JEE Mathematics Complex Numbers MCQs Set 09

Question. If \( (1 + i)z = (1 - i)\bar{z} \) then \( z \) is
(a) \( t(1 - i), t \in \mathbb{R} \)
(b) \( t(1 + i), t \in \mathbb{R} \)
(c) \( \frac{t}{1 + i}, t \in \mathbb{R}^+ \)
(d) None of the options
Answer: (a) \( t(1 - i), t \in \mathbb{R} \)

Question. If \( |z + 4| \leq 3 \), then the maximum value of \( |z + 1| \) is
(a) 4
(b) 10
(c) 6
(d) 0
Answer: (c) 6

Question. The value of \( \sum_{k=1}^{10} \left(\sin \frac{2k\pi}{11} + i \cos \frac{2k\pi}{11}\right) \)
(a) 1
(b) -1
(c) -i
(d) i
Answer: (c) -i

Question. If the cube roots of unity are \( 1, \omega, \omega^2 \), then roots of the equation \( (x - 1)^3 + 8 = 0 \) are
(a) \( -1, 1 + 2\omega, 1 + 2\omega^2 \)
(b) \( -1, 1 - 2\omega, 1 - 2\omega^2 \)
(c) \( -1, -1, -1 \)
(d) \( -1, -1 + 2\omega, -1 - 2\omega^2 \)
Answer: (b) \( -1, 1 - 2\omega, 1 - 2\omega^2 \)

Question. If \( z_1 \) and \( z_2 \) are two non-zero complex numbers such that \( |z_1 + z_2| = |z_1| + |z_2| \) then \( \text{arg} z_1 - \text{arg} z_2 \) is equal to
(a) \( -\frac{\pi}{2} \)
(b) 0
(c) \( -\pi \)
(d) \( \frac{\pi}{2} \)
Answer: (b) 0

Question. If \( w = \frac{z}{z - \frac{1}{3}i} \) and \( |w| = 1 \), then \( z \) lies on
(a) a parabola
(b) a straight line
(c) a circle
(d) an ellipse
Answer: (b) a straight line

Question. Let \( z, w \) be complex numbers such that \( \bar{z} + i\bar{w} = 0 \) and \( \text{arg} zw = \pi \). Then \( \text{arg} z \) equals
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{3\pi}{4} \)
(d) \( \frac{5\pi}{4} \)
Answer: (c) \( \frac{3\pi}{4} \)

Question. If \( |z^2 - 1| = |z^2| + 1 \), then \( z \) lies on
(a) the real axis
(b) the imaginary axis
(c) a circle
(d) an ellipse
Answer: (b) the imaginary axis

Question. Let \( z_1 \) and \( z_2 \) be two roots of the equation \( z^2 + az + b = 0 \), \( z \) being complex. Further, assume that the origin \( z_1 \) and \( z_2 \) form an equilateral triangle. Then
(a) \( a^2 = b \)
(b) \( a^2 = 2b \)
(c) \( a^2 = 3b \)
(d) \( a^2 = 4b \)
Answer: (c) \( a^2 = 3b \)

Question. If \( z \) and \( \omega \) are two non-zero complex numbers such that \( |z\omega| = 1 \), and \( \text{arg}(z) - \text{arg}(\omega) = \frac{\pi}{2} \) then \( \bar{z}\omega \) is equal to
(a) 1
(b) -1
(c) i
(d) -i
Answer: (d) -i

Question. If \( z_r = \cos \left(\frac{\pi}{2^r}\right) + i \sin \left(\frac{\pi}{2^r}\right) \), \( r = 1, 2, \dots \) then \( z_1 z_2 z_3 \dots \infty \) is equal to
(a) -1
(b) i
(c) -i
(d) 1
Answer: (a) -1

Question. \( \left(1 + \cos \frac{\pi}{8}\right)\left(1 + \cos \frac{3\pi}{8}\right)\left(1 + \cos \frac{5\pi}{8}\right)\left(1 + \cos \frac{7\pi}{8}\right) \) is equal to
(a) \( \frac{1 + \sqrt{2}}{2\sqrt{2}} \)
(b) \( \frac{1}{8} \)
(c) \( \cos \frac{\pi}{8} \)
(d) \( \frac{1}{2} \)
Answer: (b) \( \frac{1}{8} \)

Question. The product of cube roots of -1 is equal to
(a) -2
(b) 0
(c) -1
(d) 4
Answer: (c) -1

Question. If the complex numbers \( iz, z \) and \( z + iz \) represent the three vertices of a triangle then the area of the triangle is
(a) \( \frac{1}{2} |z - 1| \)
(b) \( |z|^2 \)
(c) \( \frac{1}{2} |z|^2 \)
(d) \( |z - 1|^2 \)
Answer: (c) \( \frac{1}{2} |z|^2 \)

Question. Complex numbers \( z_1, z_2 \) and \( z_3 \) in AP
(a) lie on ellipse
(b) lie on a parabola
(c) lie on line
(d) lie on circle
Answer: (c) lie on line

