JEE Main 2021 (Online) 26th August Evening ShiftQuestion 1
A hall has a square floor of dimension $10\text{ m}\times10\text{ m}$ (see the figure) and vertical walls. If the angle $GPH$ between the diagonals $AG$ and $BH$ is $\cos^{-1}\frac{1}{5}$, then the height of the hall (in meters) is : [FIGURE NOT TRANSCRIBED: cuboid with vertices $A,B,C,D,E,F,G,H$, base $10\text{ m}\times10\text{ m}$, diagonals $AG$ and $BH$ meeting at $P$; see the paper image]
3D Geometry
Solution
Take the cuboid with base $10\times10$, height $h$. The space diagonals along $AG$ and $BH$ have direction vectors $(10,10,h)$ and $(-10,10,h)$ (they differ by a flip in one base coordinate). $\cos\theta=\frac{h^2}{200+h^2}=\frac15\Rightarrow5h^2=200+h^2\Rightarrow h^2=50\Rightarrow h=5\sqrt2$. (The other pairing would give $h^2=400/3$, not an option.) Answer: (A).