#### unit test

12th Standard EM

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Maths

HFTRGRTGGTH
Time : 00:03:00 Hrs
Total Marks : 90
DSWEFDCSXX
51 x 1 = 51
1. If |adj(adj A)| = |A|9, then the order of the square matrix A is

(a)

3

(b)

4

(c)

2

(d)

5

2. If A is a 3 × 3 non-singular matrix such that AAT = ATA and B = A-1AT, then BBT =

(a)

A

(b)

B

(c)

I

(d)

BT

3. If A = $\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right]$, B = adj A and C = 3A, then $\frac { \left| adjB \right| }{ \left| C \right| }$

(a)

$\frac { 1 }{ 3 }$

(b)

$\frac { 1 }{ 9 }$

(c)

$\frac { 1 }{ 4 }$

(d)

1

4. If A = $\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right]$ and B = $\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right]$ then |adj (AB)| =

(a)

-40

(b)

-80

(c)

-60

(d)

-20

5. If P = $\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right]$ is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

(a)

15

(b)

12

(c)

14

(d)

11

6. If A = $\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right]$ and A-1 = $\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right]$ then the value of a23 is

(a)

0

(b)

-2

(c)

-3

(d)

-1

7. If A, B and C are invertible matrices of some order, then which one of the following is not true?

(a)

(b)

(c)

det A-1 = (det A)-1

(d)

(ABC)-1 = C-1B-1A-1

8. If (AB)-1 = $\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right]$ and A-1 = $\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right]$, then B-1 =

(a)

$\left[ \begin{matrix} 2 & -5 \\ -3 & 8 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 8 & 5 \\ 3 & 2 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 8 & -5 \\ -3 & 2 \end{matrix} \right]$

9. If ATA−1 is symmetric, then A2 =

(a)

A-1

(b)

(AT)2

(c)

AT

(d)

(A-1)2

10. If A is a non-singular matrix such that A-1 = $\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right]$, then (AT)−1 =

(a)

$\left[ \begin{matrix} -5 & 3 \\ 2 & 1 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} -1 & -3 \\ 2 & 5 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 5 & -2 \\ 3 & -1 \end{matrix} \right]$

11. If A = $\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right]$ and AT = A−1 , then the value of x is

(a)

$\frac { -4 }{ 5 }$

(b)

$\frac { -3 }{ 5 }$

(c)

$\frac { 3 }{ 5 }$

(d)

$\frac { 4 }{ 5 }$

12. If A = $\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right]$ and AB = I , then B =

(a)

$\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) A$

(b)

$\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) { A }^{ T }$

(c)

$\left( \cos ^{ 2 }{ \theta } \right) I$

(d)

(Sin2$\frac { \theta }{ 2 }$)A

13. If A = $\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right]$ and A(adj A) =  $\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right]$ then adj (AB) is

(a)

0

(b)

sin θ

(c)

cos θ

(d)

1

14. If A = $\left[ \begin{matrix} 2 & 3 \\ 5 & -2 \end{matrix} \right]$ be such that λA−1 =A, then λ is

(a)

17

(b)

14

(c)

19

(d)

21

15. If adj A = $\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right]$ and adj B = $\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right]$ then adj (AB) is

(a)

$\left[ \begin{matrix} -7 & -1 \\ 7 & -9 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} -6 & 5 \\ -2 & -10 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} -7 & 7 \\ -1 & -9 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} -6 & -2 \\ 5 & -10 \end{matrix} \right]$

16. The rank of the matrix $\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right]$ is

(a)

1

(b)

2

(c)

4

(d)

3

17. If ρ(A) = ρ([A | B]), then the system AX = B of linear equations is

(a)

consistent and has a unique solution

(b)

consistent

(c)

consistent and has infinitely many solution

(d)

inconsistent

18. If 0 ≤ θ  ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is

(a)

$\frac { 2\pi }{ 3 }$

(b)

$\frac { 3\pi }{ 4 }$

(c)

$\frac { 5\pi }{ 6 }$

(d)

$\frac { \pi }{ 4 }$

19. The augmented matrix of a system of linear equations is $\left[ \begin{matrix} 1 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 1 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 7 \\ \begin{matrix} 4 \\ \lambda -7 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ \mu +5 \end{matrix} \end{matrix} \right]$. The system has infinitely many solutions if

(a)

