TNPSC MATHS IMPORTANT FORMULASappolosupport.com/wp-content/uploads/2019/08/IMPORTANT... · 2019. 8....
Transcript of TNPSC MATHS IMPORTANT FORMULASappolosupport.com/wp-content/uploads/2019/08/IMPORTANT... · 2019. 8....
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TNPSC – MATHS IMPORTANT FORMULAS
S.No Name of the figure (ngah;)
Figure (glk;)
Area (Sq. Units) gug;gsT (r.myF)
Perimetre (P) Rw;wsT
1. Triangle Kf;Nfhzk;
1
2 x b x h AB + BC + CA
2. Right triangle Nrq;Nfhz Kf;Nfhzk;
1
2 x b x h
(base + height + hypotenuse)
3.
Equilateral triangle rkgf;f
Kf;Nfhzk;
3
4 a2 where
( 3 = 1.732)
AB + BC + CA= 3A; Altitude ,
h = 3
2a units
4.
Isosceles triangle
,U rkgf;f Kf;Nfhzk;
h x 2 2a h 2a + 2 2 2a h
5.
Scalene triangle
(mrkgf;f Kf;Nfhzk;)
( )( )( )s s a s b s c
Where a = 2
a b c
AB + BC + CA = (a + b + c)
6. Quadrilateral
(ehw;fuk;)
1
2 x d x (h1 + h2) AB + BC + CD + DA
7. Parallelogram (,izfuk;)
b x h 2 x (a + b)
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8. Rectangle (nrt;tfk;)
l x b 2 x (l + b)
9. Trapezium
(rhptfk;)
1
2 x h x (a + b) AB + BC + CD + DA
10. Rhombus (rha;rJuk;)
1
2 x d1 x d2 where
d1, d2 are diagonals
(%iytpl;lk;)
4a
11. Square
a2 4a
Square (%iytpl;lk;) (d) (Diagonal length) = 2a
No.
Name of the figure
figure Surface
area tisgug;G
Total surface area
Volume
1. Right circular
cylinder(Neh;tl;l jpz;k cUis)
2πrh 2πr(h + r) πr2h
2.
Right circular hollow
cylinder(Neh;tl;l cs;sPlw;w cUis)
2πh(R + r) 2π(R + r) (R – r + h)
πR2h – πr2h =
πh(R2 – r2) = πh(R + r)(R – r)
3. Right circular cone (Neh;tl;l jpzkf; $k;G)
πrl πr(l + r) 21
3r h
4. Frustum of a
cone (,ilf;fz;lk;)
---------
-------- 2 21 ( )
3h R r Rr
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5. Solid sphere
(jpz;kf;Nfhsk;)
4πr2 ______ 34
3r
6. Hollow Sphere
(cs;sPlw;w Nfhsk;)
-------
______ 3 34 ( )
3R r
7.
Solid hemisphere
(jpz;k miuf;Nfhsk;)
2πr2 3πr2
32
3r
8.
Hollow Hemisphere (cs;sPlw;w
miuf;Nfhsk;)
2π(R2 + r2) 2π(R2 + r2) + π(R2 - r2) =
π(3R2 + r2)
gad;gLj;jg;gl;l cNyhfj;jpd;
fd msT = 3 32 (
3R r )
9. tl;lf;Nfhzg;gFjp $k;ghf khw;wg;gLfpwJ
tisgug;G = tl;lf;Nfhzg;gFjpapd; gug;G
2
360rl r
tpy;ypd; ePsk;( l ) = $k;gpd; mbr;Rw;wsT(2πr)
10. Volume of water flows out through a pipe = (Cross section area x Speed x Time) Foha; topNa ghAk; jz;zPh; fd msT = (FWf;F ntl;Lg; gug;G X Ntfk; X Neuk;)
11.
