If $$\int \frac{x^3 \mathrm{~d} x}{\sqrt{1+x^2}}=\mathrm{a}\left(1+x^2\right) \sqrt{1+x^2}+\mathrm{b} \sqrt{1+x^2}+\mathrm{c}$$ (where $$\mathrm{c}$$ is a constant of integration), then the value of $$3 \mathrm{ab}$$ is

The perpendiculars are drawn to lines $$L_1$$ and $$L_2$$ from the origin making an angle $$\frac{\pi}{4}$$ and $$\frac{3 \pi}{4}$$ respectively with positive direction of $$\mathrm{X}$$-axis. If both the lines are at unit distance from the origin, then their joint equation is

The function $$\mathrm{f}(x)=[x] \cdot \cos \left(\frac{2 x-1}{2}\right) \pi$$, where $$[\cdot]$$ denotes the greatest integer function, is discontinuous at

A line with positive direction cosines passes through the point $$\mathrm{P}(2,-1,2)$$ and makes equal angles with the co-ordinate axes. The line meets the plane $$2 x+y+z=9$$ at point $$\mathrm{Q}$$. The length of the line segment $$P Q$$ equals