If $(3 + i)z = (3 - i)\bar{z}$,then the complex number $z$ is

If for $z=\alpha+i \beta$,$|z+2|=z+4(1+i)$,then $\alpha+\beta$ and $\alpha \beta$ are the roots of the equation

Let a complex number $z$,$|z| \neq 1$,satisfy $\log_{\frac{1}{\sqrt{2}}} \left( \frac{|z|+11}{(|z|-1)^2} \right) \leq 2$. Then,the largest value of $|z|$ is equal to ............

Find real $\theta$ such that $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ is purely real.

The value of the expression $\left( \cos \frac{\pi }{2} + i\sin \frac{\pi }{2} \right) \left( \cos \frac{\pi }{{{2^2}}} + i\sin \frac{\pi }{{{2^2}}} \right) \dots$ to $\infty$ is

$\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right) \times \left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) \times \left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) \times \ldots \infty =$