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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · Oct/Nov 2024 · Q310 Marks

It is given that α + β + γ + δ = 2, α² + β² + γ² + δ² = 3, α³ + β³ + γ³ + δ³ = 4. It is given that α, β, γ, δ are the roots of the equation 6x⁴-12x³+3x²+2x+6 = 0.

  1. (a)Find the value of αβ + αγ+αδ+ βγ + βδ+γδ.[2]
  2. (b)Find the value of α²β+α²γ +α²δ+β²α+β²γ+β²δ+γ²α+γ²β+γ²δ+δ²α+δ²β+δ²γ.[3]
  3. (c)(i)Find the value of α⁴ + β⁴ + γ⁴ +δ⁴.[3]
  4. (c)(ii)Find the value of α⁵ + β⁵ + γ⁵ +δ⁵.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

  1. [M1] Substitutes into formula for sum of squares.
  2. [A1]
  3. [M1 A1] Factorises and substitutes.
  4. [A1]
  5. [M1] Sums and substitutes.
  6. [A1]
  7. [A1]
  8. [M1] Multiplies equation through by x, sums and substitutes. S₅ - 2S₄ + ½S₃ + ⅓S₂ + S₁ = 0
  9. [A1]

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About this question

Topic
Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 10 marks.
Source
A-Level Further Mathematics, Oct/Nov 2024 Paper 1 Variant 2, question 3.

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