If $\omega$ is a complex cube root of unity,then find the value of $\sum_{x=1}^{10} ((\omega x+2)(\omega^2 x+2)-3)$.

Let $\omega$ be a cube root of unity not equal to $1$. Then,the maximum possible value of $|a + b\omega + c\omega^2|$,where $a, b, c \in \{+1, -1\}$ is

The general value of the real angle $\theta$, which satisfies the equation $(\cos \theta + i \sin \theta)(\cos 2\theta + i \sin 2\theta) \dots (\cos n\theta + i \sin n\theta) = 1$ is given by (assuming $k$ is an integer):

If $\omega$ is an imaginary cube root of unity,then the value of $(1-\omega+\omega^{2}) \cdot(1-\omega^{2}+\omega^{4}) \cdot(1-\omega^{4}+\omega^{8}) \cdot \ldots$ ($2n$ factors) is

If $1, \omega, \omega^2$ are the three cube roots of unity,then $(3 + \omega^2 + \omega^4)^6 = $

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: