N2022 P1 Q10

A curve \(~C\) has equation \(y=ax+b+\frac{a+2b}{x-1}\), where \(a\) and \(b\) are real constants such that \(a>0\), \(b\ne -\frac{1}{2}a\) and \(x\ne 1\).

1. Given that \(~C\) has no stationary points, use differentiation to find the relationship between \(a\) and \(b\).[3]

It is now given that \(b=-2a\).

2. Sketch \(~C\) on the axes below stating the equations of any asymptotes and the coordinates of the points where\(~C\) crosses the axes.[4]
3. On the same axes, sketch the graph of \(y=ax-a\).[1]
4. Hence solve the inequality \(x-2-\frac{3}{x-1}\le x-1\).[2]