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4PM1/01 Jun 2022, 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

Triangle OAB has OA = a, OB = b; P divides AB with AP : PB = 3 : 1, M is midpoint of OA, N is midpoint of OP. Find vectors, then OC and an area ratio K.

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

Part (a) · 4 marks

Find simplified vector expressions in terms of a and b for (i) OP and (ii) MN.

  1. 1OP = OA + (3/4)AB = a + (3/4)(b − a) = (1/4)a + (3/4)b
  2. 2MN = MO + (1/2)OP = −(1/2)a + (1/2)((1/4)a + (3/4)b)
  3. 3MN = −(3/8)a + (3/8)b

AnswerOP = (1/4)a + (3/4)b; MN = −(3/8)a + (3/8)b

Part (b) · 6 marks

Given C on OB with ANC a straight line, use vectors to find OC as a simplified expression in b.

  1. 1C on OB: OC = λb
  2. 2AN = AO + ON = −a + (1/2)((1/4)a + (3/4)b) = −(7/8)a + (3/8)b
  3. 3C on line AN: OC = OA + μAN = (1 − (7/8)μ)a + (3/8)μb
  4. 4Compare a: 1 − (7/8)μ = 0 ⇒ μ = 8/7
  5. 5Compare b: λ = (3/8)μ = 3/7, so OC = (3/7)b

AnswerOC = (3/7)b

Part (c) · 4 marks

Given area of quadrilateral AMNP divided by area of triangle OAB equals K, find the exact value of K.

  1. 1Area OAP = (3/4) × area OAB (AP : AB = 3 : 4)
  2. 2M, N are midpoints of OA, OP: area OMN = (1/4) × area OAP = (3/16) × area OAB
  3. 3Area AMNP = (3/4 − 3/16) × area OAB = (9/16) × area OAB
  4. 4K = 9/16

AnswerK = 9/16

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