If $$\mathrm{A}=\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{ccc}5 ! & 6 ! & 7 ! \\ 6 ! & 7 ! & 8 ! \\ 7 ! & 8 ! & 9 !\end{array}\right]$$, then $$|\operatorname{adj}(\operatorname{adj}(2 \mathrm{~A}))|$$ is equal to :

Let the tangent at any point P on a curve passing through the points (1, 1) and $$\left(\frac{1}{10}, 100\right)$$, intersect positive $$x$$-axis and $$y$$-axis at the points A and B respectively. If $$\mathrm{PA}: \mathrm{PB}=1: k$$ and $$y=y(x)$$ is the solution of the differential equation $$e^{\frac{d y}{d x}}=k x+\frac{k}{2}, y(0)=k$$, then $$4 y(1)-6 \log _{\mathrm{e}} 3$$ is equal to ____________.

Let the foot of perpendicular from the point $$\mathrm{A}(4,3,1)$$ on the plane $$\mathrm{P}: x-y+2 z+3=0$$ be N. If B$$(5, \alpha, \beta), \alpha, \beta \in \mathbb{Z}$$ is a point on plane P such that the area of the triangle ABN is $$3 \sqrt{2}$$, then $$\alpha^{2}+\beta^{2}+\alpha \beta$$ is equal to ___________.

If the area of the region $$\left\{(x, \mathrm{y}):\left|x^{2}-2\right| \leq y \leq x\right\}$$ is $$\mathrm{A}$$, then $$6 \mathrm{A}+16 \sqrt{2}$$ is equal to __________.