CBSE Class 12 Mathematics Determinants MCQs Set 02

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

Question. The area of a triangle with vertices (-3, 0), (3, 0) and (0, k) is 9 sq. units. The value of k will be
(a) 9
(b) 3
(c) -9
(d) 6
Answer: (b) 3

Question. If A, B and C are angles of a triangle, then the determinant \( \begin{vmatrix} -1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{vmatrix} \) is equal to
(a) 0
(b) -1
(c) 1
(d) None of the options
Answer: (a) 0

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) \( f(0) = 0 \)
Solution: We have, \( f(x) = \begin{vmatrix} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{vmatrix} \)
\( \implies \) \( f(a) = \begin{vmatrix} 0 & 0 & a-b \\ 2a & 0 & a-c \\ a+b & a+c & 0 \end{vmatrix} = [(a-b)\{2a . (a+c)\}] \neq 0 \)
and \( f(b) = \begin{vmatrix} 0 & b-a & 0 \\ b+a & 0 & b-c \\ 2b & b+c & 0 \end{vmatrix} = -(b-a)[-2b(b-c)] = 2b(b-a)(b-c) \neq 0 \)
and \( f(0) = \begin{vmatrix} 0 & -a & -b \\ a & 0 & -c \\ b & c & 0 \end{vmatrix} = a(bc) - b(ac) = abc - abc = 0 \)

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

Question. If A and B are invertible matrices, then which of the following is not correct?
(a) \( \text{adj } A = |A| . A^{-1} \)
(b) \( \det(A)^{-1} = [\det(A)]^{-1} \)
(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 x, y, z are all different from zero and \( \begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{vmatrix} = 0 \), then value of \( x^{-1} + y^{-1} + z^{-1} \) is
(a) \( xyz \)
(b) \( x^{-1} y^{-1} z^{-1} \)
(c) \( -x -y -z \)
(d) -1
Answer: (d) -1

Question. The value of the determinant \( \begin{vmatrix} x & x+y & x+2y \\ x+2y & x & x+y \\ x+y & x+2y & x \end{vmatrix} \) is 
(a) \( 9x^2(x+y) \)
(b) \( 9y^2(x+y) \)
(c) \( 3y^2(x+y) \)
(d) \( 7x^2(x+y) \)
Answer: (b) \( 9y^2(x+y) \)

Question. There are two values of \( a \) which makes determinant \( \Delta = \begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86 \), then sum of these numbers is
(a) 4
(b) 5
(c) -4
(d) 9
Answer: (c) -4

Question. If a, b, c are in AP, then the determinant \( \Delta = \begin{vmatrix} x+2 & x+4 & x+2a \\ x+3 & x+4 & x+2b \\ x+4 & x+5 & x+2c \end{vmatrix} \) is
(a) 0
(b) 1
(c) \( x \)
(d) \( 2x \)
Answer: (a) 0

Question. If x, y, z are non-zero real numbers, then the inverse of matrix \( A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} \) is
(a) \( \begin{bmatrix} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1} \end{bmatrix} \)
(b) \( xyz \begin{bmatrix} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1} \end{bmatrix} \)
(c) \( \frac{1}{xyz} \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} \)
(d) \( \frac{1}{xyz} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
Answer: (a) \( \begin{bmatrix} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1} \end{bmatrix} \)

Question. If area of 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. Let A be a square matrix of order \( 3 \times 3 \), 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. Let A be a nonsingular square matrix of order \( 3 \times 3 \). Then \( |\text{adj. } A| \) is equal to
(a) \( |A| \)
(b) \( |A|^2 \)
(c) \( |A|^3 \)
(d) \( 3|A| \)
Answer: (b) \( |A|^2 \)

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

Question. The value of \( \begin{vmatrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \end{vmatrix} \) is
(a) 0
(b) 1
(c) -1
(d) None of the options
Answer: (a) 0

Question. The value of \( \begin{vmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{vmatrix} \) is
(a) 1
(b) 0
(c) \( a+b \)
(d) \( a-b \)
Answer: (b) 0

Question. The value of \( \begin{vmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & \omega & 1 \end{vmatrix} \) is
(a) 1
(b) -1
(c) 0
(d) \( \omega \)
Answer: (c) 0

