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0606/12 May/Jun 2025, Question 11: Expressing Vectors in Terms of a and b

Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 9 marks

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The question, in short

Triangle OAB with OA = a and OB = b; P on OA with OP = (3/4)OA, Q on AB with AQ = (1/3)AB; line PQ meets OB extended at R, with OR = λb and PR = μPQ. Find OR in terms of a, b, μ, then λ and μ.

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

Part (a) · 6 marks

Express OR as a vector using a, b and μ, starting from OP and PQ.

  1. 1OQ = a + (1/3)(b − a) = (2/3)a + (1/3)b
  2. 2OP = (3/4)a
  3. 3PQ = −(3/4)a + (2/3)a + (1/3)b = (1/3)b − (1/12)a
  4. 4OR = OP + PR = OP + μPQ
  5. 5OR = (3/4)a + μ((1/3)b − (1/12)a)

AnswerOR = (3/4)a + μ((1/3)b − (1/12)a)

Part (b) · 3 marks

Use your expression for OR to find the constants λ and μ.

  1. 1OR = (3/4 − μ/12)a + (μ/3)b = λb
  2. 2b: λ = μ/3
  3. 3a: 3/4 − μ/12 = 0, so μ = 9
  4. 4λ = 9/3 = 3

Answerλ = 3, μ = 9

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Also searched as: 0606/12/M/J/25 · 0606_s25_qp_12 · 0606 s25 12 · question 11.
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