Solve the given two equations and select the correct answer from the given options.
$I.$ $x - \sqrt{169} = 0$
$II.$ $y^2 - 169 = 0$

If the equation $\frac{1}{x} + \frac{1}{x - 1} + \frac{1}{x - 2} = 3x^3$ has $k$ real roots,then $k$ is equal to -

Two positive distinct numbers $a$ and $b$ each differ from their reciprocal by $1$. The value of $a + b$ is

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2} = 64$
$II.$ $2y^{2} + 25y + 72 = 0$

$Quantity$ $1$: 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 $2 \, cm$.

Solve the given equations and choose the correct option.
$I.$ $x = \sqrt{(36)^{1/2} \times (1296)^{1/4}}$
$II.$ $2y + 3z = 33$
$III.$ $6y + 5z = 71$