JEE Mathematics Complex Numbers MCQs Set 06

Question. If \( z_r = \cos\frac{2r\pi}{5} + i\sin\frac{2r\pi}{5} \), \( r = 0, 1, 2, 3, 4, \dots \) then \( z_1z_2z_3z_4z_5 \) is equal to
(a) -1
(b) 0
(c) 1
(d) none of the options
Answer: (c) 1

Question. If \( e^{i\theta} = \cos\theta + i\sin\theta \) then for the \( \Delta ABC \), \( e^{iA} \cdot e^{iB} \cdot e^{iC} \) is
(a) –i
(b) 1
(c) -1
(d) none of the options
Answer: (c) -1

Question. If \( (\sqrt{3} + i)^n = (\sqrt{3} - i)^n \), \( n \in N \) then the least value of n is
(a) 3
(b) 4
(c) 6
(d) none of the options
Answer: (c) 6

Question. If the fourth roots of unity are \( z_1, z_2, z_3, z_4 \) then \( z_1^2 + z_2^2 + z_3^2 + z_4^2 \) is equal to
(a) 1
(b) 0
(c) i
(d) none of the options
Answer: (b) 0

Question. If \( x^3 - 1 = 0 \) has the nonreal complex roots \( \alpha, \beta \) then the value of \( (1 + 2\alpha + \beta)^3 – (3 + 3\alpha + 5\beta)^3 \) is
(a) -7
(b) 6
(c) -5
(d) 0
Answer: (a) -7

Question. If \( i = \sqrt{-1} \) then \( 4 + 5\left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right)^{334} - 3\left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right)^{365} \) is equal to
(a) \( 1 - i\sqrt{3} \)
(b) \( -1 + i\sqrt{3} \)
(c) \( 4\sqrt{3}i \)
(d) \( -i\sqrt{3} \)
Answer: (c) \( 4\sqrt{3}i \)

Question. If \( (\sqrt{3} - i)^n = 2^n \), \( n \in Z \), the set of integers, then n is a multiple of
(a) 6
(b) 10
(c) 9
(d) 12
Answer: (d) 12

Question. If \( z(2 – i2\sqrt{3})^2 = i(\sqrt{3} + i)^4 \) the amplitude of z is
(a) \( \frac{5\pi}{6} \)
(b) \( -\frac{\pi}{6} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{7\pi}{6} \)
Answer: (b) \( -\frac{\pi}{6} \)

Question. If z is a nonreal root of \( \sqrt[7]{-1} \) then \( z^{86} + z^{175} + z^{289} \) is equal to
(a) 0
(b) -1
(c) 3
(d) 1
Answer: (b) -1

Question. If \( \alpha \) is nonreal and \( \alpha = \sqrt[5]{1} \) then the value of \( 2^{|1+\alpha+\alpha^2+\alpha^{-2}-\alpha^{-1}|} \) is equal to
(a) 4
(b) 2
(c) 1
(d) none of the options
Answer: (a) 4

Question. The value of amp (i\( \omega \)) + amp (i\( \omega^2 \)), where \( i = \sqrt{-1} \) and \( \omega = \sqrt[3]{1} \) = nonreal, is
(a) 0
(b) \( \frac{\pi}{2} \)
(c) \( \pi \)
(d) none of the options
Answer: (c) \( \pi \)

Question. If \( \alpha, \beta \) be two complex numbers then \( |\alpha|^2 + |\beta|^2 \) is equal to
(a) \( \frac{1}{2}(|\alpha + \beta|^2 - |\alpha - \beta|^2) \)
(b) \( \frac{1}{2}(|\alpha + \beta|^2 + |\alpha - \beta|^2) \)
(c) \( |\alpha + \beta|^2 + |\alpha - \beta|^2 \)
(d) none of the options
Answer: (b) \( \frac{1}{2}(|\alpha + \beta|^2 + |\alpha - \beta|^2) \)

