If $z_1=1-2 i$,$z_2=1+i$,and $z_3=3+4 i$,then $\left(\frac{1}{z_1}+\frac{3}{z_2}\right) \frac{z_3}{z_2}=$

For a complex number $Z = a + ib$,let $\hat{Z} = b + ia$. If $Z_1$ and $Z_2$ are such complex numbers,then $\widehat{Z_1 Z_2} = $

Two numbers $k_1$ and $k_2$ are randomly chosen from the set of natural numbers. Then,the probability that the value of $i^{k_1} + i^{k_2}$ (where $i = \sqrt{-1}$) is non-zero,equals:

The complex number $z = \frac{i-1}{\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}}$ is equal to $.....$

$z_1$ and $z_2$ are two complex numbers such that $|z_1 + z_2| = 1$ and $|z_1^2 + z_2^2| = 25$. Then the minimum value of $|z_1^3 + z_2^3|$ is

If $\theta \in \mathbb{R}$ and $\frac{1-i \cos \theta}{1+2 i \cos \theta}$ is a real number,then $\theta$ will be (where $I$ is the set of integers):