Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}+5x+6=0$
$II.$ $y^{2}+7y+12=0$

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

The least positive value of $a$ for which the equation $2x^{2} + (a - 10)x + \frac{33}{2} = 2a$ has real roots is

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}+13x+40=0$
$II.$ $y^{2}+7y+10=0$

Given that $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2 - px + q = 0$,then the value of $\sin^2(\alpha + \beta)$ is:

If one root of the quadratic equation $ax^2 + bx + c = 0$ is equal to the $n^{th}$ power of the other root,then the value of $(ac^n)^{\frac{1}{n+1}} + (a^nc)^{\frac{1}{n+1}} = $