If $x+iy = \frac{3}{2+\cos \theta + i \sin \theta}$,then $x^2+y^2 =$

If $-i$ and $\alpha$ are the roots of the equation $iz^2 - 2(i+1)z + (2-i) = 0$,$\tan \theta = \frac{-1}{2}$ and $\theta \in 4^{\text{th}}$ quadrant,then $5^3 \cos 6\theta =$

The least value of $|z|$ where $z$ is a complex number satisfying the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{|z|+1} \log _{ e } 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i |$,where $i=\sqrt{-1}$,is equal to:

If the real part of the complex number $z = \frac{3 + 2i \cos \theta}{1 - 3i \cos \theta}$, where $\theta \in (0, \frac{\pi}{2})$, is zero, then the value of $\sin^2 3\theta + \cos^2 \theta$ is equal to:

Find the complex number $z$ satisfying the equations $\left| \frac{z - 12}{z - 8i} \right| = \frac{5}{3}$ and $\left| \frac{z - 4}{z - 8} \right| = 1$.

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