For composite functions, if $$T1$$, $$T2$$, … be the fundamental periods of the various functions involved, then the period of the composite function is the L.C.M. of (T1$$, $$T2$$, …$$). But in the case of functions where modulus is involved, the L.C.M. rule gives the period of the function but it may not be the fundamental period. For example, according to the L.C.M. rule,Period of |$$sin$$ x| $$+$$ |$$cos$$ x| $$=$$ L.C.M. of π,π $$=$$ p, but it is not the fundamental period since |$$sin$$⁡(x+$$π$$2)|$$+$$|$$cos$$⁡(x+$$π$$2)|$$=$$|$$cos$$⁡x|$$+$$|$$sin$$⁡x|which shows that the fundamental period is π$$2Thus$$, the period of |$$sin$$⁡px|$$+$$|$$cos$$⁡qx| $$=L$$.C.M. of πp,πq if p≠q $$=12L$$.C.M. of πp,πq if $$p=qThe$$ function $$f(x)=k$$|$$cos$$⁡x|$$+k2$$|$$sin$$⁡x|$$+$$ϕ(k) has period π$$2$$ if k is equal toThe period of the function $$f(x)=3x+3$$−[$$3x+3$$]$$+$$$$sin$$⁡π$$x2$$ where [x] denotes the greatest integer ≤ x, isπ is the period of the function

Question

For composite functions, if $$T1$$, $$T2$$, … be the fundamental periods of the various functions involved, then the period of the composite function is the L.C.M. of (T1$$, $$T2$$, …$$). But in the case of functions where modulus is involved, the L.C.M. rule gives the period of the function but it may not be the fundamental period. For example, according to the L.C.M. rule,Period of |$$sin$$ x| $$+$$ |$$cos$$ x| $$=$$ L.C.M. of π,π $$=$$ p, but it is not the fundamental period since |$$sin$$⁡(x+$$π$$2)|$$+$$|$$cos$$⁡(x+$$π$$2)|$$=$$|$$cos$$⁡x|$$+$$|$$sin$$⁡x|which shows that the fundamental period is π$$2Thus$$, the period of |$$sin$$⁡px|$$+$$|$$cos$$⁡qx| $$=L$$.C.M. of πp,πq if p≠q $$=12L$$.C.M. of πp,πq if $$p=qThe$$ function $$f(x)=k$$|$$cos$$⁡x|$$+k2$$|$$sin$$⁡x|$$+$$ϕ(k) has period π$$2$$ if k is equal toThe period of the function $$f(x)=3x+3$$−[$$3x+3$$]$$+$$$$sin$$⁡π$$x2$$ where [x] denotes the greatest integer ≤ x, isπ is the period of the function

Infinity Learn · Accepted Answer

Since |$$sin$$ x| $$+$$ |$$cos$$ x| is a periodic function with period π$$2$$ , therefore period of $$f(x)$$ will be π$$2when$$ k $$=$$ $$1$$.The correct option is (A)3x$$ $$+$$ $$3$$ – [$$3x$$ $$+$$ $$3$$] has the period $$1$$ and π$$x2$$ has the period $$2$$ππ$$/2$$. Therefore, the period of $$f(x) is L.C.M. (1$$, $$4) $$=$$ $$4$$.The correct option is (A)The$$ period of |$$sin$$ x| $$+$$ |$$cos$$ x| and $$sin4$$⁡$$x+cos4$$⁡xis$$π$$$$2sin$$⁡$$($$$$sin$$⁡$$x)+$$$$sin$$⁡$$($$$$cos$$⁡$$x) has period $$2$$π. The function $$1+2cos$$⁡xsin⁡$$x(2+$$$$sec$$⁡$$x)can$$ be written in a simplified form as $$cos$$⁡xsin⁡$$x=$$$$cot$$⁡x has period π.The correct option is (D)

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