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 $..........$.

If $z_1 \cdot z_2 \cdot \dots \cdot z_n = z$,then $arg(z_1) + arg(z_2) + \dots + arg(z_n)$ and $arg(z)$ differ by a

If $\frac{\pi}{2} < \alpha < \frac{3\pi}{2}$,then the modulus and argument of $(1 + \cos 2\alpha) + i \sin 2\alpha$ are respectively:

If $z = \frac{-2}{1 + \sqrt{3}i}$,then the value of $arg(z)$ is

If $z=x+iy$,where $x, y \in \mathbb{R}$ and the point $P$ in the Argand plane represents $z$,then the locus of $P$ satisfying the condition $\arg \left(\frac{z-1}{z-3i}\right)=\frac{\pi}{2}$ is:

The amplitude of $\left( \frac{1 - i}{1 + i} \right)$ is