If $p$ and $q$ are the roots of the equation ${x^2} + pq = (p + 1)x$,then $q=$

If $c > 0$ and the equation $3ax^2 + 4bx + c = 0$ has no real root,then :-

Solve the given two equations and select the correct answer from the given options.
$I. \quad x - 7\sqrt{x} + 12 = 0$
$II. \quad y - 5\sqrt{y} + 6 = 0$

If $\alpha, \beta$ are the roots of the equation $a x^{2}+b x+b=0,$ then $\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{b}{a}}=$

The number of solutions of $\log_4(x - 1) = \log_2(x - 3)$ is:

Let $a, b \in R, a \neq 0$ be such that the equation $a x^{2}-2 b x+5=0$ has a repeated root $\alpha,$ which is also a root of the equation $x^{2}-2 b x-10=0$. If $\beta$ is the other root of this equation,then $\alpha^{2}+\beta^{2}$ is equal to