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4PM1/02 Jun 2025, Question 8: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 11 marks

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The question, in short

Triangle OAB has OA = 3u + v and OB = −u + 2v. Work with vectors to find AB, a scalar μ, and BD given an area ratio.

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

Part (a) · 2 marks

Express the side AB as a simplified vector in terms of u and v.

  1. 1AB = AO + OB
  2. 2= −(3u + v) + (2v − u) = −4u + v

AnswerAB = −4u + v

Part (b) · 4 marks

Given AC is parallel to OB and OC = μv, determine the value of μ.

  1. 1AC = λ(2v − u)
  2. 2OC = OA + AC = (3 − λ)u + (1 + 2λ)v = μv
  3. 3u: 3 − λ = 0, so λ = 3
  4. 4v: μ = 1 + 2λ = 7

Answerμ = 7

Part (c) · 5 marks

With OC and AB meeting at X, and D on OC where the areas of triangles BOX and BXD are in ratio 2:3, use vectors to find BD.

  1. 1OX = OA + αAB = (3 − 4α)u + (1 + α)v, which is a multiple of v, so α = 3/4
  2. 2OX = (7/4)v
  3. 3Triangles BOX and BXD share the height from B, so XD = (3/2)OX and OD = (5/2)OX = (35/8)v
  4. 4BD = BO + OD = u − 2v + (35/8)v
  5. 5BD = u + (19/8)v

AnswerBD = u + (19/8)v

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