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

If the roots of $ax^2 + bx + c = 0$ are $\alpha, \beta$ and the roots of $Ax^2 + Bx + C = 0$ are $\alpha - k, \beta - k$,then $\frac{B^2 - 4AC}{b^2 - 4ac}$ is equal to

$PBA$ and $PDC$ are two secants. $AD$ is the diameter of the circle with center at $O$. $\angle A = 40^{\circ}$,$\angle P = 20^{\circ}$. Find the relationship between Quantity $I$ $(\angle DBC)$ and Quantity $II$ $(\angle ADB)$.

If exactly one root of the equation $x^2 + (a - 1)x + 2a = 0$ lies in the interval $(0, 3)$,then the set of values of $a$ is given by:

If the roots of the equation $ax^2 + bx + c = 0$ are real and of the form $\frac{\alpha}{\alpha - 1}$ and $\frac{\alpha + 1}{\alpha}$,then the value of $(a + b + c)^2$ is

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $ax^2 + bx + c = 0$. If $a, b, c$ are in $A.P.$ and $\alpha + \beta = 15$,then $\alpha\beta$ equals: