Question

The period of $e^{co{s}^4\pi x+x-\left[x\right]+co{s}^2\pi x}$ is ([.] denotes the greatest integer function )

• A. 2
• B. 1
• C. 0
• D. -1

Answer 1

The correct option is B 1

${\cos }^4\pi x=\frac{1}{4}(1+{\cos }^{2}2\pi x+2\cos 2\pi x)$

${\cos }^2\pi x=\frac{1}{2}(1+\cos 2\pi x)$

${\cos }^4\pi x+x-\left[x\right]+{\cos }^2\pi x=\frac{1}{4}+x-\left[x\right]+\frac{1}{4}{\cos }^{2}2\pi x+\cos 2\pi x$

Period of ${\cos }^{2}2\pi x$ is $\frac{\pi }{2\pi }=\frac{1}{2}$

Period of $x-\left[x\right]$ is $1$

Period of $\cos 2\pi x$ is $\frac{2\pi }{2\pi }=1$

So total period is LCM of $1,1,\frac{1}{2}$ which is $1$