Verified Solution • 2025–26 Session moderate

QJEEM24Maa080: JEE Main 2024 Mathematics - Class 12 JEEM24Maa080 Solution: Considering only the principal values of inverse trigonometric…

Considering only the principal values of inverse trigonometric functions, the number of positive real values of xx satisfying tan⁡−1(x)+tan⁡−1(2x)=π4\tan ^{-1}(x)+\tan ^{-1}(2 x)=\frac{\pi}{4} is :

Multiple Choice Options

  • A more than 2
  • B 2
  • C 0
  • D 1

Step-by-Step Verified Solution

amp;tan⁡−1x+tan⁡−12x=π4;xgt;0amp;⇒tan⁡−12x=π4−tan⁡−1x\begin{aligned}& \tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4} ; x>0 \\& \Rightarrow \tan ^{-1} 2 x=\frac{\pi}{4}-\tan ^{-1} x\end{aligned}

Taking tan both sides

amp;⇒2x=1−x1+xamp;⇒2x2+3x−1=0amp;x=−3±9+88=−3±178\begin{aligned}& \Rightarrow 2 \mathrm{x}=\frac{1-\mathrm{x}}{1+\mathrm{x}} \\& \Rightarrow 2 \mathrm{x}^2+3 \mathrm{x}-1=0 \\& \mathrm{x}=\frac{-3 \pm \sqrt{9+8}}{8}=\frac{-3 \pm \sqrt{17}}{8}\end{aligned}

Only possible x=−3+178x=\frac{-3+\sqrt{17}}{8}

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