Let $z = \cos \theta + i \sin \theta$. Then,the value of $\sum_{m=1}^{15} \text{Im}(z^{2m-1})$ at $\theta = 2^{\circ}$ 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 $\alpha, \beta$ are the roots of the equation $x^2-2x+4=0$ and for any $n \in N, \alpha^n+\beta^n=k \cos \frac{n \pi}{3}$, then $k=$

$\sum_{r=1}^{16}\left(\sin \frac{2 r \pi}{17}+i \cos \frac{2 r \pi}{17}\right)=$

If $\omega$ is a complex cube root of unity and $(1+\omega)^7=A+B \omega$,then the values of $A$ and $B$ are,respectively.

If $\alpha, \beta$ are the roots of the equation $1+x+x^2=0$,then $(2-\alpha)(2-\beta)(2-\alpha^{10})(2-\alpha^{20})=$