The domain of the function $$f(x)=\frac{1}{\sqrt{[x]^{2}-3[x]-10}}$$ is : ( where $$[\mathrm{x}]$$ denotes the greatest integer less than or equal to $$x$$ )
Let $$\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$$ and $$\vec{b}=\hat{i}+\hat{j}-\hat{k}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \cdot \vec{c}=11, \vec{b} \cdot(\vec{a} \times \vec{c})=27$$ and $$\vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}|$$, then $$|\vec{a} \times \vec{c}|^{2}$$ is equal to _________.
Let the line $$l: x=\frac{1-y}{-2}=\frac{z-3}{\lambda}, \lambda \in \mathbb{R}$$ meet the plane $$P: x+2 y+3 z=4$$ at the point $$(\alpha, \beta, \gamma)$$. If the angle between the line $$l$$ and the plane $$P$$ is $$\cos ^{-1}\left(\sqrt{\frac{5}{14}}\right)$$, then $$\alpha+2 \beta+6 \gamma$$ is equal to ___________.
Let $$\mathrm{A}=\{1,2,3,4,5\}$$ and $$\mathrm{B}=\{1,2,3,4,5,6\}$$. Then the number of functions $$f: \mathrm{A} \rightarrow \mathrm{B}$$ satisfying $$f(1)+f(2)=f(4)-1$$ is equal to __________.