The value of $\sum_{k=1}^{6}\left(\sin \frac{2 k \pi}{7}-i \cos \frac{2 k \pi}{7}\right)$ is

Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{6}x+3=0$ such that $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Let $a, b$ be integers not divisible by $3$ and $n$ be a natural number such that $\frac{\alpha^{99}}{\beta}+\alpha^{98}=3^n(a+ib)$,where $i=\sqrt{-1}$. Then $n+a+b$ is equal to:

If $w = \frac{-1 + i \sqrt{3}}{2}$,where $i = \sqrt{-1}$,then the value of $(3 + w + 3 w^2)^4$ is

$\sum_{k=1}^6 \left[ \sin \frac{2 k \pi}{7} - i \cos \frac{2 k \pi}{7} \right]$ is equal to

If $z^2+z+1=0$,then find the value of $\left(z^3+\frac{1}{z^3}\right)^2+\left(z^4+\frac{1}{z^4}\right)^2$,where $z$ is a complex cube root of unity.

If $a = \sqrt{2i}$,then which of the following is correct?