If $x+\frac{1}{x}=2,$ then the value of $x^{12}-\frac{1}{x^{12}}$ is:

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}+20=9x$
$II.$ $y^{2}+42=13y$

$\alpha_r$ and $\beta_r$ $(\alpha_r < \beta_r)$ are the roots of $x^2 - r^2(r + 1)x + r^5 = 0$. The value of $\sum_{r=1}^n (3\alpha_r + 2\beta_r)$ is:

Sum of roots is $-1$ and sum of their reciprocals is $\frac{1}{6}$,then the equation is

The harmonic mean of the roots of the equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$ is

Let $S$ be the set of all $\alpha \in R$ such that the equation $\cos 2x + \alpha \sin x = 2\alpha - 7$ has a solution. Then $S$ is equal to