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

Let $z, w \in \mathbb{C}$ satisfy $z^2 + \bar{w} = z$ and $w^2 + \bar{z} = w$. Then,the number of ordered pairs of complex numbers $(z, w)$ is equal to

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:

Sum of the squares of the imaginary roots of the equation $z^8-20z^4+64=0$ is

If $(1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni) = a + ib$,then $2 \times 5 \times 10 \times \dots \times (1 + n^2)$ is equal to

The values of $x$ for which $\sin x + i \cos 2x$ and $\cos x - i \sin 2x$ are conjugate to each other are