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Application of Matrices and Determinants

12th Standard EM

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 50
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 = $\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right]$ and B = $\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right]$ then |adj (AB)| =

(a)

-40

(b)

-80

(c)

-60

(d)

-20

4. 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

5. 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

6. If ATA−1 is symmetric, then A2 =

(a)

A-1

(b)

(AT)2

(c)

AT

(d)

(A-1)2

7. 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)

8. 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

9. 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)

$\frac { \pi }{ 4 }$

10. 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

11. 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

12. 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

13. 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

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

(a)

0

(b)

2n

(c)

2n

(d)

n2

15. 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

16. 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)

17. 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

18. 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

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

(a)

Ri ↔️ Rj

(b)

Ri ⟶ 2Ri + Rj

(c)

Cj ⟶ Cj + Ci

(d)

Ri ⟶ Ri + Cj

20. Every homogeneous system ______

(a)

Is always consistent

(b)

Has only trivial solution

(c)

Has infinitely many solution

(d)

Need not be consistent

21. 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

22. Cramer's rule is applicable only when ______

(a)

Δ ≠ 0

(b)

Δ = 0

(c)

Δ =0, Δx =0

(d)

Δx = Δy = Δz =0

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

(a)

1

(b)

2

(c)

3

(d)

0

24. 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)

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

25. 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