Question

# Which of the following functions has a period of $$2 \pi$$ ?

A
f(x)=sin(2πx+π3)+2sin(3πx+π4)+3sin5πx
B
f(x)=sinπx3+sinπx4
C
f(x)=sinx+cos2x
D
None

Solution

## The correct option is B $$f(x)\, =\, \sin\, x\, +\, \cos\, 2x$$Option A, $$f(x)\, =\, \sin\, \left( 2\pi x\, +\, \displaystyle \frac{\pi}{3} \right) + 2 \sin \left( 3\pi x\, +\, \displaystyle \frac{\pi}{4} \right)\, +\, 3\, \sin\, 5 \pi x$$Period of $$\sin \, \left( 2\pi x\, +\, \frac { \pi }{ 3 } \right)$$ is $$\dfrac{2\pi}{2\pi}=1$$Period of $$2 \sin \left( 3\pi x\, +\, \displaystyle \frac{\pi}{4} \right)$$is $$\dfrac{2\pi}{3\pi}=\dfrac{2}{3}$$Period of $$3\sin 5x$$ is $$\dfrac{2\pi}{5\pi}=\dfrac{2}{5}$$LCM of $$1,\dfrac{2}{3},\dfrac{2}{5}=\dfrac{LCM \quad of \quad 1,2,2}{HCF \quad of \quad 3,5}=2$$So, period of $$f(x)$$ is $$2$$.B) $$f(x)\, =\, \sin\, \displaystyle \frac{\pi x}{3}\, +\, \sin\, \frac{\pi x}{4}$$Period of $$\sin \displaystyle \frac{\pi x}{3}$$ is $$\dfrac{2\pi}{\displaystyle \frac{\pi x}{3}}=6$$Period of $$\sin \displaystyle \frac{\pi x}{4}$$ is $$\dfrac{2\pi}{\displaystyle \frac{\pi x}{4}}=8$$LCM of $$6$$ and $$8$$ is $$24.$$So, period of $$f(x)$$ is $$24$$.C) $$f(x)=\sin x+\cos 2x$$Period of $$\sin x$$ is $$2\pi$$Period of $$\cos 2x$$ is $$\dfrac{2\pi}{2}=\pi$$LCM of $$2\pi ,\pi$$ is $$2\pi$$Hence, period of $$f(x)$$ is $$2\pi$$$$\therefore$$ Period of $$f(x)=\sin x+\cos 2x=2\pi$$Mathematics

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