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2. inverse trigonometric function — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths2. inverse trigonometric function1 Mark
MCQ
Write the function in the simplest form: $\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right), a>0 ; \frac{-a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}}$
  • ✓
    $3 \tan ^{-1} \frac{x}{a}$
  • B
    $3 \tan ^{-1} \frac{a}{x}$
  • C
    $ \tan ^{-1} \frac{x}{a}$
  • D
    $3 \cot ^{-1} \frac{x}{a}$
✓

Answer

Correct option: A.
$3 \tan ^{-1} \frac{x}{a}$
a
Consider, $\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right)$

Let $x=a \tan \theta \Rightarrow \frac{x}{a}=\tan \theta$ $\Rightarrow \theta=\tan ^{-1}\left(\frac{x}{a}\right)$

$\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right)$

$=\tan ^{-1}\left(\frac{3 a^{2} \cdot a \tan \theta-a^{3} \tan ^{3} \theta}{a^{3}-3 a \cdot a^{2} \tan ^{2} \theta}\right)$

$=\tan ^{-1}\left(\frac{3 a^{3} \tan \theta-a^{3} \tan ^{3} \theta}{a^{3}-3 a^{3} \tan ^{2} \theta}\right)$

$=\tan ^{-1}(\tan 3 \theta)$

$=3 \theta$

$=3 \tan ^{-1} \frac{x}{a}$

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