If $x-\frac{1}{x}=\sqrt{21},$ then the value of $\left(x^{2}+\frac{1}{x^{2}}\right)\left(x+\frac{1}{x}\right)$ will be

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

If $x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots \infty}}}$,then

If for a positive integer $n$,the quadratic equation $x(x + 1) + (x + 1)(x + 2) + \dots + (x + n - 1)(x + n) = 10n$ has two consecutive integral solutions,then $n$ is equal to:

Solve the given two equations and select the correct answer from the given options.
$I.$ $36 x^{4}+369 x^{2}+900=0$
$II.$ $144 y^{4}+337 y^{2}+144=0$

$I. \quad p^{2}-18 p+77=0$
$II. \quad 3 q^{2}-25 q+28=0$
$Quantity \, 1$: Value of $p$
$Quantity \, 2$: Value of $q$