If the point $(x, y)$ satisfies the equation $\frac{x+i(x-2)}{3+i}-i=\frac{2y+i(1-3y)}{i-3}$,then $x+y=$

The number of solutions for $z^3+\bar{z}=0$ 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

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):

If $z = \frac{\sqrt{3}}{2} + \frac{i}{2}$ where $i = \sqrt{-1}$,then $(1 + iz + z^5 + iz^8)^9$ is equal to:

If $x_n = \cos \left(\frac{\pi}{4^n}\right) + i \sin \left(\frac{\pi}{4^n}\right)$,then the product $x_1 x_2 x_3 \ldots \infty$ is equal to