Q51 / 120 · JEE Advanced 2006 · Mathematics

Let $\theta \in\left(0, \frac{\pi}{4}\right)$ and $t_1=(\tan \theta)^{\tan \theta}, t_2=(\tan \theta)^{\cot \theta}$ and $t_4=(\cot \theta)^{\tan \theta}$, then

  • A
    $t_1>t_2>t_3>t_4$
  • Correct answer
    B
    $t_4>t_3>t_1>t_2$
  • C
    $t_3>t_1>t_2>t_4$
  • D
    $t_2>t_3>t_1>t_4$

Solution

As when $\theta \in\left(0, \frac{\pi}{4}\right)$ $\tan \theta < \cot \theta$ Since, $\tan \theta < 1$ and $\cot \theta>1$ $\therefore(\tan \theta)^{\cot \theta} < 1$ and $(\cot \theta)^{\tan \theta}>1$ $\therefore t_4>t_1$ which only holds in option (b).

Hence, the correct answer is option: 'B'.

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