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MHTCET 10 May 2024 Shift 1 Paper

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Question : 112 of 150

Marks: +1, -0

The numerical value of

\tan \left(2 \tan ^{-1}\left(\frac{1}{5}\right)+\frac{\pi}{4}\right)

$\frac{-7}{17}$
$\frac{-17}{7}$
$\frac{17}{7}$
$\frac{7}{17}$

Solution:

Concept:

Use the double-angle formula for tangent, then apply the tangent addition formula.

Explanation:

Let

x = 2\tan^{-1}\left(\frac{1}{5}\right)

.

Then

\tan^{-1}\left(\frac{1}{5}\right) = \frac{x}{2}

, so

\tan\left(\frac{x}{2}\right) = \frac{1}{5}

.

Using

\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}

:

\tan x = \frac{2 \times \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2} = \frac{\frac{2}{5}}{1 - \frac{1}{25}} = \frac{\frac{2}{5}}{\frac{24}{25}} = \frac{5}{12}

Now,

\tan\left(2\tan^{-1}\left(\frac{1}{5}\right) + \frac{\pi}{4}\right) = \tan\left(x + \frac{\pi}{4}\right)

.

Using

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

:

\tan\left(x + \frac{\pi}{4}\right) = \frac{\frac{5}{12} + 1}{1 - \frac{5}{12} \cdot 1} = \frac{\frac{17}{12}}{\frac{7}{12}} = \frac{17}{7}

Answer:

\frac{17}{7}

Correct option: C.

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