If $x = a^{1/2} + a^{-1/2}$ and $y = a^{1/2} - a^{-1/2}$,then find the value of $(x^4 - x^2 y^2 - 1) + (y^4 - x^2 y^2 + 1)$.

The value of $k$ for which the equation $(k - 2)x^2 + 8x + (k + 4) = 0$ has both roots real,distinct,and negative is

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation $x^2 + (2 - \lambda) x + (10 - \lambda) = 0$ is minimum,then the magnitude of the difference of the roots of this equation is

Solve the given two equations and select the correct answer from the given options.
$I.$ $15 x^{2}-41 x+14=0$
$II.$ $2 y^{2}-13 y+20=0$

Let $p, q, r \in \mathbb{R}$ and $r > p > 0.$ If the quadratic equation $px^2 + qx + r = 0$ has two complex roots $\alpha$ and $\beta,$ then $|\alpha| + |\beta|$ is:

Solve the given two equations and choose the correct answer from the given options.
$I.$ $x^{2}+12x+36=0$
$II.$ $y^{2}+15y+56=0$