CBSE Class 12 Mathematics Determinants MCQs Set 03

Question. If \( \begin{vmatrix} 2 & 3 & 2 \\ x & x & x \\ 4 & 9 & 1 \end{vmatrix} + 3 = 0 \), then the value of x is : 
(a) 3
(b) 0
(c) – 1
(d) 1
Answer: (c) – 1

Question. Let A = \( \begin{bmatrix} 200 & 50 \\ 10 & 2 \end{bmatrix} \) and B = \( \begin{bmatrix} 50 & 40 \\ 2 & 3 \end{bmatrix} \), then |AB| is equal to : 
(a) 460
(b) 2000
(c) 3000
(d) – 7000
Answer: (d) – 7000

Question. If A = \( \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{bmatrix} \), then det (adj A) equals: 
(a) \( a^{27} \)
(b) \( a^9 \)
(c) \( a^6 \)
(d) \( a^2 \)
Answer: (c) \( a^6 \)

Question. If A is a square matrix of order 3, such that A (adj A) = 10 I, then |adj A| is equal to : 
(a) 1
(b) 10
(c) 100
(d) 101
Answer: (c) 100

Question. If A is a 3 × 3 matrix such that |A| = 8, then |3A| equals : 
(a) 8
(b) 24
(c) 72
(d) 216
Answer: (d) 216

Question. If A is a skew symmetric matrix of order 3, then the value of |A| is : 
(a) 3
(b) 0
(c) 9
(d) 27
Answer: (b) 0

Question. If \( \begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix} \), then the value of x is:
(a) ± 2
(b) 0
(c) ± 3
(d) ± 6
Answer: (d) ± 6

Question. The determinant \( \begin{vmatrix} x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x \end{vmatrix} \) is :
(a) Independent of \( \theta \) only
(b) Independent of x only
(c) Independent of both \( \theta \) and x
(d) None of the options
Answer: (a) Independent of \( \theta \) only

Question. The area of triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \) and \( (x_3, y_3) \) is :
(a) \( \Delta = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \)
(b) \( \Delta = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ 1 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \)
(c) \( \Delta = \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \)
(d) None of the options
Answer: (a) \( \Delta = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \)

Question. The area of the triangle formed by 3 collinear points is :
(a) one
(b) two
(c) zero
(d) four
Answer: (c) zero

Question. Minor of an element of a determinant of order \( n(n \ge 2) \) is a determinant of order :
(a) n
(b) n – 1
(c) n – 2
(d) n + 1
Answer: (b) n – 1

Question. If \( \Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix} \), then the minor \( M_{31} \) is :
(a) \( - c(a^2 - b^2) \)
(b) \( c(b^2 - a^2) \)
(c) \( c(a^2 + b^2) \)
(d) \( c(a^2 - b^2) \)
Answer: (d) \( c(a^2 - b^2) \)

Question. If \( \Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} \), then the cofactor \( A_{21} \) is :
(a) – (hc + fg)
(b) fg – hc
(c) fg + hc
(d) hc – fg
Answer: (b) fg – hc

Question. If \( M_{11} = - 40 \), \( M_{12} = - 10 \) and \( M_{13} = 35 \) of the determinant \( \Delta = \begin{vmatrix} 1 & 3 & -2 \\ 4 & -5 & 6 \\ 3 & 5 & 2 \end{vmatrix} \), then the value of \( \Delta \) is :
(a) – 80
(b) 60
(c) 70
(d) 100
Answer: (a) – 80

Question. If \( \Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} \) and \( A_{ij} \) is cofactor of \( a_{ij} \), then value of \( \Delta \) is given by :
(a) \( a_{11}A_{31} + a_{12}A_{32} + a_{13}A_{33} \)
(b) \( a_{11}A_{11} + a_{12}A_{21} + a_{13}A_{31} \)
(c) \( a_{21}A_{11} + a_{22}A_{12} + a_{23}A_{13} \)
(d) \( a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31} \)
Answer: (d) \( a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31} \)

Question. If \( A = \begin{bmatrix} 2 & 3 \\ -4 & -6 \end{bmatrix} \), then which of the following is true ?
(a) \( A(\text{adj } A) \neq |A|I \)
(b) \( A(\text{adj } A) \neq (\text{adj } A)A \)
(c) \( A(\text{adj } A) = (\text{adj } A)A = |A|I = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
(d) None of the options
Answer: (c) \( A(\text{adj } A) = (\text{adj } A)A = |A|I = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)

Question. If A and B are invertible matrices, then which of the following is not correct ?
(a) \( \text{adj } A = |A| \cdot A^{-1} \)
(b) \( \text{det } (A)^{-1} = [\text{det } (A)]^{-1} \)
(c) \( (AB)^{-1} = B^{-1}A^{-1} \)
(d) \( (A + B)^{-1} = B^{-1} + A^{-1} \)
Answer: (d) \( (A + B)^{-1} = B^{-1} + A^{-1} \)

