If $\alpha, \beta$ are the roots of the quadratic equation $x^{2}-8x+k=0$,find the value of $k$ such that $\alpha^{2}+\beta^{2}=40$.

If $p$ and $q$ are non-zero real numbers and $\alpha^3 + \beta^3 = -p$,$\alpha \beta = q$,then a quadratic equation whose roots are $\frac{\alpha^2}{\beta}$ and $\frac{\beta^2}{\alpha}$ is

Let $p, q \in \{1, 2, 3, 4\}$. The number of equations of the form $px^2 + qx + 1 = 0$ having real roots is

If $x$ is a solution of the equation $\sqrt{2x + 1} - \sqrt{2x - 1} = 1$ where $x \ge \frac{1}{2}$,then $\sqrt{4x^2 - 1}$ is equal to:

If $\alpha$ and $\beta$ are the roots of the equation $6x^2 - 5x + 1 = 0$,then the value of $\tan^{-1} \alpha + \tan^{-1} \beta$ is:

Solve the given two equations and select the correct option.
$I.$ $x^{2}-19x+84=0$
$II.$ $y^{2}-25y+156=0$