Question

The period of $f\left(x\right)=3\sin \left(\frac{\mathrm{\pi x}}{3}\right)+4\cos \left(\frac{\mathrm{\pi x}}{4}\right)$ is

• A. $6$
• B. $24$
• C. $8$
• D. $2\mathrm{\pi }$

Answer 1

The correct option is B

$24$

Explanation for the correct option:

Step 1: Calculate the period of the first term.

The given function is,

$f\left(x\right)=3\sin \left(\frac{\mathrm{\pi x}}{3}\right)+4\cos \left(\frac{\mathrm{\pi x}}{4}\right)$

We know that,

Period of $\sin x=2\mathrm{\pi }$

$\Rightarrow$ Period of $\sin \left(x\times \frac{\mathrm{\pi }}{3}\right)=2\mathrm{\pi }÷\frac{\mathrm{\pi }}{3}$

$\Rightarrow$ Period of $\sin \left(\frac{\mathrm{\pi x}}{3}\right)=2\mathrm{\pi }\times \frac{3}{\mathrm{\pi }}$

$\Rightarrow$ Period of $\sin \left(\frac{\mathrm{\pi x}}{3}\right)=6$

Step 2: Calculate the period of the second term.

Again, we know that,

Period of $\cos x=2\mathrm{\pi }$

$\Rightarrow$ Period of $\cos \left(x\times \frac{\mathrm{\pi }}{4}\right)=2\mathrm{\pi }÷\frac{\mathrm{\pi }}{4}$

$\Rightarrow$ Period of $\cos \left(\frac{\mathrm{\pi x}}{4}\right)=2\mathrm{\pi }\times \frac{4}{\mathrm{\pi }}$

$\Rightarrow$ Period of $\cos \left(\frac{\mathrm{\pi x}}{4}\right)=8$

Step 3: Calculate the period of the given function.

Now, the period of the first term is $6$ and that of the second term is $8$.

So, the period of the given function $f\left(x\right)$ can be calculated as,

Period of $f\left(x\right)=LCM\left(6,8\right)$

$\Rightarrow$ Period of $f\left(x\right)=24$ $\left[\because LCM\left(6,8\right)=24\right]$

Hence, option $B$ is the correct option.