One root of the equation $3x^2 - 10x + 3 = 0$ is $\frac{1}{3}$. Find the other root.

Solve the given two equations and select the correct option.
$I. 13x - 8y + 81 = 0$
$II. 15x - 5y + 65 = 0$

If $m - 5n = 2$,then the value of $(m^3 - 125n^3 - 30mn)$ is

Let $\alpha, \beta$ be the roots of the quadratic equation $x^2 + px + p^3 = 0$ $(p \neq 0)$. If $(\alpha, \beta)$ is a point on the parabola $y^2 = x$,then the roots of the quadratic equation are:

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 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: