2024 MI PRU1 Q6

The curve \(C\) has equation \(y = \frac{x^2 - bx + b}{x - a}\), where \(a\) and \(b\) are constants. \(C\) has a vertical asymptote at \(x = 1\) and a turning point at \((0, -2)\).

1. Determine the values of \(a\) and \(b\).[2]

Use the values of \(a\) and \(b\) found in part (i) for the following parts.

3. Sketch \(C\), stating the coordinates of any points of intersection with the axes and of any turning points and the equations of the asymptotes.[4]
4. By adding a suitable curve to your diagram in part (ii), solve the inequality
   \[\frac{x^2 - bx + b}{x - a} - \ln(5 - x) \ge 2.\][3]