The number of solutions for $z^3+\bar{z}=0$ is

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:

If $\alpha$ and $\beta$ are the roots of the equation $2z^2 - 3z - 2i = 0$,where $i = \sqrt{-1}$,then $16 \cdot \operatorname{Re}\left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right) \cdot \operatorname{Im}\left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right)$ is equal to

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2+z+1|=1$. Then which of the following statements is/are $TRUE$?
$(A) |z+\frac{1}{2}| \leq \frac{1}{2}$ for all $z \in S$
$(B) |z| \leq 2$ for all $z \in S$
$(C) |z+\frac{1}{2}| \geq \frac{1}{2}$ for all $z \in S$
$(D)$ The set $S$ has exactly four elements

If $z$ is a complex number such that $\left|z-\frac{4}{z}\right|=2$,then the greatest value of $|z|$ is

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} = $