Question

The period of $\sin \frac{\pi }{4}\left[x\right]+\cos \frac{\pi x}{2}+\cot \frac{\pi }{3}\left[x\right]$, where $\left[x\right]$ denotes the integral part of $x$ is -

• A. $8$
• B. $4$
• C. $3$
• D. $24$

Answer 1

The correct option is D $24$

We have $\sin \frac{\pi }{4}\left[x\right]$ is periodic of period $8$, $\cos \frac{\pi x}{2}$ is periodic of period $4$ and $\cot \frac{\pi }{3}\left[x\right]$ is periodic of period $3$. They can be easily verified.

As $\sin \frac{\pi }{4}[x+8]=\sin (2\pi +\frac{\pi }{4}[x\left]\right)=\sin \frac{\pi }{4}\left[x\right]$ and $\cos \frac{\pi }{2}(4+x)=\cos (2\pi +\frac{\pi }{2}x)=\cos \frac{\pi x}{2}$ similar proof for $\cot \frac{\pi }{3}\left[x\right]$.

Now the period of the function $f\left(x\right)=\sin \frac{\pi }{4}\left[x\right]+\cos \frac{\pi x}{2}+\cot \frac{\pi }{3}\left[x\right]$ will be LCM of $8,4,3$ and it is $24$.