NIMCET 2009 — Mathematics PYQ

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Question

NIMCET 2009 — Mathematics PYQ

NIMCET | Mathematics | 2009

If $\tan^{-1} 2x + \tan^{-1} 3x = \frac{\pi}{4}$ , then $x$ is:

Choose the correct answer:

Explanation

To find $x$ , we apply the identity for the sum of two inverse tangents:

$tan^{- 1} A + tan^{- 1} B = tan^{- 1} (\frac{A + B}{1 - A B})$

1. Equation Setup

Substituting $A = 2 x$ and $B = 3 x$ :

$tan^{- 1} (\frac{2 x + 3 x}{1 - ( 2 x ) ( 3 x )}) = \frac{π}{4}$

$\frac{5 x}{1 - 6 x ^{2}} = tan (\frac{π}{4})$

2. Solving for $x$

Since $tan (\frac{π}{4}) = 1$ :

$\frac{5 x}{1 - 6 x ^{2}} = 1$

$5 x = 1 - 6 x^{2}$

$6 x^{2} + 5 x - 1 = 0$

3. Factoring

$6 x^{2} + 6 x - x - 1 = 0$

$6 x (x + 1) - 1 (x + 1) = 0$

$(6 x - 1) (x + 1) = 0$

4. Final Values

$x = \frac{1}{6}$ (Valid)

$x = - 1$ (Rejected because it results in a negative value for the sum)

Correct Option:(a) $\frac{1}{6}$

Choose the correct answer:

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