If $z$ is a complex number, then $(\overline {{z^{ - 1}}} )(\overline z ) = $

The amplitude of $\frac{{1 + \sqrt 3 \,i}}{{\sqrt 3 + i}}$ is

If $\frac{{z - i}}{{z + i}}(z \ne - i)$ is a purely imaginary number, then $z.\bar z$ is equal to

Find the modulus and argument of the complex number $\frac{1+2 i}{1-3 i}$

Argument of $ - 1 - i\sqrt 3 $ is

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