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A-LevelMathematicsComplex numbers9709/33 · May/June 2021 · Q1010 Marks

Verify that −1 + √2i is a root of the equation z⁴ + 3z² + 2z + 12 = 0.

  1. (a)Verify that −1 + √2i is a root of the equation z⁴ + 3z² + 2z + 12 = 0.[3]
  2. (b)Find the other roots of this equation.[7]

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

  1. [M1] Substitute -1 + √2i and attempt expansions of the z² and z⁴ terms
  2. [M1] Use i² = -1 at least once
  3. [A1] Complete the verification correctly
  4. [A1] State second root -1 - √2i
  5. [M1] Carry out a method to find a quadratic factor with zeros −1±√2i
  6. [A1] Obtain z² + 2z + 3
  7. [M1] Commence division and reach partial quotient z² + kz
  8. [A1] Obtain second quadratic factor z² - 2z + 4
  9. [M1] Solve a 3-term quadratic and use i² = -1
  10. [A1] Obtain roots 1+√3i and 1-√3i

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About this question

Topic
Complex numbers, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source
A-Level Mathematics, May/June 2021 Paper 3 Variant 3, question 10.

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