JEE Mathematics Complex Numbers MCQs Set 08

Question. If \( z \) is a complex number such that \( |z| = 4 \) and \( \text{arg}(z) = \frac{5\pi}{6} \), then \( z \) is equal to
(a) \( -2\sqrt{3} + 2i \)
(b) \( 2\sqrt{3} + i \)
(c) \( 2\sqrt{3} - 2i \)
(d) \( -\sqrt{3} + i \)
Answer: (a) \( -2\sqrt{3} + 2i \)

Question. The argument of the complex number \( \sin \left(\frac{6\pi}{5}\right) + i \left(1 + \cos \frac{6\pi}{5}\right) \) is
(a) \( \frac{6\pi}{5} \)
(b) \( \frac{5\pi}{6} \)
(c) \( \frac{9\pi}{10} \)
(d) \( \frac{2\pi}{5} \)
Answer: (c) \( \frac{9\pi}{10} \)

Question. The points \( z_1, z_2, z_3, z_4 \) in the complex plane are the vertices of a parallelogram taken in order if and only if
(a) \( z_1 + z_4 = z_2 + z_3 \)
(b) \( z_1 + z_3 = z_2 + z_4 \)
(c) \( z_1 + z_2 = z_3 + z_4 \)
(d) None of the options
Answer: (b) \( z_1 + z_3 = z_2 + z_4 \)

Question. The curve represented by \( \text{Re}(z^2) = 4 \) is
(a) a parabola
(b) an ellipse
(c) a circle
(d) a rectangular hyperbola
Answer: (d) a rectangular hyperbola

Question. The inequality \( |z - 4| < |z - 2| \) represents
(a) \( \text{Re}(z) > 0 \)
(b) \( \text{Re}(z) < 0 \)
(c) \( \text{Re}(z) > 2 \)
(d) \( \text{Re}(z) > 3 \)
Answer: (d) \( \text{Re}(z) > 3 \)

Question. The number of solutions of the system of equations \( \text{Re}(z^2) = 0, |z| = 2 \) is
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (a) 4

Question. If \( z (\neq -1) \) is a complex number such that \( \frac{z - 1}{z + 1} \) is purely imaginary, then \( |z| \) is equal to
(a) 1
(b) 2
(c) 3
(d) 5
Answer: (a) 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) \( z_1 z_2 z_3 = 1 \)
(c) \( z_1 z_2 + z_2 z_3 + z_3 z_1 = 0 \)
(d) None of the options
Answer: (a) \( z_1 + z_2 + z_3 = 0 \)

Question. If \( z_1, z_2, z_3 \) are vertices of an equilateral triangle inscribed in the circle \( |z| = 2 \) and if \( z_1 = 1 + i\sqrt{3} \) then
(a) \( z_2 = -2, z_3 = 1 + i\sqrt{3} \)
(b) \( z_2 = 2, z_3 = 1 - i\sqrt{3} \)
(c) \( z_2 = -2, z_3 = 1 - i\sqrt{3} \)
(d) \( z_2 = 1 - i\sqrt{3}, z_3 = -1 - i\sqrt{3} \)
Answer: (c) \( z_2 = -2, z_3 = 1 - i\sqrt{3} \)

Question. If \( (\cos \theta + i \sin \theta) (\cos 2\theta + i \sin 2\theta) \dots (\cos n\theta + i \sin n\theta) = 1 \), then the value of \( \theta \) is
(a) \( 4m\pi, m \in \mathbb{Z} \)
(b) \( \frac{2m\pi}{n(n + 1)}, m \in \mathbb{Z} \)
(c) \( \frac{4m\pi}{n(n + 1)}, m \in \mathbb{Z} \)
(d) \( \frac{m\pi}{n(n + 1)}, m \in \mathbb{Z} \)
Answer: (c) \( \frac{4m\pi}{n(n + 1)}, m \in \mathbb{Z} \)

Question. If \( x = a + b + c \), \( y = a\alpha + b\beta + c \) and \( z = a\beta + b\alpha + c \), where \( \alpha \) and \( \beta \) are complex cube roots of unity then \( xyz \) equals
(a) \( 2(a^3 + b^3 + c^3) \)
(b) \( 2(a^3 - b^3 - c^3) \)
(c) \( a^3 + b^3 + c^3 - 3abc \)
(d) \( a^3 - b^3 - c^3 \)
Answer: (c) \( a^3 + b^3 + c^3 - 3abc \)

Question. The equation \( |z - 1|^2 + |z + 1|^2 = 2 \) represents
(a) a circle of radius '1'
(b) a straight line
(c) the ordered pair (0, 0)
(d) None of the options
Answer: (c) the ordered pair (0, 0)

Question. The region of Argand diagram defined by \( |z - 1| + |z + 1| \leq 4 \) is
(a) interior of an ellipse
(b) exterior of a circle
(c) interior and boundary of an ellipse
(d) None of the options
Answer: (c) interior and boundary of an ellipse

