If $x$ and $y$ are two positive real numbers such that $x+iy = \frac{13 \sqrt{-5+12i}}{(2-3i)(3+2i)}$,then $13y-26x=$

If $z_n = (1 + i \sqrt{2})^n$, $n \in Z$, then $\frac{1}{9} \operatorname{Re}(z_4 \bar{z}_5) = $

For positive integers $n_1, n_2$, the value of the expression $(1 + i)^{n_1} + (1 + i^3)^{n_1} + (1 + i^5)^{n_2} + (1 + i^7)^{n_2}$, where $i = \sqrt{-1}$, is a real number if and only if:

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

If $z \in \mathbb{C}$,then the minimum value of $|z| + |2z - 3| + |z - 1|$ is

The value of $\left(\frac{-1+i \sqrt{3}}{1-i}\right)^{30}$ is