If $x+\frac{1}{x}=2 \cos \theta$,then for any integer $n$,$x^n+\frac{1}{x^n}=$

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 $f(x)$ 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

If $n$ is an integer which leaves a remainder of $1$ when divided by $3$,then $(1+\sqrt{3}i)^n + (1-\sqrt{3}i)^n$ equals

If $\alpha$ is a non-real root of $x^7=1$,then $\alpha(1+\alpha)(1+\alpha^2+\alpha^4) = $

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