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

If the complex numbers $z_1, z_2, z_3$ represent the vertices of an equilateral triangle such that $|z_1| = |z_2| = |z_3|$,then $z_1 + z_2 + z_3 = $

Let $S = \{z = x + iy : \frac{2z - 3i}{4z + 2i} \text{ is a real number} \}$. Then which of the following is $NOT$ correct?

The points $z_1, z_2, z_3, z_4$ in the complex plane are the vertices of a parallelogram taken in order,if and only if

If $z$ is a complex number such that $|z| \ge 2$,then the minimum value of $|z + \frac{1}{2}|$ is:

Let $S = \{ z \in \mathbb{C} : |z - 2| \leq 1, z(1 + i) + \overline{z}(1 - i) \leq 2 \}$. Let $|z - 4i|$ attain minimum and maximum values,respectively,at $z_1 \in S$ and $z_2 \in S$. If $5(|z_1|^2 + |z_2|^2) = \alpha + \beta \sqrt{5}$,where $\alpha$ and $\beta$ are integers,then the value of $\alpha + \beta$ is equal to