Solve the given two equations and select the correct option.
$I.$ $x = (1331)^{1/3}$
$II.$ $2y^2 - 17y + 36 = 0$

Solve the given two equations and select the correct option.
$I.$ $x^{3} - 468 = 1729$
$II.$ $y^{2} - 1733 + 1564 = 0$

Solve the given system of three linear equations and select the correct relationship between $x, y,$ and $z$ from the given options.
$I. 7x + 6y + 4z = 122$
$II. 4x + 5y + 3z = 88$
$III. 9x + 2y + z = 78$

If $x = \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ and $y = \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$,then the value of $\frac{x^{2}+xy+y^{2}}{x^{2}-xy+y^{2}} = ?$

If $3$ distinct real numbers $a, b, c$ satisfy $a^2(a + p) = b^2(b + p) = c^2(c + p)$ where $p \in R$,then the value of $bc + ca + ab$ is

The value of $c$ for which $|{\alpha ^2} - {\beta ^2}| = \frac{7}{4}$,where $\alpha$ and $\beta$ are the roots of $2{x^2} + 7x + c = 0$,is