If the roots of the equation $\frac{x^2 - bx}{ax - c} = \frac{m - 1}{m + 1}$ are equal but opposite in sign,then the value of $m$ is:

If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant $G.P.$ such that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x^2 + x - 1 = 0$ have a common root,then $\alpha(\beta + \gamma)$ is equal to

If $x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots \infty}}}$,then

Consider the quadratic equation $(c - 5)x^2 - 2cx + (c - 4) = 0$,where $c \ne 5$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0, 2)$ and its other root lies in the interval $(2, 3)$. Then the number of elements in $S$ is

Solve the given two equations and select the correct answer from the given options.
$I.$ $2x + 5y = 6$
$II.$ $5x + 11y = 9$

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}-7x+12=0$
$II.$ $y^{2}+y-12=0$