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4PM1/01 Jan 2023, Question 11: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 10 marks

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The question, in short

Triangle OAB has OA = a and OB = b, with M the midpoint of OA and N on OB where ON : NB = 3 : 1; lines AN and BM meet at X. Find vector expressions and a ratio.

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

Part (a) · 3 marks

Give vectors for (i) AN and (ii) BM in terms of a and b.

  1. 1ON = (3/4)b, so AN = AO + ON = −a + (3/4)b
  2. 2OM = (1/2)a, so BM = BO + OM = (1/2)a − b

Answer(i) AN = −a + (3/4)b (ii) BM = (1/2)a − b

Part (b) · 7 marks

Use vectors to determine the ratio AX : XN.

  1. 1OX = OA + λAN = (1 − λ)a + (3/4)λb
  2. 2OX = OB + μBM = (μ/2)a + (1 − μ)b
  3. 3Compare a: 1 − λ = μ/2; compare b: (3/4)λ = 1 − μ
  4. 4μ = 1 − (3/4)λ, so 1 − λ = ½ − (3/8)λ
  5. 5λ = 4/5
  6. 6AX = (4/5)AN, so AX : XN = 4 : 1

AnswerAX : XN = 4 : 1

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