If $z $ is a complex number of unit modulus and  argument $\theta$, then ${\rm{arg}}\left( {\frac{{1 + z}}{{1 + (\bar z)}}} \right)$ equals.

If $0 < amp{\rm{ (z)}} < \pi {\rm{,}}$then $amp(z)-amp ( - z) = $

If $\sqrt 3 + i = (a + ib)(c + id)$, then ${\tan ^{ - 1}}\left( {\frac{b}{a}} \right) + $ ${\tan ^{ - 1}}\left( {\frac{d}{c}} \right)$ has the value

The solutions of equation in $z$, $| z |^2 -(z + \bar{z}) + i(z - \bar{z})$ + $2$ = $0$ are $(i = \sqrt{-1})$

If $a > 0$ and $z = \frac{{{{\left( {1 + i} \right)}^2}}}{{a - i}}$, has magnitude $\sqrt {\frac{2}{5}} $, then $\bar z$ is equal to:

The values of $z$for which $|z + i|\, = \,|z - i|$ are