Let $A =\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1- i \sin \theta}\right.$ is purely imaginary $\}$. Then the sum of the elements in $A$ is

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

If $z$ and $\omega $ are two non-zero complex numbers such that $|z\omega |\, = 1$ and $arg(z) – arg(\omega ) = \frac{\pi }{2},$ then $\bar z\omega $ is equal to

If $\frac{{z – \alpha }}{{z + \alpha }}\left( {\alpha  \in R} \right)$ is a purely imaginary number and $\left| z \right| = 2$, then a value of $\alpha $ is

Let $z,w$be complex numbers such that $\overline z + i\overline w = 0$and $arg\,\,zw = \pi $. Then arg z equals