If $i=\sqrt{-1}$,then $\sum_{n=0}^{\infty}\left(\frac{i}{3}\right)^n=$

If $z$ is a complex number satisfying $|z|^2 - |z| - 2 < 0$,then the value of $|z^2 + z \sin \theta|$,for all values of $\theta$,is

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 $\pi$)

If $a, b, c$ and $d \in \mathbb{R}$ such that $a^2+b^2=4$ and $c^2+d^2=2$ and if $(a+ib)^2=(c+id)^2(x+iy)$,then $x^2+y^2$ is equal to

If $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely real,then $\cos ^3 \theta+\sin ^2 \theta+\cos \theta+1=$

If $z = x - iy$ and $z^{1/3} = p + iq$ $(x, y, p, q \in R)$,then $\frac{(\frac{x}{p} + \frac{y}{q})}{(p^2 + q^2)}$ is equal to