Q233 / 270 · NDA 2022 (Phase 2) · Mathematics

If $f(x)=\ln \left(x+\sqrt{1+x^2}\right)$, then which one of the following is correct?

  • Correct answer
    A
    $f(x)+f(-x)=0$
  • B
    $f(x)-f(-x)=0$
  • C
    $2 f(x)=2 f(-x)$
  • D
    $f(x)=2 f(-x)$

Solution

$\because \quad f(x)=\ln \left(x+\sqrt{1+x^2}\right)$ ...(i) $f(-x)=\ln \left(-x+\sqrt{1+x^2}\right)$ ...(ii) Adding equations (i) and (ii) $\begin{aligned} & f(x)+f(-x)=\ln \left(\sqrt{1+x^2}+x\right)+\ln \left(\sqrt{1+x^2}-x\right) \\ & =\ln 1=0\end{aligned}$

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

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