JEE Mathematics Complex Numbers MCQs Set E

Question. If \( a < 0, b > 0 \) then \( \sqrt{a} \cdot \sqrt{b} \) is equal to
(a) \( -\sqrt{|a|b} \)
(b) \( \sqrt{|a|b} i \)
(c) \( \sqrt{|a|b} \)
(d) none of the options
Answer: (b) \( \sqrt{|a|b} i \)

Question. The value of the sum \( \sum_{n=1}^{13} (i^n + i^{n+1}) \), where \( i = \sqrt{-1} \), is
(a) i
(b) i – 1
(c) –i
(d) 0
Answer: (b) i – 1

Question. If \( n_1, n_2 \) are positive integers then \( (1+i)^{n_1} + (1+i^3)^{n_1} + (1+i^5)^{n_2} + (1+i^7)^{n_2} \) is a real number if and only if
(a) \( n_1 = n_2 + 1 \)
(b) \( n_1 + 1 = n_2 \)
(c) \( n_1 = n_2 \)
(d) \( n_1, n_2 \) are any two positive integers
Answer: (d) \( n_1, n_2 \) are any two positive integers

Question. The complex number \( \frac{2^n}{(1+i)^{2n}} + \frac{(1+i)^{2n}}{2^n} \), \( n \in Z \)
(a) 0
(b) 2
(c) \( \{1 + (-1)^n\} \cdot i^n \)
(d) none of the options
Answer: (c) \( \{1 + (-1)^n\} \cdot i^n \)

Question. The smallest positive integral value of n for which \( \left(\frac{1-i}{1+i}\right)^n \) is purely imaginary with positive imaginary part, is
(a) 1
(b) 3
(c) 5
(d) none of the options
Answer: (b) 3

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 \( i = \sqrt{-1} \), the number of values of \( i^n + i^{-n} \) for different \( n \in Z \) is
(a) 3
(b) 2
(c) 4
(d) 1
Answer: (a) 3

Question. Im(z) is equal to
(a) \( \frac{1}{2}(z + \bar{z})i \)
(b) \( \frac{1}{2}(z - \bar{z}) \)
(c) \( -\frac{1}{2}(z - \bar{z})i \)
(d) none of the options
Answer: (c) \( -\frac{1}{2}(z - \bar{z})i \)

Question. The value of \( (1 + i)^3 + (1 – i)^6 \) is
(a) i
(b) \( 2(-1 + 5i) \)
(c) \( 1 – 5i \)
(d) none of the options
Answer: (b) \( 2(-1 + 5i) \)

Question. Taking the value of a square root with positive real part only, the value of \( \sqrt{-3-4i} + \sqrt{3+4i} \) is
(a) 1 + i
(b) 1 – 3i
(c) 1 + 3i
(d) none of the options
Answer: (d) none of the options

Question. \( \sin^{-1} \left\{ \frac{1}{i}(z-1) \right\} \), where z is nonreal, can be the angle of a triangle if
(a) Re(z) = 1, Im(z) = 2
(b) Re(z) = 1, -1 ≤ Im(z) ≤ 1
(c) Re(z) + Im(z) = 0
(d) none of the options
Answer: (b) Re(z) = 1, -1 ≤ Im(z) ≤ 1

Question. If n is an odd integer, \( i = \sqrt{-1} \) then \( (1 + i)^{6n} + (1 – i)^{6n} \) is equal to
(a) 0
(b) 2
(c) -2
(d) none of the options
Answer: (a) 0

Question. If \( z_1 = 9y^2 – 4 – 10ix \), \( z_2 = 8y^2 – 20i \), where \( z_1 = \bar{z}_2 \), then z = x + iy is equal to
(a) -2 + 2i
(b) -2 ± 2i
(c) -2 ± i
(d) none of the options
Answer: (b) -2 ± 2i

Question. The complex numbers \( \sin x – i\cos 2x \) and \( \cos x – i\sin 2x \) are conjugate to each other for
(a) \( x = n\pi \)
(b) x = 0
(c) \( x = (2n + 1)\frac{\pi}{2} \)
(d) no value of x
Answer: (d) no value of x

Question. If \( z = 1 + i\tan \alpha \), where \( \pi < \alpha < \frac{3\pi}{2} \), then |z| is equal to
(a) \( \sec \alpha \)
(b) \( – \sec \alpha \)
(c) \( \text{cosec } \alpha \)
(d) none of the options
Answer: (b) \( – \sec \alpha \)

Question. If z is a complex number satisfying the reaction \( |z + 1| = z + 2(1 + i) \) then z is
(a) \( \frac{1}{2}(1 + 4i) \)
(b) \( \frac{1}{2}(3 + 4i) \)
(c) \( \frac{1}{2}(1 - 4i) \)
(d) \( \frac{1}{2}(3 - 4i) \)
Answer: (c) \( \frac{1}{2}(1 - 4i) \)

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

Question. If \( z_1, z_2 \) are two nonzero complex numbers such that \( |z_1 + z_2| = |z_1| + |z_2| \) then \( \text{amp} \frac{z_1}{z_2} \) is equal to
(a) \( \pi \)
(b) \( -\pi \)
(c) 0
(d) none of the options
Answer: (c) 0

Question. The complex number z is purely imaginary if
(a) \( z\bar{z} \) is real
(b) \( z = \bar{z} \)
(c) \( z + \bar{z} = 0 \)
(d) none of the options
Answer: (c) \( z + \bar{z} = 0 \)

