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4PM1/01 Nov 2021, Question 8: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 15 marks

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The question, in short

Regular hexagon OABCDE has OA = a and AB = b; express vectors in a and b, use a vector method for a ratio, and find a triangle area given the hexagon's area is 60 cm².

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

Part (a) · 1 mark

Write the vector OB using a and b.

  1. 1OB = OA + AB = a + b

AnswerOB = a + b

Part (b) · 3 marks

Give BC as a simplified vector in a and b.

  1. 1OC = 2b (OC is parallel to AB and twice as long)
  2. 2BC = BO + OC = −(a + b) + 2b
  3. 3BC = b − a

AnswerBC = b − a

Part (c) · 2 marks

Point M splits BC in ratio 2 : 1; express OM in simplified form using a and b.

  1. 1OM = OB + (2/3)BC
  2. 2= a + b + (2/3)(b − a)
  3. 3= (1/3)a + (5/3)b

AnswerOM = (1/3)a + (5/3)b

Part (d) · 5 marks

Given OMY and ABY are straight lines, use vectors to determine the ratio AB : BY.

  1. 1OY = μOM = (μ/3)a + (5μ/3)b
  2. 2OY = OA + AY = a + λb
  3. 3Compare a: μ/3 = 1, so μ = 3
  4. 4Compare b: λ = 5μ/3 = 5, so AY = 5b
  5. 5BY = AY − AB = 4b, so AB : BY = 1 : 4

AnswerAB : BY = 1 : 4

Part (e) · 4 marks

With the hexagon's area 60 cm², work out the area of triangle OAY.

  1. 1Triangle OAB is 1/6 of the hexagon: area 10 cm²
  2. 2(1/2)ab sin 120° = 10 with |OA| = a, |AB| = b
  3. 3AY = 5AB, so area OAY = (1/2) × a × 5b × sin 120° = 5 × 10
  4. 4Area OAY = 50 cm²

Answer50 cm²

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