Let ${z_1}$ and ${z_2}$ be two complex numbers with $\alpha $ and $\beta $ as their principal arguments such that $\alpha + \beta > \pi ,$ then principal $arg\,({z_1}\,{z_2})$ is given by
$\alpha + \beta + \pi $
$\alpha + \beta - \pi $
$\alpha + \beta - 2\pi $
$\alpha + \beta $
The number of solutions of the equation ${z^2} + \bar z = 0$ is
If $z = \frac{{ - 2}}{{1 + \sqrt 3 \,i}}$ then the value of $arg\,(z)$ is
The argument of the complex number $ - 1 + i\sqrt 3 $ is ............. $^\circ$
If $Arg(z)$ denotes principal argument of a complex number $z$, then the value of expression $Arg\left( { - i{e^{i\frac{\pi }{9}}}.{z^2}} \right) + 2Arg\left( {2i{e^{-i\frac{\pi }{{18}}}}.\overline z } \right)$ is
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