The sum of all possible values of $\theta \in[-\pi, 2 \pi]$, for which $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely imaginary, is equal to (in $\pi$)

Let $z$ be a complex number with $\operatorname{Im}(z)=10$ and satisfying $\frac{2z-n}{2z+n}=2i-1$, where $i=\sqrt{-1}$, for some natural number $n$. Then:

Let $z = a + ib, b \neq 0$ be a complex number satisfying $z^{2} = \overline{z} \cdot 2^{1-|z|}$. Then the least value of $n \in N$ such that $z^{n} = (z + 1)^{n}$ is equal to:

If $\theta \in \mathbb{R}$ and $\frac{1-i \cos \theta}{1+2 i \cos \theta}$ is a real number,then $\theta$ will be (where $I$ is the set of integers):

Given that the equation $z^2 + (p + iq)z + r + is = 0$,where $p, q, r, s$ are real and non-zero,has a real root,then:

If the complex numbers $z_1, z_2$ and the origin form an equilateral triangle,then $z_1^2 + z_2^2 = $