The number of roots of `x^2-2=[sinx],w h e r e[dot]` stands for the greatest integer function is

Options

A. 0
B. 1
C. 2
D. 3

Answer

C

Explanation

Given equation is `x^(2) -2=[sinx].`
There are three possibilities:
`[sinx]= -1, [sinx] =0,[sin x] =1`
Case I: If `[sinx] = -1,`
`x^(2)=1`
or `x= +-1`
When `x=1, [sin x]=0.`
But ` x=-1 implies [sinx]= -1`
`x= -1` is a solution.
Case II: If `[sinx] =0,` the equation is
`x^(2) =2`
or ` x= +-sqrt(2)`
`[sin sqrt(2)]=0`
But ` [sin (-sqrt(2))]=-1.`
So, `x=sqrt(2)` is a solution.
Case III: If `[sinx] =1`
`x=(pi)/(2),(5pi)/(2),(9pi)/(2)`
Clearly, these values do not satisfy the original equation.
Thus, number of solutions `=2`