Question
Period of the function $$\left| {{{\sin }^3}\frac{x}{2}} \right| + \left| {{{\cos }^5}\frac{x}{5}} \right|{\text{ is :}}$$
A.
$$2\pi $$
B.
$$10\pi $$
C.
$$8\pi $$
D.
$$5\pi $$
Answer :
$$10\pi $$
Solution :
$$\eqalign{
& \Rightarrow {\text{Period of}}\,\,\sin x = 2\pi \cr
& \Rightarrow {\text{Period of}}\,\,{\sin ^3}x = 2\pi \cr
& {\text{Period of}}\,\,\left| {{{\sin }^3}x} \right| = \pi \cr
& \Rightarrow {\text{Period of}}\,\,\left| {{{\sin }^3}\frac{x}{2}} \right| = 2\pi \cr
& {\text{Period of}}\,\,{\cos ^5}x = 2\pi \cr
& \Rightarrow {\text{Period of}}\,\,\left| {{{\cos }^5}x} \right| = \pi \cr
& \Rightarrow {\text{Period of}}\,\,\left| {{{\cos }^5}\frac{x}{5}} \right| = 5\pi \cr} $$
Thus required period $$ = {\text{LCM of }}2\pi \,\,\& \,\,5\pi = 10\pi $$