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 $\left(\frac{3}{2}+i \frac{\sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)$ where $x$ and $y$ are real, then the ordered pair $(x, y)$ is

If $1, \alpha_1, \alpha_2, \ldots, \alpha_{n-1}$ are the $n^{\text{th}}$ roots of unity,then $\sum_{1 \leq i < j \leq n-1} \alpha_i \alpha_j =$

For a non-$0$ complex number $z$,let $\arg (z)$ denote the principal argument of $z$,with $-\pi < \arg (z) \leq \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg (\omega) < \pi$. Let $\alpha = \arg \left(\sum_{n=1}^{2025} (-\omega)^n\right)$. Then the value of $\frac{3 \alpha}{\pi}$ is $.....$ .

If $\alpha$ is a non-real root of $x^6=1$,then $\frac{\alpha^5+\alpha^3+\alpha+1}{\alpha^2+1}$ is equal to

$(1-i \sqrt{3})^{2025}=$