JEE Mathematics Determinants and Cramers Rule MCQs Set 01

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

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

Question. \( \begin{vmatrix} 0 & p-q & p-r \\ q-p & 0 & q-r \\ r-p & r-q & 0 \end{vmatrix} \) is equal to
(a) \( p + q + r \)
(b) 0
(c) \( p - q - r \)
(d) \( -p + q + r \)
Answer: (b) 0

Question. If \( a \neq b \neq c \) such that \( \begin{vmatrix} a^3-1 & b^3-1 & c^3-1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} = 0 \) then
(a) \( ab + bc + ca = 0 \)
(b) \( a + b + c = 0 \)
(c) \( abc = 1 \)
(d) \( a + b + c = 1 \)
Answer: (c) \( abc = 1 \)

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

Question. If \( \begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix} = x + iy \) then
(a) \( x = 3, y = 1 \)
(b) \( x = 1, y = 3 \)
(c) \( x = 0, y = 3 \)
(d) \( x = 0, y = 0 \)
Answer: (d) \( x = 0, y = 0 \)

Question. The determinant \( \begin{vmatrix} xp+y & x & y \\ yp+z & y & z \\ 0 & xp+y & yp+z \end{vmatrix} = 0 \) for all \( p \in \mathbb{R} \) if
(a) \( x, y, z \) are in AP
(b) \( x, y, z \) are in GP
(c) \( x, y, z \) are in HP
(d) \( xy, yz, zx \) are in AP
Answer: (b) \( x, y, z \) are in GP

Question. The determinant \( \begin{vmatrix} a & a+d & a+2d \\ a^2 & (a+d)^2 & (a+2d)^2 \\ 2a+3d & 2(a+d) & 2a+d \end{vmatrix} = 0 \). Then
(a) \( d = 0 \)
(b) \( a + d = 0 \)
(c) \( d = 0 \) or \( a + d = 0 \)
(d) None of the options
Answer: (c) \( d = 0 \) or \( a + d = 0 \)

Question. The value of the determinant \( \begin{vmatrix} bc & ca & ab \\ p & q & r \\ 1 & 1 & 1 \end{vmatrix} \), where \( a, b, c \) are the \( p^{th}, q^{th} \) and \( r^{th} \) terms of a HP, is
(a) \( ap + bq + cr \)
(b) \( (a + b + c)(p + q + r) \)
(c) 0
(d) None of the options
Answer: (c) 0

Question. The sum of two nonintegral roots of \( \begin{vmatrix} x & 2 & 5 \\ 3 & x & 3 \\ 5 & 4 & x \end{vmatrix} = 0 \) is
(a) 5
(b) -5
(c) -18
(d) None of the options
Answer: (b) -5

Question. If \( x, y, z \) are integers in AP, lying between 1 and 9, and \( x51, y41 \) and \( z31 \) are three-digit numbers then the value of \( \begin{vmatrix} 5 & 4 & 3 \\ x51 & y41 & z31 \\ x & y & z \end{vmatrix} \) is
(a) \( x + y + z \)
(b) \( x - y + z \)
(c) 0
(d) None of the options
Answer: (c) 0

Question. If \( \Delta_1 = \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} \), \( \Delta_2 = \begin{vmatrix} 1 & bc & a \\ 1 & ca & b \\ 1 & ab & c \end{vmatrix} \) then
(a) \( \Delta_1 + \Delta_2 = 0 \)
(b) \( \Delta_1 + 2\Delta_2 = 0 \)
(c) \( \Delta_1 = \Delta_2 \)
(d) None of the options
Answer: (a) \( \Delta_1 + \Delta_2 = 0 \)

Question. Two nonzero distinct numbers \( a, b \) are used as elements to make determinants of the third order. The number of determinants whose value is zero for all \( a, b \) is
(a) 24
(b) 32
(c) \( a + b \)
(d) None of the options
Answer: (b) 32

Question. The value of \( \begin{vmatrix} a_1x+b_1y & a_2x+b_2y & a_3x+b_3y \\ b_1x+a_1y & b_2x+a_2y & b_3x+a_3y \\ b_1x+a_1 & b_2x+a_2 & b_3x+a_3 \end{vmatrix} \) is equal to
(a) \( x^2 + y^2 \)
(b) 0
(c) \( a_1a_2a_3x^2 + b_1b_2b_3y^2 \)
(d) None of the options
Answer: (b) 0

Question. If \( \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = \begin{vmatrix} 1 & 1 & 1 \\ b_1 & b_2 & b_3 \\ a_1 & a_2 & a_3 \end{vmatrix} \) then the two triangles whose vertices are \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) and \( (a_1, b_1), (a_2, b_2), (a_3, b_3) \) are
(a) congruent
(b) similar
(c) equal in area
(d) None of the options
Answer: (c) equal in area

Question. If \( \alpha, \beta \) are nonreal numbers satisfying \( x^3 - 1 = 0 \) then the value of \( \begin{vmatrix} \lambda+1 & \alpha & \beta \\ \alpha & \lambda+\beta & 1 \\ \beta & 1 & \lambda+\alpha \end{vmatrix} \) is equal to
(a) 0
(b) \( \lambda^3 \)
(c) \( \lambda^3 + 1 \)
(d) None of the options
Answer: (b) \( \lambda^3 \)

