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 :

Let $$z$$ be a complex number such that $$|z+2|=1$$ and $$\operatorname{lm}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$$. Then the value of $$|\operatorname{Re}(\overline{z+2})|$$ is

If the set $$R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\}$$ has $$m$$ elements and $$\sum_\limits{n=1}^m\left(1-i^{n !}\right)=x+i y$$, where $$i=\sqrt{-1}$$, then the value of $$m+x+y$$ is

If $$z_1, z_2$$ are two distinct complex number such that $$\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$$, then