The maximum value of the modulus of $e^{z^2}$ on the set $\{z \in \mathbb{C} : 0 \leq \operatorname{Re}(z) \leq 1, 0 \leq \operatorname{Im}(z) \leq 1\}$ is

The locus of the complex number $z$ such that $\arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{3}$ is:

The locus of $z=x+iy$, such that $\operatorname{Im}\left(\frac{z-3i}{iz+4}\right)=0$ is

Suppose $z \in \mathbb{C}$ has an argument $\theta$ such that $0 < \theta < \frac{\pi}{2}$ and satisfies the equation $|z - 3i| = 3$. What is the value of $\cot \theta - \frac{6}{z}$?

If ${z_1}, {z_2}, {z_3}, {z_4}$ are the affixes of four points in the Argand plane and $z$ is the affix of a point such that $|z - z_1| = |z - z_2| = |z - z_3| = |z - z_4|$,then ${z_1}, {z_2}, {z_3}, {z_4}$ are

$A(z_1=2+2i)$,$B(z_2)$,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