JEE Mathematics Complex Numbers MCQs Set 07

Question. The equation \( z\bar{z} + (4 – 3i)z + (4 + 3i)\bar{z} + 5 = 0 \) represents a circle whose radius is
(a) 5
(b) \( 2\sqrt{5} \)
(c) \( \frac{5}{2} \)
(d) none of the options
Answer: (b) \( 2\sqrt{5} \)

Question. If z is a complex number such that \( \left|\frac{z - 3i}{z + 3i}\right| = 1 \) then z lies on
(a) the real axis
(b) the line Im(z) = 3
(c) a circle
(d) none of the options
Answer: (a) the real axis

Question. Let \( z_1 \) and \( z_2 \) be two nonreal complex cube roots of unity and \( |z – z_1|^2 + |z – z_2|^2 = \lambda \) be the equation of a circle with \( z_1, z_2 \) as ends of a diameter then the value of \( \lambda \) is
(a) 4
(b) 3
(c) 2
(d) \( \sqrt{2} \)
Answer: (b) 3

Question. Let \( \lambda \in R \). If the origin and the nonreal roots of \( 2z^2 + 2z + \lambda = 0 \) form the three vertices of an equilateral triangle in the Argand plane then \( \lambda \) is
(a) 1
(b) \( \frac{2}{3} \)
(c) 2
(d) 1
Answer: (b) \( \frac{2}{3} \)

Question. The equation \( |z – i| + |z + i| = k \), k > 0, can represent an ellipse if k is
(a) 1
(b) 2
(c) 4
(d) none of the options
Answer: (c) 4

Question. The equation \( |z + i| - |z – i| = k \) represents a hyperbola if
(a) -2 < k < 2
(b) k > 2
(c) 0 < k < 2
(d) none of the options
Answer: (a) -2 < k < 2

Question. Let OP.OQ = 1 and let O, P, Q be three collinear points. If O and Q represent the complex numbers 0 and z then P represents
(a) \( \frac{1}{z} \)
(b) \( \bar{z} \)
(c) \( \frac{1}{\bar{z}} \)
(d) none of the options
Answer: (c) \( \frac{1}{\bar{z}} \)

Question. Let \( z = 1 - t + i\sqrt{t^2 + t + 2} \), where t is a real parameter. Then locus of z in the Argand plane is
(a) a hyperbola
(b) an ellipse
(c) a straight line
(d) none of the options
Answer: (a) a hyperbola

Question. The area of the triangle whose vertices are i, \( \alpha, \beta \), where \( i = \sqrt{-1} \) and \( \alpha, \beta \) are the nonreal cube roots of unity, is
(a) \( \frac{3\sqrt{3}}{2} \)
(b) \( \frac{3\sqrt{3}}{4} \)
(c) 0
(d) \( \frac{\sqrt{3}}{4} \)
Answer: (d) \( \frac{\sqrt{3}}{4} \)

Question. The nonzero real value of x for which \( \frac{(1 + ix)(1 + 2ix)}{1 - ix} \) is purely real is
(a) \( \sqrt{2} \)
(b) 1
(c) \( -\sqrt{2} \)
(d) none of the options​
Answer: (a) \( \sqrt{2} \), (c) \( -\sqrt{2} \)

Question. If \( z_1 = \frac{1}{a + i}, a \ne 0 \) and \( z_2 = \frac{1}{1 + bi}, b \ne 0 \) such that \( z_1 = \bar{z}_2 \) then
(a) a = 1, b = 1
(b) a = -1, b = 1
(c) a = 1, b = -1
(d) none of the options
Answer: (c) a = 1, b = -1

