Question

Let [x] = the greatest integer less than or equal to x and let f(x) = sin x + cos x. Then the most general solution of $f\left(x\right)=\left[f\right(\frac{\pi }{10}\left)\right]$ is

• A.
  
  
• B.
• C.
• D.
  
  

Answer 1

The correct option is C

$f\left(\frac{\pi }{10}\right)=sin{18}^{\circ }+cos{18}^{\circ }=\frac{\sqrt{5}-1}{4}+\frac{\sqrt{5}+1}{4}=\frac{\sqrt{5}}{2}\Rightarrow \left[f\left(\frac{\pi }{10}\right)\right]=1$
So, the equation reduces to$sinx+cosx=1\Rightarrow cos(x-\frac{\pi }{4})=\frac{1}{\sqrt{2}}$
$\therefore x-\frac{\pi }{4}=2n\pi +\frac{\pi }{4}\Rightarrow x=2n\pi or2n\pi \pm \frac{\pi }{2}$