If $\alpha, \beta$ are the roots of the equation $x^2-2x+4=0$ and for any $n \in N, \alpha^n+\beta^n=k \cos \frac{n \pi}{3}$, then $k=$

Let $\omega=\operatorname{cis}\left(\frac{2 \pi}{3}\right)=\cos \left(\frac{2 \pi}{3}\right)+i \sin \left(\frac{2 \pi}{3}\right)$ and $f(x)=x^7-2 x^4-4 x^3+8$. Which of the following options is correct?

If $\theta = \frac{\pi}{6}$,then the $10^{th}$ term of the series $1 + (\cos \theta + i \sin \theta) + (\cos \theta + i \sin \theta)^2 + (\cos \theta + i \sin \theta)^3 + \ldots$ is equal to:

If the cube root of $1$ is $\omega$,then the value of $(3 + \omega + 3\omega^2)^4$ is

If $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^m=1$ and $2022 < m < 2029$,then $m=$

If $1, a, a^2, \ldots, a^{n-1}$ are the $n$th roots of unity,then $\sum_{i=1}^{n-1} \frac{1}{2-a^i}$ is equal to