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  • The set of all values of k for which <img alt="\mathrm{\left ( \tan^{-1}x \right )^{3}+ \left (\cot^{-1} x \right )^{3}= k\pi ^{3},x\in R}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cm

The set of all values of k for which \mathrm{\left ( \tan^{-1}x \right )^{3}+ \left (\cot^{-1} x \right )^{3}= k\pi ^{3},x\in R}, is the interval :

Option: 1

\mathrm{\left [ \frac{1}{32},\frac{7}{8} \right )}


Option: 2

\mathrm{\left ( \frac{1}{24},\frac{13}{16} \right )}


Option: 3

\mathrm{\left [\frac{1}{48},\frac{13}{16}\right ]}


Option: 4

\mathrm{\left [\frac{1}{32},\frac{9}{8}\right )}


Answers (1)

best_answer

\mathrm{\left ( \tan^{-1}x \right )^{3}+\left ( \cot ^{-1}x \right )^{3}= k\, \pi^{3}}
Using complementary formulae for inverse trigonometric functions
\mathrm{\frac{\pi}{2}\left [\left ( \frac{\pi}{2} \right )^{2}-3\tan^{-1}x\: \cot^{-1} x\right ]= k\, \pi^{3}}
\mathrm{\frac{\pi^{2}}{2}-3 \tan^{-1}x\left ( \frac{\pi}{2}-\tan^{-1}x \right )= 2\, k\, \pi^{2}}
\mathrm{3\left ( \tan^{-1}x \right )^{2}-\frac{3\, \pi}{2}\left ( \tan^{-1}x \right )+\pi^{2}\left ( -2\, k+\frac{1}{4} \right )= 0}
\mathrm{\left ( \tan^{-1}x-\frac{\pi}{4} \right )^{2}= \left ( \frac{2\, k}{3}-\frac{1}{48} \right )\pi^{2}}
\mathrm{0\leq \left ( \frac{2\, k}{3}-\frac{1}{48} \right )\pi^{2}< \frac{9\, \pi^{2}}{16}}
\mathrm{\frac{1}{48}\leq \frac{2\, k}{3}< \frac{9}{16}+\frac{1}{48}}
\mathrm{\frac{1}{32}\, \leq\, k\, <\, \frac{7}{8}}


The correct answer is (A)

Posted by

Riya

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