If $z =2+3 i$, then $z ^{5}+(\overline{ z })^{5}$ is equal to.

The complex numbers $sin\ x + i\ cos\ 2x$ and $cos\ x\ -\ i\ sin\ 2x$ are conjugate to each other, for

The number of solutions of the equation ${z^2} + \bar z = 0$ is

If $z$ is a purely real number such that ${\mathop{\rm Re}\nolimits} (z) < 0$, then $arg(z)$ is equal to

If $z$ is a complex number such that $\frac{{z - 1}}{{z + 1}}$ is purely imaginary, then

For any two complex numbers ${z_1}$and${z_2}$ and any real numbers $a$ and $b$; $|(a{z_1} - b{z_2}){|^2} + |(b{z_1} + a{z_2}){|^2} = $