If $x+iy = \frac{3}{2+\cos \theta + i \sin \theta}$,then $x^2+y^2 =$

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 ............

If $z_1=x_1+i y_1$, $z_2=x_2+i y_2$, $z_3=x_1+\frac{i x_2}{2}$, and $z_4=2 y_1+i y_2$ are complex numbers such that $|z_1|=1$, $|z_2|=2$, and $\operatorname{Re}(z_1 \bar{z}_2)=0$, then:

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2+z+1|=1$. Then which of the following statements is/are $TRUE$?
$(A) |z+\frac{1}{2}| \leq \frac{1}{2}$ for all $z \in S$
$(B) |z| \leq 2$ for all $z \in S$
$(C) |z+\frac{1}{2}| \geq \frac{1}{2}$ for all $z \in S$
$(D)$ The set $S$ has exactly four elements

The expression $\frac{(1+i)^{n}}{(1-i)^{n-2}}$ equals

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