Jee Main 2024 27 01 2024 Shift 2 - Question8
Question 8
Considering the principal values of inverse trigonometric functions, the positive real values of ’ $x$ ’ satisfying $\tan ^{-1} x+\tan ^{-1}(2 x)=\frac{\pi}{4}$ is
(1) $\frac{\sqrt{5}-1}{2}$
(2) $\frac{\sqrt{17}+3}{4}$
(3) $\frac{\sqrt{17}-3}{4}$
(4) $\frac{\sqrt{5}+1}{2}$
Show Answer
Answer: (3)
Solution:
$\tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4}$
$$ \begin{aligned} & \Rightarrow \tan ^{-1}\left(\frac{3 x}{1-2 x^{2}}\right)=\frac{\pi}{4} \\ & \Rightarrow \quad \frac{3 x}{1-2 x^{2}}=1 \\ & \Rightarrow \quad 2 x^{2}+3 x-1=0 \\ & \Rightarrow \quad x _1, x _2=\frac{-3 \pm \sqrt{9+8}}{4} \\ & \Rightarrow \quad x _1, x _2=\frac{-3 \pm \sqrt{17}}{4} \\ & \therefore \quad x _1=\frac{\sqrt{17}-3}{4}>0 \end{aligned} $$
BETA
