If $\omega$ is a complex cube root of unity,then $(1 + \omega - \omega^2)(1 - \omega + \omega^2) = $

Let $\alpha = \frac{-1 + i \sqrt{3}}{2}$. If $a = (1 + \alpha) \sum_{k=0}^{100} \alpha^{2k}$ and $b = \sum_{k=0}^{100} \alpha^{3k}$,then $a$ and $b$ are the roots of the quadratic equation:

If $z$ is a non-real root of $x^7=1$,then $1+3z+5z^2+7z^3+9z^4+11z^5+13z^6=$

The square of either of the two imaginary cube roots of unity is:

If $x = a + b$,$y = a\alpha + b\beta$,and $z = a\beta + b\alpha$,where $\alpha$ and $\beta$ are complex cube roots of unity,then $xyz$ =

If $(1 + x)^n = C_0 + C_1x + C_2x^2 + ..... + C_nx^n$,then the value of $C_0 - C_2 + C_4 - C_6 + .....$ is