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4PM1/02 Jan 2022, Question 9: Vector Geometry

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

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The question, in short

Triangle OAB has OA = 2a and OB = 4b. Find vectors along AB, OP, AQ, and the ratio in which line AQR cuts OB.

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

Part (a) · 2 marks

Find vector AB in terms of a and b.

  1. 1AB = AO + OB
  2. 2= −2a + 4b

AnswerAB = −2a + 4b

Part (b) · 2 marks

P is the midpoint of AB; find vector OP in simplified form using a and b.

  1. 1OP = OA + ½AB
  2. 2= 2a + ½(−2a + 4b)
  3. 3= a + 2b

AnswerOP = a + 2b

Part (c) · 3 marks

Q on OP with OQ : QP = 3 : 1; find vector AQ in simplified form.

  1. 1OQ = ¾OP = ¾(a + 2b)
  2. 2AQ = AO + OQ = −2a + ¾(a + 2b)
  3. 3AQ = −(5/4)a + (3/2)b

AnswerAQ = −(5/4)a + (3/2)b

Part (d) · 6 marks

R on OB with AQR straight; find the ratio OR : RB in simplest form.

  1. 1OR = λ(4b)
  2. 2OR = OA + μAQ = 2a + μ(−(5/4)a + (3/2)b)
  3. 3Compare a: 2 − (5/4)μ = 0, so μ = 8/5
  4. 4Compare b: (3/2)μ = 4λ, so 4λ = 12/5, λ = 3/5
  5. 5OR = (3/5)OB, so OR : RB = 3 : 2

AnswerOR : RB = 3 : 2

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