Let m and $$\mathrm{n}$$ be the numbers of real roots of the quadratic equations $$x^{2}-12 x+[x]+31=0$$ and $$x^{2}-5|x+2|-4=0$$ respectively, where $$[x]$$ denotes the greatest integer $$\leq x$$. Then $$\mathrm{m}^{2}+\mathrm{mn}+\mathrm{n}^{2}$$ is equal to __________.

Let $$\mathrm{R}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\}$$ and $$\mathrm{S}=\{1,2,3,4\}$$. Total number of onto functions $$f: \mathrm{R} \rightarrow \mathrm{S}$$ such that $$f(\mathrm{a}) \neq 1$$, is equal to ______________.

Let $$\mathrm{k}$$ and $$\mathrm{m}$$ be positive real numbers such that the function $$f(x)=\left\{\begin{array}{cc}3 x^{2}+k \sqrt{x+1}, & 0 < x < 1 \\ m x^{2}+k^{2}, & x \geq 1\end{array}\right.$$ is differentiable for all $$x > 0$$. Then $$\frac{8 f^{\prime}(8)}{f^{\prime}\left(\frac{1}{8}\right)}$$ is equal to ____________.

Let the solution curve $$x=x(y), 0 < y < \frac{\pi}{2}$$, of the differential equation $$\left(\log _{e}(\cos y)\right)^{2} \cos y \mathrm{~d} x-\left(1+3 x \log _{e}(\cos y)\right) \sin \mathrm{y} d y=0$$ satisfy $$x\left(\frac{\pi}{3}\right)=\frac{1}{2 \log _{e} 2}$$. If $$x\left(\frac{\pi}{6}\right)=\frac{1}{\log _{e} m-\log _{e} n}$$, where $$m$$ and $$n$$ are coprime, then $$m n$$ is equal to __________.