If $\omega$ is an $n^{th}$ root of unity,other than unity,then the value of $1 + \omega + \omega^2 + ... + \omega^{n-1}$ is

If $\alpha$ and $\beta$ are non-real roots of $x^3-x^2-x-2=0$,then $\alpha^{2020}+\beta^{2020}+\alpha^{2020} \cdot \beta^{2020}=$

Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, where $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is . . . . . . .

If $z^2+z+1=0$,where $z$ is a complex number,then $\left(z+\frac{1}{z}\right)^3+\left(z^4+\frac{1}{z^4}\right)^3$ is equal to

If $1, \omega, \omega^{2}$ are three cube roots of unity,then $(1-\omega+\omega^{2})(1+\omega-\omega^{2})$ is

If $z$ is a complex number,then the number of common roots of the equations $z^{1985}+z^{100}+1=0$ and $z^3+2z^2+2z+1=0$ is equal to: