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

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}+20=9x$
$II.$ $y^{2}+42=13y$

If the quadratic equation $x^2 + (2 - \tan \theta)x - (1 + \tan \theta) = 0$ has $2$ integral roots,then the sum of all possible values of $\theta$ in the interval $(0, 2\pi)$ is $k\pi$. Then $k$ equals:

Two candidates attempt to solve the equation $x^2 + px + q = 0$. One starts with a wrong value of $p$ and finds the roots to be $2$ and $6$,and the other starts with a wrong value of $q$ and finds the roots to be $2$ and $-9$. The roots of the original equation are

Solve the given two equations and select the correct answer from the given options:
$I.$ $679 x^{2} - 168 x^{2} = 3066$
$II.$ $\sqrt{144} y^{3} - 9 y^{3} = 1536$

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals,then $a/c, b/a, c/b$ are in