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

Solve: $i x^2 - 3 x - 2 i = 0$

Suppose $z$ is any root of $11 z^8 + 21 i z^7 + 10 i z - 22 = 0$ where $i = \sqrt{-1}$. Then,$S = |z|^2 + |z| + 1$ satisfies

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:

Let $S$ be the set of all $(\alpha, \beta)$ such that $\pi < \alpha, \beta < 2\pi$,for which the complex number $\frac{1-i \sin \alpha}{1+2i \sin \alpha}$ is purely imaginary and $\frac{1+i \cos \beta}{1-2i \cos \beta}$ is purely real. Let $Z_{\alpha \beta} = \sin 2\alpha + i \cos 2\beta$ for $(\alpha, \beta) \in S$. Then $\sum_{(\alpha, \beta) \in S} \left(i Z_{\alpha \beta} + \frac{1}{i \bar{Z}_{\alpha \beta}}\right)$ is equal to:

The values of $x$ for which $\sin x + i \cos 2x$ and $\cos x - i \sin 2x$ are conjugate to each other are