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

If $\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2020}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2021} = x+i y$,then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

Let $z=x+iy$ and $w=u+iv$ be complex numbers on the unit circle such that $z^2+w^2=1$. Then the number of ordered pairs $(z, w)$ is

If $(x-iy)^{1/3} = a-ib$,then the value of $\frac{x}{a} + \frac{y}{b}$ is

Let $S = \{z \in \mathbb{C} : z^2 + \sqrt{6}iz - 3 = 0\}$. Then $\sum_{z \in S} z^8$ is equal to:

If one of the roots of the equation $4x^4 - 24x^3 + 57x^2 + 18x - 45 = 0$ is $3 + i\sqrt{6}$,find the other roots.