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4PM1/01 Jan 2023, Question 5: When a Line Meets, Touches or Misses a Curve

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 8 marks

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The question, in short

Curve C: kx² − xy + (k + 1)x = 1 and line l: y = (k/2)x + 1 meet only at point A, with k non-zero. Find k, then the coordinates of A.

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

Part (a) · 6 marks

Substitute the line equation into the curve equation and use the condition that the resulting quadratic has exactly one solution to find the non-zero value of k.

  1. 1Substitute y = (k/2)x + 1 into C
  2. 2kx² − (k/2)x² − x + (k + 1)x = 1
  3. 3(k/2)x² + kx − 1 = 0
  4. 4One point only, so b² − 4ac = 0: k² − 4(k/2)(−1) = 0
  5. 5k² + 2k = 0, k(k + 2) = 0
  6. 6k ≠ 0, so k = −2

Answerk = −2

Part (b) · 2 marks

Using the value of k found in part a, solve for the single intersection point and state the coordinates of A.

  1. 1k = −2: −x² − 2x − 1 = 0, so (x + 1)² = 0
  2. 2x = −1, y = −(−1) + 1 = 2

AnswerA(−1, 2)

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