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Guess Paper – 2012
Class – XII
Subject – Mathematics
MATRICES-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 (adj A) = 10
0

0,
10

then find |A|
1
 
3. If A= 2  3  1andB   2  findAB
 3
 
4. If f(x) = [ x ] and g (x) = │x │, then evaluate (f o g) (1/2) – (g o f) (-1/2)
5. How many relations can be defined from a non- empty sets A to B if n(A)=2and n(B)=3
6. A matrix A of order 3 x 3 has determinant 7, what is the value of |3A|?
7. If A is a square matrix such that A² = A, then find (I + A)² - 3A.
2x

2x  1 is a skew symmetric, find value of x.
3x  8 x  8
0 

0
8. If 1  2x
x2
0
1
9. If the function f : R  R defined by f ( x)  2 x 3  7 , find the inverse of f and f- (23) .
10. If * be a binary operation defined on the set of integers as a*b=a+b+1 for a, b € I, find inverse element.
 5 2
3 6 
andX  Y  
 ,Find X and Y
0 9
 0  1
11. If X+Y= 
12. If A is 3x4 matrix and B is a matrix such that A T B and BA T are both defined, find order of mat B.
13. For what value of x if
2
x
3
3

x
1
6 3
14. If A is a square matrix of order 3 such that adj A = 25, find A
15. If f(x) = 8x3 and g(x) = x⅓ , find g o f.
 0 a b


16. Show that the matrix A=   a 0 c  is Skew-symmetric.
  b  c 0


SECTION-B
11.By using properties of determinants show that
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abc
2a
2a
2b
bca
2b
2c
2c
cab
12. Prove by using properties of determinant
 ( a  b  c)3
b 
a
2
1  bc  , then prove that aA-1 = a  bc  1 I  aA
 c
a 

13. If A = 

2  4
1  1 0 
 2



14. If A = 2 3 4 and B=  4 2  4 ,Find BA and hence solve equations x-y =3, 2x +3y +4z =17,y+2z =7




0 1 2
 2  1 5 
15.Let A =NXN and Let * be a binary operation on A defined by (a,b)*(c,d) = (ad+bc,bd) for all(a,b),(c,d) € NXN.
Determine if * is commutative, associative. Also find the identity and inverse if they exist.
1 a a 2  bc
1 b b 2  ca = 0
16. Using properties of determinants show that
1 c
1 2 3
17. Let A = 1 2 1 , B =


2 1 3
c 2  ab
1  1 0 
0 1 1 Verify that ( AB)T = BT AT.


2 1 3
(b  c) 2
a2
a2
b2
(c  a) 2
b2
c2
c2
(a  b) 2
18 Using properties of determinants, show that
19. If a, b, c are in AP, show that
= 2abc(a+b+c) 3
=0.
a
ab
20. By using properties of determinants show that: 2a 3a  2b
a bc
4a  3b  2c  a 3 .
3a 6a  3b 10a  6b  3c
21. Using properties of determinants, prove that:
  8 5
 , satisfies the equation x² + 4x – 42 = 0.Hence find A-1.
2
4


22. Show that A = 
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23.Using properties of determinants, prove that:
a  b  2c
a
b
c
b  c  2a
b
c
a
c  a  2b
 2(a  b  c) 3
x  3 x  4 x  5a
24.If a,b,c are in A.P then find the value of the determinant x  4
x  5 x  5b
x  5 x  6 x  5c
-a b -a c
3a
25. Prove using properties that - b  a
-ca
3b
-cb
- b  c = 3(a+b+c)(ab+bc+ca) .
3c
26. Show that the function F: R - {-1} R - {-1}, given by f(x) =
x3
is a bijection.Also find f 1
x 1
0 0 
n
, then by induction show that aI  bA  a n I  n.a n 1 .bA , if a& b are constants.

0 1 
27. If A= 
a  bx c  dx
28
. Prove that : ax  b cx  d
u

p  qx
a
c
p
px  q = (1 – x ) b d
q
2

u 

.
29.
 2 1 2 


30. Express A =  3 4
0  as sum of a symmetric and skew symmetric matrix.
 0  3  2


SECTION-C (6x6=36 Marks)
2  3 5 


31. If A =  3 2  4  find A-1 and hence solve the equations: 2x – 3y + 5z = 11, 3x + 2y – 4z = -5, x + y - 2z = - 3.
 1 1  2


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1 2
1
32. If A =  1  2 1 find A-1. Hence solve the equations: x – y + z = 4,


 1  2 3
33. Show that
(y  z) 2
xy
xy
(x  z)
xz
yz
x

34. Prove that  y
z

x2
y2
z2
x – 2y – 2z = 9, 2x + y + 3z = 1
zx
2
yz
= 2 xyz (x + y + z) 3
(z  y) 2
1  px 3 

1  py 3   1  pxyz x  y  y  z  z  x  x  y  z
1  pz 3 
35. Consider f : R+ --> [-5, ∞) given by f(x) = 9x² + 6x – 5. Show that f is bijective. Also find the inverse of f.
 3 1 2 
36. Using elementary row transformation find the inverse of the matrix  2 0 1 .


 3 5 0 
2
6
4
2 3 10
6 9 20
4 5 5
37. A= 3
9
 5 . findA 1 andhencesolve  
 4,  
 2,    1
x y z
x y z
x y z
10  20 5
38. Prove using properties that
39.Using matrices solve system of equations:x+3y+4z = 8,2x+y+2z =5,5x+y+z =7
40. Let f : N → R be a function defined as f(x) = 4x2 + 12 x + 15. Show that f : N → S where, S is the range
of f , is invertible. Find f -1.
 5 0 4
1 3 3 




1
41. If A =  2 3 2 andB  1 4 3 compute AB 1
1 2 1
1 3 4 




Paper Submitted By:
Name
ESTHER P
Email
[email protected]
Phone No. 0505342999
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