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4PM1/01R Jun 2022, Question 6: The Sine Rule

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 8 marks

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The question, in short

Solve three linked problems: find a from a surd equation, prove a side length in triangle PQR, then find its exact area using sin 105°.

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

Part (a) · 2 marks

Without a calculator, find the value of a in (a + √5)/(√5 − 2) = 11 + 5√5, showing working.

  1. 1a + √5 = (11 + 5√5)(√5 − 2)
  2. 2= 11√5 − 22 + 25 − 10√5 = 3 + √5
  3. 3a = 3

Answera = 3

Part (b) · 3 marks

In triangle PQR with PR = x + 3, QR = x, angle QPR = 30° and angle PQR = 45°, show that x = 3 + 3√2.

  1. 1Sine rule: (x + 3)/sin 45° = x/sin 30°
  2. 2(x + 3)/(√2/2) = x/(1/2) ⇒ x + 3 = √2x
  3. 33 = x(√2 − 1) ⇒ x = 3/(√2 − 1)
  4. 4x = 3(√2 + 1)/((√2 − 1)(√2 + 1)) = 3 + 3√2

Answerx = 3 + 3√2 shown

Part (c) · 3 marks

Given sin 105° = (√6 + √2)/4, find the exact area A of triangle PQR in the stated form with integers p, q, r, s.

  1. 1Angle PRQ = 180° − 30° − 45° = 105°
  2. 2A = (1/2) × (3 + 3√2) × (6 + 3√2) × sin 105°
  3. 3= (1/2)(3 + 3√2)(6 + 3√2)(√6 + √2)/4
  4. 4= (9/8)(4√6 + 4√2 + 6√3 + 6)

AnswerA = (9/8)(4√6 + 4√2 + 6√3 + 6)

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