Question. The set of values of \( a \in R \) for which \( x^2 + i(a – 1)x + 5 = 0 \) will have a pair conjugate complex roots is
(a) R
(b) {1}
(c) \( \{a | a^2 – 2a + 21 > 0\} \)
(d) none of the options
Answer: (b) {1}

Question. Nonreal complex numbers z satisfying the equation \( z^3 + 2z^2 + 3z + 2 = 0 \) are
(a) \( \frac{-1 \pm \sqrt{-7}}{2} \)
(b) \( \frac{1 + \sqrt{7}i}{2}, \frac{1 - \sqrt{7}i}{2} \)
(c) \( -i, \frac{-1 + \sqrt{7}i}{2}, \frac{-1 - \sqrt{7}i}{2} \)
(d) none of the options
Answer: (a) \( \frac{-1 \pm \sqrt{-7}}{2} \)

Question. For a complex number z, the minimum value of |z| + |z - 2| is
(a) 1
(b) 2
(c) 3
(d) none of the options
Answer: (b) 2

Question. If |z| = 1 then \( \frac{1+z}{1+\bar{z}} \) is equal to
(a) z
(b) \( \bar{z} \)
(c) \( z + \bar{z} \)
(d) none of the options
Answer: (a) z

Question. If \( \alpha \) is a nonreal cube root of unity then \( |\alpha^n| \), \( n \in Z \), is equal to
(a) 1
(b) 3
(c) 0
(d) none of the options
Answer: (a) 1

Question. If z be a complex number satisfying \( z^4 + z^3 + 2z^2 + z + 1 = 0 \) then |z| is
(a) \( \frac{1}{2} \)
(b) \( \frac{3}{4} \)
(c) 1
(d) none of the options
Answer: (c) 1

Question. Let \( z_1 = a + ib, z_2 = p + iq \) be two unimodular complex numbers such that \( \text{Im}(z_1\bar{z}_2) = 1 \). If \( \omega_1 = a + ip, \omega_2 = b + iq \) then
(a) \( \text{Re}(\omega_1\omega_2) = 1 \)
(b) \( \text{Im}(\omega_1\omega_2) = 1 \)
(c) \( \text{Re}(\omega_1\omega_2) = 0 \)
(d) \( \text{Im}(\omega_1\bar{\omega}_2) = 1 \)
Answer: (d) \( \text{Im}(\omega_1\bar{\omega}_2) = 1 \)

Question. If \( |z_1 – 1| < 1, |z_2 – 2| < 2, |z_3 – 3| < 3 \) then \( |z_1 + z_2 + z_3| \)
(a) is less than 6
(b) is more than 3
(c) is less than 12
(d) lies between 6 and 12
Answer: (c) is less than 12

Question. If \( |z – i| \le 2 \) and \( z_0 = 5 + 3i \) then the maximum value of \( |iz + z_0| \) is
(a) \( 2 + \sqrt{31} \)
(b) 7
(c) \( \sqrt{31} - 2 \)
(d) none of the options
Answer: (b) 7

Question. If \( |z| = \max \{|z – 1|, |z + 1|\} \) then
(a) \( |z_1 + \bar{z}| = \frac{1}{2} \)
(b) \( z_1 + \bar{z} = 1 \)
(c) \( |z_1 + z| = 1 \)
(d) none of the options
Answer: (c) \( |z_1 + z| = 1 \)

Question. \( |z – 4| < |z – 2| \) represents the region given by
(a) Re(z) > 0
(b) Re(z) < 0
(c) Re(z) > 2
(d) none of the options
Answer: (d) none of the options

Question. If \( \log_{1/2} \frac{|z|^2 + 2|z| + 4}{2|z|^2 + 1} < 0 \) then the region traced by z is
(a) |z| < 3
(b) 1 < |z| < 3
(c) |z| > 1
(d) |z| < 2
Answer: (a) |z| < 3

Question. \( \left| \frac{z-1}{z+1} \right| = 1 \) represents
(a) a circle
(b) an ellipse
(c) a straight line
(d) none of the options
Answer: (c) a straight line

