Question

Considering only the principal values of inverse functions, the set $A=\{x\ge 0:{\tan }^{-1}(2x)+{\tan }^{-1}(3x)=\frac{\pi }{4}\}$

• A. $\text{is an empty set}$
• B. $\text{is a singleton}$
• C. $\text{contains two elements}$
• D. $\text{Contains more than two elements}$

Answer 1

The correct option is B $\text{is a singleton}$

${\tan }^{-1}2x+{\tan }^{-1}3x=\frac{\pi }{4}$
$\Rightarrow {\tan }^{-1}\frac{2x+3x}{1-2x.3x}=\frac{\pi }{4}$
$\Rightarrow \frac{5x}{1-6x^2}=\tan \frac{\pi }{4}$
$5x=1-6x^2$
$\Rightarrow 6x^2+5x-1=0$
$\Rightarrow (x+1)(x-\frac{1}{6})=0$
As, $x\ge 0\Rightarrow x=\frac{1}{6}$