th§f¥gL« neh£L¥ò¤jf§fë‹ v©â¡if 11.

3.

x 1 3 5 7 8y 2 6 10 14 16

nk‰f©l m£ltizæš cŸs étu¤Â‰F tiugl« tiuªJ, mj‹ _y«

(i) x = 4 våš y-‹ kÂ¥ig¡ fh©f. (ii) y = 12 våš x-‹ kÂ¥ig¡ fh©f..

Ô®Î:

x 1 3 5 7 8y 2 6 10 14 16


fhzyh«.



y x\ & y kx= xy

k& =


k = 212

36

510

714

816= = = = =

& y = 2x .

tiugl¤ÂèUªJ, (i) 4x = vD« nghJ 8y = , (ii) 12y = vD« nghJ 6x = .

4. xU è£l® ghè‹ éiy 15 v‹f. ghè‹ msÎ¡F« éiy¡F« cŸs¤ bjhl®Ãid¡

fh£L« tiugl« tiuf. mjid¥ ga‹gL¤Â,

(i) é»jrk kh¿èia¡ fh©f. (ii) 3 è£l® ghè‹ éiyia¡ fh©f.

Ô®Î:

è£l®fë‹

v©â¡if (x)1 2 3 4 5

éiy ` (y) 15 30 45 60 75


fhzyh«.


y x\ , y kx= xy

k& =


Ï§F, k -v‹gJ é»jrk kh¿è. Mfnt,

k = 15115

210

345

460

575= = = = =

k` = 15

& y = 15x

ÏJ ne®¡nfh£L tiugl¤ij mik¡»wJ. nkY« tiugl¤ÂèUªJ

(i) é»jrk kh¿è = 15, (ii) 3 è£l® ghè‹ éiy ` 45.

5. xy 20= , ,x y > 0 v‹gj‹ tiugl« tiuf. mjid¥ ga‹gL¤Â. 5x = våš y -‹

kÂ¥igÍ«, 10y = våš x -‹ kÂ¥igÍ« fh©f.

Ô®Î: xy = 20, ,x y 0>

y = x20

x 1 2 4 5 10 20y 20 10 5 4 2 1

m£ltizæèUªJ, x-‹ kÂ¥ò mÂfç¡F« nghJ, y-‹ kÂ¥ò Fiwtij

fhzyh«. Mfnt, Ïªj khWghL vÂ®khWghL MF«

tiugl¤ÂèUªJ, 5x = vD« nghJ 4y = k‰W« 10y = vD«nghJ 2x = .


6.

ntiyah£fë‹ v©â¡if (x) 3 4 6 8 9 16

eh£fë‹ v©â¡if (y) 96 72 48 36 32 18

m£ltizæš bfhL¡f¥g£LŸs étu¤Â‰fhd tiugl« tiuf. mj‹ _y«

12 ntiyah£fŸ m›ntiyia KGtJkhf brŒJ Ko¡f MF« eh£fë‹

v©â¡ifia¡ fh©f.

Ô®Î:

ntiyah£fë‹ v©â¡if (x) 3 4 6 8 9 16eh£fë‹ v©â¡if (y) 96 72 48 36 32 18

m£ltizæèUªJ, x-‹ kÂ¥ò mÂfç¡F« nghJ, y-‹ kÂ¥ò Fiwtij

fhzyh«.

Mfnt, Ïªj khWghL vÂ®khWghL MF«

yx

xy k1 &\ =

Ï§F, k v‹gJ é»jrk kh¿è. vdnt,

k = 3 96 4 72# #= = 6 48 8 36# #=

= 9 32 16 18 288# #= =

xy = 288

y = x

288

tiugl¤ÂèUªJ, 12 ntiy M£fŸ m›ntiyia KGtJkhf Ko¡f MF«

eh£fŸ 24 MF«.

Ô®Î - òŸëæaš 273

gæ‰Á 11.1 1. Ã‹tU« kÂ¥òfS¡F Å¢R k‰W« Å¢R¡ bfG fh©f.

(i) 59, 46, 30, 23, 27, 40, 52, 35, 29. (ii) 41.2, 33.7, 29.1, 34.5, 25.7, 24.8, 56.5, 12.5

Ô®Î: (i) Û¥bgU kÂ¥ò L = 59, Û¢ÁW kÂ¥ò S = 23

` Å¢R = L-S = 59 -23 = 36

Å¢R¡ bfG = L SL S+- =

2323

5959

+- =

8236

4118=

= 0.4390 = 0.44

(ii) bfhL¡f¥g£l òŸë étu§fëèUªJ

Û¥bgU kÂ¥ò L = 56.5 Û¢ÁW kÂ¥ò S = 12.5 Å¢R = L-S = 56.5 -12.5 = 44

Å¢R¡ bfG = L SL S+- =

12.512.5

56.556.5

+- =

6944

= 0.6376 = 0.64 2. xU òŸë étu¤Â‹ Û¢ÁW kÂ¥ò 12. mj‹ Å¢R 59 våš m¥òŸë étu¤Â‹ Û¥bgU

kÂ¥ig¡ fh©f.

Ô®Î: Û¢ÁW kÂ¥ò S = 12, Å¢R R = 59. R = L – S& L = S + R = 12 + 59 = 71 Mfnt, Û¥bgU kÂ¥ò L = 71

3. 50 msÎfëš äf¥bgça kÂ¥ò 3.84 ».». mj‹ Å¢R 0.46 ».» våš, mitfë‹

Û¢ÁW kÂ¥ig¡ fh©f.

Ô®Î: Û¥bgU kÂ¥ò L = 3.84 ».», Å¢R R = 0.46 ».»

Û¢ÁW kÂ¥ò S = L - R = 3.84 – 0.46 = 3.38 = 3.38 ».».

4. f©l¿ªj òŸë étu¤ bjhF¥ÃYŸs 20 kÂ¥òfë‹ Â£l éy¡f« 5 v‹f.

òŸë étu¤Â‹ x›bthU kÂ¥igÍ« 2 Mš bgU¡»dhš »il¡F« òÂa òŸë

étu§fë‹ Â£l éy¡f« k‰W« éy¡f t®¡f¢ ruhrç fh©f.

Ô®Î: f©l¿ªj òŸë étu¤ bjhF¥ÃYŸs 20 kÂ¥òfë‹ Â£l éy¡f« 5 vd¡

bfhL¡f¥g£LŸsJ. òŸë étu¤Â‹ x›bthU kÂ¥igÍ« 2 Mš bgU¡F« nghJ,

»il¡F« òŸë étu¤Â‹ òÂa Â£l éy¡f« (v ) 2 5 MF«.

éy¡f t®¡f¢ ruhrç = SD 2^ h = 2 52^ h = 4 5 20# = .

òŸëæaš 11


5. Kjš 13 Ïaš v©fë‹ Â£l éy¡f¤ij¡ fz¡»Lf.

Ô®Î:

Kjš n Ïaš v©fë‹ Â£l éy¡f« v = n12

12-

Mfnt, Kjš 13 Ïaš v©fë‹ Â£l éy¡f«

v = n12

12- =

1213 12

- = .12168 14 3 74= =

6. Ñœ¡fhQ« òŸë étu§fë‹ Â£l éy¡f¤ij¡ fz¡»Lf.

(i) 10, 20, 15, 8, 3, 4 (ii) 38, 40, 34 ,31, 28, 26, 34.

Ô®Î: (i) bfhL¡f¥g£l òŸë étu§fis VWtçiræš mik¡f, 3,4,8,10,15,20

T£L¢ruhrç, x = nxR = 10

63 4 8 10 15 20

660+ + + + + = =

d = 10x x x- = -

x 10d x= - d2

348

101520

-7-6-2 0 5 10

49 36 4 0 25100

d2

R = 214

(ii) bfhL¡f¥g£l òŸë étu§fis VWtçiræš mik¡f, 26,28,31,34,34,38,40

T£L¢ruhrç, x = nxR

= 7

26 28 31 34 34 38 407

231 33+ + + + + + = =

x 33d x x x= - = - d2

26283134343840

-7-5-2 1 1 5 7

4925 4 1 12549

d2

R = 154

v = nd2R

= 6

214

b 5.97

v = nd2R

= 7

154

= 22

b 4.69


7. Ñœf©l m£ltizæš bfhL¡f¥g£LŸs òŸë étu¤Â‹ Â£l éy¡f¤ij¡

fz¡»Lf.

x 3 8 13 18 23

f 7 10 15 10 8

Ô®Î: Cf¢ ruhrç Kiwæš Â£léy¡f« fh©ngh«.

Cf¢ ruhrç A = 13 v‹f. d x A x 13= - = -

x f d = x – 13 d2 fd fd2

38

131823

7101510 8

-10 -5 0 5 10

100 25 0 25100

-70-50 0 50 80

700250 0250800

fR =50 fdR =10 fd2

R =2000

Â£léy¡f« v = f

fd

f

fd2 2

- e o//

//

= 50

20005010 2

- ` j

= 40251- =

25999 = .

531 61

` v - 6.321

8. xU gŸëæYŸs 200 khzt®fŸ xU ò¤jf¡ f©fh£Áæš th§»a ò¤jf§fë‹

v©â¡ifia¥ g‰¿a étu« Ñœ¡fhQ« m£ltizæš bfhL¡f¥g£LŸsJ.

ò¤jf§fë‹ v©â¡if 0 1 2 3 4khzt®fë‹ v©â¡if 35 64 68 18 15

Ï¥òŸë étu¤Â‹ Â£l éy¡f¤ij¡ fz¡»Lf.

Ô®Î: Cf¢ ruhrç Kiwæš Â£léy¡f« fh©ngh«.

Cf¢ ruhrç A = 2 v‹f. d x A x 2= - = -

x f d = x – 2 d2 fd fd2

01234

3564681815

-2-1 0 1 2

41014

-70 -64 0 18 30

140 64 0 18 60

fR = 200 fdR = – 86 fd2

R = 282


Â£léy¡f« v = ffd

ffd

2 2

R

R

R

R- c m

= 200282

20086 2

- -` j = ( )200

282

200

73962

-

= ( )200

282 200 73962

# - = 200

490042^ h

= .200221 4

Mfnt, Â£léy¡f« v - 1.107

9. Ã‹tU« òŸë étu¤Â‹ éy¡f t®¡f¢ ruhrçia¡ fz¡»Lf.

x 2 4 6 8 10 12 14 16f 4 4 5 15 8 5 4 5

Ô®Î: Cf¢ruhrç Kiwæš éy¡f t®¡f ruhrçia fh©ngh«. A = 10 v‹f

x f d = x – 10 d2 fd fd2

2468

10121416

445

158545

-8-6-4-2 0 2 4 6

643616 4 0 41636

– 32 – 24 – 20 – 30 0 10 16 30

256 144 80 60 0 20 64 180

fR =50 50fdR =- fd2

R = 804

2v =

ffd

ffd2 2

R

R

R

R- c m

= 50804

5050 2

- -` j

= 15.0850804 1

50754- = =

Mfnt, éy¡f t®¡f ruhrç = 15.08

10. xU ghjrhç FW¡F¥ ghijia (pedestrian crossing) fl¡f¢ Áy® vL¤J¡ bfh©l

neu étu« Ñœ¡f©l m£ltizæš bfhL¡f¥g£LŸsJ.

neu«

(éehoæš)5-10 10-15 15-20 20-25 25-30

eg®fë‹

v©â¡if4 8 15 12 11

Ï¥òŸë étu¤Â‰F éy¡f t®¡f¢ ruhrç k‰W« Â£l éy¡f¤ij¡ fz¡»Lf.


Ô®Î: ika Ïilbtë 15 – 20, c = 5, vdnt, A = 17.5 v‹f.

d = .c

x A x517 5- = -

ÃçÎ Ïilbtë

ika kÂ¥ò

x f x – Ad =

.x517 5- d2 fd fd2

5-1010-1515-2020-2525-30

7.512.517.522.527.5

4 8151211

–10 –5 0 5 10

-2-1 0 1 2

41014

-8-8 0 12 22

16 8 0 12 44

f 50R = fd 18R = 80fd2

R =

éy¡f t®¡f ruhrç, 2v =

ffd

ffd2 2

R

R

R

R- c m= G c2

#

= 55080

5018 2 2

#- ` j; E = 5080

50

324 252 #-c m

= 2550

4000 3242

#- = 2550 50

4000 324#

#- = 1003676

2v = 36.76

Mfnt, Â£léy¡f« v = 36.76 - 6.063 11. Å£L cçikahs®fŸ 45 ng® mt®fSila bjUé‹ ‘gRik¢ NHš’ Â£l¤Â‰fhf

ãÂ më¤jd®. tNè¡f¥g£l ãÂ¤ bjhif étu« Ã‹tU« m£ltizæš

bfhL¡f¥g£LŸsJ.

bjhif (`) 0-20 20-40 40-60 60-80 80-100

Å£L cçikahs®fë‹ v©â¡if 2 7 12 19 5

Ï›étu¤Â‰F éy¡f t®¡f¢ ruhrç k‰W« Â£léy¡f¤ij¡ fz¡»Lf.

Ô®Î: Cf¢ ruhrç A = 50 k‰W« c = 20 v‹f. d = c

x A x2050- = -

ÃçÎ Ïilbtë

ika kÂ¥ò

x f x – A

d= x2050- d2 fd fd

2

0-2020-4040-6060-80

80-100

1030507090

2 71219 5

– 40– 20 0 20 40

-2-1 0 1 2

41014

-4-7 0 19 10

8 7 01920

f 45R = fd 18R = fd 542

R =



ffd

ffd

c2 2

2#

R

R

R

R- c m= G

= 004554

4518 4

2

#- ` j; E = . ( . ) 4001 2 0 42#-^ h

= . . 4001 2 0 16 #-^ h = 1.04 400 416# =

Mfnt, Â£léy¡f«, v = 416 - 20.396.

12. Ñœ¡fhQ« gutè‹ (distribution) éy¡f t®¡f¢ ruhrç fh©f

ÃçÎ

Ïilbtë20-24 25-29 30-34 35-39 40-44 45-49

ãfœbt©fŸ 15 25 28 12 12 8

Ô®Î: ÃçÎ Ïilbtëæ‹ ika Ïilbtë 30 – 34 vdnt Cf¢ ruhrç A = 32,

c = 5 v‹f.

d = c

x A x532- = -

ÃçÎ Ïilbtë

ika kÂ¥ò

x (f) x – A

d= x

532- d2 fd fd

2

20-2425-2930-3435-3940-4445-49

222732374247

1525281212 8

– 10 – 5 0 5 10 15

-2-1 0 1 2 3

410149

-30-25 0 12 24 24

6025 0124872

f 100R = fdR =5 fd 2172

R =


ffd

ffd

c2 2

2#

R

R

R

R- c m= G = 5

100217

1005 2 2

#- ` j; E

= . ( . 252 17 0 052#-^ h = . ( . 252 17 0 0025 #-^ h

= . 252 1675 #

` 2v = 54.1875 - 54.19

13. xU òŸë étu¤ bjhF¥ÃYŸs 100 kÂ¥òfë‹ ruhrç k‰W« Â£l éy¡f« Kiwna

48 k‰W« 10 MF«. mid¤J kÂ¥òfë‹ T£L¤ bjhif k‰W« mitfë‹

t®¡f§fë‹ T£L¤ bjhif M»at‰iw¡ fh©f.

Ô®Î: 100 kÂ¥òfë‹ T£L¢ ruhrç, x = 48

100 kÂ¥òfë‹ T£L¤ bjhif x/ = 48 × 100 = 4800 xnxa R=; E

Â£l éy¡f«, v = 10


vdnt, éy¡f t®¡f ruhrç, 2v =

nx

nx2 2R R- c m = 100

( x100 100

48002 2R - ` j = 100 ( 2304x100

2R - = 100

( x100

2R = 100 + 2304 = 2404

Mfnt, x2

R = 2404 × 100 = 2,40,400

14. 20 kÂ¥òfë‹ ruhrç k‰W« Â£l éy¡f« Kiwna 10 k‰W« 2 vd fz¡»l¥g£ld.

Ã‹ò rçgh®¡F« nghJ 12 v‹w kÂ¥ghdJ jtWjyhf 8 v‹W vL¤J¡bfhŸs¥g£lJ

bjça tªjJ. rçahd ruhrç k‰W« rçahd Â£l éy¡f« M»adt‰iw¡ fh©f.

Ô®Î: rçahd T£L¢ ruhrçia Kjèš fh©ngh«.

20 òŸë étu§fë‹ ruhrç, xnx/

= = 10

( x20/ = 10

( x/ = 10 × 20 = 200

ÂU¤j¥g£l xR = 200 + 4 = 204

vdnt,ÂU¤j¥g£l ruhrç = 20204 = 10.2

éy¡f t®¡f¢ ruhrç, 2v =

nx

nx

2 2R R- c m = 4 (bfhL¡f¥g£LŸsJ) 2a v =6 @

( 10x20

22R - = 4

( x20

2R = 4 + 100 = 104

x2

R = 104 × 20 = 2080

ÂU¤j¥g£l x2

R = 2080 12 82 2+ -

= 2080 + 144 – 64 = 2160

vdnt, ÂU¤j¥g£l 2v = ˆÂU j¥g£l

n

x2/ – (ÂU¤j¥g£l ruhrç)2

= (10.2)20

2160 2-

= 108 – 104.04 = 3.96

ÂU¤j¥g£l v = .3 96 - 1.99

Mfnt, ÂU¤j¥g£l ruhrç = 10.2 k‰W« ÂU¤j¥g£l Â£l éy¡f« - 1.99 .

15. n = 10, x = 12 k‰W« x2/ = 1530 våš, khWgh£L¡ bfGit¡ fz¡»Lf.

Ô®Î: n = 10, x = nx/ = 12 k‰W« x

2R = 1530 vd¡ bfhL¡f¥g£LŸsJ.

Ï§F Â£l éy¡f«, v = nx

nx

2 2R R- c m


= 10

1530 122

- = 1153 44-

= 9 = 3

khWgh£L¡ bfG, C.V = x

100#vr

= 100123

#

= 10041# = 25

16. Ã‹tU« kÂ¥òfë‹ khWgh£L¡ bfGit¡ fz¡»Lf : 20, 18, 32, 24, 26.

Ô®Î:

x d = 24x - d2

2018322426

-4-6 8 0 2

163664 0 4

120xR = d2

R = 120

T£L¢ ruhrç, x = nx

5120 24R = =

Â£l éy¡f«, v = nd2R C.V. = 100

x#v

v = 5

120 = . 100244 9

24490

# =

v = 24 = 4.9 = . .20 416 20 42b

17. xU òŸë étu¤Â‹ khWgh£L¡ bfG 57 k‰W« Â£l éy¡f« 6.84 våš, mj‹

T£L¢ ruhrçia¡ fh©f.

Ô®Î: khWgh£L¡ bfG = 57, v = 6.84 vd¡ bfhL¡f¥g£LŸsJ

100x#v = 57 & . 100

x6 84

# = 57

x57684= = 12

18. xU FGéš 100 ng® cŸsd®, mt®fë‹ cau§fë‹ T£L¢ ruhrç 163.8 br.Û

k‰W« khWgh£L¡ bfG 3.2 våš, mt®fSila cau§fë‹ Â£l éy¡f¤ij¡

fh©f.

Ô®Î: xU FGéš cŸs 100 ngç‹ cau§fë‹ T£L¢ ruhrç x = 163.8,

khWgh£L¡ bfG = 3.2

( 100x#v = 3.2


( .

100163 8

#v = 3.2

v = . .100

3 2 163 8# = 5.2416 b 5.24

19. x/ = 99, n = 9 k‰W« x 10 2/ -^ h = 79 våš, k‰W«x x x2 2/ / -^ h M»at‰iw¡

fh©f.

Ô®Î: xR = 99 k‰W« n = 9 vd¡ bfhL¡f¥g£LŸsJ.

x = nxR =

999 11= .

x2

R -I¡ fz¡»Lnth«.

Ï§F, ( 10)x 2R - = 79

( )x x20 1002R - + = 79

20 100 1x x2R R R- + = 79 1 9a / =^ h

x 20 99 100 92

# #R - + = 79

x2

` R = 1159

( )x x 2R - = ( )x 112

R -

= x x22 1212

R - +^ h

= 20 1 1x x 212R R R- +

= 1159 2 121178 9#- +

= 1159 2178 1089- + = 70

` x2R = 1159 k‰W« x x 2R -^ h = 70.

20. xU tF¥ÃYŸs A, B v‹w ÏU khzt®fŸ bg‰w kÂ¥bg©fŸ Ã‹tUkhW:

A 58 51 60 65 66B 56 87 88 46 43

Ït®fëš ah® äFªj Ó®ik¤ j‹ikia bfh©LŸsh®?

Ô®Î:

khzt‹ A khzt‹ B

x = nx/ =

5300 60= x =

nx/ = 6

5320 4=

x d = 60x - d2 x d = 64x - d

2

5158606566

-9-2 0 5 6

81 4 02536

4346568788

-21-18 -8 23 24

441324 64529576

300xR = d2

R = 146 320xR = d2

R = 1934


v = nd2/ =

5146 = .29 2 = 5.4 v =

nd2/ = . 19.67

51934 386 8= =

C.V = 100x#v = .

605 4 100# = 9 ...(1) C.V = 100

x#v = . 100

6419 67

# = 30.73 ...(2)

(1) k‰W« (2) M»at‰¿èUªJ khzt‹ A-æ‹ khWgh£L¡ bfG, khzt‹ B-‹

khWgh£L¡bfGit él¡FiwÎ. vdnt, khzt‹ A v‹gt® mÂf Ó®ik¤

j‹ikia¡ bfh©LŸsh®.

gæ‰Á 11.2rçahd éilia¤ nj®ªbjL¡fÎ«.

1. 2, 3, 5, 7, 11, 13, 17, 19, 23 , 29 v‹w Kjš 10 gfh v©fë‹ Å¢R (A) 28 (B) 26 (C) 29 (D) 27

Ô®Î: R = 29 2 27L S = - =- (éil: (D) )

2. bjhF¥ÃYŸs étu§fëš äf¢ Á¿a kÂ¥ò 14.1 k‰W« m›étu¤Â‹ Å¢R 28.4

våš, bjhF¥Ã‹ äf¥ bgça kÂ¥ò (A) 42.5 (B) 43.5 (C) 42.4 (D) 42.1

Ô®Î: S = 14.1, R = 28.4, R = L – S & L = R + S

L& = 28.4 14.1 42.5+ = (éil: (A) )

3. bjhF¥ÃYŸs étu§fëš äf¥bgça kÂ¥ò 72 k‰W« äf¢Á¿a kÂ¥ò 28 våš,

m¤bjhF¥Ã‹ Å¢R¡ bfG

(A) 44 (B) 0.72 (C) 0.44 (D) 0.28

Ô®Î: L = 72, S = 28

Å¢R¡ bfG = 0.4472 2872 28

10044

L SL S =

+- = =

+- (éil. (C))

4. 11 kÂ¥òfë‹ x 132R = våš, mt‰¿‹ T£L¢ ruhrç (A) 11 (B) 12 (C) 14 (D) 13

Ô®Î: n = 11 xR = 132

xr = 12nx

11132R = = (éil: (B))

5. n cW¥òfŸ bfh©l vªj xU v©fë‹ bjhF¥Ã‰F« ( )x xR - = (A) xR (B) x (C) nx (D) 0

Ô®Î:n cW¥òfŸ bfh©l vªj xU v©fë‹ bjhF¥Ã‰F« ( )x x 0R - = .

(éil: (D))


6. n cW¥òfŸ bfh©l vªj xU v©fë‹ bjhF¥Ã‰F« ( )x xR - =

(A) nx (B) ( 2)n x- (C) ( 1)n x- (D) 0

Ô®Î: ( )x xR - = ( 1)nx x x n- = - (éil: (C) )

7. x, y, z- ‹ Â£l éy¡f« t våš, x + 5, y + 5, z + 5-‹ Â£l éy¡f«

(A) t3

(B) t + 5 (C) t (D) x y z

Ô®Î:xU gutè‹ x›bthU kÂ¥òlD« xnu v©iz¡ T£odhnyh mšyJ

fê¤jhnyh mj‹ Â£l éy¡f« khwhkš ÏU¡F«. t` v= . (éil: (C) )

8. xU òŸë étu¤Â‹ Â£léy¡f« 1.6 våš, mj‹ éy¡f t®¡f¢ ruhrç (gut‰go)

(A) 0.4 (B) 2.56 (C) 1.96 (D) 0.04

Ô®Î: éy¡f t®¡f¢ ruhrç = 1.6 2.562 2v = = (éil: (B) )

9. xU òŸë étu¤Â‹ éy¡f t®¡f ruhrç 12.25 våš, mj‹ Â£l éy¡f«

(A) 3.5 (B) 3 (C) 2.5 (D) 3.25

Ô®Î: Â£l éy¡f« SD = . 3.5éy¡f t®¡f¢ ruhrç 12 25= = (éil: (A) )

10. Kjš 11 Ïaš v©fë‹ éy¡f t®¡f¢ ruhrç

(A) 5 (B) 10 (C) 5 2 (D) 10

Ô®Î: Kjš 11 Ïaš v©fë‹ éy¡f t®¡f¢ ruhrç

2v = 10n12

112

11 1121202 2

- = - = = (éil: (D) )

11. 10, 10, 10, 10, 10-‹ éy¡f t®¡f¢ ruhrç

(A) 10 (B) 10 (C) 5 (D) 0

Ô®Î: x = 10, d x x 0= - =r

éy¡f t®¡f¢ ruhrç 2v = 0nd2R = ( éil: (D) )

12. 14, 18, 22, 26, 30-‹ éy¡f t®¡f¢ ruhrç 32 våš, 28, 36, 44, 52, 60-‹ éy¡f

t®¡f¢ ruhrç

(A) 64 (B) 128 (C) 32 2 (D) 32

Ô®Î: 14, 18, 22, 26, 30 ‹éy¡f t®¡f¢ ruhrç 32. ` S.D. = 32

14, 18, 22, 26, 30 -‹ x›bthU kÂ¥igÍ« 2 Mš bgU¡F« nghJ

28, 36, 44, 52, 60 vd »il¡»‹wJ.

Mfnt, òÂa Â£l éy¡f« = 2 32

òÂa éy¡f t®¡f¢ ruhrç = 2 322^ h = 4 × 32 = 128. (éil: (B) )


13. étu§fë‹ bjhF¥ò x‹¿‹ Â£léy¡f« 2 2 . mÂYŸs x›bthU kÂ¥ò«

3 Mš bgU¡f¡ »il¡F« òÂa étu¤ bjhF¥Ã‹ Â£léy¡f«

(A) 12 (B) 4 2 (C) 6 2 (D) 9 2

Ô®Î: Â£l éy¡f« = 2 2 . x›bthU kÂ¥igÍ« 3 Mš bgU¡F« nghJ

òÂa Â£l éy¡f« = 3 2 2 6 2# = (éil: (C) )

14. ( ) ,x x x48 202/ - = = k‰W« n = 12 våš, khWgh£L¡bfG

(A) 25 (B) 20 (C) 30 (D) 10

Ô®Î: v = nd

2R =

1248 4 2= =

khWgh£L¡bfG, C.V = 100x#v = 100

202 10# = . (éil: (D) )

15. Áy étu§fë‹ T£L¢ ruhrç k‰W« Â£léy¡f« Kiwna 48, 12 våš,

khWgh£L¡bfG

(A) 42 (B) 25 (C) 28 (D) 48

Ô®Î: v = 12, x = 48

khWgh£L¡bfG, C.V = 100x#v = 100

4812 25# = . (éil: (B) )

Ô®Î - ãfœjfÎ 285

gæ‰Á 12.1

1. xU igæš cŸs 1 Kjš 100 tiu v©fshš F¿¡f¥g£l 100 Ó£LfëèUªJ

xU Ó£L vL¡f¥gL»wJ. m›thW vL¡f¥gL« Ó£o‹ v© 10 Mš tFgL«

v©zhf ÏU¥gj‰fhd ãfœjféid¡ fh©f?

Ô®Î: 100 Ó£Lfëš 1 Kjš 100 tiu v©fshš F¿¡f¥g£LŸsd.

TWbtë, S = {1,2,3, ,100} ; ( ) 100.n Sg =

10 Mš tFgL« v© cŸs Ó£L vL¥gj‰fhd ãfœ¢Áia A v‹f.

{10,20, ,100} ; ( ) 10.A n Ag= =

vdnt, ( )( )( )

P An Sn A

10010

101= = =

2. xU Óuhd gfil Ïu©L Kiw cU£l¥gL»wJ. Kf v©fë‹ TLjš 9

»il¡f¥ bgWtj‰fhd ãfœjfÎ fh©f?

Ô®Î: xU gfil ÏU Kiw cU£L« nghJ »il¡F« TWbtë

, , , , , , , , , , , , , , , , , , , , ,S 1 1 1 2 1 6 2 1 2 2 2 6 6 1 6 2 6 6g g g g= ^ ^ ^ ^ ^ ^ ^ ^ ^h h h h h h h h h" ,

36n S =^ h .

Kf v©fë‹ TLjš 9 Mf ÏU¡F« ãfœ¢Áia A v‹f.

A = { (3,6), (4,5), (5,4), (6,3) }; n(A) = 4.

P(A) = ( )( )

n Sn A

364

91= = .

3. ÏU gfilfŸ xU nru cU£l¥gL»‹wd. Kf v©fis¡ bfh©L mik¡f¥gL«

<çy¡f v© 3 Mš tFgL« v©zhf ÏU¥gj‰fhd ãfœjfÎ fh©f.

Ô®Î: ÏU gfilfis cU£L« nghJ »il¡F« TWbtë,

, , , , , , , , , , , , , , , , , , , , ,S 1 1 1 2 1 6 2 1 2 2 2 6 6 1 6 2 6 6g g g g= ^ ^ ^ ^ ^ ^ ^ ^ ^h h h h h h h h h" ,

36n S =^ h .Kf v©fis¡ bfh©L mik¡f¥gL« <çy¡f v© 3 Mš tFgL« v©zhf

ÏU¡F« ãfœ¢Á A v‹f . A = { 12, 15, 21, 24, 33, 36, 42, 45, 51, 54, 63, 66}

( )n A = 12

( )P A = ( )( )

n Sn A

3612

31= = .

ãfœjfÎ 12


4. 12 ešy K£ilfSl‹ 3 mG»a K£ilfŸ fyªJŸsd. rkthŒ¥ò Kiwæš

nj®ªbjL¡f¥gL« xU K£il, mG»ajhf ÏU¥gj‰fhd ãfœjfÎ v‹d?

Ô®Î: ešy K£ilfë‹ v©â¡if=12;

mG»a K£ilfë‹ v©â¡if 3=

vdnt, K£ilfë‹ bkh¤j v©â¡if 51= .

mG»a K£il »il¥gj‰fhd ãfœ¢Á A v‹f. n(A) = 3

vdnt, ( )( )( )

.P An Sn A

153

51= = =

5. ÏU ehza§fis xnu rka¤Âš R©L«nghJ, mÂfg£rkhf xU jiy

»il¥gj‰fhd ãfœjféid¡ fh©f.

Ô®Î: ÏU ehza§fis xnu rka¤Âš R©Ltjhš V‰gL« TWbtë { , , , } ; ( ) 4.S HH HT TH TT n S= =

mÂfg£rkhf xU jiy »il¥gj‰fhd ãfœ¢Á A v‹f .

A = , ,HT TH TT" , ; 3n A =^ h .

( )P A = n S

n A

43=

^^

hh .

6. e‹F fiy¤J mL¡»a 52 Ó£Lfis¡ bfh©l f£oèUªJ rkthŒ¥ò Kiwæš

xU Ó£L vL¡f¥gL»wJ. Ã‹tUtdt‰¿‰F ãfœjfÎfis¡ fh©f.

(i) vL¤j Ó£L lak©£ Mf ÏU¡f

(ii) vL¤j Ó£L lak©£ Ïšyhkš ÏU¡f

(iii) vL¤j Ó£L V° Ó£lhf Ïšyhkš ÏU¡f.

Ô®Î: Ï§F TWbtë ( )n S 52=

(i) A v‹gJ lak©£ Ó£L »il¡F« ãfœ¢Á v‹f. vdnt, ( ) 13n A =

( )( )( )

.P An Sn A

5213

41= = =

(ii) B v‹gJ lak©£ Ó£L »il¡fhkš ÏU¡F« ãfœ¢Á v‹f.

vdnt, B A= l

( ) 1 ( ) 1P B P A41

43= - = - = .

(iii) C v‹gJ V° Ó£lhf ÏU¡F« ãfœ¢Á v‹f. vdnt, ( ) 4n C =

( )( )( )

P Cn Sn C

524

131= = = .

Mfnt, V° Ó£lhf Ïšyhkš ÏU¡f ãfœjfÎ ( ) 1 .P C131

1312= - =l

7. _‹W ehza§fŸ xnu neu¤Âš R©l¥gL»‹wd. Ã‹tU« ãfœ¢ÁfS¡F

ãfœjféid¡ fh©f.

(i) FiwªjJ xU jiy »il¥gJ (ii) ÏU ó¡fŸ k£L« »il¥gJ

(iii) FiwªjJ ÏU jiyfŸ »il¥gJ.


Ô®Î: TWbtë,

{ , , , , , , , } ; ( ) 8.S HHH HHT HTH THH HTT THT TTH TTT n S= =

(i) FiwªjJ xU jiy »il¡F« ãfœ¢Á A v‹f .

{ , , , , , , } ; ( ) 7A HHH HHT HTH THH HTT THT TTH n A= =

Mfnt, ( )( )( )

.P An Sn A

87= =

(ii) rçahf ÏU ó¡fŸ k£L« »il¡F« ãfœ¢Á B v‹f .

{ , , } ; ( ) 3B HTT THT TTH n B= =

Mfnt, ( )( )( )

.P Bn Sn B

83= =

(iii) FiwªjJ ÏU jiyfŸ »il¡F« ãfœ¢Á C v‹f .

{ , , , } ; ( ) 4C HHH HHT HTH THH n C= =

Mfnt, ( )( )( )

.P Cn Sn C

84

21= = =

8. xU igæš 1 Kjš 6 tiu v©fŸ F¿¡f¥g£l 6 btŸis ãw¥ gªJfS« k‰W«

7Kjš 10 tiu v©fŸ F¿¡f¥g£l 4 Át¥ò ãw¥ gªJfS« cŸsd. rk thŒ¥ò

Kiwæš xU gªJ vL¡f¥gL»wJ våš, Ã‹tU« ãfœ¢ÁfS¡F ãfœjféid¡

fh©f.

(i) vL¡f¥g£l gªJ xU Ïu£il v© bfh©l gªjhf ÏU¤jš

(ii) vL¡f¥g£l gªJ xU btŸis ãw¥ gªjhf ÏU¤jš.

Ô®Î: { , , , , , , , , , }S W W W W W W R R R R1 2 3 4 5 6 7 8 9 10

= ; ( )n S 10= .

(i) Ïu£il v© bfh©l gªJ »il¡F« ãfœ¢Á A v‹f.

{ , , , , }A W W W R R2 4 6 8 10

= ; ( )n A 5= .

Mfnt, ( )( )( )

.P An Sn A

105

21= = =

(ii) btŸis ãw gªJ »il¡F« ãfœ¢Á B v‹f.

{ , , , , , }B W W W W W W1 2 3 4 5 6

= ; ( )n B 6= .

Mfnt, ( )( )( )

P Bn Sn B

106

53= = = .

9. 1 Kjš 100 tiuæyhd KG v©fëèUªJ rk thŒ¥ò Kiwæš

nj®ªbjL¡f¥gL« xU v© (i) xU KG t®¡fkhf (perfect square) ÏU¡f

(ii) KG fdkhf Ïšyhkš (not a cube) ÏU¡f M»adt‰¿‹ ãfœjfÎfis¡ fh©f

Ô®Î: ( )n S 100= .

(i) xU KG t®¡fv© »il¡F« ãfœ¢Á A v‹f.

{ , , , , , , , , , }A 1 4 9 16 25 36 49 64 81 100= ; ( )n A 10= .

Mfnt, ( )( )( )

P An Sn A

10010

101= = = .


(ii) xU KG fdv© »il¡F« ãfœ¢Á B v‹f.

{ , , , }B 1 8 27 64= ; ( )n B 4= .

( )( )( )

P Bn Sn B

1004

251= = = .

Mfnt, xU KG fdv© »il¡fhkèU¡f ãfœjfÎ

( ) 1 ( )P B P B= -l 1251

2524= - = .

10. m®b#©odh, g§fshnjZ, Ódh, m§nfhyh, UZah k‰W« mšÉçah M»a

ehLfë‹ bga®fis¡ bfh©l g£oaèUªJ xU R‰Wyh¥gaâ rkthŒ¥ò

Kiwæš xU eh£o‹ bgaiu¤ nj®ªbjL¡»wh®. “m” v‹w vG¤Âš Mu«gkhF«

eh£o‹ bgaiu nj®ªbjL¥gj‰fhd ãfœjfÎ v‹d?

Ô®Î: S = {m®b#©odh, g§fshnjZ, Ódh, m§nfhyh, UZah, mšÉçah};

( )n S 6= .

“m” v‹w vG¤Âš Mu«gkhF« eh£o‹ bga® »il¡F« ãfœ¢Á A v‹f.

A = {m®b#©odh, m§nfhyh, mšÉçah}; ( ) 3n A = .

Mfnt, ( )( )( )

P An Sn A

63

21= = = .

11. xU bg£oæš 4 g¢ir, 5 Úy« k‰W« 3 Át¥ò ãw¥ gªJfŸ cŸsd. rkthŒ¥ò

Kiwæš xU gªij¤ nj®ªbjL¡f mJ

(i) Át¥ò ãw¥ gªjhf ÏU¡f (ii) g¢ir ãw¥ gªjhf ÏšyhkèU¡f

M»adt‰¿‹ ãfœjfÎfis¡ fh©f.

Ô®Î: gªJfë‹ bkh¤j v©â¡if, ( )n S 4 5 3= + + 12=

(i) Át¥ò ãw gªJ »il¡F« ãfœ¢Á A v‹f. ( )n A 3= .

Mfnt, ( )( )( )

P An Sn A

123

41= = =

(ii) g¢ir ãw gªJ »il¡F« ãfœ¢Á B v‹f. ( )n B 4= .

( )( )( )

P Bn Sn B

124

31= = = .

Mfnt, g¢ir ãw gªJ »il¡fhkèU¡f ãfœjfÎ

( ) 1 ( )P B P B= - 131

32= - = .

12. 20 Ó£Lfëš 1 Kjš 20 tiuÍŸs KG v©fŸ F¿¡f¥g£LŸsd. rkthŒ¥ò

Kiwæš xU Ó£L vL¡f¥gL»‹wJ. m›thW vL¡f¥g£l Ó£oYŸs v©

(i) 4-‹ kl§fhf ÏU¡f (ii) 6-‹ kl§fhf Ïšyhkš ÏU¡f

M»a ãfœ¢Áfë‹ ãfœjfÎfis¡ fh©f.

Ô®Î: ( )n S 20= .


(i) Ó£oYŸs v© 4 -‹ kl§fhf »il¡F« ãfœ¢Á A v‹f.

{4, 8,12,16, 20} ; ( ) 5.A n A= =

Mfnt, ( )( )( )

.P An Sn A

205

41= = =

(ii) Ó£oYŸs v© 6 -‹ kl§fhf »il¡F« ãfœ¢Áia B v‹f.

{6,12,18} ; ( ) 3.B n B= =

( )( )( )

.P Bn Sn B

203= =

Mfnt, Ó£oYŸs v© 6 -‹ kl§fhf Ïšyhkš ÏU¡f ãfœjfÎ

( ) 1 ( ) 1 .P B P B203

2017= - = - =

13. 3,5,7 M»a v©fis Ïy¡f§fshf¡ bfh©L xU Ïu©oy¡f v©

mik¡f¥gL»‹wJ. m›bt© 57-I él¥ bgçajhf ÏU¥gj‰fhd ãfœjfÎ

fh©f. (m›bt©âš xnu Ïy¡f¤ij Û©L« ga‹gL¤j¡ TlhJ).

Ô®Î: TWbtë, { 35, 37, 53, , 73, 75 }S 57= ; ( ) 6.n S =

57 -I él¥ bgçajhf cŸs v© »il¡F« ãfœ¢Áia A v‹f.

A = {73,75} ; ( ) 2n A = .

Mfnt, ( )P A = ( )( )

.nn A5 6

231= =

14. _‹W gfilfŸ xnu neu¤Âš cU£l¥gL«nghJ, _‹W gfilfëY« xnu v©

»il¥gj‰fhd ãfœ¢Áæ‹ ãfœjféid¡ fh©f.

Ô®Î: _‹W gfilfŸ cU£L« nghJ »il¡F« TWbtë S ,

( ) 6 216n S 3` = = .

_‹W gfilfëY« xnu v© »il¡F« ãfœ¢Áia A v‹f.

{ ( , , ), ( , , ), ( , , ), ( , , ), ( , , ), ( , , )}A 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6= ; ( ) 6.n A =

Mfnt, ( )( )( )

P An Sn A

2166

361= = = .

15. ÏU gfilfŸ xnu neu¤Âš cU£l¥gL«nghJ »il¡F« Kf v©fë‹

bgU¡f‰gy‹ xU gfh v©zhf ÏU¥gj‰fhd ãfœjféid¡ fh©f.

Ô®Î: ÏU gfilfis cU£L« nghJ »il¡F« TWbtë, , , , , , , , , , , , , , , , , , , , , ,S 1 1 1 2 1 6 2 1 2 2 2 6 6 1 6 2 6 6g g g g= ^ ^ ^ ^ ^ ^ ^ ^ ^h h h h h h h h h" ,

36n S =^ h .Kf v©fë‹ bgU¡f‰gy‹ gfh v©zhf »il¡F« ãfœ¢Áia A v‹f.

{( , ), (2, ), ( ,3), ( , ), ( , ), (5, )}A 1 2 1 1 3 1 1 5 1= ; ( )n A 6=

Mfnt, ( )( )( )

.P An Sn A

366

61= = =


16. xU Kfitæš Úy«, g¢ir k‰W« btŸis ãw§fëyhd 54 gë§F¡f‰fŸ

cŸsd. xU gë§F¡ fšiy vL¡F«nghJ, Úy ãw¥ gë§F¡fš »il¥gj‰fhd

ãfœjfÎ 31 k‰W« g¢ir ãw¥ gë§F¡fš »il¥gj‰fhd ãfœjfÎ

94

våš, m«Kfitæš cŸs btŸis ãw¥ gë§F¡ f‰fë‹ v©â¡ifia¡

fh©f.

Ô®Î: ( )n S 54= .

Úy ãw gë§F¡fš »il¥gj‰fhd ãfœjfÎ ( )P B31= .

g¢ir ãw gë§F¡fš »il¥gj‰fhd ãfœjfÎ ( )P G94=

btŸis ãw gë§F¡fš »il¥gj‰fhd ãfœjfÎ ( )( )

P Wn W54

= .

( ) ( ) ( )P B P G P W 1+ + = ( )1

n W54 3

194& = - +` j.

( )1

n54 9

7W& = - & ( ) 54n

92W #= ( ) 12.n W& =

vdnt, m«Kfitæš 12 btŸis ãw¥ gë§F f‰fŸ cŸsJ .

17. xU igæš cŸs 100 r£ilfëš, 88 r£ilfŸ ešy ãiyæY«, 8 r£ilfŸ

Á¿a Fiwgh£LlD« k‰W« 4 r£ilfŸ bgça Fiwgh£LlD« cŸsd. A v‹w tâf® ešy ãiyæš cŸs r£ilfis k£Lnk V‰»wh®. Mdhš B v‹w

tâf® mÂf FiwghL cila r£ilfis k£L« V‰f kW¡»wh®. rkthŒ¥ò

Kiwæš VnjD« X® r£ilia nj®ªbjL¡f mJ (i) A-¡F V‰òilajhf mika

(ii) B-¡F V‰òilajhf mika. M»adt‰¿‰F ãfœjfÎfis¡ fh©f.

Ô®Î: ( ) 100n S = . ešy ãiyæš cŸs r£ilia nj®ªbjL¥gj‰fhd ãfœ¢Á G v‹f.

( )n G 88= . mÂf FiwghLila r£ilia nj®ªbjL¡fhkèU¡F« ãfœ¢Á M v‹f.

( ) 88 8 .n M 96= + =

(i) nj®ªbjL¡f¥gL« r£il A-¡F V‰òilajhf ÏU¡f ãfœjfÎ

( )( )

n Sn G

10088

2522= = = .

(ii) nj®ªbjL¡f¥gL« r£il B-¡F V‰òilajhf ÏU¡f ãfœjfÎ

( )( )n Sn M

= .10096

2524= =

18. xU igæš cŸs 12 gªJfëš x gªJfŸ btŸis ãwKilait. (i) rkthŒ¥ò

Kiwæš xU gªJ nj®ªbjL¡f, mJ btŸis ãwkhf ÏU¥gj‰fhd ãfœjfÎ

fh©f. (ii) 6 òÂa btŸis ãw¥ gªJfis m¥igæš it¤jÃ‹d®, xU

btŸis ãw¥ gªij¤ nj®bjL¥gj‰fhd ãfœjfÎ MdJ (i)-š bgw¥g£l

ãfœjféid¥ nghy ÏUkl§F våš, x-‹ kÂ¥Ãid¡ fh©f.


Ô®Î: S1 k‰W« S

2 v‹gd Kiwna btŸis ãw¥ gªJfis igæš it¥gj‰F

K‹D« Ã‹D« cŸs TWbtëfŸ v‹f.

W1 k‰W« W

2 v‹gd Kiwna 6 btŸis ãw¥ gªJfis igæš it¥gj‰F

K‹D« Ã‹D« xU btŸis ãw¥ gªij vL¥gj‰fhd ãfœ¢Á v‹f.

( ) 12, ( ) 12 6 18n S n S1 2= = + = .

( ) , ( ) 6.n W x n W x1 2= = +

( ) , ( ) .P W x P W x12 18

61 2= = +

( ) 2 ( )P W P W2 1= x x

186 2

12& + = x x

36& + = 3x& = .

Mfnt, ( )P W123

41

1= = k‰W« 3x =

19. Ã¡» c©oaèš (Piggy bank)100 I«gJ igrh ehz§fS« 50 xU %ghŒ

ehza§fS« 20 Ïu©L %ghŒ ehza§fS« k‰W« 10 IªJ %ghŒ

ehza§fS« cŸsd. rkthŒ¥ò Kiwæš xU ehza« nj®ªbjL¡F« nghJ

(i) I«gJ igrh ehzakhf ÏU¡f (ii) IªJ %ghŒ ehzakhf Ïšyhkš ÏU¡f


Ô®Î: S v‹gJ mid¤J ehza§fisÍ« bfh©l TWbtë v‹f.

( ) 100 50 20 10 180.n S = + + + =

F1, O, T k‰W« F

2 v‹gd Kiwna I«gJ igrh, xU %ghŒ, Ïu©L %ghŒ k‰W«

IªJ %ghŒ ehza§fŸ »il¥gj‰fhd ãfœ¢ÁfŸ v‹f.

( ) 100 ; ( ) 50;n F n O1= = ( ) 20; ( ) 10n T n F

2= = .

(i) I«gJ igrh ehza« »il¥gj‰fhd ãfœjfÎ,

( )( )

( ).P F

n S

n F

180100

95

11= = =

(ii) IªJ %ghŒ ehza« »il¥gj‰fhd ãfœjfÎ,

( )( )

( ).P F

n S

n F

18010

181

22= = =

nj®ªbjL¡f¥gL« ehza« IªJ %ghŒ ehzakhf Ïšyhkš ÏU¡f ãfœjfÎ,

( ) 1 ( ) 1 .P F P F181

1817

2 2= - = - =

gæ‰Á 12.2

(Ï¥gæ‰Áæš cŸs édh¡fis T£lš nj‰w¤ij¥ ga‹gL¤jhkš,ãfœjfé‹

tiuaiwia neuoahf ga‹gL¤ÂÍ« Ô®Î fhzyh«.)

1. A k‰W« B v‹gd x‹iwbah‹W éy¡F« ãfœ¢ÁfŸ. nkY«

( ) ( )k‰W«P A P B53

51= = våš, ( )P A B, -I¡ fh©f .

Ô®Î:

A k‰W« B v‹gd x‹iwbah‹W éy¡F« ãfœ¢ÁfŸ våš, ( )P A B 0+ =


( ) ( ) ( ) ( )P A B P A P B P A B, += + -

= 0 .53

51

54+ - =

2. A k‰W« B v‹w Ïu©L ãfœ¢Áfëš ( ) , ( )P A P B41

52= = k‰W« ( )P A B

21, =

våš, ( )P A B+ -I¡ fh©f.

Ô®Î: ( ) ( ) ( ) ( )P A B P A P B P A B, += + - .

( ) ( ) ( ) ( )P A B P A P B P A B+ ,= + -

= .41

52

21

203+ - =

3. A k‰W« B v‹w Ïu©L ãfœ¢Áfëš ( ) , ( ) ( )k‰W«P A P B P A B21

107 1,= = =

våš, (i) ( )P A B+ (ii) ( )P A B,l l M»at‰iw¡ fh©f.

Ô®Î: (i) ( ) ( ) ( ) ( )P A B P A P B P A B+ ,= + -

= 121

107+ - .

105 7 10

102

51= + - = =

(ii) ( ) ( )P A B P A B, +=l l l ( ( )A B A Ba + ,=l l l)

= 1 ( )P A B+- ( ( ) 1 ( )P A P Aa = -l )

= 151

54- = .

4. xU gfil ÏUKiw cU£l¥gL»wJ. Kjyhtjhf cU£l¥gL«nghJ xU

Ïu£il¥gil v© »il¤jš mšyJ m›éU cU£lèš Kf v©fë‹

TLjš 8 Mf ÏU¤jš vD« ãfœ¢Áæ‹ ãfœjféid¡ fh©f.

Ô®Î: xU gfil ÏU Kiw cU£L« nghJ »il¡F« TWbtë

{(1,1), (1,2), . . , (1,6), (2,1), (2,2), . ., (2,6), (6,1), (6,2), . . , (6,6)}S = ; ( ) 36n S =

Kjš cU£lèš Ïu£il v© »il¡F« ãfœ¢Áia A v‹f.

{ ( , ), ( , ), ( , ), ( , ), ( , ), ( , ),

( , ), ( , ), ( , ), ( , ), ( , ), ( , ),

(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} ; ( ) 18.

A

n A

2 1 2 2 2 3 2 4 2 5 2 6

4 1 4 2 4 3 4 4 4 5 4 6

=

=

( )P A = ( )( )

n Sn A

3618

21= = .

Kf v©fë‹ TLjš 8 »il¡F« ãfœ¢Áia B v‹f.

B= { (2,6), (3,5), (4,4), (5,3), (6,2) } vdnt , ( ) 5.n B =

( )( )( )

P Bn Sn B

365= = .

nkY«, { ( , ), ( , ), ( , ) }A B 2 6 4 4 6 2+ = k‰W« ( )P A B363+ =


vdnt, Kjš cU£lèš Ïu£il v© mšyJ Kf v©fë‹ TLjš 8 Mf

»il¡f ãfœjfÎ, ( ) ( ) ( ) ( )P A B P A P B P A B, += + -

= 3618

365

363+ - .

3620

95= =

5. 1 Kjš 50 tiuæyhd KG¡fëèUªJ rkthŒ¥ò Kiwæš X® v© nj®ªbjL¡f¥

gL«nghJ m›bt©4 mšyJ 6 Mš tFgLtj‰fhd ãfœjfÎ fh©f.

Ô®Î: TWbtë, { , , , , } .S 1 2 3 50g= ( )n S 50= .

4 Mš tFgL« v©fŸ »il¡F« ãfœ¢Áia A v‹f.

A = {4,8,12, ,48}g ; ( )n A 12= .

( )P A = ( )( )

n Sn A

5012= .

6 Mš tFgL« v©fŸ »il¡F« ãfœ¢Áia B v‹f.

B = { 6,12,18,24,30,36,42,48} ; ( )n B 8= .

( )P B = ( )( )

n Sn B

508= .

nkY«, { , , , }A B 12 24 36 48+ = ; ( )n A B 4+ = .

( )P A B+ = ( )

( )n S

n A B504+

= .

Mfnt, 4 mšyJ 6 Mš tFgL« v© »il¡f ãfœjfÎ

( ) ( ) ( ) ( )P A B P A P B P A B, += + -

= 5012

508

504+ -

258= .

6. xU igæš 50 kiu MâfS« (bolts), 150 ÂUF kiufS« (nuts) cŸsd.

mt‰WŸ ghÂ kiu MâfS«, ghÂ ÂUF kiufS« JU¥Ão¤jit. rkthŒ¥ò

Kiwæš VnjD« x‹iw¤ nj®ªbjL¡F« nghJ mJ JU¥Ão¤jjhf mšyJ

xU kiu Mâahf ÏU¥gj‰fhd ãfœjféid¡ fh©f.

Ô®Î: bkh¤j« cŸs bghU£fŸ, ( )n S 200= .

kiu Mâfë‹ bkh¤j v©â¡if 50= .

ÂUF kiufë‹ bkh¤j v©â¡if 150= .

JU¥Ão¤j kiu MâfŸ k‰W« ÂUF kiufë‹bkh¤j v©â¡if

25= + 75 = 100.JU¥Ão¤j bghUŸ »il¡F« ãfœ¢Áia A v‹f. ( )n A = 75 25 100+ =

( )( )( )

.P An Sn A

200100= =

kiu MâfŸ »il¡F« ãfœ¢Áia B v‹f. ( )n B = 50 .

( )( )( )

.P Bn Sn B

20050= =


JU¥Ão¤j kiu MâfŸ »il¡F« ãfœ¢Á, ( )n A B 25+ = .

( )P A B+ = ( )

( ).

n Sn A B

20025+

=

nj®ªbjL¡f¥gL« xU bghUŸ kiu Mâahfnth mšyJ JU¥Ão¤jhfnth

ÏU¡f ãfœjfÎ,

( )P A B, = ( ) ( ) ( )P A P B P A B++ -

= 200100

20050

20025

85+ - = .

7. ÏU gfilfŸ xnu neu¤Âš nru cU£l¥gL«nghJ »il¡F« Kf v©fë‹

TLjš 3 Mš k‰W« 4 Mš tFglhkèU¡f ãfœjfÎ fh©f.

Ô®Î: TWbtë, {(1,1), (1,2), , (1,6), (2,1), (2,2), , (6,6)};S g g= ( ) 36.n S =

A v‹gJ Kf v©fë‹ TLjš 3 Mš tFgL« v©fŸ k‰W« B v‹gJ Kf

v©fë‹ TLjš 4 Mš tFgL« v©fŸ »il¡F« ãfœ¢ÁfŸ v‹f.

{(1,2), (2,1), (1,5), (5,1), (2,4), (4,2), (3,3), (3,6), (6,3), (4, ), (5,4), (6,6)}A 5= ( ) 12.n A =

{(1,3), (3,1), (2,2), (2,6), (6,2), (3,5), (5,3), (4,4), (6,6)}B = ; ( )n B 9= k‰W«

( , )A B 6 6+ = ; n A B 1+ =^ h .

Kf v©fë‹ TLjš 3 k‰W« 4 Mš tFgl ãfœjfÎ

( ) ( ) ( ) ( )P A B P A P B P A B, += + - .3612

369

361

3620= + - =

Mfnt, Kf v©fë‹ TLjš 3 k‰W« 4 Mš tFglhkèU¡f ãfœjfÎ

( ' ') ( ) 'P A B P A B+ ,= 1 ( )P A B,= - 13620

3616

94= - = =

8. xU Tilæš 20 M¥ÃŸfS« 10 MuŠR¥ gH§fS« cŸsd. mt‰WŸ 5M¥ÃŸfŸ k‰W« 3 MuŠRfŸ mG»ait. rkthŒ¥ò Kiwæš xUt® xU

gH¤ij vL¤jhš, mJ M¥Ãshfnth mšyJ ešy gHkhfnth ÏU¥gj‰fhd


Ô®Î: Tilæš cŸs gH§fë‹ bkh¤j v©â¡if ( )n S 30= .mG»a gH§fë‹ v©â¡if 8= ; ešy gH§fë‹ v©â¡if 22= .

M¥ÃŸ gH§fŸ »il¡F« ãfœ¢Áia A v‹f. n A 20=^ h

.P An S

n A

3020= =^

^^

hhh

ešy gH§fŸ »il¡F« ãfœ¢Áia B v‹f. ( ) 2n B 2=

( )( )( )

P Bn Sn B

3022= = .

ešy M¥ÃŸ gH§fë‹ v©â¡if, ( )n A B 15+ = .

( )( )

( )P A B

n Sn A B

3015+

+= = .


nj®ªbjL¡f¥gL« xU gH« M¥Ãshfnth mšyJ ešy gHkhfnth ÏU¡f

ãfœjfÎ,

( ) ( ) ( ) ( )P A B P A P B P A B, += + -

= .3020

3022

3015

109+ - =

9. xU tF¥Ãš cŸs khzt®fëš 40% ng® fâj édho édh ãfœ¢ÁæY«,

30% ng® m¿éaš édho édh ãfœ¢ÁæY«, 10% ng® m›éu©L édho

édh ãfœ¢ÁfëY« fyªJ bfh©ld®. m›tF¥ÃèUªJ rkthŒ¥ò Kiwæš

xU khzt‹ nj®ªbjL¡f¥g£lhš, mt® fâj édho édh ãfœ¢Áænyh

mšyJ m¿éaš édho édh ãfœ¢Áænyh mšyJ ÏU ãfœ¢ÁfëYnkh

fyªJ bfh©lj‰fhd ãfœjfÎ fh©f.

Ô®Î: TWbtë S v‹f. M k‰W« S1 v‹gd Kiwna nj®ªbjL¡f¥gL«

khzt‹ fâj édho édh k‰W« m¿éaš édho édhéš g§nf‰F«

ãfœ¢ÁfŸ v‹f.

( )n S 100= , n M^ h = 40, ( )n S 401=

( )P M10040= , ( )P S

10030

1=

( ) 10n M S1

+ = . ( )P M S10010

1+ = .

Mfnt, njitahd ãfœjfÎ ( )P M S1

, = ( ) ( ) ( )P M P S P M S1 1

++ -

= 10040

10030

10010+ -

10060

53= =

10. e‹F fiy¤J mL¡» it¡f¥g£l52 Ó£Lfis¡ bfh©l Ó£L¡ f£oèUªJ

rkthŒ¥ò Kiwæš xU Ó£L vL¡f¥gL»wJ. mªj¢ Ó£L °nglhfnth (Spade)

mšyJ Ïuhrhthfnth (King) ÏU¥gj‰fhd ãfœjféid¡ fh©f.

Ô®Î: TWbtë ( )n S 52= .

A v‹gJ °ngL Ó£L »il¡F« ãfœ¢Á v‹f. ( )n A 13= .

( )( )( )

P An Sn A

5213= =

B v‹gJ Ïuhrh Ó£L »il¡F« ãfœ¢Á v‹f. ( )n B 4= .

( )( )( )

P Bn Sn B

524= =

°ngL Ïuhrh Ó£Lfë‹ v©â¡if ( )n A B+ = 1

( )P A B+ = ( )

( )n S

n A B521+

=

vdnt,njitahd ãfœjfÎ ( )P A B, = ( ) ( ) ( )P A P B P A B++ -

= .5213

524

521

134+ - =


11. xU igæš 10 btŸis, 6 Át¥ò k‰W« 10 fU¥ò ãw¥ gªJfŸ cŸsd. rkthŒ¥ò

Kiwæš xU gªÂid vL¡F«nghJ mJ btŸis mšyJ Át¥ò ãw¥ gªjhf

ÏU¥gj‰fhd ãfœjféid¡ fh©f.

Ô®Î: TWbtë S v‹f. W, R k‰W« B v‹gd Kiwna btŸis, Át¥ò k‰W«

fU¥ò gªJfŸ »il¡F« ãfœ¢ÁfŸ v‹f.

( )n S 26= .

( )n W 10= , ( )( )( )

P Wn Sn W

2610= =

( )n R 6= , ( )( )( )

P Rn Sn R

266= =

( )n B 10= , ( )( )( )

P Bn Sn B

2610= = .

Ï§F Ïitæu©L« x‹iwbah‹W éy¡F« ãfœ¢ÁfŸ, ( )P W R 0+ = .

vdnt,njitahd ãfœjfÎ ( )P W R, = ( ) ( )P W P R+

= .2610

266

138+ =

12. 2, 5, 9 v‹w v©fis¡ bfh©L, X® Ïu©oy¡f v© mik¡f¥gL»wJ. mªj

v© 2 mšyJ 5 Mš tFgLkhW mika ãfœjfÎ fh©f.

(mik¡f¥gL« v©âš xnu Ïy¡f« Û©L« tuyh«)

Ô®Î: TWbtë S v‹f. A k‰W« B v‹gd Kiwna mªj v© 2 k‰W« 5 Mš

tFgLkhW mikÍ« ãfœ¢ÁfŸ v‹f.

{ , , , , , , , , }S 22 25 29 55 52 59 99 92 95= ; ( ) 9n S =

A = {22, 52, 92}; ( ) 3n A = .

( ) .P A93=

{25, 55, 95}B = ; ( ) 3n B = .

( ) .P B93=

A B+ Q= k‰W« ( )n A B 0+ = ., ( ) .P A B 0+ =

vdnt, ( ) ( ) ( ) ( )P A B P A P B P A B, += + -

= 093

93+ - .

32=

13. “ACCOMMODATION” v‹w brhšè‹ x›bthU vG¤J« jå¤jåna

Á¿a fh»j§fëš vGj¥g£L, mªj 13 Á¿a fh»j§fS« xU Kfitæš

it¡f¥g£LŸsd. rkthŒ¥ò Kiwæš KfitæèUªJ xU fh»j¤ij¤

nj®Î brŒÍ« nghJ, mÂš Ïl« bgW« vG¤J

(i) ‘A’ mšyJ ‘O’ Mfnth

(ii) ‘M’ mšyJ ‘C’ Mfnth ÏU¥gj‰fhd ãfœjfÎfis¡ fh©f.


Ô®Î: TWbtë S v‹f. S v‹gJ 13 fh»j J©Lfis¡ bfh©LŸsJ.

( )n S 13=

(i) H v‹gJ‘A’ v‹w vG¤JŸs fh»j¤ij nj®Î brŒÍ« ãfœ¢Á v‹f.

( )n H 2= .

( )( )( )

P Hn Sn H

132= = .

B v‹gJ‘O’ v‹w vG¤JŸs fh»j¤ij nj®Î brŒÍ« ãfœ¢Á v‹f. ( )n B 3= .

( )( )( )

P Bn Sn B

133= = .

vdnt, njitahd ãfœjfÎ ( ) ( ) ( )P H B P H P B, = + ( )H Ba + Q=

132

133

135= + =

(ii) C v‹gJ‘M’ v‹w vG¤JŸs fh»j¤ij nj®Î brŒÍ« ãfœ¢Á v‹f.

( )n C 2= .

( )( )( )

P Cn Sn C

132= = .

D v‹gJ‘C’ v‹w vG¤JŸs fh»j¤ij nj®Î brŒÍ« ãfœ¢Á v‹f.

( )n D 2= .

( )P D = ( )( )n Sn D

132= .

vdnt,njitahd ãfœjfÎ ( )P C D, = ( ) ( )P C P D+ ( )C Da + Q=

= .132

132

134+ =

14. xU òÂa k»œÎªJ (car) mjDila totik¥Ã‰fhf éUJ bgW« ãfœjfÎ

0.25 v‹f. Áwªj Kiwæš vçbghUŸ ga‹gh£o‰fhd éUJ bgW« ãfœjfÎ

0.35 k‰W« ÏU éUJfS« bgWtj‰fhd ãfœjfÎ 0.15 våš, m«k»œÎªJ

(i) FiwªjJ VjhtJ xU éUJ bgWjš

(ii) xnu xU éUJ k£L« bgWjš M»a ãfœ¢ÁfS¡fhd ãfœjfÎfis¡

fh©f.

