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

If $Z=x+iy$ is a complex number,then the number of distinct solutions of the equation $z^3+\bar{z}=0$ is

Consider the regions for complex number $z$ defined by $A: \frac{1}{\log_2 |z|} - \frac{1}{\log_2 |z| - 1} - 1 < 0$ and $B: \operatorname{Im}(z) = 0$. The range of values of $\operatorname{Re}(z)$ lying in the region $A \cap B$ is

If $P, Q$ and $R$ are angles of an isosceles triangle and $\angle P = \frac{\pi}{2}$, then the value of $\left(\cos \frac{P}{3} - i \sin \frac{P}{3}\right)^3 + (\cos Q + i \sin Q) (\cos R - i \sin R) + (\cos P - i \sin P) (\cos Q - i \sin Q) (\cos R - i \sin R)$ is:

$\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right) \times \left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) \times \left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) \times \ldots \infty =$

Let $z$ be a complex number (not lying on the $X$-axis) of maximum modulus such that $\left| z + \frac{1}{z} \right| = 1$. Then: