First slide

Properties of ITF

Question

The solution of $\tan^{- 1} 2 x + \tan^{- 1} 3 x = \frac{π}{4} is$

Moderate

A

1/6
B

-1
C

$(\frac{1}{6}, - 1)$
D

None of these

Solution

$\tan^{- 1} 2 x + \tan^{- 1} 3 x = \frac{π}{4} \begin{array}{l} \Rightarrow \frac{3 x + 2 x}{1 - 6 x^{2}} = \tan \frac{π}{4} \\ \Rightarrow 5 x = 1 - 6 x^{2} \\ \Rightarrow 6 x^{2} + 5 x - 1 = 0 \\ \Rightarrow x = - 1, \frac{1}{6} \end{array}$
But when x = -1,
$\begin{array}{l} \tan^{- 1} 2 x = \tan^{- 1} (- 2) < 0 \\ and \tan^{- 1} 3 x = \tan^{- 1} (- 3) < 0 \end{array}$
This, value will not satisfy the given equation.
$Hence, x = \frac{1}{6}$

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