If $z_1, z_2, z_3 \in \mathbb{C}$ such that $|z_1| = |z_2| = |z_3| = 2$,then the greatest value of the expression $|z_1 - z_2||z_2 - z_3| + |z_2 - z_3||z_3 - z_1| + |z_3 - z_1||z_1 - z_2|$ is

The locus of the point $z=x+iy$ satisfying the equation $\left|\frac{z-1}{z+1}\right|=1$ is given by :

$\sinh(ix)$ is equal to

Let $S=S_1 \cap S_2 \cap S_3$,where $S_1=\{z \in \mathbb{C}:|z|<4\}$,$S_2=\{z \in \mathbb{C}: \operatorname{Im}[\frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}]>0\}$,and $S_3=\{z \in \mathbb{C}: \operatorname{Re} z>0\}$.
$1.$ Area of $S=$
$(A) \frac{10 \pi}{3} \quad (B) \frac{20 \pi}{3} \quad (C) \frac{16 \pi}{3} \quad (D) \frac{32 \pi}{3}$
$2.$ $\min _{z \in S}|1-3 i-z|=$
$(A) \frac{2-\sqrt{3}}{2} \quad (B) \frac{2+\sqrt{3}}{2} \quad (C) \frac{3-\sqrt{3}}{2} \quad (D) \frac{3+\sqrt{3}}{2}$

The locus of the points represented by $|z+3|-|z-3|=6$,where $z$ is a complex number,is ....

Let $a, b \in \mathbb{R}$ and the roots $\alpha, \beta$ of the equation $z^2+az+b=0$ be complex. If the origin,$\alpha$ and $\beta$ represent the vertices of an equilateral triangle on the Argand plane,then