If $z = \frac{\sqrt{3} + i}{-2}$,then $z^{69}$ is equal to

If $n$ is an integer and $Z = \cos \theta + i \sin \theta$,where $\theta \neq (2n + 1) \frac{\pi}{2}$,then $\frac{1 + Z^{2n}}{1 - Z^{2n}} = $

If $1, \alpha_1, \alpha_2, \alpha_3, \alpha_4$ are the roots of $z^5-1=0$ and $\omega$ is a cube root of unity,then $(\omega-1)(\omega-\alpha_1)(\omega-\alpha_2)(\omega-\alpha_3)(\omega-\alpha_4)+\omega$ is equal to

If $z = \frac{\sqrt{3} + i}{2}$,then the value of $z^{69}$ is

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

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: