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4PM1/01R Jan 2020, Question 10: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 11 marks

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The question, in short

Quadrilateral OABC has OA = 2a − b, OB = 3a + b and BC = −a + 3b. Work with vectors to find AB, show OC is parallel to AB, then find the ratio AX : XC where diagonals OB and AC meet at X.

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

Part (a) · 2 marks

Express the vector AB in terms of a and b, giving it in simplified form.

  1. 1AB = AO + OB = −(2a − b) + (3a + b)
  2. 2= a + 2b

AnswerAB = a + 2b

Part (b) · 2 marks

Show, using vectors, that OC and AB are parallel.

  1. 1OC = OB + BC = 3a + b − a + 3b = 2a + 4b
  2. 22a + 4b = 2(a + 2b) = 2AB
  3. 3OC is a positive multiple of AB, so they are parallel

AnswerOC = 2a + 4b = 2AB, so OC is parallel to AB

Part (c) · 7 marks

Using a vector method, determine the ratio in which X divides AC, i.e. AX : XC.

  1. 1AC = AB + BC = a + 2b − a + 3b = 5b
  2. 2AX = λAC = 5λb
  3. 3Also AX = AO + OX = −(2a − b) + μ(3a + b)
  4. 45λb = (3μ − 2)a + (1 + μ)b
  5. 5a: 3μ − 2 = 0, so μ = 2/3; b: 5λ = 1 + 2/3, so λ = 1/3
  6. 6AX : XC = 1 : 2

AnswerAX : XC = 1 : 2

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