Number of integral value(s) of `m` for which the equation `sinx-sqrt(3)cosx=(4m-6)/(4-m)` has solutions, `x in [0,2pi]` , is ________

Answer

4

Explanation

Since `-2 le sin x-sqrt(3) cos x le 2`, we have
`-2 le (4m-6)/(4-m) le 2`
or `-1 le (2m-3)/(4-m) le 1`
If `(2m-3)/(4-m) le 1`, we have
`((2m-3)-(4-m))/(4-m) le 0`
or `(3m-7)/(m-4) ge 0` ...(i)
Also, `-1 le (2m-3)/(4-m)`
`rArr (m+1)/(m-4) le 0`
From Eqs. (i) and (ii), we get `m in [-1, 7/3]`
Therefore, the possible integers are `-1, 0, 1, 2`.