Question

Which of the following is an odd function?

• A. $ |x|+1 $
• B. $ \sin x+\cos x $
• C. $ {{x}^{2}}\,\,\sec x+x\,{{\tan }^{2}}x $
• D. $ {{x}^{2}}\cot x+4{{x}^{4}}\,\text{cosec x+}{{\text{x}}^{5}} $

Answer 1

Answer: (D)

Let $ f(x)=|x|+1 $
$ \therefore $ $ f(-x)=|-x|+1 $
$=f(x), $ even function [b] Let $ f(x)=\,\sin x+\cos x $
$ \therefore $ $ f(-x)=\sin \,(-x)+\cos (-x) $
$=-\sin \,x+\,\cos \,x $ neither even nor odd function
Let $ f(x)={{x}^{2}}\,\,\sec x+x\,{{\tan }^{2}}x $
$ \therefore $ $ f(-x)={{x}^{2}}\,\sec \,\,x-x\,{{\tan }^{2}}x $ neither even nor odd function
Let $ f(x)={{x}^{2}}\,\cot \,x+4{{x}^{2}}\,\text{cosec x+}{{\text{x}}^{5}} $
$ \therefore $ $ f(-x)=-{{x}^{2}}\,\cot \,x-4{{x}^{4}}\,\text{cosec x-}{{\text{x}}^{5}} $
$=-f(x), $ odd function
Hence, option is correct.