Question

The period of the function $f\left(x\right)=[6x+7]+\cos \pi x-6x$, where [.] denotes the greatest integer function, is

• A. $3$
• B. $2\pi$
• C. $2$
• D. none of these

Answer 1

Answer: (C)

$2$

$f\left(x\right)=[6x+7]+\cos \pi x-6x$
$\Rightarrow f\left(x\right)=[6x+7]+\cos \pi x-(6x+7)+7$
$\Rightarrow f\left(x\right)=-\{6x+7\}+\cos \pi x+7$, $\left\{\right\}$ is fractional part function
Thus period of $\{6x+7\}$ is $\frac{1}{6}$ and period of $\cos \pi x$ is $2$
Hence period of f is $=$ L.C.M. of $\frac{1}{6}$ and $2$ $=2$