The product $(1 - \omega + {\omega ^2})(1 - {\omega ^2} + {\omega ^4})(1 - {\omega ^4} + {\omega ^8}) \dots$ up to $2n$ factors is:

If $n$ is a positive integer and $\omega \neq 1$ is a cube root of unity,the number of possible values of $\left|e^{\sum_{k=0}^n {^nC_k} \omega^k}\right|$ is

If $\omega$ is a complex cube root of unity,then $225+(3 \omega+8 \omega^2)^2+(3 \omega^2+8 \omega)^2$ is equal to:

If $\omega$ is a complex cube root of unity,then for any $n>1$,$\sum_{r=1}^{n-1} r(r+1-\omega)(r+1-\omega^2) =$

The value of $\left(\frac{1+\sqrt{3} i}{1-\sqrt{3} i}\right)^{64}+\left(\frac{1-\sqrt{3} i}{1+\sqrt{3} i}\right)^{64}$ is

If $Z_r = \cos \left(\frac{\pi}{2^r}\right) + i \sin \left(\frac{\pi}{2^r}\right)$ for $r = 1, 2, 3, \ldots$,then the product $Z_1 Z_2 Z_3 \ldots \infty$ is equal to: