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. (1) contains two elements

2. (2) contains more than two elements

3. (3) is a singleton

4. (4) is an empty set

Correct Option: , 3

Solution:

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

$\Rightarrow \quad \tan ^{-1}\left(\frac{5 x}{1-6 x^{2}}\right)=\frac{\pi}{4}$

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

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

$\Rightarrow \quad(6 x-1)(x+1)=0$

$\Rightarrow \quad x=\frac{1}{6}($ as $x \geq 0)$

Therefore, $A$ is a singleton set.

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