The sum of all the real values of $x$ satisfying the equation $2^{(x - 1)(x^2 + 5x - 50)} = 1$ is

If $x$ is a rational number and $\frac{(x+1)^{3}-(x-1)^{3}}{(x+1)^{2}-(x-1)^{2}}=2$,then the sum of the numerator and denominator of $x$ is

Find the number of natural number solutions for the equation $x_1 + x_2 = 100$,such that $x_1$ and $x_2$ are not multiples of $5$.

If $|{x^2} - x - 6| = x + 2$,then the values of $x$ are

Solve the given two equations and select the correct answer from the given options.
$I.$ $(441)^{\frac{1}{2}} x^{2} - 111 = (15)^{2}$
$II.$ $\sqrt{121} y^{2} + (6)^{3} = 260$

Let $\alpha, \beta$ be the roots of $x^2 - x + p = 0$ and $\gamma, \delta$ be the roots of $x^2 - 4x + q = 0$. If $\alpha, \beta, \gamma, \delta$ are in $G.P.$,then the integral values of $p, q$ are respectively: