A-LevelFurther MathematicsRational functions and graphs9231/12 · Oct/Nov 2011 · Q711 Marks

The curve C has equation `y = (x² + px + 1)/(x - 2)`, where p is a constant. Given that C has two asymptotes, find the equation of each asymptote. [3] Find the set of values of p for which C has two distinct turning points. [5] Sketch C in the case p = −1. Your sketch should indicate the coordinates of any intersections with the axes, but need not show the coordinates of any turning points. [3]

(a)Given that C has two asymptotes, find the equation of each asymptote.[3]
(b)Find the set of values of p for which C has two distinct turning points.[5]
(c)Sketch C in the case p = −1. Your sketch should indicate the coordinates of any intersections with the axes, but need not show the coordinates of any turning points.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2011). The mark or mark type is shown in brackets.

[B1] Vertical asymptote. x=2
[M1] Divides by (x – 2) y = x + p + 2 + (2p+5)/(x-2)
[A1] Oblique asymptote. y = x + p + 2
[M1A1] Differentiates. dy/dx = (x²-4x+4-2p-5) / (x-2)²
[M1] y'= 0 ⇒ x² - 4x - (2p + 1) = 0
[M1] B² – 4AC > 0 ⇒ 16 + 4(2p + 1) > 0
[A1] ⇒ p > -5/2
[B1] Sketches graph. Axes and (0,-0.5) marked..
[B1] Upper Branch with minimum.
[B1] Lower with maximum below x-axis. (Deduct at most 1 for poor forms at infinity.)

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