If `alpha` be the number of solutions of the equation [sin x] =|x| ) and `beta ` be the greatest value of the function
`f(x) =cos(x^2-[x^2))` in the interval ][-1,1] the

Options

A. `alpha=beta`
B. `alpha gt beta`
C. `alpha lt beta `
D. non of these

Answer

A

Explanation

Since `-1 le sin x le " for all " x ` therefore
`[sin x]= {{:(-1"," " when " -1 le sin x lt 0 ),(0 "," " when " sin x=1),(1"," " when " sin x =1):}`
The given
[sin x[=|x|
`rArr [ sin x ] gt 0`
`therefore [ sin x ] =0,1`
Thus [sin x ] = `|x| rArr |x| =0 ,1 rArr x =0 ,Pm 1`
But , only x=0 satisfies the equation (i) Therefore `alpha=1`
Now
`beta = " Greatest value cos " (x^2-[x^2]) " in " [-1,1]`
`beta =1 " "[because cos (x^2 - [x^2])=cos =0=1 at x =0 `
Hence `alpha =beta`