Ô®Î: A v‹gJ totik¥Ã‰fhf éUJ bgW« ãfœ¢Á k‰W« B v‹gJ Áwªj

Kiwæš vçbghUŸ ga‹gh£o‰fhd éUJ bgW« ãfœ¢ÁfŸ v‹f.

( ) .P A 0 25= , ( ) .P B 0 35= k‰W« ( ) .P A B 0 15+ =

(i) FiwªjJ VjhtJ xU éUJ bgw ãfœjfÎ ,

( ) ( ) ( ) ( )P A B P A P B P A B, += + - . . .0 25 0 35 0 15= + - .0 45=

(ii) xnu xU éUJ k£L« bgw ãfœjfÎ,

( ) ( )P A B P A B+ ++ [ ( ) ( )] [ ( ) ( )]P A P A B P B P A B+ += - + -

( . . ) ( . . )0 25 0 15 0 35 0 15= - + - . . .0 10 0 20 0 3= + =


15. A, B, C M»nah® xU édhé‰F¤ Ô®Î fh©gj‰fhd ãfœjfÎfŸ Kiwna

, ,54

32

73 v‹f A k‰W« B ÏUtU« nr®ªJ Ô®Î fh©gj‰fhd ãfœjfÎ

158

B k‰W« C ÏUtU« nr®ªJ Ô®Î fh©gj‰fhd ãfœjfÎ 72 . A k‰W« C ÏUtU«

nr®ªJ Ô®Î fhz ãfœjfÎ 3512 , _tU« nr®ªJ Ô®Î fhz ãfœjfÎ

358 våš,

ahnuD« xUt® m›édhé‹ Ô®Î fh©gj‰fhd ãfœjféid¡ fh©f.

Ô®Î: ( ) , ( ) , ( )P A P B P C54

32

73= = = , ( ) ,P A B

158+ = ( ) ,P B C

72+ =

( )P A C3512+ = k‰W« ( )P A B C

358+ + = .

Mfnt, ( )P A B C, , = ( ) ( ) ( ) ( )P A P B P C P A B++ + -

( ) ( ) ( ) .P B C P A C P A B C+ + + +- - +

= 54

32

73

158

72

3512

358+ + - - - + .

105101=

gæ‰Á 12.3


1. Q v‹gJ xU Ïayh ãfœ¢Á våš, ( )P Q =

(A) 1 (B) 41 (C) 0 (D)

21

Ô®Î: Ïayh ãfœ¢Áæ‹ ãfœjfÎ 0. ( éil: (C) )

2. S v‹gJ xU rkthŒ¥ò nrhjidæ‹ TWbtë våš, P (S) =

(A) 0 (B) 81 (C)

21 (D) 1 ( éil: (D) )

Ô®Î: x›bthU ãfœ¢ÁÍ« S ‹ c£fdkhF«.Mfnt ãfœ¢Á S v¥bghGJ«

ãfG«. vdnt, P(S) = 1.

3. A v‹w ãfœ¢Áæ‹ ãfœjfÎ p våš, Ã‹tUtdt‰¿š p vij ãiwÎ brŒÍ«

(A) p0 11 1 (B) p0 1# # (C) p0 11# (D) p0 11 #

( éil: (B) ) 4. A k‰W« B v‹gd VnjD« ÏU ãfœ¢ÁfŸ. nkY« S v‹gJ rkthŒ¥ò¢

nrhjidæ‹ TWbtë våš, ( )P A B+ =

(A) ( ) ( )P B P A B+- (B) ( ) ( )P A B P B+ -

(C) ( )P S (D) P A B, l^ h6 @ ( éil: (A) )


5. xU khzt‹ fâj¤Âš 100 kÂ¥bg© bgWtj‰fhd ãfœjfÎ 54 . mt® 100

kÂ¥bg© bgwhkš ÏU¥gj‰fhd ãfœjfÎ

(A) 51 (B)

52 (C)

53 (D)

54

Ô®Î: ( ) 1 ( )P A P A= - 154= -

51= ( éil: (A) )

6. A k‰W« B v‹w ÏU ãfœ¢Áfëš ( ) . , ( ) .P A P B0 25 0 05= = k‰W«

( ) .P A B 0 14+ = våš, ( )P A B, =

(A) 0.61 (B) 0.16 (C) 0.14 (D) 0.6

Ô®Î: ( ) ( ) ( ) ( ) 0.25 0.05 0.14P A B P A P B P A B, += + - = + - .0 16= ( éil: (B) )

7. 20 bghU£fëš 6 bghU£fŸ FiwghLilait. rkthŒ¥ò Kiwæš xU bghUŸ

nj®ªbjL¡F«nghJ mJ Fiwa‰wjhf¡ »il¥gj‰fhd ãfœjfÎ

(A) 107 (B) 0 (C)

103 (D)

32

Ô®Î: Fiwghl‰w bghU£fë‹ v©â¡if 20 6= - 14= .

Fiwghl‰w bghU£fŸ »il¡f ãfœjfÎ2014=

107= . ( éil: (A) )

8. A k‰W« B v‹gd Ïu©L x‹iwbah‹W éy¡F« ãfœ¢ÁfŸ v‹f.

mªãfœ¢Áæ‹ TWbtë S, ( ) ( )P A P B31= k‰W« S A B,= våš, ( )P A =

(A) 41 (B)

21 (C)

43 (D)

83

Ô®Î: ( ) ( ) ( )P A B P A P B, = + (A , B x‹iwbah‹W éy¡F« ãfœ¢ÁfŸ )

( ) ( ) ( )P S P A P A3= + 4 ( ) .P A 1& = ( )P A41= . ( éil: (A) )

9. A, B k‰W« C v‹gdx‹iwbah‹W éy¡F« _‹W ãfœ¢ÁfŸ v‹f. mt‰¿‹

ãfœjfÎfŸ Kiwna , k‰W«31

41

125 våš, P A B C, ,^ h =

(A) 1219 (B)

1211 (C)

127 (D) 1

Ô®Î: ( ) ( ) ( ) ( )P A B C P A P B P C, , = + + 31

41

125 1= + + = . ( éil: (D) )

10. ( ) . , ( ) . , ( ) .P A P B P A B0 25 0 50 0 14+= = = våš P(A Í« mšy B Í« mšy) =

(A) 0.39 (B) 0.25 (C) 0.11 (D) 0.24

Ô®Î: ( ) 0.25 0.50 0.14P A B, = + - .0 61=

( ) 1 ( )P A B P A B+ ,= - .1 0 61= - .0 39= . ( éil: (A) )


11. xU igæš 5 fU¥ò, 4 btŸis k‰W« 3 Át¥ò ãw¥ gªJfŸ cŸsd. rkthŒ¥ò

Kiwæšnj®ªbjL¡f¥gL« xU gªJ Át¥ò ãwkhf ÏšyhkèU¥gj‰fhd ãfœjfÎ.

(A) 125 (B)

124 (C)

123 (D)

43

Ô®Î: ( ) 1 ( )P R P R= - 1123

43= - = . ( éil: (D) )

12. xnu neu¤Âš ÏU gfilfŸ cU£l¥gL»‹wd. gfilæ‹ Ïu©L Kf§fëY«

xnu v©zhf ÏU¡f ãfœjfÎ

(A) 361 (B)

31 (C)

61 (D)

32

Ô®Î: ( )n S 36= . ÏU gfilfëY« xnu v© »il¡F« ãfœ¢Á A v‹f. ( )n A 6=

{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} ; ( ) .A n A 6= = ( ) .P A366

61= =

( éil: (C) )

13. xU Óuhd gfil xU Kiw cU£l¥gL«nghJ »il¡F« v© gfh v© mšyJ

gF v©zhf ÏU¥gj‰fhd ãfœjfÎ

(A) 1 (B) 0 (C) 65 (D)

61

Ô®Î: S = {1, 2, 3, 4, 5, 6}. 1 v‹gJ gF v©Q« mšy gfh v©Q« mšy,

vdnt,njitahd ãfœjfÎ = 65 . ( éil: (C) )

14. xU ehza¤ij _‹W Kiw R©L« nrhjidæš 3 jiyfŸ mšyJ 3 ó¡fŸ

»il¡f ãfœjfÎ

(A) 81 (B)

41 (C)

83 (D)

21

Ô®Î: S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.

n (S) = 8. vdnt,njitahd ãfœjfÎ = {HHH, TTT} = 82

41= ( éil: (B) )

15. 52 Ó£LfŸ bfh©l xU Ó£L¡f£oèUªJ xU Ó£L vL¡F« nghJ mJ xU V°

(ace)Mf ÏšyhkY« k‰W« xU Ïuhrhthf (king) ÏšyhkèU¥gj‰fhd ãfœjfÎ.

(A) 132 (B)

1311 (C)

134 (D)

138

Ô®Î: n(V° ) = 4; n(Ïuhrh) = 4; n(V° k‰W« Ïuhrh Ïšyhkš) = 52 – 8 = 44

P (V° k‰W« Ïuhrh Ïšyhkš) = 5244

1311= ( éil: (B) )


16. xU be£lh©oš (Leap year) 53 btŸë¡»HikfŸ mšyJ 53 rå¡»HikfŸ

tUtj‰fhd ãfœjfÎ

(A) 72 (B)

71 (C)

74 (D)

73

Ô®Î: be£lh©oš 52 thu§fŸ k‰W« 2 eh£fŸ cŸsJ.

S = { (PhæW, Â§fŸ), (Â§fŸ, br›thŒ), (br›thŒ, òj‹), (òj‹,éahH‹), (éahH‹,btŸë), (btŸë, rå), (rå, PhæW)}

n(S) = 7. vdnt, njitahd ãfœjfÎ = 72

72

71

73+ - = . ( éil: (D) )

17. xU rhjuz tUlkhdJ 53 Phæ‰W¡»HikfŸ k‰W« 53 Â§f£»HikfŸ

bfh©oU¥gj‰fhd ãfœjfÎ.

(A) 71 (B)

72 (C)

73 (D) 0

Ô®Î: rhjuz tUl« 52 thu§fŸ k‰W« 1 ehŸ bfh©LŸsJ. vdnt, 53

Phæ‰W¡»HikfŸ k‰W« 53 Â§f£»HikfŸ bfh©oU¡f thŒ¥Ãšiy. vdnt,njitahd ãfœjfÎ = 0. ( éil: (D) )

18. 52 Ó£LfŸ bfh©l xU Ó£L¡f£oèUªJ xU Ó£L vL¡F«nghJ, mJ Ah®£

muÁahf (Heart queen) ÏU¥gj‰fhd ãfœjfÎ.

(A) 521 (B)

5216 (C)

131 (D)

261

Ô®Î: 52 Ó£Lfëš Ah®£ muÁ 1 cŸsJ. n(A) = 1, n(S) = 52, P(A) = 521

( éil: (A) ) 19. xU cWÂ ãfœ¢Áæ‹ ãfœjfÎ

(A) 1 (B) 0 (C) 100 (D) 0.1

Ô®Î: cWÂahd ãfœ¢Áæ‹ ãfœjfÎ 1. ( éil: (A) )

20. xU rkthŒ¥ò¢ nrhjidæ‹ KothdJ bt‰¿ahfnth mšyJ njhšéahfnth

ÏU¡F«. m¢nrhjidæš bt‰¿ bgWtj‰fhd ãfœjfÎ njhšé¡fhd

ãfœjféid¥ nghš ÏU kl§F våš, bt‰¿ bgWtj‰fhd ãfœjfÎ

(A) 31 (B)

32 (C) 1 (D) 0

Ô®Î: bt‰¿ bgWtj‰fhd ãfœjfÎ p k‰W« njhšé¡fhd ãfœjfÎ q v‹f.

1 2 .k‰W«p q p q p32&+ = = = ( éil: (B) )


tif¥gL¤j¥g£l édh¡fŸ

1. fz§fS« rh®òfS«

Ïu©L kÂ¥bg© édh¡fŸ

v.fh. 1.1 { 10,0,1, 9, 2, 4, 5} { 1, 2, 5, 6, 2,3,4}k‰W«A B= - = - - v‹w

fz§fS¡F Ã‹tUtdt‰iw rçgh®¡fÎ«.

(x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) fz§fë‹ nr®¥ò gçkh‰W¥ g©ò cilaJ

(ii) fz§fë‹ bt£L gçkh‰W¥ g©ò cilaJ

v.fh. 1.2 {1, 2, 3, 4, 5}, {3, 4, 5, 6}, {5, 6, 7, 8}A B C= = = våš,

(i) A B C A B C, , , ,=^ ^h h vd¡fh£Lf.

v.fh. 1.3 { , , , }, { , , } { , }k‰W«A a b c d B a c e C a e= = = våš,

(i) A B C+ +^ h = A B C+ +^ h vd¡ fh£Lf.

gæ‰Á 1.1

1. A B1 våš, A B B, = vd¡ fh£Lf (bt‹gl¤ij¥ ga‹gL¤jÎ«).

2. A B1 våš, A B+ k‰W« \A B M»at‰iw¡ fh©f. (bt‹gl¤ij¥ ga‹gL¤Jf).

3. { , , }, { , , , } { , , , }k‰W«P a b c Q g h x y R a e f s= = = våš, Ã‹tUtdt‰iw¡ fh©f.

(iii) \R P Q+^ h.

4. {4,6,7,8,9}, {2,4,6} {1,2,3,4,5,6}k‰W«A B C= = = våš,

Ã‹tUtdt‰iw¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ) (i) A B C, +^ h (ii) A B C+ ,^ h (iii) \ \A C B^ h

5. { , , , , }, {1,3,5,7, 10}A a x y r s B= = - vd¡ bfhL¡f¥g£LŸs fz§fS¡F,

fz§fë‹ nr®¥ò brayhdJ, gçkh‰W¥ g©ò cilaJ v‹gij rçgh®¡fÎ«.

6. { , , , , 2, 3, 4, 7}A l m n o= k‰W« {2, 5, 3, 2, , , , }B m n o p= - M»at‰¿‰F

fz§fë‹ bt£L, gçkh‰W¥ g©ò cilaJ v‹gij rçgh®¡fÎ«.

gæ‰Á 1.2

1. ÑnH bfhL¡f¥g£LŸs fz§fis, bt‹gl§fë‹ _y« fh£Lf.


(i) {5,6,7,8, ,13}, {5,8,10,11} {5,6,7,9,10}k‰W«U A Bg= = =

(ii) { , , , , , , , }, { , , , } { , , , , }k‰W«U a b c d e f g h M b d f g N a b d e g= = =

3. , k‰W«A B C M»a _‹W fz§fS¡F Ã‹ tUtdt‰iw és¡F« bt‹gl§fŸ

tiuf. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) A B C+ + (ii) A k‰W« B v‹gd C -æ‹ c£fz§fŸ. nkY« mitfŸ bt£lh¡ fz§fŸ.

tif¥gL¤j¥g£l édh¡fŸ - fz§fS« rh®òfS« 303

(iii) \A B C+ ^ h (iv) \B C A,^ h (v) A B C, +^ h

(vi) \C B A+ ^ h (vii) C B A+ ,^ h

5. U = { , , , , , , }4 8 12 16 20 24 28 , A= { , , }8 16 24 k‰W« B= { , , , }4 16 20 28 våš, ( ) ' ( )k‰W«i iiA B A B, + l^ ^h h M»at‰iw¡ fh©f.


gæ‰Á 1.3

1. A, B v‹gd ÏU fz§fŸ k‰W« U v‹gJ mid¤J¡ fz« v‹f. nkY« 700n U =^ h , 200, 300 100 ,k‰W« våšn A n B n A B n A B+ += = = l l^ ^ ^ ^h h h hI¡ fh©f.

2. 285, 195, 500 410 ,k‰W« våšn A n B n U n A B n A B, ,= = = = l l^ ^ ^ ^ ^h h h h h- I¡

fh©f. 3. A, B k‰W« C VnjD« _‹W fz§fŸ v‹f. nkY«,

n A 17=^ h , 17, 17, 7n B n C n A B+= = =^ ^ ^h h h , ( ) 6n B C+ = , 5n A C+ =^ h

k‰W« 2n A B C+ + =^ h våš, n A B C, ,^ h - I¡ fh©f.

v.fh. 1.14 { , , , }A 1 2 3 4= k‰W« { , , , , , , , , , , }B 1 2 3 4 5 6 7 9 10 11 12= - v‹f. R = {(1, 3), (2, 6), (3, 10), (4, 9)} A B#3 xU cwÎ våš, R I xU rh®ò

vd¡ fh£Lf. mj‹ kÂ¥gf«, Jiz kÂ¥gf« k‰W« Å¢rf« M»adt‰iw¡

fh©f.

v.fh. 1.16 X = { 1, 2, 3, 4 } v‹f. Ã‹tU« x›bthU cwÎ«, X-èUªJ X -¡F xU

rh®ghFkh vd MuhŒf. c‹ éil¡F V‰w és¡f« jUf.


(i) f = { (2, 3), (1, 4), (2, 1), (3, 2), (4, 4) } (ii) g = { (3, 1), (4, 2), (2, 1) } (iii) h = { (2, 1), (3, 4), (1, 4), (4, 3) }v.fh. 1.17 A = { 1, 4, 9, 16 }-èUªJ B = { –1, 2, –3, –4, 5, 6 }-¡F Ã‹tU« cwÎfëš

vit rh®ghF«? m›thW rh®ò våš, mj‹ Å¢rf¤ij¡ fh©f.


(i) f1 = { (1, –1), (4, 2), (9, –3), (16, –4) }

(ii) f2 = { (1, –4), (1, –1), (9, –3), (16, 2) }

(iii) f3 = { (4, 2), (1, 2), (9, 2), (16, 2) }

(iv) f4 = { (1, 2), (4, 5), (9, –4), (16, 5) }

v.fh. 1.18 , 0

, 0

vD«nghJ

vD«nghJx

x x

x x 1

$=

-) ,

{ ( ,x y) | y = | x |, x R! } v‹w cwÎ, rh®ig tiuaW¡»wjh? mj‹ Å¢rf«

fh©f.

v.fh. 1.19 F¤J¡nfhL nrhjidia¥ ga‹gL¤Â Ã‹tU« tiugl§fëš vit

rh®Ãid¡ F¿¡F« vd¤ Ô®khå¡fÎ«. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)


v.fh. 1.21 A = { 1, 2, 3, 4, 5 }, B = N k‰W« :f A B" MdJ ( )f x x2

= vd

tiuaW¡f¥g£LŸsJ f -‹ Å¢rf¤ij¡ fh©f. nkY«, rh®Ã‹ tifia¡

fh©f.

gæ‰Á 1.4

1. Ã‹tU« m«ò¡F¿¥ gl§fŸ rh®ig¡ F¿¡»‹wdth vd¡ TWf. c‹ éil¡F¤

jFªj fhuz« TWf. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

2. bfhL¡f¥g£LŸs F = { (1, 3), (2, 5), (4, 7), (5, 9), (3, 1) } vD« rh®Ã‰F, kÂ¥gf«

k‰W« Å¢rf« M»at‰iw¡ fh©f.

3. A = { 10, 11, 12, 13, 14 }; B = { 0, 1, 2, 3, 5 } k‰W« :f A Bi " , i = 1,2,3. v‹f. ÑnH bfhL¡f¥g£LŸsit v›tif¢ rh®Ãid¡ F¿¡F«? éil¡fhd jFªj

fhuz« jUf. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) f1 = { (10, 1), (11, 2), (12, 3), (13, 5), (14, 3) }

(ii) f2 = { (10, 1), (11, 1), (12, 1), (13, 1), (14, 1) }

(iii) f3 = { (10, 0), (11, 1), (12, 2), (13, 3), (14, 5) }


4. X = { 1, 2, 3, 4, 5 }, Y = { 1, 3, 5, 7, 9 } v‹f. X-èUªJ Y-¡fhd cwÎfŸ ÑnH

bfhL¡f¥g£LŸsd. Ït‰¿š vit rh®ghF«? c‹ éil¡fhd jFªj fhuz«

jUf. nkY«, mit rh®ò våš, v›tif¢ rh®ghF«?


(i) R1 = { ,x y^ h|y x 2= + , x X! , y Y! }

(ii) R2 = { (1, 1), (2, 1), (3, 3), (4, 3), (5, 5) }

(iii) R3 = { (1, 1), (1, 3), (3, 5), (3, 7), (5, 7) }

(iv) R4 = { (1, 3), (2, 5), (4, 7), (5, 9), (3, 1) }

5. R {( , 2), ( 5, ), (8, ), ( , 1)}a b c d= - - - v‹gJ rkå¢ rh®ig¡ F¿¡Fbkåš, , ,a b c k‰W« d M»at‰¿‹ kÂ¥òfis¡ fh©f.

6. A = { –2, –1, 1, 2 } k‰W« , :f xx

x A1 != ` j$ . våš, f -‹ Å¢rf¤ij¡ fh©f. nkY«, f v‹gJ A-æèUªJ A-¡F xU rh®ghFkh?

8. f = { (12, 2), (13, 3), (15, 3), (14, 2), (17, 17) } v‹w rh®Ãš 2 k‰W« 3 M»at‰¿‹

K‹cU¡fis¡ fh©f.

9. ÑnH bfhL¡f¥g£LŸs m£ltiz MdJ, A= { 5, 6, 8, 10 }-æèUªJ

B = { 19, 15, 9, 11 }-¡F f x^ h = x2 1- v‹wthW mikªj xU rh®ò våš, a k‰W« b M»adt‰¿‹ kÂ¥òfis¡ fh©f?

x 5 6 8 10

f(x) a 11 b 19

11. ÑnH bfhL¡f¥g£LŸs tiugl§fëš vit rh®Ãid¡ F¿¡»‹wd? éil¡fhd

jFªj fhuz« jUf. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)


12. bfhL¡f¥g£LŸs rh®ò f = { (–1, 2), (– 3, 1), (–5, 6), (– 4, 3) } I (i) m£ltiz (ii) m«ò¡F¿ gl« M»at‰¿‹ _y« F¿¡fÎ«.

15. rh®ò f : ,3 7- h6 "R Ñœ¡ f©lthW tiuaW¡f¥g£LŸsJ.

f x^ h = ;

;

;

x x

x x

x x

4 1 3 2

3 2 2 4

2 3 4 7

2 1

1 1

#

# #

- -

-

-

*

Ã‹tUtdt‰iw¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) f f5 6+^ ^h h (ii) f f1 3- -^ ^h h (iii) f f2 4- -^ ^h h 16. rh®ò f : ,7 6- h6 "R Ñœ¡ f©lthW tiuaW¡f¥g£LŸsJ.

( )f x = ;

;

;

x x x

x x

x x

2 1 7 5

5 5 2

1 2 6

2 1

1 1

#

# #

+ + - -

+ -

-

*

Ã‹tUtdt‰iw¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) ( ) ( )f f2 4 3 2- + (ii) ( ) ( )f f7 3- - -

IªJ kÂ¥bg© édh¡fŸ

v.fh. 1.2 {1, 2, 3, 4, 5}, {3, 4, 5, 6}, {5, 6, 7, 8}A B C= = = våš,

(ii) bt‹gl§fis¥ ga‹gL¤Â A B C A B C, , , ,=^ ^h h v‹gij rçgh®¡fÎ«.

v.fh. 1.3 { , , , }, { , , } { , }k‰W«A a b c d B a c e C a e= = = våš,

(ii) bt‹gl§fis¥ ga‹gL¤Â A B C+ +^ h = A B C+ +^ h vd rçgh®¡fÎ«.

v.fh. 1.4 { , , , , }, { , , , , }, { , , , }A a b c d e B a e i o u C c d e u= = = vd¡ bfhL¡f¥g£LŸsJ

våš, (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) \ \ \ \A B C A B C!^ ^h h vd¡ fh£Lf.

(ii) bt‹gl§fis¥ ga‹gL¤Â \ \ \ \A B C A B C!^ ^h h vd rçgh®¡fÎ«.

v.fh. 1.5 {0,1,2,3,4}, {1, 2, 3,4,5,6} {2,4,6,7}k‰W«A B C= = - = v‹f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) A B C, +^ h = A B A C, + ,^ ^h h vd¡fh£Lf. (ii) bt‹gl§fis¥ ga‹gL¤Â A B C, +^ h = A B A C, + ,^ ^h h vd

rçgh®¡fÎ«.

v.fh. 1.6 { 3 4, } , { 5, }A x x x B x x xR N1 1; # ! ; != - = k‰W«

{ 5, 3, 1,0,1,3}C = - - - våš, A B C A B A C+ , + , +=^ ^ ^h h h vd¡ fh£Lf.

gæ‰Á 1.1

7. A= { 42v‹gJx x; -‹ gfh¡ fhuâ}, { 5 12, }B x x x N1; # != k‰W«

C = { , , , }1 4 5 6 våš, A B C A B C, , , ,=^ ^h h v‹gij rçgh®¡fÎ«.


8. { , , , , }, { , , , , } { , , , }k‰W«P a b c d e Q a e i o u R a c e g= = = M»a fz§fë‹ bt£L,

nr®¥ò¥ g©ò cilaJ v‹gij rçgh®¡fÎ«.

9. {5,10,15, 20}, {6,10,12,18,24} {7,10,12,14,21,28}k‰W«A B C= = =

M»a fz§fS¡F \ \ \ \A B C A B C=^ ^h h v‹gJ bkŒahFkh vd MuhŒf.

c‹ éil¡F j¡f fhuz« TWf.

10. { 5, 3, 2, 1}, { 2, 1,0} { 6, 4, 2}k‰W«A B C= - - - - = - - = - - - v‹f.

\ \ ( \ ) \k‰W«A B C A B C^ h M»at‰iw¡ fh©f. ÏÂèUªJ »il¡F« fz

é¤Âahr¢ brašgh£o‹ g©Ãid¡ TWf.

11. { 3, 1, 0, 4,6,8,10}, { 1, 2, 3,4,5,6} { 1, 2,3,4,5,7}k‰W«A B C= - - = - - = -

M»at‰¿‰F Ã‹tUtdt‰iw rçgh®¡fÎ«. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) A B C, +^ h= A B A C, + ,^ ^h h

(ii) A B C+ ,^ h= A B A C+ , +^ ^h h

(iii) bt‹gl§fis¥ ga‹gL¤Â A B C, +^ h= A B A C, + ,^ ^h h

(iv) A B C+ ,^ h= A B A C+ , +^ ^h h M»adt‰iw rçgh®¡fÎ«.

v.fh. 1.7 bt‹gl§fis¥ ga‹gL¤Â A B A B+ ,=l l l^ h v‹gij¢ rçgh®¡f.

v.fh. 1.8 bt‹gl§fis¥ ga‹gL¤Â \ \ \A B C A B A C+ ,=^ ^ ^h h h v‹D«

o kh®få‹ fz é¤Âahr éÂæid¢ rçgh®¡fÎ«.

v.fh. 1.9 { 2, 1, 0,1, 2, 3, ,10}, { 2, 2,3,4,5}U Ag= - - = - k‰W« {1,3,5, ,9}B 8= v‹f. o kh®få‹ fz ãu¥Ã éÂfis¢ rçgh®¡fÎ«.

v.fh. 1.10 A = { , , , , , , , , , }a b c d e f g x y z , B = { , , , , }c d e1 2 k‰W« C = { , , , , , }d e f g y2 v‹f. \ \ \A B C A B A C, +=^ ^ ^h h h v‹gij rçgh®¡fÎ«.

gæ‰Á 1.2

4. bt‹gl§fis¥ ga‹gL¤Â \A B A B A+ , =^ ^h h v‹gij¢ rçgh®¡fÎ«.

6. U = { , , , , , , , }a b c d e f g h , { , , , } { , , }k‰W«A a b f g B a b c= = våš, o kh®få‹ fz

ãu¥Ã éÂfis¢ rçgh®¡fÎ«. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

7. Ã‹tU« fz§fis¡ bfh©L o kh®få‹ fz é¤Âahr éÂfis¢ rçgh®¡fÎ«. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

{1, 3, 5, 7, 9,11,13,15}, {1, 2, 5, 7} {3,9,10,12,13}k‰W«A B C= = = .

8. A = {10,15, 20, 25, 30, 35, 40, 45, }50 , B = { , ,10,15, 20, 30}1 5 k‰W«

C = { , ,15,2 ,35,45, }7 8 0 48 M»a fz§fS¡F \ \ \A B C A B A C+ ,=^ ^ ^h h h v‹gij¢ rçgh®¡fÎ«.

9. bt‹gl§fis¥ ga‹gL¤Â Ã‹tUtdt‰iw¢ rçah vd¢ nrhÂ¤J¥ gh®¡fÎ«.

(x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)


(i) A B C, +^ h = A B A C, + ,^ ^h h (ii) A B C+ ,^ h = A B A C+ , +^ ^h h

(iii) A B, l^ h = A B+l l (iv) \A B C,^ h = \ \A B A C+^ ^h h

v.fh. 1.11 xU FGéš 65 khzt®fŸ fhšgªJ«, 45 ng® Ah¡»Í«, 42 ng® »ç¡bf£L«

éisahL»wh®fŸ. 20 ng® fhšgªjh£lK« Ah¡»Í«, 25 ng® fhšgªjh£lK«

»ç¡bf£L«, 15 ng® Ah¡»Í« »ç¡bf£L« k‰W« 8 ng® _‹W éiah£LfisÍ«

éisahL»wh®fŸ. m¡FGéš cŸs khzt®fë‹ v©â¡ifia¡ fh©f.

(x›bthU khztD« FiwªjJ xU éisah£oid éisahLth® vd¡

bfhŸf.)

v.fh. 1.12 gšfiy¡fHf khzt®fë‹ fz¡bfL¥Ãš, 64 ng® fâj«, 94 ng® fâ¥bgh¿

m¿éaš, 58 ng® Ïa‰Ãaš M»a ghl§fis¡ f‰»‹wd®. 28 ng® fâjK«

Ïa‰ÃaY«, 26 ng® fâjK« fâ¥bgh¿ m¿éaY«, 22 ng® fâ¥bgh¿

m¿éaY« Ïa‰ÃaY« k‰W« 14 ng® _‹W ghl§fisÍ« f‰»‹wd®.

fz¡bfL¥Ãš fyªJ¡ bfh©l khzt®fë‹ v©â¡ifia¡ fh©f.

nkY«, xU ghl¤ij k£L« f‰»‹w khzt®fë‹ v©â¡ifia¡ fh©f.v.fh. 1.13 xU thbdhè ãiya« 190 khzt®fël« mt®fŸ éU«ò« Ïiræ‹

tiffis¤ Ô®khå¡f xU fz¡bfL¥ò el¤ÂaJ. 114 ng® nk‰f¤Âa

ÏiriaÍ«, 50 ng® »uhäa ÏiriaÍ«, 41 ng® f®ehlf ÏiriaÍ«, 14 ng®

nk‰f¤Âa ÏiriaÍ« »uhäa ÏiriaÍ«, 15 ng® nk‰f¤Âa ÏiriaÍ«

f®ehlf ÏiriaÍ«, 11 ng® f®ehlf ÏiriaÍ« »uhäa ÏiriÍ« k‰W« 5 ng®

Ï«_‹W ÏirfisÍ« éU«ò»‹wd® vd¡ fz¡bfL¥Ãš btë¥g£lJ.

Ï¤jftšfëèUªJ Ã‹tUtdt‰iw¡ fh©f.

(i) _‹W tif ÏirfisÍ« éU«ghj khzt®fë‹ v©â¡if. (ii) ÏU tif Ïirfis k£L« éU«ò« khzt®fë‹ v©â¡if. (iii) »uhäa Ïiria éU«Ã nk‰f¤Âa Ïiria éU«ghj khzt®fë‹

v©â¡if.

gæ‰Á 1.3

4. Ã‹tU« fz§fS¡F n A B C, , =^ h n A n B n C n A B++ + - -^ ^ ^ ^h h h h n B C n A C n A B C+ + + +- +^ ^ ^h h h

v‹gij rçgh®¡fÎ«. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) {4,5,6}, {5,6,7,8} {6,7,8,9}k‰W«A B C= = =

(ii) { , , , , }, { , , } { , , }k‰W«A a b c d e B x y z C a e x= = = .

5. xU fšÿçæš nrUtj‰F 60 khzt®fŸ ntÂæaèY«, 40 ng® Ïa‰ÃaèY«,

30 ng® cæçaèY« gÂÎ brŒJŸsd®. 15 ng® ntÂæaèY« Ïa‰ÃaèY«,10

ng® Ïa‰ÃaèY« cæçaèY« k‰W« 5 ng® cæçaèY« ntÂæaèY« gÂÎ

brŒJŸsd®. _‹W ghl§fëY« xUtUnk gÂÎ brŒaéšiy våš, VnjD« xU

ghl¤Â‰fhtJ gÂÎ brŒJŸst®fë‹ v©â¡if ahJ? 6. xU efu¤Âš 85% ng® jäœ bkhê, 40% ng® M§»y bkhê k‰W« 20% ng® ÏªÂ

bkhê ngR»wh®fŸ. 32% ng® jäG« M§»yK«, 13% ng® jäG« ÏªÂÍ« k‰W«

10% ng® M§»yK« ÏªÂÍ« ngR»wh®fŸ våš, _‹W bkhêfisÍ« ngr¤

bjçªjt®fë‹ rjÅj¤Âid¡ fh©f.




ãWtd« f©l¿ªjJ. nkY«, 85ng® bjhiy¡fh£Á k‰W« g¤Âç¡ifiaÍ«, 75 ng®

bjhiy¡fh£Á k‰W« thbdhèiaÍ«, 95 ng® thbdhè k‰W« g¤Âç¡ifiaÍ«,


étu§fis¢ F¿¤J, Ã‹tUtdt‰iw¡ fh©f.

(i) thbdhèia k£L« ga‹gL¤Jgt®fë‹ v©â¡if.

(ii) bjhiy¡fh£Áia k£L« ga‹gL¤Jgt®fë‹ v©â¡if.


ga‹gL¤jhjt®fë‹ v©â¡if.

8. 4000 khzt®fŸ gæY« xU gŸëæš , 2000 ngU¡F ÃbuŠR, 3000 ngU¡F¤ jäœ k‰W«

500 ngU¡F ÏªÂ bjçÍ«. nkY«, 1500 ngU¡F ÃbuŠR k‰W« jäœ, 300 ngU¡F

ÃbuŠR k‰W« ÏªÂ, 200 ngU¡F jäœ k‰W« ÏªÂ, 50 ngU¡F Ï«_‹W bkhêfS«

bjçÍ« våš, Ã‹tUtdt‰iw¡ fh©f.

(i) _‹W bkhêfS« bjçahjt®fë‹ v©â¡if. (ii) VnjD« xU bkhêahtJ bjçªjt®fë‹ v©â¡if.

(iii) ÏU bkhêfŸ k£Lnk bjçªjt®fë‹ v©â¡if.

9. 120 FL«g§fŸ cŸs xU »uhk¤Âš 93 FL«g§fŸ rikaš brŒtj‰F éwif¥

ga‹gL¤J»‹wd®. 63 FL«g§fŸ k©bz©bzæid¥ ga‹gL¤J»wh®fŸ.

45 FL«g§fŸ rikaš vçthÍit¥ ga‹gL¤J»wh®fŸ. 45 FL«g§fŸ éwF k‰W«

k©bz©bzŒ, 24 FL«g§fŸ k©bz©bzŒ k‰W« vçthÍ, 27 FL«g§fŸ

vçthÍ k‰W« éwF M»at‰iw¥ ga‹gL¤J»‹wd®. éwF, k©bz©bzŒ k‰W«

rikaš vçthÍ Ï«_‹iwÍ« ga‹gL¤J« FL«g§fë‹ v©â¡ifia¡ fh©f.

v.fh. 1.20 A= { 0, 1, 2, 3 } k‰W« B = { 1, 3, 5, 7, 9 } v‹gd ÏU fz§fŸ v‹f. :f A B" v‹D« rh®ò ( )f x x2 1= + vd¡ bfhL¡f¥g£LŸsJ. Ï¢rh®Ãid

(i) tçir¢ nrhofë‹ fz« (ii) m£ltiz (iii) m«ò¡F¿¥ gl« (iv) tiugl«

M»at‰whš F¿¡f.

v.fh. 1.22 rh®ò : [1, 6)f R$ MdJ Ã‹tUkhW tiuaW¡f¥g£LŸsJ.

,

,

,

f x

x x

x x

x x

1 1 2

2 1 2 4

3 10 4 62

1

1

1

#

#

#

=

+

-

-

^ h * ( [1 , 6) = { :1 6x xR 1! #" ,)

(i) ( )f 5 (ii) f 3^ h (iii) f 1^ h (iv) f f2 4-^ ^h h (v) 2 3f f5 1-^ ^h h M»at‰¿‹ kÂ¥òfis¡ fh©f.

gæ‰Á 1.4

7. f = { (2, 7), (3, 4), (7, 9), (–1, 6), (0, 2), (5, 3) } v‹gJ

A = { –1, 0, 2, 3, 5, 7 } -æèUªJ B = { 2, 3, 4, 6, 7, 9 } -¡F xU rh®ò v‹f. f v‹w rh®ò

(i) x‹W¡F x‹whd rh®ghFkh? (ii) nkš rh®ghFkh?

(iii) x‹W¡F x‹whd k‰W« nkš rh®ghFkh?


10. A = { 5, 6, 7, 8 }; B = { –11, 4, 7, –10,–7, –9,–13 } v‹f.

f = {( ,x y) : y = x3 2- , x A! , y B! } vd tiuaW¡f¥g£LŸsJ.

(i) f -‹ cW¥òfis vGJf (ii) mj‹ Jiz kÂ¥gf« ahJ? (iii) Å¢rf« fh©f (iv) v›tif¢ rh®ò vd¡ fh©f.

13. A = { 6, 9, 15, 18, 21 }; B = { 1, 2, 4, 5, 6 } k‰W« :f A B" v‹gJ

f x^ h = x33- vd tiuaW¡f¥g£oU¥Ã‹ rh®ò f -I

(i) m«ò¡F¿ gl« (ii) tçir¢ nrhofë‹ fz«

(iii) m£ltiz (iv) tiugl« M»at‰¿‹ _y« F¿¡fÎ«.

14. A = {4, 6, 8, 10 } k‰W« B = { 3, 4, 5, 6, 7 } v‹f. :f A B" v‹gJ f x x21 1= +^ h

vd tiuaW¡f¥g£LŸsJ. rh®ò f -I

(i) m«ò¡F¿ gl« (ii) tçir¢ nrhofë‹ fz« (iii) m£ltiz M»at‰¿‹ _y«

F¿¡fÎ«.


f x^ h = ;

;

;

x x

x x

x x

4 1 3 2

3 2 2 4

2 3 4 7

2 1

1 1

#

# #

- -

-

-

*

Ã‹tUtdt‰iw¡ fh©f. (iv) ( ) ( )

( ) ( )f f

f f2 6 13 1

-+ - .


( )f x = ;

;

;

x x x

x x

x x

2 1 7 5

5 5 2

1 2 6

2 1

1 1

#

# #

+ + - -

+ -

-

*

Ã‹tUtdt‰iw¡ fh©f. (iii) ( ) ( )( ) ( )

f ff f

6 3 14 3 2 4

- -- +

gl¤Â‹ _y« ã%gz«

1 2 3( )

nn n

21#g+ + + + =

+ v‹w thŒgh£oid gl¤Â‹ _ykhf és¡Fnth«

vdnt, 1 2 3 92

9 10#g+ + + + =

tif¥gL¤j¥g£l édh¡fŸ - bkŒba©fë‹ bjhl®tçirfS« bjhl®fS« 311

2. bkŒba©fë‹ bjhl®tçirfS« bjhl®fS«


v.fh. 2.1 -MtJ cW¥ò bfhL¡f¥g£LŸs Ã‹tU« bjhl®tçiræ‹ Kjš _‹W

cW¥òfis¡ fh©f.

cn n n

6

1 2 1n=

+ +^ ^h h , n N6 !

v.fh. 2.2 Ã‹tU« bjhl®tçirfë‹ Kjš IªJ cW¥òfis fh©f.


(i) 1,a1=- , 1a

n

an

2nn 1 2=+- k‰W« n N6 !

(ii) 1F F1 2= = k‰W« , 3,4, .F F F n

n n n1 2g= + =

- -

gæ‰Á 2.1

1. n-MtJ cW¥ò bfhL¡f¥g£l Ã‹tU« bjhl®tçir x›bth‹¿Y« Kjš

_‹W cW¥òfis¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) a n n

3

2n=

-^ h (ii) 3c 1n

n n 2= -

+^ h (iii) z n n

4

1 2n

n

=- +^ ^h h

2. x›bthU bjhl®tçiræ‹ n-MtJ cW¥ò ÑnH bfhL¡f¥g£LŸsJ. mit

x›bth‹¿Y« F¿¥Ãl¥g£LŸs cW¥òfis¡ fh©f.


(i) ; ,ann a a2 3

2n 7 9=

++ (ii) 2 ; ,a n a a1 1

n

n n 3

5 8= - +

+^ ^h h

(iii) 2 3 1; ,a n n a a.n

2

5 7= - + (iv) ( 1) (1 ); ,a n n a an

n 2

5 8= - - +

3. ( ),

,

k‰W« Ïu£il v©

k‰W« x‰iw v©

¥gil vD«nghJ

¥gil vD«nghJa

n n n n

n

n n n

3

1

2N

Nn 2

!

!=

+

+*

vd tiuaW¡f¥g£l bjhl®tçiræ‹ 18- tJ k‰W« 25-tJ cW¥òfis¡ fh©f.

4. ,

( 2),

k‰W« Ïu£il v©k‰W« x‰iw v©

¥gil vD«nghJ¥gil vD«nghJ

bn n n

n n n n

N

Nn

2!

!=

+)

vd tiuaW¡f¥g£l bjhl®tçiræ‹ 13 MtJ k‰W« 16MtJ cW¥òfis¡ fh©f.

5. 2, 3a a a1 2 1= = + k‰W« 2 5, 2,a a n

n n 12= +

- vd¡ bfh©l¤ bjhl®tçiræ‹


6. 1a a a1 2 3= = = k‰W« a a a

n n n1 2= +

- -, n 32 , vd¡ bfh©l¤ bjhl®tçiræ‹



v.fh. 2.2 Ã‹tUtdt‰WŸ vit¡ T£L¤ bjhl®tçiræš (A.P.) cŸsJ?


(i) , , ,32

54

76 g . (ii) , , , .m m m3 1 3 3 3 5 g- - -

v.fh. 2.4 Ã‹tU« T£L¤ bjhl®tçirfë‹ Kjš cW¥ò k‰W« bghJ

é¤Âahr¤ij¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) 5, 2, 1, 4,g- - . (ii) , , , , ,21

65

67

23

617g

v.fh. 2.5 20,19 ,18 ,41

21 g v‹w T£L¤ bjhl®tçiræš cW¥ò tn xU Fiw v©zhf

mika n-‹ äf¢Á¿a äif KG kÂ¥ò ahJ?

v.fh. 2.6 xU óªnjh£l¤Âš Kjš tçiræš 23 nuh#h¢ brofŸ, Ïu©lh« tçiræš 21

nuh#h¢ brofŸ _‹wh« tçiræš 19 nuh#h¢ brofŸ v‹w Kiwæš nuh#h¢

brofŸ xU bjhl®tçirmik¥Ãš cŸsd. filÁ tçiræš 5 nuh#h¢

brofŸ ÏU¥Ã‹, m¥óªnjh£l¤Âš v¤jid tçirfŸ cŸsd?

v.fh. 2.7 2010-š xUt® M©L CÂa« 30,000 vd¥ gâæš nrU»wh®. nkY« x›bthU

tUlK« `600-I M©L CÂa ca®thf¥ bgW»wh®. mtUila M©L CÂa«

vªj tUl¤Âš `39,000-Mf ÏU¡F«?

v.fh. 2.8 _‹W v©fë‹ é»j« 2 : 5 : 7 v‹f. Kjyh« v©, Ïu©lh« v©âèUªJ

7-I¡ fê¤J¥ bgw¥gL« v© k‰W« _‹wh« v© M»ad xU T£L¤

bjhl®tçiria V‰gL¤Âdhš, m›bt©fis¡ fh©f.

gæ‰Á 2.2

1. xU T£L¤ bjhl®tçiræ‹ Kjš cW¥ò 6 k‰W« bghJ é¤Âahr« 5 våš,

m¤bjhl®tçirÍ«, mj‹ bghJ cW¥igÍ« fh©f.

2. 125, 120, 115, 110, g v‹w T£L¤ bjhl®tçiræ‹ bghJ é¤Âahr¤ijÍ«

15 MtJ cW¥igÍ« fh©f.

3. 24, 23 , 22 , 21 ,41

21

43 g v‹w T£L¤ bjhl® tçiræš 3 v‹gJ v¤jidahtJ

cW¥ò MF«?

4. , 3 , 5 ,2 2 2 g v‹w T£L¤ bjhl®tçiræ‹ 12 MtJ cW¥ò ahJ?

5. 4, 9, 14, g v‹w T£L¤ bjhl®tçiræ‹ 17 MtJ cW¥ig¡ fh©f.

6. Ã‹tU« T£L¤ bjhl®tçiræš cŸs bkh¤j cW¥òfis¡ fh©f.


(i) 1, , , , .65

32

310g- - - (ii) 7, 13, 19, g , 205.

10. 13Mš tFgL« <çy¡f äif KG v©fë‹ v©â¡ifia¡ fh©f.


12. xUt®, Kjš khj« `640, 2M« khj« `720, 3M« khj« `800-I nrä¡»wh®. mt®

j‹Dila nrä¥ig Ïnj bjhl®tçiræš bjhl®ªjhš, 25MtJ khj« mt®

nrä¡F« bjhifia¡ fh©f.

16. xUt® tUl¤Â‰F jåt£o 14% jU« KjÄ£oš 25,000-I KjÄL brŒjh®. x›bthU

tUl KoéY« »il¡F« mrš k‰W« jåt£o nr®ªj bkh¤j¤ bjhif xU T£L¤

bjhl®tçiria mik¡Fkh? m›thbwåš, 20 M©LfS¡F¥ ÃwF KjÄ£oš cŸs

bjhifia¡ fh©f.

17. a, b, c M»ad T£L¤ bjhl®tçiræš ÏU¥Ã‹ ( ) 4( )a c b ac2 2

- = - vd ãWÎf.

18. a, b, c M»ad T£L¤ bjhl®tçiræš ÏU¥Ã‹ , ,bc ca ab1 1 1 M»ad xU T£L¤

bjhl®tçiræš ÏU¡F« vd ãWÎf.

v.fh. 2.9 Ã‹tUtdt‰¿š vit bgU¡F¤ bjhl®tçir MF«?


a) (i) 5, 10, 15, 20, g . (ii) 0.15, 0.015, 0.0015, g .

b) (i) 5, 10, 15, 20, g . (iii) , , 3 , 3 , .7 21 7 21 g

c) (ii) 0.15, 0.015, 0.0015, g . (iii) , , 3 , 3 , .7 21 7 21 g

v.fh. 2.10 Ã‹tU« bgU¡F¤ bjhl®tçirfë‹ bghJ é»j¤ijÍ« k‰W« mj‹ bghJ

cW¥igÍ« fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) , , ,52

256

12518 g . (ii) 0.02, 0.006, 0.0018, g .

gæ‰Á 2.3

2. , ,1, 2,41

21 g- - v‹w bgU¡F¤ bjhl®tçiræš 10 MtJ cW¥igÍ«, bghJ

é»j¤ijÍ« fh©f.

3. xU bgU¡F¤ bjhl®tçiræš 4MtJ k‰W« 7 MtJ cW¥òfŸ Kiwna 54 k‰W«

1458 våš, m¤bjhl®tçiria¡ fh©f.

4. xU bgU¡F¤ bjhl®tçiræš Kjš k‰W« MwhtJ cW¥òfŸ Kiwna 31 ,

7291 våš,

m¥bgU¡F¤ bjhl®tçiria¡ fh©f.

5. Ã‹tU« bgU¡F¤ bjhl® tçiræš bfhL¡f¥g£l cW¥ò v¤jidahtJ cW¥ò

vd¡ fh©f.

(i) 5, 2, , ,54

258 g -š

15625128 v‹w cW¥ò (ii) 1, 2, 4, 8,g -š 1024 v‹w cW¥ò

7. xU bgU¡F¤ bjhl®tçiræ‹ Kjš cW¥ò 3 k‰W« IªjhtJ cW¥ò 1875 våš, mj‹

bghJ é»j« fh©f.

12. xUt® M©o‰F 5% T£L t£o jU« xU t§»æš 1000-I it¥ò ãÂahf it¤jhš, 12 M« tUlKoéš »il¡F« bkh¤j¤ bjhifia¡ fh©f.


13. xU ãWtd« 50,000-¡F xU m¢R¥ÃuÂ ÏaªÂu¤ij th§F»wJ. m›éaªÂu«

x›bthU M©L« j‹ kÂ¥Ãš 45% ÏH¡»wJ vd kÂ¥Ãl¥gL»wJ.

15 M©LfS¡F¥ ÃwF mªj m¢R¥ÃuÂ ÏaªÂu¤Â‹ kÂ¥ò v‹d?

v.fh. 2.16 5 11 17 95g+ + + + v‹w T£L¤ bjhlç‹ TLjš fh©f.

gæ‰Á 2.4

1. Ã‹tUtdt‰¿‹ TLjš fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) Kjš 75 äif KG¡fŸ (ii) Kjš 125 Ïaš v©fŸ

2. n MtJ cW¥ò n3 2+ v‹wthW mikªj xU T£L¤ bjhl®tçiræ‹ Kjš 30

cW¥òfë‹ T£l‰gyid¡ fh©f.

3. Ã‹tU« T£L¤bjhl®fë‹ T£l‰gyid¡ fh©f.


(i) 38 35 32 2g+ + + + . (ii) 6 5 4 2541

21 g+ + + cW¥òfŸ tiu.

4. Ã‹tU« étu§fis¡ bfh©l T£L¤ bjhl®fë‹ TLjš Sn fh©f.


(i) ,a 5= ,n 30= l 121= (ii) ,a 50= ,n 25= d 4=-

7. 60, 56, 52, 48,g v‹w T£L¤ bjhl®tçiræ‹ Kjš cW¥ÃèUªJ bjhl®¢Áahf

v¤jid cW¥òfis¡ T£odhš TLjš 368 »il¡F«?

13. xU f£Lkhd FGk«, xU ghy¤ij f£o Ko¡f¤ jhkjkhF« nghJ, jhkjkhF«

x›bthU ehS¡F« mguhj¤bjhif¡ f£lnt©L«. Kjš ehŸ jhkj¤Â‰F

mguhj« `4000 nkY« mL¤JtU« x›bthU ehS¡F« Kªija ehis él `1000

mÂf« brY¤j nt©oæU¡F«. tuÎ bryÎ¤Â£l¤Â‹go m¡FGk« bkh¤j

mguhj¤bjhifahf `1,65,000 brY¤j ÏaY« våš, v¤jid eh£fS¡F ghy«

Ko¡F« gâia jhkj¥gL¤jyh«?

14. 8% Åj« jåt£o jU« ãWtd¤Âš x›bthU M©L« `1000 it¥ò¤ bjhifahf

brY¤j¥gL»wJ. x›bthU M©o‹ ÏWÂæš bgW« t£oia¡ fz¡»Lf. bgW«

t£o¤bjhiffŸ xU T£L¤ bjhl®tçiria mik¡Fkh? m›thW mik¡f

KoÍkhdhš, 30 M©Lfë‹ Koéš »il¡F« bkh¤j t£oia¡ fh©f.

15. xU bjhlç‹ Kjš n cW¥òfë‹ TLjš 3 2n n2- våš, m¤bjhluhdJ xU T£L¤

bjhl® vd ãWÎf.

16. xU fofhu« xU kâ¡F xU Kiw, 2 kâ¡F ÏU Kiw, 3 kâ¡F _‹W Kiw

v‹wthW, bjhl®ªJ rçahf x›bthU kâ¡F« xè vG¥ò« våš, xU ehëš

m¡fofhu« v¤jid Kiw xè vG¥ò«?

v.fh. 2.22 16 48 144 432 g- + - + v‹w bgU¡F¤ bjhlçš cŸs Kjš 25

cW¥òfë‹ TLjiy¡ fh©f.


v.fh. 2.23 Ã‹tU« étu§fis¡ bfh©l x›bthU bgU¡F¤ bjhlU¡F« Sn I¡ fh©f.


(i) ,a 2= t6 = 486, n = 6 (ii) a = 2400, r = – 3, n = 5

v.fh. 2.28 xU bjh©L ãWtd« xU efu¤ÂYŸs 25 ÅÂfëš ku¡f‹Wfis

eL«bghU£L, Kjš ÅÂæš xU ku¡f‹W«, Ïu©lh« ÅÂæš ÏU ku¡f‹WfŸ,

_‹wh« ÅÂæš 4 ku¡f‹WfŸ, eh‹fhtJ ÅÂæš 8 ku¡f‹WfŸ v‹w Kiwæš

eLtj‰F Â£läL»wJ. m›ntiyia Ko¡f¤ njitahd ku¡f‹Wfë‹

v©â¡ifia¡ fh©f.

gæ‰Á 2.5

1. 25

65


2. 91

271


3. Ã‹tU« étu§fis¡ bfh©l bgU¡F¤ bjhlç‹ TLjš Sn fh©f.


(i) ,a 3= 384,t8= n 8= . (ii) ,a 5= r 3= , n 12= .

5. Ã‹tU« bjhl®fëš, v¤jid cW¥òfis¡ T£odhš


(i) 3 9 27 g+ + + TLjš 1092 »il¡F«? (ii) 2 6 18 g+ + + TLjš 728 »il¡F« ?

6. xU bgU¡F¤ bjhlçš Ïu©lhtJ cW¥ò 3 k‰W« mj‹ bghJ é»j« 54 våš, T£L¤

bjhlçYŸs Kjš cW¥ÃèUªJ bjhl®¢Áahf 23 cW¥òfë‹ TLjš fh©f.

v.fh. 2.29 Ã‹tU« bjhl®fë‹ TLjš fh©f.


(i) 26 27 28 60g+ + + + (iii) 31 33 53.g+ + +

v.fh. 2.30 Ã‹tU« bjhlç‹ TLjiy¡ fh©f. (i) 1 2 3 252 2 2 2

g+ + + +

v.fh. 2.31 Ã‹tU« bjhl®fë‹ TLjiy¡ fh©f. (i) 1 2 3 203 3 3 3

g+ + + +

v.fh. 2.33 (ii) 1 2 3 36100n3 3 3 3

g+ + + + = våš, 1 2 3 ng+ + + + -‹ kÂ¥ig¡

fh©f

gæ‰Á 2.6

1. Ã‹tU« bjhl®fë‹ TLjiy¡ fh©f.


(i) 1 + 2 + 3 + g + 45 (iii) 2 + 4 + 6 + g + 100 (iv) 7 + 14 +21 g + 490


2. Ã‹tUtdt‰¿‰F k-‹ kÂ¥ò¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) 1 2 3 6084k3 3 3 3

g+ + + + = (ii) 1 2 3 2025k3 3 3 3

g+ + + + =

3. 1 2 3 171pg+ + + + = våš, 1 2 3 p3 3 3 3

g+ + + + -æ‹ kÂ¥ig¡ fh©f.


gæ‰Á 2.2

7. xU T£L¤ bjhl®tçiræ‹ 9 MtJ cW¥ò ó¢Áa« våš, 19 MtJ cW¥Ã‹ ÏUkl§F

29 MtJ cW¥ò vd ã%Ã.

8. xU T£L¤ bjhl®tçiræš 10 k‰W« 18 MtJ cW¥òfŸ Kiwna 41 k‰W« 73 våš,

27 MtJ cW¥ig¡ fh©f.

9. 1, 7, 13, 19,g k‰W« 100, 95, 90, g M»a T£L¤ bjhl®tçirfë‹ n MtJ cW¥ò

rkbkåš, n-‹ kÂ¥ig¡ fh©f.

11. xU bjhiy¡fh£Á¥ bg£o¤ jahç¥ghs® VHhtJ M©oš 1000 bjhiy¡fh£Á¥

bg£ofisÍ«, g¤jhtJ M©oš 1450 bjhiy¡fh£Á¥ bg£ofisÍ« jahç¤jh®.

x›bthU M©L« jahç¡F« bjhiy¡fh£Á¥ bg£ofë‹ v©â¡if ÓuhfÎ«

xU kh¿è v© msÎ« mÂfç¤jhš, Kjyh« M©oY«, 15 MtJ M©oY«

jahç¡f¥g£l bjhiy¡fh£Á¥ bg£ofë‹ v©â¡ifia¡ fh©f.

13. xU T£L¤ bjhl®tçiræš mL¤jL¤j _‹W cW¥òfë‹ TLjš 6 k‰W« mt‰¿‹

bgU¡F¤ bjhif –120 våš, m«_‹W v©fis¡ fh©f.


m›ÎW¥òfë‹ t®¡f§fë‹ TLjš 140 våš, m«_‹W v©fis¡ fh©f.

15. xU T£L¤ bjhl®tçiræ‹ m-MtJ cW¥Ã‹ m kl§F mj‹ n-MtJ cW¥Ã‹ n kl§F¡F¢ rkbkåš, m¡T£L¤ bjhltçiræ‹ (m+n)-MtJ cW¥ò ó¢Áa«

vd¡fh£Lf.

19. , ,a b c2 2 2 M»ad T£L¤ bjhl®tçiræš ÏU¥Ã‹ , ,

b c c a a b1 1 1+ + +

M»adÎ«

T£L¤ bjhl®tçiræ‹ ÏU¡F« vd¡fh£Lf.

20. , 0, 0, 0a b c x y zx y z

! ! != = k‰W« b ac2= våš, , ,

x y z1 1 1 M»ad xU T£L¤

bjhl®tçiræš ÏU¡F« vd¡ fh£Lf.

v.fh. 2.11 xU bgU¡F¤ bjhl®tçiræ‹ eh‹fhtJ cW¥ò 32 k‰W« mj‹ VHhtJ

cW¥ò 8116 våš, m¥bgU¡F¤ bjhl®tçiria¡ fh©f.

v.fh. 2.12 xU E©Qæ® gçnrhjidæš x›bthU kâ neuK« gh¡Oçah¡fë‹

v©â¡if Ïu£o¥gh»wJ. gçnrhjidæ‹ bjhl¡f¤Âš 30 gh¡Oçah¡fŸ

ÏUªjd. 14 MtJ kâ neu Koéš cŸs gh¡Oçah¡fë‹ v©â¡ifia¡

fh©f.


v.fh. 2.13 M©L¡F 10% Åj« T£L t£o më¡F« xU t§»æš, xUt® `500-I it¥ò¤

bjhifahf brY¤J»wh®. 10 M©L Koéš mtU¡F¡ »il¡F« bkh¤j

bjhif v›tsÎ?

v.fh. 2.13 xU bgU¡F¤ bjhl®tçiræ‹ Kjš _‹W cW¥òfë‹ TLjš 1213 k‰W«

mt‰¿‹ bgU¡f‰gy‹ -1 våš, bghJ é»j¤ijÍ« nkY« m›ÎW¥òfisÍ«

fh©f.

v.fh. 2.15 , , ,a b c d v‹gd xU bgU¡F¤ bjhl®tçiræš cŸsd våš

b c c a d b a d2 2 2 2- + - + - = -^ ^ ^ ^h h h h vd ãWÎf.

gæ‰Á 2.3

6. 162, 54, 18,g k‰W« , , ,812

272

92 g M»a bgU¡F¤ bjhl®tçirfë‹ n MtJ

cW¥ò rkbkåš, n-‹ kÂ¥ò fh©f.


mt‰¿‹ bgU¡f‰gy‹ 1 våš, m¤bjhl®tçiræ‹ bghJ é»j¤ijÍ«, m«_‹W

cW¥ò¡fisÍ« fh©f.

9. xU bgU¡F¤ bjhl®tçiræš mL¤jL¤j 3 cW¥òfë‹ bgU¡F¤ bjhif 216 k‰W«

mitfëš Ïu©ou©L cW¥ò¡fë‹ bgU¡f‰gy‹fë‹ TLjš 156 våš,

mªj cW¥òfis¡ fh©f.


mt‰¿‹ jiyÑêfë‹ TLjš 47 våš, m›ÎW¥òfis¡ fh©f.

11. xU bgU¡F¤ bjhl®tçiræš Kjš _‹W cW¥òfë‹ TLjš 13 k‰W« mt‰¿‹

t®¡f§fë‹ TLjš 91 våš, m¤bjhl®tçiria¡ fh©f.

14. , , ,a b c d M»ad xU bgU¡F¤ bjhl®tçiræš mikªjhš,

a b c b c d ab bc cd- + + + = + +^ ^h h vd¡fh£Lf.

15. , , ,a b c d M»ad xU bgU¡F¤ bjhl®tçiræš mikªjhš , ,a b b c c d+ + + v‹gitÍ« bgU¡F¤ bjhl®tçiræš mikÍ« vd ãWÎf.

v.fh. 2.17 1 2 3 4 ...2 2 2 2- + - + v‹w bjhlç‹ Kjš 2n cW¥òfë‹ TLjš fh©f.

v.fh. 2.18 xU T£L¤ bjhlçš Kjš 14 cW¥òfë‹ TLjš 203- k‰W« mL¤j 11 cW¥òfë‹ TLjš –572 våš, m¤bjhliu¡ fh©f.

v.fh. 2.19 24 21 18 15 g+ + + + v‹w T£L¤ bjhlçš bjhl®¢Áahf v¤jid

cW¥òfis¡ T£odhš TLjš –351 »il¡F«?

v.fh. 2.20 8 Mš tFgL« mid¤J _‹¿y¡f Ïaš v©fë‹ TLjš fh©f.


v.fh. 2.21 xU gynfhz¤Â‹ c£nfhz§fë‹ msÎfis tçir¥go vL¤J¡

bfh©lhš, mit xU T£L¤ bjhl®tçiria mik¡»‹wd. m¡T£L¤

bjhl®tçiræš äf¡ Fiwªj nfhzmsÎ 85c k‰W« äf ca®ªj nfhz

msÎ 215c våš, mªj¥ gynfhz¤Â‹ g¡f§fë‹ v©â¡ifia¡ fh©f.

gæ‰Á 2.4

5. 1 2 3 42 2 2 2

g- + - + v‹w bjhlç‹ Kjš 40 cW¥òfë‹ T£l‰gyid¡ fh©f.

6. xU T£L¤ bjhlçš Kjš 11 cW¥òfë‹ TLjš 44 k‰W« mj‹ mL¤j

11 cW¥òfë‹ TLjš 55 våš, m¤bjhliu¡ fh©f.

8. 9 Mš tFgL« mid¤J _‹¿y¡f Ïaš v©fë‹ TLjš fh©f.

9. xU T£L¤ bjhlç‹ 3 MtJ cW¥ò 7 k‰W« mj‹ 7 MtJ cW¥ghdJ 3 MtJ

cW¥Ã‹ _‹W kl§if él 2 mÂf«. m¤bjhlç‹ Kjš 20 cW¥òfë‹


10. 300-¡F« 500-¡F« ÏilnaÍŸs 11 Mš tFgL« mid¤J Ïaš v©fë‹ T£l‰gy‹

fh©f.

11. 1 6 11 16 148xg+ + + + + = våš, x-‹ kÂ¥Ãid¡ fh©f.

12. 100-¡F« 200-¡F« ÏilnaÍŸs 5 Mš tFglhj mid¤J Ïaš v©fë‹


17. Kjš cW¥ò a, Ïu©lh« cW¥ò b k‰W« filÁ cW¥ò c vd¡ bfh©l xU T£L¤

bjhlç‹ T£l‰gy‹ b a

a c b c a

2

2

-

+ + -

^^ ^

hh h vd¡fh£Lf.