Question. If z = x + iy such that |z + 1| = |z – 1| and \( \text{amp } \frac{z-1}{z+1} = \frac{\pi}{4} \) then
(a) \( x = \sqrt{2} + 1, y = 0 \)
(b) \( x = 0, y = \sqrt{2} + 1 \)
(c) \( x = 0, y = \sqrt{2} - 1 \)
(d) \( x = \sqrt{2} - 1, y = 0 \)
Answer: (b) \( x = 0, y = \sqrt{2} + 1 \)

Question. Let \( z = \frac{\cos\theta + i\sin\theta}{\cos\theta - i\sin\theta}, \frac{\pi}{4} < \theta < \frac{\pi}{2} \). Then arg z is
(a) \( 2\theta \)
(b) \( 2\theta - \pi \)
(c) \( \pi + 2\theta \)
(d) none of the options
Answer: (a) \( 2\theta \)

Question. If \( z = \frac{\sqrt{3} + i}{\sqrt{3} - i} \) then the fundamental amplitude of z is
(a) \( -\frac{\pi}{3} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{6} \)
(d) none of the options
Answer: (b) \( \frac{\pi}{3} \)

Question. If \( \frac{1+2i}{2+i} = r(\cos\theta + i\sin\theta) \) then
(a) \( r = 1, \theta = \tan^{-1}\frac{3}{4} \)
(b) \( r = \sqrt{5}, \theta = \tan^{-1}\frac{4}{3} \)
(c) \( r = 1, \theta = \tan^{-1}\frac{4}{3} \)
(d) none of the options
Answer: (a) \( r = 1, \theta = \tan^{-1}\frac{3}{4} \)

Question. If z = x + iy satisfies amp (z – 1) = amp (z + 3i) then the value of (x – 1) : y is equal to
(a) 2 : 1
(b) 1 : 3
(c) -1 : 3
(d) none of the options
Answer: (b) 1 : 3

Question. Let z be a complex number of constant modulus such that \( z^2 \) is purely imaginary then the number of possible values of z is
(a) 2
(b) 1
(c) 4
(d) infinite
Answer: (c) 4

Question. If \( \omega \) is an imaginary cube root of unity then \( (1 + \omega - \omega^2)^7 \) equals
(a) \( 128\omega \)
(b) \( -128\omega \)
(c) \( 128\omega^2 \)
(d) \( -128\omega^2 \)
Answer: (d) \( -128\omega^2 \)

Question. If \( \omega \) is a nonreal cube root of unity then the expression \( (1 - \omega)(1 - \omega^2)(1 + \omega^4)(1 + \omega^8) \) is equal to
(a) 0
(b) 3
(c) 1
(d) 2
Answer: (b) 3

Question. If \( 3^{49}(x + iy) = \left(\frac{3}{2} + \frac{\sqrt{3}}{2}i\right)^{100} \) and x = ky then k is
(a) \( -\frac{1}{3} \)
(b) \( \sqrt{3} \)
(c) \( -\sqrt{3} \)
(d) \( -\frac{1}{\sqrt{3}} \)
Answer: (d) \( -\frac{1}{\sqrt{3}} \)

Question. \( x^{3m} + x^{3n-1} + x^{3r-2} \), where \( m, n, r, \in N \), is divisible by
(a) \( x^2 – x + 1 \)
(b) \( x^2 + x + 1 \)
(c) \( x^2 + x – 1 \)
(d) \( x^2 – x – 1 \)
Answer: (b) \( x^2 + x + 1 \)

Question. If \( x^2 – x + 1 = 0 \) then the value of \( \sum_{n=1}^5 \left(x^n + \frac{1}{x^n}\right)^2 \) is
(a) 8
(b) 10
(c) 12
(d) none of the options
Answer: (a) 8

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

Question. The smallest positive integral value of n for which \( (1 + \sqrt{3}i)^{n/2} \) is real is
(a) 3
(b) 6
(c) 12
(d) 0
Answer: (b) 6

Question. If \( i = \sqrt{-1} \), \( \omega \) = nonreal cube root of unity then \( \frac{(1+i)^{2n} - (1-i)^{2n}}{(1+\omega^4 - \omega^2)(1-\omega^4+\omega^2)} \) is equal to
(a) 0 if n is even
(b) 0 for all \( n \in Z \)
(c) \( 2^{n-1} \cdot i \) for all \( n \in N \)
(d) none of the options
Answer: (a) 0 if n is even

Question. If \( z^2 – z + 1 = 0 \) then \( z^n – z^{-n} \), where n is a multiple of 3, is
(a) \( 2(-1)^n \)
(b) 0
(c) \( (-1)^{n+1} \)
(d) none of the options
Answer: (b) 0

Question. If \( \omega \) is a nonreal cube root of unity then \( \frac{1 + 2\omega + 3\omega^2}{2 + 3\omega + \omega^2} + \frac{2 + 3\omega + \omega^2}{3 + \omega + 2\omega^2} \) is equal to
(a) -1
(b) \( 2\omega \)
(c) 0
(d) \( -2\omega \)
Answer: (b) \( 2\omega \)

Question. If \( (x – 1)^4 – 16 = 0 \) then the sum of nonreal complex values of x is
(a) 2
(b) 0
(c) 4
(d) none of the options
Answer: (a) 2