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

Let $z$, $w \in C$ satisfy ${z^2} + \bar w = z$ and ${w^2} + \bar z = w$ then number of ordered pairs of  complex numbers $(z, w)$ is equal to

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

If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and $\beta=\left(\frac{|z|}{\arg (z)}\right)$, where $z=\frac{\pi}{4}(1+i)^4\left(\frac{1-\sqrt{\pi i}}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} \mathrm{i}}\right), i=\sqrt{-1}$, then the distance of the point $(\alpha, \beta)$ from the line $4 x-3 y=7$ is

If $z_{1}=2-i, z_{2}=1+i,$ find $\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+1}\right|$

If the equation, $x^{2}+b x+45=0(b \in R)$ has conjugate complex roots and they satisfy $|z+1|=2 \sqrt{10},$ then