Question

$f\left(x\right)=\cos \left[2{\pi }^{}\right]x+\cos [-2{\pi }^{}]x$, where $\left[x\right]$ stands for the greatest integer function, then

• A. $f\left(\pi /2\right)=-1$
• B. $f\left(\pi \right)=+1$
• C. $f(-\pi )=0$
• D. $f\left(\pi /4\right)=1$

Answer 1

Answer: (A), (C)

The correct options are
A $f\left(\pi /2\right)=-1$
C $f(-\pi )=0$

$\left\{x\right\}$ gives greatest integer less than or equal to $x$ As $2\pi \approx 6.28,\left[2\pi \right]=6$ And $2\pi \approx -6.28\Rightarrow [-6,28]=-7$

$\Rightarrow f\left(x\right)=\cos 6x+\cos 7x$ $\left\{\cos \right(-\theta )=\cos \theta \}$

$f\left(\pi /2\right)=\cos \frac{6\pi }{2}+\cos \frac{7\pi }{2}=-1+0=-1$

$f\left(\pi \right)=\cos 6\pi +\cos 7\pi =1-1=0$

$f(-\pi )=\cos (-6\pi )+\cos (-7\pi )=1-1=0$

$f\left(\pi /4\right)=\cos \frac{6\pi }{4}+\cos \frac{7\pi }{4}=0+\frac{1}{\sqrt{2}}$

$\Rightarrow \left(A\right),\left(C\right)$