JEE Mathematics Determinants and Cramers Rule MCQs Set 02

Choose the most appropriate option (a, b, c or d).

Question. The value of \( \begin{vmatrix} x & x^2-yz & 1 \\ y & y^2-zx & 1 \\ z & z^2-xy & 1 \end{vmatrix} \) is
(a) 1
(b) -1
(c) 0
(d) \( -xyz \)
Answer: (c) 0

Question. If \( \sqrt{-1} = i \), and \( \omega \) is a nonreal cube root of unity then the value of \( \begin{vmatrix} 1 & \omega^2 & 1+i+\omega^2 \\ -i & -1 & -1-i+\omega \\ 1-i & \omega^2-1 & -1 \end{vmatrix} \) is equal to
(a) 1
(b) \( i \)
(c) \( \omega \)
(d) 0
Answer: (d) 0

Question. If \( f(x) = \begin{vmatrix} 1 & x & x+1 \\ 2x & x(x-1) & x(x+1) \\ 3x(x-1) & x(x-1)(x-2) & x(x^2-1) \end{vmatrix} \) then \( f(100) \) is equal to
(a) 0
(b) 1
(c) 100
(d) -100
Answer: (a) 0

Question. The value of \( \begin{vmatrix} i^m & i^{m+1} & i^{m+2} \\ i^{m+5} & i^{m+4} & i^{m+3} \\ i^{m+6} & i^{m+7} & i^{m+8} \end{vmatrix} \), where \( i = \sqrt{-1} \), is
(a) 1 if m is a multiple of 4
(b) 0 for all real m
(c) –i if m is a multiple of 3
(d) None of the options
Answer: (b) 0 for all real m

Question. If \( \Delta_1 = \begin{vmatrix} 7 & x & 2 \\ -5 & x+1 & 3 \\ 4 & x & 7 \end{vmatrix} \), \( \Delta_2 = \begin{vmatrix} x & 2 & 7 \\ x+1 & 3 & -5 \\ x & 7 & 4 \end{vmatrix} \) then \( \Delta_1 - \Delta_2 = 0 \) for
(a) \( x = 2 \)
(b) all real x
(c) \( x = 0 \)
(d) None of the options
Answer: (b) all real x

Question. If \( \Delta_1 = \begin{vmatrix} 10 & 4 & 3 \\ 17 & 7 & 4 \\ 4 & -5 & 7 \end{vmatrix} \), \( \Delta_2 = \begin{vmatrix} 4 & x+5 & 3 \\ 7 & x+12 & 4 \\ -5 & x-1 & 7 \end{vmatrix} \) such that \( \Delta_1 + \Delta_2 = 0 \) then
(a) \( x = 5 \)
(b) \( x \) has no real value
(c) \( x = 0 \)
(d) None of the options
Answer: (a) \( x = 5 \)

Question. Let \( \begin{vmatrix} \lambda^2+3\lambda & \lambda-1 & \lambda+3 \\ \lambda+1 & -2\lambda & \lambda-4 \\ \lambda-3 & \lambda+4 & 3\lambda \end{vmatrix} = p\lambda^4 + q\lambda^3 + r\lambda^2 + s\lambda + t \) be an identity in \( \lambda \), where \( p, q, r, s, t \) are independent of \( \lambda \). Then the value of \( t \) is
(a) 4
(b) 0
(c) 1
(d) None of the options
Answer: (b) 0

Question. Let \( \begin{vmatrix} 1+x & x & x^2 \\ x & 1+x & x^2 \\ x^2 & x & 1+x \end{vmatrix} = ax^5 + bx^4 + cx^3 + dx^2 + \lambda x + \mu \) be an identity in \( x \), where \( a, b, c, d, \lambda, \mu \) are independent of \( x \). Then the value of \( \lambda \) is
(a) 3
(b) 2
(c) 4
(d) None of the options
Answer: (a) 3

Question. Using the factor theorem it is found that \( b + c \), \( c + a \) and \( a + b \) are three factors of the determinant \( \begin{vmatrix} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & c+b & -2c \end{vmatrix} \). The other factor in the value of the determinant is
(a) 4
(b) 2
(c) \( a + b + c \)
(d) None of the options
Answer: (a) 4