No. of new solids obtained by recasting = Volumeof thesolid whichismelted
Volumeof onesolid whichismade
cUf;fp jahhpf;fg;gLk; Gjpa fd cUtq;fspd; vz;zpf;if
= ; ; ; ; ;
; ; ; ; ;
cUffggll fd cUtjjpd fdmsT
cUthffggll xU fd cUtjjpd fdmsT
12. Conversions: 1 m3 = 1000 litres, 1 d.m3 = 1 litre, 1000 cm3 = 1 litre, 1000 litres = 1 kl 1 kP3 = 1000yp> 1nlrp kP3 = 1yp> 1000 nr.kP3 = 1yp> 1000yp = 1 fp.yp
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CUBE (fdr;rJuq;fs;)
CUBIOD (fdr; nrt;tfk;)
tl;lf;Nfhzg;gFjpapd; tpy;ypd; ePsk; xU tl;lf;Nfhzg;gFjpapd; ikaf;Nfhzk; kw;Wk; Muk; r vdpy;>
tpy;ypd; ePsk; 2360
l r
myFfshFk;.
tl;lf;Nfhzg;gFjpapd; gug;gsT xU tl;lf;Nfhzg;gFjpapd; ikaf;Nfhzk; kw;Wk; Muk; r vdpy;>
tl;lf;Nfhzg;gFjpapd; gug;gsT 2360
r
rJu myFfshFk;.
tl;lf;Nfhzg;gFjpapd; gug;gsT = 2
lr rJu myFfs;
tl;lf;Nfhzg;gFjpapd; Rw;wsT tpy;ypd; ePsk; l, tl;lf;Nfhzg;gFjpapd; Muk; r vdpy;> mjd; Rw;wsT
P=l + 2r myFfs;.
Length of Arc If is the central angle and r is the radius of a sector, then its arc length
is given by 2360
l r
units.
Area of a Sector If is the central angle and r is the radius of a sector, then the area of
the sector is 2360
r
square units.
Area of sector = 2
lr square units.
Key concept:
Volume of a cube (fd msT )= 3a cubic units The Total surface area of a cube (T.S.A) (nkhj;jg;gug;G)
= 26a square units. The Lateral surface area of a cube (L.S.A) (gf;fg;gug;G)
= 24a square units. diagonal = 3a
Volume of a cuboid (fd msT )= l b h cubic units
The Total surface area of a cuboid (nkhj;jg; gug;G) = 2( )lb bh lh sq.units.
The Lateral surface area of a cuboid (gf;fg;gug;G) = 2( )l b h sq.units.
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2 2 ( )( )a b a b a b 2 2 2( ) 2a b a b ab
2 2 2( ) 2a b a b ab 2 2( ) ( ) 4a b a b ab
3 3 2 2( ) ( )( )a b a b a ab b 3 3 2 2( ) ( )( )a b a b a ab b
3 3 3 2 2 3 3( ) 3 3 3 ( )a b a b a b ab a b ab a b 3 3 3 2 2 3 3( ) 3 3 3 ( )a b a b a b ab a b ab a b
2 2 2 2( ) 2( )a b c a b c ab bc ca 3 3 3 2 2 23 ( )( ) a b c abc a b c a b c ab bc ca
If a + b + c = 0 then 3 3 3 3 a b c abc
2( )( ) ( )x a x b x a b x ab 3 2( )( )( ) ( ) ( )x a x b x c x a b c x ab bc ca x abc
n(n+1)1 + 2 + 3 + ...... + n =
2
2
2 Last Number+11 3 5 ........ (2 1)2
n n
2 2 2 2 ( 1)(2 1)1 2 3 ......6
n n nn
2 2 2 2 (2 1)(2 1)1 3 5 ...... (2 1)3
n n nn
2
3 3 3 31
1 2 3 .......2
n nn
2 2 2 2 -n(n+1)1 2 3 4 ......... up to 'n' terms =2
2 1
2 4 6 ....... 2 2 1 2 3 ..... 12
n nn n n n
Logarithms Rule i) Product rule:ngUf;fy; tpjp: loga (mn) = loga m + loga n
ii) Quotient rule:tFj;jy; tpjp: loga m
n
= loga m - loga n.
iii) Power rule: gb tpjp: loga mn = nloga m. iv) loga 1 = 0. & v) loga a = 1
Perimeter of a Sector If l is the arc length and r is the radius of a sector, then its perimeter P
is given by the formula P=l + 2r units.
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APPRECIATION AND DEPRECIATION [cah;T kw;Wk; tPo;r;rp] kf;fs; njhif cah;T> thfdq;fspd; kjpg;gPL Nghd;w gioa nghUl;fspd; tpiy
eph;zaj;jpid fPNo Fwpg;gpl;l tpjpiag; gad;gLj;jpf; fhzyhk;.
