2025 TJC Promo Q8

A curve \(C\) is given by the equation \(y=x+1+\frac{k}{x-2}\), where \(k\) is a real constant.

1. Find the range of values of \(k\) for which \(C\) has two \(x\)-intercepts and two stationary points.[6]

It is now given that \(k=1\).

3. Sketch the graph of \(C\), giving the equations of the asymptotes and the coordinates of the stationary points.[2]

A line \(l\) with equation \(y=mx\), where \(m\) is a real constant, is sketched on the same axes.

4. Explain algebraically why the curve \(C\) and the line \(l\) will not intersect more than twice.[1]
5. Write down the range of values of \(m\) for which \(C\) and \(l\) intersect exactly twice.[1]