First slide

Properties of ITF

Question

The number of Solutions of the equation $T a n^{- 1} (x - 1) + T a n^{- 1} x + T a n^{- 1} (x + 1) = T a n^{- 1} (3 x)$ is /are

Moderate

A

4
B

3
C

2
D

1

Solution

$W e h a v e T a n^{- 1} (x - 1) + T a n^{- 1} x + T a n^{- 1} (x + 1) = T a n^{- 1} (3 x)$
$\Rightarrow$ $T a n^{- 1} (x - 1) + T a n^{- 1} (x + 1) = T a n^{- 1} (3 x) - T a n^{- 1} x$
$\Rightarrow T a n^{- 1} (\frac{x - 1 + x + 1}{1 - (x - 1) (x + 1)}) = T a n^{- 1} (\frac{3 x - x}{1 + (3 x) x}) \Rightarrow \frac{2 x}{2 - x^{2}} = \frac{2 x}{1 + 3 x^{2}} \Rightarrow 2 x (1 + 3 x^{2} - 2 + x^{2}) = 0 \Rightarrow x = 0, x^{2} = \frac{1}{4} \Rightarrow x = 0, x = \pm \frac{1}{2}$

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