18. xU T£L¤ bjhlçš n2 1+^ h cW¥òfŸ ÏU¥Ã‹ x‰iw¥gil cW¥òfë‹

T£l‰gyD¡F«, Ïu£il¥gil cW¥òfë‹ T£l‰gyD¡F« ÏilnaÍŸs é»j« :n n1+^ h vd ãWÎf.

19. xU T£L¤ bjhlçš Kjš m cW¥òfë‹ T£l‰gyD¡F«, Kjš n cW¥òfë‹

T£l‰gyD¡F« ÏilnaÍŸs é»j« :m n2 2 våš, m MtJ cW¥ò k‰W« n MtJ

cW¥ò M»aitfŸ :m n2 1 2 1- -^ ^h h v‹w é»j¤Âš mikÍ« vd¡ fh£Lf.

20. xU njh£l¡fhu® rçtf toéš Rt® x‹¿id mik¡f Â£läL»wh®. rçtf¤Â‹

Ú©l Kjš tçir¡F 97 br§f‰fŸ njit¥gL»wJ. Ã‹ò x›bthU tçiræ‹

ÏUòwK« Ïu©ou©L br§f‰fŸ Fiwthf it¡f nt©L«. m›totik¥Ãš

25 tçirfëU¥Ã‹, mt® th§f nt©oa br§f‰fë‹ v©â¡if v¤jid?

v.fh. 2.24 2 4 8 g+ + + v‹w bgU¡F¤ bjhlçš, Kjš cW¥ÃèUªJ bjhl®¢Áahf

v¤jid cW¥òfis¡ T£odhš, TLjš 1022 »il¡F«?


v.fh. 2.25 xU bgU¡F¤ bjhlç‹ Kjš cW¥ò 375 k‰W« mj‹ 4 MtJ cW¥ò 192 våš,

mj‹ bghJ é»j¤ijÍ«, Kjš 14 cW¥òfë‹ TLjiyÍ« fh©f.

v.fh. 2.26 bghJ é»j« äif v©zhf ÏU¡F« xU bgU¡F¤ bjhlçš 4 cW¥òfŸ

cŸsd. Kjš Ïu©L cW¥òfë‹ TLjš 8 k‰W« mj‹ filÁ Ïu©L

cW¥òfë‹ TLjš 72 våš, m¤bjhliu¡ fh©f.

v.fh. 2.27 6 + 66 + 666 +g vD« bjhlçš Kjš n cW¥òfë‹ TLjš fh©f.

gæ‰Á 2.5

4. Ã‹tU« KoÎW bjhl®fë‹ TLjš fh©f.


(i) 1 0.1 0.01 0.001 .0 1 9g+ + + + + ^ h (ii) 1 11 111 g+ + + 20 cW¥òfŸ tiu.

7. bghJ é»j« äif v©zhf cŸs xU bgU¡F¤ bjhlçš 4 cW¥òfŸ cŸsd. mj‹

Kjš Ïu©L cW¥òfë‹ TLjš 9 k‰W« filÁ Ïu©L cW¥òfë‹ TLjš 36

våš, m¤bjhliu¡ fh©f.

8. Ã‹tU« bjhl®fë‹ Kjš n cW¥òfë‹ TLjš fh©f.


(i) 7 77 777 g+ + + . (ii) 0.4 0.94 0.994 g+ + + . 9. bjh‰WnehŒ guÎ« fhy¤Âš, Kjš thu¤Âš 5 ngU¡F clšey¡ FiwÎ V‰g£lJ.

clšey¡FiwÎ‰w x›bthUtU« Ïu©lhtJ thu ÏWÂæš 4 ngU¡F m¤bjh‰W

nehia¥ gu¥òt®. Ï›tifæš bjh‰WnehŒ guédhš 15 MtJ thu ÏWÂæš

v¤jid ng®, m¤bjh‰Wnehædhš ghÂ¡f¥ggLt®?

10. e‰gâ brŒj xU ÁWtD¡F¥ gçrë¡f éU«Ãa njh£l¡fhu® Áy kh«gH§fis gçrhf

më¡f K‹tªjh®. m¢ÁWt‹ cldoahf 1000 kh«gH§fis¥ bg‰W¡ bfhŸsyh«

mšyJ Kjš ehëš 1 kh«gH«, Ïu©lh« ehëš 2 kh«gH§fŸ, _‹wh« ehëš

4 kh«gH§fŸ, eh‹fh« ehëš 8 kh«gH§fŸ g vDkhW 10 eh£fS¡F¥ bg‰W¡

bfhŸsyh« vd ÏU thŒ¥òfŸ më¤jh®. m¢ÁWt‹ mÂf v©â¡ifÍŸs

kh«gH§fis¥ bgw vªj thŒ¥Ãid nj®ªbjL¡f nt©L«?

11. Ïu£il¥gil v©â¡ifæš cW¥òfis¡ bfh©l xU bgU¡F¤ bjhlç‹ x‰iw¥

gil v©fshš F¿¡f¥gL« cW¥òfë‹ TLjè‹ _‹W kl§F, m¥bgU¡F¤

bjhlçYŸs mid¤J cW¥òfë‹ TLjY¡F¢ rkbkåš, mj‹ bghJ é»j¤ij¡

fh©f.

12. xU bgU¡F¤ bjhlç‹ Kjš n, 2n k‰W« 3n M»a cW¥òfë‹ TLjšfŸ Kiwna

, k‰W«S S S1 2 3

våš, S S S S S1 3 2 2 1

2- = -^ ^h h vd ãWÎf.

v.fh. 2.29 Ã‹tU« bjhlç‹ TLjš fh©f. (ii) 1 3 5 25g+ + + cW¥òfŸ tiu


v.fh. 2.30 Ã‹tU« bjhl®fë‹ TLjiy¡ fh©f.


(ii) 12 13 14 352 2 2 2

g+ + + + (iii) 1 3 5 512 2 2 2

g+ + + + .

v.fh. 2.31 Ã‹tU« bjhlç‹ TLjiy¡ fh©f. (ii) 11 12 13 283 3 3 3

g+ + + +

v.fh. 2.32 1 2 3 k3 3 3 3

g+ + + + = 4356 våš, k-‹ kÂ¥ig¡ fh©f.

v.fh. 2.34 11 br.Û, 12 br.Û, 13 br.Û, g , 24 br.Û M»adt‰iw Kiwna g¡f msÎfshf¡

bfh©l 14 rJu§fë‹ bkh¤j¥ gu¥ò fh©f.

gæ‰Á 2.6

1. Ã‹tU« bjhl®fë‹ TLjiy¡ fh©f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(ii) 16 17 18 252 2 2 2

g+ + + + (v) 5 7 9 392 2 2 2

g+ + + +

(vi) 16 17 353 3 3

g+ + +

4. 1 2 3 8281k3 3 3 3

g+ + + + = våš, 1 2 3 kg+ + + + -æ‹ kÂ¥ig¡ fh©f.

5. 12 br.Û, 13br.Û, g , 23 br.Û M»adt‰iw Kiwna g¡f msÎfshf¡ bfh©l

12 rJu§fë‹ bkh¤j¥ gu¥gsÎ¡ fh©f.

6. 16 br.Û, 17 br.Û, 18 br.Û, g , 30 br.Û M»adt‰iw Kiwna g¡f msÎfshf¡

bfh©l 15 fd¢rJu§fë‹ fdmsÎfë‹ TLjš fh©f.

gŸë khzt®fS¡fhf cyf mséš fâj¤Âš eilbgW»w ngh£o r®tnjr fâj xè«Ã¡

ngh£oahF« (International Mathematical Olympiad-IMO). 1959 M« M©L Unkåah eh£oš

KjyhtJ IMO eilbg‰wJ. mÂèUªJ x›bthU tUlK« eilbg‰W tU»wJ. 100 ¡F« nk‰g£l

ehLfëèUªJ khzt®fŸ ÏÂš fyªJ bfhŸ»‹wd®. x›bthU eh£oèUªJ« 6 khzt®fŸ

bfh©l FG fyªJ bfhŸsyh«. Ï¥ngh£oæš g§FbgW« khzt®fŸ fâj ÃçÎfëèUªJ

nf£f¥gL« 6 édh¡fS¡F Ô®Î fhz nt©L«. Ï›édh¡fŸ tH¡fkhf gŸëfëš el¤j¥gL«

ghl§fis x£o mikahJ. kÂ¥bg©fŸ mo¥gilæš khzt®fŸ ju¥gL¤j¥gLt®.

e« eh£oš, fâj xè«Ã¡ ngh£o bjhl®ghd brašghLfis njÁa ca®fâj thça« (National Board for Higher Mathematics-NBHM) ftå¤J¡bfhŸ»wJ. eh£oYŸs khzt®fëilna fâj

M‰wiy nk«gL¤Jtnj Ï¥ngh£oæ‹ neh¡fkhF«.

x›bthU M©L« IMO-š fyªJ bfhŸséU¡F« khzt®fis NBHM nj®Î brŒJ mt®fS¡F

njitahd gæ‰ÁæidÍ« më¡»wJ. g‹åu©lh« tF¥ò tiuæš gæY« gŸë khzt®fŸ

xè«Ã¡ ngh£oæš fyªJ bfhŸs jFÂÍilatuht®.

IMO g‰¿a jftšfis http:\\www.imo-official.org v‹w Ïiza jsKftçæš bgwyh«.

tif¥gL¤j¥g£l édh¡fŸ - Ïa‰fâj« 321

3. Ïa‰fâj«


v.fh. 3.1 Ô®: x y3 5- = –16 , x y2 5+ = 31

v.fh. 3.2 11 bg‹ÁšfŸ k‰W« 3 mê¥gh‹fë‹ bkh¤j éiy ` 50. nkY«, 8 bg‹ÁšfŸ

k‰W« 3 mê¥gh‹fë‹ bkh¤j éiy 38 våš, xU bg‹Áš k‰W« xU mê¥gh‹

éiyia¡ fh©f.

v.fh. 3.3 Ú¡fš Kiwæš Ô® : 3x y4+ = –25, x y2 3- = 6

gæ‰Á 3.1

Ú¡fš Kiwia¥ ga‹gL¤Â Ã‹tU« rk‹gh£L¤ bjhF¥òfis¤ Ô®.

1. x y2 7+ = , x y2 1- = 2. x y3 8+ = , x y5 10+ =

3. xy2

4+ = , x y3

2 5+ = 4. x y xy11 7- = , x y xy9 4 6- =

v.fh. 3.6 FW¡F¥ bgU¡fš Kiwia¥ ga‹gL¤Â Ô®¡f.

2x + 7y – 5 = 0 –3x + 8y = –11

v.fh. 3.7 FW¡F bgU¡fš Kiwia¥ ga‹gL¤Â Ã‹tU« bjhF¥Ãid Ô®¡f:

3x + 5y = 25, 7x + 6y = 30

gæ‰Á 3.2

1. FW¡F¥ bgU¡fš Kiwia¥ ga‹gL¤Â Ã‹tU« rk‹ghLfë‹

bjhF¥òfis¤ Ô®¡f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) x y3 4 24+ = , x y20 11 47- = (ii) . . .x y0 5 0 8 0 44+ = , . . .x y0 8 0 6 0 5+ =

(iii) ,x y x y23

35

23 2 6

13- =- + =

v.fh. 3.11 9 20x x2+ + v‹w ÏUgo gšYW¥ò¡nfhitæ‹ ó¢Áa§fis¡ fh©f.

ó¢Áa§fS¡F« bfG¡fS¡F« ÏilnaÍŸs mo¥gil¤ bjhl®òfis¢

rçgh®¡f.

v.fh. 3.12 xU ÏUgo gšYW¥ò¡nfhitæ‹ ó¢Áa§fë‹ TLjš –4 k‰W« mj‹ bgU¡f‰gy‹ 3 våš, m¡nfhitia¡ fh©f.

v.fh. 3.13 x = 41 k‰W« x = –1 v‹w ó¢Áa§fis¡ bfh©l ÏUgo gšYW¥ò¡nfhitia¡

fh©f.


gæ‰Á 3.3

1. Ã‹tU« ÏUgo gšYW¥ò¡nfhitfë‹ ó¢Áa§fis¡ fh©f. ó¢Áa§fS¡F«

bfG¡fS¡F« Ïilna cŸs bjhl®òfis¢ rçgh®¡f.


(i) 2 8x x2- - (ii) 4 4 1x x

2- + (iii) 6 3 7x x

2- - (iv) 4 8x x

2+

(v) 15x2- (vi) 3 5 2x x

2- + (vii) 2 2 1x x2

2- + (viii) 2 143x x

2+ -

2. Ã‹tU« x›bthU ÃçéY« bfhL¡f¥g£LŸs nrho v©fis Kiwna

ó¢Áa§fë‹ TLjyhfÎ« k‰W« mitfë‹ bgU¡f‰gydhfÎ« bfh©l

gšYW¥ò¡nfhitia¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) 3, 1 (ii) 2, 4 (iii) 0, 4 (iv) ,251

(v) ,31 1 (vi) ,

21 4- (vii) ,

31

31- (viii) ,3 2

v.fh. 3.14 7 3x x x3 2+ - - v‹gij x 3- Mš tF¡F« nghJ »il¡F« <Î k‰W« ÛÂ

fh©f.

gæ‰Á 3.4

1. bjhFKiw tF¤jiy¥ ga‹gL¤Â Ã‹tUtdt‰¿‰F <Î k‰W« ÛÂ fh©f.


(i) ( 3 5x x x3 2+ - + ) ' (x 1- ) (ii) (3 2 7 5x x x

3 2- + - ) ' (x 3+ )

(iii) (3 4 10 6x x x3 2+ - + )'( x3 2- ) (iv) (3 4 5x x

3 2- - ) ' ( 1x3 + )

(v) (8 2 6 5x x x4 2- + - )'( 1x4 + ) (vi) (2 7 13 63 48x x x x

4 3 2- - + - )'( 1x2 - )

v.fh. 3.16 (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) 6 11 6x x x3 2- + - v‹w gšYW¥ò¡nfhit¡F x 1- xU fhuâ vd ãWÎf.

(ii) 6 11 6x x x3 2+ + + v‹w gšYW¥ò¡nfhit¡F x 1+ xU fhuâ vd ãWÎf.

v.fh. 3.19 Ã‹tUtdt‰¿‹ Û.bgh.t fh©f: (ii) 15x y z4 3 5 , 12x y z

2 7 2

gæ‰Á 3.6

2. Ã‹tUtdt‰¿‹ Û. bgh. t fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ) (i) c d

2 2- , c c d-^ h (ii) 27x a x

4 3- , x a3 2-^ h

(iii) 3 18m m2- - , 5 6m m

2+ + (iv) 14 33x x

2+ + , 10 11x x x

3 2+ -

(v) 3 2x xy y2 2+ + , 5 6x xy y

2 2+ + (vi) 2 1x x

2- - , 4 8 3x x

2+ +

(x) a a1 35 2- +^ ^h h , a a a2 1 32 3 4- - +^ ^ ^h h h

v.fh. 3.22 Ã‹tUtdt‰¿‹ Û. bgh. k fh©f (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(ii) 35a c b2 3 , 42a cb

3 2 , 30ac b2 3 (iii) a a1 35 2- +^ ^h h , a a a2 1 32 3 4- - +^ ^ ^h h h


gæ‰Á 3.7


4. 66a b c4 2 3 , 44a b c

3 4 2 , 24a b c2 3 4 6. x y xy

2 2+ , x xy

2+

7. a3 1-^ h, 2 a 1 2-^ h , a 12-^ h 9. x x4 32 3+ -^ ^h h , x x x1 4 3 2- + -^ ^ ^h h h

gæ‰Á 3.8


(i) 5 6x x2- + , 4 12x x

2+ - , Ït‰¿‹ Û. bgh. t x 2- .

2. Ã‹tUtdt‰¿š Kiwna p x^ h k‰W« q x^ h M»at‰¿‹ Û. bgh. k, k‰W«

Û. bgh. t nkY« p x^ h M»ad bfhL¡f¥g£LŸsd. q x^ h v‹w k‰bwhU

gšYW¥ò¡nfhitia¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) x x1 22 2+ +^ ^h h , x x1 2+ +^ ^h h, x x1 22+ +^ ^h h. (ii) x x4 5 3 73 3+ -^ ^h h , x x4 5 3 7 2+ -^ ^h h , x x4 5 3 73 2+ -^ ^h h .

v.fh. 3.25 Ã‹tU« é»jKW nfhitfis vëa toé‰F¢ RU¡Ff.


(i) xx

7 285 20

++ (ii)

x x

x x

3 2

53 4

3 2

+

- (iii) x x

x x

9 12 5

6 5 12

2

+ -

- + (iv) x x x

x x x

1 2 3

3 5 42

2

- - -

- - +

^ ^

^ ^

h h

h h

gæ‰Á 3.9

Ã‹tUtdt‰iw vëa toé‰F¢ RU¡Ff. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) x x

x x

3 12

6 92

2

-

+ (ii) x

x

1

14

2

-

+ (iii) x x

x

1

12

3

+ +

-

(iv) x

x

9

272

3

-

- (v) x x

x x

1

12

4 2

+ +

+ + (F¿¥ò: 1x x4 2+ + = x x1

2 2 2+ -^ h vd¡bfhŸf)

(vi) x x

x

4 16

84 2

3

+ +

+ (vii) x x

x x

2 5 3

2 32

2

+ +

+ - (viii) x x

x

9 2 6

2 1622

4

+ -

-^ ^h h

(ix) x x x

x x x

4 2 3

3 5 42

2

- - -

- - +

^ ^

^ ^

h h

h h (x)

x x x

x x x

10 13 40

8 5 502

2

+ - +

- + -

^ ^

^ ^

h h

h h (xi)

x x

x x

8 6 5

4 9 52

2

+ -

+ +

(xii) x x x

x x x x

7 3 2

1 2 9 142

2

- - +

- - - +

^ ^

^ ^ ^

h h

h h h

v.fh. 3.26 (ii) a ab b

a b

22 2

3 3

+ +

+ v‹gij a ba b2 2

-- Mš bgU¡Ff.


(i) x

x

1

4 42-

- v‹gij xx

11

+- Mš tF¡f. (ii)

xx

31

3

+- v‹gij

xx x3 9

12

++ + Mš tF¡f.


gæ‰Á 3.10

1. Ã‹tU« é»jKW nfhitfis¥ bgU¡», éilia¢ RU¡»a toéš vGJf. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) x

x xxx

22

23 6

2

#+-

-+ (ii)

x

x

x x

x x

4

81

5 36

6 82

2

2

2

#-

-

- -

+ +

(iii) x x

x x

x

x x

20

3 10

8

2 42

2

3

2

#- -

- -

+

- +

2. Ã‹tUtd‰iw vëa toéš RU¡Ff:


(i) x

x

x

x1 1

2

2

'+ -

(ii) x

xxx

49

3676

2

2

'-

-++


RU¡Ff: (i) xx

xx

32

21

++ +

-- (ii)

xx

11

2-+

^ h +

x 11+

v.fh. 3.29 x

x

2

12

3

+

- cl‹ vªj é»jKW nfhitia¡ T£odhš x

x x

2

2 32

3 2

+

- + »il¡F«?

gæ‰Á 3.11


nfhitahf) vëa toéš RU¡Ff. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) x

xx2 2

83

-+

- (ii)

x x

x

x x

x

3 2

2

2 3

32 2+ +

+ +- -

-

(iii) x

x x

x x

x x

9

6

12

2 242

2

2

2

-

- - +- -

+ - (iv) x x

x

x x

x

7 10

2

2 15

32 2- +

- +- -

+

(v) x x

x x

x x

x x

3 2

2 5 3

2 3 2

2 7 42

2

2

2

- +

- + -- -

- -

2. x

x

2

12

3

+

- cl‹ vªj é»jKW¡ nfhitia¡ T£l x

x x

2

3 2 42

3 2

+

+ + »il¡F«?

3. vªj é»jKW nfhitia x

x x2 1

4 7 53 2

-- + -èUªJ fê¡f x x2 5 1

2- + »il¡F«?

v.fh. 3.31 t®¡f_y« fh©f. (iii) (2 3 ) 24x y xy2

+ -

v.fh. 3.32 t®¡f_y« fh©f. (ii) 2xx

16

6+ -

gæ‰Á 3.12

1. Ã‹tUtdt‰¿‰F t®¡f_y« fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(iii) 44x x11 2+ -^ h (iv) 4x y xy2- +^ h (v) 121x y8 6 ' 81x y

4 8

(vi) x y a b b c

a b x y b c

25

644 6 10

4 8 6

+ - +

+ - -

^ ^ ^

^ ^ ^

h h h

h h h


2. Ã‹tUtdt‰¿‰F t®¡f_y« fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) 16 24 9x x2- + (iii) 4 9 25 12 30 20x y z xy yz zx

2 2 2+ + - + -

(iv) 2xx

14

4+ +

v.fh. 3.36 Ô®¡f : 6 5 25x x2- - = 0

gæ‰Á 3.14

fhuâ¥gL¤J« Kiwæš ÑnH bfhL¡f¥g£l ÏUgo¢ rk‹ghLfis¤ Ô®¡f.


(i) 81x2 3 2+ -^ h = 0 (ii) 3 5 12x x2- - = 0 (iii) 3x x5 2 5

2+ - = 0

(v) xx

3 8- = 2 (vi) xx1+ =

526 (viii) 1a b x a b x

2 2 2 2 2- + +^ h = 0

(ix) 2 5x x1 12+ - +^ ^h h = 12 (x) 3 5x x4 42- - -^ ^h h = 12

gæ‰Á 3.15

2. ÏUgo¢ N¤Âu¤ij¥ ga‹gL¤Â Ã‹tU« rk‹ghLfis¤ Ô®¡f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) 7 12x x2- + = 0 (ii) 15 11 2x x

2- + = 0

(iii) xx1+ = 2

21 (iv) 3 2a x abx b

2 2 2- - = 0

v.fh. 3.42 xU v© k‰W« mj‹ jiyÑê M»at‰¿‹ TLjš 551 våš, mªj v©iz¡

fh©f.

v.fh. 3.45 ÑnH bfhL¡f¥g£LŸs ÏUgo¢ rk‹ghLfë‹ _y§fë‹ j‹ikia MuhŒf.


(i) 11 10 0x x2- - = (ii) 4 28 49 0x x

2- + = (iii) 2 5 5 0x x

2+ + =

gæ‰Á 3.17

1. rk‹ghLfë‹ _y§fë‹ j‹ikia MuhŒf.


(i) 8 12 0x x2- + = (ii) 2 3 4 0x x

2- + =

(iii) 9 12 4 0x x2+ + = (iv) 3 2 2 0x x6

2- + =

(v) 1 0x x53

322

- + = (vi) x a x b ab2 2 4- - =^ ^h h


kÂ¥òfis¡f©LÃo. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) 2 10 0x x k2- + = (ii) 12 4 3 0x kx

2+ + =


3. 2 2 0x a b x a b2 2 2+ + + + =^ ^h h v‹w rk‹gh£o‹ _y§fŸ bkŒba©fŸ mšy

vd¡ fh£Lf.

4. 3 2 0p x pqx q2 2 2

- + = v‹w rk‹gh£o‹ _y§fŸ bkŒba©fŸ mšy vd¡

fh£Lf.

v.fh. 3.48 3 10 0x x k2- + = v‹w rk‹gh£o‹ xU _y«

31 våš, k‰bwhU _y¤ij¡

fh©f. nkY« k-‹ kÂ¥igÍ« fh©f.

v.fh. 3.49 5 0ax x c2- + = v‹w ÏUgo¢ rk‹gh£o‹ _y§fë‹ TLjš 10 k‰W«

bgU¡f‰gy‹ 10 våš, a k‰W« c M»at‰¿‹ kÂ¥òfis¡ fh©f.

v.fh. 3.50 2 3 1 0x x2- - = v‹w rk‹gh£o‹ _y§fŸ a k‰W« b våš, Ã‹tUtdt‰¿‹

kÂ¥òfis¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(ii) ba

ab

+ (iii) a b- , Ï§F >a b (iv) 2 2

ba

ab

+e o

(v) 1 1ab a

b+ +c `m j (vi) 4 4a b+ (vii)

3 3

ba

ab

+

v.fh. 3.51 7 3+ k‰W« 7 3- M»at‰iw _y§fshf¡ bfh©l ÏUgo¢ rk‹ghL

x‹¿id mik¡f.

gæ‰Á 3.18

1. ÑnH bfhL¡f¥g£LŸs rk‹ghLfë‹ _y§fë‹ TLjš k‰W« bgU¡f‰gy‹

M»at‰iw¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) 6 5 0x x2- + = (ii) 0kx rx pk

2+ + =

(iii) 3 5 0x x2- = (iv) 8 25 0x

2- =

2. bfhL¡f¥g£LŸs _y§fis¡ bfh©l ÏUgo¢ rk‹ghLfis mik¡fÎ«.


(i) 3 , 4 (ii) 3 7+ , 3 7- (iii) ,2

4 72

4 7+ -

3. 3 5 2x x2- + = 0 v‹w rk‹gh£o‹ _y§fŸ a , b våš, Ã‹tUtdt‰¿‹

kÂ¥òfis¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) ba

ab

+ (ii) a b- (iii) 2 2

ba

ab

+

4. a , b v‹gd 3 6 4x x2- + = 0 v‹D« rk‹gh£o‹ _y§fŸ våš,

2 2a b+ -‹ kÂ¥ò¡

fh©f.



v.fh. 3.4 Ú¡fš Kiwia¥ ga‹gL¤Â Ô®¡f: x y101 99+ = 499, x y99 101+ = 501

v.fh. 3.5 Ú¡fš Kiwæš Ô®: x y3 2 +^ h = xy7 ; x y3 3+^ h = xy11

gæ‰Á 3.1

Ú¡fš Kiwia¥ ga‹gL¤Â Ã‹tU« rk‹gh£L¤ bjhF¥òfis¤ Ô®.

5. x y xy3 5 20+ = ,

x y xy2 5 15+ = , 0,x y 0! ! 6. x y xy8 3 5- = , x y xy6 5 2- =-

7. x y13 11 70+ = , x y11 13 74+ = 8. x y65 33 97- = , x y33 65 1- =

9. x y15 2 17+ = , , ,

x yx y1 1

536 0 0! !+ =

10. x y2

32

61+ = , 0, 0, 0

x yx y3 2 ! !+ =

v.fh. 3.8 X® <çy¡f v©âš, x‹wh« Ïl Ïy¡f v©, g¤jh« Ïl Ïy¡f v©iz¥

nghš ÏU kl§fhf cŸsJ. Ïy¡f§fŸ Ïl« kh¿dhš »il¡F« òÂa v©,

bfhL¡f¥g£l v©izél 27 mÂf« våš, bfhL¡f¥g£l <çy¡f v©iz¡

f©LÃo¡f.

v.fh. 3.9 xU Ã‹d¤Â‹ bjhFÂia 3 Mš bgU¡»Í« gFÂæèUªJ 3 I¡ Fiw¤jhš

»il¡f¥bgW« Ã‹d« 1118 . Mdhš, mnj Ã‹d¤Â‹ bjhFÂÍl‹ 8 I¡ T£o,

gFÂia ÏUkl§fh¡»dhš »il¡f¥ bgW« Ã‹d« 52 våš, m¥Ã‹d¤ij¡

f©LÃo.

v.fh. 3.10 8 M©fŸ k‰W« 12 ÁWt®fŸ nr®ªJ xU ntiyia 10 eh£fëš brŒJ Ko¥g®.

mnj ntiyia 6 M©fŸ k‰W« 8 ÁWt®fŸ nr®ªJ 14 eh£fëš brŒJ

Ko¥g®. xU M© jåahf m›ntiyia v¤jid eh£fëš brŒJ Ko¥gh®?

xU ÁWt‹ jåahf m›ntiyia v¤jid eh£fëš brŒJ Ko¥gh‹?

gæ‰Á 3.2

1. FW¡F¥ bgU¡fš Kiwia¥ ga‹gL¤Â Ã‹tU« rk‹gh£o‹ bjhF¥òfis¤

Ô®¡f. (iv) ,x y x y5 4 2 2 3 13- =- + =

2. Ã‹tU« fz¡FfS¡fhd¢ rk‹ghLfis mik¤J, mt‰¿‹ Ô®Îfis¡ fh©f.

(i) xU v© k‰bwhU v©â‹ _‹W kl§ifél 2 mÂf«. Á¿a v©â‹

4 kl§fhdJ bgça v©izél 5 mÂf« våš, m›bt©fis¡ fh©f.

(ii) ÏU eg®fë‹ tUkhd§fë‹ é»j« 9 : 7. mt®fë‹ bryÎfë‹ é»j« 4 : 3. x›bthUtU« khjbkh‹W¡F ` 2000 nrä¡f Koªjhš, mt®fSila

khjhªÂu tUkhd¤ij¡ fh©f.


(iii) xU <çy¡f v©â‹ kÂ¥ò mj‹ Ïy¡f§fë‹ TLjš nghš 7 kl§F

cŸsJ. Ïy¡f§fis ÏlkhWjš brŒa »il¡F« v© bfhL¡f¥g£l

v©izél 18 FiwÎ våš, m›bt©iz¡ fh©f.

(iv) 3 eh‰fhèfŸ k‰W« 2 nkirfë‹ bkh¤j éiy ` 700. nkY« 5 eh‰fhèfŸ

k‰W« 3 nkirfë‹ bkh¤jéiy ` 1100 våš, 2 eh‰fhèfŸ k‰W« 3

nkirfë‹ bkh¤jéiyia¡ fh©f.

(v) xU br›tf¤Â‹ Ús¤ij 2 br. Û mÂfç¤J mfy¤ij 2 br.Û Fiw¤jhš, mj‹

gu¥ò 28 r. br.Û Fiw»wJ. Ús¤ij 1 br.Û Fiw¤J mfy¤ij 2 br.Û mÂfç¤jhš,

br›tf¤Â‹ gu¥ò 33 r. br. Û mÂfç¡F« våš, br›tf¤Â‹ gu¥ig¡ fh©f.

(vi) Óuhd ntf¤Âš xU F¿¥Ã£l öu¤ij xU bjhl® t©o F¿¥Ã£l neu¤Âš

flªjJ. bjhl® t©oæ‹ ntf« kâ¡F 6 ».Û vd mÂfç¡f¥g£oUªjhš

m¤öu¤ij¡ fl¡f, F¿¥Ãl¥g£oUªj neu¤ij él 4 kâ neu« Fiwthf

m¤bjhl® t©o vL¤J¡ bfh©oU¡F«. bjhl® t©oæ‹ ntf« kâ¡F

6 ».Û vd Fiw¡f¥g£oUªjhš, mnj öu¤ij¡ fl¡f F¿¥Ãl¥g£oUªj

neu¤ijél 6 kâneu« mÂfç¤ÂU¡F« våš, gaz öu¤ij¡ f©LÃo.

v.fh. 3.15 2 14 19 6x x x x4 3 2+ - - + -I x2 1+ Mš tF¡F« nghJ, 6x ax bx

3 2+ - -

v‹gJ <thdhš, a k‰W« b M»at‰¿‹ kÂ¥òfisÍ« k‰W« ÛÂiaÍ«

fh©f.

gæ‰Á 3.4

2. 10 35 50 29x x x x4 3 2+ + + + vD« gšYW¥ò¡nfhitia x 4+ Mš tF¡f¡

»il¡F« <Î 6x ax bx3 2- + + våš a, b M»at‰¿‹ kÂ¥òfisÍ« k‰W«

ÛÂiaÍ« fh©f

3. 8 2 6 7x x x4 2- + - v‹gij x2 1+ Mš tF¡f¡ »il¡F« <Î 4 3x px qx

3 2+ - +

våš, p, q M»at‰¿‹ kÂ¥òfisÍ«, ÛÂiaÍ« fh©f.

v.fh. 3.17 2 3 3 2x x x3 2- - + vD« gšYW¥ò¡nfhitia xUgo¡ fhuâfshf

fhuâ¥gL¤Jf.

v.fh. 3.18 fhuâ¥gL¤Jf: 3 10 24x x x3 2- - +

gæ‰Á 3.5

1. Ã‹tU« gšYW¥ò¡nfhitfis fhuâ¥gL¤Jf.


(i) 2 5 6x x x3 2- - + (ii) 4 7 3x x3

- + (iii) 23 142 120x x x3 2- + -

(iv) 4 5 7 6x x x3 2- + - (v) 7 6x x

3- + (vi) 13 32 20x x x

3 2+ + +

(vii) 2 9 7 6x x x3 2- + + (viii) 5 4x x

3- + (ix) 10 10x x x

3 2- - +

(x) 2 11 7 6x x x3 2+ - - (xi) 14x x x

3 2+ + - (xii) 5 2 24x x x

3 2- - +


v.fh. 3.19 Ã‹tUtdt‰¿‹ Û.bgh.t fh©f:

(iii) 6 x x2 3 22- -^ h, 8 x x4 4 1

2+ +^ h, 12 x x2 7 3

2+ +^ h

v.fh. 3.20 3 3x x x4 3+ - - k‰W« 5 3x x x

3 2+ - + M»a gšYW¥ò¡nfhitfë‹

Û.bgh.t fh©f.

v.fh. 3.21 3 6 12 24x x x x4 3 2+ - - k‰W« 4 14 8 8x x x x

4 3 2+ + - M»a gšYW¥ò¡

nfhitfë‹ Û. bgh. t fh©f.

gæ‰Á 3.6

2. Ã‹tUtdt‰¿‹ Û. bgh. t fh©f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(vii) 2x x2- - , 6x x

2+ - , 3 13 14x x

2- + (viii) 1x x x

3 2- + - , 1x

4-

(ix) 24 x x x6 24 3 2- -^ h, 20 x x x2 3

6 5 4+ +^ h

3. ÑnH bfhL¡f¥g£LŸs gšYW¥ò¡nfhitfë‹ nrhofS¡F tF¤jš go Kiwæš

Û. bgh. t fh©f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) 9 23 15x x x3 2- + - , 4 16 12x x

2- +

(ii) 3 18 33 18x x x3 2+ + + , 3 13 10x x

2+ +

(iii) 2 2 2 2x x x3 2+ + + , 6 12 6 12x x x

3 2+ + +

(iv) 3 4 12x x x3 2- + - , 4 4x x x x

4 3 2+ + +

v.fh. 3.22 Ã‹tUtdt‰¿‹ Û. bgh. k fh©f (iv) x y3 3+ , x y

3 3- , x x y y

4 2 2 4+ +

gæ‰Á 3.7


8. 2 18x y2 2- , 5 15x y xy

2 2+ , 27x y

3 3+

10. 10 x xy y9 62 2+ +^ h, 12 x xy y3 5 2

2 2- -^ h, 14 x x6 2

4 3+^ h.

v.fh. 3.23 3 5 26 56x x x x4 3 2+ + + + k‰W« 2 4 28x x x x

4 3 2+ - - + M»at‰¿‹ Û.bgh.t

5 7x x2+ + våš, mt‰¿‹ Û. bgh.k-it¡ fh©f.

v.fh. 3.24 ÏU gšYW¥ò¡nfhitfë‹ Û. bgh. t k‰W« Û. bgh. k Kiwna x 1+ k‰W«

1x6- . nkY«, xU gšYW¥ò¡nfhit 1x

3+ våš, k‰bwh‹iw¡ fh©f.

gæ‰Á 3.8



(ii) 3 6 5 3x x x x4 3 2+ + + + , 2 2x x x

4 2+ + + , Ït‰¿‹ Û. bgh. t 1x x

2+ + .

(iii) 2 15 2 35x x x3 2+ + - , 8 4 21x x x

3 2+ + - , Ït‰¿‹ Û. bgh. t x 7+ .

(iv) 2 3 9 5x x x3 2- - + , 2 10 11 8x x x x

4 3 2- - - + , Ït‰¿‹ Û. bgh. t x2 1- .


2. Ã‹tUtdt‰¿š Kiwna p x^ h k‰W« q x^ h M»at‰¿‹ Û. bgh. k, k‰W« Û. bgh. t nkY« p x^ h M»ad bfhL¡f¥g£LŸsd. q x^ h v‹w k‰bwhU gšYW¥ò¡nfhitia¡

fh©f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(iii) x y x x y y4 4 4 2 2 4- + +^ ^h h, x y

2 2- , x y

4 4- .

(iv) x x x4 5 13- +^ ^h h, x x5

2+^ h, x x x5 9 2

3 2- -^ h.

(v) x x x x1 2 3 32

- - - +^ ^ ^h h h, x 1-^ h, x x x4 6 33 2- + -^ h.

(vi) 2 x x1 42

+ -^ ^h h, x 1+^ h, x x1 2+ -^ ^h h.

v.fh. 3.26 (iii) x

x

4

82

3

-

- v‹gij x x

x x

2 4

6 82

2

+ +

+ + Mš bgU¡Ff.

v.fh. 3.27 (iii) x

x

25

12

2

-

- v‹gij x x

x x

4 5

4 52

2

+ -

- - Mš tF¡f.

gæ‰Á 3.10

1. Ã‹tU« é»jKW nfhitfis¥ bgU¡», éilia¢ RU¡»a toéš vGJf


(iv) 4 16x x

xxx

x xx x

3 216

644

2 8

2

2

2

3

2

2# #- +-

+-

- -- +

(v) x x

x x

x x

x x

2

3 2 1

3 5 2

2 3 22

2

2

2

#- -

+ -

+ -

- -

(vi) x x

x

x x

x x

x x

x

2 4

2 1

2 5 3

8

2

32 2

4

2# #+ +

-

+ -

-

-

+

2. Ã‹tUtd‰iw vëa toéš RU¡Ff: (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(iii) x

x x

x x

x x

25

4 5

7 10

3 102

2

2

2

'-

- -

+ +

- - (iv) x x

x x

x x

x x

4 77

11 28

2 15

7 122

2

2

2

'- -

+ +

- -

+ +

(v) x x

x x

x x

x x

3 10

2 13 15

4 4

2 62

2

2

2

'+ -

+ +

- +

- - (vi) x

x x

x x

x

9 16

3 4

3 2 1

4 42

2

2

2

'-

- -

- -

-

(vii) x x

x x

x x

x x

2 9 9

2 5 3

2 3

2 12

2

2

2

'+ +

+ -

+ -

+ -

v.fh. 3.28 RU¡Ff: (iii) x

x x

x x

x x

9

6

12

2 242

2

2

2

-

- - +- -

+ -

v.fh. 3.30 xx

xx

xx

12 1

2 11

12

-- -

++ +

++c m v‹gij ÏU gšYW¥ò¡nfhitfë‹ xU

Ã‹dkhf (é»jKW nfhitahf) vëa toéš RU¡Ff.


gæ‰Á 3.11


nfhitahf) vëa toéš RU¡Ff. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(vi) x x

x

x x

x x

6 8

4

20

11 302

2

2

2

+ +

- -- -

- + (vii) xx

x

xxx

12 5

1

11

3 22

2

++ +

-

+ ---` j= G

(viii) x x x x x x3 2

15 61

4 32

2 2 2+ +

++ +

-+ +

.

4. P = x yx+

, Q = x yy

+ våš,

P Q P Q

Q1 22 2-

--

I¡ fh©f.

v.fh. 3.32 t®¡f_y« fh©f. (iii) x x x x x x6 2 3 5 2 2 12 2 2- - - + - -^ ^ ^h h h

gæ‰Á 3.12

2. Ã‹tUtdt‰¿‰F t®¡f_y« fh©f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(ii) x x x x x25 8 15 2 152 2 2- + + - -^ ^ ^h h h

(v) x x x x x x6 5 6 6 2 4 8 32 2 2+ - - - + +^ ^ ^h h h

(vi) x x x x x x2 5 2 3 5 2 6 12 2 2- + - - - -^ ^ ^h h h

v.fh. 3.33 10 37 60 36x x x x4 3 2- + - + -‹ t®¡f_y¤ij¡ fh©f

v.fh. 3.34 x x x x6 19 30 254 3 2- + - + -‹ t®¡f_y¤ij¡ fh©f.

v.fh. 3.35 28 12 9m nx x x x2 3 4

- + + + MdJ xU KG t®¡f« våš, m, n M»at‰¿‹

kÂ¥òfis¡ fh©f.

gæ‰Á 3.13

1. Ã‹tU« gšyW¥ò¡ nfhitfë‹ t®¡f_y¤ij tF¤jš Kiw_y« fh©f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) 4 10 12 9x x x x4 3 2- + - + (ii) 4 8 8 4 1x x x x

4 3 2+ + + +

(iii) 9 6 7 2 1x x x x4 3 2- + - + (iv) 4 25 12 24 16x x x x

2 3 4+ - - +

2. Ã‹tU« gšYW¥ò¡nfhitfŸ KGt®¡f§fŸ våš, a, b M»at‰¿‹ kÂ¥òfis¡

fh©f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) 4 12 37x x x ax b4 3 2- + + + (ii) 4 10x x x ax b

4 3 2- + - +

(iii) 109ax bx x x60 364 3 2+ + - + (iv) 40 24 36ax bx x x

4 3 2- + + +

v.fh. 3.37 Ô®¡f : x x x x7 216

6 9

1

9

12 2-

-- +

+-

= 0

v.fh. 3.38 Ô®¡f : x24 10- = x3 4- , x3 4 0>-


gæ‰Á 3.14

fhuâ¥gL¤J« Kiwæš ÑnH bfhL¡f¥g£l ÏUgo¢ rk‹ghLfis¤ Ô®¡f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(iv) 3 x 62-^ h = x x 7 3+ -^ h (vii)

xx

xx

11

++ + =

1534

v.fh. 3.39 t®¡f¥ ó®¤Â Kiwæš 5 6 2x x2- - = 0-I¤ Ô®¡f.

v.fh. 3.40 t®¡f¥ ó®¤Â Kiwæš rk‹gh£il¤ Ô®¡f: 3 2a x abx b2 2 2

- + = 0

v.fh. 3.41 ÏUgo¢ N¤Âu¤ij¥ ga‹gL¤Â Ã‹tU« rk‹gh£il¤ Ô®.

x x11

22

++

+ =

x 44+

Ï§F x 1 0!+ , x 2 0!+ k‰W« x 4 0!+ .

gæ‰Á 3.15

1. t®¡f¥ ó®¤Â Kiwæš Ã‹tU« rk‹ghLfis¤ Ô®¡f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) 6 7x x2+ - = 0 (ii) 3 1x x

2+ + = 0

(iii) 2 5 3x x2+ - = 0 (iv) 4 4x bx a b

2 2 2+ - -^ h = 0

(v) x x3 1 32- + +^ h = 0 (vi)

xx

15 7-+ = x3 2+

2. ÏUgo¢ N¤Âu¤ij¥ ga‹gL¤Â Ã‹tU« rk‹ghLfis¤ Ô®¡f. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(v) a x 12+^ h = x a 1

2+^ h (vi) 36 12x ax a b

2 2 2- + -^ h = 0

(vii) xx

xx

11

43

+- +

-- =

310 (viii) a x a b x b

2 2 2 2 2+ - -^ h = 0

v.fh. 3.43 xU K¡nfhz¤Â‹ mo¥g¡f« mj‹ F¤Jau¤ijél 4 br.Û mÂf«.

K¡nfhz¤Â‹ gu¥ò 48 r.br.Û våš, m«K¡nfhz¤Â‹ mo¥g¡f¤ijÍ«

cau¤ijÍ« fh©f.

v.fh. 3.44 xU k»GªJ òw¥gl nt©oa neu¤ÂèUªJ 30 ãäl« jhkjkhf¥ òw¥g£lJ.

150 ».Û öu¤Âš cŸs nrUäl¤ij rçahd neu¤Âš br‹wila mjDila

tH¡fkhd ntf¤ij kâ¡F 25 ».Û mÂf¥gL¤j nt©oæUªjJ våš,

k»GªÂ‹ tH¡fkhd ntf¤ij¡ fh©f.

gæ‰Á 3.16

1. xU v© k‰W« mj‹ jiyÑê M»at‰¿‹ TLjš 865 våš, mªj v©iz¡

fh©f.

2. Ïu©L äif v©fë‹ t®¡f§fë‹ é¤Âahr« 45. Á¿a v©â‹ t®¡f«

MdJ, bgça v©â‹ eh‹F kl§»‰F¢ rk« våš, mªj v©fis¡ fh©f.


3. xU étrhæ 100 r.Û gu¥gséš xU br›tf tot¡ fhŒf¿¤ njh£l¤ij mik¡f

éU«Ãdh®. mtçl« 30 Û ÚsnkÍŸs KŸf«Ã ÏUªjjhš Å£o‹ kÂšRtiu¤

njh£l¤Â‹ eh‹fhtJ g¡f ntèahf it¤J¡ bfh©L m¡f«Ãahš _‹W

g¡fK« ntèia mik¤jh®. njh£l¤Â‹ g¡f msÎfis¡ fh©f.

4. xU br›tf tot ãy« 20 Û Ús« k‰W« 14 Û mfy« bfh©lJ. mij¢ R‰¿

btë¥òw¤Âš mikªJŸs Óuhd mfyKŸs ghijæ‹ gu¥ò 111 r.Û våš,

ghijæ‹ mfy« v‹d?

5. Óuhd ntf¤Âš xU bjhl® t©oahdJ (train) 90 ».Û öu¤ij¡ flªjJ.

mjDila ntf« kâ¡F 15 ».Û mÂfç¡f¥g£oUªjhš, gaz« brŒÍ« neu« 30 ãäl§fŸ FiwªÂU¡F« våš, bjhl® t©oæ‹ Óuhd ntf« fh©f.

6. mirt‰w Úçš xU ÏaªÂu¥gl»‹ ntf« kâ¡F 15 ».Û. v‹f. m¥glF

Únuh£l¤Â‹ Âiræš 30 ».Û öu« br‹W, ÃwF vÂ®¤ Âiræš ÂU«Ã 4 kâ 30 ãäl§fëš Û©L« òw¥g£l Ïl¤Â‰F ÂU«Ã tªjhš Úç‹ ntf¤ij¡ fh©f.

7. xU tUl¤Â‰F K‹ò, xUtç‹ taJ mtUila kfå‹ taij¥nghš 8 kl§F.

j‰nghJ mtUila taJ, kfå‹ taÂ‹ t®¡f¤Â‰F¢ rk« våš, mt®fSila

j‰nghija taij¡ fh©f.

8. xU rJu§f¥ gyifæš 64 rk rJu§fŸ cŸsd. x›bthU rJu¤Â‹ gu¥ò

6.25 r.brÛ. v‹f. rJu§f¥ gyifæš eh‹F¥ g¡f§fëY« btë¥òw rJu§fis

x£o 2 br.Û mfy¤Âš g£ilahd Xu« cŸsJ våš, rJu§f¥ gyifæ‹ g¡f¤Â‹

Ús¤Âid¡ fh©f.

9. xU ntiyia¢ brŒa A-¡F B-ia él 6 eh£fŸ Fiwthf¤ njit¥gL»wJ.

ÏUtU« nr®ªJ m›ntiyia¢ brŒjhš mij 4 eh£fëš Ko¡f ÏaY« våš, B

jåna m›ntiyia v¤jid eh£fëš Ko¡f ÏaY«?

10. xU Ïuæš ãiya¤ÂèUªJ Ïu©L bjhl® t©ofŸ xnu neu¤Âš òw¥gL»‹wd.

Kjš t©o nk‰F Âiria neh¡»Í«, Ïu©lh« t©o tl¡F Âiria neh¡»Í«

gaz« brŒ»‹wd. Kjš t©oahdJ Ïu©lhtJ t©oia él kâ¡F 5 ». Û

mÂf ntf¤Âš brš»wJ. Ïu©L kâ neu¤Â‰F¥ ÃwF mt‰¿‰F ÏilnaÍŸss

bjhiyÎ 50 ».Û. våš, x›bthU t©oæ‹ ruhrç ntf¤Âid¡ fh©f.

v.fh. 3.46 a k‰W« b M»ad bkŒba©fŸ. nkY« c xU é»jKW v© vd mikªj

rk‹ghL ( ) ( ) ( )a b c x a b x a b c22- + + - + - - = 0 ‹ _y§fŸ MdJ

é»jKW v©fŸ vd ãWÎf.

v.fh. 3.47 2 7 0x x k k1 3 3 22- + + + =^ ^h h v‹w rk‹gh£o‹ _y§fŸ bkŒba©fŸ

k‰W« rk« våš k-‹ kÂ¥òfis¡ fh©f.

gæ‰Á 3.17


kÂ¥òfis¡f©LÃo. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(iii) 5 0x k x2 22+ - + =^ h (iv) 2 1 0k x k x1 1

2+ - - + =^ ^h h


5. 0, 0, 0, 0a b c d! ! ! ! vd mikªj 2 0a b x ac bd x c d2 2 2 2 2+ - + + + =^ ^h h

v‹w rk‹gh£o‹ _y§fŸ rkbkåš, ba

dc= vd ãWÎf.

6. x a x b x b x c x c x a 0- - + - - + - - =^ ^ ^ ^ ^ ^h h h h h h -‹ _y§fŸ v¥bghGJ«

bkŒba©fŸ v‹W«, a b c= = vd Ïšyhéoš k£Lnk m«_y§fŸ rkk‰wit

v‹W« ãWÎf.

7. rk‹ghL 2 0m x mcx c a12 2 2 2

+ + + - =^ h -‹ _y§fŸ rk« våš, c a m12 2 2= +^ h

vd ãWÎf.

v.fh. 3.50 2 3 1 0x x2- - = v‹w rk‹gh£o‹ _y§fŸ a k‰W« b våš, Ã‹tUtdt‰¿‹

kÂ¥òfis¡ fh©f. (mid¤J c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) 2 2a b+ (vi) 4 4

a b+ (vii) 3 3

ba

ab

+

v.fh. 3.52 k‰W«a b v‹gd 3 4 1x x2- + = 0 v‹D« rk‹gh£o‹ _y§fŸ våš,

2

ba k‰W«

2

ab M»at‰iw _y§fshf¡ bfh©l ÏUgo¢ rk‹gh£oid

mik¡f.

gæ‰Á 3.18

5. a , b v‹gd 2 3 5x x2- - = 0-‹ _y§fŸ våš,

2a k‰W«

2b M»at‰iw


6. 3 2x x2- + = 0-‹ _y§fŸ a , b våš, a- k‰W« b- M»at‰iw _y§fshf¡

bfh©l ÏUgo¢ rk‹ghL x‹¿id mik¡f.

7. a ,b v‹gd 3 1x x2- - = 0-‹ _y§fŸ våš, 1

2a

k‰W« 12b

M»at‰iw


8. 3 6 1x x2- + = 0 v‹w rk‹gh£o‹ _y§fŸ a ,b våš, Ñœ¡fhQ« _y§fis¡

bfh©l rk‹ghLfismik¡f (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) ,1 1a b

(ii) ,2 2a b b a (iii) 2 , 2a b b a+ +

9. 4 3 1x x2- - = 0 v‹w rk‹gh£o‹ _y§fë‹ jiyÑêfis _y§fshf¡ bfh©l

rk‹ghL x‹¿id mik¡f.

10. 3 81x kx2+ - = 0 v‹w rk‹gh£o‹ xU _y« k‰bwhU _y¤Â‹ t®¡fbkåš, k-‹

kÂ¥ig¡ fh©f.

11. x ax2 64 02- + = v‹w rk‹gh£o‹ xU _y« k‰bwhU _y¤Â‹ ÏUkl§F våš,

a-‹ kÂ¥ig¡ fh©f.

12. 5 1x px2- + = 0 v‹w rk‹gh£o‹ _y§fŸ a k‰W« b v‹f. nkY« a b- = 1

våš, p-‹ kÂ¥ig¡ fh©f.

tif¥gL¤j¥g£l édh¡fŸ - mâfŸ 335

4. mâfŸ


v.fh. 4.1 m£ltizæš 5 eh£fë‹ ca®ªj

(H) bt¥gãiy k‰W« Fiwªj (L) bt¥gãiy (ghu‹Ë£oš) ju¥g£LŸsd.

5 eh£fë‹ ca®ªj k‰W« Fiwªj

bt¥gãiyfis mâ tot¤Âš Kiwna

Kjš ãiu ca®ªj bt¥gãiyiaÍ«,

2-M« ãiu Fiwªj bt¥gãiyiaÍ«

F¿¥gjhf mik¡fÎ«. mÂf bt¥gãiy cila ehis¡ fh©f.

v.fh. 4.2 eh‹F czÎ¥ bghU£fëš x›bth‹¿Y« cŸs bfhG¥ò¢ r¤J,

fh®nghiAonu£ k‰W« òuj¢ r¤J M»t‰¿‹ msÎfŸ (»uhäš) Ñœ¡f©lthW

m£ltiz¥gL¤j¥g£LŸsJ.

bghUŸ 1 bghUŸ 2 bghUŸ 3 bghUŸ 4

bfhG¥ò¢ r¤J 5 0 1 10

fh®nghiAonu£ 0 15 6 9

òuj¢ r¤J 7 1 2 8

nk‰f©l étu§fis 3 4# k‰W« 4 3# mâfshf vGJf.

v.fh. 4.3 A aij

= =6 @

1

6

3

9

4

2

7

2

8

5

0

1- -

J

L

KKKKK

N

P

OOOOO våš, (i) mâæ‹ tçir (ii) a

13 k‰W« a

42 cW¥òfŸ

(iii) 2 v‹w cW¥ò mikªJŸs ãiy M»adt‰iw¡ fh©f.

v.fh. 4.4 a i j2 3ij= - v‹w cW¥òfis¡ bfh©l, tçir 2 3# cŸs mâ

A aij

= 6 @-æid mik¡fÎ«.

v.fh. 4.5 A8

1

5

3

2

4=

-e o våš, A

T k‰W« ( )A

T T M»at‰iw¡ fh©f.

gæ‰Á 4.1

1. xU bghGJ ngh¡F Ú® éisah£L¥ ó§fhé‹ f£lz é»j« Ã‹tUkhW.

thu eh£fëš f£lz«

(`)éLKiw eh£fëš f£lz«

(`)taJ tªnjh® 400 500ÁWt® 200 250_¤j¡ Fokf‹ 300 400


taJ tªnjh®, ÁWt® k‰W« _¤j¡FokfD¡fhd f£lz é»j§fS¡fhd

mâfis vGJf. nkY« mt‰¿‹ tçirfis vGJf.

2. xU efu¤Âš 6 nkšãiy¥ gŸëfŸ, 8 ca®ãiy¥ gŸëfŸ k‰W« 13 bjhl¡f¥ gŸëfŸ

cŸsd. Ïªj étu§fis 3 1# k‰W« 1 3# tçirfis¡ bfh©l mâfshf

F¿¡fÎ«.

4. 8 cW¥òfŸ bfh©l xU mâ¡F v›tif tçirfŸ ÏU¡f ÏaY«?

5. 30 cW¥òfŸ bfh©l mâ¡F v›tif tçirfŸ ÏU¡f ÏaY«?

6. Ã‹tUtdt‰iw¡ bfh©L 2 2# tçirÍila mâ A aij

= 6 @-ia¡ fh©f


(i) a ijij

= (ii) 2a i jij= - (iii) a

i ji j

ij=

+-

7. Ã‹tUtdt‰iw¡ bfh©L 3 2# tçiria¡ bfh©l mâ A aij

= 6 @-æid¡ fh©f.


(i) aji

ij= (ii) ( )

ai j

22

ij

2

=- (iii) a i j

2

2 3ij=

-

8. A

1

5

6

1

4

0

3

7

9

2

4

8

=

-

-f p våš, (i) mâæ‹ tçiria¡ fh©f. (ii) a24

k‰W« a32

M»a

cW¥òfis vGJf (iii) cW¥ò 7 mikªJŸs ãiu k‰W« ãuiy¡ fh©f. [VnjD«

Ïu©L c£ÃçÎfS¡F Ïu©L kÂ¥bg© (v-f. (i, ii), (ii, iii), (i, iii)) ]

9. A

2

4

5

3

1

0

= f p våš, A-æ‹ ãiu ãuš kh‰W mâia¡ fh©f.

10. A

1

2

3

2

4

5

3

5

6

=

-

-f p våš, ( )A AT T

= v‹gjid¢ rçgh®¡f.

v.fh. 4.6 x

y

z

5

5

9

4

1

3

5

5

1=c cm m våš, x, y k‰W« z M»adt‰¿‹ kÂ¥òfis¡

fh©f.

v.fh. 4.7 Ô®Î fh© : y

x

x

y3

6 2

31 4=

-

+c em o

v.fh. 4.10 A5

1

6

0

2

4

3

2=

-c m k‰W« B

3

2

1

8

4

2

7

3=

-c m våš, A + B I¡ fh©f.

v.fh. 4.11 vil¡ Fiw¥ò¡fhd czÎ¡ f£L¥ghL¤ Â£l¤Â‹ bjhl¡f¤Âš 4 khzt®fŸ k‰W« 4 khzéfë‹ vil (».». Ïš) Kiwna mâ A-‹ Kjš

ãiu k‰W« Ïu©lh« ãiuahf¡ bfhL¡f¥g£LŸsJ. Ïªj czÎ¡ f£L¥ghL

Â£l¤Â‰F¥ Ã‹ò mt®fSila vil mâ B-š bfhL¡f¥g£LŸsJ.


A35

42

40

38

28

41

45

30= c m

B32

40

35

30

27

34

41

27= c m

khzt®fŸ k‰W« khzéfSila Fiw¡f¥g£l vil mséid mâæš

fh©f.

gæ‰Á 4.2

1. Ã‹tU« mâ¢rk‹gh£oèUªJ x, y k‰W« z-fë‹ kÂ¥òfis¡ fh©f.

x y

z

5 2

0

4

4 6

12

0

8

2

+ -

+=

-e co m

2. x y

x y

2

3

5

13

+

-=e co m våš, x k‰W« y-fë‹ Ô®Îfis¡ fh©f.

3. A2

9

3

5

1

7

5

1=

--

-e eo o våš, A-‹ T£lš ne®khW mâia¡ fh©f.

4. A3

5

2

1= c m k‰W« B

8

4

1

3=

-c m våš, C A B2= + v‹w mâia¡ fh©f.

5. k‰W«A B4

5

2

9

8

1

2

3=

-

-=

- -e eo o våš, A B6 3- v‹w mâia¡ fh©f.

6. a b2

3

1

1

10

5+

-=c c cm m m våš, a k‰W« b M»adt‰¿‹ kÂ¥òfis¡ fh©f.

9. , k‰W«A B O3

5

2

1

1

2

2

3

0

0

0

0= =

-=c c cm m m

våš, Ã‹tUtdt‰iw rçgh®¡f :(x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) A B B A+ = + (ii) ( ) ( )A A O A A+ - = = - + .

v.fh. 4.12 Ã‹tUtdt‰¿‰F mâfë‹ bgU¡f‰gy‹ tiuaW¡f¥g£LŸsjh vd¤

Ô®khå¡fÎ«. m›thW tiuaW¡f¥go‹, bgU¡» tU« mâæ‹ tçiria¡

fh©f.

(i) k‰W«A B2 5 5 4# #

(ii) k‰W«A B1 3 4 3# #

v.fh. 4.13 Ô®Î fh©f : x

y

3

4

2

5

8

13=c c cm m m

v.fh. 4.17 A1

9

3

6=

-e o våš, AI IA A= = v‹gij¢ rçgh®¡f. Ï§F I v‹gJ tçir 2

bfh©l myF mâ.

khzt®fŸ

khzéfŸ

k‰W«khzt®fŸ

khzéfŸ


v.fh. 4.18 k‰W«3

1

5

2

2

1

5

3-

-c em o M»ad mâ¥ bgU¡fiy¥ bghU¤J

x‹W¡bfh‹W ne®khW mâ vd ãWÎf.

gæ‰Á 4.3

2. Ã‹tUtdt‰¿‰F mâfë‹ bgU¡fš tiuaW¡f¥gLkhdhš, mt‰iw¡ fh©f


(i) 2 15

4-^ ch m (ii) 3

5

2

1

4

2

1

7

-c cm m

(iii) 2

4

9

1

3

0

4

6

2

2

7

1-

--

-

e fo p (iv) 6

32 7

--e ^o h

3. xU gH éahghç j‹Dila filæš gH§fis é‰gid brŒ»wh®. M¥ÃŸ, kh«gH«

k‰W« MuŠR M»at‰¿‹ x›bth‹¿‹ é‰gid éiy Kiwna ` 20, ` 10 k‰W«

` 5 MF«. 3 eh£fëš é‰gidahF« gH§fë‹ v©â¡iffë‹ étu§fŸ ÑnH

ju¥g£LŸsd.

ehŸ M¥ÃŸ kh«gH« MuŠR

1 50 60 302 40 70 203 60 40 10

x›bthU ehëY« »il¤j bkh¤j é‰gid¤ bjhifia¡ F¿¥ÃL« xU mâia

vGJf. ÏÂèUªJ gH§fë‹ é‰gidæš »il¤j bkh¤j¤ bjhifia¡ fz¡»Lf.

4. x

y

x1

3

2

3 0

0

9

0

0=c c cm m m våš, x k‰W« y-fë‹ kÂ¥òfis¡ fh©f.

5. , k‰W«A Xx

yC

5

7

3

5

5

11= = =

-

-c c em m o k‰W« AX C= våš, x k‰W« y-fë‹

kÂ¥òfis¡ fh©f.

10. k‰W«A B5

7

2

3

3

7

2

5= =

-

-c em o v‹w mâfŸ x‹W¡bfh‹W bgU¡fš ne®khW

mâ vd ãWÎf.


gæ‰Á 4.2

7. 2 3X Y2

4

3

0+ = c m k‰W« 3 2X Y

2

1

2

5+ =

-

-e o våš, X k‰W« Y M»a mâfis¡

fh©f.

8. Ô®¡f : 3x

y

x

y

2 9

4

2

2 +-

=-

e e co o m.


10. ,A B

4

1

0

1

2

3

2

3

2

2

6

2

0

2

4

4

8

6

= - =f fp p k‰W« C

1

5

1

2

0

1

3

2

1

=

-

-

f p

våš, ( ) ( )A B C A B C+ + = + + v‹gjid¢ rçgh®¡f.

11. xU ä‹dQ FGk« jdJ é‰gid¡fhf th§F« ä‹dQ¥ bghU£fis

f©fhâ¡F« bghU£L jdJ _‹W é‰gid¡ Tl§fëš é‰gid brŒa¥gL«

bghGJngh¡F¢ rhjd§fŸ g‰¿a étu§fis¥ gÂÎ brŒjJ. Ïu©L thu§fëš

eilbg‰w é‰gid étu§fŸ Ã‹tU« m£ltidæš F¿¥Ãl¥g£LŸsd.

T.V. DVD Videogames CD Players

thu« Ifil I 30 15 12 10fil II 40 20 15 15fil III 25 18 10 12

thu« IIfil I 25 12 8 6fil II 32 10 10 12fil III 22 15 8 10

mâfë‹ T£liy¥ ga‹gL¤Â Ïu©L thu§fëš é‰gid brŒa¥g£l

rhjd§fë‹ TLjiy¡ fh©f.

12. xU Ú¢rš Fs¤Â‰F xU ehS¡fhd EiHÎ¡ f£lz« Ã‹tUkhW

xU ehS¡fhd EiHÎ¡ f£lz« ( ` )

cW¥Ãd® ÁWt® bgçat®


cW¥Ãd® mšyhjt®


cW¥Ãd® mšyhjt®fS¡F V‰gL« TLjš f£lz¤ij¡ F¿¡F« mâia¡ fh©f.

v.fh. 4.14 k‰W«Aa

c

b

dI

1

0

0

12= =c cm m våš, ( ) ( )A a d A bc ad I

2

2- + = - vd

ãWÎf.

v.fh. 4.15 k‰W«A B

8

2

0

7

4

3

9

6

3

1

2

5= -

-

=-

- -f ep o våš, AB k‰W« BA M»a

mâfis¡ fh©f. (mitfŸ ÏU¥Ã‹)

v.fh. 4.16 , k‰W«A B C3

1

2

4

2

6

5

7

1

5

1

3=

-=

-=

-e c eo m o våš, ( )A B C AB AC+ = +

v‹gij rç¥gh®¡fÎ«.


v.fh. 4.19 k‰W«A B

2

4

5

1 3 6=

-

= -f ^p h v‹w mâfS¡F ( )AB B AT T T= v‹gij

rç¥gh®¡f.

gæ‰Á 4.3

6. A1

2

1

3=

-c m våš, 4 5A A I O

2

2- + = vd ãWÎf.