Question. The value of \( \begin{vmatrix} 0 & a-b & a-c \\ b-a & 0 & b-c \\ c-a & c-b & 0 \end{vmatrix} \) is
(a) a
(b) b
(c) 0
(d) None of the options
Answer: (c) 0

Question. The value of \( \begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix} = \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} \) is
(a) \( (a-b)(b-c)(c-a) \)
(b) \( (b-a)(c-b)(c-a) \)
(c) \( a(b-c)(c-a) \)
(d) None of the options
Answer: (a) \( (a-b)(b-c)(c-a) \)

Assertion-Reason Questions 

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : Determinant is a number associated with a square matrix.
Reason (R) : Determinant is a square matrix.
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (c) A is true but R is false.
Solution: Clearly, Assertion (A) is true and Reason (R) is false.
Hence, (c) is the correct option.

Question. Assertion (A) : If \( A = \begin{bmatrix} 5-x & x+1 \\ 2 & 4 \end{bmatrix} \), then the matrix A is singular if x = 3.
Reason (R) : A square matrix is a singular matrix if its determinant is zero.
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution: For singular matrix, we have \( \begin{vmatrix} 5-x & x+1 \\ 2 & 4 \end{vmatrix} = 0 \)
\( \implies \) \( 20 - 4x - 2x - 2 = 0 \)
\( \implies \) \( 18 - 6x = 0 \)
\( \implies \) \( 6x = 18 \)
\( \implies \) \( x = \frac{18}{6} = 3 \)
\( \implies \) \( x = 3 \)
So, Assertion (A) and Reason (R) both are true and Reason (R) is the correct explanation of Assertion(A).
Hence, (a) is the correct option.

Question. Assertion (A) : If A is a \( 3 \times 3 \) matrix, \( |A| \neq 0 \) and \( |5A| = K|A| \), then the value of K = 125.
Reason (R) : If A be any square matrix of order \( n \times n \) and k be any scalar then \( |KA| = K^n|A| \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution: We have, A is a square matrix of order \( 3 \times 3 \)
\( \therefore |5A| = 5^3|A| = 125|A| \)
\( \implies \) \( 125|A| = K|A| \)
\( \implies \) \( K = 125 \)
So, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Hence, (a) is the correct option.

Question. Assertion (A) : If \( \begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix} \) then \( x = \pm 6 \).
Reason (R) : If A is a skew-symmetric matrix of odd order, then \( |A| = 0 \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.
Solution: We have, \( \begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix} \)
\( \implies \) \( x^2 - 36 = 36 - 36 \)
\( \implies \) \( x^2 - 36 = 0 \)
\( \implies \) \( x^2 = 36 \)
\( \implies \) \( x = \pm 6 \)
Clearly, both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Hence, (b) is the correct option.

Question. Assertion (A) : If \( A = \begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix} \), then \( C_{22} = 1 \), where \( C_{ij} \) denotes the co-factor of \( i^{\text{th}} \) row and \( j^{\text{th}} \) column.
Reason (R) : The co-factor \( C_{ij} \) of \( a_{ij} \) in the matrix \( A = [a_{ij}]_{n \times n} \) equal to \( (-1)^{i+j}M_{ij} \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution: We have, \( A = \begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix} \)
\( \therefore M_{22} = -9 + 10 = 1 \)
\( \therefore C_{22} = (-1)^{2+2} . M_{22} = 1 \times 1 = 1 \)
Clearly, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Hence, (a) is the correct option.

Case Study

The sum of three numbers x, y, z is 6. If we multiply third number by 3 and add second number to it, we get 11. By adding first and third numbers, we get double of the second number. On the basis of above information, answer the following questions :

Question. If we represent the above information in matrix form, we can write it as :
(a) \( \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 11 \\ 0 \end{bmatrix} \)
(b) \( \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 11 \\ 0 \end{bmatrix} \)
(c) \( \begin{bmatrix} 1 & 1 & 1 \\ 0 & -1 & 3 \\ 1 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 11 \\ 0 \end{bmatrix} \)
(d) \( \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & -3 \\ 1 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 11 \\ 0 \end{bmatrix} \)
Answer: (b) \( \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 11 \\ 0 \end{bmatrix} \)

