The value of $\sum_{n=0}^{\infty}\left(\frac{2 i}{3}\right)^n$ is

If $\frac{1-10 i \cos \theta}{1-10 \sqrt{3} i \sin \theta}$ is purely real,then one of the values of $\theta$ is

For a non-real root $z$ of the equation $z^4+z^2+1=0$,find the value of $\left(z+\frac{1}{z}\right)^3+\left(z^2+\frac{1}{z^2}\right)^2+\left(z^3+\frac{1}{z^3}\right)^3$.

If $i z^3+z^2-z+i=0$ (where $z$ is a complex number),then the value of $|z|$ is

If $z_{1}, z_{2}, z_{3}$ are complex numbers such that $|z_{1}|=|z_{2}|=|z_{3}|=|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}|=1$, then $|z_{1}+z_{2}+z_{3}|$ is

Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$,where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be non-zero.
Match each entry in List-$I$ to the correct entries in List-$II$.

List-$I$	List-$II$
$(P)$ $|z|^2$ is equal to	$(1)$ $12$
$(Q)$ $|z-\bar{z}|^2$ is equal to	$(2)$ $4$
$(R)$ $|z|^2+|z+\bar{z}|^2$ is equal to	$(3)$ $8$
$(S)$ $|z+1|^2$ is equal to	$(4)$ $10$
	$(5)$ $7$