If three complex numbers are in $A.P.$,then they lie on

If $\left| z - \frac{1 + 3i}{2} \right| = \frac{\sqrt{10}}{2}$ and $P$,$Q$,and $R$ are points representing the complex numbers $z$,$z e^{i \pi / 3}$,and $z(1 + e^{i \pi / 3})$ respectively in the Argand plane,then the area of the triangle $PQR$ is:

The polar coordinates of the point,whose Cartesian coordinates are $(-2 \sqrt{3}, 2)$,are

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 area of the triangle whose vertices are complex numbers $z, iz, z + iz$ in the Argand diagram is

In the Argand diagram,if $O, P$ and $Q$ represent the origin,the complex number $z$ and the complex number $z + iz$ respectively,then the angle $\angle OPQ$ is