If $\omega$ is a complex root of the equation $z^3 = 1$,then $\omega + \omega^{\left( \frac{1}{2} + \frac{3}{8} + \frac{9}{32} + \frac{27}{128} + \dots \right)}$ is equal to

$\left(\frac{1+\cos (3 \theta)+i \sin (3 \theta)}{1+\cos (3 \theta)-i \sin (3 \theta)}\right)^{20} = ?$

If $x+iy=(-1+i\sqrt{3})^{2010}$,then $x$ is

If $1, \omega, \omega^2, \ldots, \omega^8$ are the roots of the equation $x^9-1=0$,then $\sum_{r=1}^8 \left(\omega^r\right)^{99} =$

For the equation $x^2 + x + 1 = 0$,if $\alpha$ and $\beta$ are the roots,then which of the following equations has roots $\alpha^{19}$ and $\beta^{7}$?

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: