A) Statement 1 is true, Statement 2 is false.
B) Both the Statements are true, but Statement2 is not the correct explanation of Statement1.
C) Both the Statements are true, and Statement2 is correct explanation of Statement 1.
D) Statement 1 is false, Statement 2 is true.
Correct Answer: D
Solution :
Statement - 2: \[{{\cos }^{3}}x\] is a periodic function. It is a true statement. Statement-1 Given \[f(x)=\int_{{}}^{{}}{{{\cos }^{3}}x\,dx}\] \[=\int_{{}}^{{}}{\left( \frac{\cos 3x}{4}+\frac{3\cos x}{4} \right)dx}\] \[=\frac{1}{4}\frac{\sin 3x}{3}+\frac{3}{4}\sin x\] \[=\frac{1}{12}\sin 3x+\frac{3}{4}\sin x\] Now, period of \[\frac{1}{12}\sin 3x=\frac{2\pi }{3}\] Period of\[\frac{3}{4}\sin x=2\pi \] Hence period of \[f(x)=\frac{L.C.M.\left( 2\pi ,2\pi \right)}{HCF\,\text{of}\,\left( 1,3 \right)}\] \[=\frac{2\pi }{1}=2\pi \] Thus,y(x) is aperiodic function of period \[2\pi .\] Hence, Statement -1 is false.
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