7. k‰W«A B3

4

2

0

3

3

0

2= =c cm m våš, AB k‰W« BA M»at‰iw¡ fh©f. mit

rkkhf ÏU¡Fkh?

8. , k‰W«A B C1

1

2

2

1

3

0

1

2

2 1=-

= =c f ^m p h våš, ( ) ( )AB C A BC= v‹gij

rç¥gh®¡fÎ«.

9. k‰W«A B5

7

2

3

2

1

1

1= =

-

-c em o våš, ( )AB B A

T T T= v‹gij rç¥gh®¡fÎ«.

11. Ô®¡f : 0xx

11

2

0

3 5- -=^ e c ^h o m h.

12. k‰W«A B1

2

4

3

1

3

6

2=

-

-=

-

-e eo o våš, ( ) 2A B A AB B

2 2 2!+ + + vd ãWÎf.

13. , k‰W«A B C3

7

3

6

8

0

7

9

2

4

3

6= = =

-c c cm m m våš, ( ) k‰W«A B C AC BC+ +

v‹w mâfis¡ fh©f. nkY«, ( )A B C AC BC+ = + v‹gJ bkŒahFkh?

tif¥gL¤j¥g£l édh¡fŸ - Ma¤bjhiy toéaš 341

5. Ma¤bjhiy toéaš


v.fh. 5.2 (3 , 5), (8 , 10) M»a òŸëfis Ïiz¡F« nfh£L¤ J©il c£òwkhf 2 : 3 v‹w é»j¤Âš Ãç¡F« òŸëia¡ fh©f.

v.fh. 5.5 A(4, -6), B(3,-2) k‰W« C(5, 2) M»at‰iw c¢Áfshf¡ bfh©l

K¡nfhz¤Â‹ eL¡nfh£L ika« fh©f.

v.fh. 5.6 , , , ,7 3 6 1^ ^h h ,8 2^ h k‰W« ,p 4^ h v‹gd X® Ïizfu¤Â‹ tçir¥go mikªj

c¢ÁfŸ våš, p-‹ kÂ¥ig¡ fh©f.

gæ‰Á 5.1

2. Ã‹tU« òŸëfis Kidfshf¡ bfh©l K¡nfhz§fë‹ eL¡nfh£L

ika§fis¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) , , , ,k‰W«1 3 2 7 12 16-^ ^ ^h h h (ii) , , , ,k‰W«3 5 7 4 10 2- - -^ ^ ^h h h

3. xU t£l¤Â‹ ika« (-6, 4). m›t£l¤Â‹ xU é£l¤Â‹ xU Kid, MÂ¥òŸë


4. òŸë (1, 3)-I eL¡nfh£L ikakhf¡ bfh©l K¡nfhz¤Â‹ ÏU KidfŸ

(-7, 6) k‰W« (8, 5) våš, K¡nfhz¤Â‹ _‹whtJ Kidia¡ fh©f.

6. (3, 4) k‰W« (–6, 2) M»a òŸëfis Ïiz¡F« nfh£L¤ J©oid btë¥òwkhf


7. (–3, 5) k‰W« (4, –9) M»a òŸëfis Ïiz¡F« nfh£L¤ J©oid c£òwkhf


v.fh. 5.8 (1, 2), (-3 , 4) k‰W« (-5 ,-6) M»at‰iw Kidfshf¡ bfh©l

K¡nfhz¤Â‹ gu¥ig¡ fh©f.

v.fh. 5.9 A(6 ,7), B(–4 , 1) k‰W« C(a , –9) M»at‰iw Kidfshf¡ bfh©l

ABCT -‹ gu¥ò 68 r. myFfŸ våš, a-‹ kÂ¥ig¡ fh©f.

v.fh. 5.10 A(2 , 3), B(4 , 0) k‰W« C(6, -3) M»a òŸëfŸ xnu ne®¡nfh£oš

mikªJŸsd vd ã%Ã.

v.fh. 5.11 ,a 0^ h, , b0^ h M»a òŸëfis Ïiz¡F« ne®¡nfh£L¤ J©o‹ nkš

mikªJŸs VnjD« xU òŸë P ,x y^ h våš, ax

by

1+ = vd ãWÎf. Ï§F a k‰W« b 0! .

gæ‰Á 5.2

1. Ñœ¡f©l òŸëfis Kidfshf¡ bfh©l K¡nfhz§fë‹ gu¥òfis¡ fh©f.


(i) (0, 0), (3, 0) k‰W « (0, 2) (ii) (5, 2), (3, -5) k‰W« (-5, -1) (iii) (-4, -5), (4, 5) k‰W« (-1, -6)


2. tçiræš mikªj K¡nfhz¤Â‹ KidfS« mit mik¡F« K¡nfhz¤Â‹

gu¥gsÎfS« ÑnH¡ bfhL¡f¥g£LŸsd. x›bth‹¿Y« a-‹ kÂ¥ig¡ fh©f.


c¢ÁfŸ gu¥ò (rJu myFfŸ)

(i) ( , )0 0 , (4, a) k‰W« (6, 4) 17

(ii) (a, a), (4, 5) k‰W« (6,-1) 9

(iii) (a, -3), (3, a) k‰W« (-1,5) 12

3. Ã‹tU« òŸëfŸ xnu ne®¡nfh£oš mikÍ« òŸëfsh vd MuhŒf.


(i) (4, 3), (1, 2) k‰W« (-2, 1) (ii) (-2, -2), (-6, -2) k‰W« (-2, 2)

(iii) 23 ,3-` j,(6, -2) k‰W« (-3, 4)

4. bfhL¡f¥g£oU¡F« òŸëfŸ xU nfhliktd våš, x›bth‹¿Y« k-‹ kÂ¥ig¡

fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) (k, -1), (2, 1) k‰W« (4, 5) (ii) , , , ,k‰W« k2 5 3 4 9- -^ ^ ^h h h

(iii) , , , ,k‰W«k k 2 3 4 1-^ ^ ^h h h

6. , , ( , ) ,k‰W«h a b k0 0^ ^h h v‹gd xU ne®¡nfh£oš mikÍ« òŸëfŸ våš,

K¡nfhz¤Â‹ gu¥Ã‰fhd N¤Âu¤ij¥ ga‹gL¤Â 1ha

kb+ = vd ãWÎf.

Ï§F 0k‰W«h k ! .

gæ‰Á 5.3

4. bfhL¡f¥g£l òŸëfŸ tê¢ bršY« ne®¡nfhLfë‹ rhŒÎ¡ nfhz§fis¡

fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) ,1 2^ h, ,2 3^ h (ii) ,3 3^ h, ,0 0^ h (iii) (a , b), (-a , -b)

5. MÂ¥ òŸë têahfÎ«, ,0 4-^ h k‰W« (8 , 0) M»a òŸëfis Ïiz¡F«

nfh£L¤J©o‹ eL¥òŸë têahfÎ« bršY« nfh£o‹ rhŒit¡ fh©f.

7. rkg¡f K¡nfhz« ABC-‹ g¡f« BC MdJ x-m¢Á‰F Ïiz våš, AB k‰W« BC M»at‰¿‹ rhŒÎfis¡ fh©f.

v.fh. 5.19 ,3 4-^ h v‹w òŸë tê¢ bršY« k‰W« Ma m¢RfS¡F Ïizahf

mikªj ne®¡nfhLfë‹ rk‹ghLfis¡ fh©f.

v.fh. 5.20 rhŒÎ¡ nfhz« 45c k‰W« y-bt£L¤J©L 52 M»at‰iw¡ bfh©l


v.fh. 5.21 ,2 3-^ h v‹w òŸë tê¢ brštJ«, rhŒÎ 31 cilaJkhd ne®¡nfh£o‹



v.fh. 5.22 ,1 1-^ h, ,2 4-^ h M»a òŸëfë‹ tê¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡

fh©f.

v.fh. 5.24 xU ne®¡nfh£o‹ x-bt£L¤J©L 32 k‰W« y-bt£L¤J©L

43 våš,


gæ‰Á 5.4

2. (-5,-2) v‹w òŸë tê¢ brštJ« Mam¢RfS¡F ÏizahdJkhd ne®¡nfhLfë‹

rk‹ghLfis¡ fh©f.

3. ÑnH¡ bfhL¡f¥g£LŸs étu§fS¡F ne®¡nfhLfë‹ rk‹ghLfis¡ fh©f.


(i) rhŒÎ -3; y-bt£L¤J©L 4.

(ii) rhŒÎ¡nfhz« 600 , y-bt£L¤J©L 3.

4. xU ne®¡nfhL y-m¢ir MÂ¥òŸë¡F nkyhf 3 myFfŸ öu¤Âš bt£L»wJ k‰W«

tan21i = (i v‹gJ ne®¡nfh£o‹ rhŒÎ¡ nfhz«) våš, mªj ne®¡nfh£o‹


5. Ã‹tU« ne®¡nfhLfë‹ rhŒÎ k‰W« y bt£L¤J©L M»adt‰iw¡ fh©f.


(i) y x 1= + (ii) x y5 3= (iii) x y4 2 1 0- + = (iv) x y10 15 6 0+ + =

6. Ã‹tU« étu§fS¡F, ne®¡nfhLfë‹ rk‹ghLfis¡ fh©f.


(i) rhŒÎ -4 ; (1, 2) v‹w òŸë tê¢ brš»wJ.

(ii) rhŒÎ 32 ; (5, -4) v‹w òŸë tê¢ brš»wJ.

7. rhŒÎ¡ nfhz« 300 bfh©l k‰W« (4, 2), (3, 1) M»a òŸëfis Ïiz¡F«

ne®¡nfh£L¤ J©o‹ eL¥òŸë tê¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡

fh©f.

8. Ã‹tU« òŸëfë‹ tê¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.


(i) (-2, 5), (3, 6) (ii) (0, -6), (-8, 2)

11. ÑnH¡ bfhL¡f¥g£LŸs x, y-bt£L¤J©Lfis¡ bfh©l ne®¡nfhLfë‹

rk‹ghLfis¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) 2, 3 (ii) 31- ,

23 (iii)

52 ,

43-


12. ÑnH¡ bfhL¡f¥g£LŸs ne®¡nfhLfë‹ rk‹ghLfëèUªJ x, y-bt£L¤

J©Lfis¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) x y5 3 15 0+ - = (ii) 2 1 0x y 6- + = (iii) x y3 10 4 0+ + =

v.fh. 5.26 x y3 2 12 0+ - = , x y6 4 8 0+ + = M»a ne®¡nfhLfŸ Ïiz vd ãWÎf.

v.fh. 5.27 x y2 1 0+ + = , x y2 5 0- + = M»a ne®¡nfhLfŸ x‹W¡F x‹W

br§F¤jhdit vd ãWÎf.

v.fh. 5.28 x y8 13 0- + = v‹w nfh£o‰F ÏizahdJ« (2, 5) v‹w òŸë tê¢


gæ‰Á 5.5

2. x y2 1 0+ + = , x y3 6 2 0+ + = M»a ne®¡nfhLfŸ Ïiz vd ãWÎf.

3. x y3 5 7 0- + = , x y15 9 4 0+ + = M»a ne®¡nfhLfŸ x‹W¡F x‹W br§F¤J

vd ãWÎf.

4. 5 3k‰W«yx p ax y

2= - + = v‹gd Ïiz våš, a -‹ kÂ¥ig¡ fh©f.

5. x y5 2 9 0- - = , 11 0ay x2+ - = M»a ne®¡nfhLfŸ x‹W¡F x‹W br§F¤J

våš, a -‹ kÂ¥ig¡ fh©f.

8. x y3 7 0- + = v‹w ne®¡nfh£o‰F ÏizahdJ« (1, -2) v‹w òŸë tê¢


9. x y2 3 0- + = v‹w ne®¡nfh£o‰F¢ br§F¤jhdJ« (1, -2) v‹w òŸë tê¢



v.fh. 5.3 A(–3, 5) k‰W« B(4, –9) M»a òŸëfis Ïiz¡F« nfh£L¤ J©il

P(–2 , 3) v‹w òŸë c£òwkhf vªj é»j¤Âš Ãç¡F«?

v.fh. 5.4 ,4 1-^ h, ,2 3- -^ h M»a òŸëfis Ïiz¡F« nfh£L¤ J©il _‹W rk


v.fh. 5.7 A(4 , 0), B(0 , 6) M»a òŸëfis Ïiz¡F« nfh£L¤ J©o‹ eL¥òŸë C k‰W« O v‹gJ MÂ våš, C MdJ 3OAB-‹ c¢ÁfëèUªJ rk bjhiyéš

mikÍ« vd¡ fh£Lf.

gæ‰Á 5.1

5. ÃçÎ¢ N¤Âu¤ij¥ ga‹gL¤Â, A(1,0), B(5,3), C(2,7) k‰W« D(–2, 4) v‹w tçiræš

vL¤J¡bfhŸs¥g£l òŸëfŸ xU Ïizfu¤Â‹ c¢ÁfshF« vd ãWÎf.


8. A (–6,–5), B (–6, 4) v‹gd ÏU òŸëfŸ v‹f. nfh£L¤J©L AB-æ‹ nkš

AP = 92 AB v‹wthW mikªJŸs òŸë P-ia¡ fh©f.

9. A(2, –2) k‰W« B(–7, 4) v‹w òŸëfis Ïiz¡F« nfh£L¤ J©il _‹W rk


10. A(–4, 0) k‰W« B(0, 6) v‹w òŸëfis Ïiz¡F« nfh£L¤ J©il eh‹F

rkghf§fshf¥ Ãç¡F« òŸëfis¡ fh©f.

11. (6, 4) k‰W« (1, –7) v‹w òŸëfis Ïiz¡F« nfh£L¤ J©oid x-m¢R Ãç¡F«

é»j¤ij¡ fh©f.

12. (–5, 1) k‰W« (2, 3) v‹w òŸëfis Ïiz¡F« nfh£L¤ J©oid y-m¢R Ãç¡F«

é»j¤ijÍ« k‰W« Ãç¡F« òŸëiaÍ« fh©f.

13. xU K¡nfhz¤Â‹ KidfŸ (1, –1), (0, 4) k‰W« (–5, 3) våš, m«K¡nfhz¤Â‹

eL¡nfhLfë‹ (medians) Ús§fis¡ fz¡»lÎ«.

v.fh. 5.12 (-4, -2), (-3, -5), (3, -2) k‰W« (2, 3) M»a òŸëfis Kidfshf¡

bfh©l eh‰fu¤Â‹ gu¥ig¡ fh©f.

gæ‰Á 5.2

5. Ã‹tUtdt‰iw Kidfshf¡ bfh©l eh‰fu§fë‹ gu¥gsÎfis¡ fh©f.


(i) , , , , , ,k‰W«6 9 7 4 4 2 3 7^ ^ ^ ^h h h h

(ii) , , , , , ,k‰W«3 4 5 6 4 1 1 2- - - -^ ^ ^ ^h h h h

(iii) , , , , , ,k‰W«4 5 0 7 5 5 4 2- - - -^ ^ ^ ^h h h h

7. xU K¡nfhz¤Â‹ KidfŸ , , , ,k‰W«0 1 2 1 0 3-^ ^ ^h h h våš, Ïj‹ g¡f§fë‹

eL¥òŸëfis Ïiz¤J cUth¡F« K¡nfhz¤Â‹ gu¥gsit¡ fh©f.

nkY«, Ï¢Á¿a K¡nfhz¤Â‹ gu¥gsé‰F«, bfhL¡f¥g£l K¡nfhz¤Â‹

gu¥gsé‰F« cŸs é»j¤ij¡ fh©f.

v.fh. 5.16 ne®¡nfh£o‹ rhŒéid¥ ga‹gL¤Â, A(5, -2), B(4, -1) k‰W« C(1, 2) M»ad xnu ne®¡nfh£oš mikªj òŸëfŸ vd ãWÎf.

v.fh. 5.17 (-2, -1), (4, 0), (3, 3) k‰W« (-3, 2) M»a òŸëfis tçirahf vL¤J¡

bfh©L rhŒéid¥ ga‹gL¤Â mit X® Ïizfu¤ij mik¡F« vd¡

fh£Lf.

v.fh. 5.18 A(1 , 2), B(-4 , 5) k‰W« C(0 , 1) M»ad 3ABC-‹ KidfŸ. Ï«K¡nfhz¤Â‹

x›bthU KidæèUªJ« mj‹ vÂ®¥ g¡f¤Â‰F tiua¥gL«

F¤J¡nfhLfë‹ (altitudes) rhŒÎfis¡ fh©f.


gæ‰Á 5.3

8. rhŒéid¥ ga‹gL¤Â, Ñœ¡f©l òŸëfŸ xnu ne®¡nfh£oš mikÍ« vd ãWÎf.


(i) (2 , 3), (3 , -1) k‰W« (4 , -5) (ii) (4 , 1), (-2 , -3) k‰W« (-5 , -5)

(iii) (4 , 4), (-2 , 6) k‰W« (1 , 5)

9. (a, 1), (1, 2) k‰W« (0, b+1) v‹gd xnu ne®¡nfh£oš mikÍ« òŸëfŸ våš,

a b1 1+ = 1 vd ãWÎf.

10 . A(–2, 3), B(a, 5) M»a òŸëfis Ïiz¡F« ne®¡nfhL k‰W« C(0, 5), D(–2, 1) M»a òŸëfis Ïiz¡F« ne®¡nfhL M»ad Ïiz nfhLfŸ våš, a-‹

kÂ¥ig¡ fh©f.

11. A(0, 5), B(4, 2) v‹w òŸëfis Ïiz¡F« ne®¡nfhlhdJ, C(-1, -2), D(5, b) M»a

òŸëfis Ïiz¡F« ne®¡nfh£o‰F¢ br§F¤J våš, b-‹ kÂ¥ig¡ fh©f.

12. 3ABC-‹ KidfŸ A(1, 8), B(-2, 4), C(8, -5). nkY«, M, N v‹gd Kiwna AB, AC Ït‰¿‹ eL¥òŸëfŸ våš, MN-‹ rhŒit¡ fh©f. Ïij¡ bfh©L MN k‰W«

BC M»a ne®¡nfhLfŸ Ïiz vd¡ fh£Lf.

13. (6, 7), (2, -9) k‰W« (-4, 1) M»ad xU K¡nfhz¤Â‹ KidfŸ våš,

K¡nfhz¤Â‹ eL¡nfhLfë‹ rhŒÎfis¡ fh©f.

14. A(-5, 7), B(-4, -5) k‰W« C(4, 5) M»ad 3ABC-‹ KidfŸ våš, K¡nfhz¤Â‹

F¤Jau§fë‹ rhŒÎfis¡ fh©f.

15. rhŒéid¥ ga‹gL¤Â (1, 2), (–2 , 2), (–4 , –3) k‰W« (–1, –3) v‹gd mnj tçiræš

X® Ïizfu¤ij mik¡F« vd ãWÎf.

16. ABCD v‹w eh‰fu¤Â‹ KidfŸ Kiwna A(–2 ,–4), B(5 , –1), C(6 , 4) k‰W« D(–1, 1) våš, Ïj‹ vÂ®¥ g¡f§fŸ Ïiz vd¡ fh£Lf.

v.fh. 5.23 A(2, 1), B(-2, 3), C(4, 5) v‹gd 3ABC-‹ c¢ÁfŸ. c¢Á A-æèUªJ

tiua¥gL« eL¡nfh£o‹ (median) rk‹gh£il¡ fh©f.

v.fh. 5.25 (6, -2) vD« òŸë tê¢ brštJ« k‰W« bt£L¤J©Lfë‹ TLjš 5

bfh©lJkhd ne®¡nfhLfë‹ rk‹ghLfis¡ fh©f.

gæ‰Á 5.4

9. P(1, -3), Q(-2, 5), R(-3, 4) M»a Kidfis¡ bfh©l 3PQR-š Kid R-ÏèUªJ

tiua¥gL« eL¡nfh£o‹ rk‹gh£il¡ fh©f.

10. ne®¡nfh£o‹ rk‹ghL fhQ« Kiwia¥ ga‹gL¤Â, Ã‹tU« òŸëfŸ xnu

ne®¡nfh£oš mikÍ« vd¡ fh£Lf. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) (4, 2), (7, 5), (9, 7) (ii) (1, 4), (3, -2), (-3, 16)


13. (3, 4) v‹w òŸë tê¢ brštJ«, bt£L¤J©Lfë‹ é»j« 3 : 2 vd cŸsJkhd


14. (2, 2) v‹w òŸë tê¢ brštJ«, bt£L¤J©Lfë‹ TLjš 9 MfÎ« bfh©l


15. (5, -3) v‹w òŸë têahf¢ bršY«, mséš rkkhfÎ«, Mdhš F¿ bt›ntwhfÎ«

cŸs bt£L¤ J©Lfis¡ bfh©l ne®¡nfh£o‹ rk‹gh£oid¡ fh©f.

16. (9, -1) v‹w òŸë tê¢ brštJ« x-bt£L¤J©lhdJ, y-bt£L¤J©o‹ msit¥

nghš K«kl§F bfh©lJkhd ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

17. xU ne®¡nfhL Mam¢Rfis A k‰W« B M»a òŸëfëš bt£L»‹wJ. AB-‹

eL¥òŸë (3, 2) våš, AB-‹ rk‹gh£il¡ fh©f.

18. x-bt£L¤J©lhdJ y-bt£L¤J©o‹ msit él 5 myFfŸ mÂfkhf¡

bfh©l xU ne®¡nfhlhdJ (22, -6) v‹w òŸë tê¢ brš»‹wJ våš,


19. ABCD v‹w rhŒrJu¤Â‹ ÏU KidfŸ A(3, 6) k‰W« C(-1, 2) våš, mj‹ _iy

é£l« BD têahf¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

20. A(-2, 6), B (3, -4) M»a òŸëfis Ïiz¡F« ne®¡nfh£L¤ J©il P v‹w

òŸë c£òwkhf 2 : 3 v‹w é»j¤Âš Ãç¡»wJ. òŸë P têahf¢ bršY«

rhŒÎ 23 cila, ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

v.fh. 5.29 ABC3 -‹ KidfŸ A(2, 1), B(6, –1), C(4, 11) v‹f. A-æèUªJ tiua¥gL«

F¤J¡nfh£o‹ rk‹gh£il¡ fh©f.

gæ‰Á 5.5

6. px p y8 2 3 1 0+ - + =^ h , px y8 7 0+ - = M»ad br§F¤J ne®¡nfhLfŸ våš,

p -‹ kÂ¥òfis¡ fh©f.

7. ,h 3^ h, (4, 1) M»a òŸëfis Ïiz¡F« ne®¡nfhL«, x y7 9 19 0- - = v‹w

ne®¡nfhL« br§F¤jhf bt£o¡ bfhŸ»‹wd våš, h -‹ kÂ¥ig¡ fh©f.

10. (3, 4), (-1, 2) v‹w òŸëfis Ïiz¡F« ne®¡nfh£L¤J©o‹ ika¡ F¤J¡nfh£o‹

(perpendicular bisector) rk‹gh£il¡ fh©f.

11. x y2 3 0+ - = , x y5 6 0+ - = M»a ne®¡nfhLfŸ rªÂ¡F« òŸë têahfÎ«,

(1, 2), (2, 1) M»a òŸëfis Ïiz¡F« ne®¡nfh£o‰F ÏizahfÎ« cŸs



12. x y5 6 1- = , x y3 2 5 0+ + = M»a ne®¡nfhLfŸ rªÂ¡F« òŸë têahfÎ«,

x y3 5 11 0- + = v‹w ne®¡nfh£o‰F br§F¤jhfÎ« mikÍ« ne®¡nfh£o‹


13. 3 0x y 9- + = , x y2 4+ = M»a ne®¡nfhLfŸ bt£L« òŸëÍl‹, 2 0x y 4+ - = ,x y2 3 0- + = M»a ne®¡nfhLfŸ bt£L« òŸëia Ïiz¡F« ne®¡nfh£o‹


14. 3ABC-‹ KidfŸ A(2, -4), B(3, 3), C(-1, 5) våš, B-èUªJ tiua¥gL«

F¤J¡nfh£L tê¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

15. 3ABC-‹ KidfŸ A(-4,4 ), B(8 ,4), C(8,10) våš, A-èUªJ tiua¥gL« eL¡nfh£L

tê¢ bršY« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

16. MÂ¥òŸëèUªJ x y3 2 13+ = v‹w ne®¡nfh£o‰F tiua¥gL« br§F¤J¡

nfh£o‹ mo¥òŸëia¡ (foot of the perpendicular) fh©f.

17. xU t£l¤Â‹ ÏU é£l§fë‹ rk‹ghLfŸ 2 7x y+ = , x y2 8+ = k‰W«

t£l¤Â‹ ÛJ mikªJŸs xU òŸë (0, -2) våš, Ï›t£l¤Â‹ Mu¤ij fh©f.

18. x y2 3 4 0- + = , x y2 3 0- + = M»a ne®¡nfhLfŸ rªÂ¡F« òŸëiaÍ«,

(3, -2), (-5, 8) M»a òŸëfis Ïiz¡F« ne®¡nfh£L¤J©o‹

eL¥òŸëiaÍ«, Ïiz¡F« nfh£L¤J©o‹ rk‹gh£il¡ fh©f.

19. ÏUrkg¡f K¡nfhz« 3PQR-š PQ = PR k‰W« mo¥g¡f« QR v‹gJ x-m¢Á‹

ÛJ mik»wJ v‹f. nkY«, Kid P MdJ y-m¢Á‹ ÛJ mik»wJ. PQ-‹

rk‹ghL x y2 3 9 0- + = våš, PR têahf bršY« ne®¡nfh£o‹ rk‹gh£il¡

fh©f.


31

3

1

3

121

2 3g+ + + = v‹w Koéid gl¤Â‹ _ykhf és¡Fnth«

tif¥gL¤j¥g£l édh¡fŸ - toéaš 349

6. toéaš


v.fh. 6.1 ABC3 -šDE BC< k‰W« DBAD

32= . AE = 3.7 br. Û våš, EC-I¡ fh©f.

v.fh. 6.2 PQR3 -‹ g¡f§fŸ PQ k‰W« PR-fë‹ ÛJ mikªj òŸëfŸ S

k‰W« T v‹f. nkY«, ST QR< , PR = 5.6 br.Û k‰W« SQPS

53= våš,

PT-I¡ fh©f.

v.fh. 6.5 ABC3 -š A+ v‹w nfhz¤Â‹ c£òw ÏUrkbt£o AD MdJ, g¡f« BC-I D-š rªÂ¡»wJ. BD = 2.5 br. Û, AB = 5 br. Û k‰W« AC = 4.2 br. Û våš, DC-I¡

fh©f.

v.fh. 6.6 ABC3 -š, A+ -‹ btë¥òw ÏUrkbt£o MdJ BC-‹ Ú£Áæid E-š

rªÂ¡»wJ. AB = 10 br. Û, AC = 6 br. Û k‰W« BC = 12 br. Û våš, CE-I¡

fh©f.

gæ‰Á 6.1

1. D k‰W« E M»a òŸëfŸ Kiwna ABCD -‹ g¡f§fŸ AB k‰W« AC-fëš DE BC<

v‹¿U¡FkhW mikªJŸsd. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) AD = 6 br. Û, DB = 9 br. Û k‰W« AE = 8 br. Û våš, AC-I fh©f.

(ii) AD = 8 br. Û, AB = 12 br. Û k‰W« AE = 12 br. Û våš, CE-I fh©f.

3. E k‰W« F v‹w òŸëfŸ Kiwna PQR3 -‹ g¡f§fŸ PQ k‰W« PR M»at‰¿‹

ÛJ mikªJŸsd. Ã‹tUtdt‰¿‰F EF QR< v‹gjid¢ rçgh®¡f. (x›bthU

c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) PE = 3.9 br. Û., EQ = 3 br.Û., PF = 3.6 br.Û k‰W« FR = 2.4 br. Û.

(ii) PE = 4 br. Û, QE = 4.5 br.Û., PF = 8 br.Û k‰W« RF = 9 br. Û.

9. AD v‹gJ ABC3 -š A+ -‹ c£òw nfhz ÏUrkbt£o. mJ BCI D-š rªÂ¡»wJ.


(i) BD = 2 br. Û, AB = 5 br. Û, DC = 3 br. Û våš, AC fh©f.

(ii) AB = 5.6 br. Û, AC = 6 br. Û k‰W« DC = 3 br. Û våš, BC fh©f.

(iii) AB = x, AC = x – 2, BD = x + 2 k‰W« DC = x – 1 våš, x-‹ kÂ¥ig¡ fh©f.

10. Ã‹tUtdt‰WŸ AD v‹gJ ABC3 -š A+ -‹ nfhz ÏUrkbt£o MFkh vd¢ nrhÂ¡f.


(i) AB = 4 br. Û, AC = 6 br. Û, BD = 1.6 br. Û, k‰W« CD = 2.4 br. Û, (ii) AB = 6 br. Û, AC = 8 br. Û, BD = 1.5 br. Û. k‰W« CD = 3 br. Û.


11. MP v‹gJ MNO3 -š M+ -‹ btë¥òw ÏUrkbt£o. nkY«, ÏJ NO-‹ Ú£Áæid

P-æš rªÂ¡»wJ. MN = 10 br.Û, MO = 6 br.Û, NO = 12 br.Û våš, OP-I fh©f.

v.fh. 6.8 A, B v‹gd PQRT -‹ g¡f§fŸ PQ, PR-fë‹ nkš mikªj òŸëfŸ v‹f.

nkY«, AB QR< , AB = 3 br. Û, PB = 2 br.Û k‰W« PR = 6 br.Û våš, QR-‹ Ús¤Âid fh©f.

nj‰w« 6.5 Ãjhfu° nj‰w¤ij (bgsja‹ nj‰w«) vGJf

nj‰w« 6.6 Ãjhfu° nj‰w¤Â‹ kWjiyia vGJf

nj‰w« 6.7 bjhLnfhL - eh© nj‰w¤ij vGJf.

nj‰w« 6.8 bjhLnfhL - eh© nj‰w¤Â‹ kWjiyia vGJf.

v.fh. 6.12 xU t£l¤Â‹ òŸë A-š tiua¥gL« bjhLnfhL

PQ v‹f. AB v‹gJ t£l¤Â‹ eh© v‹f. nkY«,

54BAC+ = c k‰W« 62BAQ+ = c v‹W mikÍkhW

t£l¤Â‹ nkš cŸs òŸë C våš, ABC+ -I fh©f.

v.fh. 6.13 Ñœ¡fhQ« x›bthU gl¤ÂY« x-‹ kÂ¥ig fh©f.

(X›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) (ii)

gæ‰Á 6.3

1. gl¤Âš TP xU bjhLnfhL. A, B v‹gd t£l¤Â‹ ÛJŸs òŸëfŸ.

+BTP = 72c k‰W« +ATB = 43c våš +ABT-I¡ fh©f.

2. xU t£l¤Âš AB, CD v‹D« ÏU eh©fŸ x‹iwbah‹W c£òwkhf P-æš bt£o¡

bfhŸ»‹wd. (X›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) CP = 4 br.Û, AP = 8 br.Û., PB = 2 br.Û våš, PD-I¡ fh©f.

(ii) AP = 12 br.Û, AB = 15 br.Û, CP = PD våš, CD-I¡ fh©f.

3. AB k‰W« CD v‹w ÏU eh©fŸ t£l¤Â‰F btëna P vD« òŸëæš bt£o¡

bfhŸ»‹wd. (X›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) AB = 4 br.Û, BP = 5 br.Û k‰W« PD = 3 br.Û våš, CD-I¡ fh©f.

(ii) BP = 3 br.Û, CP = 6 br.Û k‰W« CD = 2 br.Û våš, AB-I¡ fh©f.

A

BC

P Q62c

62c

54c

A B

T P27 c

43c



nj‰w« 6.1 mo¥gil é»jrk (njš°) nj‰w¤ij vGÂ ãWÎf. (mšyJ) xU ne®¡nfhL

xU K¡nfhz¤Â‹ xU g¡f¤Â‰F ÏizahfÎ« k‰w Ïu©L g¡f§fis

bt£LkhW« tiua¥g£lhš m¡nfhL m›éU¥ g¡f§fisÍ« rké»j¤Âš

Ãç¡F« vd ãWÎf.

nj‰w« 6.2 mo¥gil¢ é»jrk (njš°) nj‰w¤Â‹ kWjiyia vGÂ ãWÎf (mšyJ) xU

ne®¡nfhL xU K¡nfhz¤Â‹ ÏU g¡f§fis xnu é»j¤Âš Ãç¡Fkhdhš,

m¡nfhL _‹whtJ g¡f¤Â‰F Ïizahf ÏU¡F« vd ãWÎf.

nj‰w« 6.3 nfhz ÏUrkbt£o¤ nj‰w¤ij vGÂ ãWÎf (mšyJ) xU K¡nfhz¤Âš

xU nfhz¤Â‹ c£òw ÏUrkbt£oahdJ m¡nfhz¤Â‹ vÂ® g¡f¤ij

c£òwkhf, m¡nfhz¤Âid ml¡»a g¡f§fë‹ é»j¤Âš Ãç¡F« vd

ãWÎf.

nj‰w« 6.4 nfhz ÏUrkbt£o¤ nj‰w¤Â‹ kWjiyia vGÂ ãWÎf (mšyJ) xU

K¡nfhz¤Â‹ xU c¢Áæ‹ tê¢ bršY« xU ne®¡nfhL, mj‹

vÂ®g¡f¤Âid c£òwkhf, k‰w ÏU g¡f§fë‹ é»j¤Âš Ãç¡Fkhdhš,

m¡nfhL c¢Áæš mikªj nfhz¤Âid c£òwkhf ÏU rkghf§fshf

Ãç¡F« vd ãWÎf.

v.fh. 6.3 ABCT -‹ g¡f§fŸ AB k‰W« AC-š M»at‰¿‹ nkš mikªj òŸëfŸ

D k‰W« E v‹f. nkY«, DBAD

ECAE= k‰W« ADE DEA+ += våš, ABC3

xU ÏUrkg¡f K¡nfhz« vd ãWÎf.

v.fh. 6.4 ABCT -‹ g¡f§fŸ AB, BC k‰W« CA M»at‰¿š mikªj òŸëfŸ D, E k‰W« F v‹f. nkY«, DE AC< k‰W« FE AB< våš,

ADAB = FC

AC vd

ãWÎf.

v.fh. 6.7 ABCT -š g¡f« BC-‹ eL¥òŸë D v‹f. P k‰W« Q v‹gd AB k‰W«

AC-fë‹ nkš mikªj òŸëfŸ MF«. nkY«, BDA+ k‰W« ADC+ M»a

nfhz§fis Kiwna DP k‰W« DQ v‹gd ÏU rkghf§fshf Ãç¡F«

våš, PQ BC< vd ãWÎf.

gæ‰Á 6.1

1. D k‰W« E M»a òŸëfŸ Kiwna ABCD -‹ g¡f§fŸ AB k‰W« AC-fëš DE BC<

v‹¿U¡FkhW mikªJŸsd.

(iii) AD = 4x–3, BD = 3x–1, AE = 8x–7 k‰W« EC = 5x–3 våš, x-‹ kÂ¥ig¡ fh©f.


2. gl¤Âš AP = 3 br. Û, AR = 4.5 br. Û, AQ = 6 br. Û, AB = 5 br. Û

k‰W« AC = 10 br. Û. våš, AD-‹ kÂ¥ig¡ fh©f.

4. gl¤Âš AC BD< k‰W« OA =12 br. Û,

AB = 9 br. Û, OC = 8 br. Û. k‰W« EF = 4.5 br. Û

våš, FO-it¡ fh©f.

5. ABCD v‹w eh‰fu¤Âš, AB-¡F Ïiz CD v‹f. AB-¡F Ïizahf tiua¥g£l

xU ne®¡nfhL AD-I P-æY« BC-I Q-æY« rªÂ¡»wJ våš, PDAP

QCBQ

= vd

ãWÎf.

6. gl¤Âš PC QK< k‰W« BC HK< . AQ = 6 br. Û, QH = 4 br. Û, HP = 5 br. Û, KC = 18 br. Û våš, AK k‰W« PB-¡fis¡ fh©f.

7. gl¤Âš DE AQ< k‰W« DF AR<

våš, EF QR< vd ãWÎf.

8. gl¤Âš DE AB< k‰W«

DF AC< våš, EF BC< vd ãWÎf.

12. ABCD v‹w eh‰fu¤Âš B+ k‰W« D+ M»at‰¿‹ ÏUrkbt£ofŸ AC-I E-š

bt£L»‹wd våš, BCAB

DCAD= vd ãWÎf.

13. ABCT -š A+ -‹ c£òw ÏUrkbt£o BC-I D-æY«, A+ -‹ btë¥òw ÏUrkbt£o

BC-‹ Ú£Áæid E-æY« rªÂ¡»‹wd våš, BEBD

CECD= vd ãWÎf.

14. ABCD v‹w eh‰fu¤Âš AB =AD. AE k‰W« AF v‹gd Kiwna BAC+ k‰W« DAC+ M»at‰¿‹ c£òw ÏUrkbt£ofŸ våš, EF BD< vd ãWÎf.

v.fh. 6.9 1.8 Û cauKŸs xUt® xU Ãuäo‹ (Pyramid) mUnf ã‹W bfh©oU¡»‹wh®.

xU F¿¥Ã£l neu¤Âš mtUila ãHè‹ Ús« 2.7Û k‰W« Ãuäo‹ ãHè‹

Ús« 210 Û våš, Ãuäo‹ cau« fh©f.

A

P Q

BCD

R

AE

B

F

D

O

C

A

K

CB

PH

Q

FPE

D

A

CB

P

E F

R

A

Q

D


v.fh. 6.10 xU nfhòu¤Â‹ c¢Áæid, xUt® nfhòu¤ÂèUªJ 87.6 Û öu¤Âš jiuæš

cŸs xU f©zhoæš gh®¡»wh®. f©zho nkš neh¡»athW cŸsJ. mt®

f©zhoæèUªJ 0.4Û öu¤ÂY« mtç‹ »ilãiy¥ gh®it¡ nfh£o‹

k£l« jiuæèUªJ 1.5Û cau¤ÂY« cŸsJ våš, nfhòu¤Â‹ cau« fh©f.

(kåjå‹ mo, f©zho k‰W« nfhòu« M»ait xnu ne®¡nfh£oš cŸsd)

v.fh. 6.11 xU ãH‰gl¡ fUéæYŸs gl¢ RUëš xU ku¤Â‹ Ã«g¤Â‹ Ús« 35 ä.Û.

by‹°¡F« gl¢RUS¡F« Ïil¥g£l öu« 42 ä.Û. nkY«, by‹ìèUªJ

ku¤J¡F cŸs öu« 6Û våš, ãH‰gl« vL¡f¥gL« ku¤Â‹ gFÂæ‹ Ús«

fh©f.

gæ‰Á 6.2

1. Ã‹tU« gl§fŸ x›bth‹¿Y« bjçahjdt‰¿‹ kÂ¥òfis¡ fh©f. všyh

Ús§fS« br‹o Û£lçš bfhL¡f¥g£LŸsd. (msÎfŸ msÎ¤Â£l¥go Ïšiy,

X›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) (ii) (iii)

2. xU ãH‰gl¡ fUéæ‹ gl¢RUëš, 1.8 Û cauKŸs xU kåjå‹ Ã«g¤Â‹ Ús«

1.5 br.Û. v‹f. fUéæ‹ by‹ìèUªJ gl¢RUŸ 3 br.Û öu¤Âš ÏUªjhš, mt®

ãH‰gl¡ fUéæèUªJ v›tsÎ öu¤Âš ÏU¥gh®?

3. 120 br.Û. cauKŸs xU ÁWä xU és¡F¡ f«g¤Â‹ moæèUªJ éy»,

mj‰F nebuÂuhf 0.6 Û./é ntf¤Âš elªJ¡ bfh©oU¡»whŸ. és¡F,

jiu k£l¤ÂèUªJ 3.6 Û. cau¤Âš cŸsJ våš, ÁWäæ‹ ãHè‹ Ús¤ij

4 édhofS¡F ÃwF fh©f.

5. ABCD -š g¡f§fŸ AB k‰W« AC-æš Kiwna P k‰W« Q v‹w òŸëfŸ cŸsd. AP = 3 br.Û, PB = 6 br.Û, AQ = 5 br.Û k‰W« QC = 10 br.Û våš, BC = 3 PQ vd

ãWÎf.

6. ABCD -š, AB = AC k‰W« BC = 6 br.Û v‹f. nkY«, AC-š D v‹gJ AD = 5 br.Û

k‰W« CD = 4 br.Û v‹W ÏU¡FkhW xU òŸë våš, BCD ACB+D D vd ãWÎf.

Ïj‹ _y« DB-ia¡ fh©f.

7. ABCD -‹ g¡f§fŸ AB k‰W« AC-fë‹ nkš mikªj òŸëfŸ Kiwna D k‰W« E v‹f. nkY«, DE || BC, AB = 3 AD k‰W« ABCD -‹ gu¥gsÎ 72 br.Û2 våš,

eh‰fu« DBCE-‹ gu¥gsit¡ fh©f.

GA

B FC

D E24

8 6

8

x

y

A

D G

HFE

B C

6

9

53

4

xy

z

A

E

B D C

F

x

y

5 7

6G


8. TABC-‹ g¡f Ús§fŸ 6 br.Û, 4 br.Û, 9 br.Û k‰W« PQR ABCT T+ . 3PQR-‹xU g¡f« 35 br.Û våš, TPQR-‹ R‰wsÎ äf mÂfkhf v›tsÎ ÏU¡f¡ TL«?

9. gl¤Âš DE || BC. nkY«, BDAD

53= våš,

(i) ‹ gu¥gsÎ‹ gu¥gsÎ

ABCADE

--

D

D ,

(ii) ‹ gu¥gsÎ

rçtf« ‹ gu¥gsÎABC

BCED-

-D

M»adt‰¿‹ kÂ¥òfis¡ fh©f.

10. muR, xU khefçš ga‹gL¤j¥glhj ãy¥gFÂ x‹¿š

òÂa bjhê‰ng£ilæid ãWt¤ Â£läL»wJ.

ãHè£l¥ gFÂ òÂajhf mik¡f¥gL« bjhê‰ng£il

gFÂæ‹ gu¥gsit¡ F¿¡»wJ. Ï¥gFÂæ‹

gu¥gsit¡ fh©f.

11. xU ÁWt‹ itu¤Â‹ FW¡F bt£L¤ njh‰w toéš, gl¤Âš

fh£oathW xU g£l« brŒjh‹.

Ï§F AE = 16 br.Û, EC = 81 br.Û. mt‹ BD v‹w FW¡F¡

F¢Áæid¥ ga‹gL¤j éU«ò»wh‹. m¡F¢Áæ‹ Ús«

v›tsÎ ÏU¡fnt©L«?

12. xU khzt‹ bfho¡f«g¤Â‹ cau¤Âid¡ fz¡»l éU«ò»wh‹.

bfho¡f«g¤Â‹ c¢Áæ‹ vÂbuhë¥ig¡ f©zhoæš fhQ« tifæš, xU

ÁW f©zhoia¤ jiuæš it¡»wh‹. m¡f©zho mtåläUªJ 0.5Û

bjhiyéš cŸsJ. f©zho¡F« bfhof«g¤Â‰F« Ïilna cŸs bjhiyÎ

3Û k‰W« mtDila »ilk£l¥ gh®it¡ nfhL jiuæèUªJ 1.5 Û cau¤Âš

cŸsJ våš, bfho¡f«g¤Â‹ cau¤ij¡ fh©f. (khzt‹, f©zho k‰W«

bfho¡ f«g« M»ad xnu ne®¡nfh£oš cŸsd.)

13. xU nk‰Tiu gl¤Âš fh£oathW FW¡F bt£L¤ njh‰w¤ij¡ bfh©LŸsJ. ÏÂš

(i) tobth¤j K¡nfhz§fis¤ bjçªbjL¡fÎ«.

(ii) Tiuæ‹ cau« h-I¡ fh©f.

v.fh. 6.14 gl¤Âš PA, PB v‹gd O-it ikakhf¡ bfh©l

t£l¤Â‰F btë¥ òŸë P-æèUªJ tiua¥g£l

bjhLnfhLfshF«. CD v‹gJ t£l¤Â‰F E v‹D«

òŸëæš tiua¥g£l bjhLnfhL. AP = 15 br.Û våš, TPCD -æ‹ R‰wsit¡ f©LÃo.

D

CA

PE

B

O


v.fh. 6.15 ABCD v‹w eh‰fu«, mj‹ všyh g¡f§fS« xU t£l¤ij bjhLkhW

mikªJŸsJ. AB = 6 br.Û, BC = 6.5 br.Û k‰W« CD = 7 br.Û våš, AD-‹

Ús¤ij¡ fh©f.

gæ‰Á 6.3

4. xU t£l«, TABC-š g¡f« BC-I P-š bjhL»wJ. m›t£l« AB k‰W« AC-fë‹

Ú£Áfis Kiwna Q k‰W« R-š bjhL»wJ våš, AQ = AR = 21 (TABC-‹ R‰wsÎ)

vd ãWÎf.


m›éizfu« xU rhŒrJukhF« vd ãWÎf.

6. xU jhkiu¥ óthdJ j©Ù® k£l¤Â‰F nkš 20 br.Û cau¤Âš cŸsJ. j©o‹

ÛÂ¥ gFÂ j©Ù® k£l¤Â‰F ÑnH cŸsJ. fh‰W ÅR«nghJ j©L jŸs¥g£L,

jhkiu¥ óthdJ j©o‹ Mu«g ãiyæèUªJ 40 br.Û öu¤Âš j©Ùiu¤

bjhL»wJ. Mu«g ãiyæš j©Ù® k£l¤Â‰F¡ ÑnH cŸs j©o‹ Ús« fh©f?

7. br›tf« ABCD-‹ c£òw òŸë O-éèUªJ br›tf¤Â‹ KidfŸ A, B, C, D Ïiz¡f¥g£LŸsd våš, OA OC OB OD

2 2 2 2+ = + vd ãWÎf.


1 ( )n n3 5 7 2 12

g+ + + + + - = v‹w thŒgh£oid gl¤Â‹ _ykhf és¡Fnth«

Mfnt, , , ,1 3 2 1 3 5 3 1 3 5 7 42 2 2

g+ = + + = + + + =


7. K¡nfhzéaš


v.fh. 7.1 1cosecsin

seccos

ii

ii+ = v‹w K‰bwhUikia ãWÎf.

v.fh. 7.2 coscos cosec cot

11

ii i i

+- = - v‹w K‰bwhUikia ãWÎf.

v.fh. 7.7 1 3sin cos sin cos6 6 2 2i i i i+ = -^ h v‹w K‰bwhUikia ãWÎf.

v.fh. 7.8 cos cos

sin sin tan2

23

3

i i

i i i-

- = v‹w K‰bwhUikia ãWÎf.

v.fh. 7.10 secsec

cossin1

1

2

ii

ii+ =

- vd ãWÎf.

gæ‰Á 7.1

1. Ã‹tUtd x›bth‹W« K‰bwhUik MFkh vd¡ fh©f. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) 2cos sec2 2i i+ = +sini (ii) cot cos sin

2 2i i i+ =

2. Ã‹tU« K‰bwhUikfis ãWÎf. (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

(i) sec cosec sec cosec2 2 22

i i i i+ = (ii) cos

sin cosec cot1 i

i i i-

= +

(iii) sinsin sec tan

11

ii i i

+- = - (iv) 1

sec tancos sini ii i

-= +

(v) sec cosec tan cot2 2i i i i+ = + (vi) 1

sin coscos sin cot

1

2

i ii i i+

+ - =^ h

(vii) 1sec sin sec tan1i i i i- + =^ ^h h (viii) cosec cot

sini ii+

= cos1 i-

v.fh. 7.14 200 Û ÚsKŸs üèdhš xU fh‰who f£l¥g£L gwªJ¡ bfh©oU¡»wJ. mªj

üš jiuk£l¤Jl‹ 300 nfhz¤ij V‰gL¤Âdhš, fh‰who jiuk£l¤ÂèUªJ

v›tsÎ cau¤Âš gw¡»wJ vd¡ fh©f. (Ï§F üš xU ne®¡nfh£oš

cŸsjhf¡ fUJf)

v.fh. 7.15 Rtçš rhŒ¤J it¡f¥g£l xU VâahdJ jiuÍl‹ 60c nfhz¤ij

V‰gL¤J»wJ. Vâæ‹ mo Rt‰¿èUªJ 3.5 Û öu¤Âš cŸsJ våš,

Vâæ‹ Ús¤ij¡ fh©f.

v.fh. 7.16 30 Û ÚsKŸs xU f«g¤Â‹ ãHè‹ Ús« 10 3 Û våš, Nçaå‹ V‰w¡

nfhz¤Â‹ (jiu k£l¤ÂèUªJ V‰w¡ nfhz«) mséid¡ fh©f.

tif¥gL¤j¥g£l édh¡fŸ - K¡nfhzéaš 357

v.fh. 7.17 xU nfhòu¤Â‹ moæèUªJ 30 3 Û bjhiyéš ã‰F« xU gh®itahs®,

m¡nfhòu¤Â‹ c¢Áæid 30c V‰w¡ nfhz¤Âš fh©»wh®. jiuk£l¤ÂèUªJ

mtUila »ilãiy¥ gh®it¡nfh£o‰F cŸs öu« 1.5 Û våš, nfhòu¤Â‹

cau¤ij¡ fh©f.

gæ‰Á 7.2

1. xU Rik C®ÂæèUªJ (truck) Rikia Ïw¡f VJthf 30c V‰w¡ nfhz¤Âš xU

rhŒÎ¤ js« (ramp) cŸsJ. rhŒÎ¤ js¤Â‹, c¢Á jiuæèUªJ 0.9 Û cau¤Âš

cŸsJ våš, rhŒÎ¤ js¤Â‹ Ús« ahJ?

2. cau« 150 br.Û cŸs xU ÁWä xU és¡F¡ f«g¤Â‹ K‹ ã‹wthW 150 3 br.Û ÚsKŸs

ãHiy V‰gL¤J»whŸ våš, és¡F¡ f«g¤Â‹ c¢Áæ‹ V‰w¡ nfhz¤ij¡

fh©f.

3. A k‰W« B v‹w ó¢ÁfS¡F Ïil¥g£l öu« 2 Û ÏU¡F« tiuæš, x‹W vG¥ò«

xèia k‰wJ nf£f ÏaY«. Rt‰¿èUªJ 1 Û öu¤Âš jiuæYŸs ó¢Á A MdJ

xU ÁyªÂahš c©z¥gL« ãiyæš cŸs ó¢Á B-ia Rt‰¿š fh©»wJ. A-æèUªJ B-¡F V‰w¡ nfhz« 30c Mf ÏU¡F«nghJ A MdJ B-¡F

v¢rç¡if xè éL¤jhš, ÁyªÂ¡F Ïiu »il¡Fkh mšyJ »il¡fhjh? (A-æ‹ v¢rç¡if xèia B nf£F«nghJ mJ j¥ÃéL« vd¡ bfhŸf)

5. 40 br.Û ÚsKŸs xU CryhdJ (pendulum), xU KG miyé‹ nghJ, mj‹ c¢Áæš 60c nfhz¤ij V‰gL¤J»wJ. mªj miyéš, Crš F©o‹ Jt¡f ãiy¡F«,

ÏWÂ ãiy¡F« Ïilna cŸs äf¡ Fiwªj öu¤ij¡ fh©f.


v.fh. 7.3 [ ] [ ]cosec sin cosec sin tan cot90 90 1i i i i i i- - - - + =c c^ ^h h6 @

v‹w K‰bwhUikia ãWÎf.

v.fh. 7.4 1tan sectan sec

cossin

11

i ii i

ii

- ++ - = + vd ãWÎf.

v.fh. 7.5 1cot

tantan

cot tan cot1 1i

iii i i

-+

-= + + v‹w K‰bwhUikia ãWÎf.

v.fh. 7.6 7sin cosec cos sec tan cot2 2 2 2i i i i i i+ + + = + +^ ^h h v‹w K‰bwhUikia

ãWÎf.

v.fh. 7.9 1 2 2sec tansec tan sec tan tan

2

i ii i i i i+- = - + v‹w K‰bwhUikia ãWÎf.

v.fh. 7.11 cosec sin sec costan cot

1i i i ii i

- - =+

^ ^h h v‹w K‰bwhUikia ãWÎf.


v.fh. 7.12 tan sin mi i+ = , tan sin ni i- = k‰W« m n! våš, 4m n mn2 2- =

vd¡ fh£Lf.

v.fh. 7.13 tan cos sin2 2 2a b b= - våš, cos sin tan2 2 2a a b- = vd ãWÎf.

gæ‰Á 7.1

3. Ã‹tU« K‰bwhUikfis ãWÎf. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) sin

sin

coscos

1

90

1 90i

i

ii

+

-+

- -

c

c^

^h

h = 2seci .

(ii) cot

tantan

cot sec cosec1 1

1ii

ii i i

-+

-= + .

(iii) tan

sincot

coscos sin

190

190

0 0

ii

ii

i i-

-+

--

= +^ ^h h .

(iv) 902 .

cosectan

cotcosec sec

11

0

ii

ii i

+-

+ + =^ h

(v) .cot coseccot cosec cosec cot

11

i ii i i i- ++ - = +

(vi) 2cot cosec tan sec1 1i i i i+ - + + =^ ^h h .

(vii) sin cossin cos

sec tan11 1

i ii i

i i+ -- + =

-.

(viii) 1 2 1tan

tan

sin

sin sin

90

900

0

2 2i

i

i

i i

-=

- -

-

^

^

h

h

(ix) cosec cot sin sin cosec cot

1 1 1 1i i i i i i-

- = -+

.

(x) ( )tan cosec

cot sec sin cos tan cot2 2

2 2

i i

i i i i i i+

+ = +^ h.

4. sec tanx a bi i= + k‰W« tan secy a bi i= + våš, x y a b2 2 2 2- = - vd ãWÎf.

5. tan tanni a= k‰W« sin sinmi a= våš, 1

1cosn

m2

2

2

i =-

- , n !+1, vd ãWÎf.

6. , k‰W«sin cos tani i i v‹gd bgU¡F¤ bjhlçš (G.P.) ÏU¥Ã‹

1cot cot6 2i i- = vd ãWÎf.

v.fh. 7.18 ne®¡F¤jhd xU ku¤Â‹ nkšghf« fh‰¿dhš K¿ªJ, m«K¿ªj gFÂ ÑnH

éGªJélhkš, ku¤Â‹ c¢Á jiuÍl‹ 30c nfhz¤ij V‰gL¤J»wJ. ku¤Â‹

c¢Á mj‹ moæèUªJ 30 Û bjhiyéš jiuia¤ bjhL»wJ våš, ku¤Â‹

KG cau¤ij¡ fh©f.


v.fh. 7.19 X® mÂntf¥ ngh® ékhd«, jiu k£l¤ÂèUªJ 3000 Û cau¤Âš, k‰bwhU

mÂntf¥ ngh® ékhd¤ij ne® nkyhf¡ fl¡»wJ. m›thW fl¡F« nghJ jiu

k£l¤Âš xU F¿¥Ã£l òŸëæèUªJ mt‰¿‹ V‰w¡ nfhz§fŸ Kiwna 60c k‰W« 45c våš, mªj neu¤Âš Ïu©lhtJ ngh® ékhd« k‰W« Kjš ngh®

ékhd« M»at‰¿‰F Ïil¥g£l öu¤ij¡ fz¡»Lf. ( .3 1 732= )

v.fh. 7.20 xU nfhòu¤Â‹ moæèUªJ xU F‹¿‹ c¢Áæ‹ V‰w¡nfhz« 60c v‹f. F‹¿‹ moæèUªJ nfhòu¤Â‹ c¢Áæ‹ V‰w¡nfhz« 30c k‰W«

nfhòu¤Â‹ cau« 50 Û våš, F‹¿‹ cau¤ij¡ fh©f.

v.fh. 7.21 xU br§F¤jhd RtU«, xU nfhòuK« xU F¿¥Ã£l Ïilbtëæš

cŸsd. nfhòu¤Â‹ c¢ÁæèUªJ gh®¡F« nghJ, Rt‰¿‹ c¢Á k‰W« mo

M»at‰¿‹ Ïw¡f¡ nfhz§fŸ Kiwna 45c k‰W« 60c MF«. nfhòu¤Â‹

cau« 90 Û våš, Rt‰¿‹ cau¤ij¡ fh©f. ( 3 .1 732= )

v.fh. 7.22 fl‰fiuæš cŸs br§F¤jhd¥ ghiw x‹¿‹ ÛJ f£l¥g£LŸs xU

fy§fiu és¡f¤Âš ã‹W¡bfh©oU¡F« xU ÁWä, »H¡FÂiræš ÏU

glFfis¥ gh®¡»whŸ. m¥glFfë‹ Ïw¡f¡nfhz§fŸ Kiwna 30c, 60c k‰W« ÏU glFfS¡»ilnaÍŸs öu« 300 Û våš, flš k£l¤ÂèUªJ

fy§fiu és¡f¤Â‹ c¢Áæ‹ öu¤ij¡ fh©f.

( glFfS«, fy§fiu és¡fK« xnu ne®¡nfh£oš cŸsd)

v.fh. 7.23 xU ÁWt‹, jiuæèUªJ 88.2 Û cau¤Âš »ilãiy¡ nfh£oš fh‰¿š

efU« xU gÿid 60c V‰w¡ nfhz¤Âš gh®¡»wh‹. jiu¡F« mtdJ

»ilãiy¥ gh®it¡ nfh£o‰F« Ïilna cŸs öu« 1.2 Û. Á¿J neu« fê¤J,

mnj Ïl¤ÂèUªJ mt‹ gÿid¥ gh®¡F« nghJ V‰w¡nfhz« 30cMf¡

Fiw»wJ våš, Ï¡fhy Ïilbtëæš gÿ‹ ef®ªj öu¤ij¡ fh©f.

v.fh. 7.24 xU f£ll¤Â‹ nkš xU bfho¡ f«g« ã‰»wJ. jiuæYŸs xU òŸëæèUªJ

bfho¡f«g¤Â‹ c¢Á k‰W« mo M»at‰¿‹ V‰w¡nfhz§fŸ Kiwna 60c k‰W« 45c v‹f. nkY« bfho¡ f«g¤Â‹ cau« 10 Û våš, f£ll¤Â‹

cau¤ij¡ fh©f. ( 3 .1 732= )

v.fh. 7.25 xUt® flšk£l¤ÂèUªJ 14 Û cauKŸs cŸs xU f¥gè‹ nkš js¤Âš ã‹W¡

bfh©L br§F¤jhd xU ghiw Kf£o‹ c¢Áæid 60c V‰w¡ nfhz¤ÂY«

mj‹ moæid 30c Ïw¡f¡ nfhz¤ÂY« fh©»wh® våš, br§F¤jhd

ghiwæ‹ cau« ahJ?

v.fh. 7.26 »ilãiyæš gwªJ bfh©oU¡F« xU Mfha ékhd¤ij A v‹D«

òŸëæèUªJ gh®¡F«nghJ mj‹ V‰w¡ nfhz« 60c. m›ékhd«

»ilãiyæš 15 édhofŸ gwªj Ã‹ mnj¥ òŸëæèUªJ mªj Mfha

ékhd¤Â‹ V‰w¡ nfhz« 30c Mf khW»wJ. Ï›ékhd« 200 Û/é ntf¤Âš

gwªJ bfh©oUªjhš, ékhd« gwªJ bfh©oU¡F« khwhj »ilãiy

cau¤ij¡ fh©f.


gæ‰Á 7.2

6. x‹W¡bfh‹W nebuÂuhf cŸs ÏU ku§fë‹ ÛJ A, B v‹w ÏU fh¡iffŸ 15 Û k‰W«

10 Û cau§fëš mk®ªJ¡ bfh©oUªjd. mit jiuæèU¡F« xU tilæid

Kiwna 45c k‰W« 60c Ïw¡f¡ nfhz¤Âš gh®¡»‹wd. mit xnu neu¤Âš »s«Ã¡

Fiwthd ÚsKŸs¥ ghijæš rkntf¤Âš gwªJ, m›tilia vL¡f Ka‰Á¤jhš

vªj gwit bt‰¿ bgW«? (ÏU ku§fë‹ mo, til M»ad xnu ne®¡nfh£oš cŸsd)

7. t£l toéš cŸs xU ó§fhé‹ ika¤Âš xU és¡F¡ f«g« ã‰»wJ. t£l¥ gçÂæš mikªj P, Q v‹D« ÏU òŸëfŸ és¡F¡ f«g¤Âdoæš 90c nfhz¤ij V‰gL¤J»‹wd. nkY«, P-æèUªJ és¡F¡ f«g¤Â‹ c¢Áæ‹

V‰w¡ nfhz« 30c MF«. PQ = 30 Û våš, és¡F¡ f«g¤Â‹ cau¤ij¡ fh©f.

8. 700 Û cau¤Âš gwªJ¡ bfh©oU¡F« xU bAèfh¥lçèUªJ xUt® X® M‰¿‹

ÏU fiufëš nebuÂuhf cŸs ÏU bghU£fis 30 , 45c c Ïw¡f¡ nfhz§fëš

fh©»wh® våš, M‰¿‹ mfy¤ij¡ fh©f. ( .3 1 732= )

9. rkjs¤Âš ã‹W bfh©oU¡F« X v‹gt®, mtçläUªJ 100 Û öu¤Âš gwªJ

bfh©oU¡F« xU gwitæid 30c V‰w¡ nfhz¤Âš gh®¡»wh®. mnj neu¤Âš,

Y v‹gt® 20 Û cauKŸs xU f£ll¤Â‹ c¢Áæš ã‹W bfh©L mnj gwitia

45c V‰w¡ nfhz¤Âš gh®¡»wh®. ÏUtU« ã‹W bfh©L mt®fS¡»ilnaÍŸs

gwitia vÂbuÂuhf¥ gh®¡»‹wd® våš, Y-æèUªJ gwit cŸs öu¤ij¡

fh©f.

10. tF¥giwæš mk®ªJ¡ bfh©oU¡F« xU khzt‹ fU«gyifæš »ilãiy¥

gh®it¡ nfh£oèUªJ 1.5 Û cau¤Âš cŸs Xéa¤ij 30c V‰w¡ nfhz¤Âš

fh©»wh‹. Xéa« mtD¡F¤ bjëthf¤ bjçahjjhš neuhf¡ fU«gyifia

neh¡» ef®ªJ Û©L« mªj Xéa¤ij 45c V‰w¡ nfhz¤Âš bjëthf¡

fh©»wh‹ våš, mt‹ ef®ªj öu¤ij¡ fh©f.

11. xU ÁWt‹ 30 Û cauKŸs f£ll¤Â‰F vÂnu F¿¥Ã£l öu¤Âš ã‰»wh‹.

mtDila¡ »ilãiy¥ gh®it¡nfhL jiuk£l¤ÂèUªJ 1.5Û cau¤Âš

cŸsJ. mt‹ f£ll¤ij neh¡» elªJ bršY« nghJ, m¡f£ll¤Â‹ c¢Áæ‹

V‰w¡ nfhz« 30c-èUªJ 60cMf ca®»wJ. mt‹ f£ll¤ij neh¡» elªJ

br‹w¤ öu¤ij¡ fh©f.

