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

Let $A = \{z : (\frac{z - \bar{z}}{2i})^2 \leqslant 2(\frac{z - \bar{z}}{2i})\}$ where $i = \sqrt{-1}$ and $B = \{z : |z| \leqslant \sqrt{5}\}$. The number of points with integral real and imaginary parts of $z$ lying in $A \cap B$ is -

If $z$ and $w$ are two complex numbers such that $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$,then

Convert the complex number $z = \frac{i-1}{\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}}$ into polar form.

The complex numbers $\sin x + i \cos 2x$ and $\cos x - i \sin 2x$ (where $i = \sqrt{-1}$) are conjugate to each other for:

The number of solutions for $z^3+\bar{z}=0$ is