Question. If \( z_1, z_2, z_3, z_4 \) are roots of the equation \( a_0z^4 + a_1z^3 + a_2z^2 + a_3z + a_4 = 0 \) where \( a_0, a_1, a_2, a_3 \) and \( a_4 \) are real, then
(a) \( \bar{z}_1, \bar{z}_2, \bar{z}_3, \bar{z}_4 \) are also roots of the equation
(b) \( z_1 \) is equal to at least one of \( \bar{z}_1, \bar{z}_2, \bar{z}_3, \bar{z}_4 \)
(c) \( -\bar{z}_1, -\bar{z}_2, -\bar{z}_3, -\bar{z}_4 \) are also roots of the equation
(d) none of the options
Answer: (a) \( \bar{z}_1, \bar{z}_2, \bar{z}_3, \bar{z}_4 \) are also roots of the equation, (b) \( z_1 \) is equal to at least one of \( \bar{z}_1, \bar{z}_2, \bar{z}_3, \bar{z}_4 \)

Question. If \( \alpha \) is a complex constant such that \( \alpha z^2 + z + \bar{\alpha} = 0 \) has a real root then
(a) \( \alpha + \bar{\alpha} = 1 \)
(b) \( \alpha + \bar{\alpha} = 0 \)
(c) \( \alpha + \bar{\alpha} = -1 \)
(d) the absolute value of the real roots is 1
Answer: (a) \( \alpha + \bar{\alpha} = 1 \), (c) \( \alpha + \bar{\alpha} = -1 \), (d) the absolute value of the real roots is 1

Question. If amp(\( z_1z_2 \)) = 0 and \( |z_1| = |z_2| = 1 \) then
(a) \( z_1 + z_2 = 0 \)
(b) \( z_1z_2 = 1 \)
(c) \( z_1 = \bar{z}_2 \)
(d) none of the options
Answer: (b) \( z_1z_2 = 1 \), (c) \( z_1 = \bar{z}_2 \)

Question. If z is a nonzero complex number then \( \frac{|z|^2}{z\bar{z}} \) is equal to
(a) \( \frac{z}{\bar{z}} \)
(b) 1
(c) \( \frac{\bar{z}}{z} \)
(d) none of the options
Answer: (a) \( \frac{z}{\bar{z}} \) (Error in option matching/key? Key says ab. But expression is 1. If ab are correct, then z must be real. Assuming general z, answer is 1.)
Note: The answer key indicates (a) and (b).

Question. If \( \omega \) is a nonreal cube root of unity then the value of \( 1.(2 - \omega)(2 - \omega^2) + 2. (3 - \omega)(3 - \omega^2) + .... + (n – 1)(n - \omega)(n - \omega^2) \) is
(a) real
(b) \( \frac{n^2(n-1)^2}{4} - n + 1 \)
(c) \( \left\{ \frac{n(n+1)}{2} \right\}^2 - n \)
(d) not real
Answer: (a) real, (b) \( \frac{n^2(n-1)^2}{4} - n + 1 \)

Question. If z is a complex number satisfying \( z + z^{-1} = 1 \) then \( z^n + z^{-n} \), \( n \in N \), has the value
(a) \( 2(-1)^n \) when n is a multiple of 3
(b) \( (-1)^{n-1} \) when n is not a multiple of 3
(c) \( (-1)^{n+1} \) when n is a multiple of 3
(d) 0 when n is not a multiple of 3
Answer: (a) \( 2(-1)^n \) when n is a multiple of 3, (b) \( (-1)^{n-1} \) when n is not a multiple of 3

Question. The value of \( \alpha^{-n} + \alpha^{-2n} \), \( n \in N \) and \( \alpha \) is a nonreal cube root of unity, is
(a) 3 if n is a multiple of 3
(b) -1 if n is not a multiple of 3
(c) 2 if n is a multiple of 3
(d) none of the options
Answer: (b) -1 if n is not a multiple of 3, (c) 2 if n is a multiple of 3

Question. The value of \( \alpha^{4n-1} + \alpha^{4n-2} + \alpha^{4n-3} \), \( n \in N \) and \( \alpha \) is a nonreal fourth root of unity, is
(a) 0
(b) -1
(c) 3
(d) none of the options
Answer: (b) -1

