If $-\pi < \arg (z) < -\frac{\pi}{2}$, then $\arg (\bar{z}) - \arg (-\bar{z})$ is

$\operatorname{Arg}\left(\frac{4+2 i}{1-2 i}+\frac{3+4 i}{2+3 i}\right)$ lies in the interval

Let $z$ satisfy $|z| = 1$ and $z = 1 - \bar{z}$.
Statement $1$: $z$ is a real number.
Statement $2$: The principal argument of $z$ is $\frac{\pi}{3}$.

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 $z = \frac{(2-i)(1+i)^3}{(1-i)^2}$,then $\operatorname{Arg}(z) = $

If $Arg(z)$ denotes the principal argument of a complex number $z$,then the value of the expression $Arg\left( -i e^{i\frac{\pi}{9}} z^2 \right) + 2Arg\left( 2i e^{-i\frac{\pi}{18}} \bar{z} \right)$ is