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

The real part of $\frac{(\cos a+i \sin a)^6}{(\sin b+i \cos b)^8}$ is

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$,then $\alpha^9+\beta^9$ is equal to

If $\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0$,then $\left(\cos ^3 \alpha+\cos ^3 \beta+\cos ^3 \gamma\right)^2+\left(\sin ^3 \alpha+\sin ^3 \beta+\sin ^3 \gamma\right)^2=$

If $1, \alpha_1, \alpha_2, \ldots, \alpha_{n-1}$ are the $n$th roots of unity and $n$ is an even natural number,then $(1+\alpha_1)(1+\alpha_2) \ldots (1+\alpha_{n-1})=$

If $f(x)$ and $g(x)$ are two polynomials such that $\phi(x) = f(x^3) + x g(x^3)$ is divisible by $x^2 + x + 1$, then