Question. If the determinant \( \begin{vmatrix} \cos 2x & \sin^2 x & \cos 4x \\ \sin^2 x & \cos 2x & \cos^2 x \\ \cos 4x & \cos^2 x & \cos 2x \end{vmatrix} \) is expanded in powers of \( \sin x \) then the constant term in the expansion is
(a) 1
(b) 2
(c) -1
(d) None of the options
Answer: (c) -1

Question. If \( \Delta(x) = \begin{vmatrix} 1 & \cos x & 1-\cos x \\ 1+\sin x & \cos x & 1+\sin x-\cos x \\ \sin x & \sin x & 1 \end{vmatrix} \) then \( \int_0^{\pi/2} \Delta(x) dx \) is equal to
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{2} \)
(c) 0
(d) \( -\frac{1}{2} \)
Answer: (d) \( -\frac{1}{2} \)

Question. If \( i = \sqrt{-1} \) and \( \sqrt[4]{1} = \alpha, \beta, \gamma, \delta \) then \( \begin{vmatrix} \alpha & \beta & \gamma & \delta \\ \beta & \gamma & \delta & \alpha \\ \gamma & \delta & \alpha & \beta \\ \delta & \alpha & \beta & \gamma \end{vmatrix} \) is equal to
(a) \( i \)
(b) \( -i \)
(c) 1
(d) 0
Answer: (d) 0

Question. The roots of \( \begin{vmatrix} x & a & b & 1 \\ \lambda & x & b & 1 \\ \lambda & \mu & x & 1 \\ \lambda & \mu & \nu & 1 \end{vmatrix} = 0 \) are independent of
(a) \( \lambda, \mu, \nu \)
(b) \( a, b \)
(c) \( \lambda, \mu, \nu, a, b \)
(d) None of the options
Answer: (b) \( a, b \)

Question. The value of \( \begin{vmatrix} 1 & 0 & 0 & 0 & 0 \\ 2 & 2 & 0 & 0 & 0 \\ 4 & 4 & 3 & 0 & 0 \\ 5 & 5 & 5 & 4 & 0 \\ 6 & 6 & 6 & 6 & 5 \end{vmatrix} \) is
(a) \( 6! \)
(b) \( 5! \)
(c) \( 1 \cdot 2^2 \cdot 3 \cdot 4^3 \cdot 5^4 \cdot 6^4 \)
(d) None of the options
Answer: (b) \( 5! \)

Question. If \( \begin{vmatrix} b^2+c^2 & ab & ac \\ ba & c^2+a^2 & bc \\ ca & cb & a^2+b^2 \end{vmatrix} = \) square of a determinant \( \Delta \) of the third order then \( \Delta \) is equal to
(a) \( \begin{vmatrix} 0 & c & b \\ c & 0 & b \\ b & a & 0 \end{vmatrix} \)
(b) \( \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} \)
(c) \( \begin{vmatrix} 0 & -c & b \\ c & 0 & -a \\ -b & a & 0 \end{vmatrix} \)
(d) None of the options
Answer: (a) \( \begin{vmatrix} 0 & c & b \\ c & 0 & b \\ b & a & 0 \end{vmatrix} \)

Question. The system of equation \( ax + 4y + z = 0, bx + 3y + z = 0, cx + 2y + z = 0 \) has nontrivial solutions if \( a, b, c \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (a) AP

Question. If the equations \( a(y + z) = x, b(z + x) = y \) and \( c(x + y) = z \), where \( a \neq -1, b \neq -1, c \neq -1 \), admit of nontrivial solutions then \( (1 + a)^{-1} + (1 + b)^{-1} + (1 + c)^{-1} \) is
(a) 2
(b) 1
(c) \( \frac{1}{2} \)
(d) None of the options
Answer: (a) 2

Question. The system of equations \( 2x - y + z = 0 \), \( x - 2y + z = 0 \), \( \lambda x - y + 2z = 0 \) has infinite number of nontrivial solutions for
(a) \( \lambda = 1 \)
(b) \( \lambda = 5 \)
(c) \( \lambda = -5 \)
(d) no real value of \( \lambda \)
Answer: (c) \( \lambda = -5 \)

