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Which one is not periodic?

JEE · Math · previous-year question

  1. A.$$\left| {\sin 3x} \right| + {\sin ^2}x$$
  2. B.$$\cos \sqrt x + {\cos ^2}x$$correct
  3. C.$$\cos \,4x + {\tan ^2}x$$
  4. D.$$cos\,2x + \sin x$$

Answer

B. $$\cos \sqrt x + {\cos ^2}x$$

Explanation

$$\sqrt x $$ is non periodic function and $$\cos \left( {something} \right)$$ is a periodic function so here in $$\cos \sqrt x $$ $$ \to $$ inside periodic function there is non periodic function which always produce non periodic function. $${{{\cos }^2}x}$$ is a periodic function with period $$\pi $$ Note : (1) When $$n$$ is odd then the period of $${\sin ^n}\theta $$, $${\cos ^n}\theta $$, $${\csc ^n}\theta $$, $${\sec ^n}\theta $$ = $$2\pi $$ (2) When $$n$$ is even then the period of $${\sin ^n}\theta $$, $${\cos ^n}\theta $$, $${\csc ^n}\theta $$, $${\sec ^n}\theta $$ = $$\pi $$ (3) When $$n$$ is even/odd then the period of $${\tan ^n}\theta $$, $${\cot ^n}\theta $$ = $$\pi $$ (3) When $$n$$ is even/odd then the period of $$\left| {{{\sin }^n}\theta } \right|$$, $$\left| {{{\cos }^n}\theta } \right|$$, $$\left| {{{\csc }^n}\theta } \right|$$, $$\left| {{{\sec }^n}\theta } \right|$$, $$\left| {{{\tan }^n}\theta } \right|$$, $$\left| {{{\cot }^n}\theta } \right|$$ = $$\pi $$ $$\cos \sqrt x + {\cos ^2}x$$ = non periodic function + periodic function = non periodic function

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