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4PM1/02R Jan 2022, Question 11: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 13 marks

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The question, in short

Triangle OAB has OA = a and OB = b, with C on OB in ratio OC : CB = 2 : 1, M the midpoint of CB, N the midpoint of AC, and AM crossing NB at P. Use vectors throughout.

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

Part (a) · 9 marks

Find OP as a simplified vector expression in a and b, using a vector method.

  1. 1AB = −a + b, AC = −a + (2/3)b, AM = −a + (5/6)b
  2. 2NB = −½AC + AB = −½a + (2/3)b
  3. 3OP = OA + λAM = (1 − λ)a + (5/6)λb
  4. 4OP = OB − μNB = (μ/2)a + (1 − 2μ/3)b
  5. 5Compare: 1 − λ = μ/2 and (5/6)λ = 1 − 2μ/3
  6. 6λ = 2/3, μ = 2/3
  7. 7OP = a + (2/3)(−a + (5/6)b) = (1/3)a + (5/9)b

AnswerOP = (1/3)a + (5/9)b

Part (b) · 4 marks

Given Q is the midpoint of AB, use a vector method to show that C, P and Q lie on one straight line.

  1. 1OC = (2/3)b, OQ = ½(a + b)
  2. 2CP = OP − OC = (1/3)a − (1/9)b
  3. 3CQ = OQ − OC = ½a − (1/6)b
  4. 4CQ = (3/2)CP
  5. 5CQ is a multiple of CP and they share point C, so C, P and Q are collinear

AnswerCP = (1/3)a − (1/9)b and CQ = (1/2)a − (1/6)b = (3/2)CP, so C, P and Q are collinear

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