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4PM1/02 Jun 2023, Question 11: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 12 marks

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The question, in short

Quadratic E has roots α and β with α > β > 0, α − β = 2√6 and α² + β² = 30. Show related identities, then find α⁴ + β⁴, α⁴ − β⁴, and P, Q.

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

Part (ai) · 4 marks

Show that αβ = 3 using the given difference and sum of squares.

  1. 1(α − β)² = 24, so (α + β)² − 4αβ = 24
  2. 2α² + β² = 30, so (α + β)² − 2αβ = 30
  3. 3Subtract: 2αβ = 6
  4. 4αβ = 3

Answerαβ = 3 shown

Part (aii) · 2 marks

Show that α + β = 6.

  1. 1(α + β)² = α² + β² + 2αβ = 30 + 6 = 36
  2. 2α > β > 0, so α + β = 6

Answerα + β = 6 shown

Part (bi) · 2 marks

Without solving E, find α⁴ + β⁴.

  1. 1α⁴ + β⁴ = (α² + β²)² − 2(αβ)²
  2. 2= 900 − 18 = 882

Answer882

Part (bii) · 2 marks

Without solving E, find the exact value of α⁴ − β⁴.

  1. 1α⁴ − β⁴ = (α² + β²)(α² − β²) = (α² + β²)(α − β)(α + β)
  2. 2= 30 × 2√6 × 6 = 360√6

Answer360√6

Part (c) · 2 marks

Given α⁴ = P + Q√6 with positive integers P and Q, find P and Q.

  1. 1Add: 2α⁴ = 882 + 360√6
  2. 2α⁴ = 441 + 180√6
  3. 3P = 441, Q = 180

AnswerP = 441, Q = 180

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