$\sinh(ix)$ is equal to

Let $z \neq -i$ be any complex number such that $\frac{z - i}{z + i}$ is a purely imaginary number. Then $z + \frac{1}{z}$ is

If $z = x + iy$ and $\arg\left( \frac{z - 2}{z + 2} \right) = \frac{\pi}{6}$,then the locus of $z$ is

If $a$ and $b$ are real numbers between $0$ and $1$ such that the points $z_1 = a + i$,$z_2 = 1 + bi$,and $z_3 = 0$ form an equilateral triangle,then

If $Z$ is a complex number such that $|Z| \leq 3$ and $-\frac{\pi}{2} \leq \operatorname{amp}(Z) \leq \frac{\pi}{2}$,then the area of the region formed by the locus of $Z$ is

Let $S_{1}, S_{2}$ and $S_{3}$ be three sets defined as:
$S_{1} = \{ z \in C : |z - 1| \leq \sqrt{2} \}$
$S_{2} = \{ z \in C : \operatorname{Re}((1 - i)z) \geq 1 \}$
$S_{3} = \{ z \in C : \operatorname{Im}(z) \leq 1 \}$
Then the set $S_{1} \cap S_{2} \cap S_{3}$