12. 200 mo cauKŸs fy§fiu és¡f¤Â‹ c¢ÁæèUªJ, mj‹ fh¥ghs® xU njhâ

k‰W« xU glF M»at‰iw gh®¡»wh®. fy§fiu és¡f¤Â‹ mo, njhâ

k‰W« glF M»ad xnu Âiræš xnu ne®¡nfh£oš mik»‹wd. njhâ, glF

M»at‰¿‹ Ïw¡f¡ nfhz§fŸ Kiwna k‰W«45 30c c v‹f. Ï›éu©L«

ghJfh¥ghf ÏU¡f nt©Lbkåš, mitfS¡F Ïil¥g£l öu« FiwªjJ 300 moahf ÏU¡f nt©L«. Ïilbtë Fiwªjhš fh¥ghs® v¢rç¡if xè vG¥g

nt©L«. mt® v¢rç¡if xè vG¥g nt©Lkh?


13. jiuæš ã‹Wbfh©oU¡F« xU ÁWt‹ fh‰¿š, »ilãiy¡ nfh£oš khwhj

cau¤Âš ef®ªJ¡ bfh©oU¡F« xU gÿid¡ fh©»wh‹. xU F¿¥Ã£l

ãiyæš ÁWt‹ 60c V‰w¡ nfhz¤Âš gÿid¡ fh©»wh‹. 2 ãäl§fŸ fêªj

Ã‹d® mnj ãiyæèUªJ ÁWt‹ Û©L« gÿid¥ gh®¡F«nghJ V‰w¡nfhz«

30c Mf¡ Fiw»wJ. fh‰¿‹ ntf« 29 3 Û/ãäl« våš, jiuæèUªJ gÿå‹

cau« fh©f.

14. xU neuhd beLŠrhiy xU nfhòu¤ij neh¡»¢ brš»wJ. nfhòu¤Â‹ c¢Áæš

ã‹W¡ bfh©oU¡F« xUt® Óuhd ntf¤Âš tªJ¡bfh©oU¡F« xU C®Âia

30c Ïw¡f¡ nfhz¤Âš fh©»wh®. 6 ãäl§fŸ fêªj Ã‹d® mªj C®Âæ‹

Ïw¡f¡ nfhz« 60cvåš, nfhòu¤ij mila C®Â nkY« v¤jid ãäl§fŸ

vL¤J¡ bfhŸS«?

15. xU bra‰if¡ nfhS¡F xnu Âiræš óäæš mikªJŸs ÏU f£L¥gh£L

ãiya§fëèUªJ m¢bra‰if¡ nfhë‹ V‰w¡ nfhz§fŸ Kiwna 30c k‰W« 60c vd cŸsd. m›éU ãiya§fŸ, bra‰if¡nfhŸ M»a Ïit _‹W«

xnu br§F¤J¤ js¤Âš mik»‹wd. ÏU ãiya§fS¡F« Ïilna cŸs

öu« 4000 ».Û våš, bra‰if¡nfhS¡F«, óä¡F« Ïilna cŸs öu¤ij¡

fh©f. ( .3 1 732= )

16. 60 Û cauKŸs xU nfhòu¤ÂèUªJ xU f£ll¤Â‹ c¢Á k‰W« mo M»at‰¿‹

Ïw¡f¡ nfhz§fŸ Kiwna 30 60k‰W«c c våš, f£ll¤Â‹ cau¤ij¡ fh©f.

17. 40 Û cauKŸs xU nfhòu¤Â‹ c¢Á k‰W« mo M»at‰¿èUªJ xU fy§fiu

és¡»‹ c¢Áæ‹ V‰w¡ nfhz§fŸ Kiwna 30ck‰W« 60c våš, fy§fiu

és¡»‹ cau¤ij¡ fh©f. fy§fiu és¡»‹ c¢ÁæèUªJ nfhòu¤Â‹

mo¡F cŸs öu¤ijÍ« fh©f.

18. xU Vçæ‹ nk‰gu¥Ãš xU òŸëæèUªJ fhQ«nghJ, 45 Û cau¤Âš gwªJ

bfh©oU¡F« xU bAèfh¥lç‹ V‰w¡ nfhz« 30c Mf cŸsJ.

m¥òŸëæèUªJ mnj neu¤Âš j©Ùçš bAèfh¥lç‹ ãHè‹ Ïw¡f¡

nfhz« 60c våš, bAèfh¥lU¡F« Vçæ‹ nk‰gu¥Ã‰F« Ïil¥g£l¤ öu¤ij¡

fh©f.


8. mséaš


v.fh. 8.1 xU Â©k ne® t£l cUisæ‹ (solid right circular cylinder) Mu« 7 brÛ k‰W« cau«

20 brÛ våš, mj‹ (i) tisgu¥ò (ii) bkh¤j¥ òw¥gu¥ò M»at‰iw¡ fh©f.

(722r = v‹f). (x›bthU c£Ãçé‰F« Ïu©L kÂ¥bg©fŸ)

v.fh. 8.6 xU Â©k ne® t£l¡ T«Ã‹ Mu« k‰W« rhÍau« Kiwna 35 br.Û k‰W«

37 br.Û. våš T«Ã‹ tisgu¥ò k‰W« bkh¤j¥ òw¥gu¥ig¡ fh©f.

(722r = v‹f)

v.fh. 8.9 7 Û cŸé£lKŸs xU cŸÇl‰w nfhs¤ÂDŸ c£òwkhf xU r®¡f°

Åu® nkh£lh® ir¡»ëš rhfr« brŒ»wh®. mªj rhfr Åu® rhfr« brŒa¡

»il¤ÂL« cŸÇl‰w¡ nfhs¤Â‹ c£ òw¥gu¥ig¡ fh©f. ( 722r = v‹f)

v.fh. 8.10 xU Â©k miu¡nfhs¤Â‹ bkh¤j òw¥gu¥ò 675r r.br.Û våš mj‹

tisgu¥ig¡ fh©f.

v.fh. 8.11 miu¡nfhs tot »©z¤Â‹ jok‹ 0.25 br.Û. mj‹ c£òw Mu« 5 br.Û

våš m¡»©z¤Â‹ btë¥òw tisgu¥ig¡ fh©f. ( 722r = v‹f)

gæ‰Á 8.1

1. xU Â©k ne® t£l cUisæ‹ Mu« 14 br.Û k‰W« cau« 8 br.Û. våš, mj‹


7. Ïu©L ne® t£l cUisfë‹ Mu§fë‹ é»j« 3 : 2 v‹f. nkY« mt‰¿‹

cau§fë‹ é»j« 5 : 3 våš, mt‰¿‹ tisgu¥òfë‹ é»j¤ij¡ fh©f.

12. xU Â©k ne® t£l¡ T«Ã‹ mo¢R‰wsÎ 236 br.Û . k‰W« mj‹ rhÍau« 12 br.Û

våš, m¡T«Ã‹ tisgu¥ig¡ fh©f.

16. 98.56 r.br.Û òw¥gu¥ò bfh©l xU Â©k¡ nfhs¤Â‹ Mu¤ij¡ fh©f.

17. xU Â©k miu¡nfhs¤Â‹ tisgu¥ò 2772 r.br.Û våš, mj‹ bkh¤j¥ òw¥gu¥ig¡

fh©f.

18. Ïu©L Â©k miu¡nfhs§fë‹ Mu§fŸ 3 : 5 v‹w é»j¤Âš cŸsd.

m¡nfhs§fë‹ tisgu¥òfë‹ é»j« k‰W« bkh¤j¥ òw¥gu¥òfë‹ é»j«

M»at‰iw¡ fh©f.

v.fh. 8.15 xU Â©k ne®t£l¡ T«Ã‹ fdmsÎ 4928 f.br.Û. k‰W« mj‹ cau«

24 br. Û våš, m¡T«Ã‹ Mu¤ij¡ fh©f. (722r = v‹f )

v.fh. 8.16 xU Ïil¡f©l toéyhd thëæ‹ nk‰òw k‰W« mo¥òw Mu§fŸ Kiwna

15 br.Û k‰W« 8 br.Û. nkY«, MH« 63 br.Û våš, mj‹ bfhŸssit è£lçš

fh©f. (722r = )

tif¥gL¤j¥g£l édh¡fŸ - mséaš 363

v.fh. 8.17 8.4 br.Û é£l« bfh©l xU nfhstot Â©k cnyhf v¿F©o‹

fdmsit¡ fh©f.(722r = v‹f )

v.fh. 8.18 xU T«ò, xU miu¡nfhs«, k‰W« xU cUis M»ad rk mo¥gu¥Ãid¡

bfh©LŸsd. T«Ã‹ cau«, cUisæ‹ cau¤Â‰F rkkhfÎ«,

nkY« Ï›Îau«, mt‰¿‹ Mu¤Â‰F rkkhfÎ« ÏUªjhš Ï«_‹¿‹

fd msÎfS¡»ilna cŸs é»j¤ij¡ fh©f.

v.fh. 8.19 xU Â©k¡ nfhs¤Â‹ fdmsÎ 7241 71 f.br.Û våš, mj‹ Mu¤ij¡

fh©f. (722r = v‹f)

v.fh. 8.20 xU cŸÇl‰w nfhs¤Â‹ fdmsÎ 7

11352 f.br.Û. k‰W« mj‹ btë Mu«

8 br.Û. våš, m¡nfhs¤Â‹ cŸMu¤ij¡ fh©f (722r = v‹f).

gæ‰Á 8.2

1. xU Â©k cUisæ‹ Mu« 14 br.Û. mj‹ cau« 30 br.Û våš, m›ÎUisæ‹

fdmsit¡ fh©f.

2. xU kU¤JtkidæYŸs nehahë xUtU¡F ÂdK« 7 br.Û é£lKŸs cUis tot

»©z¤Âš to¢rhW (Soup) tH§f¥gL»wJ. m¥gh¤Âu¤Âš 4 br.Û cau¤Â‰F

to¢rhW xU nehahë¡F tH§f¥g£lhš, 250 nehahëfS¡F tH§f¤ njitahd

to¢rh¿‹ fdmséid¡ fh©f.

4. 62.37 f.br.Û. fdmsÎ bfh©l xU Â©k ne®t£l cUisæ‹ cau« 4.5 br.Û våš,

m›ÎUisæ‹ Mu¤ij¡ fh©f.

5. Ïu©L ne® t£l cUisfë‹ Mu§fë‹ é»j« 2 : 3. nkY« cau§fë‹ é»j«

5 : 3 våš, mt‰¿‹ fdmsÎfë‹ é»j¤ij¡ fh©f.

8. xU bg‹ÁyhdJ xU ne® t£l cUis toéš cŸsJ. bg‹Áè‹ Ús« 28 br.Û k‰W«

mj‹ Mu« 3 ä.Û. bg‹ÁèDŸ mikªj ikæ‹ (»uh~ig£)-‹ Mu« 1 ä.Û våš,

bg‹Áš jahç¡f ga‹gL¤j¥g£l ku¥gyifæ‹ fdmsit¡ fh©f.

10. ku¤Âdhyhd xU Â©k¡ T«Ã‹ mo¢R‰wsÎ 44 br.Û k‰W« mj‹ cau«

12 br.Ûvåš m¤Â©k¡ T«Ã‹ fdmsit¡ fh©f.

13. 5 br.Û, 12 br.Û. k‰W« 13 br.Û. g¡f msÎfŸ bfh©l xU br§nfhz TABC MdJ

12 br.Û. ÚsKŸs mj‹ xU g¡f¤ij m¢rhf¡ bfh©L RH‰w¥gL«nghJ cUthF«

T«Ã‹ fdmsit¡ f©LÃo.

15. xU ne® t£l¡ T«Ã‹ fdmsÎ 216r f.br.Û k‰W« m¡T«Ã‹ Mu« 9 br.Û våš,

mj‹ cau¤ij¡ fh©f.

17. xU cŸÇl‰w nfhs¤Â‹ btë k‰W« cŸ Mu§fŸ Kiwna 12 br.Û k‰W« 10 br. Û

våš, m¡nfhs¤Â‹ fd msit¡ fh©f.


19. 14 br.Û g¡f msÎfŸ bfh©l xU fd¢rJu¤Âš ÏUªJ bt£obaL¡f¥gL«

äf¥bgça T«Ã‹ fdmsit¡ fh©f.

20. 7 br.Û Mu« bfh©l nfhs tot gÿåš fh‰W brY¤j¥gL«nghJ mj‹ Mu«

14 br.Û Mf mÂfç¤jhš, m›éU ãiyfëš gÿå‹ fdmsÎfë‹ é»j¤ij¡

fh©f.


v.fh. 8.2 xU Â©k ne® t£l cUisæ‹ bkh¤j¥ òw¥gu¥ò 880 r.br.Û k‰W« mj‹ Mu«

10 br.Û våš, m›ÎUisæ‹ tisgu¥ig¡ fh©f (722r = v‹f).

v.fh. 8.3 xU Â©k ne® t£l cUisæ‹ MuK« cauK« 2 : 5 v‹w é»j¤Âš cŸsd.

mj‹ tis¥gu¥ò 7

3960 r.br.Û våš, cUisæ‹ Mu« k‰W« cau« fh©f.

(722r = v‹f)

v.fh. 8.4 120 br.Û ÚsK«, 84 br.Û é£lK« bfh©l xU rhiyia rk¥gL¤J« cUisia¡ (road roller) bfh©L xU éisah£L¤Âlš rk¥gL¤j¥gL»wJ.

éisah£L¤ Âliy rk¥gL¤j Ï›ÎUis 500 KG¢ R‰W¡fŸ RHy nt©L«.

éisah£L¤Âliy rk¥gL¤j xU r. Û£lU¡F 75 igrh Åj«, Âliy¢ rk¥gL¤j

MF« bryit¡ fh©f. (722r = v‹f)

v.fh. 8.5 xU cŸÇl‰w cUisæ‹ cŸ k‰W« btë Mu§fŸ Kiwna 12 br.Û. k‰W« 18 br.Û. v‹f. nkY« mj‹ cau« 14 br.Û våš m›ÎUisæ‹ tisgu¥ò

k‰W« bkh¤j¥ òw¥gu¥ig¡ fh©f. (722r = v‹f.)

v.fh. 8.7 O k‰W« C v‹gd Kiwna xU ne® t£l¡ T«Ã‹ mo¥gFÂæ‹ ika« k‰W«

c¢Á v‹f. B v‹gJ mo¥gFÂæ‹ t£l¢ R‰W éë«Ãš VnjD« xU òŸë

v‹f. T«Ã‹ mo¥gFÂæ‹ Mu« 6 br.Û k‰W« OBC 60o+ = våš, T«Ã‹

cau« k‰W« tisgu¥ig¡ fh©f.

v.fh. 8.8 21 br.Û MuKŸs xU t£l¤ÂèUªJ 120c ika¡ nfhz« bfh©l xU

t£l¡ nfhz¥gFÂia bt£obaL¤J, mj‹ Mu§fis x‹¿iz¤J xU

T«gh¡»dhš, »il¡F« T«Ã‹ tisgu¥ig¡ fh©f (722r = ).

gæ‰Á 8.1

2. xU Â©k ne® t£l cUisæ‹ bkh¤j¥ òw¥gu¥ò 660 r. br.Û. mj‹ é£l« 14 br.Û.

våš, m›ÎUisæ‹ cau¤ijÍ«, tisgu¥igÍ« fh©f.

3. xU Â©k ne® t£l cUisæ‹ tisgu¥ò k‰W« mo¢R‰wsÎ Kiwna 4400 r.br.Û

k‰W« 110 br.Û. våš, m›ÎUisæ‹ cau¤ijÍ«, é£l¤ijÍ« fh©f.

4. xU khëifæš, x›bth‹W« 50 br.Û. MuK«, 3.5 Û cauK« bfh©l 12 ne® t£l

cUis tot¤ ö©fŸ cŸsd. m¤ö©fS¡F t®z« ór xU rJu Û£lU¡F

` 20 Åj« v‹d brythF«?


5. xU Â©k ne® t£l cUisæ‹ bkh¤j¥ òw¥gu¥ò 231 r. br.Û. mj‹ tisgu¥ò bkh¤j

òw¥gu¥Ãš _‹¿š Ïu©L g§F våš, mj‹ Mu« k‰W« cau¤ij¡ fh©f.

6. xU Â©k ne® t£l cUisæ‹ bkh¤j òw¥gu¥ò 1540 br.Û 2. mj‹ caukhdJ,

mo¥g¡f Mu¤ij¥nghš eh‹F kl§F våš, cUisæ‹ cau¤ij¡ fh©f.

8. xU cŸÇl‰w cUisæ‹ btë¥òw tisgu¥ò 540r r.br.Û. mj‹ cŸé£l« 16 br.Û

k‰W« cau« 15 br.Û. våš, mj‹ bkh¤j òw¥gu¥ig¡ fh©f.

9. xU cUis tot ÏU«ò¡FHhæ‹ btë¥òw é£l« 25 br.Û, mj‹ Ús« 20 br.Û. k‰W«

mj‹ jok‹ 1 br.Û våš, m¡FHhæ‹ bkh¤j¥ òw¥gu¥ig¡ fh©f.

10. xU Â©k ne®t£l¡ T«Ã‹ Mu« k‰W« cau« Kiwna 7 br.Û k‰W« 24 br.Û. våš,

mj‹ tisgu¥ò k‰W« bkh¤j¥ òw¥gu¥ig¡ fh©f.

11. xU Â©k ne® t£l¡ T«Ã‹ c¢Á¡nfhz« k‰W« Mu« Kiwna 60ck‰W« 15 br.Û våš, mj‹ cau« k‰W« rhÍau¤ij¡ fh©f.

13. ne®t£l T«ò toéš Fé¡f¥g£l be‰Féaè‹ é£l« 4.2 Û k‰W« mj‹ cau«

2.8 Û. v‹f. Ïªbe‰Féaiy kiHæèUªJ ghJfh¡f »¤jh‹ Jâahš äf¢rçahf

_l¥gL»wJ våš, njitahd »¤jh‹ Jâæ‹ gu¥ig¡ fh©.

14. 180c ika¡ nfhzK« 21 br.Û. MuK« bfh©l t£lnfhz toéyhd ÏU«ò¤ jf£o‹

Mu§fis Ïiz¤J xU T«ò cUth¡f¥gL»wJ våš, m¡T«Ã‹ Mu¤ij¡

fh©f.

15. xU ne®t£l Â©k¡ T«Ã‹ MuK« rhÍauK« 3 : 5 v‹w é»j¤Âš cŸsd. m¡T«Ã‹

tisgu¥ò 60r r.br.Û våš, mj‹ bkh¤j¥ òw¥gu¥ig¡ fh©f.

19. xU cŸÇl‰w miu¡nfhs¤Â‹ btë Mu« k‰W« cŸ Mu« Kiwna 4.2 br.Û k‰W«

2.1 br.Û våš mj‹ tisgu¥ò k‰W« bkh¤j òw¥gu¥ig¡ fh©f.

20. miu¡nfhs tot nk‰Tiuæ‹ c£òw tisgu¥Ã‰F t®z« ór nt©oÍŸsJ. mj‹

c£òw mo¢R‰wsÎ 17.6 Û våš, xU rJu Û£lU¡F ` 5 Åj«, t®z« ór MF«

bkh¤j bryit¡ fh©f.

v.fh. 8.12 xU ne®t£l cUisæ‹ tisgu¥ò 704 r.br.Û k‰W« mj‹ cau« 8 br.Û våš,

m›ÎUisæ‹ fdmsit è£lçš fh©f. ( 722r = )

v.fh. 8.13 xU cŸÇl‰w ÏU«ò FHhæ‹ Ús« 28 br.Û., mj‹ btë k‰W« cŸé£l§fŸ

Kiwna 8 br.Û k‰W« 6 br.Û. våš, ÏU«ò¡ FHhæ‹ fdmsit¡ fh©f.

nkY« 1 f.br.Û ÏU«Ã‹ vil 7 »uh« våš, ÏU«ò¡ FHhæ‹ vilia¡

fh©f. ( 722r = )

v.fh. 8.14 xU Â©k ne® t£l cUisæ‹ mo¥g¡f¥ gu¥ò k‰W« fdmsÎ Kiwna

13.86 r.br.Û k‰W« 69.3 f.br.Û. våš, m›ÎUisæ‹ cau« k‰W« tisgu¥ig¡

fh©f. (722r = )


gæ‰Á 8.2

3. xU Â©k cUisæ‹ Mu« k‰W« cau¤Â‹ TLjš 37 br.Û. v‹f. nkY«, mj‹

bkh¤j òw¥gu¥ò 1628 r.br.Û våš, m›ÎUisæ‹ fdmsit¡ fh©f.

6. xU cUisæ‹ Mu« k‰W« cau¤Â‹ é»j« 5 : 7. nkY« mj‹ fdmsÎ

4400 f.br.Û våš, m›ÎUisæ‹ Mu¤ij¡ fh©f.

7. 66 br.Û # 12 br.Û vD« msÎ¡ bfh©l xU cnyhf¤ jf£oid 12 br.Û cauKŸs xU

cUisahf kh‰¿dhš »il¡F« cUisæ‹ fdmsit¡ fh©f.

9. xU Â©k¡ T«Ã‹ Mu« k‰W« rhÍau« Kiwna 20 br.Û k‰W« 29 br.Û. våš

m¤Â©k¡ T«Ã‹ fdmsit¡ fh©f.

11. xU gh¤Âu« Ïil¡f©l« toéš cŸsJ. mj‹ nk‰òw Mu« k‰W« cau« Kiwna

8 br.Û k‰W« 14 br.Û v‹f. m¥gh¤Âu¤Â‹ fdmsÎ 3

5676 f.br.Û våš,

mo¥g¡f¤ÂYŸs t£l¤Â‹ Mu¤Âid¡ fh©f.

12. xU ne®t£l¡ T«Ã‹ Ïil¡f©l¤Â‹ ÏUòwK« mikªj t£l éë«òfë‹

R‰wsÎfŸ Kiwna 44 br.Û k‰W« 8.4r br.Û v‹f. mj‹ cau« 14 br.Û våš,

m›éil¡f©l¤Â‹ fdmsit¡ fh©f.

14. xU Â©k ne® t£l¡ T«Ã‹ MuK« cauK« 2 : 3 v‹w é»j¤Âš cŸsJ. mj‹

fdmsÎ 100.48 f.br.Û våš, m¡T«Ã‹ rhÍau¤ij¡ f©LÃo. (r = 3.14 v‹f)

16. nfhs toéyikªj 200 ÏU«ò F©LfŸ (ball bearings) x›bth‹W« 0.7 br.Û Mu«

bfh©lJ. ÏU«Ã‹ ml®¤Â 7.95 »uh« /br.Û 3 våš ÏU«ò¡ F©Lfë‹

ãiwia¡ fh©f. (ãiw = fdmsÎ × ml®¤Â)

18. X® miu¡nfhs¤Â‹ fd msÎ 1152r f.br. Û. våš, mj‹ tisgu¥ò fh©f.

v.fh. 8.21 xU Â©k ku¥bgh«ikahdJ miu¡nfhs¤Â‹ nkš T«ò Ïizªj

toéš cŸsJ. miu¡nfhs« k‰W« T«ò M»at‰¿‹ Mu« 3.5 br.Û.

nkY« bgh«ikæ‹ bkh¤j cau« 17.5 br.Û våš m¥bgh«ik jahç¡f¥

ga‹gL¤j¥g£l ku¤Â‹ fd msit¡ fh©f. (r= 722 )

v.fh. 8.22 xU nfh¥igahdJ miu¡nfhs¤Â‹ ÛJ cUis Ïizªj toéš cŸsJ.

cUis¥ gFÂæ‹ cau« 8 br.Û k‰W« nfh¥igæ‹ bkh¤j cau« 11.5 br.Û.

våš m¡nfh¥igæ‹ bkh¤j¥ òw¥gu¥ig¡ fh©f. (r = 722 v‹f.)

v.fh. 8.23 xU r®¡f° TlhukhdJ cUisæ‹ ÛJ T«ò Ïizªj toéš

mikªJŸsJ. Tlhu¤Â‹ bkh¤j cau« 49 Û. mj‹ mo¥ghf¤Â‹ é£l«

42 Û. cUis¥ghf¤Â‹ cau« 21 Û. nkY« 1 r.Û »¤jh‹ Jâæ‹ éiy 12.50 våš, Tlhu« mik¡f¤ njitahd »¤jh‹ Jâæ‹ éiyia¡ fh©f.

( r = 722 )

v.fh. 8.24 xU cŸÇl‰w nfhs¤Â‹ btë k‰W« cŸ é£l§fŸ Kiwna 8 br.Û k‰W«

4 br.Û. Ï¡nfhskhdJ cU¡f¥g£L 8 br.Û é£lKŸs ne® t£l Â©k¡

T«ghf kh‰w¥g£lhš T«Ã‹ cau¤ij¡ fh©f.


v.fh. 8.25 7 br.Û é£lKŸs cUis tot Kfitæš Á¿jsÎ j©Ù® cŸsJ. mÂš

x›bth‹W« 1.4 br.Û é£lKŸs Áy nfhs tot gë§F¡ f‰fŸ nghl¥gL»wJ.

cUisæYŸs Úç‹ k£l« 5.6 br.Û cau v¤jid gë§F f‰fis KfitæDŸ

nghlnt©L« vd¡ fh©f?

v.fh. 8.2 14 br.Û é£lKŸs xU cUis tot FHhŒ têahf, j©Ùiu kâ¡F 15 ».Û

ntf¤Âš 50 Û ÚsK« k‰W« 44 Û mfyKŸs xU br›tf tot bjh£o¡FŸ

brY¤Âdhš, bjh£oæš 21 br.Û cau¤Â‰F j©Ù® ãu«g v¤jid kâ

neukhF«? (r = 722 v‹f.)

v.fh. 8.27 55 br.Û#40 br.Û#15 br.Û v‹w msÎfŸ bfh©l fd¢br›tf tot Â©k

ÏU«ò¥ gyif cU¡f¥g£L xU FHhahf th®¡f¥gL»wJ. m¡FHhæ‹ btë

é£l« k‰W« jok‹ Kiwna 8 br.Û k‰W« 1 br.Û våš, m¡FHhæ‹ Ús¤ij¡

fh©f. (r = 722 v‹f.)

gæ‰Á 8.3

1. xU éisah£L g«gukhdJ (Top) T«Ã‹ ÛJ miu¡nfhs« Ïizªj toéš cŸsJ.

miu¡nfhs¤Â‹ é£l« 3.6 br.Û k‰W« g«gu¤Â‹ bkh¤j cau« 4 . 2 br.Û våš,

mj‹ bkh¤j¥ òw¥gu¥ig¡ fh©f.

2. xU fd cUt«, miu¡nfhs¤Â‹ ÛJ cUis Ïizªj toéš cŸsJ.

m¡fdÎUt¤Â‹ é£l« k‰W« bkh¤j cau« Kiwna 21 br.Û k‰W« 25.5 br.Û

våš, mj‹ fd msit¡ fh©f.

3. xU kUªJ¡ F¥ÃahdJ xU cUisæ‹ ÏUòwK« miu¡nfhs§fŸ Ïizªj toéš

cŸsJ. kUªJ¡ F¥Ãæ‹ bkh¤j Ús« 14 ä.Û k‰W« é£l« 5 ä.Û våš m«kUªJ¡

F¥Ãæ‹ òw¥gu¥ig¡ fh©f.

4. xU TlhukhdJ cUisæ‹ ÛJ T«ò Ïizªj toéš cŸsJ. Tlhu¤Â‹ bkh¤j

cau« 13.5 Û k‰W« é£l« 28 Û. nkY« cUis¥ ghf¤Â‹ cau« 3 Û våš,

Tlhu¤Â‹ bkh¤j òw¥gu¥ig¡ fh©f.

5. fëk©iz¥ ga‹gL¤Â xU khzt‹ 48 br.Û cauK« 12 br.Û MuK« bfh©l ne®

t£lÂ©k¡ T«ig¢ brŒjh®. m¡T«ig k‰bwhU khzt® xU Â©k¡ nfhskhf

kh‰¿dh®. m›thW kh‰w¥g£l òÂa nfhs¤Â‹ Mu¤ij¡ fh©f.

6. 24 br.Û MuKŸs xU Â©k cnyhf nfhskhdJ cU¡f¥g£L 1.2 ä.Û MuKŸs Óuhd

cUis¡ f«Ãahf kh‰w¥g£lhš, m¡f«Ãæ‹ Ús¤ij¡ fh©f.

7. 5 br.Û cŸt£lMuK« 24 br.Û cauK« bfh©l T«ò tot gh¤Âu¤Âš KG mséš

j©Ù® cŸsJ. Ï¤j©ÙuhdJ 10 br.Û cŸMuKŸs cUis tot fhè¥

gh¤Âu¤Â‰F¥ kh‰w¥gL«nghJ, cUis¥ gh¤Âu¤Âš cŸs Ú® k£l¤Â‹ cau¤ij¡

fh©f.

8. Á¿jsÎ j©Ù® ãu¥g¥g£l 12 br.Û é£lKŸs cUis tot¥ gh¤Âu¤Âš 6 br.Û

é£lKŸs xU Â©k¡ nfhs¤ij KGtJkhf _œf¢ brŒjhš, cUis tot¥

gh¤Âu¤Âš ca®ªj Ú® k£l¤Â‹ cau¤ij¡ fh©f.


9. 7 br.Û cŸ Mu« bfh©l cUis tot FHhæ‹ têna 5 br.Û / édho ntf¤Âš

j©Ù® ghŒ»wJ. miu kâ neu¤Âš m¡FHhŒ têna ghŒªj j©Ùç‹

fd msit¡ (è£lçš) fh©f.

10. 4 Û é£lK« 10 Û cauKŸs cUis tot¤ bjh£oæYŸs j©ÙuhdJ 10 br.Û

é£lKŸs xU cUis tot FHhŒ têna kâ¡F 2.5 ».Û ntf¤Âš

btëna‰w¥gL»wJ. bjh£oæš ghÂasÎ j©Ù® btëna‰w¥gl MF« neu¤ij¡

fh©f. (Mu«g ãiyæš bjh£o KGtJ« j©Ù® ãu¥g¥g£LŸsJ vd¡ bfhŸf.)

11. 18 br.Û MuKŸs Â©k cnyhf¡ nfhskhdJ cU¡f¥g£L _‹W Á¿a bt›ntW


nfhs§fë‹ Mu§fŸ Kiwna 2 br.Û k‰W« 12 br.Û våš _‹whtJ nfhs¤Â‹

Mu¤ij¡ fh©f.

12. xU cŸÇl‰w cUis tot¡ FHhæ‹ Ús« 40 br.Û. mj‹ cŸ k‰W« btë Mu§fŸ

Kiwna 4 br.Û k‰W« 12 br.Û. m›ÎŸÇl‰w cUis¡ FHhŒ cU¡f¥g£L 20 br.Û

ÚsKŸs Â©k ne® t£l cUisahf kh‰W«nghJ »il¡F« òÂa cUisæ‹

Mu¤ij¡ fh©f.

13. 8 br.Û é£lK« 12 br.Û cauK« bfh©l xU ne® t£l Â©k ÏU«ò¡ T«ghdJ

cU¡f¥g£L 4 ä.Û MuKŸs Â©k¡ nfhs tot F©Lfshf th®¡f¥g£lhš

»il¡F« nfhs tot F©Lfë‹ v©â¡ifia¡ fh©f.

14. 12 br.Û é£lK« 15 br.Û cauK« bfh©l ne®t£l cUis KGtJ« gå¡Têdhš

(ice cream) ãu¥g¥g£LŸsJ. Ï¥gå¡THhdJ 6 br.Û é£lK«, 12 br.Û cauK«

bfh©l nk‰òw« miu¡nfhs« Ïizªj toéyikªj T«Ãš ãu¥g¥gL»wJ.

v¤jid T«òfëš gå¡Têid KGtJkhf ãu¥gyh« vd¡ fh©f.

15. 4.4 Û ÚsK« 2 Û mfyK« bfh©l xU fd¢ br›tf tot¤ bjh£oæš kiHÚ®

nrfç¡f¥ gL»wJ. Ï¤bjh£oæš 4 br.Û cau¤Â‰F nrfç¡f¥g£l kiH ÚuhdJ

40br.Û MuKŸs cUis tot fhè¥ gh¤Âu¤Â‰F kh‰w¥gL«nghJ m¥gh¤Âu¤Âš

cŸs j©Ù® k£l¤Â‹ cau¤ij¡ fh©f.

16. kzyhš ãu¥g¥g£l xU cUis tot thëæ‹ cau« 32 br.Û k‰W« Mu«

18 br.Û. m«kzš KGtJ« jiuæš xU ne®t£l¡ T«ò toéš bfh£l¥gL»wJ.

m›thW bfh£l¥g£l kz‰ T«Ã‹ cau« 24 br.Û våš, m¡T«Ã‹ Mu« k‰W«

rhÍau¤ij¡ fh©f.

17. 14 Û é£lK« k‰W« 20 Û MHKŸs xU »zW cUis toéš bt£l¥gL»wJ.

m›thW bt£L«nghJ njh©obaL¡f¥g£l k© Óuhf gu¥g¥g£L 20 Û # 14 Û

msÎfëš mo¥g¡fkhf¡ bfh©l xU nkilahf mik¡f¥g£lhš, m«nkilæ‹

cau« fh©f.

tif¥gL¤j¥g£l édh¡fŸ - òŸëæaš 369

11. òŸëæaš


v.fh. 11.1 43, 24, 38, 56, 22, 39, 45 M»a òŸë étu§fë‹ Å¢R k‰W« Å¢R¡bfG

fh©f.

v.fh. 11.2 xU tF¥ÃYŸs 13 khzt®fë‹ vil (».») Ã‹tUkhW.

42.5, 47.5, 48.6, 50.5, 49, 46.2, 49.8, 45.8, 43.2, 48, 44.7, 46.9, 42.4

Ït‰¿‹ Å¢R k‰W« Å¢R¡ bfGit¡ fh©f.

v.fh. 11.3 xU òŸë étu¤ bjhF¥Ã‹ Û¥bgU kÂ¥ò 7.44 k‰W« mj‹ Å¢R 2.26 våš,

m¤bjhF¥Ã‹ Û¢ÁW kÂ¥ig¡ fh©f.

v.fh. 11.11 Kjš 10 Ïaš v©fë‹ Â£l éy¡f« fh©f.

v.fh. 11.19 xU òŸë étu¤Âš 30 kÂ¥òfë‹ T£L¢ ruhrç k‰W« Â£l éy¡f«

Kiwna 18 k‰W« 3 MF«. mt‰¿‹ T£L¤ bjhifiaÍ«, nkY«

mt‰¿‹ t®¡f§fë‹ T£L¤ bjhifiaÍ« fh©f.

gæ‰Á 11.1

1. Ã‹tU« kÂ¥òfS¡F Å¢R k‰W« Å¢R¡ bfG fh©f.


(i) 59, 46, 30, 23, 27, 40, 52, 35, 29.

(ii) 41.2, 33.7, 29.1, 34.5, 25.7, 24.8, 56.5, 12.5.

2. xU òŸë étu¤Â‹ Û¢ÁW kÂ¥ò 12. mj‹ Å¢R 59 våš m¥òŸë étu¤Â‹ Û¥bgU

kÂ¥ig¡ fh©f.

3. 50 msÎfëš äf¥bgça kÂ¥ò 3.84 ».». mj‹ Å¢R 0.46 ».» våš, mitfë‹ Û¢ÁW kÂ¥ig¡ fh©f.

4. f©l¿ªj òŸë étu¤ bjhF¥ÃYŸs 20 kÂ¥òfë‹ Â£l éy¡f« 5 v‹f.

òŸë étu¤Â‹ x›bthU kÂ¥igÍ« 2 Mš bgU¡»dhš »il¡F« òÂa òŸë

étu§fë‹ Â£l éy¡f« k‰W« éy¡f t®¡f¢ ruhrç fh©f.


13. xU òŸë étu¤ bjhF¥ÃYŸs 100 kÂ¥òfë‹ ruhrç k‰W« Â£l éy¡f« Kiwna

48 k‰W« 10 MF«. mid¤J kÂ¥òfë‹ T£L¤ bjhif k‰W« mitfë‹

t®¡f§fë‹ T£L¤ bjhif M»at‰iw¡ fh©f.

15. n = 10, x = 12 k‰W« x2

R = 1530 våš, khWgh£L¡ bfGit¡ fz¡»Lf.

17. xU òŸë étu¤Â‹ khWgh£L¡ bfG 57 k‰W« Â£l éy¡f« 6.84 våš, mj‹ T£L¢

ruhrçia¡ fh©f.

18. xU FGéš 100 ng® cŸsd®, mt®fë‹ cau§fë‹ T£L¢ ruhrç 163.8 br.Û k‰W«

khWgh£L¡ bfG 3.2 våš, mt®fSila cau§fë‹ Â£l éy¡f¤ij¡ fh©f.



v.fh. 11.4 xU khj¤Âš 8 khzt®fŸ go¤j ò¤jf§fë‹ v©â¡if Ã‹tUkhW.

2, 5, 8, 11, 14, 6, 12, 10. Ï¥òŸë étu¤Â‹ Â£l éy¡f¤ij¡ fz¡»Lf.

v.fh. 11.5 xU tF¥Ã‰F el¤j¥g£l bghJ m¿Î¤nj®éš bkh¤j kÂ¥bg©fŸ 40-¡F,

6 khzt®fŸ bg‰w kÂ¥bg©fŸ 20, 14, 16, 30, 21 k‰W« 25. Ï¥òŸë

étu¤Â‹ Â£l éy¡f« fh©f.

v.fh. 11.6 62, 58, 53, 50, 63, 52, 55 M»a v©fS¡F Â£l éy¡f« fh©f.

v.fh. 11.7 10 khzt®fŸ fâj¤ nj®éš bg‰w kÂ¥bg©fŸ Ã‹tUkhW,

80, 70, 40, 50, 90, 60, 100, 60, 30, 80. Ï«kÂ¥òfS¡F Â£l éy¡f« fh©f

v.fh. 11.18 3, 5, 6, 7 M»a òŸë étu¤Â‰F¤ Â£l éy¡f« fh©f. x›bthU

kÂ¥òlD« 4 I¡ T£l »il¡F« òÂa kÂ¥òfS¡fhd Â£l éy¡f¤ijÍ«

fh©f.

v.fh. 11.9 40, 42, 48 vD« Ï¥òŸë étu¤Â‰F¤ Â£l éy¡f« fh©f. x›bthU kÂ¥ò« 3 Mš bgU¡f¥gL«nghJ »il¡F« òÂa kÂ¥òfS¡fhdÂ£l éy¡f« fh©f.

v.fh. 11.10 Kjš n Ïaš v©fë‹ Â£l éy¡f« v = n12

12- vd ã%Ã¡f.

v.fh. 11.12 xU fâj édho édh¥ ngh£oæš 48 khzt®fŸ bg‰w kÂ¥bg©fŸ

Ã‹tU« m£ltizæš ju¥g£LŸsd.

kÂ¥bg©fŸ x 6 7 8 9 10 11 12ãfœbt©fŸ f 3 6 9 13 8 5 4

Ï›étu¤Â‰fhd Â£l éy¡f¤ij¡ fz¡»Lf.

v.fh. 11.13 Ã‹tU« òŸë étu¤Â‰fhd Â£l éy¡f« fh©f.

x 70 74 78 82 86 90f 1 3 5 7 8 12

v.fh. 11.14 Ã‹tU« gutè‹ éy¡f t®¡f ruhrçia¡ fh©f.

ÃçÎ Ïilbtë 3.5-4.5 4.5-5.5 5.5-6.5 6.5-7.5 7.5-8.5ãfœbt©fŸ 9 14 22 11 17

v.fh. 11.15 cyf¡ fhšgªJ ngh£ofëš 71 K‹dâ Åu®fŸ mo¤j nfhšfë‹

v©â¡ifæ‹ étu§fŸ Ã‹tU« m£ltizæš bfhL¡f¥g£LŸsJ.

Ï›étu¤Â‹ Â£l éy¡f« fh©f.

ÃçÎ Ïilbtë 0-10 10-20 20-30 30-40 40-50 50-60 60-70ãfœbt©fŸ 8 12 17 14 9 7 4

tif¥gL¤j¥g£l édh¡fŸ - òŸëæaš 371

v.fh. 11.16 40 f«Ã¤ J©Lfë‹ Ús§fŸ (br.Û-¡F ÂU¤jkhf) ÑnH bfhL¡f¥g£LŸsd. Ï›étu¤Â‰fhd éy¡f t®¡f¢ ruhrçia¡

fz¡»Lf.

Ús« (br.Û) 1-10 11-20 21-30 31-40 41-50 51-60 61-70f«Ã¤

J©Lfë‹

v©â¡if

2 3 8 12 9 5 1

v.fh. 11.17 18, 20, 15, 12, 25 v‹w étu§fS¡F khWgh£L¡ bfGit¡ fh©f.

v.fh. 11.18 5 »ç¡bf£ éisah£L¥ ngh£ofëš Ïu©L k£il Åu®fŸ vL¤j X£l§fŸ

(runs) Ã‹tUkhW. mt®fëš X£l§fŸ vL¥gÂš ah® mÂf Ó®ik¤ j‹ik

cilat®?

k£il Åu® A 38 47 34 18 33k£il Åu® B 37 35 41 27 35

v.fh. 11.20 xU òŸë étu¤Âš, 20 kÂ¥òfë‹ T£L¢ ruhrç k‰W« Â£l éy¡f«

Kiwna 40 k‰W« 15 vd fz¡»l¥g£ld. mitfis¢ rçgh®¡F«nghJ

43 v‹w kÂ¥ò jtWjyhf 53 vd vGj¥g£lJ bjça tªjJ. m›étu¤Â‹

rçahd T£L¢ ruhrç k‰W« rçahd Â£l éy¡f« M»at‰iw¡ fh©f.

v.fh. 11.21 xU òŸë étu¤ bjhF¥Ãš xR =35, n = 5 , ( )x 9 822

R - = våš,

k‰W«x x x2 2R R -^ h M»at‰iw¡ fh©f.

v.fh. 11.22 ÏU étu¤ bjhl®fë‹ khWgh£L¡ bfG¡fŸ 58 k‰W« 69 v‹f. nkY«,

mt‰¿‹ Â£l éy¡f§fŸ Kiwna 21.2 k‰W« 15.6 våš, mt‰¿‹ T£L¢

ruhrçfis¡ fh©f.

gæ‰Á 11.1

6. Ñœ¡fhQ« òŸë étu§fë‹ Â£l éy¡f¤ij¡ fz¡»Lf. (x›bthU c£Ãçé‰F« IªJ kÂ¥bg©fŸ)

(i) 10, 20, 15, 8, 3, 4. (ii) 38, 40, 34 ,31, 28, 26, 34.


fz¡»Lf.

x 3 8 13 18 23

f 7 10 15 10 8

8. xU gŸëæYŸs 200 khzt®fŸ xU ò¤jf¡ f©fh£Áæš th§»a ò¤jf§fë‹

v©â¡ifia¥ g‰¿a étu« Ñœ¡fhQ« m£ltizæš bfhL¡f¥g£LŸsJ.

ò¤jf§fë‹ v©â¡if 0 1 2 3 4khzt®fë‹ v©â¡if 35 64 68 18 15

Ï¥òŸë étu¤Â‹ Â£l éy¡f¤ij¡ fz¡»Lf.


9. Ã‹tU« òŸë étu¤Â‹ éy¡f t®¡f¢ ruhrçia¡ fz¡»Lf.

x 2 4 6 8 10 12 14 16f 4 4 5 15 8 5 4 5

10. xU ghjrhç FW¡F¥ ghijia fl¡f¢ Áy® (pedestrian crossing) vL¤J¡ bfh©l neu

étu« Ñœ¡f©l m£ltizæš bfhL¡f¥g£LŸsJ.

neu« (éehoæš) 5-10 10-15 15-20 20-25 25-30eg®fë‹ v©â¡if 4 8 15 12 11

Ï¥òŸë étu¤Â‰F éy¡f t®¡f¢ ruhrç k‰W« Â£l éy¡f¤ij¡ fz¡»Lf.

11. Å£L cçikahs®fŸ 45 ng® mt®fSila bjUé‹ ‘gRik¢ NHš’ Â£l¤Â‰fhf

ãÂ më¤jd®. tNè¡f¥g£l ãÂ¤ bjhif étu« Ã‹tU« m£ltizæš

bfhL¡f¥g£LŸsJ.

bjhif (`) 0-20 20-40 40-60 60-80 80-100Å£L cçikahs®fë‹

v©â¡if2 7 12 19 5

Ï›éu¤Â‰F éy¡f t®¡f¢ ruhrç k‰W« Â£léy¡f¤ij¡ fz¡»Lf.

12. Ñœ¡fhQ« gutè‹ (distribution) éy¡f t®¡f¢ ruhrç fh©f.

ÃçÎ Ïilbtë 20-24 25-29 30-34 35-39 40-44 45-49ãfœbt©fŸ 15 25 28 12 12 8

14. 20 kÂ¥òfë‹ ruhrç k‰W« Â£l éy¡f« Kiwna 10 k‰W« 2 vd fz¡»l¥g£ld.

Ã‹ò rçgh®¡F« nghJ 12 v‹w kÂ¥ghdJ jtWjyhf 8 v‹W vL¤J¡bfhŸs¥g£lJ

bjça tªjJ. rçahd ruhrç k‰W« rçahd Â£l éy¡f« M»adt‰iw¡ fh©f.

16. Ã‹tU« kÂ¥òfë‹ khWgh£L¡ bfGit¡ fz¡»Lf : 20, 18, 32, 24, 26.

19. xR = 99, n = 9 k‰W« x 10 2R -^ h = 79 våš, k‰W«x x x2 2R R -^ h M»at‰iw¡

fh©f.

20. xU tF¥ÃYŸs A, B v‹w ÏU khzt®fŸ bg‰w kÂ¥bg©fŸ Ã‹tUkhW: A 58 51 60 65 66B 56 87 88 46 43

Ït®fëš ah® äFªj Ó®ik¤ j‹ikia bfh©LŸsh®?

tif¥gL¤j¥g£l édh¡fŸ - ãfœjfÎ 373

12. ãfœjfÎ


v.fh. 12.3 Kjš ÏUgJ Ïaš v©fëèUªJ xU KG v© rkthŒ¥ò Kiwæš

nj®ªbjL¡f¥gL»wJ. mªj v© xU gfh v©zhf ÏU¥gj‰fhd


v.fh. 12.4 35 bghU£fŸ ml§»a bjhF¥ò x‹¿š 7 bghU£fŸ FiwghLilad.

m¤bjhF¥ÃèUªJ xU bghUŸ rkthŒ¥ò Kiwæš nj®ªbjL¡F«

nghJ mJ Fiwghl‰w bghUshf ÏU¥gj‰fhd ãfœjfÎ ahJ?

v.fh. 12.7 xU tF¥Ãš cŸs 35 khzt®fëš 20 ng® M©fŸ k‰W« 15 ng®

bg©fŸ. rkthŒ¥ò Kiwæš xU khzt® nj®ªbjL¡f¥ gL»wh® våš,

Ã‹tU« ãfœ¢Áfë‹ ãfœjfÎfis¡ fh©f. (i) nj®ªbjL¡f¥gLgt®

khztdhf ÏU¤jš (ii) nj®ªbjL¡f¥gLgt® khzéahf ÏU¤jš

v.fh. 12.8 xU F¿¥Ã£l ehëš kiH tUtj‰fhd ãfœjfÎ 0.76. m¡F¿¥Ã£l

ehëš kiH tuhkš ÏU¥gj‰fhd ãfœjfÎ ahJ?.

v.fh. 12.9 xU igæš 5 Át¥ò k‰W« Áy Úy ãw¥ gªJfŸ cŸsd. m¥igæèUªJ

xU Úy ãw¥ gªij vL¥gj‰fhd ãfœjfÎ, xU Át¥ò ãw¥ gªij

vL¥gj‰fhd ãfœjfé‹ _‹W kl§F våš, m¥igæYŸs Úy ãw¥

gªJfë‹ v©â¡ifia¡ fh©f.

v.fh. 12.10 Ã‹tUtdt‰¿‰fhd ãfœjféid¡ fh©f.

(x›bthU c£ÃçéwF« Ïu©L kÂ¥bg©fŸ)

(i) rkthŒ¥ò Kiwæš nj®ªbjL¡f¥gL« be£lh©oš (leap year)53 btŸë¡»HikfŸ ÏU¤jš.

(ii) rkthŒ¥ò Kiwæš nj®ªbjL¡f¥gL« be£lh©oš 52 btŸë¡

»HikfŸ k£Lnk ÏU¤jš.

(iii) rkthŒ¥ò Kiwæš nj®ªbjL¡f¥gL« rhjhuz tUl¤Âš

(Non-leap year) 53 btŸë¡»HikfŸ ÏU¤jš.

v.fh. 12.11 xU rkthŒ¥ò¢ nrhjidæš xU ãfœ¢Á A v‹f. mªãfœ¢Áæ‹ ãu¥ò

ãfœ¢Á A v‹f. ( ) : ( ) 7 :12P A P A = våš, P(A) I¡ fh©f.

gæ‰Á 12.1

1. xU igæš cŸs 1 Kjš 100 tiu v©fshš F¿¡f¥g£l 100 Ó£LfëèUªJ xU

Ó£L vL¡f¥gL»wJ. m›thW vL¡f¥gL« Ó£o‹ v© 10 Mš tFgL« v©zhf


2. xU Óuhd gfil Ïu©L Kiw cU£l¥gL»wJ. Kf v©fë‹ TLjš 9 »il¡f¥

bgWtj‰fhd ãfœjfÎ fh©f?

3. ÏU gfilfŸ xU nru cU£l¥gL»‹wd. Kf v©fis¡ bfh©L mik¡f¥gL«

<çy¡f v© 3 Mš tFgL« v©zhf ÏU¥gj‰fhd ãfœjfÎ fh©f.


4. 12 ešy K£ilfSl‹ 3 mG»a K£ilfŸ fyªJŸsd. rkthŒ¥ò Kiwæš

nj®ªbjL¡f¥gL« xU K£il, mG»ajhf ÏU¥gj‰fhd ãfœjfÎ v‹d?

5. ÏU ehza§fis xnu rka¤Âš R©L«nghJ, mÂfg£rkhf xU jiy »il¥gj‰fhd


6. e‹F fiy¤J mL¡»a 52 Ó£Lfis¡ bfh©l f£oèUªJ rkthŒ¥ò Kiwæš xU

Ó£L vL¡f¥gL»wJ. Ã‹tUtdt‰¿‰F ãfœjfÎfis¡ fh©f.


(i) vL¤j Ó£L lak©£ Mf ÏU¡f (ii) vL¤j Ó£L lak©£ Ïšyhkš ÏU¡f

(iii) vL¤j Ó£L V° Ó£lhf Ïšyhkš ÏU¡f.

8. xU igæš 1 Kjš 6 tiu v©fŸ F¿¡f¥g£l 6 btŸis ãw¥ gªJfS« k‰W« 7 Kjš 10 tiu v©fŸ F¿¡f¥g£l 4 Át¥ò ãw¥ gªJfS« cŸsd. rk thŒ¥ò Kiwæš xU

gªJ vL¡f¥gL»wJ våš, Ã‹tU« ãfœ¢ÁfS¡F ãfœjféid¡ fh©f.


(i) vL¡f¥g£l gªJ xU Ïu£il v© bfh©l gªjhf ÏU¤jš

(ii) vL¡f¥g£l gªJ xU btŸis ãw¥ gªjhf ÏU¤jš.

9. 1 Kjš 100 tiuæyhd KG v©fëèUªJ rk thŒ¥ò Kiwæš nj®ª bjL¡f¥gL«

xU v© (i) xU KG t®¡fkhf (perfect square) ÏU¡f (ii) KG fdkhf Ïšyhkš

(not a cube) ÏU¡f M»adt‰¿‹ ãfœjfÎfis¡ fh©f. (x›bthU c£ÃçéwF«

Ïu©L kÂ¥bg©fŸ)

11. xU bg£oæš 4 g¢ir, 5 Úy« k‰W« 3 Át¥ò ãw¥ gªJfŸ cŸsd. rkthŒ¥ò Kiwæš

xU gªij¤ nj®ªbjL¡f mJ

(i) Át¥ò ãw¥ gªjhf ÏU¡f (ii) g¢ir ãw¥ gªjhf ÏšyhkèU¡f


12. 20 Ó£Lfëš 1 Kjš 20 tiuÍŸs KG v©fŸ F¿¡f¥g£LŸsd. rkthŒ¥ò Kiwæš

xU Ó£L vL¡f¥gL»‹wJ. m›thW vL¡f¥g£l Ó£oYŸs v©

(i) 4-‹ kl§fhf ÏU¡f

(ii) 6-‹ kl§fhf Ïšyhkš ÏU¡f M»a ãfœ¢Áfë‹ ãfœjfÎfis¡ fh©f.

13. 3, 5, 7 M»a v©fis Ïy¡f§fshf¡ bfh©L xU Ïu©oy¡f v©

mik¡f¥gL»‹wJ. m›bt© 57 I él¥ bgçajhf ÏU¥gj‰fhd ãfœjfÎ

fh©f. (m›bt©âš xnu Ïy¡f¤ij Û©L« ga‹gL¤j¡ TlhJ).

14. _‹W gfilfŸ xnu neu¤Âš cU£l¥gL«nghJ, _‹W gfilfëY« xnu v©

»il¥gj‰fhd ãfœ¢Áæ‹ ãfœjféid¡ fh©f.

17. xU igæš cŸs 100 r£ilfëš, 88 r£ilfŸ ešy ãiyæY«, 8 r£ilfŸ Á¿a

Fiwgh£LlD« k‰W« 4 r£ilfŸ bgça Fiwgh£LlD« cŸsd. A v‹w

tâf® ešy ãiyæš cŸs r£ilfis k£Lnk V‰»wh®. Mdhš B v‹w

tâf® mÂf FiwghL cila r£ilfis k£L« V‰f kW¡»wh®. rkthŒ¥ò

Kiwæš VnjD« X® r£ilia nj®ªbjL¡f mJ (i) A-¡F V‰òilajhf mika (ii) B-¡F V‰òilajhf mika M»adt‰¿‰F ãfœjfÎfis¡ fh©f.


19. Ã¡» c©oaèš (Piggy bank) 100 I«gJ igrh ehz§fS« 50 xU %ghŒ ehza§fS«

20 Ïu©L %ghŒ ehza§fS« k‰W« 10 IªJ %ghŒ ehza§fS« cŸsd.

rkthŒ¥ò Kiwæš xU ehza« nj®ªbjL¡F« nghJ (i) I«gJ igrh ehzakhf

ÏU¡f (ii) IªJ %ghŒ ehzakhf Ïšyhkš ÏU¡f M»adt‰¿‹ ãfœjfÎfis¡

fh©f.

v.fh. 12.15 “ENTERTAINMENT” v‹w brhšèYŸs vG¤JfëèUªJ

rkthŒ¥ò Kiwæš xU vG¤ij¤ nj®Î brŒa, m›btG¤J

M§»y cæbuG¤jhfnth mšyJ vG¤J T Mfnth ÏU¥gj‰fhd

ãfœjféid¡ fh©f. (vG¤JfŸ ÂU«g¤ ÂU«g tuyh«).

gæ‰Á 12.2

1. A k‰W« B v‹gd x‹iwbah‹W éy¡F« ãfœ¢ÁfŸ. nkY« ( ) ( )k‰W«P A P B53

51= =

våš, ( )P A B, -I¡ fh©f .

2. A k‰W« B v‹w Ïu©L ãfœ¢Áfëš ( ) , ( )P A P B41

52= = k‰W« ( )P A B

21, =

våš, ( )P A B+ -I¡ fh©f.

11. xU igæš 10 btŸis, 6 Át¥ò k‰W« 10 fU¥ò ãw¥ gªJfŸ cŸsd. rkthŒ¥ò

Kiwæš xU gªÂid vL¡F«nghJ mJ btŸis mšyJ Át¥ò ãw¥ gªjhf


12. 2, 5, 9 v‹w v©fis¡ bfh©L, X® Ïu©oy¡f v© mik¡f¥gL»wJ. mªj v© 2 mšyJ 5 Mš tFgLkhW mika ãfœjfÎ fh©f.

(mik¡f¥gL« v©âš xnu Ïy¡f« Û©L« tuyh« )

13. “ACCOMMODATION” v‹w brhšè‹ x›bthU vG¤J« jå¤jåna Á¿a fh»j§fëš

vGj¥g£L, mªj 13 Á¿a fh»j§fS« xU Kfitæš it¡f¥g£LŸsd.

rkthŒ¥ò Kiwæš KfitæèUªJ xU fh»j¤ij¤ nj®Î brŒÍ« nghJ, mÂš

Ïl« bgW« vG¤J (x›bthU c£ÃçéwF« Ïu©L kÂ¥bg©fŸ)

(i) ‘A’ mšyJ ‘O’ Mfnth

(ii) ‘M’ mšyJ ‘C’ Mfnth ÏU¥gj‰fhd ãfœjfÎfis¡ fh©f.


v.fh. 12.1 xU Óuhd gfil xU Kiw cU£l¥gL»wJ. Ã‹tU« ãfœ¢ÁfS¡fhd

ãfœjfÎfis¡ fh©f.

(i) v© 4 »il¤jš

(ii) xU Ïu£il¥gil v© »il¤jš

(iii) 6-‹ gfh fhuâfŸ »il¤jš

(iv) 4-I él¥ bgça v© »il¤jš


v.fh. 12.2 xU Óuhd ehza« Ïu©L Kiw R©l¥gL»wJ. Ñœ¡fhQ«

ãfœ¢ÁfS¡fhd ãfœjféid¡ fh©f.

(i) ÏU jiyfŸ »il¤jš (ii) FiwªjJ xU jiy »il¤jš

(iii) xU ó k£L« »il¤jš.

v.fh. 12.5 ÏU Óuhd gfilfŸ xU Kiw cU£l¥gL»‹wd. Ñœ¡fhQ«

ãfœ¢ÁfS¡fhd ãfœjféid¡ fh©f.

(i) Kf v©fë‹ TLjš 8 Mf ÏU¤jš

(ii) Kf v©fŸ xnu v©fshf (doublet) ÏU¤jš=

(iii) Kf v©fë‹ TLjš 8-I él mÂfkhf ÏU¤jš

v.fh. 12.6 e‹F fiy¤J it¡f¥g£l 52 Ó£Lfis¡ bfh©l Ó£L¡ f£oèUªJ

rkthŒ¥ò¢ nrhjid Kiwæš xU Ó£L vL¡f¥gL»wJ. mªj¢ Ó£L

Ã‹tUtdthf ÏU¡f ãfœjfÎfis¡ fh©f.

(i) Ïuhrh (ii) fU¥ò Ïuhrh

(iii) °ngL (iv) lak©£ 10

gæ‰Á 12.1

7. _‹W ehza§fŸ xnu neu¤Âš R©l¥gL»‹wd. Ã‹tU« ãfœ¢ÁfS¡F


(i) FiwªjJ xU jiy »il¥gJ (ii) ÏU ó¡fŸ k£L« »il¥gJ (iii) FiwªjJ ÏU

jiyfŸ »il¥gJ.

15. ÏU gfilfŸ xnu neu¤Âš cU£l¥gL«nghJ »il¡F« Kf v©fë‹ bgU¡f‰gy‹

xU gfh v©zhf ÏU¥gj‰fhd ãfœjféid¡ fh©f.

16. xU Kfitæš Úy«, g¢ir k‰W« btŸis ãw§fëyhd 54 gë§F¡f‰fŸ cŸsd.

xU gë§F¡ fšiy vL¡F«nghJ, Úy ãw¥ gë§F¡fš »il¥gj‰fhd ãfœjfÎ

31 k‰W« g¢ir ãw¥ gë§F¡fš »il¥gj‰fhd ãfœjfÎ

94 våš, m«Kfitæš

cŸs btŸis ãw¥ gë§F¡ f‰fë‹ v©â¡ifia¡ fh©f.

18. xU igæš cŸs 12 gªJfëš x gªJfŸ btŸis ãwKilait. (i) rkthŒ¥ò Kiwæš

xU gªJ nj®ªbjL¡f, mJ btŸis ãwkhf ÏU¥gj‰fhd ãfœjfÎ fh©f.

(ii) 6 òÂa btŸis ãw¥ gªJfis m¥igæš it¤jÃ‹d®, xU btŸis ãw¥

gªij¤ nj®bjL¥gj‰fhd ãfœjfÎ MdJ (i)-š bgw¥g£l ãfœjféid¥ nghy

ÏUkl§F våš, x-‹ kÂ¥Ãid¡ fh©f.

v.fh. 12.12 _‹W ehza§fŸ xnu neu¤Âš R©l¥gL»‹wd. ãfœjfé‹ T£lš

nj‰w¤ij ga‹gL¤Â, rçahf ÏU ó¡fŸ mšyJ Fiwªjg£r« xU

jiyahtJ »il¡F« ãfœ¢Áæ‹ ãfœjféid¡ fh©f.


v.fh. 12.13 xU gfil ÏUKiw cU£l¥gL»wJ. FiwªjJ xU cU£lèyhtJ

v© 5 »il¥gj‰fhd ãfœjféid¡ fh©f. (T£lš nj‰w¤ij¥

ga‹gL¤Jf)

v.fh. 12.14 xU khzé¡F kU¤Jt¡ fšÿçæš nr®¡if »il¥gj‰fhd ãfœjfÎ

0.16 v‹f. bgh¿æaš fšÿçæš nr®¡if »il¥gj‰fhd ãfœjfÎ

0.24 k‰W« ÏU fšÿçfëY« nr®¡if »il¥gj‰fhd ãfœjfÎ 0.11 våš,

(i) kU¤Jt« k‰W« bgh¿æaš fšÿçfëš VnjD« xU fšÿçæš

nr®¡if »il¥gj‰fhd ãfœjfÎ fh©f.

(ii) kU¤Jt¡ fšÿçæš k£Lnkh mšyJ bgh¿æaš fšÿçæš

k£Lnkh nr®¡if »il¥gj‰fhd ãfœjfÎ fh©f.

v.fh. 12.16 A, B k‰W« C v‹gd x‹iwbah‹W éy¡F« k‰W« ãiwÎbrŒ

ãfœ¢ÁfŸ v‹f. nkY«, ( )P B = ( )P A23 k‰W« ( ) ( )P C P B

21= våš,

P(A)-I fh©f.

v.fh. 12.17 52 Ó£Lfis¡ bfh©l xU Ó£L¡f£oèUªJ rkthŒ¥ò Kiwæš xU

Ó£L vL¡f¥gL« nghJ, m¢Ó£L xU Ïuhrh (King) mšyJ xU Ah®£

(Heart) mšyJ xU Át¥ò ãw¢ Ó£lhf¡ »il¥gj‰fhd ãfœjféid¡

fh©f.

v.fh. 12.18 xU igæš 10 btŸis, 5 fU¥ò, 3 g¢ir k‰W« 2 Át¥ò ãw¥ gªJfŸ

cŸsd. rkthŒ¥ò Kiwæš nj®ªbjL¡f¥gL« xU gªJ, btŸis

mšyJ fU¥ò mšyJ g¢ir ãwkhf ÏU¥gj‰fhd ãfœjféid¡

fh©f.

gæ‰Á 12.2

3. A k‰W« B v‹w Ïu©L ãfœ¢Áfëš ( ) , ( ) ( ) 1k‰W«P A P B P A B21

107 ,= = =

våš, (i) ( )P A B+ (ii) ( )P A B,l l M»at‰iw¡ fh©f.

