If $z$ is a complex number such that $|z| = 4$ and $\text{arg}(z) = \frac{5\pi}{6}$,then $z$ is equal to

If $z$ is a purely real number such that $\text{Re}(z) < 0$,then $\text{arg}(z)$ is equal to

The locus of the variable point $z=x+iy$ whose amplitude is always equal to $\theta$,is

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 $z_1, z_2$ be two complex numbers such that $\bar{z}_1 - i \bar{z}_2 = 0$ and $\arg(z_1 z_2) = \frac{3 \pi}{4}$,then $\arg(z_1) =$

$x$ and $y$ are two complex numbers such that $|x|=|y|=1$. If $\operatorname{Arg}(x)=2 \alpha$,$\operatorname{Arg}(y)=3 \beta$,and $\alpha+\beta=\frac{\pi}{36}$,then $x^6 y^4+\frac{1}{x^6 y^4}=$