The growth or Appreciation (A): After ‘n’ years
kf;fs; njhif tsh;r;rp (m) tpiyNaw;wk; n Mz;LfSf;F gpwF 1100
nr
A p
After ‘n’ years Depreciation tPo;r;rp 1100
nr
D p
The growth or Appreciation (A): Before ‘n’ years or ‘n’ years ago
kf;fs; njhif tsh;r;rp (m) tpiyNaw;wk; n Mz;LfSf;F Kd;G
1100
n
pA
r
Before ‘n’ years or ‘n’ years ago Depreciation tPo;r;rp n Mz;LfSf;F Kd;
1100
n
pD
r
Arithmetic Progression (A.P.) [$l;Lj;njhlh; ] General form of an A.P. is a, a + d, a + 2d …with first term a, and
Common Difference = d
The general term or the nth term of an A.P. is tn = a + (n – 1) d →
1l a
nd
Sum to n terms of an A.P / Sn = 2 ( 1)2
na n d
Sn = 2
n [a + a + (n-1)d] or Sn = 2
n [a + l]
Where l = tn = a + (n – 1) d = last term
GEOMETRIC PROGRESSION[ngUf;Fj;njhlh; thpir]
ngUf;Fj;njhlh; thpirapd; nghJ tbtk; a, ar, ar2, ar3, … a ≠ 0 ngUf;Fj;njhlh; thpirapd; n - Mk; cWg;G tn = arn-1 Sum of n terms of a G.P[xU ngUf;Fj;njhlhpd; n cWg;Gfspd; $Ljy;]
Sn = ( 1)
1
na r
r
The sum of infinite geometric series[Kbtwh ngUf;Fj;njhlh; njhFg;gpd; $Ljy;]
S= 1
a
r
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MATHS FORMULAS
For the system of equations
1 1 1
2 2 2
0
0
a x b y C
a x b y c
where 2 2 2 21 1 2 20, 0a b a b
If 1 2 1 2 0a b b a or 1 1
2 2
a b
a b then the system of equations has a unique solution
If 1 1 12 2 2
a b c
a b c then the system of equations has infinitely many solutions
If 1 1 12 2 2
a b c
a b c then the system of equations has no solution
1 1 1
2 2 2
0
0
a x b y C
a x b y c
,q;F 2 2 2 21 1 2 20, 0a b a b Mfpa rkd;ghLfspd; njhFg;gpw;F
1 2 1 2 0a b b a mjhtJ 1 1
2 2
a b
a b vdpy; xNunahU jPh;T (unique solution) cz;L
1 1 1
2 2 2
a b c
a b c vdpy; Kbtpyp vz;zpf;ifapy; jPh;Tfs; (infinitely many solutions)
cz;L
1 1 1
2 2 2
a b c
a b c vdpy; jPh;T VJkpy;iy (no solution)
The Basic relationship between the zeros and the coefficients of 2( )p x ax bx c
sum of zeros: 2
b coefficient of xa
a coefficient of x
product of zeros 2
tanc cons t terma
a coefficient of x
2( )p x ax bx c -d; nfOf;fSf;Fk;> g+r;rpaq;fSf;Fk; ,ilNaahd mbg;gilj; njhFg;G
G+r;rpaq;fspd; $Ljy;> 2
b xa
a x
d; nfO
d; nfO
G+r;rpaq;fspd; ngUf;fw;gyd;
2
ca
a x
khwpyp cWg; G
d; nfO
Nature of roots of a quadratic equation
The roots of the equation 2 0ax bx c are given by 2 4
2
b b acx
a
If 2 4 0b ac we get two distinct real roots 2 4
2
b b acx
a
; &
2 4
2
b b acx
a
If 2 4 0b ac , then the equation has two equal roots 2
bx
a
If 2 4 0b ac , then 2 4b ac is not a real number. Therefore there is no real root
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for the given quadratic equation.
