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WMA11/01 Jan 2025, Question 7: When a Line Meets, Touches or Misses a Curve

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 8 marks

The question, in short

For the curve y = 2/x − k with k positive, sketch it and find the range of k for which the line y = −kx − 6 meets the curve at two distinct points.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 3 marks

Draw the curve y = 2/x − k, marking axis crossings and the horizontal asymptote, each expressed using k.

  1. 1y = 2/x − k is y = 2/x translated down by k
  2. 2y = 0: 2/x = k, so x = 2/k: crosses the x-axis at (2/k, 0)
  3. 3As x → ±∞, 2/x → 0, so the horizontal asymptote is y = −k
  4. 4x = 0 is a vertical asymptote, so there is no y-intercept

AnswerGraph of y = 2/x moved down k units: right branch comes down from the top near the y-axis, crosses the x-axis at (2/k, 0) and levels off towards y = −k; left branch lies entirely in the third quadrant, below y = −k. Horizontal asymptote y = −k (vertical asymptote x = 0); no y-intercept.

Part (b) · 5 marks

Using that the line y = −kx − 6 cuts the curve at two distinct points, determine the set of possible k values.

  1. 12/x − k = −kx − 6
  2. 2Multiply by x: 2 − kx = −kx² − 6x, so kx² + (6 − k)x + 2 = 0
  3. 3Two distinct points: b² − 4ac > 0, so (6 − k)² − 8k > 0
  4. 4k² − 20k + 36 > 0, so (k − 2)(k − 18) > 0
  5. 5k < 2 or k > 18; with k positive: 0 < k < 2 or k > 18

Answer0 < k < 2 or k > 18

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Also searched as: WMA11 January 2025 · WMA11/01 · wma11 jan25 · IAS Maths Pure Mathematics 1 January 2025 · question 7.
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