Question. The value of \( \begin{vmatrix} ^{10}C_4 & ^{10}C_5 & ^{11}C_m \\ ^{11}C_6 & ^{11}C_7 & ^{12}C_{m+2} \\ ^{12}C_8 & ^{12}C_9 & ^{13}C_{m+4} \end{vmatrix} \) is equal to zero when \( m \) is
(a) 6
(b) 4
(c) 5
(d) None of the options
Answer: (c) 5

Question. If \( x > 0 \) and \( x \neq 1, y > 0 \) and \( y \neq 1, z > 0 \) and \( z \neq 1 \) then the value of \( \begin{vmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 1 & \log_y z \\ \log_z x & \log_z y & 1 \end{vmatrix} \) is
(a) 0
(b) 1
(c) -1
(d) None of the options
Answer: (a) 0

Question. The value of \( \begin{vmatrix} 1 & 1 & 1 \\ (2^x+2^{-x})^2 & (3^x+3^{-x})^2 & (5^x+5^{-x})^2 \\ (2^x-2^{-x})^2 & (3^x-3^{-x})^2 & (5^x-5^{-x})^2 \end{vmatrix} \) is
(a) 0
(b) \( 30^x \)
(c) \( 30^{-x} \)
(d) None of the options
Answer: (a) 0

Question. The value of the determinant \( \begin{vmatrix} ^5C_0 & ^5C_3 & 14 \\ ^5C_1 & ^5C_4 & 1 \\ ^5C_2 & ^5C_5 & 1 \end{vmatrix} \) is
(a) 0
(b) \( -(6!) \)
(c) 80
(d) None of the options
Answer: (b) \( -(6!) \)

Question. \( \begin{vmatrix} \cos C & \tan A & 0 \\ \sin B & 0 & -\tan A \\ 0 & \sin B & \cos C \end{vmatrix} \) has the value
(a) 0
(b) 1
(c) \( \sin A \sin B \cos C \)
(d) None of the options
Answer: (a) 0

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

Question. Let \( \{ \Delta_1, \Delta_2, \Delta_3, \dots, \Delta_k \} \) be the set of third order determinants that can be made with the distinct nonzero real numbers \( a_1, a_2, a_3, \dots, a_9 \). Then
(a) \( k = 9! \)
(b) \( \sum_{i=1}^{k} \Delta_i = 0 \)
(c) at least one \( \Delta_i = 0 \)
(d) None of the options
Answer: (a) \( k = 9! \) (b) \( \sum_{i=1}^{k} \Delta_i = 0 \)

Question. \( \begin{vmatrix} x^2 & (y+z)^2 & yz \\ y^2 & (z+x)^2 & zx \\ z^2 & (x+y)^2 & xy \end{vmatrix} \) is divisible by
(a) \( x^2 + y^2 + z^2 \)
(b) \( x - y \)
(c) \( x - y - z \)
(d) \( x + y + z \)
Answer: (a) \( x^2 + y^2 + z^2 \) (b) \( x - y \) (d) \( x + y + z \)

Question. The equation \( \begin{vmatrix} 1 & x & x^2 \\ x^2 & 1 & x \\ x & x^2 & 1 \end{vmatrix} = 0 \) has
(a) exactly two distinct roots
(b) one pair of equal real roots
(c) modulus of each root 1
(d) three pairs of equal roots
Answer: (b) one pair of equal real roots (c) modulus of each root 1 (d) three pairs of equal roots

Question. Let \( f(n) = \begin{vmatrix} n & n+1 & n+2 \\ ^nP_n & ^{n+1}P_{n+1} & ^{n+2}P_{n+2} \\ ^nC_n & ^{n+1}C_{n+1} & ^{n+2}C_{n+2} \end{vmatrix} \), where the symbols have their usual meanings. The \( f(n) \) is divisible by
(a) \( n^2 + n + 1 \)
(b) \( (n + 1)! \)
(c) \( n! \)
(d) None of the options
Answer: (a) \( n^2 + n + 1 \) (c) \( n! \)

Question. Let \( x \neq -1 \) and let \( a, b, c \) be nonzero real numbers. Then the determinant \( \begin{vmatrix} a(1+x) & b & c \\ a & b(1+x) & c \\ a & b & c(1+x) \end{vmatrix} \) is divisible by
(a) \( abcx \)
(b) \( (1 + x)^2 \)
(c) \( (1 + x)^3 \)
(d) \( x(1 + x)^2 \)
Answer: (a) \( abcx \) (b) \( (1 + x)^2 \) (d) \( x(1 + x)^2 \)

Question. The arbitrary constant on which the value of the determinant \( \begin{vmatrix} 1 & \alpha & \alpha^2 \\ \cos(p-d)a & \cos pa & \cos(p+d)a \\ \sin(p-d)a & \sin pa & \sin(p+d)a \end{vmatrix} \) does not depend is
(a) \( \alpha \)
(b) \( p \)
(c) \( d \)
(d) \( a \)
Answer: (b) \( p \)