If $z_1, z_2, z_3, z_4$ are the roots of the equation $z^4 + z^3 + z^2 + z + 1 = 0$,then $\prod_{i=1}^{4} (z_i + 2)$ is equal to:

If ${z_1}$ and ${z_2}$ are two complex numbers,then $Re({z_1}{z_2}) = $

Find real $\theta$ such that $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ is purely real.

The product of the real roots of the equation $4x^4 - 24x^3 + 57x^2 + 18x - 45 = 0$,given that one of the roots is $3 + i\sqrt{6}$,is:

$\omega$ is a complex cube root of unity and if $Z$ is a complex number satisfying $|Z-1| \leq 2$ and $|\omega^2 Z-1-\omega|=a$,then the set of possible values of $a$ is

If $z \in \mathbb{C}$,then the minimum value of $|z| + |2z - 3| + |z - 1|$ is