List of practice Questions

Which ONE of the following is the value of \( \frac{1}{i^n} \)?

where \( i = \sqrt{-1} \), and \( n \) is an even positive integer.

Let \(Z_1\) and \(Z_2\) be two complex numbers, then:

Let \(Z_1 = -2 + 2i\) and \(Z_2 = 3i\), then:

Let \(x + iy = (2 - i)^2 + \left(\frac{7-4i}{2-i}\right) - 8\), the values of \(x\) and \(y\) are:

(a) If \( y = \sqrt{\cos x + y} \), prove that \[ \frac{dy}{dx} = \frac{\sin x}{1 - 2y}. \]

Let \(Z_1 = r_1(\cos \theta_1 + i\sin \theta_1)\) and \(Z_2 = r_2(\cos \theta_2 + i\sin \theta_2)\) be two complex numbers, then value of \(Z_1Z_2\) is

If \(\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}\) and \(\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}\), then \(\cos \theta - i\sin \theta\) is equal to

If A and B be orthogonal matrices of same size, then

Let \(z_1\) and \(z_2\) be two complex numbers, then the dot product of \(z_1\) and \(z_2\) is :

The value of \(\frac{5+5i}{3-4i} + \frac{20}{4+3i}\) is given by :

Let \(z_1\) and \(z_2\) be two complex numbers, then :

Polar form of the complex number \(z = -4 + i4\sqrt{3}\) is :