If $x = -2 - \sqrt{3} i$,where $i = \sqrt{-1}$,then the value of $2x^4 + 5x^3 + 7x^2 - x + 41$ is

If $z = x + iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$,then

The complex numbers $\sin x + i\cos 2x$ and $\cos x - i\sin 2x$ are conjugate to each other for

Suppose $z$ is any root of $11 z^8 + 21 i z^7 + 10 i z - 22 = 0$ where $i = \sqrt{-1}$. Then,$S = |z|^2 + |z| + 1$ satisfies

Let $A = \{\theta \in (0, 2\pi) : \frac{1+2i \sin \theta}{1-i \sin \theta} \text{ is purely imaginary} \}$. Then the sum of the elements in $A$ is

If $z = 2 + 3i$,then $z^{5} + (\bar{z})^{5}$ is equal to: