Suppose $z_1, z_2, z_3$ are the vertices of an equilateral triangle inscribed in the circle $|z| = 2$. If $z_1 = 1 + i\sqrt{3}$,then the values of $z_3$ and $z_2$ are respectively:

If the locus of $z \in \mathbb{C}$, such that $\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)+\operatorname{Re}\left(\frac{\bar{z}-1}{2 \bar{z}-i}\right)=2$, is a circle of radius $r$ and center $(a, b)$, then $\frac{15 a b}{r^2}$ is equal to :

Let $z_1$ and $z_2$ be two distinct complex numbers and let $z = (1-t)z_1 + tz_2$ for some real number $t$ with $0 < t < 1$. If $\operatorname{Arg}(w)$ denotes the principal argument of a non-zero complex number $w$,then which of the following are true?
$(A)$ $|z-z_1| + |z-z_2| = |z_1-z_2|$
$(B)$ $\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z-z_2)$
$(C)$ $\left|\begin{array}{cc} z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1 \end{array}\right| = 0$
$(D)$ $\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z_2-z_1)$

The maximum value of $|z|$ where $z$ satisfies the condition $\left| z + \frac{2}{z} \right| = 2$ is

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

If $A, B, C$ are represented by $3 + 4i, 5 - 2i, -1 + 16i$,then $A, B, C$ are