Q57 / 100 · BITSAT 2024 (Memory Based Paper 1) · Mathematics

Let the function $g:(-\infty, \infty) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ be given by $g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}$. Then, g is

  • A
    even and is strictly increasing in $(0, \infty)$
  • B
    odd and is strictly decreasing in $(-\infty, \infty)$
  • Correct answer
    C
    odd and is strictly increasing in $(-\infty, \infty)$
  • D
    neither even nor odd, but is strictly increasing in $(-\infty, \infty)$

Solution

Given that $g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}$ $\therefore \quad g(-u)=2 \tan ^{-1}\left(e^{-u}\right)-\frac{\pi}{2}=$ $2 \tan ^{-1}\left(\frac{1}{e^u}\right)-\frac{\pi}{2}$ $=2 \cot ^{-1}\left(e^u\right)-\frac{\pi}{2}=2\left[\frac{\pi}{2}-\tan ^{-1}\left(e^u\right)\right]-\frac{\pi}{2}$ $=\pi-2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}$ $=\frac{\pi}{2}-2 \tan ^{-1}\left(e^u\right)=-g(u)$ $\therefore \quad g$ is an odd function. Also $g^{\prime}(u)=\frac{2 e^u}{1+e^{2 u}}\gt0, \forall u \in(-\infty, \infty)$ $\therefore g$ is strictly increasing on $(-\infty, \infty)$.

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

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