If $2i$ is a root of $f(z) = z^4 + z^3 + 2z^2 + 4z - 8 = 0$,then which among the following cannot be a root of $f(z) = 0$?

If $x+iy = (1+i)^6 - (1-i)^6$,then which one of the following is true?

If $\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2024}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2025}=x+i y$,then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

If $z = 3 - 4i$,then ${z^4} - 3{z^3} + 3{z^2} + 99z - 95$ is equal to

If $a = \frac{1 - i \sqrt{3}}{2}$, then the correct matching of List-$I$ with List-$II$ is:

List-$I$	List-$II$
$(i)$ $a \bar{a}$	$(A)$ $-\frac{\pi}{3}$
$(ii)$ $\arg \left(\frac{1}{\bar{a}}\right)$	$(B)$ $-i \sqrt{3}$
$(iii)$ $a - \bar{a}$	$(C)$ $2i / \sqrt{3}$
$(iv)$ $\operatorname{Im}\left(\frac{4}{3a}\right)$	$(D)$ $1$
	$(E)$ $\pi / 3$
	$(F)$ $\frac{2}{\sqrt{3}}$

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