The value of $\frac{a + b\omega + c\omega^2}{b + c\omega + a\omega^2} + \frac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2}$ is

If $\omega$ is a complex cube root of unity,then $(1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8) \dots$ to $2n$ factors is equal to:

The number of real values of $(-1-\sqrt{3} i)^{3/4}$ is

If $z \in \mathbb{C}$ and $i z^3+4 z^2-z+4 i=0$,then a complex root of this equation having minimum magnitude is

If $1, \omega, \omega^2$ are the cube roots of unity,$k$ is a positive integer and $(1-\omega+\omega^2)^{3k} + (1-\omega^2+\omega)^{3k} = (1-\omega+\omega^2)^{3k+1} + (1+\omega-\omega^2)^{3k+1}$,then $k=$

If $\omega$ is a complex cube root of unity,then $\sin \left\{\left(\omega^{10}+\omega^{23}\right) \pi-\frac{\pi}{4}\right\}=$