Solve the given two equations and select the correct answer from the given options.
$I.$ $(441)^{\frac{1}{2}} x^{2} - 111 = (15)^{2}$
$II.$ $\sqrt{121} y^{2} + (6)^{3} = 260$

The value of $a$ for which one root of the quadratic equation $(a^2 - 5a + 3)x^2 + (3a - 1)x + 2 = 0$ is twice as large as the other,is

$Quantity$ $1$: Height of the tank if the volume of a cylindrical tank is $12320 \, cm^3$. Its radius and height are in the ratio of $7:10$ respectively. $Quantity$ $2$: Height of the kerosene level in a jar if a cone of radius $2 \, cm$ and height $3 \, cm$ is melted and recast into a cylinder of radius $2 \, cm$.

The range of $a$ for which the roots of $x^2 - 2x - a^2 + 1 = 0$ lie between the roots (exclusive) of the equation $x^2 - 2(a + 1)x + a(a - 1) = 0$ is:

Find the quadratic equation whose roots are reciprocals of the roots of the equation $3x^{2}-20x+17=0$.

Solve the given two equations and select the correct answer from the given options.
$I.$ $7x = 4y + 85$
$II.$ $y = \sqrt[3]{17576}$