If $a = \operatorname{Im}\left(\frac{1+z^2}{2iz}\right)$ and $z$ is any non-zero complex number such that $|z|=1$,then $a=$

Let ${z_1}, {z_2}, {z_3}$ be three vertices of an equilateral triangle circumscribing the circle $|z| = \frac{1}{2}$. If ${z_1} = \frac{1}{2} + \frac{\sqrt{3}i}{2}$ and ${z_1}, {z_2}, {z_3}$ are in anticlockwise sense,then ${z_2}$ is

$z_1$ and $z_2$ are two complex numbers such that $\left|z_1-z_2\right| < k$. If a complex number $z$ satisfies the condition $\left|z-z_1\right|+\left|z-z_2\right|=k$,then $z$ lies on:

If $\cos \alpha + \cos \beta + \cos \gamma = 0$ and $\sin \alpha + \sin \beta + \sin \gamma = 0$,then $\sin 2 \alpha + \sin 2 \beta + \sin 2 \gamma = $

Let $P=\{z \in C:|z+2-3 i| \leq 1\}$ and $Q=\{z \in C: z(1+i)+\bar{z}(1-i) \leq-8\}$. Let in $P \cap Q, |z-3+2 i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $|z_1|^2+2|z_2|^2=\alpha+\beta \sqrt{2}$,where $\alpha, \beta$ are integers,then $\alpha+\beta$ equals . . . . . . .

If $\mu = \frac{2z + 5i}{z - 3}$ and $|\mu| = 2$,then the locus of $z$ is: