$A$ value of $n$ such that $\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)^n=1$ is

If $\alpha$ and $\beta$ are imaginary cube roots of unity,then the value of $\alpha^4 + \beta^{28} + \frac{1}{\alpha \beta}$ is

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 ${\left( {\frac{{1 + \cos \theta + i\sin \theta }}{{i + \sin \theta + i\cos \theta }}} \right)^4} = \cos n\theta + i\sin n\theta $,then $n$ is equal to

If $\omega$ is a complex cube root of unity,then for any $n>1$,$\sum_{r=1}^{n-1} r(r+1-\omega)(r+1-\omega^2) =$

If $1, \omega, \omega^2$ are the cube roots of unity,then $\omega^2(1 + \omega)^3 - (1 + \omega^2)\omega = $