λ = 7, μ ≠ -5

(b)

λ = 7, μ = 5

(c)

λ ≠ 7, μ ≠ -5

(d)

λ = 7, μ = -5

20. Let A = $\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right]$ and 4B = $\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right]$. If B is the inverse of A, then the value of x is

(a)

2

(b)

4

(c)

3

(d)

1

21. If A = $\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right]$, then adj(adj A) is

(a)

$\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 6 & -6 & 8 \\ 4 & -6 & 8 \\ 0 & -2 & 2 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} -3 & 3 & -4 \\ -2 & 3 & -4 \\ 0 & 1 & -1 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 3 & -3 & 4 \\ 0 & -1 & 1 \\ 2 & -3 & 4 \end{matrix} \right]$

22. The system of linear equations x + y + z  = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ $\in$ R) is consistent with unique solution if

(a)

λ = 8

(b)

λ = 8, μ ≠ 36

(c)

λ ≠ 8

(d)

none

23. If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then

(a)

a2 + b2 + c2 = 1

(b)

abc ≠ 1

(c)

a + b + c =0

(d)

a2 + b2 + c2 + 2abc =1

24. Let A be a 3 x 3 matrix and B its adjoint matrix If |B|=64, then |A|=

(a)

±2

(b)

±4

(c)

±8

(d)

±12

25. If AT is the transpose of a square matrix A, then

(a)

|A| ≠ |AT|

(b)

|A| = |AT|

(c)

|A| + |AT| =0

(d)

|A| = |AT| only

26. The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is

(a)

0

(b)

1

(c)

2

(d)

infinitely many

27. If A is a square matrix that IAI = 2, than for any positive integer n, |An| =

(a)

0

(b)

2n

(c)

2n

(d)

n2

28. The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = has a unique solution if

(a)

k ≠ 0

(b)

-1 < k < 1

(c)

-2 < k < 2

(d)

k=0

29. If A is a square matrix of order n, then |adj A| =

(a)

|A|n-1

(b)

|A|n-2

(c)

|A|n

(d)

None

30. If the system of equations x + 2y - 3x = 2, (k + 3) z = 3, (2k + 1) y + z = 2. is inconsistent then k is

(a)

-3, -$\frac{1}{2}$

(b)

-$\frac{1}{2}$

(c)

1

(d)

2

31. If A =$\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right)$ and A(adj A) =$\lambda$ $\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right)$ then $\lambda$ is

(a)

sinx cosx

(b)

1

(c)

2

(d)

none

32. If A is a matrix of order m x n, then $\rho$(A) is

(a)

m

(b)

n

(c)

≤ min (m,n)

(d)

≥ min (m,n)

33. The system of equations x + 2y + 3z = 1, x - y + 4z = 0, 2x + y + 7z = 1 has

(a)

One solution

(b)

Two solution

(c)

No solution

(d)

Infinitely many solution

34. If $\rho$(A) = $\rho$([A/B]) = number of unknowns, then the system is

(a)

consistent and has infinitely many solutions

(b)

consistent

(c)

inconsistent

(d)

consistent and has unique solution

35. Which of the following is not an elementary transformation?

(a)

Ri ↔️ Rj

(b)

Ri ⟶ 2Ri + Rj

(c)

Cj ⟶ Cj + Ci

(d)

Ri ⟶ Ri + Cj

36. If $\rho$(A) = r then which of the following is correct?

(a)

all the minors of order n which do not vanish

(b)

'A' has at least one minor "of order r which does not vanish and all higher order minors vanish

(c)

'A' has at least one (r + 1) order minor which vanish

(d)

all (r + 1) and higher order minors should not vanish

37. Every homogeneous system ______

(a)

Is always consistent

(b)

Has only trivial solution

(c)

Has infinitely many solution

(d)

Need not be consistent

38. If $\rho$(A) ≠ $\rho$([AIB]), then the system is

(a)

consistent and has infinitely many solutions

(b)

consistent and has a unique solution

(c)

consistent

(d)

inconsistent

39. In the non - homogeneous system of equations with 3 unknowns if $\rho$(A) = $\rho$([AIB]) = 2, then the system has _______

(a)

unique solution

(b)

one parameter family of solution

(c)

two parameter family of solutions

(d)

in consistent

40. Cramer's rule is applicable only when ______

(a)

