A) \[|x|+1\]
B) \[\sin x+\cos x\]
C) \[{{x}^{2}}\,\,\sec x+x\,{{\tan }^{2}}x\]
D) \[{{x}^{2}}\cot x+4{{x}^{4}}\,\text{cosec x+}{{\text{x}}^{5}}\]
Correct Answer: D
Solution :
[a] 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 [c] 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 [d] 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 [d] is correct.
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