Question. If \( \sin^3 x \sin 3x = \sum_{m=0}^n C_m \cos mx \) is an identity in \( x \), where \( C_0, C_1, \dots, C_n \) are constants and \( C_n \neq 0 \) then the value of \( n \) equals
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (c) 6

Question. If magnitude of a complex number \( 4 - 3i \) is tripled and is rotated by an angle \( \pi \) anticlockwise then resulting complex number would be
(a) \( -12 + 9i \)
(b) \( 12 + 9i \)
(c) \( 7 - 6i \)
(d) \( 7 + 6i \)
Answer: (a) \( -12 + 9i \)

Question. If \( |z - 2 - 3i| + |z + 2 - 6i| = 4 \) where \( i = \sqrt{-1} \) then locus of \( P(z) \) is
(a) an ellipse
(b) \( \phi \)
(c) segment joining the point \( 2 + 3i; -2 + 6i \)
(d) None of the options
Answer: (b) \( \phi \)

Question. For all complex numbers \( z_1, z_2 \) satisfying \( |z_1| = 12 \) and \( |z_2 - 3 - 4i| = 5 \), the minimum value of \( |z_1 - z_2| \) is
(a) 0
(b) 2
(c) 7
(d) 13
Answer: (b) 2

Question. If \( z_1, z_2 \) and \( z_3 \) are complex numbers such that \( |z_1| = |z_2| = |z_3| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1 \), then \( |z_1 + z_2 + z_3| \) is
(a) equal to 1
(b) less than 1
(c) greater than 3
(d) equal to 3
Answer: (a) equal to 1

Question. If \( 1, \alpha, \alpha^2, \dots, \alpha^{n - 1} \) are nth roots of unity. The value of \( (3 - \alpha) (3 - \alpha^2) (3 - \alpha^3) \dots (3 - \alpha^{n - 1}) \) is
(a) \( n \)
(b) 0
(c) \( \frac{3^n - 1}{2} \)
(d) \( \frac{3^n + 1}{2} \)
Answer: (c) \( \frac{3^n - 1}{2} \)

Question. In one root of the quadratic equation \( (1 + i)x^2 - (7 + 3i)x + (6 + 8i) = 0 \) is \( 4 - 3i \), then the other root must be
(a) \( 1 + i \)
(b) \( 4 + 3i \)
(c) \( 1 - i \)
(d) None of the options
Answer: (a) \( 1 + i \)

Question. If \( P, P' \) represent the complex number \( z_1 \) and its additive inverse respectively then the complex equation of the circle with \( PP' \) as a diameter is
(a) \( \frac{z}{z_1} = \left(\overline{\frac{z_1}{z}}\right) \)
(b) \( z\bar{z} + z_1\bar{z}_1 = 0 \)
(c) \( z\bar{z} + z\bar{z}_1 = 0 \)
(d) None of the options
Answer: (a) \( \frac{z}{z_1} = \left(\overline{\frac{z_1}{z}}\right) \)

Question. If \( z = x + iy \) satisfies \( \text{amp} (z - 1) = \text{amp} (z + 3) \) then the value of \( (x - 1) : y \) is equal to
(a) 2 : 1
(b) 1 : 3
(c) -1 : 3
(d) does not exist
Answer: (d) does not exist

Question. Let \( z (\neq 2) \) be a complex number such that \( \log_{1/2} |z - 2| > \log_{1/2} |z| \), then
(a) \( \text{Re}(z) > 1 \)
(b) \( \text{Im}(z) > 1 \)
(c) \( \text{Re}(z) = 1 \)
(d) \( \text{Im}(z) = 1 \)
Answer: (a) \( \text{Re}(z) > 1 \)

Question. The number of solutions of \( z^3 + \bar{z} = 0 \) is
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (d) 5

Question. If \( iz^3 + z^2 - z + i = 0 \), then \( |z| \) equals
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (d) 1

Question. If \( a > 0 \) and the equation \( |z - a^2| + |z - 2a| = 3 \) represents an ellipse then \( a \) lies in
(a) (1, 3)
(b) \( (\sqrt{2}, \sqrt{3}) \)
(c) (0, 3)
(d) \( (1, \sqrt{3}) \)
Answer: (c) (0, 3)

Question. If \( w \neq 1 \) is nth root of unity, then value of \( \sum_{k=0}^{n-1} |z_1 + w^k z_2|^2 \) is
(a) \( n(|z_1|^2 + |z_2|^2) \)
(b) \( |z_1|^2 + |z_2|^2 \)
(c) \( (|z_1| + |z_2|)^2 \)
(d) \( n(|z_1| + |z_2|)^2 \)
Answer: (a) \( n(|z_1|^2 + |z_2|^2) \)

Question. If \( |z_1| = 2, |z_2| = 3, |z_3| = 4 \) and \( |2z_1 + 3z_2 + 4z_3| = 4 \) then absolute value of \( 8z_2 z_3 + 27z_3 z_1 + 64z_1 z_2 \) equals
(a) 24
(b) 48
(c) 72
(d) 96
Answer: (d) 96

Question. If \( z_1, z_2, z_3 \) are three complex numbers such that \( 4z_1 - 7z_2 + 3z_3 = 0 \), then \( z_1, z_2, z_3 \) are
(a) vertices of a scalene triangle
(b) vertices of a right triangle
(c) points on a circle
(d) collinear points
Answer: (d) collinear points