Question. If \( 2z_1 – 3z_2 + z_3 = 0 \) then \( z_1, z_2, z_3 \) are represented by
(a) three vertices of a triangle
(b) three collinear points
(c) three vertices of a rhombus
(d) none of the options
Answer: (b) three collinear points

Question. If A, B, C are three points in the Argand plane representing the complex numbers \( z_1, z_2, z_3 \) such that \( z_1 = \frac{\lambda z_2 + z_3}{\lambda + 1} \), where \( \lambda \in R \), then the distance of A from the line BC is
(a) \( \lambda \)
(b) \( \frac{\lambda}{\lambda + 1} \)
(c) 1
(d) 0
Answer: (d) 0

Question. The roots of the equation \( 1 + z + z^3 + z^4 = 0 \) are represented by the vertices of
(a) a square
(b) an equilateral triangle
(c) a rhombus
(d) none of the options
Answer: (b) an equilateral triangle

Question. If \( \text{Re}\left(\frac{z + 4}{2z - i}\right) = \frac{1}{2} \) then z is represented by a point lying on
(a) a circle
(b) an ellipse
(c) a straight line
(d) none of the options
Answer: (c) a straight line

Question. The angle that the vector representing the complex number \( \frac{1}{(\sqrt{3}-i)^{25}} \) makes with the positive direction of the real axis is
(a) \( \frac{2\pi}{3} \)
(b) \( -\frac{\pi}{6} \)
(c) \( \frac{5\pi}{6} \)
(d) \( \frac{\pi}{6} \)
Answer: (d) \( \frac{\pi}{6} \)

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) \( \frac{z}{z_1} = \frac{\bar{z}_1}{\bar{z}} \)
(b) \( z\bar{z} + z_1\bar{z}_1 = 0 \)
(c) \( z\bar{z}_1 + \bar{z}z_1 = 0 \)
(d) none of the options
Answer: (a) \( \frac{z}{z_1} = \frac{\bar{z}_1}{\bar{z}} \)

Question. If \( |z_1| = |z_2| = |z_3| = |z_4| \) then the points representing \( z_1, z_2, z_3, z_4 \) are
(a) concyclic
(b) vertices of a square
(c) vertices of a rhombus
(d) none of the options
Answer: (a) concyclic

Question. Suppose \( z_1, z_2, z_3 \) are the vertices of an equilateral triangle inscribed in the circle |z| = 2. If \( z_1 = 1 + \sqrt{3}i \) and \( z_1, z_2, z_3 \) are in the clockwise sense then
(a) \( z_1 = 1 - \sqrt{3}i, z_3 = -2 \)
(b) \( z_2 = 2, z_3 = 1 - \sqrt{3}i \)
(c) \( z_2 = -1 + \sqrt{3}i, z_3 = -2 \)
(d) none of the options
Answer: (a) \( z_1 = 1 - \sqrt{3}i, z_3 = -2 \)

Question. Suppose \( z_1, z_2, z_3 \) are the vertices of an equilateral triangle circumscribing the circle |z| = 1. If \( z_1 = 1 + \sqrt{3}i \) and \( z_1, z_2, z_3 \) are in the anticlockwise sense then \( z_2 \) is
(a) \( 1 - \sqrt{3}i \)
(b) 2
(c) \( -\frac{1}{2}(1 - \sqrt{3}i) \)
(d) none of the options
Answer: (d) none of the options

Question. If \( \text{amp } \frac{z-1}{z+1} = \frac{\pi}{3} \) then z represents a point on
(a) a straight line
(b) a circle
(c) a pair of lines
(d) none of the options
Answer: (b) a circle

Question. If the roots of \( z^3 + iz^2 + 2i = 0 \) represent the vertices of a \( \Delta ABC \) in the Argand plane then the area of the triangle is
(a) \( \frac{3\sqrt{7}}{2} \)
(b) \( \frac{3\sqrt{7}}{4} \)
(c) 2
(d) none of the options
Answer: (c) 2