The number of solutions for the equations $|z - 1| = |z - 2| = |z - i|$ is

Let the minimum value $v_{0}$ of $v = |z|^{2} + |z-3|^{2} + |z-6i|^{2}$,where $z \in \mathbb{C}$,be attained at $z = z_{0}$. Then $|2z_{0}^{2} - \bar{z}_{0}^{3} + 3|^{2} + v_{0}^{2}$ is equal to:

The largest value of $r$ for which the region represented by the set $\{ \omega \in \mathbb{C} : |\omega - 4 - i| \le r \}$ is contained in the region represented by the set $\{ z \in \mathbb{C} : |z - 1| \le |z + i| \}$ is equal to

If $z$ is a complex number such that $\frac{z - 1}{z + 1}$ is purely imaginary,then

$A(z_1=2+2i)$,$B(z_2)$,and $C(z_3)$ are three points on the Argand plane satisfying $|z_k-2i|=2$ for $k=1, 2, 3$. If $\triangle ABC$ encloses the maximum area,then the sum of the imaginary parts of $z_2$ and $z_3$ is

The real part of ${e^{e^{i\theta }}}$ is