#### creative multiple choice questions

12th Standard

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Maths

Do not write anything on the question paper
Time : 00:20:00 Hrs
Total Marks : 25

Part - A

25 x 1 = 25
1. 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

2. 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

3. 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

4. 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

5. 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

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

(a)

0

(b)

2n

(c)

2n

(d)

n2

7. 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

8. 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

9. 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

10. 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

11. 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)

12. 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

13. 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

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

(a)

Ri ↔️ Rj

(b)

Ri ⟶ 2Ri + Rj

(c)

Cj ⟶ Cj + Ci

(d)

Ri ⟶ Ri + Cj

15. 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

16. Every homogeneous system ______

(a)

Is always consistent

(b)

Has only trivial solution

(c)

Has infinitely many solution

(d)

Need not be consistent

17. 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

18. 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

19. Cramer's rule is applicable only when ______

(a)

Δ ≠ 0

(b)

Δ = 0

(c)

Δ =0, Δx =0

(d)

Δx = Δy = Δz =0

20. 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

21. 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

22. 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.

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