If $1, \omega, \omega^2$ are the cube roots of unity,then $\omega^2(1 + \omega)^3 - (1 + \omega^2)\omega = $

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 $\alpha, \beta$ are the roots of $x^2-2x+4=0$, for $n \in N$, what is the value of $\alpha^n+\beta^n$?

If $\omega$ is a complex cube root of unity and $a, b, c$ are distinct real numbers,then $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2} = $

If $\omega$ is a complex cube root of unity, then the value of the expression $2(1 + \frac{1}{\omega})(1 + \frac{1}{\omega^2}) + 3(2 + \frac{1}{\omega})(2 + \frac{1}{\omega^2}) + ... + (n + 1)(n + \frac{1}{\omega})(n + \frac{1}{\omega^2})$ is...

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