If $x+\frac{1}{x}=2,$ then the value of $x^{12}-\frac{1}{x^{12}}$ is:

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}-25=0$
$II.$ $y^{2}-9y+20=0$

The quadratic equation whose one root is $\frac{1}{2 + \sqrt{5}}$ will be

Solve the given system of three equations and select the correct relationship between $x, y,$ and $z$ from the given options.
$I. x + 3y + 4z = 96$
$II. 2x + 8z = 80$
$III. 2x + 6y = 120$

Find the value of $k$ so that the sum of the roots of the equation $3 x^{2}+(2 k+1) x-k-5=0$ is equal to the product of the roots:

If $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root and $a, b, c$ are non-zero real numbers,then $\frac{a^3 + b^3 + c^3}{abc} = $