If $1, \omega, \omega^2$ are the cube roots of unity,then their product is

If $\alpha \neq 1$ is any $n^{th}$ root of unity,then $S = 1 + 3\alpha + 5\alpha^2 + \dots$ up to $n$ terms,is equal to

If ${x_n} = \cos \left( \frac{\pi }{4^n} \right) + i \sin \left( \frac{\pi }{4^n} \right)$,then ${x_1} \cdot {x_2} \cdot {x_3} \dots \infty = $

If $Z_{r} = \sin \frac{2 \pi r}{11} - i \cos \frac{2 \pi r}{11}$,then $\sum_{r=0}^{10} Z_{r}$ is equal to

If $\alpha_1, \alpha_2, \alpha_3, \ldots, \alpha_n$ are real numbers,$\alpha_1 \neq 0$ and $z = \cos \theta + i \sin \theta$ is a root of the equation $\alpha_1 + \alpha_2 z + \alpha_3 z^2 + \ldots + \alpha_n z^{n-1} + z^n = 0$,then $\alpha_1 \cos n \theta + \alpha_2 \cos (n-1) \theta + \ldots + \alpha_n \cos \theta =$

If $\omega$ is the complex cube root of unity and $\left(\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}\right)^k+\left(\frac{a+b \omega+c \omega^2}{b+a \omega^2+c \omega}\right)^l=2$,then $2k+l$ is always