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

Assertion $(A)$: If the arguments of $\bar{z}_1$ and $z_2$ are $\frac{\pi}{5}$ and $\frac{\pi}{3}$ respectively,then $\arg(z_1 z_2)$ is $\frac{2\pi}{15}$. Reason $(R)$: For any complex number $z$,$\arg(\bar{z}) = \frac{\pi}{2} + \arg(z)$. The correct option among the following is:

If $z_1, z_2$ and $z_3, z_4$ are $2$ pairs of complex conjugate numbers,then $\arg \left( \frac{z_1}{z_4} \right) + \arg \left( \frac{z_2}{z_3} \right)$ equals

Let $z$ satisfy $|z|=1$, $z=1-\bar{z}$ and $\operatorname{Im}(z) > 0$.
Statement-$I$: $z$ is a real number.
Statement-$II$: Principal argument of $z$ is $\frac{\pi}{3}$.
Then

Let $z$ and $w$ be two complex numbers such that $\bar{z}+i \bar{w}=0$ and $\operatorname{Arg}(z w)=\pi$. Then,$\operatorname{Arg} z=$

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