If the points with position vectors $$\alpha \hat{i}+10 \hat{j}+13 \hat{k}, 6 \hat{i}+11 \hat{j}+11 \hat{k}, \frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{k}$$ are collinear, then $$(19 \alpha-6 \beta)^{2}$$ is equal to :

Let $$[t]$$ denote the greatest integer $$\leq t$$. Then $$\frac{2}{\pi} \int_\limits{\pi / 6}^{5 \pi / 6}(8[\operatorname{cosec} x]-5[\cot x]) d x$$ is equal to __________.

Let $$A=\{0,3,4,6,7,8,9,10\}$$ and $$R$$ be the relation defined on $$A$$ such that $$R=\{(x, y) \in A \times A: x-y$$ is odd positive integer or $$x-y=2\}$$. The minimum number of elements that must be added to the relation $$R$$, so that it is a symmetric relation, is equal to ____________.

If the solution curve of the differential equation $$\left(y-2 \log _{e} x\right) d x+\left(x \log _{e} x^{2}\right) d y=0, x > 1$$ passes through the points $$\left(e, \frac{4}{3}\right)$$ and $$\left(e^{4}, \alpha\right)$$, then $$\alpha$$ is equal to ____________.