Solve the given two equations and select the correct answer from the given options.
$I.$ $x = \sqrt[3]{2197}$
$II.$ $y^2 = 169$

$\alpha_r$ and $\beta_r$ $(\alpha_r < \beta_r)$ are the roots of $x^2 - r^2(r + 1)x + r^5 = 0$. The value of $\sum_{r=1}^n (3\alpha_r + 2\beta_r)$ is:

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).

The number of real solutions of the equation $|x^2 + 4x + 3| + 2x + 5 = 0$ is:

Find the value of $(a+b+c)^{3}-3(b+c)(c+a)(a+b)$ if $a=5, b=3$ and $c=2$.

Consider the equation $x^2 + \alpha x + \beta = 0$ having roots $\alpha, \beta$ such that $\alpha \neq \beta$. Also consider the inequality $| |y - \beta| - \alpha | < \alpha$,then: