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

Options

A. 0
B. 4
C. infinite
D. 1

Answer

C

Explanation

`[sin x + cos x]=3+[-sin x]+[-cos x]`
The maximum value of the L.H.S. is 1 and the minimum of the R.H.S. is also 1. Thus,
`[sin x+ cos x]=3+[- sin x]+[-cos x]=1`
or `[sin x+cos x]=1, [-sin x]=-1, [-cos x]=-1`
`[sin x+cos x]=1`
`:. 1 le sin x + cos x le sqrt(2)`
`:. x in [0, pi/2]` ...(i)
`[-sin x]=-1`
`:. -1 le -sin x lt 0`
`:. 0 lt sin x le 1` ...(ii)
`[-cos x]=-1`
`:. -1 le -cos x lt 0`
`:. 0 lt cos x le 1` ...(iii)
From Eqs. (i), (ii), (iii), `x in (0, pi/2)`.