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

For what value of $\lambda$ is the sum of the squares of the roots of ${x^2} + (2 + \lambda )x - \frac{1}{2}(1 + \lambda ) = 0$ minimum?

The roots of the given equation $(p - q){x^2} + (q - r)x + (r - p) = 0$ are

Solve the given two equations and select the correct option.
$I.$ $\frac{12}{\sqrt{x}} - \frac{23}{\sqrt{x}} = 5\sqrt{x}$
$II.$ $\frac{\sqrt{y}}{12} - \frac{5\sqrt{y}}{12} = -\frac{1}{\sqrt{y}}$

If one root of the equation ${x^2} + px + q = 0$ is $2 + \sqrt{3}$,then the values of $p$ and $q$ are:

If $P(x) = ax^2 + bx + c$ and $Q(x) = -ax^2 + dx + c$ where $ac \neq 0$,then $P(x) \cdot Q(x) = 0$ has at least