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4PM1/01 Jun 2020, Question 11: Expressing Vectors in Terms of a and b

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 13 marks

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The question, in short

Using vectors OA = a and OB = b with C dividing OB in ratio 1:3 and D the midpoint of AC, find vector expressions, a collinearity parameter, and an area ratio.

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

Part (a) · 5 marks

Find simplified vector expressions in terms of a and b for AC, OD and BD.

  1. 1OC = ¼b, so AC = −a + ¼b
  2. 2OD = OA + ½AC = a + ½(−a + ¼b) = ½a + ⅛b
  3. 3BD = BO + OD = −b + ½a + ⅛b = ½a − ⅞b

AnswerAC = −a + ¼b, OD = ½a + ⅛b, BD = ½a − ⅞b

Part (b) · 4 marks

Given E on OA with OE = λOA and E, D, B collinear, find λ.

  1. 1BE = k BD = k(½a − ⅞b)
  2. 2BE = BO + OE = −b + λa
  3. 3Compare b: (7/8)k = 1 ⇒ k = 8/7
  4. 4Compare a: λ = k/2 = 4/7

Answerλ = 4/7

Part (c) · 4 marks

Given the ratio of area OAC to area OEB equals μ, find μ.

  1. 1Triangle OAC = ¼ triangle OAB (OC = ¼OB)
  2. 2Triangle OEB = (4/7) triangle OAB (OE = 4/7 OA)
  3. 3μ = (1/4) ÷ (4/7) = 7/16

Answerμ = 7/16

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