Q75 / 150 · MHT CET 2022 (05 Aug Shift 1) · Mathematics

Considering only the Principal values of inverse function, the set $$ \left\{x \geq 0 / \tan ^{-1}(2 x)+\tan ^{-1} 3 x=\frac{\pi}{4}\right\} $$

  • Correct answer
    A
    is a singleton set.
  • B
    contains more than two elements.
  • C
    contains two elements.
  • D
    is an empty set.

Solution

$\begin{aligned} & \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4} \\ & \Rightarrow \tan ^{-1} \frac{2 x+3 x}{1-2 x \times 3 x}=\tan ^{-1}(1) \\ & \Rightarrow \frac{5 x}{1-6 x^2}=1 \\ & \Rightarrow 5 x=1-6 x^2 \\ & \Rightarrow 6 x^2+5 x-1=0 \\ & \Rightarrow(x+1)(6 x-1)=0\end{aligned}$ $\Rightarrow x=-1$ or $x=\frac{1}{6}$ But $x \geq 0$ Hence $x=\frac{1}{6}$ (only) i.e., single ton set

Hence, the correct answer is option: 'A'.

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