If $\omega$ is an imaginary cube root of unity,$(1 + \omega - \omega^2)^7$ equals

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:

$(27)^{1/3} = $

If $1, \alpha_1, \alpha_2, \alpha_3, \alpha_4$ are the roots of $z^5-1=0$ and $\omega$ is a cube root of unity,then $(\omega-1)(\omega-\alpha_1)(\omega-\alpha_2)(\omega-\alpha_3)(\omega-\alpha_4)+\omega$ is equal to

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 27 = 0$,find the quadratic equation whose roots are $\left( \frac{\gamma}{\alpha} \right)^2$ and $\left( \frac{\beta}{\alpha} \right)^2$.

The value of $\frac{(\cos \theta + i \sin \theta)^4}{(\sin \theta + i \cos \theta)^5}$ is equal to,where $i = \sqrt{-1}$: