# SAMPLE PAPERS

MATHEMATICS

CLASS- XII

Matrices

1. Find a, b, c when f(x) = ax2 + bc + c, f(2) = 11 and f(-3) = 6  =  f(0) Determine the quadratic function f(x) and find its value when x = 1.

2.         Using determinants solve the following system of equations :

(i) 2x – 4y = -3                        (b) 4x + 3y = 3

4x + 2y = 9                                8x – 9y = 1.

3.         Solve the following system of equations using Cramer’s rule :

(i) x + 2y = 1                           (b) 9x + 5y = 10

3x + y = 4                                  3y – 2x = 8

1. Solve the following system of equations by using Cramer’s rule :

(a) x + y + z = 6                      (b) 3x + y + z = 10

x – y + z = 2                            x + y – z   = 0

2x + y – z = 1                               5x – 9y = 1

(c) 2x – y + 3z = 9                  (d) 3x + y + 2z   = 3

x + y + z = 6                             x + y – z   = -3

x – y + z = 2                             x – 2y + z = 4

1. Solve the following system of equations by using Cramer’s rule :

(a) x – y + z – 4 = 0                (b)        x + y + z = 1

2x + y – 3z = 0                        3x + 5y + 6z = 4

x + y + z – 2 = 0                       9x + 2y – 36z = 17

1. Solve the following system of equations by using Cramer’s rule :

(a)   5t – s + 4u = 5                 (b)      x + y + z + w  = 1

2t + 3s + 5u = 2                        x – 2y + 2z + 2w = -6

5t – 3s + 6u = -1                       2x + y + 2z – 2w = -5

3x – y + 3z – 3w = -3.

1. Adjoint of a Square Matrix : The adjoint of a square matrix is the transpose of the matrix obtained by replacing each element of A by its co-factor in | A |.

1. Theorem : If A be any n-rowed square matrix : then  (Adj. A) A = A(Adj. A) = | A | ln where ln is the n-rowed matrix.

1. For the following matrix A ; prove that

1          -1         1

A =  2          3          0

18        2          10

1. Find the adjoint of the matrix

1          0          -1

A =   3          4          5

0          -6         -7

1. Singular Matrix : A square A is called a singular matrix of a non-singular matrix according as | A |  or  | A | ¹ 0, respectively.

1. Theorem: If A, B, be two n-rowed non-singular matrices, then A B is also non-singular and  (AB) –1 = b –1 A –1 i.e. the inverse of a product is the product of the inverses taken in the order.

3     8

1. Let A be the matrix                   Find A –1  and verify that A –1  = 1/13  A – 4/13 I

2     1

where I is 2 ´ 2 unit matrix.

3     1                      4      0

14.       If             A  =                and B =                 verify that (AB) –1 = B –1 A –1

4     0                      2      5

1      2

15.       Find the adjoint of the matrix   A =                  and verify A (Adj.A) A = | A | I2

3       -5

a          b

16.       If          A  =                     , find Adj. A.

c          d

2          -3

17.       Given  A =                       , compute A –1 and show that  2A –1 9I – A.

-4         7

1          0          0

18.       Find Adj. A and A –1, if it exits where     A =   3          3          0

5          2          -1

1          -1         1

19.       If  A =  2          -1         0   , find A2 and show that A2 = A –1

1          0          0

3          -1                     2          1

20.       If  A =  -4         0    and  B =    -1         -2    . Find (A’B) –1

2          1                      1          1

1       2       5

21.       Compute the inverse of the matrix  A =    2      3       1   and verify that A-1 A = 1

-1       1       1

1   2   2

22.       Let       A = 2   1    2  . Prove that A2 – 4A – 5I = 0, Hence obtain A –1

2   2    1

2          0          -1

23.       If  A =  5          1          0   Prove that A –1 = A2 – 6A + 11I.

0          1          3

-4         -3         -3

24.       If          A  =    1          0          1    Show that Adj. A = A

4          4          3

1    1    1

24.       If  A =  1    2   -3     Verify the theorem A (Adj. A) = (Adj. A ) A = | A | I.

2   -2    1

1     -2      3

25.       Find A (Adj. A) for the matrix   A  =     0      2     -1

-4      5      2

1. Compute the inverse of each of the following matrices.

1          2          3                      cos q   -sin q   0

(i)         2          3          2          (ii)        sin q     cosq    0

3          3          4                      q          0          0

1. Verify that (A B) –1= B –1 A –1 for the matrices A and B

2       1                     4      5

Where A =                  and B =

5       3                     3      4

2       0                    0      1

28.       Where A =                  and B =                               Verify that  (AB) –1= B –1 A –1

5       3                     2      4

2     5

29.       If          A =                , find A-1  and verify that A –1 = -1/7 A + 8/7 I.

1        6

1     1     2                      1        2          0

30.       If  A =  1     9     3     and  B  =  1        3          -1    , verify that  (AB) –1 = B –1 A –1

1     4     2                      1        -1         3

4      5

31.       If          A =               then, show that  A – 3I = 2[I + 3A –1]

2      1

1. Find the inverse of each of the following matrices and verify : A –1 A = I

2          0          -1                     2          3          1

(i)         5          1          0          (ii)        3          4          1

0          1          3                      3          7          2

-8         1          4

33.       (a) If  A = 1/9   4          4          7     Prove that A –1= A’.

1          -8         4

0          -1         2               0       1

(b) Given A =                               , B –1    1       0

2          -2         0               1       1

From the product C = AB and find C –1. What is the matrix BA?

cos x   -sin x   0

34.       (a) If F(x) =     sin x     cos x   0

0          0          0

then show that F(x)F(y) = F(x + y), Hence prove that [F(x)] –1 = F(-x).

5          0          4                      1          3          3

(b) Given A = 2          3          2  , B –1 =        1          4          3    compute (AB) -1

1          2          1                      1          3          4

cos a   sin a

35.       If A =                           , verify that   (i) (A –1) –1 = A           (ii) (A’) –1  = (A –1)’

sin a    cos a