9758/2022/P1/Q10

10 A curve $C$ has equation $y=a x+b+\frac{a+2 b}{x-1}$, where $a$ and $b$ are real constants such that $a>0, b \neq-\frac{1}{2} a$ and $x \neq 1$.
(a) Given that $C$ has no stationary points, use differentiation to find the relationship between $a$ and $b$.

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It is now given that $b=-2 a$.
(b) Sketch $C$ on the axes on page 19 stating the equations of any asymptotes and the coordinates of the points where $C$ crosses the axes.

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(c) On the same axes, sketch the graph of $y=a x-a$.

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(d) Hence solve the inequality $x-2-\frac{3}{x-1} \leqslant x-1$.

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