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4024/12 Oct/Nov 2025, Question 21: 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

In parallelogram OPQR with OP = a and OR = 2b, M is midpoint of QR and N divides PQ with PN:NQ = 1:3. Find vectors and the position vector of X on extended OP.

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

Part (ai) · 1 mark

Find vector MQ in terms of a and/or b, simplified.

  1. 1RQ = OP = a and M is the midpoint of QR
  2. 2→MQ = 1/2 a

Answer1/2 a

Part (aii) · 2 marks

Find vector NM in terms of a and/or b, simplified.

  1. 1PQ = OR = 2b, so →NQ = 3/4 × 2b = 3/2 b
  2. 2QR is parallel to PO, so →QR = −a and →QM = −1/2 a
  3. 3→NM = →NQ + →QM = 3/2 b − 1/2 a

Answer3/2 b − 1/2 a

Part (b) · 2 marks

Given OP extended to X and MNX straight, find the position vector of X.

  1. 1Triangles NPX and NQM are similar with scale factor PN/NQ = 1/3
  2. 2→PX = 1/3 × →MQ = 1/3 × 1/2 a = 1/6 a
  3. 3→OX = a + 1/6 a = 7/6 a

Answer7/6 a

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