Question. Let x be a nonreal complex number satisfying \( (x – 1)^3 + 8 = 0 \) then x is
(a) \( 1 + 2\omega \)
(b) \( 1 - 2\omega \)
(c) \( 1 - 2\omega^2 \)
(d) none of the options
Answer: (b) \( 1 - 2\omega \), (c) \( 1 - 2\omega^2 \)

Question. If \( z = \frac{1+3i}{1+i} \) then
(a) Re(z) = 2Im(z)
(b) Re(z) + 2Im(z) = 0
(c) \( |z| = \sqrt{5} \)
(d) amp z = tan⁻¹2
Answer: (a) Re(z) = 2Im(z), (c) \( |z| = \sqrt{5} \)

Question. If z is different from \( \pm i \) and |z| = 1 then \( \frac{z+i}{z-i} \) is
(a) purely real
(b) nonreal, whose real and imaginary parts are equal
(c) purely imaginary
(d) none of the options
Answer: (c) purely imaginary

Question. If \( z_1, z_2 \) are two compelx numbers then
(a) \( |z_1 + z_2| \le |z_1| + |z_2| \)
(b) \( |z_1 - z_2| \ge |z_1| - |z_2| \)
(c) \( |z_1 + z_2| \ge |z_1 \cdot z_2| \)
(d) \( |z_1 - z_2| \le |z_1 + z_2| \)
Answer: (a) \( |z_1 + z_2| \le |z_1| + |z_2| \), (b) \( |z_1 - z_2| \ge |z_1| - |z_2| \)

Question. Let \( z_1, z_2 \) be two complex numbers represented by points on the circle |z| = 1 and |z| = 2 respectively then
(a) \( \max |2z_1 + z_2| = 4 \)
(b) \( \min |z_1 - z_2| = 1 \)
(c) \( \left| z_2 + \frac{1}{z_1} \right| \le 3 \)
(d) none of the options
Answer: (a) \( \max |2z_1 + z_2| = 4 \), (b) \( \min |z_1 - z_2| = 1 \), (c) \( \left| z_2 + \frac{1}{z_1} \right| \le 3 \)

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 \( \bar{z} \)
(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 \( z(\bar{z} + \alpha) + \bar{z}(z + \alpha) = 0 \), where \( \alpha \) is a complex constant, then z is represented by a point on
(a) a straight line
(b) a circle
(c) a parabola
(d) none of the options
Answer: (b) a circle

Question. If \( z_1, z_2, z_3, z_4 \) are the four complex numbers represented by the vertices of a quadrilateral taken in order such that \( z_1 – z_4 = z_2 – z_3 \) and \( \text{amp } \frac{z_4-z_1}{z_2-z_1} = \frac{\pi}{2} \) then the quadrilateral is a
(a) rhombus
(b) square
(c) rectangle
(d) a cyclic quadrilateral
Answer: (c) rectangle, (d) a cyclic quadrilateral

Question. If \( z_0, z_1 \) represent point P, Q on the locus |z – 1| = 1 and the line segment PQ subtends and angle \( \pi/2 \) at the point z = 1 then \( z_1 \) is equal to
(a) \( 1 + i(z_0 - 1) \)
(b) \( 1 + \frac{i}{z_0-1} \)
(c) \( 1 - i(z_0 - 1) \)
(d) \( i(z_0 - 1) \)
Answer: (a) \( 1 + i(z_0 - 1) \), (c) \( 1 - i(z_0 - 1) \)

Question. If \( |z_1| = |z_2| = |z_3| = 1 \) and \( z_1, z_2, z_3 \) are represented by the vertices of an equilateral triangle then
(a) \( z_1 + z_2 + z_3 = 0 \)
(b) \( z_1z_2z_3 = 1 \)
(c) \( z_1z_2 = z_2z_3 + z_3z_1 = 0 \)
(d) none of the options
Answer: (a) \( z_1 + z_2 + z_3 = 0 \), (b) \( z_1z_2z_3 = 1 \)

