If $z$ is a complex number, then $z.\,\overline z = 0$ if and only if

Let $z$ be a complex number, then the equation ${z^4} + z + 2 = 0$ cannot have a root, such that

The modulus and amplitude of $\frac{{1 + 2i}}{{1 - {{(1 - i)}^2}}}$ are

For the complex number $z$, one from $z + \bar z$ and $z\,\bar z$ is

Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of

$(\bar{z})^2+\frac{1}{z^2}$

are integers, then which of the following is/are possible value($s$) of $|z|$ ?

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