Question

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 1

`asinx+bcosx`
`sinx-sqrt3cosx=(4m-6)/(4-m)`
`2(1/2sinx-sqrt3/2cosx)=(4m-6)/(4-m)`
`2(cospi/3*sinx-sinpi/3*cosx)=(4m-6)/(4-m)`
`2(sin(x-pi/3))=(4m-6)/(4-m)`
`sin(x-pi/3)=(2m-3)/(4-m)`
`-1<=sin(x-pi/3)<=1`<br> `-1<=(2m-3)/(4-m)<=1`<br> `(2m-3)/(4-m)+1>=0`
`(2m-3+4-m)/(4-m)>=0`
`(m+1)/(4-m)>=0`
`m in[-1,4)`
`(2m-3)/(4-m)-1<=0`<br> `(2m-3-4+m)/(4-m)<=0`<br> `(3m-7)/(4-m)<=0`<br> `m in(-oo,7/3]uu(4,oo)`
`m in[-1,7/3]`
`m=-1,0,1,2`.