If $z=x+iy$, $x^2+y^2=1$ and $z_1=ze^{i\theta}$, then $\frac{z_1^{2n}-1}{z_1^{2n}+1}=$

Let $\alpha = \frac{-1 + i \sqrt{3}}{2}$. If $a = (1 + \alpha) \sum_{k=0}^{100} \alpha^{2k}$ and $b = \sum_{k=0}^{100} \alpha^{3k}$,then $a$ and $b$ are the roots of the quadratic equation:

Consider a regular $10$-gon with its vertices on the unit circle. With one vertex fixed,draw straight lines to the other $9$ vertices. Call them $L_1, L_2, \ldots, L_9$ and denote their lengths by $l_1, l_2, \ldots, l_9$ respectively. Then,the product $l_1 \times l_2 \times \ldots \times l_9$ is

If $\omega$ is a complex cube root of unity,then the value of $\omega^{99} + \omega^{100} + \omega^{101}$ is

If $Z_{r} = \sin \frac{2 \pi r}{11} - i \cos \frac{2 \pi r}{11}$, then $\sum_{r=0}^{10} Z_{r}$ is equal to

The modulus of the complex number $\left(\frac{2+i \sqrt{5}}{2-i \sqrt{5}}\right)^{10}+\left(\frac{2-i \sqrt{5}}{2+i \sqrt{5}}\right)^{10}$ is