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WMA11/01 Jan 2024, Question 7: When a Line Meets, Touches or Misses a Curve

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

Sketch curve y = 4/(x − k) for positive k, marking axis intercepts and vertical asymptote, then find the range of k for which line y = 9 − x misses the curve.

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

Part (a) · 4 marks

Sketch y = 4/(x − k), k positive, showing where the curve meets the axes and the vertical asymptote's equation.

  1. 1y = 4/x translated k units right: vertical asymptote x = k, horizontal asymptote y = 0
  2. 2x = 0: y = 4/(0 − k) = −4/k, so it cuts the y-axis at (0, −4/k)
  3. 3y = 0 has no solution, so no x-intercept
  4. 4Right branch in quadrant 1; left branch below the x-axis through (0, −4/k)

AnswerReciprocal-type curve with vertical asymptote x = k (k > 0) and horizontal asymptote the x-axis (y = 0): one branch to the right of x = k in the first quadrant, decreasing towards y = 0; the other branch to the left of x = k, below the x-axis, crossing the y-axis at (0, −4/k) and going down to −∞ as x → k; no x-intercept

Part (b) · 5 marks

Given y = 9 − x neither crosses nor touches C, determine the possible values of k.

  1. 14 = (9 − x)(x − k) ⇒ 4 = 9x − 9k − x² + kx
  2. 2x² − (9 + k)x + 9k + 4 = 0
  3. 3No intersection: b² − 4ac < 0 ⇒ (9 + k)² − 4(9k + 4) < 0
  4. 4k² − 18k + 65 < 0 ⇒ (k − 5)(k − 13) < 0
  5. 55 < k < 13

Answer5 < k < 13

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Also searched as: WMA11 January 2024 · WMA11/01 · wma11 jan24 · IAS Maths Pure Mathematics 1 January 2024 · question 7.
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