A-LevelFurther MathematicsIntegration9231/12 · Oct/Nov 2013 · Q1012 Marks
The curve C has equation y = (px² + 4x + 1) / (x + 1), where p is a positive constant and p ≠ 3.
- (i)Obtain the equations of the asymptotes of C.[3]
- (ii)Find the value of p for which the x-axis is a tangent to C, and sketch C in this case.[4]
- (iii)For the case p = 1, show that C has no turning points, and sketch C, giving the exact coordinates of the points of intersection of C with the x-axis.[5]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.
- [B1] (i) Vertical asymptote. Asymptotes: x = −1
- [M1A1] Oblique asymptote. y = px + 4 − p + (p − 3)(x+1)⁻¹ ⇒ y = px + 4 − p
- [B1] (ii) Obtains value of p. p = 4 ⇒ x-axis is a tangent
- [B1] Sketches curve. Correct location of turning points and asymptotes.
- [B1B1] Each branch.
- [M1A1] (iii) Proves required result. p = 1 ⇒ y = x + 3 − 2(x+1)⁻¹ ⇒ y' = 1 + 2(x+1)⁻² (≥ 1)
- [B1] Sketches graph. Intersections on x-axis at (−2 ± √3, 0)
- [B1B1] Each branch.
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