CBSE Class 12 Mathematics Determinants Important Questions Set 02

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 \( xyz(x - y)(y - z)(z - x) \).