Question. In inverse of matrix \( \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & -2 & 1 \end{bmatrix} \) is :
(a) \( \frac{1}{9} \begin{bmatrix} 7 & -3 & 2 \\ -3 & 0 & -3 \\ -1 & 3 & 1 \end{bmatrix} \)
(b) \( \frac{1}{9} \begin{bmatrix} 7 & -3 & 2 \\ 3 & 0 & 3 \\ 1 & 3 & -1 \end{bmatrix} \)
(c) \( \frac{1}{9} \begin{bmatrix} 7 & -3 & 2 \\ 3 & 0 & -3 \\ 1 & -3 & 1 \end{bmatrix} \)
(d) \( \frac{1}{9} \begin{bmatrix} 7 & -3 & 2 \\ -3 & 0 & 3 \\ 1 & 3 & -1 \end{bmatrix} \)
Answer: (d) \( \frac{1}{9} \begin{bmatrix} 7 & -3 & 2 \\ -3 & 0 & 3 \\ 1 & 3 & -1 \end{bmatrix} \)

Question. The value of x is :
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. The value of y is :
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The value of z is :
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Case Study

A school wants to award their students for the values of honesty, regularity and hardwork with a total cash award of Rs. 6000. Three times the award money for hardwork added to that given for Honesty amounts to Rs. 11,000. The ward money given for honesty and hardwork together is double for regularity. The amount of award for honesty, regularity and hardwork be Rs. x, Rs. y, Rs. z respectively. On the basis of above information, answer the following questions :

Question. If we represent the above situation in matric form as AX = X, then A =
(a) \( \begin{bmatrix} 6000 \\ 11000 \\ 0 \end{bmatrix} \)
(b) \( \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{bmatrix} \)
(c) \( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \)
(d) None of the options
Answer: (b) \( \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{bmatrix} \)

Question. \( \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{bmatrix}^{-1} = \)
(a) \( \frac{1}{6} \begin{bmatrix} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & -1 \end{bmatrix} \)
(b) \( \frac{1}{6} \begin{bmatrix} 6 & 2 & -2 \\ -3 & 0 & 3 \\ 3 & -2 & -1 \end{bmatrix} \)
(c) \( \frac{1}{10} \begin{bmatrix} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & -1 \end{bmatrix} \)
(d) \( \frac{1}{10} \begin{bmatrix} 6 & 2 & -2 \\ -3 & 0 & 3 \\ 3 & -2 & -1 \end{bmatrix} \)
Answer: (c) \( \frac{1}{10} \begin{bmatrix} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & -1 \end{bmatrix} \)

Question. The value of x is :
(a) 3500
(b) 2000
(c) 1000
(d) 500
Answer: (d) 500

Question. The value of y is :
(a) 3500
(b) 2000
(c) 1000
(d) 500
Answer: (b) 2000

Question. The value of z is :
(a) 3500
(b) 2000
(c) 1000
(d) 500
Answer: (a) 3500

Case Study

Manjit wants to donate a rectangular plot of land for a school in his village. When he was asked to give dimensions of the plot, he told that if its length is decreased by 50 m and breadth is increased by 50 m, then its area will remain same but if length is decreased by 10 m and breadth is decreased by 20 m, then its area will decrease by 5300 \( m^2 \). Based on the information given above, answer the following questions :

Question. The equations in terms of x and y are :
(a) x – y = 50, 2x –y = 550
(b) x – y = 50, 2x + y = 550
(c) x + y = 50, 2x + y = 550
(d) x + y = 50, 2x + y = 550
Answer: (b) x – y = 50, 2x + y = 550

Question. Which of the following matrix equation represents the given information :
(a) \( \begin{bmatrix} 1 & -1 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 550 \end{bmatrix} \)
(b) \( \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 550 \end{bmatrix} \)
(c) \( \begin{bmatrix} 1 & 1 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 550 \end{bmatrix} \)
(d) \( \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} - 50 \\ - 550 \end{bmatrix} \)
Answer: (a) \( \begin{bmatrix} 1 & -1 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 550 \end{bmatrix} \)

Question. The value of x (length of rectangular field) is :
(a) 150 m
(b) 400 m
(c) 200 m
(d) 320 m
Answer: (c) 200 m

Question. The value of y (breadth of rectangular field) is :
(a) 150 m
(b) 200 m
(c) 430 m
(d) 350 m
Answer: (a) 150 m

Question. How much is the area of rectangular field ?
(a) 60000 sq. m.
(b) 30000 sq. m.
(c) 30000 m.
(d) 3000 m.
Answer: (b) 30000 sq. m.