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 ____________.

The ordinates of the points P and $$\mathrm{Q}$$ on the parabola with focus $$(3,0)$$ and directrix $$x=-3$$ are in the ratio $$3: 1$$. If $$\mathrm{R}(\alpha, \beta)$$ is the point of intersection of the tangents to the parabola at $$\mathrm{P}$$ and $$\mathrm{Q}$$, then $$\frac{\beta^{2}}{\alpha}$$ is equal to _______________.

Let $$0 < z < y < x$$ be three real numbers such that $$\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$$ are in an arithmetic progression and $$x, \sqrt{2} y, z$$ are in a geometric progression. If $$x y+y z+z x=\frac{3}{\sqrt{2}} x y z$$ , then $$3(x+y+z)^{2}$$ is equal to ____________.