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4PM1/02 Jun 2019, Question 10: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 15 marks

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The question, in short

For the quadratic x² + 3x − 5 = 0 with roots α and β, use root properties without solving to find symmetric expressions, factorise α⁴ − β⁴, and determine p and q in β⁴ = p + q√29.

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

Part (a) · 5 marks

Without solving, find α² + β² and then α⁴ + β⁴ using sum and product of roots.

  1. 1α + β = −3, αβ = −5
  2. 2α² + β² = (α + β)² − 2αβ = 9 + 10 = 19
  3. 3α⁴ + β⁴ = (α² + β²)² − 2(αβ)² = 361 − 50 = 311

Answer(i) 19 (ii) 311

Part (b) · 2 marks

Given α > β, show without solving that α − β equals √29.

  1. 1(α − β)² = (α + β)² − 4αβ = 9 + 20 = 29
  2. 2α > β so α − β = √29

Answerα − β = √29 shown

Part (c) · 3 marks

Factorise α⁴ − β⁴ completely.

  1. 1α⁴ − β⁴ = (α² − β²)(α² + β²)
  2. 2= (α − β)(α + β)(α² + β²)

Answer(α − β)(α + β)(α² + β²)

Part (d) · 2 marks

Using earlier results, find the exact value of α⁴ − β⁴.

  1. 1α⁴ − β⁴ = √29 × (−3) × 19
  2. 2= −57√29

Answer−57√29

Part (e) · 3 marks

Given β⁴ = p + q√29 with p, q positive, find p and q.

  1. 12β⁴ = (α⁴ + β⁴) − (α⁴ − β⁴)
  2. 22β⁴ = 311 + 57√29
  3. 3β⁴ = 311/2 + (57/2)√29: p = 311/2, q = 57/2

Answerp = 311/2, q = 57/2

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