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

If $\omega \neq 1$ is a cube root of unity, then the sum of the series $S = 1 + 2\omega + 3\omega^2 + \dots + 3n\omega^{3n-1}$ is

The sum of the $162^{\text{th}}$ power of the roots of the equation $x^{3}-2x^{2}+2x-1=0$ is

Let $1, \omega$,and $\omega^2$ be the cube roots of unity. The least possible degree of a polynomial with real coefficients,having $2\omega, 2\omega^2, 3+4\omega, 3+4\omega^2$,and $5-\omega-\omega^2$ as roots is

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 $\beta^{7}$?

$\alpha, \beta$ are the roots of the equation $x^2+2x+4=0$. If the point representing $\alpha$ in the Argand diagram lies in the $2^{nd}$ quadrant and $\alpha^{2024}-\beta^{2024}=ik, (i=\sqrt{-1})$,then $k=$