,Ugbr; rkd;ghl;bd; %yq;fspd; jd;ik
2 0ax bx c vDk; ,Ugbr; rkd;ghl;bd; %yq;fs; 2 4
2
b b acx
a
vd mwpNthk;
2 4 0b ac vdpy; ,U ntt;Ntwhd nka;naz; %yq;fs; cs;sd. mitfs;>
2 4
2
b b acx
a
kw;Wk;
2 4
2
b b acx
a
2 4 0b ac vdpy; rkd;ghl;bw;F ,U rkkhd nka;naz; %yq;fs; cs;sd. rk
%yk; 2
bx
a
MFk;
2 4 0b ac vdpy; 2 4b ac xU nka;naz; my;y. Mifahy;> ,Ugbr; rkd;ghl;bw;F nka;naz; %yq;fs; ,y;iy
Therefore, if ,a are the roots of
2 0ax bx c then
the sum of the roots b
aa
the product of roots, c
aa
2 0ax bx c -d; %yq;fs; ,a vdpy;
%yq;fspd; $Ljy;> b
aa
%yq;fspd; ngUf;fw;gyd;> c
aa
Area of a quadrilateral ehw;fuj;jpd; gug;G
1 2 2 3 3 4 4 1 2 1 3 2 4 3 1 4
1{( ) ( )}
2x y x y x y x y x y x y x y x y sq.units / r. myFfs;
The distance between 1 1 2 2( , ), ( , )P x y Q x y is 2 2
2 1 2 1( ) ( )x x y y
1 1 2 2( , ), ( , )P x y Q x y Mfpa Gs;spfSf;F ,ilNaAs;s njhiyT 2 2
2 1 2 1( ) ( )x x y y
Area of Triangle Kf;Nfhzj;jpy; gug;G If 1 1 2 2( , ), ( , )A x y B x y and 3 3( , )C x y are the vertices of a ABC then the area of the
ABC is 1 2 3 2 3 1 3 1 2
1{ ( ) ( ) ( )}
2x y y x y y x y y sq.units
1 1 2 2( , ), ( , )A x y B x y kw;Wk; 3 3( , )C x y Mfpatw;iw Kidfshff; nfhz;l ABC -d;
gug;G 1 2 3 2 3 1 3 1 2
1{ ( ) ( ) ( )}
2x y y x y y x y y r. myFfs;
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Equation of straight lines (Neh;f;Nfhl;bd; rkd;ghLfs;) Straight Line Equation
x – axis y = 0
y – axis x = 0
Parallel to x - axis y = k
Parallel to y – axis x = k
Parallel to ax + by + c = 0 ax + by + k = 0
Perpendicular to ax + by + c = 0 bx – ay + k = 0
Given Equation
Passing through the origin y mx
Slope m, y-intercept c y mx c
Slope m a point 1 1( , )x y 1 1( )y y m x x
Passing through two points 1 1 2 2( , ), ( , )x y x y 1 12 1 2 1
y y x x
y y x x
x- intercept a and y-intercept b 1
x y
a b
Neh;f;Nfhl;bd; rkd;ghLfs;
Neh;f;NfhL rkd;ghL x – mr;R y = 0 y – mr;R x = 0
x – mr;rpw;F ,iz y = k
y – mr;rpw;F ,iz x = k ax + by + c = 0 f;F ,iz ax + by + k = 0 ax + by + c = 0 f;F nrq;Fj;J bx – ay + k = 0
nfhLf;fg;gl;lit rkd;ghL Mjp topr; nry;Yk; Neh;f;NfhL y mx
rha;T m kw;Wk; y-ntl;Lj;Jz;L c y mx c
rha;T m kw;Wk; xUGs;sp 1 1( , )x y 1 1( )y y m x x
1 1 2 2( , ), ( , )x y x y Mfpa ,U Gs;spfs; topr; nry;Yk; Neh;f;NfhL
1 1
2 1 2 1
y y x x
y y x x
x- ntl;Lj;Jz;L a kw;Wk; y-ntl;Lj;Jz;L b 1
x y
a b
,g;nghOJ (x1, y1) kw;Wk; (x2, y2) Mfpa ,UGs;spfis ,izf;Fk; Nfhl;Lj;Jz;bid m : n vd;w nfhLf;fg;gl;l tpfpjj;jpy; cl;Gwkhfg; gphpf;Fk; Gs;spapd; Maj;njhiyTj; J}uq;fisf; fhz;Nghk;.
To find the coordinates of the point which divides internally the line segment joining two given points (x1, y1) and (x2, y2) in the given ratio m : n
2 1 2 1,mx nx my ny
m n m n
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AB I ntspg;Gwkhf m : n (m > n) vd;fpw tpfpjj;jpy; gphpf;Fk; Gs;sp D ,d; Maj;njhiyTj;J}uk;
Hence the point which divides AB externally in the ratio m : n (m > n) is given by
2 1 2 1,mx nx my ny
m n m n
Middle Point Formula (or) Mid – Point Formula ikag;Gs;sp #j;jpuk; my;yJ eLg;Gs;sp #j;jpuk;
1 2 1 2,2 2
x x y y
1 1 2 2( , ),( , )x y x y kw;Wk; 3 3( , )x y Mfpa cr;rpg; Gs;spfisf; nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;Nghk;.
xU Kf;Nfhzj;jpw;F %d;W eLf;NfhLfs; cz;L. mit G vd;fpw Gs;spapy; re;jpf;Fk;. me;jg;Gs;sp, xU Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; (Centroid) vdg;gLk;.