Question. If \( f(x) = \begin{vmatrix} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{vmatrix} \), then : 
(a) f(a) = 0
(b) f(b) = 0
(c) f(0) = 0
(d) f(1) = 0
Answer: (c) f(0) = 0

Question. If \( A = \begin{bmatrix} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{bmatrix} \), then \( A^{-1} \) exist if :
(a) \( \lambda = 2 \)
(b) \( \lambda \neq 2 \)
(c) \( \lambda = - 2 \)
(d) None of the options
Answer: (d) None of the options

Question. Let A be a square matrix of order 3 × 3, then |kA| is equal to :
(a) k|A|
(b) \( k^2|A| \)
(c) \( k^3|A| \)
(d) 3k|A|
Answer: (c) \( k^3|A| \)

Question. Which of the following is correct ?
(a) Determinant is a square matrix
(b) Determinant is a number associated to a matrix
(c) Determinant is a number associated to a square matrix
(d) None of the options
Answer: (c) Determinant is a number associated to a square matrix

Question. If area of a triangle is 35 sq. units with vertices (2, – 6), (5, 4) and (k, 4), then k is :
(a) 12
(b) – 2
(c) – 12, 2
(d) 12, – 2
Answer: (d) 12, – 2

Question. If \( A = \begin{bmatrix} -3 & 5 \\ 2 & 4 \end{bmatrix} \), then which of the following is true ?
(a) \( A (\text{adj } A) \neq |A|I \)
(b) \( A (\text{adj } A) \neq (\text{adj } A) A \)
(c) \( A (\text{adj } A) = (\text{adj } A) A = |A|I = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
(d) None of the options
Answer: (c) \( A (\text{adj } A) = (\text{adj } A) A = |A|I = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)

Question. If A is an invertible matrix of order 2, then \( \text{det } (A^{-1}) \) is equal to :
(a) det (A)
(b) \( \frac{1}{\text{det } (A)} \)
(c) 1
(d) 0
Answer: (b) \( \frac{1}{\text{det } (A)} \)

Question. Find the adjoint of the matrix \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) :
(a) \( \begin{bmatrix} 4 & 2 \\ 3 & 1 \end{bmatrix} \)
(b) \( \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \)
(c) \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)
(d) \( \begin{bmatrix} 1 & -2 \\ -3 & 4 \end{bmatrix} \)
Answer: (b) \( \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \)

Question. Find x, if \( \begin{vmatrix} x & 2 \\ 1 & 1 & 1 \\ 2 & 1 & 1 \end{vmatrix} \) is singular :
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4

Question. If \( A = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix} \), then \( \frac{A^2 - 3I}{2} = \)
(a) \( A^{-1} \)
(b) 2A
(c) \( 2A^{-1} \)
(d) \( \frac{3}{2} A^{-1} \)
Answer: (a) \( A^{-1} \)

Question. Value of \( \begin{vmatrix} \cos 15^\circ & \sin 15^\circ \\ \sin 15^\circ & \cos 15^\circ \end{vmatrix} \) is :
(a) 1
(b) \( \frac{1}{2} \)
(c) \( \frac{\sqrt{3}}{2} \)
(d) None of the options
Answer: (c) \( \frac{\sqrt{3}}{2} \)

Question. The value of \( \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} \) is :
(a) \( abc(a + b + c) \)
(b) \( a^3 + b^3 + c^3 - 3abc \)
(c) \( - a^3 - b^3 - c^3 + 3abc \)
(d) None of the options
Answer: (c) \( - a^3 - b^3 - c^3 + 3abc \)

Question. Find the value of \( \begin{vmatrix} a + ib & c + id \\ -c + id & a - ib \end{vmatrix} \) :
(a) \( a^2 + b^2 - c^2 - d^2 \)
(b) \( a^2 - b^2 + c^2 - d^2 \)
(c) \( a^2 + b^2 + c^2 + d^2 \)
(d) None of the options
Answer: (c) \( a^2 + b^2 + c^2 + d^2 \)

Question. If A is square matrix x, such that \( A^2 = I \), then \( A^{-1} \) is equal to :
(a) 2A
(b) 0
(c) A
(d) A + 1
Answer: (c) A

Question. If A(3, 4), B(– 7, 2), C(x, y) are collinear, then :
(a) x + 5y + 17 = 0
(b) x + 5y + 13 = 0
(c) x – 5y + 17 = 0
(d) None of the options
Answer: (c) x – 5y + 17 = 0

