If $$\int_0^{\frac{\pi}{4}} \frac{\sin ^2 x}{1+\sin x \cos x} \mathrm{~d} x=\frac{1}{\mathrm{a}} \log _{\mathrm{e}}\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi}{\mathrm{b} \sqrt{3}}$$, where $$\mathrm{a}, \mathrm{b} \in \mathrm{N}$$, then $$\mathrm{a}+\mathrm{b}$$ is equal to _________.

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $$m$$ and $$n$$ respectively be the least and the most number of students who studied all the three subjects. Then $$\mathrm{m}+\mathrm{n}$$ is equal to ___________.

Let the length of the focal chord PQ of the parabola $$y^2=12 x$$ be 15 units. If the distance of $$\mathrm{PQ}$$ from the origin is $$\mathrm{p}$$, then $$10 \mathrm{p}^2$$ is equal to __________.

Let $$\mathrm{ABC}$$ be a triangle of area $$15 \sqrt{2}$$ and the vectors $$\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \overrightarrow{\mathrm{BC}}=\mathrm{a} \hat{i}+\mathrm{b} \hat{j}+\mathrm{c} \hat{k}$$ and $$\overrightarrow{\mathrm{AC}}=6 \hat{i}+\mathrm{d} \hat{j}-2 \hat{k}, \mathrm{~d}>0$$. Then the square of the length of the largest side of the triangle $$\mathrm{ABC}$$ is _________.