Question. If \( z = x + iy \) then the equation of a straight line \( Ax + By + C = 0 \) where \( A, B, C \in \mathbb{R} \), can be written on the complex plane in the form \( \bar{a}z + a\bar{z} + 2C = 0 \) where 'a' is equal to
(a) \( \frac{(A + iB)}{2} \)
(b) \( \frac{A - iB}{2} \)
(c) \( A + iB \)
(d) None of the options
Answer: (c) \( A + iB \)

Question. If \( z_1, z_2, z_3, \dots, z_n \) lie on the circle \( |z| = 2 \), then the value of \( E = |z_1 + z_2 + \dots + z_n| - 4 \left| \frac{1}{z_1} + \frac{1}{z_2} + \dots + \frac{1}{z_n} \right| \) is
(a) 0
(b) n
(c) -n
(d) None of the options
Answer: (a) 0

Question. The number of solutions of the equation in \( z, z\bar{z} - (3 + i)z - (3 - i)\bar{z} - 6 = 0 \) is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (d) infinite

Question. If \( 1 + x^2 = \sqrt{3}x \) then \( \sum_{n=1}^{24} \left(x^n - \frac{1}{x^n}\right) \) is equal to
(a) 48
(b) -48
(c) \( \pm 48 (\omega - \omega^2) \)
(d) None of the options
Answer: (d) None of the options

Question. If \( w (\neq 1) \) is a cube root of unity, then
\( \begin{vmatrix} 1 & 1 + i + \omega^2 & \omega^2 \\ 1 - i & -1 & \omega^2 - 1 \\ -i & -i + \omega - 1 & -1 \end{vmatrix} \) equals
(a) 0
(b) 1
(c) i
(d) \( \omega \)
Answer: (a) 0

Question. If z is a complex number then the equation \( z^2 + z |z| + |z|^2 = 0 \) is satisfied by (\( \omega \) and \( \omega^2 \) are imaginary cube roots of unity)
(a) \( z = k\omega \) where \( k \in R \)
(b) \( z = k\omega^2 \) where \( k \) is non negative real
(c) \( z = k\omega \) where \( k \) is positive real
(d) \( z = k \omega^2 \) where \( k \in R \)
Answer: (b) \( z = k\omega^2 \) where \( k \) is non negative real, (c) \( z = k\omega \) where \( k \) is positive real

Question. If \( 2 \cos \theta = x + \frac{1}{x} \) and \( 2 \cos \varphi = y + \frac{1}{y} \), then
(a) \( x^n + \frac{1}{x^n} = 2 \cos (n\theta) \)
(b) \( \frac{x}{y} + \frac{y}{x} = 2 \cos (\theta - \varphi) \)
(c) \( xy + \frac{1}{xy} = 2 \cos (\theta + \varphi) \)
(d) None of the options
Answer: (a) \( x^n + \frac{1}{x^n} = 2 \cos (n\theta) \), (b) \( \frac{x}{y} + \frac{y}{x} = 2 \cos (\theta - \varphi) \), (c) \( xy + \frac{1}{xy} = 2 \cos (\theta + \varphi) \)

Question. The value of \( i^n + i^{-n} \), for \( i = \sqrt{-1} \) and \( n \in \mathbb{I} \) is
(a) \( \frac{2^n}{(1 - i)^{2n}} + \frac{(1 + i)^{2n}}{2^n} \)
(b) \( \frac{(1 + i)^{2n}}{2^n} + \frac{(1 - i)^{2n}}{2^n} \)
(c) \( \frac{(1 + i)^{2n}}{2^n} + \frac{2^n}{(1 - i)^{2n}} \)
(d) \( \frac{2^n}{(1 + i)^{2n}} + \frac{2^n}{(1 - i)^{2n}} \)
Answer: (b) \( \frac{(1 + i)^{2n}}{2^n} + \frac{(1 - i)^{2n}}{2^n} \), (d) \( \frac{2^n}{(1 + i)^{2n}} + \frac{2^n}{(1 - i)^{2n}} \)

Question. ABCD is a square, vertices being taken in the anticlockwise sense. If A represents the complex number z and the intersection of the diagonals is the origin then
(a) B represents the complex number \( iz \)
(b) D represents the complex number \( i\bar{z} \)
(c) B represents the complex number \( i\bar{z} \)
(d) D represents the complex number \( -iz \)
Answer: (a) B represents the complex number \( iz \), (d) D represents the complex number \( -iz \)

Question. If \( g(x) \) and \( h(x) \) are two real polynomials such that the polynomial \( g(x^3) + xh (x^3) \) is divisible by \( x^2 + x + 1 \), then
(a) \( g(1) = h(1) = 0 \)
(b) \( g(1) = h(1) \neq 0 \)
(c) \( g(1) = -h(1) \)
(d) \( g(1) + h(1) = 0 \)
Answer: (a) \( g(1) = h(1) = 0 \), (c) \( g(1) = -h(1) \), (d) \( g(1) + h(1) = 0 \)