Let $$S$$ be the area of the region enclosed by $$y = {e^{ - {x^2}}}$$, $$y=0$$, $$x=0$$, and $$x=1$$; then

If $$y(x)$$ satisfies the differential equation $$y' - y\,tan\,x = 2x\,secx$$ and $$y(0)=0,$$ then

The point $$P$$ is the intersection of the straight line joining the points $$Q(2, 3, 5)$$ and $$R(1, -1, 4)$$ with the plane $$5x-4y-z=1.$$ If $$S$$ is the foot of the perpendicular drawn from the point $$T(2, 1, 4)$$ to $$QR,$$ then the length of the line segment $$PS$$ is