Question. The equations \( x + y + z = 6, x + 2y + 3z = 10, x + 2y + mz = n \) give infinite number of values of the triplet \( (x, y, z) \) if
(a) \( m = 3, n \in \mathbb{R} \)
(b) \( m = 3, n \neq 10 \)
(c) \( m = 3, n = 10 \)
(d) None of the options
Answer: (c) \( m = 3, n = 10 \)

Question. The system of equations \( 2x + 3y = 8 \), \( 7x - 5y + 3 = 0 \), \( 4x - 6y + \lambda = 0 \) is
(a) 6
(b) 8
(c) -8
(d) -6
Answer: (b) 8

Question. If the system of equations \( ax + by + c = 0, bx + cy + a = 0, cx + ay + b = 0 \) has a solution then the system of equations \( (b + c)x + (c + a)y + (a + b)z = 0 \), \( (c + a)x + (a + b)y + (b + c)z = 0 \), \( (a + b)x + (b + c)y + (c + a)z = 0 \) has
(a) only one solution
(b) no solution
(c) infinite number of solutions
(d) None of the options
Answer: (c) infinite number of solutions

Choose the correct options. One or more options may be correct.

Question. Let \( \Delta(x) = \begin{vmatrix} x+a & x+b & x+a-c \\ x+b & x+c & x-1 \\ x+c & x+d & x-b+d \end{vmatrix} \) and \( \int_0^2 \Delta(x) dx = -16 \), where \( a, b, c, d \) are in AP, then the common difference of the AP is
(a) 1
(b) 2
(c) -2
(d) None of the options
Answer: (b) 2 (c) -2

Question. If \( A + B + C = \pi \), \( e^{i\theta} = \cos \theta + i\sin \theta \) and \( z = \begin{vmatrix} e^{2iA} & e^{-iC} & e^{-iB} \\ e^{-iC} & e^{2iB} & e^{-iA} \\ e^{-iB} & e^{-iA} & e^{2iC} \end{vmatrix} \) then
(a) \( \text{Re}(z) = 4 \)
(b) \( \text{Im}(z) = 0 \)
(c) \( \text{Re}(z) = -4 \)
(d) \( \text{Im}(z) = -1 \)
Answer: (b) \( \text{Im}(z) = 0 \) (c) \( \text{Re}(z) = -4 \)

Question. If \( \begin{vmatrix} a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x \end{vmatrix} = 0 \) then \( x \) is
(a) 0
(b) a
(c) 3a
(d) 2a
Answer: (a) 0 (c) 3a

Question. A value of \( c \) for which the system of equations \( x + y = 1 \), \( (c + 2)x + (c + 4)y = 6 \), \( (c + 2)^2x + (c + 4)^2y = 36 \) is
(a) 1
(b) 2
(c) 4
(d) None of the options
Answer: (b) 2 (c) 4

Question. Eliminating \( a, b, c \) from \( x = \frac{a}{b-c}, y = \frac{b}{c-a}, z = \frac{c}{a-b} \) we get
(a) \( \begin{vmatrix} 1 & -x & x \\ 1 & -y & y \\ 1 & -z & z \end{vmatrix} = 0 \)
(b) \( \begin{vmatrix} 1 & -x & x \\ 1 & 1 & -y \\ 1 & z & 1 \end{vmatrix} = 0 \)
(c) \( \begin{vmatrix} 1 & -x & x \\ y & 1 & -y \\ -z & z & 1 \end{vmatrix} = 0 \)
(d) None of the options
Answer: (b) \( \begin{vmatrix} 1 & -x & x \\ 1 & 1 & -y \\ 1 & z & 1 \end{vmatrix} = 0 \) (c) \( \begin{vmatrix} 1 & -x & x \\ y & 1 & -y \\ -z & z & 1 \end{vmatrix} = 0 \)

Question. The system of equations \( 6x + 5y + \lambda z = 0, 3x - y + 4z = 0, x + 2y - 3z = 0 \) has
(a) only a trivial solution for \( \lambda \in \mathbb{R} \)
(b) exactly one nontrivial solution for some real \( \lambda \)
(c) infinite number of nontrivial solutions for one value of \( \lambda \)
(d) only one solution for \( \lambda \neq -5 \)
Answer: (c) infinite number of nontrivial solutions for one value of \( \lambda \) (d) only one solution for \( \lambda \neq -5 \)