Question. If the point A(3, – 2), B(k, 2) and C(8, 8) are collinear, then the value of k is :
(a) 2
(b) – 3
(c) 5
(d) – 4
Answer: (c) 5

Question. Find the minor of the element of second row and third column in the following determinant : \( \begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix} \)
(a) 13
(b) 4
(c) 5
(d) 0
Answer: (c) 13

Choose the correct option :

(a) Both (A) and (B) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : \( \begin{vmatrix} a^2+x^2 & ab-cx & ac+bx \\ ab+cx & b^2+x^2 & bc-ax \\ ac-bx & bc+ax & c^2+x^2 \end{vmatrix} = \begin{vmatrix} x & c & -b \\ -c & x & a \\ b & -a & x \end{vmatrix}^2 \)
Reason (R) : \( \Delta^c = \Delta^{n-1} \) where n is order of determinant, and \( \Delta^c \) is the determinant of cofactors of \( \Delta \).
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion (A) : \( \begin{vmatrix} \cos (\theta + \alpha) & \cos (\theta + \beta) & \cos (\theta + \gamma) \\ \sin (\theta + \alpha) & \sin (\theta + \beta) & \sin (\theta + \gamma) \\ \sin (\beta - \gamma) & \sin (\gamma - \alpha) & \sin (\alpha - \beta) \end{vmatrix} \) is independent of \( \theta \).
Reason (R) : If \( f (\theta) = c \), then \( f (\theta) \) is independent of \( \theta \).
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion (A) : If \( \Delta (x) = \begin{vmatrix} f_1(x) & f_2(x) \\ g_1(x) & g_2(x) \end{vmatrix} \), then \( \Delta' (x) \neq \begin{vmatrix} f'_1(x) & f'_2(x) \\ g'_1(x) & g'_2(x) \end{vmatrix} \)
Reason (R) : \( \frac{d}{dx} \{ f(x) g (x)\} \neq \frac{d}{dx} f(x) \frac{d}{dx} g (x) \).
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion (A) : If \( \Delta (x) = \begin{vmatrix} f(x) & g(x) \\ a & b \end{vmatrix} \), then \( \int_{a}^{b} \Delta(x) dx = \begin{vmatrix} \int_{a}^{b} f(x) dx & \int_{a}^{b} g(x) dx \\ a & b \end{vmatrix} \)
Reason (R) : \( \int \lambda f(x) dx = \lambda \int f(x) dx \)
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion : If a, b, c are even natural numbers then \( \Delta = \begin{vmatrix} a-1 & a & a+1 \\ b-1 & b & b+1 \\ c-1 & c & c+1 \end{vmatrix} \) is an even natural number.
Reason : Sum and product of two even natural number is also an even natural number.
Answer: (d) A is false but R is true.

Question. Assertion : The matrix \( A = \begin{bmatrix} 2 & 3 & -1/2 \\ 7 & 3 & 2 \\ 3 & 1 & 1 \end{bmatrix} \) is singular.
Reason : The value of determinant of matrix A is zero.
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion : For a matrix \( A = [a_{ij}]_3 \), if \( \text{det } (\text{adj } A) = 49 \), then \( \text{det } (A) = \pm 7 \).
Reason : For a square matrix A of order n. \( |\text{adj } A| = |A|^{n-1} \).
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion : Value of x for which the matrix \( \begin{bmatrix} 2 & 1 & 0 \\ 0 & 1 & 2 \\ 1 & -2 & x \end{bmatrix} \) is singular is – 5.
Reason : A matrix A is singular if \( |A| \neq 0 \).
Answer: (c) A is true but R is false.

Question. Assertion : Minor of the element 6 in the matrix \( \begin{bmatrix} 0 & 2 & 6 \\ 1 & 2 & -1 \\ 2 & 1 & 3 \end{bmatrix} \) is 3.
Reason : Minor of an element \( a_{ij} \) of a matix is the determinant obtained by deleting it \( i^{th} \) row.
Answer: (d) A is false but R is true.

Question. Assertion : For two matrices A & B of order 3, |A| = 3, |B| = – 4, then |2AB| is – 96.
Reason : For a matrix A of order n & a scalar k, |kA| = \( k^n|A| \).
Answer: (b) Both (A) and (R) are true but R is not correct explanation of A.

Question. Assertion : For a matrix A = \( A (\text{adj } A) = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} \).
Reason : For a square matrix A, \( A (\text{adj } A) = (\text{adj } A)A = |A|I \).
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion : Values of k for which area of the triangle with vertices (1, 1), (0, 2), (k, 0) is 3 sq. units are 4 and 8.
Reason : Area of the triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), \( (x_3, y_3) \) is \( \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \).
Answer: (d) A is false but R is true.