Question

The values of $x$ satisfying the inequality ${\left[{\tan }^{-1}x\right]}^2-\left[{\tan }^{-1}x\right]-2\le 0$ where [ ] denote integer part, is

• A. $[-\tan 1,\infty )$
• B. $[-\frac{\pi }{4},\tan 2]$
• C. $[-\tan 1,\tan 2]$
• D. none of these

Answer 1

Answer: (C)

$[-\tan 1,\tan 2]$

${\left[{\tan }^{-1}x\right]}^2-\left[{\tan }^{-1}x\right]-2\le 0$
Put ${\tan }^{-1}x=y$
$\Rightarrow y^2-y-2\le 0$
$\Rightarrow (y-2)(y+1)\le 0$
$\Rightarrow y\in [-1,2]$
$-1\le {\tan }^{-1}x\le 2$
$\Rightarrow \tan (-1)\le x\le \tan \left(2\right)$

$\Rightarrow -\tan 1\le x\le \tan 2$