Question. Let \( z_1 \) and \( z_2 \) be two non real 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. The curve represented by \( |z| = \text{Re}(z) + 2 \) is
(a) a straight line
(b) a circle
(c) an ellipse
(d) None of the options
Answer: (d) None of the options

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

Question. If \( z_1 = -3 + 5i \); \( z_2 = -5 - 3i \) and \( z \) is a complex number lying on the line segment joining \( z_1 \) & \( z_2 \), then \( \text{arg}(z) \) can be
(a) \( -\frac{3\pi}{4} \)
(b) \( -\frac{\pi}{4} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{5\pi}{6} \)
Answer: (d) \( \frac{5\pi}{6} \)

Question. In G.P. the first term and common ratio are both \( \frac{1}{2}(\sqrt{3} + i) \), then the absolute value of its n^th term is
(a) 1
(b) \( 2^n \)
(c) \( 4^n \)
(d) None of the options
Answer: (a) 1

Question. If \( z = x + iy \) and \( z^{1/3} = a - ib \) then \( \frac{x}{a} - \frac{y}{b} = k(a^2 - b^2) \) where \( k \) equals
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4

Question. Let A, B, C represent the complex numbers \( z_1, z_2, z_3 \) respectively on the complex plane. If the circumcentre of the triangle ABC lies at the origin, then the orthocentre is represented by the complex number
(a) \( z_1 + z_2 - z_3 \)
(b) \( z_2 + z_3 - z_1 \)
(c) \( z_3 + z_1 - z_2 \)
(d) \( z_1 + z_2 + z_3 \)
Answer: (d) \( z_1 + z_2 + z_3 \)

Question. Find the least value of \( n \) (\( n \in \mathbb{N} \)), for which \( \left(\frac{1 + i}{1 - i}\right)^n \) is real
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. If \( (a + ib)^5 = \alpha + i\beta \) then \( (b + ia)^5 \) is equal to
(a) \( \beta + i\alpha \)
(b) \( \alpha - i\beta \)
(c) \( \beta - i\alpha \)
(d) \( -\alpha - i\beta \)
Answer: (a) \( \beta + i\alpha \)

Question. If \( |z| = \max\{|z - 1|, |z + 1|\} \) then
(a) \( |z + \bar{z}| = 1/2 \)
(b) \( z + \bar{z} = 1 \)
(c) \( |z + \bar{z}| = 1 \)
(d) None of the options
Answer: (d) None of the options

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. The vector \( z = -4 + 5i \) is turned counter clockwise through an angle of 180° & stretched 1.5 times. The complex number corresponding to the newly obtained vector is
(a) \( 6 - \frac{15}{2}i \)
(b) \( -6 + \frac{15}{2}i \)
(c) \( 6 + \frac{15}{2}i \)
(d) None of the options
Answer: (a) \( 6 - \frac{15}{2}i \)

Question. Points \( z_1 \) & \( z_2 \) are adjacent vertices of a regular octagon. The vertex \( z_3 \) adjacent to \( z_2 \) (\( z_3 \neq z_1 \)) is represented by
(a) \( z_2 + \frac{1}{\sqrt{2}}(1 \pm i)(z_1 + z_2) \)
(b) \( z_2 + \frac{1}{\sqrt{2}}(1 + i)(z_1 - z_2) \)
(c) \( z_2 + \frac{1}{\sqrt{2}}(1 \pm i)(z_2 - z_1) \)
(d) None of the options
Answer: (c) \( z_2 + \frac{1}{\sqrt{2}}(1 \pm i)(z_2 - z_1) \)

Question. If \( z_1 \) & \( z_2 \) are two complex numbers & if \( \text{arg} \frac{z_1 + z_2}{z_1 - z_2} = \frac{\pi}{2} \) but \( |z_1 + z_2| \neq |z_1 - z_2| \) then the figure formed by the points represented by \( 0, z_1, z_2 \) & \( z_1 + z_2 \) is
(a) a parallelogram but not a rectangle or a rhombus
(b) a rectangle but not a square
(c) a rhombus but not a square
(d) a square
Answer: (c) a rhombus but not a square

Question. The expression \( \left[ \frac{1 + i \tan \alpha}{1 - i \tan \alpha} \right]^n - \frac{1 + i \tan n\alpha}{1 - i \tan n\alpha} \) when simplified reduces to
(a) zero
(b) \( 2 \sin n\alpha \)
(c) \( 2 \cos n\alpha \)
(d) None of the options
Answer: (a) zero

Question. If \( p = a + b\omega + c\omega^2 \); \( q = b + c\omega + a\omega^2 \) and \( r = c + a\omega + b\omega^2 \) where \( a, b, c \neq 0 \) and \( \omega \) is the complex cube root of unity then
(a) \( p + q + r = a + b + c \)
(b) \( p^2 + q^2 + r^2 = a^2 + b^2 + c^2 \)
(c) \( p^2 + q^2 + r^2 = 2(pq + qr + rp) \)
(d) None of the options
Answer: (c) \( p^2 + q^2 + r^2 = 2(pq + qr + rp) \)

