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4MA1/2HR Jun 2024, Question 25: Vector Geometry

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2HR (Higher) · Calculator · 6 marks

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The question, in short

Parallelogram OPQR has OP→ = 2a and OR→ = 3b, with M on PQ and N on RQ in given ratios. Find vectors ON→ and MR→, then use a vector method to find k where OY = k × ON.

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

Part (ai) · 1 mark

Find ON→ in terms of a and b in simplest form.

  1. 1RQ is parallel and equal to OP, so RQ→ = 2a
  2. 2RN→ = 4/5 × 2a = 8/5 a
  3. 3ON→ = OR→ + RN→ = 8/5 a + 3b

Answer8/5 a + 3b

Part (aii) · 1 mark

Find MR→ in terms of a and b in simplest form.

  1. 1PQ is parallel and equal to OR, so PQ→ = 3b
  2. 2PM = ¼ PQ, so MQ→ = ¾ × 3b = 9/4 b
  3. 3MR→ = MQ→ + QR→ = 9/4 b + (−2a) = 9/4 b − 2a

Answer9/4 b − 2a

Part (b) · 4 marks

Given MR and ON meet at Y and OY = k × ON, use vectors to find k.

  1. 1Route 1: OY→ = k × ON→ = k(8/5 a + 3b)
  2. 2Route 2: OY→ = OM→ + λMR→ = 2a + ¾b + λ(9/4 b − 2a)
  3. 3Match a: 8/5 k = 2 − 2λ, so λ = 1 − 4/5 k
  4. 4Match b: 3k = ¾ + 9/4 λ = 3 − 9/5 k, so 24/5 k = 3
  5. 5k = 15/24 = 5/8

Answer5/8

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