4PM1/02 Jan 2023, Question 8: Roots α and β of a Quadratic
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 11 marks
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The question, in short
Quadratic x² − 4k√2 x + 2k⁴ − 1 = 0 has roots α, β with α² + β² = 66 and α³ + β³ = p√2; find integer p.
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Solution · 11 marks
- 1α + β = 4√2 k, αβ = 2k⁴ − 1
- 2(α + β)² = α² + β² + 2αβ: 32k² = 66 + 2(2k⁴ − 1)
- 34k⁴ − 32k² + 64 = 0, so k⁴ − 8k² + 16 = 0
- 4(k² − 4)² = 0, k > 0 so k = 2
- 5α + β = 8√2, αβ = 31
- 6α³ + β³ = (α + β)³ − 3αβ(α + β)
- 7= 1024√2 − 3 × 31 × 8√2 = 280√2
- 8p = 280
Answerp = 280
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