Question

The number of solutions of $[\sin x+\cos x]=3+[-\sin x]+[-\cos x]$ in the interval $[0,2\pi ]$ is (where $[.]$ denotes the greatest integer function.)

• A. $0$
• B. $4$
• C. $3$
• D. Infinitely many

Answer 1

The correct option is D Infinitely many

Let us consider different intervals of $x$.
Case 1 : $0<x<\frac{\pi }{2}$
$[\sin x+\cos x]=1$
$[-\sin x]=-1$
$[-\cos x]=-1$
Hence, the equation is reduced to :
$1=1$
This is true for all $x$ in the interval. Hence, there are infinitely many solutions