Complex Numbers Mcq-02

Complex Numbers Quiz

1. The value of \( \left(\frac{-1 + i\sqrt{3}}{1 - i}\right)^{30} \) is

(a) \(-2^{15}\) (b) \(2^{15}i\) (c) \(-2^{15}i\) (d) \(6^5\)

2. Let \( z=\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)^5+\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)^5 \). If \(R(z)\) and \(I(z)\) respectively denote the real and imaginary parts of \(z\), then:

(a) \(R(z) > 0\) and \(I(z) > 0\) (b) \(I(z) = 0\) (c) \(R(z) < 0\) and \(I(z) > 0\) (d) \(R(z) = -3\)

3. Let \(\left(-2-\frac{1}{3}i\right)^3=\frac{x+iy}{27}\), where \(x\) and \(y\) are real numbers, then \(y - x\) equals:

(a) -91 (b) -85 (c) 85 (d) 91

4. The least positive integer \(n\) for which \(\left(\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)^n = 1\), is

(a) 3 (b) 5 (c) 2 (d) 6

5. Let \(\omega\) be a complex number such that \(2\omega + 1 = z\), where \(z = \sqrt{-3}\). If

\[ \text{det}\begin{bmatrix}1 & 1 & 1 \\ 1 & - \omega^2 - 1 & \omega^2 \\ 1 & \omega^2 & \omega^7\end{bmatrix} = 3k \]

(a) z (b) -1 (c) 1 (d) -z

6. A value of \(\theta\) for which \(\frac{2+3i\sin\theta}{1-2i\sin\theta}\) is purely imaginary is

(a) \(\frac{\pi}{3}\) (b) \(\frac{\pi}{6}\) (c) \(\sin^{-1}\left(\frac{\sqrt{3}}{4}\right)\) (d) \(\sin^{-1}\left(\frac{1}{\sqrt{3}}\right)\)

7. If \(\omega \ne 1\) is a cube root of unity, and \((1+\omega)^7 = A + B\omega\). Then \((A, B)\) equals

(a) (1, 0) (b) (-1, 1) (c) (0, 1) (d) (1, 1)

8. If \(z^2 + z + 1 = 0\), then the value of \((z + \frac{1}{z})^2 + (z^2 + \frac{1}{z^2})^2 + ... + (z^6 + \frac{1}{z^6})^2\) is

(a) 18 (b) 54 (c) 6 (d) 12

9. If the cube roots of unity are \(1, \omega, \omega^2\), then the roots of the equation \((x-1)^3 + 8 = 0\) are

(a) -1, -1, -1 (b) -1, \(-1 + 2\omega\), \(-1 - 2\omega^2\) (c) -1, \(1 + 2\omega\), \(1 + 2\omega^2\) (d) -1, \(1 - 2\omega\), \(1 - 2\omega^2\)

10. If \(z = x - iy\) and \(z^{1/3} = p + iq\), then \(\frac{\frac{x}{p} + \frac{y}{q}}{p^2 + q^2}\) is equal to

(a) 2 (b) -1 (c) 1 (d) -2

11. If \(\left(\frac{1+i}{1-i}\right)^x = 1\), then

(a) \(x = 2n\), where \(n\) is any positive integer (b) \(x = 4n + 1\), where \(n\) is any positive integer (c) \(x = 2n + 1\), where \(n\) is any positive integer (d) \(x = 4n\), where \(n\) is any positive integer