$\cos (x + iy)$ is equal to

If $\left|z-\frac{2}{z}\right|=2$,then the greatest value of $|z|$ is

The real part of $(1-\cos \theta+i \sin \theta)^{-1}$ is

$z_1$ 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

The number of complex numbers $z$ satisfying $\overline{z} = i z^2$ is

If the real part of the complex number $z = \frac{3 + 2i \cos \theta}{1 - 3i \cos \theta}$, where $\theta \in (0, \frac{\pi}{2})$, is zero, then the value of $\sin^2 3\theta + \cos^2 \theta$ is equal to: