Let $z = 1 + i$ and $z_1 = \frac{1 + i \overline{z}}{\overline{z}(1 - z) + \frac{1}{z}}$. Then $\frac{12}{\pi} \arg(z_1)$ is equal to $..........$.

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\overline{z}_{1} = i \overline{z}_{2}$ and $\arg \left( \frac{z_{1}}{\overline{z}_{2}} \right) = \pi$. Then:

If $-1 + \sqrt{-3} = re^{i\theta}$, then the value of $\theta$ is

If the complex number $z = 2 - i(2 \tan \frac{5 \pi}{8})$ has modulus $r$ and argument $\theta$,then what are $(r, \theta)$?

Argument of the complex number $z = \frac{13-5i}{4-9i}$,where $i = \sqrt{-1}$,is

If complex numbers $z_1$ and $z_2$ are such that $|z_1| = \sqrt{2}$,$|z_2| = \sqrt{3}$ and $|z_1 + z_2| = \sqrt{5 - 2\sqrt{3}}$,then the value of $|Arg(z_1) - Arg(z_2)|$ is