CBSE Class 12 Mathematics Determinants Important Questions Set B

Objective Type Questions

Choose and write the correct option in each of the following questions.

Question. The maximum value of \( \Delta = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 + \cos \theta & 1 & 1 \end{vmatrix} \) is (\( \theta \) is real number)
(a) \( \frac{1}{2} \)
(b) \( \frac{\sqrt{3}}{2} \)
(c) \( \sqrt{2} \)
(d) \( \frac{2\sqrt{3}}{4} \)
Answer: (a)

Question. The value of \( \begin{vmatrix} 5^2 & 5^3 & 5^4 \\ 5^3 & 5^4 & 5^5 \\ 5^4 & 5^5 & 5^6 \end{vmatrix} \) is
(a) 0
(b) \( 5^2 \)
(c) \( 5^9 \)
(d) \( 5^{13} \)
Answer: (a)

Question. Let \( A = \begin{bmatrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{bmatrix} \), when \( 0 \leq \theta \leq 2\pi \). Then
(a) \( \text{Det}(A) = 0 \)
(b) \( \text{Det}(A) \in (2, \infty) \)
(c) \( \text{Det}(A) \in (2, 4) \)
(d) \( \text{Det}(A) \in [2, 4] \)
Answer: (d)

Question. If \( \begin{vmatrix} x & 4 \\ 2 & 2x \end{vmatrix} = 0 \), then the value \( x \) is
(a) 0
(b) \( \pm 2 \)
(c) 2
(d) \( -2 \)
Answer: (b)

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

Question. If \( A + B + C = \pi \), then the value of \( \begin{vmatrix} \sin(A + B + C) & \sin(A + C) & \cos C \\ -\sin B & 0 & \tan A \\ \cos(A + B) & \tan(B + C) & 0 \end{vmatrix} \) is equal to
(a) 0
(b) 1
(c) \( 2 \tan A \sin B \cos C \)
(d) none of these
Answer: (a)

Question. The determinant \( \begin{vmatrix} b^2-ab & b-c & bc-ac \\ ab-a^2 & a-b & b^2-ab \\ bc-ac & c-a & ab-a^2 \end{vmatrix} \) equals
(a) \( abc(b - c)(c - b)(a - b) \)
(b) \( (b - c)(c - b)(a - b) \)
(c) \( (a + b + c)(b - c)(c - a)(a - b) \)
(d) None of these
Answer: (d)

Fill in the blanks.

Question. If \( \begin{vmatrix} 2x & -9 \\ -2 & x \end{vmatrix} = \begin{vmatrix} -4 & 8 \\ 1 & -2 \end{vmatrix} \), then value of \( x \) is _____________ . 
Answer: \( \pm 3 \)

Question. If \( A \) and \( B \) are square matrices of order 3 and \( |A| = 5, |B| = 3 \), then the value of \( |3AB| \) is _____________ . 
Answer: 405

Very Short Answer Questions: 

Question. For what value of \( x \), the following matrix is singular?
\( \begin{vmatrix} 5 - x & x + 1 \\ 2 & 4 \end{vmatrix} = 0 \)
Answer: \( x = 3 \)

Question. Write the value of the following determinant: \( \begin{vmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{vmatrix} \) 
Answer: 0

Question. If \( A_{ij} \) is the cofactor of the element \( a_{ij} \) of the determinant \( \begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix} \), then write the value of \( a_{32} \cdot A_{32} \). 
Answer: 110

Question. If \( \begin{vmatrix} 3x & 7 \\ -2 & 4 \end{vmatrix} = \begin{vmatrix} 8 & 7 \\ 6 & 4 \end{vmatrix} \), then find the value of \( x \).
Answer: \( x = -2 \)

Question. If \( A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \), then for any natural number \( n \), find the value of \( \text{det}(A^n) \).
Answer: \( |A^n| = 1 \)

Question. If \( A \) is a square matrix of order 3 and \( |3A| = k|A| \), then write the value of \( k \). 
Answer: \( k = 27 \)

Question. If \( A = [a_{ij}] \) is a matrix of order \( 2 \times 2 \), such that \( |A| = -15 \) and \( C_{ij} \) represents the cofactor of \( a_{ij} \), then find \( a_{21}C_{21} + a_{22}C_{22} \). 
Answer: \( -15 \)

Question. Find the cofactors of all the elements of \( \begin{bmatrix} 1 & -2 \\ 4 & 3 \end{bmatrix} \). 
Answer: \( C_{11} = 3, C_{21} = 2, C_{12} = -4, C_{22} = 1 \)

Short Answer Questions

Question. Using the properties of determinant, evaluate \( \begin{vmatrix} a + x & y & z \\ x & a + y & z \\ x & y & a + z \end{vmatrix} \) 
Answer: \( a^2(a + x + y + z) \)

Question. Show that \( \begin{vmatrix} a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z \end{vmatrix} = 0 \), using properties of determinant.
Answer: Applying \( R_2 \to R_2 - 2R_3 \), we get \( R_2 = [a, b, c] \). Since \( R_1 \) and \( R_2 \) are identical, the determinant is 0.

Question. Find the equation of line joining \( (3, 1) \) and \( (9, 3) \) using determinant.
Answer: \( x - 3y = 0 \)

Question. Using co-factors of elements of third column, evaluate \( \Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix} \)
Answer: \( (x - y)(y - z)(z - x) \)

Long Answer Questions: 

Question. Using properties of determinant, solve for \( x \): \( \begin{vmatrix} a + x & a - x & a - x \\ a - x & a + x & a - x \\ a - x & a - x & a + x \end{vmatrix} = 0 \) 
Answer: \( x = 0, 3a \)

Question. In a triangle \( ABC \), if \( \begin{vmatrix} 1 & 1 & 1 \\ 1 + \sin A & 1 + \sin B & 1 + \sin C \\ \sin A + \sin^2 A & \sin B + \sin^2 B & \sin C + \sin^2 C \end{vmatrix} = 0 \), then prove that \( \Delta ABC \) is an isosceles triangle.
Answer: Expanding the determinant leads to \( (\sin A - \sin B)(\sin B - \sin C)(\sin C - \sin A) = 0 \). This implies \( \sin A = \sin B \) or \( \sin B = \sin C \) or \( \sin C = \sin A \). Thus, at least two angles are equal, proving the triangle is isosceles.

Question. Using properties of determinant, prove the following: \( \begin{vmatrix} x & y & z \\ x^2 & y^2 & z^2 \\ x^3 & y^3 & z^3 \end{vmatrix} = xyz(x - y)(y - z)(z - x) \) 
Answer: Taking out \( x, y, z \) from \( C_1, C_2, C_3 \) respectively, we get \( xyz \begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{vmatrix} \). Applying \( C_2 \to C_2 - C_1 \) and \( C_3 \to C_3 - C_1 \) and expanding leads to the result \( xyz(x - y)(y - z)(z - x) \).