2022 TYS

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]