Δ ≠ 0

(b)

Δ = 0

(c)

Δ =0, Δx =0

(d)

Δx = Δy = Δz =0

41. In a homogeneous system if $\rho$ (A) =$\rho$([A|0]) < the number of unknouns then the system has ________

(a)

trivial solution

(b)

only non - trivial solution

(c)

no solution

(d)

trivial solution and infinitely many non - trivial solutions

42. In the system of equations with 3 unknowns, if Δ = 0, and one of Δx, Δy of Δz is non zero then the system is ______

(a)

Consistent

(b)

inconsistent

(c)

consistent with one parameter family of solutions

(d)

consistent with two parameter family of solutions

43. In the system of liner equations with 3 unknowns If $\rho$(A) = $\rho$([A|B]) =1, the system has ________

(a)

unique solution

(b)

inconsistent

(c)

consistent with 2 parameter -family of solution

(d)

consistent with one parameter family of solution.

44. If A = [2 0 1] then the rank of AAT is ______

(a)

1

(b)

2

(c)

3

(d)

0

45. If A is a non-singular matrix then IA-1|= ______

(a)

$\left| \frac { 1 }{ { A }^{ 2 } } \right|$

(b)

$\frac { 1 }{ |A^{ 2 }| }$

(c)

$\left| \frac { 1 }{ A } \right|$

(d)

$\frac { 1 }{ |A| }$

46. _____________ redirect here.

(a)

matices

1. If A$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right]$, then A =

(a)

$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 1 & 2 \\ -1 & 4 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} 4 & 2 \\ -1 & 1 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 4 & -1 \\ 2 & 1 \end{matrix} \right]$

2. If A = $\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right]$, then 9I - A =

(a)

A-1

(b)

$\frac { { A }^{ -1 } }{ 2 }$

(c)

3A-1

(d)

2A-1

47. In a square matrix the minor Mij and the' co-factor Aij of and element aij are related by _____

(a)

Aij = -Mij

(b)

Aij = Mij

(c)

Aij = (-1)i+j Mij

(d)

Aij =(-1)i-j Mij

48. Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.

1. If xayb = em, xcyd = en, Δ1 = $\left| \begin{matrix} m & b \\ n & d \end{matrix} \right|$, Δ2 = $\left| \begin{matrix} a & m \\ c & n \end{matrix} \right|$, Δ3 = $\left| \begin{matrix} a & b \\ c & d \end{matrix} \right|$, then the values of x and y are respectively,

(a)

e21), e31)

(b)

log (Δ13), log (Δ23)

(c)

log (Δ21), log(Δ31)

(d)

e(Δ13),e(Δ23)

2. Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λn adj(A).

(a)

Only (i)

(b)

(ii) and (iii)

(c)

(iii) and (iv)

(d)

(i), (ii) and (iv)

49. DO OR DIE
10 x 3 = 30
50. Decrypt the received encoded message $\left[ \begin{matrix} 2 & -3 \end{matrix} \right] \left[ \begin{matrix} 20 & 4 \end{matrix} \right]$ with the encryption matrix $\left[ \begin{matrix} -1 & -1 \\ 2 & 1 \end{matrix} \right]$
and the decryption matrix as its inverse, where the system of codes are described by the numbers 1 - 26 to the letters A - Z respectively, and the number 0 to a blank space.

51. Solve: 3x+ay =4, 2x+ay=2, a≠0 by Cramer's rule.

52. Verify (AB)-1 =B-1 A-1 for A=$\left[ \begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix} \right]$ and B=$\left[ \begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix} \right]$.

53. Use pencil only
54. Under what co.nditions will the rank of the matrix $\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right]$ be less than 3?

55. Find,the rank of the matrix math $\left[ \begin{matrix} 4 \\ -2 \\ 1 \end{matrix}\begin{matrix} 4 \\ 3 \\ 4 \end{matrix}\begin{matrix} 0 \\ -1 \\ 8 \end{matrix}\begin{matrix} 3 \\ 5 \\ 7 \end{matrix} \right]$.

56. Solve 2x - 3y = 7, 4x - 6y = 14 by Gaussian Jordan method.

57. Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y +  z = 10.

58. Verify that (A-1)T = (AT)-1 for A=$\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right]$.

59. If the rank of the matrix $\left[ \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right]$ is 2, then find ⋋.