Find the number of real solutions for the equation $\left( \frac{5}{7} \right)^x = -x^2 + 2x - 3$.

If $\sin A, \sin B, \cos A$ are in $G.P.$,then the roots of the equation $x^2 + 2x \cot B + 1 = 0$ are always

The number which exceeds its positive square root by $12$ is

If $f(x) = ax^{2} + bx + 2$ and $f(1) = 4, f(3) = 38$,then $a - b = $

Solve the equation $-x^{2}+x-2=0$.

Let $a, b \in \mathbb{R}, a \neq 0$ 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: