If ${z_1},{z_2},{z_3}$ are complex numbers such that $|{z_1}|\, = \,|{z_2}|\, = $ $\,|{z_3}|\, = $ $\left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + \frac{1}{{{z_3}}}} \right| = 1\,,$ then${\rm{ }}|{z_1} + {z_2} + {z_3}|$ is

If $z=\frac{1}{2}-2 i$, is such that $|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$ and $\alpha, \beta \in R \quad$, then $\alpha+\beta$ is equal to

The conjugate of complex number $\frac{{2 - 3i}}{{4 - i}},$ is

For any complex number $z,\bar z = \left( {\frac{1}{z}} \right)$if and only if

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 for complex numbers ${z_1}$ and ${z_2}$, $arg({z_1}/{z_2}) = 0,$ then $|{z_1} - {z_2}|$ is equal to