UPSEE Solved Model Paper 7

Given f (x) = sin $[\log (x + \sqrt{x^2+1]})]$ ... (i) Now. replacing x by —x in (I), we get f (- x) = sin [log (- x + $\sqrt{(-x)^2+1}$] = = sin $\left[ \log \left( \frac{x^2-x^2-1}{-(x+\sqrt{x^2+1})} \right) \right]$ = sin $\left[ \log \left( \frac{1}{x+\sqrt{x^2+1}} \right) \right]$ = sin [log 1 - log (x + $\sqrt{x^{+}1}$)] [Properties of logarithmic functlon-2] = sin $[-\log (x + \sqrt{x^2+1})]$ [Properties of logarithmic function-4] = - sin [log (x + $\sqrt{x^2+1}$)] = - f (x) Since, f (- x) = - f (x) ∴ f (x) is an odd function Hence, option 'B' is correct.