Question

If ${\sin }^{-1}\left(\sin p\right)=3\pi -p$ and the point of intersection of the lines $x+y=6$ and $px-y=3$ has integral coordinates, then the sum of value(s) of $p$ is

Answer 1

Given : ${\sin }^{-1}\left(\sin p\right)=3\pi -p$


We know that
${\sin }^{-1}\left(\sin x\right)=3\pi -x$ iff
$\frac{5\pi }{2}\le x\le \frac{7\pi }{2}\Rightarrow p\in [\frac{5\pi }{2},\frac{7\pi }{2}]$

The point of intersection of $x+y=6$ and $px-y=3$ is
$(\frac{9}{1+p},\frac{6p-3}{1+p})\cdots \left(1\right)$
As the coordinates are integer, so $p$ has to be integer
As $p\in [\frac{5\pi }{2},\frac{7\pi }{2}]$, so possible values of $p$ are $8,9,10$
The coordinates are integer only when $p=8$