Question. If \( x^2 + x + 1 = 0 \) then the numerical value of the expression \( \left(x + \frac{1}{x}\right)^2 + \left(x^2 + \frac{1}{x^2}\right)^2 + \left(x^3 + \frac{1}{x^3}\right)^2 + \dots + \left(x^{27} + \frac{1}{x^{27}}\right)^2 \) is
(a) 54
(b) 36
(c) 27
(d) 18
Answer: (a) 54

Question. If \( \alpha \) is non real 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. Number of roots of the equation \( z^{10} - z^5 - 992 = 0 \) with real part negative is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (c) 5

Question. The points \( z_1 = 3 + \sqrt{3}i \) and \( z_2 = 2\sqrt{3} + 6i \) are given on a complex plane. The complex number lying on the bisector of the angle formed by the vectors \( z_1 \) and \( z_2 \) is
(a) \( z = \frac{(3 + 2\sqrt{3})}{2} + \frac{\sqrt{3} + 2}{2}i \)
(b) \( z = 5 + 5i \)
(c) \( z = -1 - i \)
(d) None of the options
Answer: (b) \( z = 5 + 5i \)

Question. The points of intersection of the two curves \( |z - 3| = 2 \) and \( |z| = 2 \) in an argand plane are
(a) \( \frac{1}{2}(7 \pm i\sqrt{3}) \)
(b) \( \frac{1}{2}(3 \pm i\sqrt{7}) \)
(c) \( \frac{3}{2} \pm i\frac{\sqrt{7}}{2} \)
(d) \( \frac{7}{2} \pm i\frac{\sqrt{3}}{2} \)
Answer: (b) \( \frac{1}{2}(3 \pm i\sqrt{7}) \)

Question. The equation of the radical axis of the two circles represented by the equations, \( |z - 2| = 3 \) and \( |z - 2 - 3i| = 4 \) on the complex plane is
(a) \( 3iz - 3i\bar{z} - 2 = 0 \)
(b) \( 3iz - 3i\bar{z} + 2 = 0 \)
(c) \( iz - i\bar{z} + 1 = 0 \)
(d) \( 2iz - 2i\bar{z} + 3 = 0 \)
Answer: (b) \( 3iz - 3i\bar{z} + 2 = 0 \)

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) \( \text{amp} \frac{z_1}{z_2} \) may be equal to \( \frac{\pi}{2} \)
Answer: (b) \( \frac{z_1}{z_2} \) is purely imaginary, (c) \( z_1 \bar{z}_2 + z_2 \bar{z}_1 = 0 \), (d) \( \text{amp} \frac{z_1}{z_2} \) may be equal to \( \frac{\pi}{2} \)

Question. The equation \( |z - i| + |z + i| = k \), \( k > 0 \), can represent
(a) an ellipse if \( k > 2 \)
(b) line segment if \( k = 2 \)
(c) an ellipse if \( k = 5 \)
(d) line segment if \( k = 1 \)
Answer: (a) an ellipse if \( k > 2 \), (b) line segment if \( k = 2 \), (c) an ellipse if \( k = 5 \)

Question. The equation \( ||z + i| - |z - i|| = k \) represents
(a) a hyperbola if \( 0 < k < 2 \)
(b) a pair of ray if \( k > 2 \)
(c) a straight line if \( k = 0 \)
(d) a pair of ray if \( k = 2 \)
Answer: (a) a hyperbola if \( 0 < k < 2 \), (c) a straight line if \( k = 0 \), (d) a pair of ray if \( k = 2 \)

Question. POQ is a straight line through the origin O, P and Q represent the complex number \( a + i b \) and \( c + i d \) respectively and OP = OQ. Then
(a) \( |a + i b| = |c + i d| \)
(b) \( a + c = b + d \)
(c) \( \arg (a + i b) = \arg (c + i d) \)
(d) None of the options
Answer: (a) \( |a + i b| = |c + i d| \), (b) \( a + c = b + d \)

Question. If z satisfies the inequality \( |z - 1 - 2i| \le 1 \), then
(a) \( \min (\arg (z)) = \tan^{-1} \left(\frac{3}{4}\right) \)
(b) \( \max (\arg(z)) = \frac{\pi}{2} \)
(c) \( \min (|z|) = \sqrt{5} - 1 \)
(d) \( \max (|z|) = \sqrt{5} + 1 \)
Answer: (a) \( \min (\arg (z)) = \tan^{-1} \left(\frac{3}{4}\right) \), (b) \( \max (\arg(z)) = \frac{\pi}{2} \), (c) \( \min (|z|) = \sqrt{5} - 1 \), (d) \( \max (|z|) = \sqrt{5} + 1 \)