Question

If $p$ and $q$ are the roots of $6x^2+10x+1=0$ then the value of $|[{\tan }^{-1}p+{\tan }^{-1}q]|$ is
(where [ ] denotes greater integer function)

Answer 1

$p+q=-\frac{5}{3},pq=\frac{1}{6}\Rightarrow p<0\&q<0$
${\tan }^{-1}p+{\tan }^{-1}q={\tan }^{-1}\left(\frac{p+q}{1-pq}\right)=-{\tan }^{-1}2\in (-\frac{\pi }{2},-\frac{\pi }{3})\text{So},[-{\tan }^{-1}2]=-2$
So,
$|[{\tan }^{-1}p+{\tan }^{-1}q]|=2$