4. xU gfil ÏUKiw cU£l¥gL»wJ. Kjyhtjhf cU£l¥gL«nghJ xU

Ïu£il¥gil v© »il¤jš mšyJ m›éU cU£lèš Kf v©fë‹ TLjš

8 Mf ÏU¤jš vD« ãfœ¢Áæ‹ ãfœjféid¡ fh©f.

5. 1 Kjš 50 tiuæyhd KG¡fëèUªJ rkthŒ¥ò Kiwæš X® v© nj®ªbjL¡f¥

gL«nghJ m›bt© 4 mšyJ 6 Mš tFgLtj‰fhd ãfœjfÎ fh©f.

6. xU igæš 50 kiu MâfS« (bolts), 150 ÂUF kiufS« (nuts) cŸsd. mt‰WŸ

ghÂ kiu MâfS«, ghÂ ÂUF kiufS« JU¥Ão¤jit. rkthŒ¥ò Kiwæš

VnjD« x‹iw¤ nj®ªbjL¡F« nghJ mJ JU¥Ão¤jjhf mšyJ xU kiu

Mâahf ÏU¥gj‰fhd ãfœjféid¡ fh©f.


7. ÏU gfilfŸ xnu neu¤Âš nru cU£l¥gL«nghJ »il¡F« Kf v©fë‹ TLjš

3 Mš k‰W« 4 Mš tFglhkèU¡f ãfœjfÎ fh©f.

8. xU Tilæš 20 M¥ÃŸfS« 10 MuŠR¥ gH§fS« cŸsd. mt‰WŸ

5 M¥ÃŸfŸ k‰W« 3 MuŠRfŸ mG»ait. rkthŒ¥ò Kiwæš xUt® xU gH¤ij

vL¤jhš, mJ M¥Ãshfnth mšyJ ešy gHkhfnth ÏU¥gj‰fhd ãfœjféid¡

fh©f.

9. xU tF¥Ãš cŸs khzt®fëš 40% ng® fâj édho édh ãfœ¢ÁæY«,

30% ng® m¿éaš édho édh ãfœ¢ÁæY«, 10% ng® m›éu©L édho édh

ãfœ¢ÁfëY« fyªJ bfh©ld®. m›tF¥ÃèUªJ rkthŒ¥ò Kiwæš xU

khzt‹ nj®ªbjL¡f¥g£lhš, mt® fâj édho édh ãfœ¢Áænyh mšyJ

m¿éaš édho édh ãfœ¢Áænyh mšyJ ÏU ãfœ¢ÁfëYnkh fyªJ

bfh©lj‰fhd ãfœjfÎ fh©f.

10. e‹F fiy¤J mL¡» it¡f¥g£l 52 Ó£Lfis¡ bfh©l Ó£L¡ f£oèUªJ

rkthŒ¥ò Kiwæš xU Ó£L vL¡f¥gL»wJ. mªj¢ Ó£L °nglhfnth (Spade)mšyJ Ïuhrhthfnth (King) ÏU¥gj‰fhd ãfœjféid¡ fh©f.

14. xU òÂa k»œÎªJ (car) mjDila totik¥Ã‰fhf éUJ bgW« ãfœjfÎ 0.25 v‹f. Áwªj Kiwæš vçbghUŸ ga‹gh£o‰fhd éUJ bgW« ãfœjfÎ 0.35 k‰W« ÏU

éUJfS« bgWtj‰fhd ãfœjfÎ 0.15 våš, m«k»œÎªJ


(ii) xnu xU éUJ k£L« bgWjš

M»a ãfœ¢ÁfS¡fhd ãfœjfÎfis¡ fh©f


, ,54

32

73 v‹f. A k‰W« B ÏUtU« nr®ªJ Ô®Î fh©gj‰fhd ãfœjfÎ

158 .

B k‰W« C ÏUtU« nr®ªJ Ô®Î fh©gj‰fhd ãfœjfÎ 72 . A k‰W« C ÏUtU«

nr®ªJ Ô®Î fhz ãfœjfÎ 3512 , _tU« nr®ªJ Ô®Î fhz ãfœjfÎ

358 våš,

ahnuD« xUt® m›édhé‹ Ô®Î fh©gj‰fhd ãfœjféid¡ fh©f.

379

jahç¡f¥g£l édh¡fŸ[ ÑnH bfhL¡f¥g£LŸs jahç¡f¥g£l édh¡fŸ khÂç édh¡fns MF«. Ïnj

ngh‹W nkY« édh¡fis jahç¤J bghJ¤nj®éš Ïl«bgw¢ brŒayh« ]

2. bkŒba©fë‹ bjhl®tçirfS« bjhl®fS«

F¿¡nfhŸ tif édh¡fŸ

1. , ,b c c a a b1 1 1+ + +

M»ad xU T£L¤bjhlçš cŸsd våš,

Ã‹tUtdt‰WŸ vJ xU T£L¤bjhl® tçiria mik¡F«?

(A) ( )( ), ( )( ), ( )( )a c a b a b b c b c c a+ + + + + +

(B) , ,a b c a b c a b c2 2 2+ + + + + +

(C) , ,a b c a b c b c a- + + - + -

(D) , ,a b c a b c b c a2 2 2- + + - + -

2. , , , , , ,a a a a a a a1 2 3 4 5 6 7

M»ad xU T£L¤bjhl®tçiræš cŸsd k‰W«

a 104= våš a a

1 7+ =

(A) 5 (B) 10 (C) 15 (D) 20

3. Ã‹tUtdt‰WŸ vJ Ãnghdh» bjhl® tçiræ‹ xU gFÂahf mikÍ«?

(A) 14, 21, 35, 56, 91 (B) 11, 19, 30, 49, 79

(C) 13, 21, 34, 55, 89 (D) 13, 20, 33, 53, 86

4. 4tJ cW¥ò 19 k‰W« 7tJ cW¥ò 28 Mf cŸs xU T£L¤bjhl®tçiræ‹ bghJ

é¤Âahr«

(A) 2 (B) 3 (C) 4 (D) 5

5. Ã‹tU« rh®òfëš vJ bjhl®tçir mšy?

(A) ( ) 2 1, 1,2,3,4,f p p p g= - =

(B) ( ) , ( , ]f x x x1 0 1!= +

(C) ( ) , , , ,f nn

n n12

3 1 2 3 g= + + =

(D) ( ) , , , ,f t t t2 1 2 3 g= + =

6. xU bgU¡F¤ bjhl® tçiræ‹ Kjš 5 cW¥òfë‹ bgU¡f‰gy‹ 32 våš mj‹

_‹whtJ cW¥ò

(A) 41 (B)

21 (C) 2 (D) 4

7. Kjš cW¥ò a 5= k‰W« bghJ é»j« 2 I¡ bfh©l xU bgU¡F¤ bjhl®

tçiræ‹ IªjhtJ cW¥ò

(A) 5 2 (B) 10 (C) 10 2 (D) 20

jahç¡f¥g£l édh¡fŸ - bkŒba©fë‹ bjhl®tçirfS« bjhl®fS«


8. x 2= våš x x x1 2 9g+ + + + ‹ kÂ¥ò

(A) 511 (B) 1023 (C) 513 (D) 1025

9. 3 6 9 60g+ + + + ‹ kÂ¥ò

(A) 510 (B) 570 (C) 600 (D) 630

10. k2 4 6 2 90g+ + + + = våš k‹ kÂ¥ò

(A) 8 (B) 9 (C) 10 (D) 11

11. mid¤J m k‰W« n v‹w Ïaš v©fS¡F« t t tm n m n

=+

k‰W« t 31= våš

, ,t t t1 2 3

MdJ

(A) xU T£L¤bjhl® tçir (B) xU bgU¡F¤bjhl® tçir

(C) T£L¤bjhl® tçir k‰W« bgU¡F¤bjhl® tçir

(D) T£L¤bjhl® tçirÍkšy k‰W« bgU¡F¤bjhl® tçirÍkšy

12. , , , ,21

43

65

87 g v‹w bjhl®tçiræ‹ nMtJ cW¥ò

(A) n

121+ (B) 1

n21- (C) 1

n 21-+

(D) nn

21

+-

13. , , , , ,a a a an1 2 3

g g v‹gJ xU T£L¤bjhl® tçir k‰W« xU bgU¡F¤bjhl®

tçir våš mj‹ bghJ é¤Âahr« k‰W« bghJ é»j« Kiwna

(A) 0, 0 (B) 0, 1 (C) 1, 0 (D) 1, 1

14. 2, , , 54x y M»ait xU bgU¡F¤ bjhl® tçiræš mik»‹wd våš Ït‰¿‹

bghJ é»j«

(A) 2 (B) 3 (C) 3 3 (D) 2 2

15. 2, , 2x 2 M»ad xU T£L¤ bjhl®tçiræš mik»‹wd våš Ït‰¿‹

bghJ é¤Âahr«

(A) 2 (B) 2 1+ (C) 2 1- (D) 22

16. , , 5 , 5 , 25 ,11 55 11 55 11 g v‹w bjhl® tçir



(D) T£L¤bjhl® tçirÍkšy k‰W« bgU¡F¤bjhl® tçirÍkšy

17. a 1! våš ( ) ( )a a a1 1 1 2+ + + + + =

(A) ( )a

a a a1

3 2 3

-- + + (B) a a3 2 2 2

+ +

(C) a a3 2+ + (D) ( )

aa a

13 1 2

-- + +

381

18. , , ca

ac

cab a c2 02

!+ = - våš , ,a b c MdJ



(D) T£L¤bjhl® tçirÍkšy bgU¡F¤bjhl® tçirÍkšy

19. xU bgU¡F¤ bjhl®tçiræš cŸs _‹W v©fëš eLéš cŸs v©iz

ÏU kl§fh¡F« nghJ mJ xU T£L¤ bjhl® tçirah»wJ våš,

bgU¡F¤bjhl®tçiræ‹ bghJ é»j«

(A) 2 3+ (B) 3 2+ (C) 5 3+ (D) 5 3+

20. , , ,a a a1 2 3

g M»ad xU T£L¤bjhl® tçiræš cŸsd k‰W«

a a a a a a 2251 5 10 15 20 24+ + + + + = våš a a

1 24+ =

(A) 150 (B) 100 (C) 75 (D) 125


1. ( )

,

,

k‰W« xUÏu£il¥gilv©

k‰W« xUx‰iw¥gilv©C

n nn N n

n

n n N n

42

1

4n

2

!

!=

+

+

Z

[

\

]]

]]

vd tiuaW¡f¥g£l bjhl®tçiræ‹ 18MtJ k‰W« 25MtJ cW¥òfis¡

fh©f. ( éil: ;C C9031350

18 25= = )

2. , ,bc ca ab1 1 1 M»ad T£L¤bjhl® tçiræš ÏU¥Ã‹ , ,a b c M»ad xU

T£L¤bjhl® tçiræš ÏU¡F« vd ãWÎf.

3. xU T£L¤ bjhl® tçiræ‹ eh‹fhtJ cW¥ò mj‹ Kjš cW¥Ã‹ 3 kl§F k‰W«

mj‹ VHhtJ cW¥ò _‹whtJ cW¥Ã‹ ÏU kl§ifél 1 mÂf« våš , mj‹

Kjš cW¥ò k‰W« bghJ é¤Âahr« M»at‰iw¡ fh©f.

( éil: ,a d3 2= = )

4. xUt® rdtç khj« ` 320 nrä¡»wh®; Ã¥utç khj« ` 360 nrä¡»wh®; kh®¢

khj« ` 400 nrä¡»wh®. mt® Ïnj bjhl® tçiræš jdJ nrä¥ig¤ bjhl®»wh®

våš, mnj tUl« et«g® khj« mtUila nrä¥ò v‹dthf ÏU¡F«? ( éil: ` 720 )

5. , ,a b c M»ad xU T£l¤bjhlçš cŸsd våš,

( )( )( )a b c b c a c a b2 2 2+ - + - + - ‹ kÂ¥ig¡ fh©f. ( éil: 0 )

6. Kjš cW¥ò 2 k‰W« IªjhtJ cW¥ò _‹whtJ cW¥Ã‹ 4 kl§F v‹wthW cŸs

bgU¡F¤ bjhl® tçiria¡ fh©f.( )r 0> . ( éil: , , , ,2 4 8 16 g )



7. xU bgU¡F¤ bjhl® tçiræ‹ Kjš k‰W« VHhtJ cW¥òfŸ Kiwna 24 k‰W«

192 våš, mj‹ 11tJ cW¥ò fh©f.( )r 0> . ( éil: 768 )

8. xU bgU¡F¤ bjhl® tçiræ‹ _‹whtJ cW¥ò 31 . mj‹ Kjš IªJ cW¥òfë‹

bgU¡f‰gy‹ fh©f. ( éil: 2431 )

9. , ,a b c M»ad xU bgU¡F¤ bjhl® tçiræš cŸsd våš,

, ,a b ab bc b c2 2 2 2+ + + M»adÎ« xU bgU¡F¤ bjhlçš cŸsd vd ãWÎf.

10. xU bgU¡F¤ bjhl® tçiræ‹ Kjš cW¥ò 1. mj‹ _‹whtJ k‰W« IªjhtJ

cW¥òfë‹ TLjš 90. mj‹ bghJ é»j« fh©f. ( éil: r 3!= )

11. , , ( )m m72

27 2- - + M»ad xU bgU¡F¤ bjhl® tçiræš cŸsd. m ‹ kÂ¥ò

fh©f. ( éil: ,m 2 1= - )

12. 1 2 3 42 2 2 2 g- + - + v‹w bjhl® tçiræ‹ Kjš 10 cW¥òfë‹ TLjš fh©f.

( éil: – 55 )

13. , ,a b c M»ad xU T£L¤ bjhl® tçiræš cŸsd k‰W« x,y,z M»ad xU

bgU¡F¤bjhl® tçiræš cŸsd. våš , 1x y zb c c a a b=- - - vd ãWÎf.

14. xU bjhl® tçiræ‹ Kjš n cW¥òfë‹ TLjš [ ]n n21 2

+ våš, 50MtJ cW¥ò

fh©f. ( éil: 50 )

15. xU bjhl® tçiræ‹ Kjš n cW¥òfë‹ TLjš [ ( 1) ]n n41 2 2

+ våš, 5MtJ

cW¥ò fh©f. ( éil: 125 ) 16. 5 156 73 3 3 3g+ + + + Ï‹ TLjš fh©f. ( éil: 14300 )

17. 92

31

94 g+ + + v‹w bjhl® tçiræ‹ Kjš 7 cW¥òfë‹ TLjš fh©f.

( éil: 935 )

18. 32

34 2

38

332g+ + + + + v‹w bjhl® tçiræ‹ TLjš fh©f.( éil: 90

32 )

19. 50 k‰W« 200 Ït‰¿‰»ilnaahd 10Mš tFgL« mid¤J KG v©fë‹

TLjiy¡ fh©f. ( éil: 1750 )

20. xU khzt® xU T£L¤ bjhl® tçiræ‹ bghJ é¤Âahr« 2 v‹gj‰F¥ gÂyhf

-2 vd¥ go¤jjhš Kjš 5 cW¥òfë‹ TLjš -5 vd¥ bg‰wh®. c©ikahd¡

TLjiy¡ fh©f. ( éil: 35 )


1. xU _‰¿y¡f äif v©â‹ Ïy¡f§fŸ xU T£L¤bjhlçš cŸsd. k‰W«

mt‰¿‹ TLjš 15. m›bt©â‹ Ïy¡f§fis vÂ®tçiræš(ÂU¥Ã)

vGJtjhš »il¡F« v© c©ikahd v©izél 594 FiwÎ. våš, mªj

v©iz¡ fh©f. ( éil: 852 )

383

2. xU T£L¤ bjhl® tçiræ‹ m MtJ cW¥ò n k‰W« n MtJ cW¥ò m våš, p MtJ cW¥ò m n p+ - vd ãWÎf.

3. xU T£L¤ bjhl® tçiræš 21 cW¥òfŸ cŸsd. eLéš cŸs _‹W cW¥òfë‹

TLjš 129 k‰W« filÁæš cŸs _‹W cW¥òfë‹ TLjš 237. Kjš cW¥ò

k‰W« bghJ é¤Âahr« M»at‰iw¡ fh©f. ( éil: ,a d3 4= = )

4. xU T£L¤ bjhl® tçiræ‹ n MtJ cW¥ò , 15t an bnn

2= + + vd

tiuaW¡f¥g£LŸsJ. t 132= k‰W« t 27

4= våš a k‰W« b M»at‰iw¡

fh©f. ( éil: 2 ; 5a b= =- )

5. xU T£L¤ bjhl® tçiræ‹ p MtJ, q MtJ k‰W« r MtJ cW¥òfŸ Kiwna

, ,a b c . våš ( ) ( ) ( ) 0q r a r p b p q c- + - + - = vd ãWÎf.

6. , ,x y y y z1

21 1

+ + M»ad xU T£L¤ bjhl® tçiræ‹ _‹W mL¤jL¤j

cW¥òfŸ våš, , ,x y z M»ad xU bgU¡F¤ bjhl® tçiræ‹ _‹W

mL¤jL¤j cW¥òfŸ vd ãWÎf.

7. , , ,a b c d M»ad xU bgU¡F¤ bjhl® tçiræš cŸsd våš

, ,a b b c c d2 2 2 2 2 2+ + + M»adÎ« xU bgU¡F¤ bjhl® tçiræš cŸsd vd

ãWÎf. [F¿¥ò: , ,b ar c ar d ar2 3= = = vd¡bfhŸf].

8. xU bgU¡F¤ bjhl® tçiræ‹ p MtJ, q MtJ k‰W« r MtJ cW¥òfŸ Kiwna

, ,a b c våš, 1a b cq r r p p q=- - - vd ãWÎf.

9. xU bgU¡F¤ bjhl® tçiræ‹ Kjš _‹W cW¥òfë‹ TLjš 1039 k‰W«

mt‰¿‹ bgU¡f‰ gy‹ 1 våš, bghJ é»j« k‰W« bgU¡F¤ bjhl® tçir

M»at‰iw¡ fh©f. ( éil: , , ,r52

25

52 1

25G.P.:or= )

10. xU bgU¡F¤ bjhl® tçiræ‹ 4MtJ, 10MtJ k‰W« 16 MtJ cW¥òfŸ Kiwna

x, y k‰W« z våš, x, y k‰W« z M»ad xU bgU¡F¤ bjhl® tçiræš cŸsd

vd ã%Ã.

11. Ïu©L T£L¤ bjhl® tçirfë‹ Kjš n cW¥òfë‹ TLjšfë‹ é»j«

( ) : ( )n n3 8 7 15+ + våš, m¤bjhl®fë‹ 3MtJ cW¥òfë‹ é»j« fh©f.

( éil: 23 : 50 )

12. xU T£L¤ bjhl® tçiræ‹ mL¤jL¤j eh‹F cW¥òfë‹ TLjš 4 k‰W«

mt‰¿‹ bgU¡f‰gy‹ 385. våš, mªj v©fis¡ fh©f.

( éil: , , , , , ,735

311 9 9

311

35 7or- - - - )

13. xU T£L¤ bjhl® tçiræ‹ mL¤jL¤j eh‹F cW¥òfë‹ TLjš 20 k‰W«

mt‰¿‹ t®¡f§fë‹ TLjš 120. mªj v©fis¡ fh©f.

( éil: 2, 4, 6, 8 mšyJ 8, 6, 4, 2 )



14. Ïu©L k»GªJ t©ofŸ Xçl¤ÂèUªJ xnu Âiræš òw¥gL»‹wd. Kjš

k»GªJ 60 ».Û./kâ v‹w Óuhd ntf¤Âš brš»wJ. Ïu©lhtJ k»GªJ Kjš

kâæš 50 ».Û./kâ v‹w ntf¤ÂY« x›bthU kâ¡F« 4 ».Û. ntf¤ij¡

T£o¡ bfh©L« brš»wJ. Ïu©L k»GªJfS« ã‰fhkš brš»wJ v‹whš

Ïu©lhtJ k»GªJ Kjš k»Gªij v¥bghGJ KªÂ bršY«. ( éil: 6 kâ. )

15. xU T£L¤ bjhl® tçiræ‹ m MtJ cW¥ò n1 k‰W« n MtJ cW¥ò

m1 våš

mn MtJ cW¥ò ( 1)mn21 + vd ã%Ã.

16. xU T£L¤ bjhl® tçiræ‹ Kjš VG cW¥òfë‹ TLjš 105 k‰W« Kjš cW¥ò

6 våš, Kjš n cW¥òfë‹ TLjš k‰W« Kjš n 3- cW¥òfë‹ TLjš

Ït‰¿‹ é»j« ( ) :n n3 3+ - vd ã%Ã.

17. 5, 5.5, 5.55, g v‹w bjhl® tçiræ‹ Kjš 50 cW¥òfë‹ TLjš fh©f.

(éil: 4490815

10

149

+; E )

18. xU bgU¡F¤ bjhl® tçiræ‹ mL¤jL¤j 4 cW¥òfë‹ TLjš 60 k‰W« , ,t t181 4

M»ad xU T£L¤ bjhl® tçiræš mik»‹wd våš m›bt©fis¡

fh©f. ( éil: 4, 8, 16, 32 or 32, 16, 8, 4 )

19. ( ) ( ) ( ) ( )2 4 3 6 5 8 7 22 212 2 2 2g+ + + + + v‹w bjhl®tçiræ‹ TLjš fh©f.

( éil: 52107 )

20. Kjš _‹W cW¥òfŸ xU T£L¤bjhl® tçiræY« k‰W« filÁ _‹W cW¥òfŸ

xU bgU¡F¤ bjhl®tçiræY« mikÍkhW eh‹F v©fŸ cŸsd. Kjš k‰W«

_‹wh« cW¥òfë‹ TLjš 2 k‰W« Ïu©lh« k‰W« eh‹fh« cW¥òfë‹

TLjš 26 våš m›bt©fis fh©f. ( éil: , , , , , ,mšyJ3 1 5 25 7 1 5 25- - )

3. Ïa‰fâj«


1. x x2 5 7 02- + = v‹w rk‹gh£o‹ xU _y« 1

a våš, 7 52a a- ‹ kÂ¥ò

(A) 2 (B) – 2 (C) 5 (D) – 5

2. y x xy

9 4 12+ = , nkY« ,x y0 0> > våš x y3 2- =

(A) 5 (B) 1 (C) 2 (D) 0

3. a x b y c 01 1 1

+ + = k‰W« a x b y c 02 2 2

+ + = v‹w rk‹ghLfë‹ xnu xU Ô®Î

(0, – 1) våš

(A) a

a

b

b

2

1

2

1= (B) a

a

c

c

2

1

2

1= (C) b

b

c

c

2

1

2

1= (D) b

b

c

c

2

1

2

1!

385

4. x kx3 5 75 02- + = v‹w rk‹gh£o‹ _y§fŸ äif k‰W« rk« våš m«_y§fŸ

(A) 3, 3 (B) 6, 6 (C) ,5 5 5 5 (D) ,5 3 5 3

5. 14a b+ = k‰W« 2 3a b- = , våš ab =

(A) 42 (B) 44 (C) 46 (D) 48

6. ,xx

x1 23 0>22

+ = våš xx1+ =

(A) 2 (B) 3 (C) 4 (D) 5

7. x x k2 02+ + = k‰W« x x k4 02

+ - = M»a rk‹ghLfë‹ bghJthd _y« a våš, k ‹ kÂ¥ò

(A) a (B) a- (C) 3a (D) 3a-

8. ( )x 3 9 02+ - = v‹w rk‹gh£o‹ _y§fŸ

(A) ( , )0 6- (B) ( , )3 6- - (C) ( , )0 3- (D) ( , )3 3-

9. x x a2 02- + = v‹w rk‹gh£o‹ _y§fŸ ,a b k‰W« 6a b- = våš a ‹ kÂ¥ò

(A) 4 (B) – 4 (C) 8 (D) – 8

10. a k‰W« 2a M»ad x bx 8 02- + = v‹w rk‹gh£o‹ _y§fŸ våš b ‹ kÂ¥ò

(A) 2 (B) 4 (C) 6 (D) 8

11. 12, 2 6 15ax y x y6- = - = v‹w rk‹ghLfë‹ bjhF¥ò¡F Ô®Î Ïšiy våš

a -‹ kÂ¥ò

(A) 1 (B) 2 (C) 3 (D) 4

12. x y2 6 m- = k‰W« x y32 2 5- = v‹w rk‹ghLfë‹ bjhF¥ò¡F Ô®Î Ïšiy

våš !m

(A) 5 (B) 9 (C) 12 (D) 15

13. Ã‹tUtdt‰WŸ vJ x x x x2 5 3 13 9 04 3 2- - + + = v‹w rk‹gh£o‹ xU _y«

MF«?

(A) 1 (B) – 1 (C) 2 (D) 0

14. x 1= MdJ ( ) 2 3p x x x x x44 3 2 m= - + - + v‹w gšYW¥ò¡ nfhitæ‹ xU

ó¢Áa« våš, m‹ kÂ¥ò

(A) 2 (B) – 2 (C) 3 (D) – 3

15. x 4= våš ( ) ( ) ( )x x x1 2 32 2 2- - - ‹ kÂ¥ò

(A) 3 (B) – 3 (C) 6 (D) – 6

16. ( )p x x 12= + v‹w gšYW¥ò¡ nfhitæ‹ bkŒba© ó¢Áa§fë‹ v©â¡if

(A) 0 (B) 1 (C) 2 (D) 3

jahç¡f¥g£l édh¡fŸ - Ïa‰fâj«


17. x x x x2 14 19 64 3 2+ - + + v‹w gšYW¥ò¡ nfhitia x2 1+ v‹w nfhitahš

tF¡F«nghJ »il¡F« ÛÂ x ax bx 63 2+ - - våš, a k‰W« b Ïitfë‹

kÂ¥òfŸ Kiwna

(A) 6, 7 (B) 0, 7 (C) 7, 0 (D) 7, 6

18. , ,x y x yz x y z6 9 122 2 2 2 M»at‰¿‹ Û.bgh.k.

(A) xy z36 2 2 (B) x y z36 2 2 (C) 6x y z3 2 2 2 (D) xy z36 2

19. ( )p x k‰W« ( )q x M»a gšYW¥ò¡ nfhitfë‹ Û.bgh.t. x2 våš, ( )p x k‰W« ( )q x M»at‰¿‹ Û.bgh.k.

(A) ( ) ( )x p x q x2 (B) ( ) ( )

x

p x q x2

(C) ( ) ( )

x

p x q x4

(D) ( ) ( )x p x q x4

20. ,a b2 3= = våš ab b

a b a b2

2 3 3 2

+

+ ‹ kÂ¥ò

(A) 12 (B) 18 (C) 24 (D) 36


1. Ô®: ( ) ( ) ( )x y x2 2 3 1 2 4 8+ = + = + . ( éil: ,2 1- - )

2. Ô®: y x y2

3 13

7 15

8 1+= + =

+ . ( éil: ,2 3 )

3. Ô®: 0.3 0.1 0.9, 0.2 0.1 0.4x y x y- =- + = . ( éil: ,1 6- )

4. Ô®: x y x y x y8 2 14 3 2 10+ - = + - = + - . ( éil: ,2 6- )

5. Ïu©L v©fë‹ TLjš 55 k‰W« mt‰¿‹ é¤Âahr« 7 våš , m›bt©fis¡

fh©f. ( éil: 31, 24 ) 6. xU ngdh k‰W« xU neh£L ò¤jf¤Â‹ éiy 60. xU ngdhé‹ éiy xU neh£L

ò¤jf¤Â‹ éiyiaél ` 10 FiwÎ våš mt‰¿‹ éiyfis¡ fh©f.

( éil: ` 25, ` 35 )

7. 1x x x3

2 5 22+ = + v‹w rk‹gh£o‹ ó¢Áa§fis¡ fh©f.

( éil: ,23 1- )

8. xx3

4 12 1

2 1+ -+

= v‹w rk‹gh£o‹ ó¢Áa§fis¡ fh©f.

( éil: – 1, 1 )

9. Ã‹tU« v©fis Kiwna ó¢Áa§fë‹ TLjš k‰W« bgU¡f‰gy‹fshf¡

bfh©l ÏUgo gšYW¥ò¡nfhitfis¡ fh©f.

(i) 9, 14 (ii) ,3

1 3 (iii) ,21

23- . (x›bthU c£ÃçÎ¡F« Ïu©L

kÂ¥bg©fŸ) ( éil: (i) x x9 142- + , (ii) x x

3

1 32- + , (iii) x x

2 232

+ + )

387

10. x x x3 17 31 123 2- + - v‹gij x3 2- Mš tF¡F« nghJ »il¡F« <Î k‰W« ÛÂ

fh©f. ( éil: <Î: x x5 72- + , ÛÂ = 2 )

11. x x x x2 7 7 34 3 2+ + - - v‹gij 2 1x + Mš tF¡F« nghJ »il¡F« <Î k‰W«

ÛÂ fh©f. ( éil: <Î: x x x3 33 2+ - - , ÛÂ = 0 )

12. x x x3 8 3 23 2+ + - v‹w gšYW¥ò¡ nfhit¡F ( )x3 1+ MdJ xU fhuâah vd

MuhŒf. ( éil: fhuâašy )

13. Ã‹tUtdt‰¿‰F Û.bgh.t. fh©f: (i) , ,a bc ab c a b c25 75 1253 3 2 2 4 2 2 ,

(ii) ,x y x y3 3 4 4+ - (iii) ,a a a a2 13 2 2

- - + .

( éil: (i) abc252 , (ii) x y+ , (iii) a 1- )

14. Ã‹tUtdt‰¿‰F Û.bgh.k. fh©f: (i) , ,xy z x y x yz10 5 22 2 4 3 4 , (ii) ,a b a b2 2 3 3- +

(iii) ( )( ), ( )( ), ( )( )a b b c b c c a c a a b+ + + + + + .

(x›bthU c£ÃçÎ¡F« Ïu©L kÂ¥bg©fŸ)

( éil: (i) x y z103 4 4 , (ii) ( )( )a b a b

3 3- + , (iii) ( )( )( )a b b c c a+ + + )

15. Ïu©L gšYW¥ò¡nfhitfë‹ Û.bgh.k. k‰W« Û.bgh.t . M»ait Kiwna x y z5 4 7 k‰W« x z2 3 . gšYW¥ò¡nfhitfëš x‹W x z2 3 våš k‰bwh‹iw¡ fh©f.

( éil: x y z5 4 7 )

16. RU¡Ff: (i) a

x

x x

a a

1

4

22

2

3 2

3

#-

-

+

- , (ii) x

x x

x x

x x

9

4 3

6 5

4 32

2

2

2

#-

- +

+ +

+ + .(x›bthU

c£ÃçÎ¡F« Ïu©L kÂ¥bg©fŸ) ( éil: (i) ( )

( )

x a

a x

1

22

2

+

- , (ii) xx

51

+- )

17. RU¡Ff: (i) xx

xx

12

132 2

++ +

+- , (ii)

a ba

b ab3 3

-+

-, (iii)

x x11

12

--

+.

(x›bthU c£ÃçÎ¡F« Ïu©L kÂ¥bg©fŸ)

( éil: (i) xx

12 1

2

+- , (ii) a ab b

2 2+ + , (iii)

x

x

1

32-

- )

18. t®¡f _y« fh©f: ( ) 4x x4 22+ + - . ( éil: | 2 |x + )

19. Ô®: x x2 8 3 02+ + = . ( éil:

4,

44 10 4 10- - - + )

20. xU äif v©Ql‹ mj‹ t®¡f¤ij T£L«nghJ 30 »il¡»wJ våš, mªj

v©iz¡ fh©f. ( éil: – 6 k‰W« 5 )

21. 0.cx bx a2+ + = v‹w rk‹gh£o‹ _y§fŸ a k‰W« b våš, a b ab+ - ‹ kÂ¥ò

fh©f. ( éil: ( )ca b- + )

22. x x 6 02+ - = k‰W« x x5 6 02

+ + = v‹w rk‹ghLfë‹ bghJ _y« fh©f.

( éil: 3- )




1. FW¡F¥ bgU¡fš Kiwiw¥ ga‹gL¤Â Ã‹tU« rk‹ghLfë‹ bjhF¥òfis¤

Ô®:

(i) ;y x xy y x xy4 1 5 2 3 13- = + = , (ii) , , , .

x y x yx y5 2 3 3 4 7 0 0! !+ =- - =-

(x›bthU c£ÃçÎ¡F« IªJ kÂ¥bg©fŸ) ( éil: (i) (2, 3), (ii) (– 1, 1) )

2. xU br›tf¤Â‹ R‰wsÎ 60 br.Û. mj‹ Ús¤ij 3 br.Û. mÂfç¤J mfy¤ij

3 br.Û. Fiw¡F« nghJ mj‹ g¡f§fë‹ é»j« 2 : 1 v‹wh»wJ våš

br›tf¤Â‹ Ús, mfy§fis¡ fh©f. ( éil: 17 br.Û, 13 br.Û )

3. xU Ã‹d¤Â‹ bjhFÂ k‰W« gFÂæ‹ TLjš 12. nkY« mj‹ gFÂÍl‹ 3 I¡

T£odhš mJ bjhFÂia¥ nghš 2 kl§fh»wJ. våš ,m›bt©iz¡ fh©f.

( éil: 75 )

4. fhuâ¥gL¤Jf: ( )x x x6 13 2+ + - . (F¿¥ò: x

2I¡ T£o fê¡f)

( éil: ( )( )( 3)x x x1 2+ + + )

5. x x x x3 6 5 34 3 2+ + + + k‰W« x x x2 24 2

+ + + M»a gšYW¥ò¡ nfhitfë‹

Û.bgh.t. x x 12+ + våš Ït‰¿‹ Û.bgh.k. fh©f.

( éil: ( )( )x x x x x2 3 2 22 4 2+ + + + + )

6. Ïu©L gšYW¥ò¡nfhitfë‹ Û.bgh.t. k‰W« Û.bgh.k. M»ait Kiwna ( )x2 3- k‰W« (2 3)(3 4)(5 4) .x x x- + + gšYW¥ò¡nfhitfëš x‹W x x6 122

- - våš, k‰bwh‹iw¡ fh©f. ( éil: ( )( )x x2 3 5 4- + )

7. RU¡Ff: a

a

a a

a a

a a

a a

8

16

2 9 4

2 3 2

2 4

3 11 43

2

2

2

2

2

# '-

-

+ +

- -

+ +

- - . ( éil: a3 11+

)

8. RU¡Ff: x

x x

x x

x x

9

4 3

6 5

2 152

2

2

2

#-

- +

+ +

- - . ( éil: ( )( )( )( )x xx x

5 11 5

+ +- - )

9. RU¡Ff: m

m m

m m

m m

16

12

8 16

62

2

2

2

'-

- -

+ +

+ - . ( éil: mm

24

-+ )

10. RU¡Ff: a a a a a a7 12

1

5 6

1

6 8

22 2 2+ +

++ +

-+ +

. ( éil: 0 )

11. RU¡Ff: a b a b a b

a1 1 22 2+

+-

--

( éil: 0 )

12. RU¡Ff: ( ) ( )

a a

a

a a

a

3 2

4 1

6

4 32 2+ +

++

- -

- . ( éil: a 28+

)

13. RU¡Ff: a a a

a14

13

1

72+

--

--

. ( éil: a 1

7+- )

14. x x x x4 10 12 94 3 2+ + + + v‹w gšYW¥ò¡nfhitæ‹ t®¡f _y« fh©f.

( éil: x x2 32+ + )

389

15. x x x x1 4 10 12 92 3 4+ + + + v‹w gšYW¥ò¡nfhitæ‹ t®¡f _y« fh©f..

( éil: 3 2 1x x2+ + )

16. x x x ax b4 12 254 3 2+ + + + v‹gJ xU KG t®¡f« våš a k‰W« b Ït‰¿‹

kÂ¥òfis¡ fh©f. ( éil: ,a b24 16= = ) 17. a bx x x x25 24 162 3 4

+ + - + v‹gJ xU KG t®¡f« våš a k‰W« b Ït‰¿‹

kÂ¥òfis¡ fh©f. ( éil: ,a b4 12= =- )

18. Ô®: x

xx

x1

13061

++ + = . ( éil: ,6 5- )

19. fhuâgL¤Jjš Kiwæš Ô®¡f: x a x a b2 02 2 4 4- + - = .

( éil: ( , )b a a b2 2 2 2- + )

20. xU v© k‰W« mj‹ jiyÑê Ït‰¿‹ TLjš 750 . m›bt©iz¡ fh©f.

( éil: ,71 7 )

21. a k‰W« b M»ait x x3 1 02- + = v‹w rk‹gh£o‹ _y§fŸ våš, 1

a b+

k‰W« 1ab

M»at‰iw _y§fshf¡ bfh©l ÏUgo¢ rk‹gh£oid¡ fh©f.

( éil: x x3 4 1 02- + = )

22. Ïu©L v©fë‹ TLjš 24 k‰W« mt‰¿‹ jiyÑêfë‹ TLjš 61 .

m›bt©fis¡ fh©f. ( éil: 12, 12 )

23. xUt® `6000 ¡F Áy bghU£fis th§»dh®. mt® éiy `10 mÂfKŸs ju«

ca®ªj bghU£fis th§»dhš K‹d® th§»a bghU£fis él 50 bghU£fis

Fiwthf th§f ÏaY«, våš mt® th§»a bghU£fë‹ v©â¡ifiaÍ«

éiyiaÍ« fh©f. ( éil: 200, ` 30 )

24. xU br§nfhz K¡nfhz¤Â‹ g¡f§f§fŸ Kiwna 2, 2 1x x+ - k‰W« x2 1+ våš, mj‹ g¡f msÎfisÍ« gu¥gsÎfisÍ« fh©f.

( éil: g¡f§fŸ 4, 3, 5 myFfŸ k‰W« gu¥gsÎ = 6 r.myFfŸ )

25. xU nkh£lh® X£Le® j‹ Ïašghd ntf¤ijél 10 ».Û./kâ Fiwthd

ntf¤Âš bršY« nghJ, 160 ».Û. bjhiyit¡ fl¡f mt® 32 ãäl§fŸmÂf«

vL¤J¡bfh©lh® våš, mtç‹ Ïašghd ntf« v›tsÎ ? ( éil: 60 ».Û./kâ )

26. x x m7 02+ + = v‹w ÏUgo¢rk‹gh£o‹ xU _y« k‰bwh‹iwél 1 mÂf«

våš, m ‹ kÂ¥ig fh©f. ( éil: m = 12 )

27. x px q 02- + = v‹w rk‹gh£o‹ _y§fë‹ é¤Âahr« 1 våš p q4 12

= + vd

ã%Ã¡f.

28. x x ax x b4 16 724 3 2- + - + v‹gJ xU KG t®¡f« våš a k‰W« b Ït‰¿‹

kÂ¥òfis¡ fh©f. ( éil: ,a b52 81= = )

29. RU¡Ff: 2x x

x x

x x

x x

3 2

2

2 5 3

2 32

2

2

2

- +

- - ++ +

+ - - . ( éil: x 1

42-

)





1. (3, 4), ( , )y1 M»a òŸëfis Ïiz¡F« nfh£L¤ J©o‹ eL¥òŸë ( , 2)x våš, x k‰W« y kÂ¥òfŸ Kiwna

(A) 1, 2 (B) 2, 0 (C) ,2 2- (D) ,1 2-

2. y 5=- k‰W« x y 1 0+ + = M»a ne®¡nfhLfŸ bt£L« òŸë

(A) (– 4, – 5) (B) (6, – 5) (C) (4, – 5) (D) (– 5, 6)

3. xU K¡nfhz¤Â‹ eL¡nfh£L ika« MÂ¥òŸë, ( , )1 2- k‰W« ( , )3 5- v‹gd

m«K¡nfhz¤Â‹ ÏUKidfŸ våš, _‹whtJ Kid

(A) (– 2, 3) (B) (2, 3) (C) (– 2, – 3) (D) (2, – 3)

4. y 5= v‹w ne®¡nfh£o‹ ÛJŸs VnjD« xU òŸë ( , )P x y våš, x -m¢ÁèUªJ

òŸë P ¡F cŸs bjhiyÎ

(A) 3 (B) 4 (C) 5 (D) 6

5. ( , )y21

, ( , )y22

M»a òŸëfS¡F ÏilnaÍŸs bjhiyÎ 7 myFfŸ våš, | |y y1 2- =

(A) 7 (B) 7 (C) 4 (D) 0

6. x y2 3 9+ = k‰W« ax y 3+ = M»ad xnu ne®¡nfh£il¡ F¿¡F« rk‹ghLfŸ

våš a -‹ kÂ¥ò

(A) 2 (B) 31 (C)

32 (D) 3

7. xU t£l¤Â‹ ika« ( , )3 4 . Ï›t£l« x -m¢ir¡ bjhLkhdhš, t£l¤Â‹ Mu«

(A) 3 (B) 4 (C) 5 (D) 7

8. A, B v‹w ÏUòŸëfis Ïiz¡F« ne®¡nfh£L¤ J©il ,P 037-` j v‹w òŸë

:2 1 v‹w é»j¤Âš c£òwkhf¥ Ãç¡»wJ. A v‹gJ ( , )2 3- våš B v‹w

òŸë

(A) (2, 1) (B) (– 2, – 1) (C) (2, – 1) (D) (– 1, – 2)

9. P v‹w òŸëia MÂ¥òŸëÍl‹ Ïiz¡F« ne®¡nfh£o‹ eL¥òŸë 21 , 1-` j

våš, òŸë P MdJ

(A) (1, – 2) (B) (4, – 8) (C) (– 2, 1) (D) (– 8, 4)

10. ( , ), ,0 0746 0` j k‰W« ,0

2321` j M»a òŸëfshš mik¡f¥gL« K¡nfhz¤Â‹

gu¥ò

(A) 1 (B) 2 (C) 3 (D) 4 11. t£l¤Â‹ ika« ( , )4 2- . xU é£l¤Â‹ xU Kid MÂ¥òŸë våš, m›é£l¤Â‹

kWKid

(A) (2, – 4) (B) (– 8, 4) (C) (4, – 8) (D) (– 4, 8)

391

12. x -m¢R¡F Ïizahd ne®¡nfh£o‹ rhŒÎ

(A) 0 (B) 1 (C) – 1 (D) tiuaW¡f¥gléšiy

13. x y2 4 5 0+ + = v‹w nfh£o‰F Ïizahd ne®¡nfh£o‹ rhŒÎ

(A) 2 (B) 21 (C)

21- (D) – 2

14. x y4 1 0- + = v‹w nfh£o‰F Ïizahf MÂ¥òŸë tê¢ bršY« ne®¡nfh£o‹

rk‹ghL

(A) x y4 0- = (B) x y4 0+ = (C) x y4 2 0- + = (D) 4 0x y- =

15. x y5 10- = v‹w nfh£o‰F¢ br§F¤jhfÎ« MÂ¥òŸë tê¢bršY«

ne®¡nfh£o‹ rk‹ghL

(A) x y5 10+ = (B) x y5 0- = (C) x y5 0- = (D) 5 0x y+ =

16. rhŒÎ 3 bfh©l ne®¡nfh£o‹ rhŒÎ¡ nfhz«

(A) 0c (B) 30c (C) 60c (D) 90c

17. rhŒÎ¡ nfhz« 90c , bfh©l ne®¡nfh£o‹ rhŒÎ

(A) 1 (B) – 1 (C) 0 (D) tiuaW¡f¥ gléšiy

18. ( , )2 3- v‹w òŸë tê¢ bršY«, y -m¢R¡F Ïizahd ne®¡nfh£o‹ rk‹ghL

(A) 0x 2- = (B) 0x 2+ = (C) 0y 2+ = (D) 0y 2- =

19. x m¢ÁèUªJ 5 myFfŸ bjhiyéY« x m¢R¡F ÏizahfÎ« cŸs

ne®¡nfh£o‹ rk‹ghLfŸ

(A) ,y y5 5= =- (B) ,x x5 5= =-

(C) ,x y0 0= = (D) ,x c y k= =

20. rhŒÎ 3- MfÎ« y bt£L¤J©L 4 MfÎ« cŸs ne®¡nfh£o‹ rk‹ghL

(A) x y3 4 0+ + = (B) x y3 4 0- + =

(C) x y3 4 0+ - = (D) x y3 4 0- - =

21. x y5 3= v‹w ne®¡nfh£o‹ y bt£L¤J©L

(A) 35 (B)

53 (C) 0 (D)

35-

22. x , y bt£L¤J©LfŸ Kiwna 2, 3 vd¡bfh©l ne®¡nfh£o‹ rk‹ghL

(A) x y2 3 6+ = (B) x y3 2 6+ = (C) x y2 3 0+ = (D) x y3 2 0+ =

23. y 7=- k‰W« x 4= v‹w ne®¡nfhLfŸ bt£L«òŸë.

(A) ( , )7 4- (B) (7, 4) (C) (4, – 7) (D) (4, 7) 24. y x5 5 10= + v‹w ne®¡nfh£o‹ rhŒÎ¡nfhz«

(A) 0c (B) 30c (C) 60c (D) 45c 25. ( , )2 1- - , ( , )k 0 M»a òŸëfis Ïiz¡F« ne®¡nfh£o‹ rhŒÎ

61 våš k -‹

kÂ¥ò

(A) – 2 (B) 1 (C) 0 (D) 4jahç¡f¥g£l édh¡fŸ - Ma¤bjhiy toéaš



1. ( , )a2 2 3+ , ( , )b4 2 1+ M»a òŸëfis Ïiz¡F« nfh£L¤J©o‹

eL¥òŸëæ‹ Ma¤bjhiyÎfŸ ( , )a b2 2 våš, a , b M»at‰¿‹ kÂ¥òfis¡

fh©f. ( éil: a = 3, b = 2 )

2. ( , )A 3 7 k‰W« ( , )B 8 2 v‹w òŸëfis Ïiz¡F« nfh£L¤J©oid c£òwkhf :2 3 v‹w é»j¤Âš Ãç¡F« òŸëia¡ fh©f. ( éil: (5, 5) )

3. ( , ), ( , ), ( , )A B a C b2 1 0 4- - k‰W« ( , )D 1 2 v‹gd Ïizfu« ABCD -‹

c¢ÁfŸ våš a ,b M»at‰¿‹ kÂ¥òfis¡ fh©f. ( éil: a = 1, b = 3 )

4. ( , )P 1 2 k‰W« ( , )Q 4 5 v‹w òŸëfis Ïiz¡F« nfh£L¤J©oid

btë¥òwkhf :5 3 v‹w é»j¤Âš Ãç¡F« òŸëia¡ fh©f.

( éil: ,217

219` j )

5. ( , )A 3 1- , ( , )B 6 5- v‹w ÏU òŸëfis Ïiz¡F« nfh£L¤ J©o‹ nkš

AP PB2 = v‹wthW mikªJŸs òŸë P I¡ fh©f. ( éil: (0, 1) )

6. ( , )2 1 k‰W« ( , )5 8- v‹w òŸëfis Ïiz¡F« nfh£L¤ J©il P, Q v‹w

òŸëfŸ _‹W rkghf§fshf¥ Ãç¡»‹wd. òŸë P MdJ x y k2 0- + = v‹w

nfh£o‹ nkš cŸsJ våš k -‹ kÂ¥ò fh©f. ( éil: k = – 8 )

7. ( , ), ( , ), ( , )4 1 6 0 7 2- v‹gd xU rhŒrJu¤Â‹ tçir¥go mikªj c¢ÁfŸ

våš, eh‹fhtJ c¢Áia¡ fh©f. ( éil: (5, 1) )

8. A ( , )1 4 , B ( , )5 3 v‹gd ABCT -‹ Ïu©L c¢ÁfŸ. mj‹ eL¡nfh£L ika« (3, 3) våš, _‹whtJ c¢Á C I¡ fh©f. ( éil: (3, 2) )

9. ( 4, 6), ( , )A B 2 1- - - k‰W« ( , )C 1 2 M»at‰iw Kidfshf¡ bfh©l

K¡nfhz¤Â‹ gu¥ig¡ fh©f. ( éil: 11.5 r.myFfŸ )

10. ( 5, 1), (3, 5)A B- - - k‰W« (5, )C k M»at‰iw Kidfshf¡ bfh©l

ABCT -‹ gu¥ò 32 r.myFfŸ våš, k-‹ kÂ¥ig¡ fh©f. ( éil: k = 2) )

11. ( , ), ( , )2 5 3 4- - k‰W« ( , )8 1 M»aòŸëfŸ xnu ne®¡nfh£oš mikªJŸsd

vd ã%Ã.

12. ( , )p 02 , ( , )q0 2 k‰W« ( , )1 1 v‹gd xnu ne®¡nfh£oš mikÍ« òŸëfŸ, våš 1

p q

1 12 2+ = vd ãWÎf.

13. xU t£l¤Â‹ ika« ( , )2 1- k‰W« AB v‹gJ xU é£l« A v‹gJ ( , )3 5- våš

BI¡ fh©f. ( éil: (7, – 7) )

14. rhŒÎ fhQ« Kiwia¥ ga‹gL¤Â ( , ), ( , )4 3 5 1 k‰W« ( , )1 9 v‹w òŸëfŸ xnu

ne®¡nfh£oš mikÍ« vd ãWÎf.

15. ( 5, ), ( , 7)P a Q b- k‰W« ( , )R 1 3- M»a òŸëfŸ xnu nfh£oš mikªJŸsd

k‰W« PQ QR= våš, a , b -‹ kÂ¥òfis¡ fh©f. ( éil: a = 17, b = – 2 )

393

16. ( , ), ( , )A B p1 3 2- k‰W« ( , )C 5 1- v‹gd xnu nfh£oš mikªJŸs òŸëfŸ

våš p-‹ kÂ¥ig¡ fh©f. ( éil: p = 1 )

17. rhŒÎ¡ nfhz« 60° k‰W« y -bt£L¤ J©L 3

1 vd¡bfh©l ne®¡nfh£o‹

rk‹gh£il¡ fh©f. ( éil: x y3 3 1 0- + = )

18. ( , )3 5- v‹w òŸë tê¢ brštJ«, rhŒÎ 52 cilaJkhd ne®¡nfh£o‹

rk‹gh£il¡ fh©f. ( éil: x y2 5 31 0- - = )

19. xU ne®¡nfh£o‹ x , y bt£L¤J©LfŸ Kiwna 72 k‰W«

31- våš, m¡nfh£o‹

rk‹gh£il¡ fh©f. ( éil: 7 6 2 0x y- - = )

20. x y3 7 21 0+ - = v‹w ne®¡nfh£o‰F ÏizahdJ« ( , )2 3 v‹w òŸë tê¢

brštJkhd ne®¡nfh£o‹ rk‹gh£il¡ fh©f. ( éil: x y3 7 27 0+ - = )

21. ( , )4 4- k‰W« ( , )10 12- - v‹w òŸëfis Ïiz¡F« ne®¡nfhL« ( , )8 16- ,( , )14 0 v‹w òŸëfis Ïiz¡F« ne®¡nfhL« ÏizahF« vd ãWÎf.

22. ( , )5 2- v‹w òŸë tê¢brštJ« ( , )A 12 2- k‰W« ( , )B 4 10- - v‹w òŸëfis

Ïiz¡F« nfh£o‰F ÏizahdJkhd nfh£o‹ rk‹ghL fh©f.

( éil: x y2 9 0- + = )

23. x y2 3 6 0+ + = , x y3 2 12 0- - = v‹w ne®¡nfhLfŸ x‹W¡bfh‹W

br§F¤ jhdit vd ãWÎf.

24. x y2 4

1+ = k‰W« x y2 5+ = v‹w ne®¡nfhLfŸ Ïizahditah vd MuhŒf.

( éil:Ïizahdit )

25. kx y3 14 0- - = , x y3 4 10 0+ + = v‹gd x‹W¡bfh‹W br§F¤jhdit våš, k -‹ kÂ¥ig¡ fh©f. ( éil: k = 4 )


1. rhŒÎ fhQ« Kiwia¥ ga‹gL¤Â ( , ), ( , ), ( , )A B C4 1 2 4 4 0- - - - k‰W« ( , )D 2 3 v‹w òŸëfŸ xU br›tf¤Â‹ c¢ÁfŸ vd ãWÎf.

2. rhŒÎ fhQ« Kiwia¥ ga‹gL¤Â ( , ), ( , ), ( , )0 5 2 2 5 0- - k‰W« ( , )7 7 v‹w

òŸëfŸ xU rhŒrJu¤Â‹ c¢ÁfŸ vd ãWÎf.

3. ( , )A a2 0 , ( , )B b0 2 M»a òŸëfis Ïiz¡F« nfh£L¤ J©o‹ eL¥òŸë M MÂ¥òŸë O våš, M MdJ ,O A k‰W« B v‹w òŸëfëèUªJ rk bjhiyéš

mikÍ« vd¡fh£Lf.

4. ( , )5 4- k‰W« ( , )3 2- M»a òŸëfis Ïiz¡F« nfh£L¤ J©il _‹W

rkghf§fshf¥ Ãç¡F« òŸëfë‹ m¢R¤ bjhiyÎfis¡ fh©f.

( éil: ,37 2-` j k‰W« ,

31 0-` j )

jahç¡f¥g£l édh¡fŸ - Ma¤bjhiy toéaš


5. ( 2, 0)- k‰W« ( , )0 8 v‹w òŸëfis Ïiz¡F« nfh£L¤J©il eh‹F

rk ghf§fshf¥ Ãç¡F« òŸëfë‹ m¢R¤bjhiyÎfis¡ fh©f.

( éil: , , ( , ) ,k‰W«23 2 1 4

21 6- - -` `j j )

6. ( , )3 4- k‰W« ( , )1 2 v‹w òŸëfis Ïiz¡F« nfh£L¤ J©il _‹W

rkghf§fshf¥ Ãç¡F« òŸëfŸ ( , )P a 2- k‰W« ,Q b35` j våš, a k‰W« b

Ït‰¿‹ kÂ¥òfis¡ fh©f. ( éil: , ba37 0= = )

7. ( , )A 2 2- k‰W« ( , )B 3 7 v‹w òŸëfis Ïiz¡F« nfh£L¤ J©il ( , 6)m v‹w òŸë Ãç¡F« é»j¤ij¡ fh©f. nkY«, m -‹ kÂ¥ig¡ fh©f.

( éil: 4 : 1, m = 2 )

8. ( , )2 3- k‰W« ( , )5 6 v‹w òŸëfis Ïiz¡F« nfh£L¤ J©oid x -m¢R

Ãç¡F« é»j¤Âid¡ fh©f. nkY«, Ãç¡F« òŸëæ‹ m¢R¤ bjhiyÎfisÍ«

fh©f. ( éil: 1 : 2, (3, 0) )

9. ( 2, 3)- - k‰W« ( , )3 7 v‹w òŸëfis Ïiz¡F« nfh£L¤ J©oid vªj

é»j¤Âš y -m¢R Ãç¡»wJ. nkY«, Ãç¡F« òŸëfis fh©f.

( éil: 2 : 3, (0, 1) )

10. xU K¡nfhz¤Â‹ c¢ÁfŸ ( , ), ( , )A B1 3 1 1- - k‰W« ( , )C 5 1 våš, ( , )1 3- k‰W« ( , )5 1 v‹w c¢Áfë‹ têna m«K¡nfhz¤Â‰F tiua¥gL«

eL¡nfhLfë‹ Ús§fis¡ fh©f. ( éil: 5, 5 )

11. ( , ), ( , ), ( , )1 6 3 9 5 8- - - - - k‰W« ( , )3 9 v‹w òŸëfis c¢Áfshf¡ bfh©l

eh‰fu¤Â‹ gu¥ig¡ fh©f. ( éil: 60 r.myFfŸ)

12. (1,2), ( 3,4),- , , (4, )k5 6- -^ h v‹w òŸëfis c¢Áfshf¡ bfh©l eh‰fu¤Â‹

gu¥ò 43 r.myFfŸ våš k -‹ kÂ¥ig¡ fh©f. ( éil: – 1)

13. ( , ), ( , )2 5 4 1- - k‰W« ( , )6 3 M»at‰iw Kidfshf¡ bfh©l K¡nfhz¤Â‹

eL¡nfhLfë‹ rhŒÎfis¡ fh©f. ( éil: , ,74

25

51- - )

14. x y4 3 12 0+ - = v‹w ne®¡nfhL ,x y m¢R¡fis Kiwna A , B v‹w òŸëfëš

bt£L»wJ. våš, AOBT -‹ gu¥ig¡ fh©f. ( éil: 6 r.myFfŸ )

15. ( , )A 2 4- k‰W« ( , )B 6 8- v‹w òŸëfis Ïiz¡F« ne®¡nfh£L¤J©o‹

ika¡F¤J¡ nfh£o‹ rk‹gh£il¡ fh©f. ( éil: 2 0x y3 1 0- - = )

16. ABCT -‹ c¢ÁfŸ ( , ), ( , )A B1 7 0 2- , ( , )C 3 3 . våš, A têna bršY« eL¡nfh£o‹

rhŒÎ k‰W« rk‹gh£il¡ fh©f. ( éil: – 13, 13 20 0x y+ - = )

17. ( , )4 5 v‹w òŸëÍl‹ x y5 3 8- = , x y2 3 5- = v‹w ne®¡nfhLfŸ rªÂ¡F«

òŸëia Ïiz¡F« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

( éil: x y2 3 0- - = )

395

18. x y 2 0+ - = , x y2 3 0+ - = M»a ne®¡nfhLfŸ rªÂ¡F« òŸëÍl‹

( , )4 2 , ( , )6 4- v‹w òŸëfis Ïiz¡F« ne®¡nfh£il ÏUrk¡ T¿L«

òŸëia Ïiz¥gjhš »il¡F« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

( éil: x y 2 0+ - = )

19. x y 2 0- - = , x y3 4 15 0+ + = M»a ne®¡nfhLfŸ rªÂ¡F« òŸëiaÍ«, x y3 3 0- + = , 2 8 0x y+ - = M»a ne®¡nfhLfŸ rªÂ¡F« òŸëiaÍ«

Ïiz¡F« ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

( éil: (– 1, – 3), (3, 2), x y5 4 7 0- - = )

20. x y 5 0+ - = , x y3 1 0- + = M»a ne®¡nfhLfŸ rªÂ¡F« òŸë tê¢brštJ« ( , )3 1 k‰W« (2,3) v‹w òŸëfis Ïiz¡F« ne®¡nfh£o‰F ÏizahdJkhd

ne®¡nfh£o‹ rk‹gh£il¡ fh©f. ( éil: x y2 6 0+ - = )

21. x y5 8 23 0- + = , 7 6 7 0x y 1+ - = M»a ne®¡nfhLfŸ rªÂ¡F« òŸë

tê¢brštJ« ( , )5 1 k‰W« ( , )2 2- v‹w òŸëfis Ïiz¡F« ne®¡nfh£o‰F

br§F¤jhdJkhd ne®¡nfh£o‹ rk‹gh£il¡ fh©f.

( éil: (5, 6), x y7 29 0- - = )

jahç¡f¥g£l édh¡fŸ - Ma¤bjhiy toéaš


6. toéaš


1. gl¤Âš DE AB< k‰W« : 3:2AD DC =

våš, ABCT^ h Ï‹ gu¥ò : DECT^ h Ï‹ gu¥ò =

(A) 4 : 25 (B) 4 : 9

(C) 9 : 4 (D) 25 : 4

2. ÏU rk g¡f br§nfhz K¡nfhz« ABC-š, C-š br§nfhz« mikªJŸsJ våš

(A) AB AC22 2= (B) AC AB22 2

=

(C) BC CA22 2= (D) AC BC22 2

=

3. ABC PQRT T+ k‰W« PQRT Ï‹ gu¥ò 4= Ï‹ gu¥òABCT^ h våš :AB PQ =

(A) 2 : 1 (B) 4 : 1 (C) 1 : 2 (D) 1 : 4

4. rçtf« ABCD-š AB DC< k‰W« AB = 2DC, våš

AOBT Ï‹ gu¥ò : CODT Ï‹ gu¥ò =

(A) 1 : 2 (B) 4 : 1

(C) 1 : 4 (D) 2 : 1

5. gl¤Âš AB = AC våš, Ã‹tUtdt‰¿š vJ rçahdJ ?

(A) TABD +TCFE

(B) TABD +TFCE

(C) TABD +TECF

(D) TABD +TEFC

6. gl¤Âš DE AC< k‰W« DF AE< våš, FEBF =

(A) ECFE (B)

ECBE

(C) BABD (D)

BEEC

7. gl¤Âš C v‹gJ t£l§fë‹ bghJ ika«. eh© PQ MdJ,

Mu« 3 br.Û cŸs Á¿a t£l¤ij R-š bjhL»wJ k‰W« PQ = 8 br.Û våš, bgça t£l¤Â‹ Mu«

(A) 3 br.Û. (B) 4 br.Û.

(C) 5 br.Û. (D) 2 br.Û.

8. TABC-š, AB = 6 br.Û k‰W« A+ æ‹ ÏUrkbt£o AD. BD : DC = 3 : 2 våš AC =

(A) 4 br.Û. (B) 6 br.Û. (C) 6 br.Û. (D) 8 br.Û.

397

9. gl¤Âš, ,AB QRAPPQ

2< = k‰W« QR = 8 br.Û, våš, AB =

(A) 10 br.Û. (B) 8 br.Û.

(C) 6 br.Û. (D) 4 br.Û.

10. gl¤Âš x-‹ kÂ¥ò

(A) 3 (B) 4

(C) 5 (D) 6

11. gl¤Âš T PTl v‹gJ t£l¤Â‰F P-š bjhLnfhL.

130QPT+ =l c våš PRQ+ =

(A) 65c (B) 50c

(C) 130c (D) 40c

12. Ïizfu« ABCD-š 110B+ = c. E v‹gJ ,ADAB

EDEB= vDkhW BD æš cŸs xU

òŸë våš, EAD+ =

(A) 35c (B) 45c

(C) 60c (D) 70c

13. gl¤Âš, PT v‹gJ P-š bjhLnfhL k‰W« PQ = QR våš QPT+ =

(A) 30c (B) 60c

(C) 45c (D) 90c

14. gl¤Âš, LM NQ< k‰W« LN PQ< våš Ã‹tUtdt‰¿š vJ rçahdJ ?

(A) TLMN +TNQP

(B) TQNP +TMNL

(C) TLNM +TQNP

(D) TLMN +TQNP

15. 6 Û, 11 Û cauKŸs Ïu©L br§F¤jhd f«g§fS¡F Ïilna cŸs bjhiyÎ

12 Û våš , mt‰¿‹ c¢ÁfS¡F Ïilna cŸs öu«

(A) 7 Û (B) 13 Û (C) 9 Û (D) 10 Û

jahç¡f¥g£l édh¡fŸ - toéaš


7. K¡nfhzæaš


1. cos tan sin cot1 12 2i i i i+ + + =

(A) 2 (B) 1 (C) cosec i (D) sec i

2. ( ) ( )cos cosec sin sec9 900 i i i i- - - =c c

(A) 1 (B) 0 (C) 2 (D) 1-

3. (90 ) (90 )tan tan cot coti i i i- + - =c c

(A) tan 45c (B) 60cosec c (C) sec 60c (D) cot 90c

4. (90 ) (90 )cos cos sin sini i i i- - - =c c

(A) sin90c (B) cot45c (C) cosec45c (D) cos90c

5. A + B = 90c, våš cos A sin B + sin A cos B =

(A) 2 (B) 1 (C) 0 (D) 45c

6. cosec tan67 232 2- =c c

(A) 0 (B) 1 (C) – 1 (D) 2

7. gl¤Âš AB BC3= våš, i =

(A) 30c (B) 45c

(C) 60c (D) 90c

8. cosecsin

seccos

ii

ii+ =

(A) sin cos2 2i i- (B) cosec cot2 2i i-

(C) sec cosec2 2i i+ (D) 0

9. 1 (90 ) 45tan cosec2 2i+ - =c c våš, i =

(A) 30c (B) 90c (C) 60c (D) 45c

10. , ( )sec cotx y5 5 90i i= = -c våš, y x2 2- =

(A) 0 (B) 5 (C) 25 (D) – 25

11. cossin sin cos

3

ii i i+ =

(A) sini (B) cosi

(C) tani (D) coti

12. gl¤Âš x -‹ kÂ¥ò

(A) 5 br.Û. (B) 20 br.Û.

