14013revision Mat Det Fns 2011 12

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Sample Paper 2012

Class XII

Subject MathematicsMATRICES-DETERMINANTS-FNS-RELNS

SECTION-A

1. How many matrices of order 3 x 3 are possible with each entry as 0 or 1?

2. For any 2 x 2 matrix A, if A (ad A! " 100010

, then find #A#

3. $f A"( ) findABandB

= 32

1

132 %. $f f(x! " & x ' and (x! " )x ), then e*al+ate (f o ! (1 2! - ( o f! ( 1 2!

/. How many relations can be defined from a non empty setsA to if n(A!"2and n( !"3

. A matrix A of order 3 x 3 has determinant , what is the *al+e of #3A#?

. $f A is a s +are matrix s+ch that A4 " A, then find ($ 5 A!4 3A.

6. $f0 x 2 2 x

1 2x 0 2x 1

3x 8 x 8 0

+

is a s7ew symmetric, find *al+e of x.

8.$f the f+nction f 9R :R defined by 2!( 3 += x x f , find the in*erse of f and f 1 (23! .

10. $f ; be a binary operation defined on the set of inte ers as a;b"a5b51 for a, b < $, find in*erse element.

11. $f =5>"

=

10

3

80

2/Y andX ,Find = and >

12. $f A is 3x% matrix and is a matrix s+ch that AT and A T are both defined, find order of mat .

13. For what *al+e of x if 31

33

2

=

x x

1%. $f A is a s +are matrix of order 3 s+ch thatAadj " 2/, find A1/. $f f(x! " 6x3 and (x! " x , find o f.

1 . @how that the matrix A"

ba

0

c

a

0

0

c

b

is @7ew symmetric.

SECTION-B

11. y +sin properties of determinants show that

!. ro*e by +sin properties of determinant 3!(22

22

22

cbabaccc

bacbbaacba

++=

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13. $f A " +

a

bcc

ba1 , then pro*e that aA1 "( aA I bca ++ 12

1%. $f A "

210

%32

011

and "

/12

%2%

%22

,Find A and hence sol*e e +ations x y "3, 2x 53y 5%B "1 ,y52B

"1/.Cet A "D=D and Cet ; be a binary operation on A defined by (a,b!;(c,d! " (ad5bc,bd! for all(a,b!,(c,d! < DEetermine if ; is comm+tati*e, associati*e. Also find the identity and in*erse if they exist.

1 . sin properties of determinants show thatabcc

cabb

bcaa

2

2

2

1

1

1

" 0

1 . Cet A "312

121

321

, "

312

110

011

Gerify that ( A ! " A .

16 sin properties of determinants, show that222

222

222

b!(acc

ba!(c b

aac!(b

++

+ " 2abc(a5b5c!3

18. $fa, b, c are in A , show that "0.

20. y +sin properties of determinants show that9 32 3 2 % 3 2 .3 3 10 3

a a b a b c

a a b a b c a

a a b a b c

+ + ++ + + =+ + +

21. sin properties of determinants, pro*e that9

22. @how that A "

%2

/6

, satisfies the e +ation x4 5 %x - %2 " 0.Hence find A1

.

23. sin properties of determinants, pro*e that9 3!(22

2

2

cbabacac

bacbcbacba

++=++

++++

2%.$f a,b,c are in A. then find the *al+e of the determinant/cxx/x

/bx/x%x

/ax%x3x

+++++++++




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2/. ro*e +sin properties that3c bcac

c b3ba b

ca ba3

++

++++a

" 3(a5b5c!(ab5bc5ca! .

2 . @how that the f+nction F9 I J 1K I J 1K, i*en by f(x! "13

+

x x is a bi ection.Also find f 1

2 . $f A"10

00, then by ind+ction show that( ) bAan I abAaI nnn .. 1+=+ , if aL b are constants.

26. ro*e that 9 u

q pxd cxbax

qx pdxcbxa+++

+++

" (1 - x2! uqd b pca

.

28.

30. Mxpress A "

230

0%3

212

as s+m of a symmetric and s7ew symmetric matrix.

SECTION-C "#$#% M'()s !

31. $f A "

211

%23

/32

find A1 and hence sol*e the e +ations9 2x - 3y 5 /B " 11, 3x 5 2y - %B " /, x 5 y 2B

32. $f A "

321

121

211

find A1. Hence sol*e the e +ations9 x - y 5 B " %, x - 2y - 2B " 8, 2x 5 y 5 3B " 1

33. @how that2

2

2

y!(ByBxB

yBB!(xxy

BxxyB!(y

++

+ " 2 xyB (x 5 y 5 B!3

3%. ro*e that( )( )( )( ) x z z y y x pxyz

pz z z

py y y px x x

+=+++

1

1

11

32

32

32

z y x 3/. Nonsider f 9 I 5 O & /, P! i*en by f (x! " 8x4 5 x - /. @how that f is bi ecti*e. Also find the in*erse of f.

3 . sin elementary row transformation find the in*erse of the matrix3 1 2

2 0 1 .

3 / 0

3 . A" 1//%,2208,%1032./2010

/83

%21 =+=+=++

z y x z y x z y xlveandhenceso findA




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36. ro*e +sin properties that38. sin matrices sol*e system of e +ations9x53y5%B " 6,2x5y52B "/,/x5y5B "%0. Cet f 9 D Q I be a f+nction defined as f(x! " %x2 5 12 x 5 1/. @how that f 9 D Q @ where, @ is the ran e of f , is in*ertible. Find f1.

%1. $f A " ( ) 11%31

3%1

331

121

232

%0/

=

ABcomputeandB

*'pe( Subm+tte, B :

N'me Esthe( p(' ee

Em'+0 esthe(.p1oo,2gemse,u.com

*ho e No. 34&&43 5



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