Find the value of $a^{2}+b^{2}+c^{2}-2ab-2bc-2ca$ when $a=17, b=15$ and $c=13$.

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 $x = \sqrt{7 + 4\sqrt{3}},$ then $x + \frac{1}{x} = $

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 $\alpha, \beta$ are the roots of the equation $a x^{2}+b x+b=0,$ then $\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{b}{a}}=$

Solve the given two equations and select the correct option.
$I. 13x - 8y + 81 = 0$
$II. 15x - 5y + 65 = 0$