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Question

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


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