Q62 / 160 · TS EAMCET 2013 · Mathematics

The period of $f(x)=\cos \left(\frac{x}{3}\right)+\sin \left(\frac{x}{2}\right)$ is

  • A
    $2 \pi$
  • B
    $4 \pi$
  • C
    $8 \pi$
  • Correct answer
    D
    $12 \pi$

Solution

Given, $f(x)=\cos \left(\frac{x}{3}\right)+\sin \left(\frac{x}{2}\right)$ Period of $\cos x$ and $\sin x$ are $2 \pi$. $$ \begin{aligned} \therefore \text { Period of } f(x) & =\text { Period of }\left[\cos \frac{x}{3}+\sin \frac{x}{2}\right] \\ & =\text { Period of } \cos \frac{x}{3}+\text { Period of } \sin \frac{x}{2} \\ = & \frac{2 \pi}{\left(\frac{1}{3}\right)}+\frac{2 \pi}{\left(\frac{1}{2}\right)}=\frac{6 \pi}{1}+\frac{4 \pi}{1} \\ = & \frac{\text { LCM of }(6 \pi, 4 \pi)}{\text { LCM of }(1,1)}=\frac{12 \pi}{1}=12 \pi \end{aligned} $$

Hence, the correct answer is option: 'D'.

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