First slide

Properties of ITF

Question

The number of positive solutions satisfying the equation $\tan^{- 1} (\frac{1}{2 x + 1}) + \tan^{- 1} (\frac{1}{4 x + 1}) = \tan^{- 1} (\frac{2}{x^{2}})$ , is

Moderate

A

1
B

2
C

8
D

9

Solution

We have,
$\begin{array}{l} \tan^{- 1} (\frac{1}{2 x + 1}) + \tan^{- 1} (\frac{1}{4 x + 1}) = \tan^{- 1} (\frac{2}{x^{2}}) \\ \Rightarrow \tan^{- 1} \{\frac{\frac{1}{2 x + 1} + \frac{1}{4 x + 1}}{1 - \frac{1}{(2 x + 1) (4 x + 1)}}\} = \tan^{- 1} (\frac{2}{x^{2}}) \\ \Rightarrow \tan^{- 1} (\frac{3 x + 1}{4 x^{2} + 3 x}) = \tan^{- 1} (\frac{2}{x^{2}}) \\ \Rightarrow \frac{3 x + 1}{4 x^{2} + 3 x} = \frac{2}{x^{2}} \Rightarrow 3 x^{2} - 7 x - 6 = 0 \Rightarrow x = - \frac{2}{3}, 3 \end{array}$

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