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WMA11/01 Jan 2021, Question 6: When a Line Meets, Touches or Misses a Curve

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 10 marks

The question, in short

Sketch two reciprocal curves involving −k/x with k > 0, give an intercept and asymptote, then find k values so one curve misses the line y = 3x + 4.

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

Part (a) · 2 marks

Sketch y = −k/x for k > 0, x ≠ 0, showing the correct shape and position of the two branches.

  1. 1y = −k/x with k > 0: for x > 0, y < 0; for x < 0, y > 0
  2. 2As x → 0, |y| → ∞; as |x| → ∞, y → 0
  3. 3So the branches lie in the second and fourth quadrants, with the axes as asymptotes

AnswerTwo-branch reciprocal curve in the second and fourth quadrants only, approaching both axes as asymptotes (x = 0 and y = 0)

Part (b) · 3 marks

Sketch y = −k/x + k, giving the x-axis intersection coordinates and the horizontal asymptote equation in terms of k.

  1. 1Adding k translates the curve of (a) up by k
  2. 2Horizontal asymptote becomes y = k
  3. 3y = 0 when k/x = k, so x = 1: intercept (1, 0)

AnswerThe curve from (a) translated up by k: horizontal asymptote y = k, vertical asymptote x = 0, crossing the x-axis at (1, 0)

Part (c) · 5 marks

Determine the range of k > 0 for which y = −k/x + k neither touches nor crosses y = 3x + 4.

  1. 13x + 4 = −k/x + k
  2. 23x² + 4x = −k + kx, so 3x² + (4 − k)x + k = 0
  3. 3No intersection: (4 − k)² − 12k < 0
  4. 4k² − 20k + 16 < 0; critical values k = 10 ± 2√21
  5. 510 − 2√21 < k < 10 + 2√21

Answer10 − 2√21 < k < 10 + 2√21

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Also searched as: WMA11 January 2021 · WMA11/01 · wma11 jan21 · IAS Maths Pure Mathematics 1 January 2021 · question 6.
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