Let $$\alpha|x|=|y| \mathrm{e}^{x y-\beta}, \alpha, \beta \in \mathbf{N}$$ be the solution of the differential equation $$x \mathrm{~d} y-y \mathrm{~d} x+x y(x \mathrm{~d} y+y \mathrm{~d} x)=0,y(1)=2$$. Then $$\alpha+\beta$$ is equal to ________

If $$\alpha=\lim _\limits{x \rightarrow 0^{+}}\left(\frac{\mathrm{e}^{\sqrt{\tan x}}-\mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$$ and $$\beta=\lim _\limits{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}$$ are the roots of the quadratic equation $$\mathrm{a} x^2+\mathrm{b} x-\sqrt{\mathrm{e}}=0$$, then $$12 \log _{\mathrm{e}}(\mathrm{a}+\mathrm{b})$$ is equal to _________.

An arithmetic progression is written in the following way

The sum of all the terms of the 10^th row is _________.

Let a ray of light passing through the point $$(3,10)$$ reflects on the line $$2 x+y=6$$ and the reflected ray passes through the point $$(7,2)$$. If the equation of the incident ray is $$a x+b y+1=0$$, then $$a^2+b^2+3 a b$$ is equal to _________.