Question. Let A, B, C be three collinear points which are such that AB.AC = 1 and the points are represented in the Argand plane by the line complex numbers 0, \( z_1, z_2 \) respectively. Then
(a) \( z_1z_2 = 1 \)
(b) \( z_1\bar{z}_2 = 1 \)
(c) \( |z_1||z_2| = 1 \)
(d) none of the options
Answer: (b) \( z_1\bar{z}_2 = 1 \), (c) \( |z_1||z_2| = 1 \)

Question. If \( z_1, z_2, z_3, z_4 \) are represented by the vertices of a rhombus taken in the anticlockwise order then
(a) \( z_1 - z_2 + z_3 - z_4 = 0 \)
(b) \( z_1 + z_2 = z_3 + z_4 \)
(c) \( \text{amp} \frac{z_2 - z_4}{z_1 - z_3} = \frac{\pi}{2} \)
(d) \( \text{amp} \frac{z_1 - z_2}{z_3 - z_4} = \frac{\pi}{2} \)
Answer: (a) \( z_1 - z_2 + z_3 - z_4 = 0 \), (c) \( \text{amp} \frac{z_2 - z_4}{z_1 - z_3} = \frac{\pi}{2} \)

Question. If \( \text{amp} \frac{z-2}{2z+3i} = 0 \) and \( z_0 = 3 + 4i \) then
(a) \( z_0\bar{z} + \bar{z}_0z = 12 \)
(b) \( z_0z + \bar{z}_0\bar{z} = 12 \)
(c) \( z_0\bar{z} + \bar{z}_0z = 0 \)
(d) none of the options
Answer: (b) \( z_0z + \bar{z}_0\bar{z} = 12 \)

Question. If \( z_1 \ne z_2 \) and \( |z_1 + z_2| = \left| \frac{1}{z_1} + \frac{1}{z_2} \right| \) then
(a) at least one of \( z_1, z_2 \) is unimodular
(b) both \( z_1, z_2 \) are unimodular
(c) \( z_1.z_2 \) is unimodular
(d) none of the options
Answer: (c) \( z_1.z_2 \) is unimodular

Question. Let \( z_1 = \frac{(\sqrt{3}+i)^2(1-\sqrt{3}i)}{1+i}, z_2 = \frac{(1+\sqrt{3}i)^2(\sqrt{3}-i)}{1-i} \). Then
(a) \( |z_1| = |z_2| \)
(b) amp \( z_1 \) + amp \( z_2 \) = 0
(c) 3\( |z_1| = |z_2| \)
(d) 3amp \( z_1 \) + amp \( z_2 \) = 0
Answer: (a) \( |z_1| = |z_2| \), (d) 3amp \( z_1 \) + amp \( z_2 \) = 0

Question. If \( |z_1 + z_2| = |z_1 – z_2| \) then
(a) \( |\text{amp } z_1 – \text{amp } z_2| = \frac{\pi}{2} \)
(b) \( |\text{amp } z_1 – \text{amp } z_2| = \pi \)
(c) \( \frac{z_1}{z_2} \) is purely real
(d) \( \frac{z_1}{z_2} \) is purely imaginary
Answer: (a) \( |\text{amp } z_1 – \text{amp } z_2| = \frac{\pi}{2} \), (d) \( \frac{z_1}{z_2} \) is purely imaginary

Question. If \( |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 \) then
(a) \( \frac{z_1}{z_2} \) is purely real
(b) \( \frac{z_1}{z_2} \) is purely imaginary
(c) \( z_1\bar{z}_2 + z_2\bar{z}_1 = 0 \)
(d) \( \text{amp } \frac{z_1}{z_2} = \frac{\pi}{2} \)
Answer: (b) \( \frac{z_1}{z_2} \) is purely imaginary, (c) \( z_1\bar{z}_2 + z_2\bar{z}_1 = 0 \), (d) \( \text{amp } \frac{z_1}{z_2} = \frac{\pi}{2} \)