Solve the given two equations and select the correct answer from the given options.
$I.$ $3x^2 - 7x - 20 = 0$
$II.$ $y^2 - 8y + 16 = 0$

Solve the given two equations and select the correct option based on the relationship between $x$ and $y$. The equations are: $8x + 7y = 135$ and $5x + 6y = 99$.

If ${x^2} + px + q = 0$ is the quadratic equation whose roots are $a - 2$ and $b - 2$,where $a$ and $b$ are the roots of ${x^2} - 3x + 1 = 0$,then

If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant $G.P.$ such that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x^2 + x - 1 = 0$ have a common root,then $\alpha(\beta + \gamma)$ is equal to

If $\left\{\frac{1}{2}(a-b)\right\}^{2}+a b=p(a+b)^{2},$ then the value of $p$ is (assume that $a \neq-b).$

If $x+\frac{1}{x}=2,$ then the value of $x^{2}+\frac{1}{x^{2}}$ will be