The product of all the values of $(\sqrt{3}-i)^{2/5}$ is

If ${x_n} = \cos \left( \frac{\pi }{4^n} \right) + i \sin \left( \frac{\pi }{4^n} \right)$,then ${x_1} \cdot {x_2} \cdot {x_3} \dots \infty = $

Let $\alpha$ and $\beta$ be the roots of $x^{2}+x+1=0$. If $n$ is a positive integer, then $\alpha^{n}+\beta^{n}$ is

If $f(x)$ and $g(x)$ are two polynomials such that $\phi(x) = f(x^3) + x g(x^3)$ is divisible by $x^2 + x + 1$, then

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?

Let $\alpha$ and $\beta$ be the roots of $x^2 - \sqrt{2}x + 1 = 0$. Then the value of $\alpha^{50} + \beta^{50}$ is: