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4PM1/01 Jun 2021, Question 10: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 14 marks

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The question, in short

In triangle OAB with D on OA and C beyond B on OB, given OA = 5p, AB = 3q, OC = (3/2)OB and OD = (3/5)OA, work with vectors to find DC, OF and OG.

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

Part (a) · 3 marks

Express vector DC in terms of p and q in simplest form.

  1. 1OB = OA + AB = 5p + 3q
  2. 2OC = (3/2)(5p + 3q), OD = 3p
  3. 3DC = DO + OC = −3p + (3/2)(5p + 3q)
  4. 4DC = (9/2)p + (9/2)q = (9/2)(p + q)

AnswerDC = (9/2)(p + q) (= (9/2)p + (9/2)q)

Part (b) · 7 marks

Given DC meets AB at F, use vectors to express OF in terms of p and q.

  1. 1OF = OD + λDC = 3p + (9/2)λ(p + q)
  2. 2OF = OA + μAB = 5p + 3μq
  3. 3p: 3 + (9/2)λ = 5 ⇒ λ = 4/9
  4. 4q: (9/2)λ = 3μ ⇒ μ = 2/3
  5. 5OF = 5p + 2q

AnswerOF = 5p + 2q

Part (c) · 4 marks

Given G on OB with FG parallel to AO, use vectors to find OG in terms of p and q.

  1. 1G on OB: OG = α(5p + 3q)
  2. 2FG parallel to AO: OG = OF + βAO = 5p + 2q − 5βp
  3. 3Compare q: 3α = 2 ⇒ α = 2/3
  4. 4OG = (2/3)(5p + 3q) = (10/3)p + 2q

AnswerOG = (10/3)p + 2q

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