We are now able to find the coordinates of the centroid of the triangle whose vertices are the given points 1 1 2 2( , ),( , )x y x y and 3 3( , )x y .
There are three medians of a triangle and they are concurrent at a point G, called the centroid of the triangle.
1 2 3 1 2 3,3 3
x x x y y y
LCM & HCF
LCM of fractions = min
LCM of numerotors
HCF of deno ators
njhFjpapYs;s vz; fspd; kP.rp.k
gpd; d vz; fspd; k.P rp.kgFjpapYs;s vz; fspd; kP.ng.t
=
HCF of fractions = min
HCF of numerotors
LCMof deno ators
njhFjpapYs;s vz; fspd; kP.ng.tgpd; d vz; fspd; k.P ng.t
gFjpapYs;s vz; fspd; kP.rp.k=
First number second number = LCM x HCF ,uz;L vz;fspd; ngUf;fw;gyd; = LCM x HCF
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STATISTICS Range and Coefficient of Range
Range (tPr;R) = L – S
Coefficient of Range (tPr;R nfO) = L SL S
Quartiles deviation (Q.D.)and Coefficient of Quartile deviation
Quartiles deviation (fhy;khd tpyf;fk;) = 3 12
Q Q
Coefficient of Quartile deviation (fhy;khd tpyf;ff;nfO) 3 13 1
Q Q
Q Q
Mean deviation and Coefficient of Mean deviation
Mean deviation (ruhrhp tpyf;fk;) = Dn
Coefficient (ruhrhp tpyf;ff;nfO) = Median or Mode
Mean deviation
Mean or
RATIO AND PROPORTION In a Proportion: Product of Extremes = Product of Means (tpfpjrkj;jpy;
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Standard deviation (jpl;ltpyf;fk;)
n
2d= d = x - x
Variance (khWghL) = 2
Remarks: It is quite interesting to note the following: The S.D. of any n successive terms of an A.P. with common difference d is,
2 1
12
nd
Thus,
S.D. i, i+1, i+2, ….., i+n is 2 1
12
n
, i
S.D. of any n consecutive even integers, is given by 2 1
2 ,12
nn
S.D. of any n consecutive odd integers, is given by 2 1
2 ,12
nn
Mutually exclusive events: (xd;iwnahd;W tpyf;Fk; epfo;r;rpfs;) ( ) ( ) ( )P A B P A P B
Not Mutually exclusive events: (xd;iwnahd;W tpyf;fh epfo;r;rpfs;) ( ) ( ) ( ) ( )P A B P A P B P A B
Dimensional Diagrams [tpsf;fg; glq;fspd; tiffs;;] One-dimensional diagrams [xU ghpkhd tpsf;fg; glq;fs;] Line Diagram [Nfhl;L tpsf;fg;glk;] Simple Diagram [rhjhuz gl;il tpsf;fg;glk;] Multiple Bar Diagram [gy; mq;fg; gl;il tpsf;fg;glk;] Sub-divided Bar Diagram [$W gl;il tpsf;fg;glk; (gFjp gl;il)] Percentage Bar Diagram [rjtPj gl;il tpsf;fg; glk;]
Two-dimensional diagrams [,U ghpkhd tpsf;fg;g glq;fs;] Rectangles [nrt;tfq;fs;] Squares [rJuq;fs;] Pie-diagrams [tl;l tpsf;fg; glq;fs;]
Three-dimensional diagrams [Kg;ghpkhd tpsf;fg; glq;fs;] Cubes, Cylinders, Spheres, Prisms, Pyramids, etc…,
Graphs: [tiuglq;fs;] Histogram [guty; nrt;tfg; glk;] – To find Mode Frequency Polygon [epfo;ntz; gy Nfhz tbtk;] Frequency Curve [epfo;ntz; tisNfhL] Ogive [tsh; epfo; tistiufs; (xift;)] - To find Median Lorenz Curve [yhud;]; tistiu] – Study about dispersion(Variability)