#### Important one mark questions

12th Standard

Reg.No. :
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Maths

Use blue pen only
Time : 00:20:00 Hrs
Total Marks : 25

Part - A

25 x 1 = 25
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$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right]$, 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)

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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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)

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

11. If ATA−1 is symmetric, then A2 =

(a)

A-1

(b)

(AT)2

(c)

AT

(d)

(A-1)2

12. 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)

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

13. 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)

$\frac { 4 }{ 5 }$

14. 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

15. 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

16. 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

17. 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)

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

18. 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

19. 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)

20. 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)

21. 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

22. 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)

$\frac { \pi }{ 4 }$

23. 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

24. 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

25. 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)

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