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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · May/June 2014 · Q15 Marks

The equation x³ + px + q = 0, where p and q are constants, with q ≠ 0, has one root which is the reciprocal of another root. Prove that p + q² = 1.

  1. Prove that p + q² = 1.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2014). The mark or mark type is shown in brackets.

  1. [M1] Let roots be α, α¯¹ and β ⇒ α + α¯¹ + β = 0 (1) Any of three for M1
  2. [A1] Product of roots ⇒ β = -q (2) A1 for another.
  3. [A1] Sum of products in pairs ⇒ 1+ β (α+α¯¹)=p (3) A1 for a third
  4. [M1] From (1) and (3) 1−β² = p Wrong sign in (2) scores M1A0A1M1A0
  5. [A1] Using (2) 1-q²=p or p + q² = 1

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About this question

Topic
Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 5 marks.
Source
A-Level Further Mathematics, May/June 2014 Paper 1 Variant 2, question 1.

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