If $(x + iy)^{1/3} = a + ib$,then $\frac{x}{a} + \frac{y}{b}$ is equal to

The value of $\left(\frac{-1+i \sqrt{3}}{1-i}\right)^{30}$ is

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

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1| = 4|z_2|$. If $z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$,then:

Let the complex number $z = x + iy$ be such that $\frac{2z - 3i}{2z + i}$ is purely imaginary. If $x + y^2 = 0$,then $y^4 + y^2 - y$ is equal to:

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