Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2} = 49$
$II.$ $y^{2} + 15y + 56 = 0$

The number of solutions of the equation $\log_{\sqrt{3}}(x^3 - 1) = \log_{\sqrt{3}}(x - 1) + 2$ is:

The value of $a (a \ge 3)$ for which the sum of the cubes of the roots of $x^2 - (a - 2)x + (a - 3) = 0$ assumes the least value is

If $\alpha$ and $\beta$ are roots of the equation $Ax^2 + Bx + C = 0$,then the value of $\alpha^3 + \beta^3$ is

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals,then $\frac{b^2}{ac} + \frac{bc}{a^2} = $

If $\left(\frac{p^{-1} q^{2}}{p^{3} q^{-2}}\right)^{\frac{1}{3}}+\left(\frac{p^{5} q^{-3}}{p^{-2} q^{3}}\right)^{\frac{1}{3}}=p^{a} q^{b},$ then the value of $a+b,$ where $p$ and $q$ are different positive positive primes,is