#### 12th Standard Maths English Medium Application of Matrices and Determinants Reduced Syllabus Important Questions With Answer key 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

Multiple Choice Questions

15 x 1 = 15
1. 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

2. 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

3. 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

4. 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)

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

5. 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)

6. 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

7. 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)

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

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

(a)

0

(b)

2n

(c)

2n

(d)

n2

9. 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

10. Every homogeneous system ______

(a)

Is always consistent

(b)

Has only trivial solution

(c)

Has infinitely many solution

(d)

Need not be consistent

11. 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

12. Cramer's rule is applicable only when ______

(a)

Δ ≠ 0

(b)

Δ = 0

(c)

Δ =0, Δx =0

(d)

Δx = Δy = Δz =0

13. 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

14. 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.

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

(a)

1

(b)

2

(c)

3

(d)

0

1. 2 Marks

10 x 2 = 20
16. If adj(A) = $\left[ \begin{matrix} 0 & -2 & 0 \\ 6 & 2 & -6 \\ -3 & 0 & 6 \end{matrix} \right]$, find A−1.

17. Find the rank of the following matrices which are in row-echelon form :
$\left[ \begin{matrix} 2 & 0 & -7 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{matrix} \right]$

18. Find the rank of the following matrices by minor method:
$\left[ \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right]$

19. Find the rank of the following matrices by minor method:
$\left[ \begin{matrix} -1 & 3 \\ 4 & -7 \\ 3 & -4 \end{matrix} \right]$

20. Find the rank of the following matrices by minor method:
$\left[ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 8 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 2 \\ 1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 2 \end{matrix} \end{matrix} \right]$

21. Solve the following system of homogenous equations.
2x + 3y − z = 0, x − y − 2z = 0, 3x + y + 3z = 0

22. For any 2 x 2 matrix, if A (adj A) =$\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right]$ then find |A|.

23. Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

24. Flod the rank of the matrix $\left[ \begin{matrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{matrix} \right]$.

25. Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.

1. 3 Marks

10 x 3 = 30
26. Solve the following systems of linear equations by Cramer’s rule:
5x − 2y +16 = 0, x + 3y − 7 = 0

27. In a competitive examination, one mark is awarded for every correct answer while $\frac { 1 }{ 4 }$ mark is deducted for every wrong answer. A student answered 100 questions and got 80 marks. How many questions did he answer correctly ? (Use Cramer’s rule to solve the problem).

28. Test for consistency of the following system of linear equations and if possible solve:
x - y + z = -9, 2x - 2y + 2z = -18, 3x - 3y + 3z + 27 = 0.

29. Solve the following system:
x + 2y + 3z = 0, 3x + 4y + 4z = 0, 7x + 10y + 12z = 0.

30. Find the adjoint of the following:
$\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right]$

31. Find the inverse (if it exists) of the following:
$\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right]$

32. Solve the following system of linear equations by matrix inversion method:
2x − y = 8, 3x + 2y = −2

33. Solve the following systems of linear equations by Cramer’s rule:
$\frac { 3 }{ x }$ + 2y = 12, $\frac { 2 }{ x }$ + 3y = 13

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

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

1. 5 Marks

7 x 5 = 35
36. Test for consistency and if possible, solve the following systems of equations by rank method.
i) x - y + 2z = 2, 2x + y + 4z = 7, 4x - y + z = 4
ii) 3x + y + z = 2, x - 3y + 2z = 1, 7x - y + 4z = 5
iii) 2x + 2y + z = 5, x - y + z = 1, 3x + y + 2z = 4
iv) 2x - y + z = 2, 6x - 3y + 3z = 6, 4x - 2y + 2z = 4

37. Investigate the values of λ and μ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y - 5z = 8, 2x + 3y + λz = μ, have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.

38. Solve the following system of linear equations by matrix inversion method:
2x + 3y − z = 9, x + y + z = 9, 3x − y − z  = −1

39. Solve the following system of linear equations by matrix inversion method:
x + y + z − 2 = 0, 6x − 4y + 5z − 31 = 0, 5x + 2y + 2z = 13.

40. Solve: $\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z }$=2

41. The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.

42. Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has (i) unique solution (ii) infinite solutions and (iii) no solution.