If $z$ is a complex number such that $\left| z \right| \geqslant 2$ , then the minimum value of $\left| {z + \frac{1}{2}} \right|$

$\frac{{3 + 2i\sin \theta }}{{1 – 2i\sin \theta }}$ will be purely imaginary, if $\theta = $ 

[Where $n$ is an integer]

Let the minimum value $v_{0}$ of $v=|=|^{2}+|z-3|^{2}+|z-60|^{2}$, $z \in C$ is attained at $z=z_{0}$. Then $\left|2 z_{0}^{2}-z_{0}^{3}+3\right|^{2}+v_{0}^{2}$ is equal to.

Let $\mathrm{n}$ denote the number of solutions of the equation $z^{2}+3 \bar{z}=0$, where $\mathrm{z}$ is a complex number. Then the value of $\sum_{k=0}^{\infty} \frac{1}{n^{k}}$ is equal to:

${\mathop{\rm Re}\nolimits} \frac{{{{(1 + i)}^2}}}{{3 – i}}$ =