Considering only the principal values of inverse functions,

Question:

Considering only the principal values of inverse functions, the set

$A=\left\{x \geq 0: \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right\}$

  1. is an empty set

  2. Contains more than two elements

  3. Contains two elements

  4. is a singleton


Correct Option: , 4

Solution:

$\tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\pi / 4$

$\Rightarrow \frac{5 x}{1-6 x^{2}}=1$

$\Rightarrow 6 x^{2}+5 x-1=0$

$x=-1$ or $x=\frac{1}{6}$

$x=\frac{1}{6} \quad \because x>0$

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