If $$\mathrm{d}_1$$ is the shortest distance between the lines $$x+1=2 y=-12 z, x=y+2=6 z-6$$ and $$\mathrm{d}_2$$ is the shortest distance between the lines $$\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}, \frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}$$, then the value of $$\frac{32 \sqrt{3} \mathrm{~d}_1}{\mathrm{~d}_2}$$ is :

Let $$\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\ldots$$ upto 10 terms and $$\beta=\sum_\limits{n=1}^{10} n^4$$. If $$4 \alpha-\beta=55 k+40$$, then $$\mathrm{k}$$ is equal to __________.

Let $$\mathrm{A}=\{1,2,3, \ldots, 7\}$$ and let $$\mathrm{P}(\mathrm{A})$$ denote the power set of $$\mathrm{A}$$. If the number of functions $$f: \mathrm{A} \rightarrow \mathrm{P}(\mathrm{A})$$ such that $$\mathrm{a} \in f(\mathrm{a}), \forall \mathrm{a} \in \mathrm{A}$$ is $$\mathrm{m}^{\mathrm{n}}, \mathrm{m}$$ and $$\mathrm{n} \in \mathrm{N}$$ and $$\mathrm{m}$$ is least, then $$\mathrm{m}+\mathrm{n}$$ is equal to _________.

If the function

$$f(x)= \begin{cases}\frac{1}{|x|}, & |x| \geqslant 2 \\ \mathrm{a} x^2+2 \mathrm{~b}, & |x|<2\end{cases}$$

is differentiable on $$\mathbf{R}$$, then $$48(a+b)$$ is equal to __________.