The real value of $\theta$ for which the expression $\frac{{1 + i\,\cos \theta }}{{1 - 2i\cos \theta }}$ is a real number is $\left( {n \in I} \right)$ 

If $z_1$ and $z_2$ are two unimodular complex numbers that satisfy $z_1^2 + z_2^2 = 5,$ then ${\left( {{z_1} - {{\bar z}_1}} \right)^2} + {\left( {{z_2} - {{\bar z}_2}} \right)^2}$ is equal to -

If $z_1 , z_2$ and $z_3, z_4$ are $2$ pairs of complex conjugate numbers, then $\arg \left( {\frac{{{z_1}}}{{{z_4}}}} \right) + \arg \left( {\frac{{{z_2}}}{{{z_3}}}} \right)$ equals 

If $z$ is a complex number such that $|z - \bar{z}| = 2$ and $|z + \bar{z}| = 4 $, then which of the following is always incorrect -

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

Let $w$ $(Im\, w \neq 0)$ be a complex number. Then the set of all complex number $z$ satisfying the equation $w - \overline {w}z  = k\left( {1 - z} \right)$ , for some real number $k$, is