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A-LevelFurther MathematicsRational functions and graphs9231/12 · Oct/Nov 2015 · Q811 Marks

The curve C has equation y = (2x² + kx)/(x+1), where k is a constant. Find the set of values of k for which C has no stationary points. For the case k = 4, find the equations of the asymptotes of C and sketch C, indicating the coordinates of the points where C intersects the coordinate axes.

  1. (a)Find the set of values of k for which C has no stationary points.[5]
  2. (b)For the case k = 4, find the equations of the asymptotes of C and sketch C, indicating the coordinates of the points where C intersects the coordinate axes.[6]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2015). The mark or mark type is shown in brackets.

  1. [M1] (a) y' = (x+1)(4x+k) - (2x²+kx) / (x+1)² = 0
  2. [A1] (a) ⇒ 2x² + 4x + k = 0
  3. [M1] (a) B² − 4AC < 0
  4. [A1] (a) ⇒ 16 − 8k < 0
  5. [A1] (a) ⇒ k > 2 for no stationary points.
  6. [B1] (b) Vertical asymptote: x = -1
  7. [M1] (b) Oblique asymptote: y = 2x+2 - 2/(x+1)
  8. [A1] (b) ⇒ y = 2x + 2
  9. [B1] (b) Axes and asymptotes (sketch)
  10. [B1] (b) Each branch (sketch)
  11. [B1] (b) Intercepts (sketch)

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About this question

Topic
Rational functions and graphs, A-Level Further Mathematics (syllabus 9231). Structured question, 11 marks.
Source
A-Level Further Mathematics, Oct/Nov 2015 Paper 1 Variant 2, question 8.

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