If $\theta = \frac{\pi}{6}$,then the $10^{th}$ term of the series $1 + (\cos \theta + i \sin \theta) + (\cos \theta + i \sin \theta)^2 + (\cos \theta + i \sin \theta)^3 + \ldots$ is equal to:

Let $z_1, z_2, \ldots, z_7$ be the vertices of a regular heptagon inscribed in the unit circle with the center at the origin in the complex plane. If $w = \sum_{1 \leq i < j \leq 7} z_i z_j$,then $|w|$ is equal to:

If $1, \omega, \omega^2$ are the cube roots of unity,then
$1(2+\frac{1}{\omega})(2+\frac{1}{\omega^2})+2(3+\frac{1}{\omega})(3+\frac{1}{\omega^2})+3(4+\frac{1}{\omega})(4+\frac{1}{\omega^2})+\ldots 10 \text{ terms} =$

When $n=8$,$(\sqrt{3}+i)^n+(\sqrt{3}-i)^n=$

Let $\omega=\operatorname{cis}\left(\frac{2 \pi}{3}\right)=\cos \left(\frac{2 \pi}{3}\right)+i \sin \left(\frac{2 \pi}{3}\right)$ and $f(x)=x^7-2 x^4-4 x^3+8$. Which of the following options is correct?

If $\alpha, \beta$ are the roots of the equation $x+\frac{4}{x}=2 \sqrt{3}$,then $\frac{2}{\sqrt{3}}\left|\alpha^{2024}-\beta^{2024}\right|=$