The curve C has equation y = (2x²-3x-2)/(x²-2x+1). State the equations of the asymptotes of C. Show that y ≤ 25/12 at all points of C. Find the coordinates of any stationary points of C. Sketch C, stating the coordinates of any intersections of C with the coordinate axes and the asymptotes.

Mark scheme

Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.

1. B1B1 States asymptotes. Vertical: x = 1 and Horizontal: y = 2
2. M1A1 Obtains quadratic form in x. yx² - 2yx + y = 2x² – 3x - 2 ⇒ (y-2)x² - (2y - 3)x + (y + 2) = 0
3. M1 Uses B² - 4AC ≥ 0 for real roots. For real x (2y-3)² - 4(y − 2)(y + 2) ≥ 0
4. A1 ⇒ 12y ≤ 25 ⇒ y ≤ 25/12
5. M1 Finds condition for y' = 0. y' = 0 ⇒ (x² – 2x + 1)(4x – 3) – (2x² – 3x – 2)(2x – 2) = 0
6. A1 Solves ⇒ x² - 8x + 7 = 0 ⇒ (x - 7)(x − 1) = 0
7. A1 Obtains stationary point. x = 7, (since x = 1 is vertical asymptote). Stationary point is (7, 25/12)
8. B1 Sketch showing: Axes and asymptotes
9. B1 (-0.5,0), (2,0), (0,-2) and (4,2)
10. B1 Left hand branch.
11. B1 Right hand branch.