Question

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

• A. $\left[\frac{1}{32},\frac{7}{8}\right)$
• B. $\left(\frac{1}{24},\frac{13}{16}\right)$
• C. $\left[\frac{1}{48},\frac{13}{16}\right]$
• D. $\left[\frac{1}{32},\frac{9}{8}\right)$

Answer 1

Answer: (A)

$LetS=\left({\tan }^{-1}x+{\cot }^{-1}x\right)-3{\tan }^{-1}x\cdot {\cot }^{-1}x\left({\tan }^{-1}x+{\cot }^{-1}x\right)$
$=\frac{{\pi }^3}{8}-\frac{3\pi }{2}{\tan }^{-1}x\left(\frac{\pi }{2}-{\tan }^{-1}x\right)$
$=\frac{3\pi }{2}{\left({\tan }^{-1}x-\frac{\pi }{4}\right)}^2+\frac{{\pi }^3}{32}$
$\Rightarrow \frac{{\pi }^3}{32}\le S<\frac{7}{8}{\pi }^3$
$=\frac{{\pi }^3}{32}\le K{\pi }^3<\frac{7}{8}{\pi }^3$
$\frac{1}{32}\le K<\frac{7}{8}$