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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · May/June 2022 · Q48 Marks

The cubic equation 2x³ + 5x² - 6 = 0 has roots α, β, γ.

  1. (a)Find a cubic equation whose roots are 1/(α³), 1/(β³), 1/(γ³).[3]
  2. (b)Find the value of 1/(α⁶) + 1/(β⁶) + 1/(γ⁶).[3]
  3. (c)Find also the value of 1/(α⁹) + 1/(β⁹) + 1/(γ⁹).[2]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

  1. [B1] Substitutes.
  2. [M1] Cubes to eliminate radical.
  3. [A1]
  4. [B1 FT] Using their answer to part (a).
  5. [M1]
  6. [A1]
  7. [M1] Using their 216a⁻⁹ – 216a⁻⁶ – 53a⁻³ – 8 = 0
  8. [A1]

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About this question

Topic
Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 8 marks.
Source
A-Level Further Mathematics, May/June 2022 Paper 1 Variant 2, question 4.

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