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4PM1/01R Jun 2025, Question 6: Unit Vectors and i, j Components

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 9 marks

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The question, in short

Vectors from origin O: A has position 9a + 12b, B has 2a + pb, with |a| = |b| = 1 and |AB| = 25. Work with p values and points on OB.

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Part (a) · 4 marks

Using |AB| = 25, show that p can equal −12 or 36.

  1. 1AB = AO + OB = −9a − 12b + 2a + pb = −7a + (p − 12)b
  2. 2(−7)² + (p − 12)² = 25²
  3. 3(p − 12)² = 576
  4. 4p = 12 ± 24, so p = −12 or 36

Answerp = −12 and p = 36 shown

Part (b) · 2 marks

When p = −12, express a unit vector in the direction of OB using a and b, simplified.

  1. 1OB = 2a − 12b
  2. 2|OB| = √(2² + 12²) = √148 = 2√37
  3. 3Unit vector = ±(2a − 12b)/(2√37) = ±(1/√37)(a − 6b)

Answer±(1/√37)(a − 6b) or ±(√37/37)(a − 6b)

Part (c) · 3 marks

When p = 36, C divides OB with OC : CB = 2 : 3; find vector CA in terms of a and b, simplified.

  1. 1OB = 2a + 36b
  2. 2OC = (2/5)(2a + 36b)
  3. 3CA = CO + OA = −(2/5)(2a + 36b) + 9a + 12b
  4. 4CA = (41/5)a − (12/5)b

AnswerCA = (41/5)a − (12/5)b

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Also searched as: 4PM1 1R June 2025 · 4PM1/01R · 4pm1 jun25 1r · question 6.
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