Solve the given two equations and select the correct option.
$I.$ $x = \sqrt[3]{2744}$
$II.$ $y = \sqrt{487}$

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

For the equation $3x^2 + px + 3 = 0, p > 0$,if one root is the square of the other,then $p$ is equal to:

Let $x_1, x_2, x_3 \in R - \{0\}$,$x_1 + x_2 + x_3 \neq 0$ and $\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} = \frac{1}{x_1 + x_2 + x_3}$. Then $\frac{1}{x_1^n + x_2^n + x_3^n} = \frac{1}{x_1^n} + \frac{1}{x_2^n} + \frac{1}{x_3^n}$ holds good for:

Solve the given two equations and select the correct answer from the given options.
$I. x = \sqrt{1764}$
$II. y^2 = 1764$

If the roots of the equation $ax^2 + bx + c = 0$ are $(\alpha - \beta)$ and $(\gamma - \delta)$,and the roots of the equation $Ax^2 + Bx + C = 0$ are $(\alpha + \delta)$ and $(\beta + \gamma)$,then $\left| \frac{a}{A} \right|$ is equal to (where $D_1$ and $D_2$ are the discriminants of the given equations respectively).