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4024/12 Oct/Nov 2022, Question 26: Expressing Vectors in Terms of a and b

Cambridge O Level Mathematics 4024 · Paper 12 (Non-calculator) · Non-calculator · 5 marks

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The question, in short

Triangle OAC has B on AC with AB:BC = 3:2, OA = a and OB = b. Express OC using a and b, then use D on OC with OD = b − (2/5)a to prove OABD is a trapezium.

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

Part (a) · 3 marks

Write the vector OC in terms of a and b, simplified as far as possible.

  1. 1→AB = →AO + →OB = b − a
  2. 2AB : BC = 3 : 2, so →AC = 5/3 →AB = 5/3(b − a)
  3. 3→OC = →OA + →AC = a + 5/3 b − 5/3 a
  4. 4→OC = (5/3)b − (2/3)a

Answer(5/3)b − (2/3)a

Part (b) · 2 marks

Given D lies on OC with OD = b − (2/5)a, demonstrate that quadrilateral OABD has one pair of parallel sides.

  1. 1→DB = →OB − →OD = b − (b − (2/5)a)
  2. 2→DB = (2/5)a
  3. 3→DB is a multiple of →OA = a, so DB is parallel to OA
  4. 4OABD has a pair of parallel sides, so it is a trapezium

Answer→DB = (2/5)a or →BD = −(2/5)a; →DB is a multiple of →OA, hence parallel

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Also searched as: 4024/12/O/N/22 · 4024_w22_qp_12 · 4024 w22 12 · question 26.
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