(C) 10 br.Û. (D) 40 br.Û.

399

13. xU nfhòu¤Â‹ ãHè‹ Ús« mj‹ Ús¤ij¥ nghš 3

1 kl§F våš Nçaå‹

V‰w¡ nfhz«

(A) 30c (B) 45c (C) 60c (D) 90c

14. 5 85tan tanc c-‹ kÂ¥ò

(A) 3

1 (B) 3 (C) 1 (D) 0

15. cossin1ii+ =

(A) sin

cos1 i

i-

(B) cos

sin1 i

i+

(C) sinsin

11

ii

-+ (D)

sinsin

11

ii

+-

16. sec tan

sec tan2 2

4 4

i i

i i

+

- =

(A) 1 (B) sec tan2 2i i (C) 2 (D) sin cos2 2i i

17. tan cot

1i i+

=

(A) sin cosi i+ (B) sin cosi i (C) sin cosi i- (D) cosec coti i+

18. sin sin tan20 70 452 2+ - =c c c

(A) 1 (B) 0 (C) 2 (D) – 1

19. br§nfhz K¡nfhz« ABC-æš, 90 , sinB A31+ = =c k‰W« BC 11= våš,

AC =

(A) 311 (B) 120 (C) 33 (D) 118

20. cos cos

sin sin3

3

i i

i i

-

- =

(A) tan2i (B) cot2i (C) tan i (D) cot i

jahç¡f¥g£l édh¡fŸ - K¡nfhzéaš


8. mséaš


1. xU Â©k ne® t£l cUisæ‹ mo¢R‰wsÎ 132 br.Û. k‰W« mj‹ cau«

10 br.Û.våš, mj‹ bkh¤j òw¥gu¥ig¡ fh©f. ( éil: 4092 br.Û 2 )

2. xU Â©k ne®t£l cUisæ‹ bkh¤j òw¥gu¥ò 600r r.br.Û. k‰W« mj‹ cau« 13 br.Û. våš, mj‹Mu¤ij¡ fh©f. ( éil: 12 br.Û. )

3. 14 Û MHKŸs xU »z‰¿‹ Mu« 5 Û. xU rJu Û£lU¡F ` 2 Åj« m¡»z‰¿‹

c£òw¢ Rt‰¿‰F Ábk©£ ór MF« bryéid¡ fh©f. ( éil: ` 880 )

4. xU ne®t£l T«Ã‹ Mu« k‰W« bkh¤j òw¥gu¥ò Kiwna 6 br.Û. k‰W«

96r r.br.Û. våš m¡T«Ã‹ cau« fh©f. ( éil: 8 br.Û. )

5. xU ne®t£l¡ T«Ã‹ bkh¤j òw¥gu¥ò 770 r.br.Û. m¡T«Ã‹ rhÍaukhdJ,

Mu¤ij¥ nghš 4 kl§F våš mo¥ghf¤Â‹ é£l« fh©f. ( éil: 14 br.Û. )

6. xU T«Ã‹ Ïil¡f©l toéyikªj thëæ‹ cau« 24 br.Û. k‰W« ÏUòwK«

mikªj t£l¥ gFÂfë‹ Mu§fŸ 12 br.Û., 4 br.Û våš, mj‹ bfhŸsit¡

fh©f. ( éil: 1664rbr.Û 3 )

7. 14 br.Û. ÚsKŸs cŸÇl‰w cUisæ‹ btë k‰W« cŸ Mu§fŸ Kiwna 4 br.Û.

k‰W« 3 br.Û våš, cUisæ‹ bkh¤j òw¥gu¥ò fh©f. ( éil: 660 br.Û 2 )

8. xU nfhs¤Â‹ bkh¤j òw¥gu¥ò 616 br.Û 2 våš, mj‹ Mu¤ij fh©f.

( éil: 7 br.Û. )

9. ãyé‹ é£lkhdJ njhuhakhf òéæ‹ é£l¤Âš 4-š 1 g§F MF«. mt‰¿‹

òw¥gu¥òfë‹ é»j¤ij¡ fh©f. ( éil: 1 : 16 )

10. xU ne® t£l cUisæ‹ fdmsÎ 744 br.Û 3 . m›ÎUisæ‹ cau« 8 br.Û. våš,

mj‹ é£l« fh©f. ( éil: 1 br.Û. )

11. xU ne®t£l¡ T«Ã‹ Mu« k‰W« rhÍau« Kiwna 5 br.Û. k‰W« 13 br.Û våš,

T«Ã‹ fdmsÎ fh©f. ( éil: 31472 br.Û 3 )

12. xU ne® t£l¡ T«Ã‹ Mu« k‰W« rhÍau« Kiwna 8 br.Û. k‰W« 12 br.Û. MF«.

4 br.Û. MuKŸs nfhs¤Â‹ fd msé‹ v¤jid kl§fhdJ bfhL¡f¥g£l

T«Ã‹ fd msé‰F rk«. ( éil: 3 kl§F )

13. 14 br.Û. g¡f msÎ bfh©l xU fd¢rJu¤ÂèUªJ, äf¥bgça (Û¥bgU fdmsÎ

cŸs) nfhs« bt£obaL¡f¥gL»wJ våš, m¡nfhs¤Â‹ fdmsÎ fh©f.

( éil: 143731 br.Û 3 )

401

14. xU Õ¥ghæ‹ c£òwK«, btë¥òwK« mj‹ òw¥gu¥òfS¡F t©z« ór nt©L«.

Ï¥Õ¥ghæ‹ Mu« k‰W« cau« Kiwna 1.4 Û k‰W« 3 Û MF«. xU r. Û£lU¡F ` 10 Åj« t©z« ór MF« bryÎ fh©f. ( éil: ` 528 )

15. nfhs¤Â‹ fdmsÎ 17932 br.Û 3 våš, nfhs¤Â‹ tisgu¥ò fh©f.

( éil: 154 br.Û 2 )IªJ kÂ¥bg© édh¡fŸ

1. 1.5 br.Û. é£lK« 0.2 br.Û. jokD« bfh©l v¤jid ehza§fis cU¡»dhš

10 br.Û. cauK« 4.5 br.Û. é£lK« bfh©l ne® t£l Â©k cUis »il¡F«? ( éil: 450 ehza§fŸ )

2. fd¢br›tf tot, cnyhf¡ f£oæ‹ Ús«, mfy« k‰W« cau« Kiwna 44 br.Û.,

21 br.Û. k‰W« 12 br.Û. Ï›Înyhf¡ f£oahdJ cU¡f¥g£L xU Â©k¡ T«ghf

kh‰w¥gL»wJ. T«Ã‹ cau« 24 br.Û. våš mj‹ mo¥g¡f¤Â‹ é£l¤Â‹

msÎ fh©f. ( éil: é£l« = 42 br.Û. ) 3. 9 br.Û. MuKŸs miu¡nfhs tot »©z« KGtJ« Âut¤jhš ãu¥g¥g£LŸsJ.

Ï¤ÂutkhdJ 3 br.Û. é£lK«, 4 br.Û. cauK« bfh©l Á¿a cUis tot

F¥ÃfS¡F kh‰w¤ njitahd F¥Ãfë‹ v©â¡if fh©f.

( éil: 54 F¥ÃfŸ)

4. xU Á¿a ÏU«ò nfhs¤Â‹ é£l« 6 br.Û. MF«. mnj msÎ bfh©l 8 ÏU«ò¡

nfhs§fshdJ Á¿jsÎ j©Ù® ãu¥g¥g£l 20 br.Û. é£lKŸs xU cUis

tot¥ gh¤Âu¤Âš nghl¥gL»wJ. Ï›éU«ò¡nfhs§fŸ mid¤ijÍ« Úçš

_œf¢ brŒjhš, caU« j©Ù® k£l¤Â‹ cau¤ij¡ fh©f. ( éil: 2 br.Û. )

5. 14 Û MuK« 50 Û rhÍauK« bfh©l 5 xnu khÂçahd T«ò tot¡ Tlhu§fŸ

mik¡f¤ njitahd Jâæ‹ mfy« 4 Û våš m¤Jâæ‹ Ús¤ij¡ fh©f.

( éil: 2750 Û. )

6. 10 Û é£lKŸs xU miu¡nfhs tot¤bjh£oæš j©ÙuhdJ 20 br.Û. MuKŸs

FHhŒ têna kâ¡F 20 ».Û. ntf¤Âš brY¤j¥gL»wJ. bjh£o KGtijÍ«

j©Ùuhš ãu¥g MF« neu¤ij¡ fh©f. ( éil: 6 kâ 15 édhofŸ. ) 7. »ç¡bf£ °l«ghdJ cUisæ‹ÛJ T«ò Ïizªj toéš cŸsJ. °l«Ã‹

é£l« k‰W« bkh¤j cau« Kiwna 18 br.Û. k‰W« 80 br.Û. MF«. T«ò ghf¤Â‹

cau« 17 br.Û. våš °l«Ã‹ bkh¤j òw¥gu¥Ãid¡ fh©f.

( éil: 234173 br.Û 2 )

8. 14 br.Û. é£lK« 10 br.Û. cauK« bfh©l xU Â©k cUis tot ku¡f£il

æèUªJ 12 br.Û. é£lKŸs xU nfhs« bt£o vL¡f¥g£l¥Ã‹ vŠÁa

ku¡f£ilæ‹ fd mséid¡ fh©f.

( éil: 63476 br.Û 3 )

jahç¡f¥g£l édh¡fŸ - mséaš


9. xU Â©k ne® t£l¡ T«Ã‹ Mu« 7 br.Û, bkh¤j òw¥gu¥ò 704 br.Û 2 våš,

mj‹ fd msÎ fh©f. ( éil: 1232 br.Û 3)

10. xU Â©k ne® t£l cUisæ‹ tisgu¥ò k‰W« bkh¤j òw¥gu¥ò Kiwna 880 r.Û.

k‰W« 1188 r.Û. MF«. Ï›ÎUisæ‹ fdmsÎ fh©f. ( éil: 3080 br.Û 3)

11. xU Â©k miu¡ nfhs¤Â‹ òw¥gu¥ò 942 br.Û 2. Ï¡nfhskhdJ cU¡f¥g£L

2 br.Û é£l« bfh©l xnu khÂçahd Á¿a nfhs¥gªJfshf kh‰w¥g£lhš,

v¤jid nfhs¥gªJfŸ »il¡F«. ( 3.14r = )

( éil: 500 nfhs§fŸ)

12. bfhL¡f¥g£l xU nfhs¤Â‹ òw¥gu¥ò 154 br.Û 2 . Ï¡nfhs¤Â‹ Mu¤ij¥nghy

ÏU kl§F Mu« bfh©l nfhs¤Â‹ fdmsÎ fh©f.

( éil: 143731 br.Û 3)

13. xU Â©k ne®t£l cUisæ‹ bkh¤j òw¥gu¥ò 96r br.Û 2. Ï›ÎUisæ‹

caukhdJ Mu¤ijél 4 br.Û mÂf« våš, mj‹ fd msÎ fh©f.

( éil: 40272 br.Û 3)

403

F¿¡nfhŸ tif édh¡fë‹ Ô®ÎfŸ

2. bkŒba©fë‹ bjhl®tçirfŸ k‰W« bjhl®fŸ

1 2 3 4 5 6 7 8 9 10A D C B B C D B D B11 12 13 14 15 16 17 18 19 20B B B B B D A D A C

3. Ïa‰fâj«

1 2 3 4 5 6 7 8 9 10B D C B C D A A D C11 12 13 14 15 16 17 18 19 20B D B A C A B B B A


1 2 3 4 5 6 7 8 9 10B C D C A C B D A C11 12 13 14 15 16 17 18 19 20B A C A D C D A A C21 22 23 24 25C B C D D

6. toéaš

1 2 3 4 5 6 7 8 9 10D A C B B B C A D C11 12 13 14 15C A C C B

7. K¡nfhzéaš

1 2 3 4 5 6 7 8 9 10A B C D B B C B D C11 12 13 14 15 16 17 18 19 20C B C C A A B B C D

jahç¡f¥g£l édh¡fŸ - F¿¡nfhŸ tif édh¡fë‹ Ô®ÎfŸ


khÂç édh¤jh£fŸ

bghJ tê fh£L« be¿KiwfŸ

1. fz¡F ghl üèš ju¥g£LŸs édh¤jhŸ totik¥ò

(Blue Print)mo¥gilæškhÂçédh¤jh£fŸmikªJŸsd.

2. SCORE BOOK üèšcŸstif¥gL¤j¥g£lédh¡fëèUªJ

2kÂ¥bg©,5kÂ¥bg©vL¡f¥g£LŸsd.ghlüèèUªJ

10kÂ¥bg©édh¡fŸvL¡f¥g£LŸsd.

3. édh¤jhëš Ïl«bgW« jahç¡f¥g£l édh¡fŸ,

ghl¤Â£l« k‰W« ghlüèš mikªj bghUsl¡f¤Â‰F

c£g£L jahç¡f¥g£LŸsd. SCORE BOOK-š ju¥g£LŸs

cUth¡f¥g£lkhÂçédh¡fisfU¤ÂšbfhŸf.

4. édh¡fëš njitahd Ïl§fëš gl§fŸ bfhL¡f¥

g£LŸsd.

405khÂçédh¤jhŸ

khÂç édh¤jhŸ - 1

fhy« : 2.30 kâ bkh¤j kÂ¥bg©fŸ : 100

bghJ F¿¥òfŸ :

(i)Ï›édh¤jhŸeh‹FÃçÎfis¡bfh©LŸsJ.éilaë¡F«K‹d®x›bthUÃçéY«

bfhL¡f¥g£LŸsF¿¥òfisftdkhfgo¡fÎ«.

(ii)éilfë‹têKiwfŸéil¤jhë‹x›bthUg¡f¤Â‹Ñœ¥gFÂæšfh£l¥glnt©L«.

(iii) fâ¥gh‹k‰W«ä‹dQrhjd§fŸga‹gL¤j¡TlhJ.

ÃçÎ - mF¿¥ò:(i)Ï¥ÃçéšcŸs15édh¡fS¡F«éilaë¡fÎ«.

(ii)bfhL¡f¥g£LŸseh‹FéilfëšäfÎ«rçahdéilia¤nj®ªbjL¤JvGjÎ«.

(iii)x›bthUédhé‰F«xUkÂ¥bg©. 15 × 1 = 15

1. ( )f x = 1 x-^ h v‹gJN-èUªJ Z- ¡FtiuaW¡f¥g£LŸsJ. f -‹Å¢rf«

(A) { 1} (B) N (C) { 1, – 1 } (D) Z 2. –3, –3, –3,g v‹wbjhl®tçirahdJ

(A) xUT£L¤bjhl®tçirk£L«

(B) xUbgU¡F¤bjhl®tçirk£L«


(D) xUT£L¤bjhl®tçirk‰W«bgU¡F¤bjhl®tçir

3. , , , , , ,1 1 0 1 1 0 g- - v‹wbjhl®tçiræ‹108tJcW¥ò

(A) 1 (B) –1 (C) 0 (D) 108

4. ax bx c 02+ + = v‹w ÏUgo¢ rk‹gh£o‹ _y§fŸ a k‰W« b våš,

1ak‰W« 1

b M»adt‰iw_y§fshf¡bfh©lÏUgo¢rk‹ghL

(A) ax bx c 02+ + = (B) 0bx ax c2

+ + =

(C) 0cx bx a2+ + = (D) 0cx ax b2

+ + =

5. x x7 2 12- + v‹wgšYW¥ò¡nfhitiax 3- MštF¡F«nghJ»il¡F«ÛÂ

(A) 58 (B) 70 (C) 0 (D) 3

6. A-‹tçir m n# k‰W«B-‹tçir p q# v‹f.nkY«, Ak‰W« B M»adt‰¿‹

TLjšfhzÏaYbkåš, (A) m p= (B) n = q (C) n = p (D) m = p, n = q 7. 3x + 6y + 7 = 0 k‰W«2x + ky = 5 M»ane®¡nfhLfŸbr§F¤jhditvåš,k-‹

kÂ¥ò

(A) 1 (B) –1 (C) 2 (D) 21

8. (1, 0), (0, 1), (–1, 0) k‰W« (0, –1) M»aòŸëfisKidfshf¡bfh©leh‰fu¤Â‹

xU_iyé£l«______ myFfŸ

(A) 1 (B) –1 (C) 2 (D) 2


9. (DIAGRAM) gl¤Âš, PA, PB v‹gdt£l¤Â‰FbtënaÍŸs òŸëP-æèUªJtiua¥g£l¤bjhLnfhLfŸ. nkY«CD v‹gJ Q v‹wòŸëæšt£l¤Â‰F bjhLnfhL. PA = 8 br.Û, CQ = 3 br.Ûvåš, PC =


10. Ïu©L tobth¤j K¡nfhz§fë‹ gu¥gsÎfŸ Kiwna 16 br.Û2, 36 br.Û2.

mitfë‹F¤Jau§fë‹é»j«2 : x våšx ‹kÂ¥ò

(A) 2 (B) 3 (C) 4 (D) 6

11. cos sinx x4 4

- =

(A) 2 1sin x2

- (B) 2 1cos x2

- (C) 1 2sin x2

+ (D) 1 2 .cos x2

-

12. sin cos sec tan cosec cot2 2 2 2 2 2i i i i i i+ + - - + =

(A) 0 (B) 1 (C) 2 (D) 3

13. xUnfhs¤Â‹tisgu¥ò36r r.br.Û våš,mj‹fdmsÎ

(A) 12r br.Û3 (B) 36r br.Û3 (C) 72r br.Û3 (D) 108r br.Û3

14. Kjš11 Ïašv©fë‹éy¡ft®¡f¢ruhrç

(A) 5 (B) 10 (C) 5 2 (D) 10

15. ( ) . , ( ) . , ( ) . ( )våš, « mšy k‰W« « mšyÍ ÍP A P B P A B P A B0 25 0 50 0 14+= = = =

(A) 0.39 (B) 0.25 (C) 0.11 (D) 0.24

ÃçÎ - MF¿¥ò:(i)g¤Jédh¡fS¡Féilaë¡fÎ«.

(ii)Kjš14édh¡fëèUªJVnjD«9édh¡fS¡Féilaë¡fÎ«.édhv©30¡F

f©o¥ghféilaë¡fÎ«.

(iii)x›bthUédhé‰F«Ïu©LkÂ¥bg©fŸ. 10 × 2 = 20 16. { , , }, { , , , } { , , , }k‰W«P a b c Q g h x y R a e f s= = = våš,Ã‹tUtdt‰iw¡fh©f.

\R P Q+^ h. 17. ÑnHbfhL¡f¥g£LŸsm«ò¡F¿¥gl«xUrh®Ãid¡F¿¡FkhvdMuhŒf.

18. 16 48 144 432 g- + - + v‹w bgU¡F¤ bjhlçš cŸs Kjš 25 cW¥òfë‹

TLjiy¡fh©f.

19. x

x

2

12

3

+

- cl‹vªjé»jKWnfhitia¡T£odhš x

x x

2

2 32

3 2

+

- + »il¡F« ?

20. ,2

4 72

4 7+ - _y§fshf¡bfh©lÏUgo¢rk‹gh£oidfh©f

A

BC

D

PQ

407khÂçédh¤jhŸ

21. a i j2 3ij= - v‹wcW¥òfis¡bfh©l,tçir2 3# cŸsmâA a

ij= 6 @-æid

mik¡fÎ«.

22. k‰W«A B4

5

2

9

8

1

2

3=

-

-=

- -e eo o våš, A B6 3- v‹wmâia¡fh©f.

23. , , , ,7 3 6 1^ ^h h ,8 2^ h k‰W« ,p 4^ h v‹gd X® Ïizfu¤Â‹ tçir¥go mikªj

c¢ÁfŸvåš,p-‹kÂ¥ig¡fh©f.

24. ABCT -š A+ v‹w nfhz¤Â‹c£òwÏUrkbt£oAD MdJ, g¡f«BCI D-š

rªÂ¡»wJ. BD = 2.5 br.Û, AB = 5 br.Û k‰W« AC = 4.2 br.Û våš,DC-Ifh©f.

25. cau« 150 br.Û cŸs xU ÁWä xUés¡F¡ f«g¤Â‹K‹ ã‹wthW 150 3 br.Û ÚsKŸs ãHiy V‰gL¤J»whŸ våš, és¡F¡ f«g¤Â‹ c¢Áæ‹ V‰w¡

nfhz¤ij¡fh©f.

26. Ã‹tU«K‰bwhUikiaãWÎf. sinsin sec tan

11

ii i i

+- = -

27. Ïu©Lne®t£lcUisfë‹Mu§fë‹é»j«2 : 3. nkY«cau§fë‹é»j«

5 : 3 våš,mt‰¿‹fdmsÎfë‹é»j¤ij¡fh©f.

28. xU òŸëétu¤Â‹ Û¢ÁW kÂ¥ò 12. mj‹ Å¢R 59 våšm¥òŸëétu¤Â‹

Û¥bgUkÂ¥ig¡fh©f.

29. xUigæšcŸs1Kjš100tiuv©fshšF¿¡f¥g£l100 Ó£LfëèUªJxU

Ó£LvL¡f¥gL»wJ.m›thWvL¡f¥gL«Ó£o‹v©10 MštFgL«v©zhf

ÏU¥gj‰fhdãfœjféid¡fh©f.

30. (a) y x xy10 9 8+ =- v‹wne®¡nfh£o‹ , x y bt£L¤J©Lfis¡fh©f

(mšyJ)

(b) xU ne®t£lcUisæ‹ bkh¤j¥gu¥òmj‹ òw¥gu¥ig nghš_‹W kl§F

våšmj‹cau¤ijmj‹Mu«tê¡fh©f.

ÃçÎ - ÏF¿¥ò:(i) 9édh¡fS¡Féilaë¡fÎ«.

(ii)Kjš14édh¡fëèUªJVnjD«8édh¡fS¡Féilaë¡fÎ«.édhv©45-¡F


(iii)x›bthUédhé‰F«IªJkÂ¥bg©fŸ. 9 × 5 = 45

31. bt‹gl§fis¥ga‹gL¤Â \A B C,^ h = \ \A B A C+^ ^h hrçahvd¢nrhÂ¡f.

32. A= { 0, 1, 2, 3 } k‰W« B = { 1, 3, 5, 7, 9 } v‹gdÏUfz§fŸv‹f. :f A B" v‹D«

rh®ò ( )f x x2 1= + vd¡bfhL¡f¥g£LŸsJ.Ï¢rh®Ãid(i)tçir¢nrhofë‹

fz« (ii) m£ltiz (iii) m«ò¡F¿¥gl« (iv) tiugl«M»at‰whšF¿¡f. 33. 7 77 777 g+ + + .v‹wbjhlç‹Kjšn cW¥òfë‹TLjšfh©f.

34. fhuâ¥gL¤Jf. 2 5 6x x x3 2- - +


35. tF¤jšKiwæšt®¡f_y«fh©f. 9 6 7 2 1x x x x4 3 2- + - +

36. k‰W«A B

2

4

5

1 3 6=

-

= -f ^p h v‹w mâfS¡F ( )AB B AT T T= v‹gij

rç¥gh®¡f.

37. , , , , , ,k‰W«6 9 7 4 4 2 3 7^ ^ ^ ^h h h h Kidfshf¡bfh©leh‰fu¤Â‹gu¥gsÎfh©f.

38. A(1 , 2), B(-4 , 5) k‰W« C(0 , 1) M»ad3ABC-‹KidfŸ. Ï«K¡nfhz¤Â‹

x›bthUKidæèUªJ«mj‹vÂ®¥g¡f¤Â‰Ftiua¥gL«F¤J¡nfhLfë‹

(altitudes)rhŒÎfis¡fh©f.


m›éizfu«xUrhŒrJukhF«vdãWÎf.

40. ne®¡F¤jhd xU ku¤Â‹ nkšghf« fh‰¿dhš K¿ªJ, m«K¿ªj gFÂ ÑnH

éGªJélhkš, ku¤Â‹c¢ÁjiuÍl‹ 30c nfhz¤ijV‰gL¤J»wJ. ku¤Â‹

c¢Ámj‹moæèUªJ30 Ûbjhiyéšjiuia¤bjhL»wJvåš,ku¤Â‹KG

cau¤ij¡fh©f.

41. 18 br.ÛMuKŸsÂ©kcnyhf¡nfhskhdJcU¡f¥g£L_‹WÁ¿abt›ntW


nfhs§fë‹Mu§fŸKiwna2 br.Ûk‰W«12 br.Ûvåš_‹whtJnfhs¤Â‹

Mu¤ij¡fh©f.

42. xUÂ©kku¥bgh«ikahdJmiu¡nfhs¤Â‹nkšT«òÏizªjtoéšcŸsJ.

miu¡nfhs«k‰W«T«òM»at‰¿‹Mu«3.5 br.Û.nkY«bgh«ikæ‹bkh¤j

cau«17.5 br.Ûvåšm¥bgh«ikjahç¡f¥ga‹gL¤j¥g£lku¤Â‹fdmsit¡

fh©f.(r= 722 )

43. xU òŸëétu¤ bjhF¥Ãš xR = 35, n = 5, 82x 9 2R - =^ h våš, x2R k‰W«

( )x x 2R - M»at‰iw¡fh©f.


, ,54

32

73 v‹f. A k‰W« B ÏUtU« nr®ªJ Ô®Î fh©gj‰fhd ãfœjfÎ

158 .

B k‰W«C ÏUtU«nr®ªJÔ®Îfh©gj‰fhdãfœjfÎ72 . A k‰W« C ÏUtU«

nr®ªJÔ®ÎfhzãfœjfÎ3512 , _tU« nr®ªJÔ®ÎfhzãfœjfÎ

358 våš,

ahnuD«xUt®m›édhé‹Ô®Îfh©gj‰fhdãfœjféid¡fh©f.

45. (a) 400-¡F« 600-¡F«Ïilna 11-MštFgL«mid¤JÏašv©fë‹TLjš

fh©f.

(mšyJ)

(b) ãWÎf.( )

xx

xx

xx x x

11

11

12 14 3 2 3

-- +

+- =

++ + +

409khÂçédh¤jhŸ

ÃçÎ - <F¿¥ò:(i)Ï¥ÃçéšcŸsx›bthUédhéY«Ïu©Lkh‰Wédh¡fŸbfhL¡f¥g£LŸsd.

(ii) x›bthUédhéY«cŸsÏu©Lkh‰Wédh¡fëèUªJxUédhitnj®ªbjL¤J

ÏUédh¡fS¡F«éilaë¡fÎ«.

(iii)x›bthUédhé‰F«g¤JkÂ¥bg©fŸ. 2 x 10 = 20

46. (a) 3.2 br.Û MuKŸst£l«tiuf.t£l¤Â‹nkš P v‹wòŸëiaia¡F¿¤J

m¥òŸëæšbjhLnfhL-eh©nj‰w¤ij¥ga‹gL¤ÂbjhLnfhLtiuf.

(mšyJ)

(b) AB = 6.5 br.Û., 110ABC+ = c, BC = 5.5 br.Û. k‰W« AB || CD v‹wthWmikÍ«

t£leh‰fu« ABCD tiuf.

47. (a) 2 6y x x2

= + - -‹ tiugl« tiuªJ, mjid¥ ga‹gL¤Â 2 10 0x x2+ - =

v‹wrk‹gh£il¤Ô®¡fÎ«.

(mšyJ)

(b) xU è£l® ghè‹ éiy ` 15 v‹f. ghè‹ msÎ¡F« éiy¡F« cŸs¤

bjhl®Ãid¡fh£L«tiugl«tiuf.mjid¥ga‹gL¤Â,

(i) é»jrkkh¿èia¡fh©f. (ii) 3 è£l®ghè‹éiyia¡fh©f.




bghJ F¿¥òfŸ :







(iii)x›bthUédhé‰F«xUkÂ¥bg©. 15 × 1 = 15 1. A= { p, q, r, s }, B = { r, s, t, u } våš, \A B =

(A) { p, q } (B) { t, u } (C) { r, s } (D){p, q, r, s }

2. 100 n +10 v‹gJxUbjhl®tçiræ‹ n MtJcW¥òvåš,mJ




3. , , ,52

256

12518 g v‹wbgU¡F¤bjhl®tçiræ‹bghJcW¥ò

(A) 53 (B)

n2 n 1-

` j (C) 52

53 n 1-

` `j j (D) 53

52 n 1-

` `j j

4. ax bx c 02+ + = v‹wrk‹gh£o‹_y§fŸrk«våš,c-‹kÂ¥ò

(A) ab2

2

(B) ab4

2

(C) ab2

2

- (D) ab4

2

-

5. x a- MdJ p x^ h-¡FxUfhuâvåš,våšk£Lnk

(A) p(a) = p(x) (B) ( )p a 0! (C) p(a) = 0 (D) p(–a) = 0

6. A k‰W« B v‹gdrJumâfŸ.nkY«AB = I k‰W«BA = I våš, B v‹gJ

(A) myFmâ (B) ó¢Áamâ

(C) A-‹bgU¡fšne®khWmâ (D) A-

7. (–2, –5), (–2, 12), (10, –1) M»aòŸëfisKidfshf¡bfh©lK¡nfhz¤Â‹

eL¡nfh£Lika«(centroid)

(A) ,6 6^ h (B) ,4 4^ h (C) ,3 3^ h (D) ,2 2^ h

8. ,1 2^ h, ,2 3^ h v‹wòŸëfŸtê¢bršY«ne®¡nfh£o‹rhŒÎ¡nfhz«.

(A) 30c (B) 45c (C) 60c (D) 90c

9. ÏU tobth¤j K¡nfhz§fë‹ g¡f§fë‹ é»j« 2:3 våš, mt‰¿‹

gu¥gsÎfë‹é»j«

(A) 9 : 4 (B) 4 : 9 (C) 2 : 3 (D) 3 : 2

411khÂçédh¤jhŸ

10. ABCD -š BC-‹ÏiznfhLDE MdJ AB-I D-æY«AC -I E-æY«bt£L»wJ

våš,

(A) ADAB

AEAC= (B)

AEAB

ADAC= (C)

ECAB

DBAC= (D)AB AC=

11. cos cot1 12 2i i- +^ ^h h =

(A) sin2i (B) 0 (C) 1 (D) tan2i

12. gl¤Âš CAB 60+ = c, .AB 3 5= ÛvåšAC =

(A) 7 Û (B) 3.5Û

(C) 1.75 Û (D) 1 Û

13. 100r r.br.Û tisgu¥òbfh©lnfhs¤Â‹Mu«

(A) 25 br.Û (B) 100 br.Û (C) 5 br.Û (D) 10 br.Û. 14. 10, 10, 10, 10, 10-‹ éy¡ft®¡f¢ruhrç

(A) 10 (B) 10 (C) 5 (D) 0 15. A k‰W« B v‹wÏUãfœ¢Áfëš

( ) 0.25, ( ) 0.05 ( ) 0.14 , ( )k‰W« våšP A P B P A B P A B+ ,= = = =

(A) 0.61 (B) 0.16 (C) 0.14 (D) 0.6




(iii)x›bthUédhé‰F«Ïu©LkÂ¥bg©fŸ. 10 × 2 = 20

16. A B1 våš, bt‹gl¤ij¥ga‹gL¤Â A B+ k‰W« \A B M»at‰iw¡fh©f.

17. A = { 1, 2, 3, 4, 5 }, B = N k‰W« :f A B" MdJ ( )f x x2

= vdtiuaW¡f¥g£LŸsJ f -‹Å¢rf¤ij¡fh©f.nkY«,rh®Ã‹tifia¡fh©f.

18, TLjiy¡fh©f.1 2 3 203 3 3 3g+ + + +

19. RU¡Ff. x

x

x x

x x

4

81

5 36

6 82

2

2

2

#-

-

- -

+ +

20. ai ji j

ij=

+- bfh©L2 2# tçirÍilamâA a

ij= 6 @-ia¡fh©f

21. k‰W«A B

8

2

0

7

4

3

9

6

3

1

2

5= -

-

=-

- -f ep o våš,KoÍ«våš BA ia¡fh©f.

22. (–3, 5) k‰W« (4, –9) M»aòŸëfisÏiz¡F«nfh£L¤J©oidc£òwkhf

1 : 6 v‹wé»j¤ÂšÃç¡F«òŸëæ‹m¢R¤bjhiyÎfis¡fh©f.

23. ,2 3-^ h v‹w òŸë tê¢ brštJ«, rhŒÎ31 cilaJkhd ne®¡nfh£o‹

rk‹gh£il¡fh©f.

Û


24. ABCT -š, A+ -‹btë¥òwÏUrkbt£oMdJBC-‹Ú£ÁæidE-šrªÂ¡»wJ. AB = 10 br.Û, AC = 6 br.Û k‰W« BC = 12 br.Û våš,CE-Ifh©f.

25. Ã‹tU«K‰bwhUikfisãWÎf. 1sec sin sec tan1i i i i- + =^ ^h h

26. 30 Û ÚsKŸs xU f«g¤Â‹ ãHè‹ Ús« 10 3 Û våš, Nçaå‹ V‰w¡

nfhz¤Â‹(jiuk£l¤ÂèUªJV‰w¡nfhz«)mséid¡fh©f.

27. xUÂ©kne®t£l¡T«Ã‹mo¢R‰wsÎ236br.Û.k‰W«mj‹rhÍau«12br.Ûvåš,m¡T«Ã‹tisgu¥ig¡fh©f.

28. xUòŸëétu¤Â‹khWgh£L¡bfG57k‰W«Â£léy¡f«6.84våš,mj‹

T£L¢ruhrçia¡fh©f.

29. _‹W ehza§fŸ xnu neu¤Âš R©l¥gL«nghJ FiwªjJ ÏU jiyfŸ

»il¥gj‰fhdãfœjféid¡fh©f.

30. (a) RU¡Ff.x

x x x2 18

4 122

3 2

-

- - (mšyJ)

(b) xUÂ©k¡nfhs¤Â‹tisgu¥ò616r.br.Ûvåšmj‹é£lij¡fh©f.





31. xUthbdhèãiya«190khzt®fël«mt®fŸéU«ò«Ïiræ‹tiffis¤

Ô®khå¡fxUfz¡bfL¥òel¤ÂaJ.114 ng®nk‰f¤ÂaÏiriaÍ«, 50ng®»uhäa

ÏiriaÍ«, 41 ng® f®ehlf ÏiriaÍ«, 14 ng® nk‰f¤Âa ÏiriaÍ« »uhäa

ÏiriaÍ«, 15 ng® nk‰f¤Âa ÏiriaÍ« f®ehlf ÏiriaÍ«, 11 ng® f®ehlfÏiriaÍ«»uhäaÏiriÍ«k‰W«5ng®Ï«_‹WÏirfisÍ«éU«ò»‹wd®

vd¡fz¡bfL¥Ãšbtë¥g£lJ.Ï¤jftšfëèUªJÃ‹tUtdt‰iw¡fh©f.

(i) _‹WtifÏirfisÍ«éU«ghjkhzt®fë‹v©â¡if. (ii) ÏUtifÏirfisk£L«éU«ò«khzt®fë‹v©â¡if. (iii) »uhäa Ïiria éU«Ã nk‰f¤Âa Ïiria éU«ghj khzt®fë‹

v©â¡if.

32. A = {4, 6, 8, 10 } k‰W« B = { 3, 4, 5, 6, 7 } v‹f. :f A B" v‹gJ f x x21 1= +^ h

vdtiuaW¡f¥g£LŸsJ.rh®ò f -I (i) m«ò¡F¿gl« (ii) tçir¢nrhofë‹

fz« (iii) m£ltizM»at‰¿‹_y«F¿¡fÎ«.

33. 6 + 66 + 666 +g vD«bjhlçšKjšn cW¥òfë‹TLjšfh©f.

34. 3 5 26 56x x x x4 3 2+ + + + k‰W« 2 4 28x x x x

4 3 2+ - - + M»at‰¿‹ Û.bgh.t

5 7x x2+ + våš,mt‰¿‹Û.bgh.k-it¡fh©f.

35. x x x ax b4 12 374 3 2- + + + v‹gJKGt®¡fbkåša k‰W« b‹kÂ¥òfisfh©f.

36. 5 1x px2- + = 0 v‹wrk‹gh£o‹_y§fŸa k‰W« b v‹f. nkY« a b- = 1

våš, p-‹kÂ¥ig¡fh©f.

413khÂçédh¤jhŸ

37. A1

2

1

3=

-c m våš, 4 5A A I O

2

2- + = vdãWÎf.

38. (3, 4), (-1, 2) v‹w òŸëfis Ïiz¡F« ne®¡nfh£L¤J©o‹ ika¡

F¤J¡nfh£o‹(perpendicular bisector)rk‹gh£il¡fh©f.

39. ABCD v‹weh‰fu¤Âš,AB-¡FÏiz CD v‹f. AB-¡FÏizahftiua¥g£lxU

ne®¡nfhL AD-I P-æY« BC-I Q-æY«rªÂ¡»wJvåš, PDAP

QCBQ

= vdãWÎf.

40. 40 ÛcauKŸsxUnfhòu¤Â‹c¢Ák‰W«moM»at‰¿èUªJxUfy§fiu

és¡»‹ c¢Áæ‹ V‰w¡ nfhz§fŸ Kiwna 30ck‰W« 60c våš, fy§fiu

és¡»‹cau¤ij¡fh©f.fy§fiués¡»‹c¢ÁæèUªJnfhòu¤Â‹mo¡F

cŸsöu¤ijÍ«fh©f. 41. xUÂ©kne®t£lcUisæ‹bkh¤j¥òw¥gu¥ò880 r.br.Ûk‰W«mj‹Mu«10

br.Ûvåš,m›ÎUisæ‹tisgu¥ig¡fh©f(722r = v‹f).

42. xUTlhukhdJcUisæ‹ÛJT«òÏizªjtoéšcŸsJ.Tlhu¤Â‹bkh¤j

cau« 13.5 Û k‰W«é£l« 28 Û. nkY« cUis¥ ghf¤Â‹ cau« 3 Û våš,

Tlhu¤Â‹bkh¤jòw¥gu¥ig¡fh©f.

43. 62, 58, 53, 50, 63, 52, 55 M»av©fS¡FÂ£léy¡f«fh©f.

44. xU gfil ÏUKiw cU£l¥gL»wJ. FiwªjJ xU cU£lèyhtJ v© 5 »il¥gj‰fhdãfœjféid¡fh©f.(T£lšnj‰w¤ij¥ga‹gL¤Jf)

45. (a) xUT£L¤bjhl®tçiræšmL¤jL¤jKjš10 cW¥òfë‹TLjš25k‰W«

bghJé¤Âahr«KjšcW¥Ã‹ÏUkl§Fvåš,10 tJcW¥ig¡fh©f.

(mšyJ)

(b) (–1, 6), (–3, –9), (5, –8) k‰W« (3, 9) M»a òŸëfisKidfshfbfh©l

eh‰fu¤Â‹gu¥ò¡fh©f.




(iii)x›bthUédhé‰F«g¤JkÂ¥bg©fŸ. 2 × 10 = 20

46. (a) AB = 6 br.Û., AD = 4.8 br.Û., BD = 8 br.Û. k‰W«CD = 5.5 br.Û. v‹wmsÎfŸ

bfh©lt£leh‰fu«ABCD tiuf.

(mšyJ)

(b) TPQR-š mo¥g¡f« PQ = 6 br.Û., R 60+ = c k‰W« c¢Á R-èUªJ PQ-¡F

tiua¥g£lF¤J¡nfh£o‹Ús« 4 br.ÛvdÏU¡FkhWTPQR tiuf.

47. (a) 2y x2

= -‹tiugl¤ijtiuªJmÂèUªJ 2 6 0x x2+ - = v‹wrk‹gh£il¤

Ô®¡fÎ«.

(mšyJ)

(b) xy = 20, x , y > 0 v‹gj‹tiugl«tiuf.mjid¥ga‹gL¤Â x 5= våš, y-‹kÂ¥igÍ«, y 10= våš,x-‹kÂ¥igÍ«fh©f.




bghJ F¿¥òfŸ :








1. ÑnHbfhL¡f¥g£LŸsitfëšjtwhdT‰WvJ?

(A) \A B = A B+ l (B) \A B A B+= (C) \ ( )A B A B B, += l (D) \ ( ) \A B A B B,=

2. a, b, c v‹gdxUbgU¡F¤bjhl®tçiræšcŸsdvåš, b ca b

-- =

(A) ba (B)

ab (C)

ca (D) b

c

3. 19, 14, 9... v‹wT£L¤bjhl®tçiræ‹17 tJcW¥ò

(A) 84 (B) – 61 (C) – 84 (D) – 51

4. p x^ h = (k +4)x2+13x+3k v‹D«gšYW¥ò¡nfhitæ‹xUó¢Áa«k‰bwh‹¿‹

jiyÑêahdhš,k-‹kÂ¥ò

(A) 2 (B) 3 (C) 4 (D) 5

5. 4 0x kx2- + = v‹wÏUgo¢rk‹gh£o‹_y§fŸrk«våšk kÂ¥ò/kÂ¥òfis¡

fh©f

(A) 4! (B) 2 (C) 3 (D) 5!

6. A aij 2 2

=#

6 @ k‰W«a i jij= + våš,A =

(A) 1

3

2

4c m (B) 2

3

3

4c m (C) 2

4

3

5c m (D)

4

6

5

7c m

7. (2, 5), (4, 6), ,a a^ h M»a òŸëfŸxnu ne®¡nfh£ošmik»‹wdvåš, a-‹

kÂ¥ò

(A) -8 (B) 4 (C) -4 (D) 8

8. x y3= v‹wne®¡nfh£o‹rhŒÎ¡nfhz«

(A) 0c (B) 60c (C) 30c (D) 45c

9. TABC-š AB k‰W« AC-fëYŸsòŸëfŸ D k‰W« E v‹gdDE < BC v‹wthW

cŸsd.nkY«, AD = 3 br.Û, DB = 2 br.Û k‰W« AE = 2.7 br.Û våš, AC =


415khÂçédh¤jhŸ

10. gl¤Âš ,DE BC< ABC ADET T+ ,AD = 1 br.Û k‰W« BD = 2.7 br.Ûvåš ABCT k‰W« ADET Ïitfë‹gu¥gsÎfë‹é»j«

(A) 1 : 9 (B) 1 : 2 (C) 9 : 1 (D) 2 : 1.

11. gl¤Âš ABC+ =

(A) 45c (B) 30c

(C) 60c (D) 05 c

12. gl¤ÂšCE-‹Ús«

(A) 15 br.Û (B) 12 br.Û

(C) 45 br.Û (D) 18 br.Û

13. Ïu©L nfhs§fë‹ tisgu¥òfë‹é»j« 9 : 25. mt‰¿‹ fd msÎfë‹

é»j«

(A) 81 : 625 (B) 729 : 15625 (C) 27 : 75 (D) 27 : 125.

14. x, y, z- ‹Â£léy¡f« t våš, x + 5, y + 5, z + 5-‹Â£léy¡f«

(A) t3

(B) t + 5 (C) t (D) x y z

15. A, B k‰W« C v‹gdx‹iwbah‹Wéy¡F«_‹Wãfœ¢ÁfŸv‹f.mt‰¿‹

ãfœjfÎfŸKiwna , k‰W«31

41

125 våš, P A B C, ,^ h =

(A) 1219 (B)

1211 (C)

127 (D) 1





16. U = { , , , , , , }4 8 12 16 20 24 28 , A = { , , }8 16 24 k‰W« B = { , , , }4 16 20 28 våš, 'A B,^ h M»at‰iw¡fh©f.

17. f = { (1, 2), (4, 5), (9, –4), (16, 5)} v‹w cwÎ A = { 1, 4, 9, 16 }-èUªJ

B = { –1, 2, –3, –4, 5, 6 }-¡FxUrh®ghFkh?rh®òvåš,mj‹Å¢rf¤ij¡fh©f.

18. bjhFKiwtF¤jiyga‹gL¤Â<Î,ÛÂfh©f.( ) ( )x x x x3 5 13 2'+ - + -

19. 5 0ax x c2- + = v‹w ÏUgo¢ rk‹gh£o‹ _y§fë‹ TLjš 10 k‰W«

bgU¡f‰gy‹ 10 våš, a k‰W« c M»at‰¿‹kÂ¥òfis¡fh©f.

20. A1

2

3

2

4

5

3

5

6

=

-

-f p våš,( )A AT T

= v‹gjid¢rçgh®¡f.


21. k‰W«3

1

5

2

2

1

5

3-

-c em o M»admâ¥ bgU¡fiy¥ bghU¤J x‹W¡bfh‹W

ne®khWmâvdãWÎf.

22. x y2 3 0- + = v‹w ne®¡nfh£o‰F¢ br§F¤jhdJ« (1, -2) v‹w òŸë tê¢

brštJkhdne®¡nfh£o‹rk‹gh£il¡fh©f.

23. AB k‰W«CD v‹wÏUeh©fŸt£l¤Â‰FbtënaP-šbt£o¡bfhŸ»‹wd. AB = 4 br.Û,BP = 5br.Ûk‰W«PD = 3br.ÛvåšCD-I¡fh©f.

24. Ã‹tU«K‰bwhUikfisãWÎf. 1sin cos

cos sin cot1

2

i ii i i+

+ - =^ h

25. xU nfhòu¤Â‹ moæèUªJ 30 3 Û bjhiyéš ã‰F« xU gh®itahs®,

m¡nfhòu¤Â‹c¢Áæid30c V‰w¡nfhz¤Âšfh©»wh®.jiuk£l¤ÂèUªJ

mtUila»ilãiy¥gh®it¡nfh£o‰FcŸsöu«1.5 Ûvåš,nfhòu¤Â‹

cau¤ij¡fh©f.

26. xU Â©k ne® t£l cUisæ‹ (solid right circular cylinder) Mu« 7 brÛ k‰W«

cau«20 brÛ våš,mj‹bkh¤j¥òw¥gu¥ig¡fh©f. (722r = v‹f).

27. xUcŸÇl‰wnfhs¤Â‹btëk‰W«cŸMu§fŸKiwna12 br.Ûk‰W«10 br.Ûvåš,m¡nfhs¤Â‹fdmsit¡fh©f.

28. Kjš10Ïašv©fë‹Â£léy¡f«fh©f.

29. A k‰W« B v‹wÏu©Lãfœ¢Áfëš ( ) , ( )P A P B41

52= = k‰W« ( )P A B

21, =

våš, ( )P A B+ -I¡fh©f.

30. (a) 1 5 52 g+ + + v‹wbjhlçš8 cW¥òfŸtiuTLjšfh©f(mšyJ)

(b) x k‰W« y bt£L¤J©LfŸ Kiwna ,72

53- våš mj‹ ne®¡nfh£L

rk‹gh£il¡fh©f.





31. A = { , , , , , , , , , }a b c d e f g x y z , B = { , , , , }c d e1 2 k‰W« C = { , , , , , }d e f g y2 v‹f. \ \ \A B C A B A C, +=^ ^ ^h h h v‹gijrçgh®¡fÎ«.

32. A = { 5, 6, 7, 8 }; B = { –11, 4, 7, –10,–7, –9,–13 } v‹f.

f = {( ,x y) : y = x3 2- , x A! , y B! } vdtiuaW¡f¥g£LŸsJvåš,

(i) f -‹cW¥òfisvGJf (ii) mj‹JizkÂ¥gf«ahJ?

(iii) Å¢rf«fh©f (iv) v›tif¢rh®òvd¡fh©f.

417khÂçédh¤jhŸ

33. xUbgU¡F¤bjhl®tçiræšmL¤jL¤j 3cW¥òfë‹bgU¡F¤bjhif 216

k‰W« mitfëš Ïu©ou©L cW¥ò¡fë‹ bgU¡f‰gy‹fë‹ TLjš 156

våš,mªjcW¥òfis¡fh©f.

34. 11 br.Û,12br.Û,13br.Û,g 24br.ÛM»adt‰iwKiwnag¡fmsÎfshf¡bfh©l

14rJu§fë‹bkh¤j¥gu¥òfh©f.

35. ÏUgšYW¥ò¡nfhitfë‹Û.bgh.k.,Û.bgh.tKiwna x x x4 5 13- +^ ^h h k‰W«

x x52+^ h nkY«,xUgšYW¥ò¡nfhit p x^ h = x x x5 9 2

3 2- -^ h våš, k‰bwhU

gšYW¥ò¡nfhitq x^ hIfh©f.

36. ÏUgo¢ N¤Âu¤ij¥ ga‹gL¤Â Ã‹tU« rk‹gh£il¤ Ô®.

x x11

22

++

+ =

x 44+

Ï§Fx 1 0!+ , x 2 0!+ k‰W« x 4 0!+ .

37. k‰W«Aa

c

b

dI

1

0

0

12= =c cm m våš, ( ) ( )A a d A bc ad I

2

2- + = - vdãWÎf.

38. (–5, 1) k‰W« (2, 3) v‹wòŸëfisÏiz¡F«nfh£L¤J©oidy-m¢RÃç¡F«

é»j¤ijÍ«k‰W«Ãç¡F«òŸëiaÍ«fh©f.

39. TABC-‹KidfŸA(1, 8), B(-2, 4), C(8, -5). nkY«, M, N v‹gdKiwnaAB, AC Ït‰¿‹eL¥òŸëfŸvåš,MN-‹rhŒit¡fh©f.Ïij¡bfh©LMN k‰W«

BC M»ane®¡nfhLfŸÏizvd¡fh£Lf.

40. xU ãH‰gl¡ fUéæYŸs gl¢ RUëš xU ku¤Â‹ Ã«g¤Â‹ Ús« 35 ä.Û.

by‹°¡F« gl¢RUS¡F« Ïil¥g£l öu« 42 ä.Û. nkY«, by‹ìèUªJ

ku¤J¡F cŸsöu« 6 Û våš, ãH‰gl« vL¡f¥gL« ku¤Â‹ gFÂæ‹ Ús«

fh©f.

41. tan tanni a= k‰W«sin sinmi a= våš,1

1cosn

m2

2

2

i =-

- , n !+1, vdãWÎf.

42. xU ne®t£l Â©k¡ T«Ã‹ MuK« rhÍauK« 3 : 5 v‹w é»j¤Âš cŸsd.

m¡T«Ã‹tisgu¥ò60r r.br.Ûvåš,mj‹bkh¤j¥òw¥gu¥ig¡fh©f.

43. xUòŸëétu¤Âš,20kÂ¥òfë‹T£L¢ruhrçk‰W«Â£léy¡f«Kiwna40 k‰W« 15 vd fz¡»l¥g£ld. mitfis¢ rçgh®¡F«nghJ 43 v‹w kÂ¥ò

jtWjyhf53 vdvGj¥g£lJbjçatªjJ.m›étu¤Â‹rçahdT£L¢ruhrç

k‰W«rçahdÂ£léy¡f«M»at‰iw¡fh©f.

44. ÏUgfilfŸxnuneu¤ÂšcU£l¥gL«nghJ»il¡F«Kfv©fë‹bgU¡f‰gy‹

xUgfhv©zhfÏU¥gj‰fhdãfœjféid¡fh©f.

45. (a) 42 br.Ûé£l«bfh©lxUÂ©k¡nfhs«7 br.Û é£lK« 3 br.ÛcauK«

bfh©l ÁW T«òfshf kh‰w¥g£lhš »il¡F« T«òfë‹ v©â¡ifia¡

fh©f.

(mšyJ)

(b) t®¡f_y«fh©f.81 72 70 24 9x x x x4 3 2- + - +


ÃçÎ - <

F¿¥ò:(i)Ï¥ÃçéšcŸsx›bthUédhéY«Ïu©Lkh‰Wédh¡fŸbfhL¡f¥g£LŸsd.




46. (a) 6 br.Û MuKŸs xU t£l« tiuªJ mj‹ ika¤ÂèUªJ 10 br.Û

bjhiyéYŸs xU òŸëia¡ F¿¡f. m¥òŸëæèUªJ t£l¤Â‰F bjhL

nfhLfŸtiuªJmj‹Ús§fisfz¡»Lf.

(mšyJ)

(b) PQ = 5 br.Û., QR = 4 br.Û., 35QPR+ = c k‰W« 70PRS+ = c M»amsÎfŸ

bfh©lt£leh‰fu«PQRS tiuf.

47. (a) xUäÂt©oX£Lgt®A v‹wÏl¤ÂèUªJB v‹wÏl¤Â‰FxUÓuhd

ntf¤Âšxnutêæšbt›ntWeh£fëšgaz«brŒ»wh®.mt®gaz«brŒj

ntf«,m¤öu¤Âid¡fl¡fvL¤J¡bfh©lneu«M»adt‰iw¥g‰¿a

étu§fŸ(ntf-fhy)Ã‹tU«m£ltizæšbfhL¡f¥g£LŸsd.

ntf«(».Û./ kâ) x 2 4 6 10 12neu«(kâæš) y 60 30 20 12 10

ntf-fhytiugl«tiuªJmÂèUªJ

(i)mt®kâ¡F5».Ûntf¤Âšbr‹whšöu¤ij¡fl¡fMF«gazneu«

(ii) mt® Ï¡F¿¥Ã£löu¤ij 40 kâneu¤Âš fl¡f vªj ntf¤Âš gaâ¡f

nt©L«

M»adt‰iw¡fh©f.

(mšyJ)

(b) 12y x x2

= + - -‹ tiugl« tiuªJ, mjid¥ ga‹gL¤Â 2 2 0x x2+ + =

v‹wrk‹gh£il¤Ô®¡fÎ«.

419khÂçédh¤jhŸ



bghJ F¿¥òfŸ :








1. A = { 5, 6, 7 }, B = { 1, 2, 3, 4, 5 } v‹f. ( )f x x 2= - v‹wthWtiuaiwbrŒa¥g£l

rh®ò :f A B" Ï‹Å¢rf«,

(A) { 1, 4, 5 } (B) { 1, 2, 3, 4, 5 } (C) { 2, 3, 4 } (D) { 3, 4, 5 }

2. , , , ,21

61

121

201 g v‹wbjhl®tçiræš,cW¥ò

201 -¡FmL¤jcW¥ò

(A) 241 (B)

221 (C)

301 (D) 18

1

3. a, b k‰W«c M»aitxUbgU¡F¤bjhl®tçiræšcŸsJvåš

(A)ac

cb= (B)

ac

ab= (C)

ac

ab 2

= ` j (D) ca

ab=

4. a ba3

- cl‹

b ab3

- I¡T£l,»il¡F«òÂanfhit

(A) a ab b2 2+ + (B) a ab b

2 2- + (C) a b

3 3+ (D) a b

3 3-

5. a ba b

b aa b

-+ -

-- =

(A) 1 (B) a b

b2-

(C) b a

b2-

(D) ( )a ba b2-+

6. x

y

1

2

2

1

2

4=c c cm m m våš, x k‰W« y-fë‹kÂ¥òfŸKiwna

(A) 2 , 0 (B) 0 , 2 (C) 0 , 2- (D) 1 , 1

7. ,2 6-^ h, ,4 8^ h M»a òŸëfis Ïiz¡F« ne®¡nfh£o‰F¢ br§F¤jhd

ne®¡nfh£o‹rhŒÎ

(A) 31 (B) 3 (C) -3 (D) 3

1-

8. (2, ) (5, 2 )k‰W«3 3 M»aòŸëfisÏiz¡F«ne®¡nfh£o‹rhŒÎ

(A) 30c (B) 45c (C) 60c (D) 90c


9. gl¤Âš ACAB

DCBD= , 40B c+ = k‰W« 60C c+ = våš, BAD+ =

(A)30c (B) 50c

(C) 80c (D) 40c

10. ABCT š k‰W«DE BCDBAD

32< = . AE = 6 våš BC =

(A) 9 (B) 18 (C) 15 (D) 12

11. secx a i= , tany b i= våš, ax

b

y2

2

2

2

- -‹kÂ¥ò

(A) 1 (B) –1 (C) tan2i (D) cosec2i

12. cosec cot

tan sec2 2

2 2

i i

i i

-

- =

(A) 1 (B) –1 (C) sini (D) cosi

13. Ïu©LcUisfë‹cau§fŸKiwna1:2 k‰W«mt‰¿‹Mu§fŸKiwna 2:1 M»aé»j§fëèU¥Ã‹,mt‰¿‹fdmsÎfë‹é»j«

(A) 4 : 1 (B) 1 : 4 (C) 2 : 1 (D) 1 : 2

14. étu§fë‹bjhF¥òx‹¿‹Â£léy¡f« 2 2 . mÂYŸsx›bthUkÂ¥ò«

3 MšbgU¡f¡»il¡F«òÂaétu¤bjhF¥Ã‹Â£léy¡f«

(A) 12 (B) 4 2 (C) 6 2 (D) 9 2

15. xUbe£lh©oš (Leap year) 53 btŸë¡»HikfŸmšyJ 53 rå¡»HikfŸ

tUtj‰fhdãfœjfÎ

(A) 72 (B)

71 (C)

74 (D) 7

3





16. , k‰W«A B C M»a_‹Wfz§fS¡FÃ‹tUtdt‰iwés¡F«bt‹gl§fŸ

tiuf. \B C A,^ h

17. , 0

, 0

vD«nghJ

vD«nghJx

x x

x x 1

$=

-) , { ( ,x y) | y = |x |, x R! } v‹w cwÎ, rh®ig

tiuaW¡»wjh?mj‹Å¢rf«fh©f.

18. 6 3 7x x2- - v‹w ÏUgo gšYW¥ò¡nfhiæ‹ ó¢Áa§fS¡F« bfG¡fS¡F«

ÏilnacŸsbjhl®òfis¢rçgh®¡f.

19. 0x x3 5 22- + = v‹wrk‹gh£o‹_y§fŸa k‰W«b våš,

2 2

ba

ab

+ ‹kÂ¥ò

fh©f

20. A3

5

2

1= c m k‰W«B

8

4

1

3=

-c m våš,C A B2= + v‹wmâia¡fh©f.

421khÂçédh¤jhŸ

21. Ô®Îfh©:y

x

x

y3

6 2

31 4=

-

+c em o

22. rkg¡f ABCT -‹g¡f«BCMdJx-m¢Á‰FÏizvåšABk‰W«BCM»at‰¿‹

rhŒÎfis¡fh©f.

23. A(a ,–3), B(3 , a) k‰W« C(–1 , 5) M»at‰iwKidfshf¡bfh©l ABCT -‹

gu¥ò12 r.myFfŸvåš,a-‹kÂ¥ig¡fh©f.

24. MPv‹gJ MNO3 -š M+ -‹btë¥òwÏUrkbt£o.nkY«, ÏJ

NO-‹Ú£ÁæidP-æšrªÂ¡»wJ. MN = 10 br.Û, MO = 6 br.Û, NO = 12 br.Û våš,OP-Ifh©f.

25. secsec

cossin1

1

2

ii

ii+ =

- vdãWÎf.

26. 40 br.ÛÚsKŸsxUCryhdJ(pendulum),xUKGmiyé‹nghJ,mj‹c¢Áæš 60c nfhz¤ijV‰gL¤J»wJ. mªjmiyéš,CršF©o‹Jt¡fãiy¡F«,

ÏWÂãiy¡F«ÏilnacŸsäf¡Fiwªjöu¤ij¡fh©f.

27. xU Ïil¡f©l toéyhd thëæ‹ nk‰òw k‰W« mo¥òw Mu§fŸ Kiwna

15 br.Ûk‰W«8 br.Û.nkY«,MH«63 br.Ûvåš,mj‹bfhŸssitè£lçšfh©f.

(722r = )

28. xUtF¥ÃYŸs13khzt®fë‹vil(».»)Ã‹tUkhW.

42.5, 47.5, 48.6, 50.5, 49, 46.2, 49.8, 45.8, 43.2, 48, 44.7, 46.9, 42.4 Ït‰¿‹Å¢R

k‰W«Å¢R¡bfGit¡fh©f.

29. xUigæš5 Át¥òk‰W«ÁyÚyãw¥gªJfŸcŸsd.m¥igæèUªJxUÚyãw¥

gªijvL¥gj‰fhdãfœjfÎ, xUÁt¥òãw¥gªijvL¥gj‰fhdãfœjfé‹

_‹Wkl§Fvåš,m¥igæYŸsÚyãw¥gªJfë‹v©â¡ifia¡fh©f.

30. (a) 6 k‰W« 40 ¡F Ïilnaahd x‰iw¥gil Ïašv©fë‹ TLjš fh©f (mšyJ)

(b) xUne®t£l¡T«Ã‹fdmsÎ,cau«Kiwna120 br.Û3r k‰W« 10 br.Û

våšmj‹tisgu¥òfh©f.





31. bt‹gl«_y«rçgh®. A B A B+ ,=l l l^ h


32. A = { 6, 9, 15, 18, 21 }; B = { 1, 2, 4, 5, 6 } k‰W« :f A B" v‹gJ

f x^ h = x33- vdtiuaW¡f¥g£oU¥Ã‹rh®ò f -I

(i) m«ò¡F¿gl« (ii) tçir¢nrhofë‹fz«

(iii) m£ltiz (iv) tiugl«M»at‰¿‹_y«F¿¡fÎ«.

33. xU T£L¤ bjhlç‹ 3 MtJ cW¥ò 7 k‰W« mj‹ 7 MtJ cW¥ghdJ 3 MtJcW¥Ã‹_‹Wkl§ifél2mÂf«.m¤bjhlç‹Kjš 20cW¥òfë‹

T£l‰gyid¡fh©f.

34. xUbgU¡F¤bjhlç‹KjšcW¥ò375k‰W«mj‹4MtJcW¥ò192våš,mj‹

bghJé»j¤ijÍ«,Kjš14cW¥òfë‹TLjiyÍ«fh©f.

35. 2 3 3 2x x x3 2- - + vD« gšYW¥ò¡nfhitia xUgo¡ fhuâfshf

fhuâ¥gL¤Jf.

36. vëatoéšRU¡Ff. xx

x

xxx

12 5

1

11

3 22

2

++ +

-

+ ---` j= G

37. , k‰W«A B C3

7

3

6

8

0

7

9

2

4

3

6= = =

-c c cm m m våš, ( ) k‰W«A B C AC BC+ +

v‹wmâfis¡fh©f.nkY«,( )A B C AC BC+ = + v‹gJbkŒahFkh?

38. A(2, 1), B(-2, 3), C(4, 5) v‹gd3ABC-‹c¢ÁfŸ.c¢ÁA-æèUªJtiua¥gL«

eL¡nfh£o‹(median)rk‹gh£il¡fh©f.

39. njš°nj‰w¤Â‹kWjiyiavGÂãWÎf

40. 60 ÛcauKŸsxUnfhòu¤ÂèUªJxUf£ll¤Â‹c¢Ák‰W«moM»at‰¿‹

Ïw¡f¡nfhz§fŸKiwna 30 60k‰W«c c våš,f£ll¤Â‹cau¤ij¡fh©f.

41. xUÂ©kne®t£lcUisæ‹bkh¤j¥òw¥gu¥ò231 r.br.Û. mj‹tisgu¥ò

bkh¤j òw¥gu¥Ãš _‹¿š Ïu©L g§F våš, mj‹ Mu« k‰W« cau¤ij¡

fh©f.

42. xU r®¡f°TlhukhdJ cUisæ‹ ÛJT«ò Ïizªj toéšmikªJŸsJ.

