If$z = \frac{{1 - i\sqrt 3 }}{{1 + i\sqrt 3 }},$then $arg(z) = $ ............. $^\circ$

Let $z$ be complex number such that $\left|\frac{z-i}{z+2 i}\right|=1$ and $|z|=\frac{5}{2} \cdot$ Then the value of $|z+3 i|$ is 

The sum of amplitude of $z$ and another complex number is $\pi $. The other complex number can be written

Number of complex numbers $z$ such that $\left| z \right| + z - 3\bar z = 0$ is equal to

Let $z$ be a purely imaginary number such that ${\mathop{\rm Im}\nolimits} (z) < 0$. Then $arg\,(z)$ is equal to

If $\alpha $ and $\beta $ are different complex numbers with $|\beta | = 1$, then $\left| {\frac{{\beta - \alpha }}{{1 - \overline \alpha \beta }}} \right|$ is equal to