JEE Advanced 2016 Paper 1Question 1
Consider a pyramid $OPQRS$ located in the first octant ($x\ge0,y\ge0,z\ge0$) with $O$ as origin, and $OP$ and $OR$ along the $x$-axis and the $y$-axis, respectively. The base $OPQR$ of the pyramid is a square with $OP=3$. The point $S$ is directly above the mid-point $T$ of diagonal $OQ$ such that $TS=3$. Then
3D Geometry
Solution
$Q=(3,3,0)$, $S=(\frac32,\frac32,3)$. $\cos\angle QOS=\frac{9}{3\sqrt2\cdot\sqrt{27/2}}=\frac1{\sqrt3}\ne\frac12$. Plane $OQS$: $x-y=0$. Distance of $P(3,0,0)$ is $\frac3{\sqrt2}$. Distance of $O$ from $RS$: $\sqrt{\frac{101.25}{13.5}}=\sqrt{\frac{15}2}$.