Let $\mathrm{P}=\{\mathrm{z} \in \mathbb{C}:|\mathrm{z}+2-3 \mathrm{i}| \leq 1\}$ and $\mathrm{Q}=\{\mathrm{z} \in \mathbb{C}: \mathrm{z}(1+\mathrm{i})+\overline{\mathrm{z}}(1-\mathrm{i}) \leq-8\}$. Let in $\mathrm{P} \cap \mathrm{Q},|\mathrm{z}-3+2 \mathrm{i}|$ be maximum and minimum at $z_1$...

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