Tlhu¤Â‹bkh¤jcau«49 Û.mj‹mo¥ghf¤Â‹é£l«42 Û.cUis¥ghf¤Â‹

cau«21 Û.nkY«1 r.Û»¤jh‹Jâæ‹éiy`12.50 våš,Tlhu«mik¡f¤

njitahd»¤jh‹Jâæ‹éiyia¡fh©f. ( r = 722 )

43. Ã‹tU«òŸëétu¤Â‰fhdÂ£léy¡f«fh©f.

x 70 74 78 82 86 90f 1 3 5 7 8 12

44. ÏUgfilfŸxnuneu¤ÂšnrucU£l¥gL«nghJ»il¡F«Kfv©fë‹TLjš

3 Mšk‰W«4 MštFglhkèU¡fãfœjfÎfh©f.

423khÂçédh¤jhŸ

45. (a) 25x x x ax b30 114 3 2- - + - v‹gJxUKGt®¡f«våš a k‰W«b kÂ¥ig¡

fh©f. (mšyJ)

(b) x y2 3 1 0+ - = , x y3 2 4+ = M»ane®¡nfhLfŸrªÂ¡F«òŸëtêahfÎ«

,83

107-` j k‰W« ,

87

103- -` jM»aòŸëfisÏiz¡F«ne®¡nfh£o‹ika¥òŸë

têahfÎ«bršY«ne®¡nfh£o‹rk‹gh£oid¡fh©f.





46. (a) 3br.ÛMuKŸst£l«tiuf.t£l¤Â‹ika¤ÂèUªJ7br.Û.bjhiyéš

xU òŸëia¡F¿¤J,m¥òŸëæèUªJt£l¤Â‰FbjhLnfhLfŸtiuf.

nkY«bjhLnfhLfë‹Ús¤ijmsªJvGJf.

(mšyJ)

(b) BC = 5 br.Û., 40BAC+ = c k‰W«c¢ÁA-èUªJBC-¡Ftiua¥g£leL¡nfh£o‹

Ús«6 br.Û.v‹wmsÎfŸbfh©l ABCT tiuf.nkY«c¢ÁA-èUªJtiua¥g£l

F¤J¡nfh£o‹Ús«fh©f.

47. (a) tiugl«_y«rk‹gh£oid¤Ô®¡fÎ«. x x2 1 3 0+ - =^ ^h h

(mšyJ)

(b) xU ngUªJ kâ¡F 40 ».Û. ntf¤Âš brš»wJ. Ïj‰Fça öu-fhy

bjhl®Ã‰fhdtiugl«tiuf. Ïij¥ga‹gL¤Â3 kâneu¤ÂšÏ¥ngUªJ

gaâ¤j¤öu¤ij¡f©LÃo.




bghJ F¿¥òfŸ :








1. ( )f x x 52= + våš, ( )f 4- =

(A) 26 (B) 21 (C) 20 (D) –20

2. 3 5t nn= - v‹gJxUT£L¤bjhl®tçiræ‹ n MtJcW¥òvåš,m¡T£L¤

bjhl®tçiræ‹Kjšn cW¥ò¡fë‹TLjš

(A) n n21 5-6 @ (B) n n1 5-^ h (C) n n

21 5+^ h (D) n n

21 +^ h

3. 4, –2, +1, –7, .... v‹wbgU¡F¤bjhlç‹bghJé»j«

(A) 4 (B) –2 (C) 21 (D)

21-

4. 0x bx c2- + = k‰W« x bx a 0

2+ - = M»ark‹ghLfë‹bghJthd_y«

(A) b

c a2+ (B)

bc a2- (C)

ac b2+ (D)

ca b2+

5. w s

x y z

64

8112 14

4 6 8

=

(A) w s

x y z

8

912 14

4 6 8

(B) w s

x y z

8

912 14

2 3 4

(C) w s

x y z

8

96 7

2 3 4

(D) x y z

w s

9

82 3 4

6 7

6. 20x5 1

2

1

3

- =^ f ^h p h våš, x-‹kÂ¥ò

(A) 7 (B) 7- (C) 71 (D) 0

7. y x7 2 11- = v‹wne®¡nfh£o‹rhŒÎ

(A) 27- (B)

27 (C)

72 (D)

72-

8. (1, –1) k‰W« (–5, 3) M»aòŸëfisÏiz¡F«nfh£L¤J©o‹eL¥òŸë

(A) (–2, 1) (B) (2, –1) (C) (–2, –1) (D) (–1, –2)

425khÂçédh¤jhŸ

9. gl¤Âš eh©fŸ AB k‰W« CD v‹gd P-š bt£L»‹wd AB = 16 br.Û, PD = 8 br.Û, PC = 6 k‰W« AP >PB våš, AP =


10. gl¤ÂšPQ bjhLnfhL, BAQ 62+ = c k‰W« BAC 52+ = c,

våš ACB+ =

(A) 64c (B) 90c (C) 54c (D) 62c

11. 11

cottan

2

2

i

i

++ =

(A) cos2i (B) tan2i (C) sin2i D) cot2i

12. sin cos tan coti i i i+ =^ h

(A) 0 (B) 2 (C) 1 (D) tani 13. xUnfhs¤Â‹MukhdJk‰bwhUnfhs¤Â‹Mu¤ÂšghÂvåšmt‰¿‹fd

msÎfë‹é»j«

(A) 1 : 8 (B) 2: 1 (C) 1 : 2 (D) 8 : 1 14. n cW¥òfŸbfh©lvªjxUv©fë‹bjhF¥Ã‰F« ( )x xR - = (A) nx (B) ( 2)n x- (C) ( 1)n x- (D) 0 15. xUÓuhdgfilxUKiwcU£l¥gL«nghJ»il¡F«v©gfhv©mšyJgF

v©zhfÏU¥gj‰fhdãfœjfÎ

(A) 1 (B) 0 (C) 65 (D) 6

1





16. { 10,0,1, 9, 2, 4, 5} { 1, 2, 5, 6, 2,3,4}k‰W«A B= - = - - v‹w fz§fS¡F fz§fë‹bt£Lgçkh‰W¥g©òcilaJv‹gijrçgh®¡fÎ«.

17. rh®òf : ,3 7- h6 "R Ñœ¡f©lthWtiuaW¡f¥g£LŸsJ.

f x^ h = ;

;

;

x x

x x

x x

4 1 3 2

3 2 2 4

2 3 4 7

2 1

1 1

#

# #

- -

-

-

* Ã‹tUtdt‰iw¡fh©f. f f1 3- -^ ^h h

18. a, b, c M»adT£L¤bjhl®tçiræšÏU¥Ã‹ ( ) 4( )a c b ac2 2

- = - vdãWÎf.

19. Ú¡fšKiwia¥ga‹gL¤ÂÃ‹tU«rk‹gh£il¤Ô®. x y2 7+ = , x y2 1- =

20. mâfë‹bgU¡fšfh©f. 3

5

2

1

4

2

1

7

-c cm m


21. vil¡Fiw¥ò¡fhdczÎ¡f£L¥ghL¤Â£l¤Â‹bjhl¡f¤Âš 4 khzt®fŸ

k‰W« 4 khzéfë‹ vil (».». Ïš) Kiwna mâ A-‹ Kjš ãiu k‰W«

Ïu©lh« ãiuahf¡ bfhL¡f¥g£LŸsJ. Ïªj czÎ¡ f£L¥ghL Â£l¤Â‰F¥

Ã‹òmt®fSilavilmâ B-šbfhL¡f¥g£LŸsJ.

A35

42

40

38

28

41

45

30= c m k‰W« B

32

40

35

30

27

34

41

27= c m

khzt®fŸk‰W«khzéfSilaFiw¡f¥g£lvilmséidmâæšfh©f.

22. (3 , 5), (8 , 10) M»aòŸëfisÏiz¡F«nfh£L¤J©ilc£òwkhf2 : 3 v‹w

é»j¤ÂšÃç¡F«òŸëia¡fh©f.

23. A, B v‹gd PQRT -‹g¡f§fŸPQ, PR-fë‹nkšmikªjòŸëfŸv‹f.nkY«, AB QR< , AB = 3 br.Û, PB = 2 br.Û k‰W« PR = 6 br.Ûvåš, QR-‹Ús¤Âid

fh©f.

24. xURikC®ÂæèUªJ (truck)RikiaÏw¡fVJthf30c V‰w¡nfhz¤ÂšxU

rhŒÎ¤js«(ramp)cŸsJ.rhŒÎ¤js¤Â‹,c¢ÁjiuæèUªJ0.9 Ûcau¤Âš

cŸsJvåš,rhŒÎ¤js¤Â‹Ús«ahJ?

25. cos

sin cosec cot1 i

i i i-

= + v‹wK‰bwhUikiaãWÎf.

26. Ïu©L ne®t£l cUisfë‹Mu§fë‹é»j« 3 : 2 v‹f. nkY« mt‰¿‹

cau§fë‹é»j«5 : 3våš,mt‰¿‹tisgu¥òfë‹é»j¤ijfh©f.

27. 7 br.Û Mu« bfh©l nfhstot gÿåš fh‰W brY¤j¥gL« nghJ mj‹

Mu« 14 br.Û Mf mÂfç¤jhš m›éU ãiyfëš gÿå‹ fdmsÎfë‹

é»j¤ij¡fh©f.

28. 43, 24, 38, 56, 22, 39, 45 M»aòŸëétu§fë‹Å¢Rk‰W«Å¢R¡bfGfh©f.

29. Kjš ÏUgJ Ïaš v©fëèUªJ xU KG v© rkthŒ¥ò Kiwæš

nj®ªbjL¡f¥gL»wJ.mªjv©xUgfhv©zhfÏU¥gj‰fhdãfœjféid¡

fh©f.

30. (a) t®¡f_y«fh©f. 2xx

11

166

+ ++

+^^

hh

(mšyJ)

(b) rhŒÎ¡nfhz« 60c k‰W« y-bt£L¤J©L3

1 bfh©l ne®¡nfh£o‹

rk‹gh£il¡fh©f.







ãWtd«f©l¿ªjJ.nkY«,85 ng®bjhiy¡fh£Ák‰W«g¤Âç¡ifiaÍ«,75ng®bjhiy¡fh£Ák‰W«thbdhèiaÍ«,95 ng®thbdhèk‰W«g¤Âç¡ifiaÍ«,


étu§fis¢F¿¤J,Ã‹tUtdt‰iw¡fh©f.

khzt®fŸ

khzéfŸ

khzt®fŸ

khzéfŸ

427khÂçédh¤jhŸ

(i) thbdhèiak£L«ga‹gL¤Jgt®fë‹v©â¡if.

(ii) bjhiy¡fh£Áiak£L«ga‹gL¤Jgt®fë‹v©â¡if.


ga‹gL¤jhjt®fë‹v©â¡if.

32. rh®ò : [1, 6)f R$ MdJÃ‹tUkhWtiuaW¡f¥g£LŸsJ.

,

,

,

f x

x x

x x

x x

1 1 2

2 1 2 4

3 10 4 62

1

1

1

#

#

#

=

+

-

-

^ h * ( [1 , 6) = { :1 6x xR 1! #" ,)

(i) ( )f 5 (ii) f 3^ h (iii) f 1^ h (iv) f f2 4-^ ^h h (v) 2 3f f5 1-^ ^h h

M»at‰¿‹kÂ¥òfis¡fh©f.

33. xUT£L¤bjhlçšKjš m cW¥òfë‹T£l‰gyD¡F«,Kjšn cW¥òfë‹

T£l‰gyD¡F«ÏilnaÍŸsé»j« :m n2 2 våš,m MtJcW¥òk‰W«n MtJ

cW¥òM»aitfŸ :m n2 1 2 1- -^ ^h h v‹wé»j¤ÂšmikÍ«vd¡fh£Lf.

34. Ú¡fšKiwæšÔ®: x y3 2 +^ h = xy7 ; x y3 3+^ h = xy11

35. t®¡f_y«fh©f. ( )( )( )x x x x x x6 5 6 6 2 4 8 32 2 2+ - - - + +

36. xUk»GªJòw¥glnt©oaneu¤ÂèUªJ30ãäl«jhkjkhf¥òw¥g£lJ.150».Û

öu¤ÂšcŸsnrUäl¤ij rçahd neu¤Âš br‹wilamjDilatH¡fkhd

ntf¤ijkâ¡F25».ÛmÂf¥gL¤jnt©oæUªjJvåš,k»GªÂ‹tH¡fkhd

ntf¤ij¡fh©f.

37. 2 3X Y2

4

3

0+ = c m k‰W« 3 2X Y

2

1

2

5+ =

-

-e o våš, X k‰W« Y M»a

mâfis¡fh©f.

38. (-4, -2), (-3, -5), (3, -2) k‰W« (2, 3) M»aòŸëfisKidfshf¡bfh©l

eh‰fu¤Â‹gu¥ig¡fh©f.

39. (6, -2) vD« òŸë tê¢ brštJ« k‰W« bt£L¤J©Lfë‹ TLjš 5

bfh©lJkhdne®¡nfhLfë‹rk‹ghLfis¡fh©f.

40. nfhzÏUrkbt£o¤nj‰w«-vGÂãWÎf.

41. fl‰fiuæšcŸsbr§F¤jhd¥ghiwx‹¿‹ÛJf£l¥g£LŸsxUfy§fiu

és¡f¤Âš ã‹W¡bfh©oU¡F« xU ÁWä, »H¡FÂiræš ÏU glFfis¥

gh®¡»whŸ. m¥glFfë‹ Ïw¡f¡nfhz§fŸ Kiwna 30c, 60c k‰W« ÏU

glFfS¡»ilnaÍŸs öu« 300 Û våš, flš k£l¤ÂèUªJ fy§fiu

és¡f¤Â‹ c¢Áæ‹öu¤ij¡ fh©f. ( glFfS«, fy§fiués¡fK« xnu

ne®¡nfh£ošcŸsd)

42. 14 Ûé£lK«k‰W«20 ÛMHKŸsxU»zWcUistoéšbt£l¥gL»wJ.

m›thWbt£L«nghJ njh©obaL¡f¥g£lk©Óuhf gu¥g¥g£L 20 Û # 14 ÛmsÎfëšmo¥g¡fkhf¡bfh©lxUnkilahfmik¡f¥g£lhš,m«nkilæ‹

cau«fh©f.


43. xUghjrhçFW¡F¥ghijiafl¡f¢Áy® (pedestrian crossing) vL¤J¡bfh©l

neuétu«Ñœ¡f©lm£ltizæšbfhL¡f¥g£LŸsJ.

neu«(éehoæš) 5-10 10-15 15-20 20-25 25-30eg®fë‹v©â¡if 4 8 15 12 11

Ï¥òŸëétu¤Â‰Féy¡ft®¡f¢ruhrçk‰W«Â£léy¡f¤ij¡fz¡»Lf.

44. xUòÂak»œÎªJ (car) mjDilatotik¥Ã‰fhféUJbgW«ãfœjfÎ0.25 v‹f. ÁwªjKiwæšvçbghUŸga‹gh£o‰fhdéUJbgW«ãfœjfÎ0.35 k‰W«

ÏUéUJfS«bgWtj‰fhdãfœjfÎ0.15 våš,m«k»œÎªJ

(i) FiwªjJVjhtJxUéUJbgWjš

(ii) xnuxUéUJk£L«bgWjšM»aãfœ¢ÁfS¡fhdãfœjfÎfis¡

fh©f

45. (a) . . .0 7 0 97 0 997 g+ + + v‹wbgU¡F¤bjhlç‹n cW¥òfë‹TLjšfh©f.

(mšyJ)

(b) 4 br.Ûé£lK«45 br.ÛcauK«bfh©lne®t£lcUisiacU¡»3 br.ÛMuKŸsÂ©k¡nfhs§fshfkh‰¿dhš,»il¡F«nfhs§fë‹v©â¡ifia

fh©f.





46. (a) mo¥g¡f« BC = 5.5 br.Û., 60A+ = c k‰W« c¢Á A-æèUªJ tiua¥g£l

eL¡nfhLAM-‹ Ús« = 4.5 br.Ûbfh©l ABCT tiuf.

(mšyJ)

(b) AB = 6 br.Û., 70ABC+ = c, BC = 5 br.Û. k‰W« 30ACD+ = cM»amsÎfŸ

bfh©lt£leh‰fu«ABCD tiuf.

47. (a) 2 3y x x2

= + - -‹tiugl«tiuªJ,mjid¥ga‹gL¤Â 6 0x x2- - = v‹w

rk‹gh£il¤Ô®¡fÎ«.

(mšyJ)

(b) th§f¥g£lneh£L¥ò¤jf§fë‹v©â¡ifk‰W«mj‰fhdéiyétu«

Ã‹tU«m£ltizæšju¥g£LŸsJ.

neh£L¥ò¤jf§fë‹v©â¡if x 2 4 6 8 10 12

éiy` y 30 60 90 120 150 180

Ïj‰fhdtiugl«tiuªJmj‹_y«

(i) VGneh£L¥ò¤jf§fë‹éiyia¡fh©f.

(ii) ` 165-¡Fth§f¥gL«neh£L¥ò¤jf§fë‹v©â¡ifia¡fh©f.

Ô®Î - muR khÂç édh¤jhŸ kÂ¥ÕL 429

SCHEME OF EVALUATION - MATHEMATICS

GENERAL GUIDELINES

(i) The answers given in the Scheme of Evaluation are based on the Text Book and SCORE Book.

(ii) Full Credit (marks) should be awarded to a student if his / her approach in giving solution to a problem is correct and leading to required answer. This approach may be Text Book oriented / SCORE Book oriented / Mathematically correct.

(iii) Marks should be allotted according to the different stages required to arrive at the appropriate answer.

(iv) Stage marks are meant for only when a student gives partial answer/commits mistakes/furnishes irrelevant information in the course of his/her answer to a particular question.

(v) While answering a question, if a student starts from a stage with correct step but reaches the next stage with a wrong result, then suitable credits should be given to the correct steps instead of denying the entire marks allotted for that stage.

(vi) If the solution to a particular question, other than questions under Section D, requires a diagram, then a rough sketch of the diagram is enough. Full Credit must be given for such a rough diagram.

(vii) Full credit must be given for an equivalent answer wherever possible.

(viii) There is no separate marks allotted for formula. If a particular stage is wrong and if the student writes the appropriate formula, then suitable mark, which is attached with that stage, should be awarded for the formula. Mark should not be deducted for not writing the formula if the student arrives at the correct answer.


10 M« tF¥ò

muR khÂç édh¤jhŸ - kÂ¥bg© g§ÑL

ÃçÎ-m ÃçÎ-M ÃçÎ-Ï

Q.No. Chapter Exercise Creative Q.No. Chapter Exercise Example Q.No. Chapter Exercise Example

1 Set ü 16 Set ü 31 Set ü

2 Seq. ü 17 Set ü 32 Set ü

3 Seq. ü 18 Seq. ü 33 Seq. ü

4 Alg. ü 19 Alg. ü 34 Alg. ü

5 Alg. ü 20 Mat. ü 35 Alg. ü

6 Mat. ü 21 Mat. ü 36 Alg. ü

7 Co-or. ü 22 Co-or. ü 37 Mat. ü

8 Co-or. ü 23 Geo. ü 38 Co-or. ü

9 Geo. ü 24 Tri. ü 39 Co-or. ü

10 Geo. ü 25 Tri. ü 40 Geo. ü

11 Tri. ü 26 Men. ü 41 Tri. ü

12 Tri. ü 27 Men. ü 42 Men. ü

13 Men. ü 28 Stat. ü 43 Stat. ü

14 Stat. ü 29 Prob. ü 44 Prob. ü

15 Prob. ü Creative Questions Creative Questions

30 a Alg. ü 45 a Seq. ü

30 b Co-or. ü 45 b Men. ü

ÃçÎ-<

Q.No. Chapter Exercise Example Q.No. Chapter Exercise Example

46 a Prac.Geo

ü 47 aGraph

ü

46 b ü 47 b ü

kÂ¥bg© g§ÑL : bjhF¥ò m£ltiz

édh tif Chapter Exercise Creative

òwta édh¡fŸ 10 - 52 kÂ¥bg© édh¡fŸ 8 6 25 kÂ¥bg© édh¡fŸ 8 6 210 kÂ¥bg© édh¡fŸ 3 1 -


muR khÂç édh¤jhŸ - kÂ¥ÕL


bghJ F¿¥òfŸ :

(i) Ï›édh¤jhŸ eh‹F ÃçÎfis¡ bfh©LŸsJ. éilaë¡F« K‹d® x›bthU ÃçéY«

bfhL¡f¥g£LŸs F¿¥òfis ftdkhf go¡fÎ«.

(ii) éilfë‹ têKiwfŸ éil¤jhë‹ x›bthU g¡f¤Â‹ Ñœ¥gFÂæš fh£l¥gl nt©L«.

(iii) fâ¥gh‹ k‰W« ä‹dQ rhjd§fŸ ga‹gL¤j¡ TlhJ.

ÃçÎ - mF¿¥ò : (i) Ï¥Ãçéš cŸs 15 édh¡fS¡F« éilaë¡fÎ«.

(ii) bfhL¡f¥g£LŸs eh‹F éilfëš äfÎ« rçahd éilia¤ nj®ªbjL¤J vGjÎ«.

(iii) x›bthU édhé‰F« xU kÂ¥bg©. [15 × 1 = 15]

1. A = { 1, 3, 4, 7, 11 } k‰W« B = {–1, 1, 2, 5, 7, 9 } v‹f. f = { (1, –1), (3, 2), (4, 1), (7, 5), (11, 9) } v‹wthW mikªj rh®ò :f A B" v‹gJ

(A) x‹W¡F x‹whd rh®ò (B) nkš rh®ò (C) ÏUòw¢ rh®ò (D) rh®ò mšy ( éil: (A) )

2. , , ,52

256

12518 g v‹w bgU¡F¤ bjhl®tçiræ‹ bghJ é»j«

(A) 52 (B) 5 (C)

53 (D)

54 ( éil: (C) )

3. , , ,a a a1 2 3

gv‹gd xU T£L¤ bjhl®tçiræYŸsd. nkY« a

a

23

7

4 = våš, 13tJ

cW¥ò

(A) 23 (B) 0 (C) a12 1 (D) a14 1

Ô®Î: 2( a + 3d ) = 3( a + 6d ) & 3a + 18d – 2a – 6d = 0

& a + 12d = 0 t 013

& = ( éil: (B) ) 4. , ,x y x yz x y z6 9 122 2 2 2 -‹ Û.bgh.k.

(A) x y z36 2 2 (B) xy z36 2 2 (C) 6x y z3 2 2 2 (D) xy z36 2

Ô®Î: 6x y2 = x y2 3 2# Û.bgh.k. = 2 3 x y z2 2 2 2

#

x yz9 2 = x yz32 2 = 4 9x y z2 2#

x y z12 2 2 = x y z2 32 2 2# = x y z36 2 2 ( éil: (A))

5. b = a + c v‹f. 0ax bx c2+ + = v‹w rk‹gh£o‹ _y§fŸ rk« våš,

(A) a = c (B) a = – c (C) a = 2 c (D) a = – 2cÔ®Î: 4 4 0 .b ac a c ac a c a c2 2 2& & &= + = - = =^ ^h h ( éil: (A) )


6. A1

0

1

21 2# =c ^m h våš, A-‹ tçir

(A) 2 1# (B) 2 2# (C) 1 2# (D) 3 2#

Ô®Î: A 1

0

1

21 2# =c ^m h, ( )A

1

0

1

21 2

m n2 2

1 2=

##

#c m

2n& = k‰W« 1m = . vdnt, A-‹ tçir ( éil: (C) )

7. y x7 2 11- = v‹w ne®¡nfh£o‹ rhŒÎ

(A) 27- (B)

27 (C)

72 (D) 7

2-

Ô®Î: rhŒÎ .mba

72

72= - =- - =` j ( éil: (C) )

8. (0, 0), (1, 0), (0,1) v‹w òŸëfshš cUth¡f¥g£l K¡nfhz¤Â‹ R‰wsÎ

(A) 2 (B) 2 (C) 2 2+ (D) 2 2-

Ô®Î: AB = 1, BC = 2 , AC = 1

R‰wsÎ = AB + BC + CA = 1 2 1+ + = 2 2+ ( éil: (C) )

9. 9PQRš RS v‹gJ R+ -‹ nfhz c£òw ÏUrkbt£o.

PQ = 6 br.Û, QR = 8 br.Û, RP = 4 br.Û våš, PS =

(A) 2 br.Û (B) 4 br.Û

(C) 3 br.Û (D) 6 br.Û

Ô®Î: PS x= br.Û.k‰W« (6 )SQ x= - br.Û v‹f.

RS v‹gJ PRQ+ -‹ nfhz c£òw ÏUrkbt£o, vdnt

QRPQ =

SQPS

84

21= =

2 6 2x

x x x x6 2

1& & &-

= = - = ( éil: (A) )

10. AB k‰W« CD v‹w eh©fŸ t£l¤ÂDŸ P v‹w òŸëæš bt£o¡bfhŸ»‹wd.

AB = 7, AP = 4, CP = 2 våš, CD =

(A) 4 (B) 8 (C) 6 (D) 10

Ô®Î: AP × PB = CP × PD & 4×3 = 2 × PD& PD212= = 6

2 6 8CD CP PD` = + = + = ( éil: (B) ) 11. xU nfhòu¤ÂèUªJ 28.5 Û öu¤Âš ã‹W bfh©oU¡F« xUt® nfhòu¤Â‹

c¢Áia 45c V‰w¡ nfhz¤Âš fh©»wh®. mtUila »ilãiy¥ gh®it¡ nfhL

jiuæèUªJ 1.5 Û cau¤Âš cŸsJ våš, nfhòu¤Â‹ cau«

(A) 30 Û (B) 27.5 Û (C) 28.5 Û (D) 27 Û

P

S

Q R8 br.Û.

6 br.Û. 4 br.Û

.


Ô®Î: nfhòu¤Â‹ cau« = tanx y i+

= 1.5 28.5 45tan o#+

= 1.5 28.5 30+ = Û ( éil: (A) ) 12.

tan cot1i i+

=

(A) sin cosi i+ (B) sin cosi i (C) sin cosi i- (D) cosec coti i+

Ô®Î: tan cot

cossin

sincos sin cos

sin cos1 12 2i i

ii

ii i i

i i+

=+

=+

= sin cosi i ( éil: (B) ) 13. 12r br.Û 2 bkh¤j¥gu¥ò bfh©l Â©k miu¡nfhs¤Â‹ tisgu¥ò

(A) 6r br.Û 2 (B) 24r br.Û 2 (C) 36r br.Û 2 (D) 8r br.Û 2

Ô®Î: bkh¤j¥gu¥ò, r3 2r = 12r br.Û 2 r 42

& =

tisgu¥ò, r2 2r = 8r br.Û 2 ( éil: (D) )

14. Áy étu§fë‹ T£L¢ ruhrç k‰W« Â£léy¡f« Kiwna 48, 12 våš,

khWgh£L¡bfG

(A) 42 (B) 25 (C) 28 (D) 48

Ô®Î: bfhL¤JŸsgo, v = 12, x = 48

khWgh£L¡bfG, C.V, = 100x#vr

= 1004812 25# = . ( éil: (B) )

15. A k‰W« B v‹gd Ïu©L x‹iwbah‹W éy¡F« ãfœ¢ÁfŸ v‹f. mªãfœ¢Áæ‹

TWbtë S, ( ) ( )P A P B31= k‰W« S A B,= våš, ( )P A =

(A) 41 (B)

21 (C)

43 (D) 8

3

Ô®Î: ( ) ( ) ( )P A B P A P B, = +

(A, B v‹gd Ïu©L x‹iwbah‹W éy¡F« ãfœ¢ÁfŸ)

( ) ( ) ( )P S P A P A3= + 4 ( ) 1P A& =

vdnt, ( )P A41= . ( éil: (A) )

ÃçÎ - MF¿¥ò : (i) g¤J édh¡fS¡F éilaë¡fÎ«.

(ii) Kjš 14 édh¡fëèUªJ VnjD« 9 édh¡fS¡F éilaë¡fÎ«. édh v© 30¡F

f©o¥ghf éilaë¡fÎ«.

(iii) x›bthU édhé‰F« Ïu©L kÂ¥bg©fŸ. [10 × 2 = 20]

16. {4,6,7,8,9}, {2,4,6} {1,2,3,4,5,6}k‰W«A B C= = = våš, A B C, +^ h-I¡

fh©f.

Ô®Î: B C+ = {2, 4, 6} + {1, 2, 3, 4, 5, 6} = {2, 4, 6}. ... 1 kÂ¥bg©

( )A B C, + = {4, 6, 7, 8, 9} , {2, 4, 6} = {2, 4, 6, 7, 8, 9}. ... 1 kÂ¥bg©


17. X = { 1, 2, 3, 4 } v‹f. g = { (3, 1), (4, 2), (2, 1) } v‹w cwÎ«, X-èUªJ X -¡F xU

rh®ghFkh vd MuhŒf. c‹ éil¡F V‰w és¡f« jUf.

Ô®Î: 1 vD« cW¥Ã‰F ãHš cU Ïšiy. g‹ kÂ¥gf« {2, 3, 4} X!= ... 1 kÂ¥bg©

g = { (3, 1), (4, 2), (2, 1)} v‹w cwÎ xU rh®gšy. ... 1 kÂ¥bg©

18. _‹W v©fë‹ é»j« 2 : 5 : 7 v‹f. Kjyh« v©, Ïu©lh« v©âèUªJ 7-I¡

fê¤J¥ bgw¥gL« v© k‰W« _‹wh« v© M»ad xU T£L¤ bjhl®tçiria

V‰gL¤Âdhš, m›bt©fis¡ fh©f.

Ô®Î: m›bt©fis 2 ,5x x k‰W« 7x v‹f. (x 0! )

bfhL¡f¥g£l étu¤Â‹go, 2 , 5 7, 7x x x- v‹gd xU T£L¤ bjhl®tçir MF«.

` 2 ( )x x x x5 7 7 5 7- - = - -^ h ( 3 7x x2 7- = + ( x = 14. ... 1 kÂ¥bg©

njitahd m›bt©fŸ 28, 70, 98 MF«. ... 1 kÂ¥bg©

19. 2 3 1 0x x2- - = v‹w rk‹gh£o‹ _y§fŸ a k‰W« b våš, a b- ‹ kÂ¥òfis¡

fh©f. Ï§F >a b

Ô®Î: bfhL¡f¥g£l rk‹ghL x x2 3 1 02- - = .

Ïij 0ax bx c2+ + = v‹w rk‹gh£Ll‹ x¥Ãl,

a 2= , b 3=- , c 1=- . a k‰W« b M»ait _y§fŸ.

` a b+ = ab- =

2

3- -^ h = 23 k‰W«

ac

21ab = = ... 1 kÂ¥bg©

a b- = 423 4

21

2172

2a b ab+ - = - - =^ ` `h j j ... 1 kÂ¥bg©

20. A2

9

3

5

1

7

5

1=

--

-e eo o våš, A-‹ T£lš ne®khW mâia¡ fh©f.

Ô®Î: A = 2

9

3

5

1

7

5

1--

-e eo o 2

9

3

5

1

7

5

1=

-+

-

-

-e eo o

= 1

16

2

6-

-e o ... 1 kÂ¥bg©

A-‹ T£lš ne®khW mâ –A. vdnt, A-‹ T£lš ne®khW mâ

A- = 1

16

2

6

1

16

2

6-

-

-=

-

-e eo o ... 1 kÂ¥bg©

21. 2

4

9

1

3

0

4

6

2

2

7

1-

--

-

e fo p M»a mâfë‹ bgU¡fiy¡ fh©f.

(tiuaW¡f¥gLkhdhš)

Ô®Î: A = 2

4

9

1

3

0-

-e o k‰W« B =

4

6

2

2

7

1

-

-

f p v‹f.


A-‹ tçir 2 # 3 k‰W« B-‹ tçir 3 # 2.vdnt bgU¡fš tiuaW¡f¥gL»wJ. ... 1 kÂ¥bg©

AB = 2

4

9

1

3

0

4

6

2

2

7

1-

--

-

e fo p = 8 54 6

16 6 0

4 63 3

8 7 0

- +

+ -

+ -

- +e o

= 40

22

64

1

-c m (éil¡F 2 kÂ¥bg© bfhL¡fyh«) ... 1 kÂ¥bg©

22. xU t£l¤Â‹ ika« (-6, 4). m›t£l¤Â‹ xU é£l¤Â‹ xU Kid MÂ¥òŸë


Ô®Î: ( , )x y é£l¤Â‹ kW Kid v‹f.

t£l¤Â‹ ika« mj‹ é£l¤Â‹ eL¥òŸë MF«.

vdnt, ,xy

20

20+ +

c m = (– 6, 4) ... 1 kÂ¥bg©

x k‰W« y m¢R¤ bjhiyÎfis rk¥gL¤j,

6 12x x2

0 &+ =- =- k‰W« 4 8.y

y2

0&

+= =

vdnt, é£l¤Â‹ kW Kid ( , )12 8- . ... 1 kÂ¥bg©

23. ABC3 -šDE BC< k‰W« DBAD

32= . AE = 3.7 br. Û våš, EC-I¡ fh©f.

Ô®Î: ABC3 -š, DE BC<

` DBAD = EC

AE (njš° nj‰w«) ... 1 kÂ¥bg©

& EC = AD

AE DB#

vdnt, EC = .2

3 7 3# = 5.55 br. Û ... 1 kÂ¥bg©

24. Rtçš rhŒ¤J it¡f¥g£l xU VâahdJ jiuÍl‹ 60c nfhz¤ij

V‰gL¤J»wJ. Vâæ‹ mo Rt‰¿èUªJ 3.5 Û öu¤Âš cŸsJ våš, Vâæ‹

Ús¤ij¡ fh©f.

Ô®Î: AC v‹gJ VâiaÍ«, B v‹gJ Rt‰¿‹ moiaÍ« F¿¡f£L«.

CAB 60+ = c k‰W«

AB =3.5Û vd¡ bfhL¡f¥g£LŸsJ.

br§nfhz3ABC-š, cos60c = ACAB ... 1 kÂ¥bg©

& AC = 60cos

ABc

= 2 3.5 7# = Û ... 1 kÂ¥bg©

vdnt, Vâæ‹ Ús« 7 Û.

B

D3.7 br.Û.

C

E

A


25. 1cosecsin

seccos

ii

ii+ = v‹w K‰bwhUikia ãWÎf.

Ô®Î: Ï§F, cosecsin

seccos

ii

ii+ =

sin

sin

cos

cos1 1i

i

i

i+` `j j

... 1 kÂ¥bg©

= sin cos2 2i i+ = 1. ... 1 kÂ¥bg©

26. xU Â©k ne® t£l cUisæ‹ Mu« 14 br.Û k‰W« cau« 8 br.Û. våš, mj‹


Ô®Î: Mu«, r = 14 br.Û k‰W« cau«, h = 8br.Û vd¡ bfhL¡f¥g£LŸsJ.

vdnt, tisgu¥ò = 2 rhr

= 2 × × ×722 14 8 = 704 r.br.Û ... 1 kÂ¥bg©

nkY«, bkh¤j¥ òw¥gu¥ò = 2 ( )r h rr +

= 2 × 722 ×14 (8 + 14)

= 88 × 22 = 1936 r.br.Û ... 1 kÂ¥bg©

27. ku¤Âdhyhd xU Â©k¡ T«Ã‹ mo¢R‰wsÎ 44 br.Û. k‰W« mj‹ cau« 12 br.Û

våš m¤Â©k¡ T«Ã‹ fdmsit¡ fh©f.

Ô®Î: xU Â©k¡ T«Ã‹ Mu« k‰W« cau« Kiwna r k‰W« h v‹f.

h = 12br.Û. k‰W« mo¢R‰wsÎ 44 br.Û vd¡ bfhL¡f¥g£LŸsJ.

mo¢R‰wsÎ, 2 rr = 44& r =

244r

= 2 2244 7## = 7 br.Û. ... 1 kÂ¥bg©

Â©k¡ T«Ã‹ fdmsÎ = r h31 2r

= 7 131

722 22

# # # f.br.Û.

= 166 f.br.Û. ... 1 kÂ¥bg©


Ô®Î:

Kjš n Ïaš v©fë‹ Â£léy¡f« v = n12

12-

` Kjš 13 Ïaš v©fë‹ Â£léy¡f« v = n12

12- =

1213 1

2- ... 1 kÂ¥bg©

= .12168 14 3 74= = ... 1 kÂ¥bg©

29. ÏU ehza§fis xnu rka¤Âš R©L«nghJ, mÂfg£rkhf xU jiy

»il¥gj‰fhd ãfœjféid¡ fh©f.

Ô®Î: TW btë { , , , } ; ( ) 4.S HH HT TH TT n S= = ... 1 kÂ¥bg©


mÂfg£rkhf xU jiy »il¥gj‰fhd ãfœ¢Áia A v‹f.

vdnt, A = , ,HT TH TT" , k‰W« 3n A =^ h .

Mfnt, ( )P A = n S

n A

43=

^^

hh . ... 1 kÂ¥bg©

30. (a) RU¡Ff. x x

x

7 12

6 542

2

+ +

- . (mšyJ)

(b) y x2 4 3= + k‰W« x y2 10+ = M»a ne®¡nfhLfŸ x‹W¡bfh‹W

br§F¤jhdit vd ãWÎf.

Ô®Î: (a) x x

x

7 12

6 542

2

+ +

- = x x

x4 3

6 92

+ +-

^ ^^

h hh

= x x

x x

4 3

6 3 3

+ +

+ -

^ ^^ ^

h hh h ... 1 kÂ¥bg©

= x

x

4

6 3

+

-^ h ... 1 kÂ¥bg©

(b) 2y = 4x + 3 & 4x – 2y + 3 = 0.

rhŒÎ m1 =

‹ bfG‹ bfG

yx- = 2

2

4

-

-=

^ h

ne®¡nfhL x y2 10 0+ - = ¡F, rhŒÎ m2=

21- ... 1 kÂ¥bg©

Ï§F, 2m m21

1 2#= - = – 1 ... 1 kÂ¥bg©

vdnt, bfhL¡f¥g£l ne®¡nfh£L¤ J©LfŸ x‹W¡bfh‹W br§F¤jhdit.

ÃçÎ - ÏF¿¥ò : (i) 9 édh¡fS¡F éilaë¡fÎ«.

(ii) Kjš 14 édh¡fëèUªJ VnjD« 8 édh¡fS¡F éilaë¡fÎ«. édh v© 45-¡F

f©o¥ghf éilaë¡fÎ«.

(iii) x›bthU édhé‰F« IªJ kÂ¥bg©fŸ. [9 × 5 = 45]

31. bt‹gl§fis¥ ga‹gL¤Â \ \ \A B C A B A C+ ,=^ ^ ^h h h v‹D« o kh®få‹

fz é¤Âahr éÂæid¢ rçgh®¡fÎ«.

Ô®Î: (x›bthU gl¤Â‰F« 1 kÂ¥bg©)


(2) k‰W« (5) èUªJ, \ \ \A B C A B A C+ ,=^ ^ ^h h h.


( )f x = ;

;

; .

x x x

x x

x x

2 1 7 5

5 5 2

1 2 6

2 1

1 1

#

# #

+ + - -

+ -

-

* ( ) ( )( ) ( )

f ff f

6 3 14 3 2 4

- -- + -I¡ fh©f.

Ô®Î: 6x =- våš, ( ) 2 1f x x x2= + +

vdnt, ( )f 6- = ( ) ( )6 2 6 1 36 12 1 252- + - + = - + = . ... (1 kÂ¥bg©)

3x =- k‰W« 1 våš ( ) 5f x x= +

nkY«, ( )f 3- = 2 k‰W« (1) 6f = . ... (2 kÂ¥bg©fŸ)

x 4= våš, ( ) 1.f x x= - Mfnt, (4) 3.f = ... (1 kÂ¥bg©)

( ) ( )( ) ( )

f ff f

6 3 14 3 2 4

- -- + =

25 3 64 2 2 3

25 188 6

714 2

## #

-+ =

-+ = = . ... (1 kÂ¥bg©)

33. 1 2 3 4 ...2 2 2 2- + - + v‹w bjhlç‹ Kjš 2n cW¥òfë‹ TLjš fh©f.

Ô®Î: 1 2 3 42 2 2 2

g- + - + n2 cW¥òfŸ

= 1 4 9 16 25 g- + - + - n2 cW¥òfŸ

= 1 4 9 16 25 36 g- + - + - +^ ^ ^h h h n cW¥òfŸ. (bjhF¥Ã‰F¥Ã‹)

= 3 7 11 g- + - + - +^ ^h h n cW¥òfŸ ... (1 kÂ¥bg©) nk‰f©l T£L¤bjhl® tçiræš Kjš cW¥ò 3a =- k‰W« bghJ é¤Âahr«

d 4=- . ... (1 kÂ¥bg©)

vdnt, njitahd TLjš = n a n d2

2 1+ -^ h6 @

= n n2

2 3 1 4- + - -^ ^ ^h h h6 @ ... (2 kÂ¥bg©fŸ)

= n n2

6 4 4- - +6 @ = n n2

4 2- -6 @

= n n22 2 1- +^ h = n n2 1- +^ h ... (1 kÂ¥bg©)

34. 2 5 6x x x3 2- - + v‹w gšYW¥ò¡nfhitia fhuâ¥gL¤Jf.

Ô®Î: ( )p x = x x x5 2 243 2- - + v‹f.

( )p x ‹ bfG¡fë‹ TLjš = 1 5 2 24 0!- - + .


vdnt, ( )x 1- MdJ ( )p x ¡F xU fhuâ mšy.nkY«, ( )p 1- = 1 5 2 24 0!- - + + . vdnt, 1x + xU fhuâašy.nrhjidæ‹ go, ( ) .p 2 0- = vdnt, ( )x 2+ xU fhuâ. ... (1 kÂ¥bg©)

– 2 1 – 5 – 2 240 – 2 14 – 241 – 7 12 0 " ÛÂ ... (2 kÂ¥bg©fŸ)

k‰bwhU fhuâ ( )( )x x x x7 12 3 42- + = - - . ... (1 kÂ¥bg©)

vdnt, 5 2 24x x x3 2- - + = ( )( )( )x x x2 3 4+ - - ... (1 kÂ¥bg©)

35. 28 12 9m nx x x x2 3 4

- + + + MdJ xU KG t®¡f« våš, m, n M»at‰¿‹

kÂ¥òfis¡ fh©f.

Ô®Î: gšYW¥ò¡ nfhitia x-‹ mL¡Ffë‹ Ïw§F tçiræš vGj,

9 12 28x x x nx m4 3 2+ + - +

3 2 4x x2+ +

3 x2

9 12 28x x x nx m4 3 2+ + - +

9x4

... (2 kÂ¥bg©fŸ)

6 2x x2+ 12 28x x

3 2+

12 4x x3 2+

... (1 kÂ¥bg©)

6 4 4x x2+ + 24x nx m

2- +

24 16 16x x2+ + ... (1 kÂ¥bg©)

0

bfhL¡f¥g£l gšYW¥ò¡nfhit xU KG t®¡fkhjyhš,

n = –16 k‰W« m = 16. ... (1 kÂ¥bg©) 36. mirt‰w Úçš xU ÏaªÂu¥gl»‹ ntf« kâ¡F 15 ».Û. v‹f. m¥glF

Únuh£l¤Â‹ Âiræš 30 ».Û öu« br‹W, ÃwF vÂ®¤ Âiræš ÂU«Ã 4 kâ 30 ãäl§fëš Û©L« òw¥g£l Ïl¤Â‰F ÂU«Ã tªjhš Úç‹ ntf¤ij¡ fh©f.

Ô®Î: Úç‹ ntf« x ».Û/kâ

mirt‰w Úçš xU ÏaªÂu¥gl»‹ ntf« = 15 ».Û./kâ

Únuh£l¤Â‹ Âiræš k‰W« Únuh£l¤Â‹ vÂ®¤ Âiræš gl»‹ ntf§fŸ

Kiwna ( )x15 + ».Û./kâ k‰W« ( )x15 - ».Û./kâ v‹f. .... (1 kÂ¥bg©)

30 ».Û öu¤ij Únuh£l¤Â‹ Âiræš fl¡f MF« neu« T1 v‹f.

30 ».Û öu¤ij Únuh£l¤Â‹ vÂ®¤ Âiræš fl¡f MF« neu« T2

v‹f.

fhy« ntf«öu«

= v‹gjhš, T1 =

x1530+

k‰W« T2

= x15

30-

. .... (1 kÂ¥bg©)


Ï§F, T1 + T

2 = 4 kâ 30 ãäl« = 4

21 kâfŸ

& x x15

301530

-+

+ =

29 .... (1 kÂ¥bg©)

& ( )( )

( ) ( )x x

x x15 15

30 15 30 15- +

+ + - = 29

& ( )x9 225 2- = 1800

& x225 2- = 200 .... (1 kÂ¥bg©)

& x = 5!

Úç‹ ntf« Fiw v©zhf ÏU¡f ÏayhJ v‹gjhš .x 5= vdnt, Úç‹ ntf« 5 ».Û./kâ. .... (1 kÂ¥bg©)

37. k‰W«A B5

7

2

3

2

1

1

1= =

-

-c em o våš, ( )AB B A

T T T= v‹gij rç¥gh®¡fÎ«.

Ô®Î: A = 5

7

2

3c m k‰W« B = 2

1

1

1-

-e o

A ‹ tçir 2 #2 k‰W« B ‹ tçir 2 #2. vdnt, AB ‹ tçir 2 #2.

vdnt, AB = 5

7

2

3

2

1

1

1-

-c em o

= 10 2

14 3

5 2

7 3

-

-

- +

- +e o = 8

11

3

4

-

-e o .... (1 kÂ¥bg©)

(AB)T = 8

3

11

4- -e o g (1) .... (1 kÂ¥bg©)

nkY«, BT = 2

1

1

1-

-e o .... (1 kÂ¥bg©)

AT = 5

2

7

3c m .... (1 kÂ¥bg©)

BT AT = 2

1

1

1

5

2

7

3-

-e co m = 10 2

5 2

14 3

7 3

-

- +

-

- +e o

= 8

3

11

4- -e o g (2) .... (1 kÂ¥bg©)

(1) k‰W« (2) èUªJ, (AB)T = BT AT

38. (-4, -2), (-3, -5), (3, -2) k‰W« (2, 3) M»a òŸëfis Kidfshf¡ bfh©l

eh‰fu¤Â‹ gu¥ig¡ fh©f.

Ô®Î: òŸëfis fofhu KŸnsh£l vÂ®¤Âiræš

mikÍ«go cjégl¤Âš F¿¡f, Kidfis

A(-4, -2), B(-3, -5), C(3, -2) k‰W« D(2, 3) v‹f.


eh‰fu« ABCD‹ gu¥ò

= 21 4

2

3

5

3

2

2

3

4

2

-

-

-

- -

-

-) 3 .... (2 kÂ¥bg©fŸ)

= 21 20 6 9 4 6 15 4 12+ + - - - - -^ ^h h" , .... (2 kÂ¥bg©fŸ)

= 21 31 25+" , = 28 r.myFfŸ .... (1 kÂ¥bg©)

39. ABC3 -‹ KidfŸ A(2, 1), B(6, –1), C(4, 11) v‹f. A-æèUªJ tiua¥gL«

F¤J¡nfh£o‹ rk‹gh£il¡ fh©f.

Ô®Î: BC‹ rhŒÎ = 4 611 1-+ = – 6 .... (1 kÂ¥bg©)

BC ¡F AD br§F¤J v‹gjhš AD‹ rhŒÎ = 61 .... (1 kÂ¥bg©)

vdnt, AD‹ rk‹ghL, y y1

- =m x x1

-^ h .... (1 kÂ¥bg©)

y 1- = x61 2-^ h ( y6 6- = x 2- .... (1 kÂ¥bg©)

vdnt, njitahd rk‹ghL, x y6 4- + = 0. .... (1 kÂ¥bg©) 40. xU ÁWt‹ itu¤Â‹ FW¡F bt£L¤ njh‰w toéš, gl¤Âš

fh£oathW xU g£l« brŒjh‹. Ï§F AE = 16 br.Û, EC = 81 br.Û. mt‹ BD v‹w FW¡F¡ F¢Áæid¥ ga‹gL¤j

éU«ò»wh‹. m¡F¢Áæ‹ Ús« v›tsÎ ÏU¡fnt©L«?

Ô®Î: bfhL¡f¥g£l gl¤Âš ADCT xU br§nfhz

K¡nfhz«. k‰W« DE AC=

[xU br§nfhz K¡nfhz¤Âš KidæèUªJ f®z¤Â‰F br§F¤J¡nfhL

tiuªjhš br§F¤J¡nfh£o‹ ÏUòwK« mikªj K¡nfhz§fŸ

x‹W¡bfh‹W tobth¤jit. nkY« jå¤jåna KGikahd K¡nfhz¤Â‰F

tobth¤jitahF« vd ek¡F bjçÍ«]

vdnt, EAD EDC+D DEDEA& =

ECED .... (1 kÂ¥bg©)

ED2 = 16 81EA EC# #= .... (1 kÂ¥bg©)

Mfnt, ED = 4 9 3616 81# #= = .... (1 kÂ¥bg©)

Ï§F, ABDT xU ÏUrkg¡f K¡nfhz« k‰W« AE BD=

vdnt, BE = ED .... (1 kÂ¥bg©)

BD& = ED2

= 2 × 36

= 72 br.Û. .... (1 kÂ¥bg©) 41. ne®¡F¤jhd xU ku¤Â‹ nkšghf« fh‰¿dhš K¿ªJ, m«K¿ªj gFÂ ÑnH

éGªJélhkš, ku¤Â‹ c¢Á jiuÍl‹ 30c nfhz¤ij V‰gL¤J»wJ. ku¤Â‹

c¢Á mj‹ moæèUªJ 30 Û bjhiyéš jiuia¤ bjhL»wJ våš, ku¤Â‹ KG

cau¤ij¡ fh©f.


Ô®Î: C v‹w òŸëæš ku« K¿ªJŸsJ vdÎ« ku¤Â‹ c¢Á jiuia¤ bjhL«

òŸë A vdÎ« bfhŸf. B v‹gJ ku¤Â‹ mo v‹f.

AB =30 Û k‰W« 30CAB+ = c.

br§nfhz ABCT š,

tan 30c = ABBC .....(1 kÂ¥bg©)

& BC = 30tanAB c

vdnt, BC = 3

30 = 01 3 Û .... (1 kÂ¥bg©)

Ï§F, 30cos c = ACAB .....(1 kÂ¥bg©)

& AC = cosAB30c

vdnt, AC = 10 2 203

30 2 3 3# #= = Û .....(1 kÂ¥bg©)

Mfnt, ku¤Â‹ cau« = 10 20BC AC 3 3+ = +

= 30 3 Û. .....(1 kÂ¥bg©) 42. fëk©iz¥ ga‹gL¤Â xU khzt‹ 48 br.Û cauK« 12 br.Û MuK« bfh©l

ne® t£lÂ©k¡ T«ig¢ brŒjh®. m¡T«ig k‰bwhU khzt® xU Â©k¡

nfhskhf kh‰¿dh®. m›thW kh‰w¥g£l òÂa nfhs¤Â‹ Mu¤ij¡ fh©f.

Ô®Î: ne® t£l T«Ã‹ Mu« k‰W« cau« Kiwna r1 k‰W« h v‹f.

Â©k¡ nfhs¤Â‹ Mu« r2 v‹f.

bfhL¡f¥g£LŸsgo, r1 = 12 br.Û.; h = 48 br.Û.

T«ig nfhskhf kh‰W«nghJ,

nfhs¤Â‹ fd msÎ = T«Ã‹ fd msÎ ..... (1 kÂ¥bg©)

r34

2

3r = r h31

1

2r

r2

3 = 31 12 48

432

# # # #r

r = 123 ..... (3 kÂ¥bg©fŸ)

vdnt, nfhs¤Â‹ Mu« = 12br.Û. ..... (1 kÂ¥bg©)


fz¡»Lf.

x 3 8 13 18 23

f 7 10 15 10 8

Ô®Î: Â£l éy¡f¤ij Cf ruhrç Kiwæš fh©ngh«.

Cf ruhrç A = 13. d x A x 13= - = -


x f d = x – 13 d2 fd fd2

38

131823

71015108

-10-5 0 5

10

100250

25100

-70-50 0 50 80

700250

0250800

fR =50 fdR =10 fd2

R =2000

Â£l éy¡f«, v = f

fd

f

fd2 2

- e o//

//

= 50

20005010 2

- ` j .... (2 kÂ¥bg©fŸ)

= 40251- =

25999 = .

531 61

vdnt, v - 6.321 .... (1 kÂ¥bg©)

44. xU òÂa k»œÎªJ (car) mjDila totik¥Ã‰fhf éUJ bgW« ãfœjfÎ 0.25 v‹f. Áwªj Kiwæš vçbghUŸ ga‹gh£o‰fhd éUJ bgW« ãfœjfÎ 0.35 k‰W«

ÏU éUJfS« bgWtj‰fhd ãfœjfÎ 0.15 våš, m«k»œÎªJ


(ii) xnu xU éUJ k£L« bgWjš M»a ãfœ¢ÁfS¡fhd ãfœjfÎfis¡ fh©f.

Ô®Î: A v‹gJ totik¥Ã‰fhf éUJ bgW« ãfœ¢Á k‰W« B v‹gJ Áwªj

Kiwæš vçbghUŸ ga‹gh£o‰fhd éUJ bgW« ãfœ¢Á v‹f.

( ) .P A 0 25= , ( ) .P B 0 35= k‰W« ( ) .P A B 0 15+ = .... (1 kÂ¥bg©)

(i) FiwªjJ VjhtJ xU éUJ bgw ãfœjfÎ,

( )P A B, = ( ) ( ) ( )P A P B P A B++ -

= .25 0.35 0.150 + -

= 0.45 .... (2 kÂ¥bg©fŸ)

(ii) xnu xU éUJ k£L« bgw ãfœjfÎ,

( ) ( )P A B P A B+ ++ = [ ( ) ( )] [ ( ) ( )]P A P A B P B P A B+ +- + -

= (0.25 0.15) (0.35 0.15)- + -

= 0.10 0.20+

= 0.3 .... (2 kÂ¥bg©fŸ) 45. (a) xU T£L¤bjhl® tçiræ‹ mL¤jL¤j _‹W cW¥òfë‹ TLjš – 6 k‰W«

mt‰¿‹ bgU¡f‰gy‹ 90. mªj v©fis fh©f.

(mšyJ)

.... (2 kÂ¥bg©fŸ)


(b) 14 br.Û. é£lK«, 20 br.Û. MHK« bfh©l xU cUis tot gh¤Âu¤Âš

ghÂasÎ Ú® cŸsJ. xnu msÎŸs 300 nfhs¡F©Lfis m¥gh¤Âu¤ÂDŸ

nghl¥gL«nghJ Úç‹ cau« 2.8 br.Û. caU»wJ, våš nfhs¤Â‹ é£l¤ij¡

fh©f.

Ô®Î: (a) a – d, a, a + d v‹gd xU T£L¤bjhl® tçiræš cŸs _‹W mL¤jL¤j

cW¥òfŸ v‹f. ... (1 kÂ¥bg©)

vdnt, a – b + a + a + d = – 6 & a = – 2 .... (1 kÂ¥bg©) (a – d) a (a + d) = 90

( ) [( ) ]d2 22 2

- - - = 90 & d 492= & d = 7! .... (2 kÂ¥bg©fŸ)

,a d2 7=- = våš, _‹W cW¥òfŸ, , ,2 7 2 2 7- - - - + & , ,9 2 5- -

,a d2 7=- =- våš, _‹W cW¥òfŸ, ( ), ,2 7 2 2 7- - - - - - & , ,5 2 9- -

vdnt, njitahd _‹W v©fŸ , ,9 2 5- - mšyJ , ,5 2 9- - . .... (1 kÂ¥bg©)

(mšyJ)

(b) cUisæ‹ é£l« 2r = 14 & r = 7 br.Û., h = 2.8 br.Û. (ca®ªj Úç‹ k£l«)

nfhs¡ F©o‹ Mu« r1v‹f.

cUistot gh¤Âu¤Âš ca®ªj Úç‹ bfhŸssÎ

= 300 (nfhs¡ F©Lfë‹ fdmsÎ) .... (1 kÂ¥bg©)

r h2r = 300 r34

1

3# r

.7 7 2 8# # = 300 r34

1

3#

r1

3 = .100 4

7 7 2 8#

# # = . . .0 7 0 7 0 7# #

r1 = 0.7 br.Û. .... (3 kÂ¥bg©fŸ)

é£l« = r21 = 2 × 0.7 br.Û.

= 1.4 br.Û.

vdnt, nfhs¡ F©o‹ é£l« = 1.4 br.Û. .... (1 kÂ¥bg©)

ÃçÎ - <F¿¥ò : (i) Ï¥Ãçéš cŸs x›bthU édhéY« Ïu©L kh‰W édh¡fŸ bfhL¡f¥g£LŸsd.

(ii) x›bthU édhéY« cŸs Ïu©L kh‰W édh¡fëèUªJ xU édhit nj®ªbjL¤J

ÏU édh¡fS¡F« éilaë¡fÎ«.

(iii) x›bthU édhé‰F« g¤J kÂ¥bg©fŸ. [2 × 10 = 20]

46. (a) 6 br.Û MuKŸs xU t£l« tiuªJ mj‹ ika¤ÂèUªJ 10 br.Û bjhiyé

YŸs xU òŸëia¡ F¿¡f. m¥òŸëæèUªJ t£l¤Â‰F bjhLnfhLfŸ

tiuªJ mj‹ Ús§fis fz¡»Lf.


cjé¥gl« .... (1 kÂ¥bg©)

Kjš t£l« .... (3 kÂ¥bg©fŸ)

nfh£L¤J©L OP .... (1 kÂ¥bg©)

ika¡F¤J¡nfhL .... (1 kÂ¥bg©)

2tJ t£l« .... (2 kÂ¥bg©fŸ)

Ïu©L bjhLnfhLfŸ .... (1 kÂ¥bg©)

bjhLnfhLfë‹ Ús§fis ms¤jš .... (1 kÂ¥bg©)


tiuKiw:



(iii) OP ‹ ika¡F¤J¡nfhL tiuf. mJ OP-ia M-š bt£l£L«. (iv) M-I ikakhfÎ«, MO (= MP)I MukhfÎ« bfh©L k‰bwhU t£l« tiuf.



rçgh®¤jš: br§nfhz OPAT -š PA = OP OA2 2

- = 10 62 2- = 100 36- = 64

bjhLnfh£o‹ Ús«, PA = 8 br.Û. (mšyJ)

(b) BC = 5 br.Û., BAC 40+ = c k‰W« c¢Á A-èUªJ BC-¡F tiua¥g£l

eL¡nfh£o‹ Ús« 6 br.Û. v‹w msÎfŸ bfh©l ABCT tiuf. nkY« c¢Á A-èUªJ

tiua¥g£l F¤J¡nfh£o‹ Ús« fh©f.


cjé¥gl« .... (1 kÂ¥bg©)

nfh£L¤J©L BC .... (1 kÂ¥bg©)

t£l« .... (5 kÂ¥bg©fŸ)

K¡nfhz« .... (2 kÂ¥bg©fŸ)

F¤J¡nfhL .... (1 kÂ¥bg©)

tiuKiw:


(ii) òŸë B têna 4CBX 0+ = c vd ÏU¡F« go BX tiuf.

(iii) BY=BX tiuf.









(ix) CB I CZ tiu.

(x) AE CZ= . (xi) F¤J¡nfhL AE ‹ Ús« 3.8 br.Û.

47. (a) 8y x x2

= - - -‹ tiugl« tiuªJ, mjid¥ ga‹gL¤Â 2 15 0x x2- - =

v‹w rk‹gh£il¤ Ô®¡fÎ«. Ô®Î: 8y x x

2= - -

x – 4 – 3 – 2 – 1 0 1 2 3 4 5

x2 16 9 4 1 0 1 4 9 16 25

x- 4 3 2 1 0 – 1 – 2 – 3 – 4 – 5– 8 – 8 – 8 – 8 – 8 – 8 – 8 – 8 – 8 – 8 – 8y 12 4 – 2 – 6 – 8 – 8 – 6 – 2 4 12

òŸëfŸ: ( , ), ( , ), ( , ), ( , ), ( , )4 12 3 4 2 2 1 6 0 8- - - - - - - ( , ), ( , ), ( , ), ( , ), ( , )1 8 2 6 3 2 4 4 5 12- - -

Ô®¡f: y = x x 82- -

0 = x x2 152- -

y = x 7+

y x 7= + v‹w ne®¡nfh£o‰fhd m£ltizia mik¥ngh«.


x – 3 – 2 – 1 0 1 2 3 4 5y x 7= + 4 5 6 7 8 9 10 11 12

òŸëfŸ: ( 3,4), ( 2,5), ( 1,6), (0,7), (1,8), (2,9), (3,10), (4,11), (5,12)- - -

ne®¡nfhL k‰W« tistiu M»ait bt£o¡bfhŸS« òŸëfŸ (– 3, 4) k‰W« (5, 12).Mfnt, x-Ma¤bjhiyÎfŸ – 3 k‰W« 5 MF«. Mfnt, Ô®Î fz«{–3, 5}.

Kjš m£ltiz .... (2 kÂ¥bg©fŸ)rk‹ghLfë‹ Ô®Î fhz .... (1 kÂ¥bg©)Ïu©lhtJ m£ltiz .... (1 kÂ¥bg©)x, y m¢R¡fŸ tiujš, k‰W« msÎ¤Â£l« .... (2 kÂ¥bg©fŸ)òŸëfis F¿¤jš .... (3 kÂ¥bg©fŸ)Ô®Î fz« .... (1 kÂ¥bg©)

(mšyJ)


(b) xU äÂt©o X£Lgt® A v‹w Ïl¤ÂèUªJ B v‹w Ïl¤Â‰F xU Óuhd ntf¤Âš

xnu têæš bt›ntW eh£fëš gaz« brŒ»wh®. mt® gaz« brŒj ntf«,

m¤öu¤Âid¡ fl¡f vL¤J¡ bfh©l neu« M»adt‰iw¥ g‰¿a

étu§fŸ (ntf-fhy) Ã‹tU« m£ltizæš bfhL¡f¥g£LŸsd.

ntf« (».Û/kâ) x 2 4 6 10 12neu« (kâæš) y 60 30 20 12 10

ntf - fhy tiugl« tiuªJ mÂèUªJ

(i) mt® kâ¡F 5 ».Û ntf¤Âš br‹whš öu¤ij¡ fl¡f MF« gaz neu«

(ii) mt® Ï¡F¿¥Ã£l öu¤ij 40 kâneu¤Âš fl¡f vªj ntf¤Âš gaâ¡f

nt©L«

M»adt‰iw¡ fh©f.

Ô®Î: m£ltizæèUªJ x-‹ kÂ¥ò mÂfkhF«nghJ y-‹ kÂ¥ò vÂ®é»j¤Âš

Fiw»wJ vd m¿»nwh«. Ïªj tifahd khWghL vÂ® khWghL MF«.

vdnt, xy = k MF«. Ï§F xy = 120Mfnt, y =

x120

(2 , 60), (4 , 30), (6 , 20), (10 , 12) k‰W« (12, 10) M»a òŸëfis tiugl¤jhëš

F¿¡fÎ«. Ï¥òŸëfis ne®¡nfhl‰w ÏiHthd tistiuahš Ïiz¡fÎ«.

tiugl¤ÂèUªJ,

(i) kâ¡F 5 ».Û ntf¤Âš br‹whš gaz neu« 24 kâ MF«.

(ii) F¿¥Ã£l öu¤ij 40 kâ neu¤Âš fl¡f, mtUila ntf« 3 ».Û / kâ

MF«.

rk‹ghL mik¤jš .... (1 kÂ¥bg©)òŸëfis¡ F¿¤J tisÎ tiujš .... (5 kÂ¥bg©fŸ)x, y m¢R¡fŸ tiujš, k‰W« msÎ¤Â£l« .... (2 kÂ¥bg©fŸ)Ô®Î fz« .... (2 kÂ¥bg©fŸ)


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