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A-LevelMathematicsComplex numbers9709/31 · May/June 2025 · Q36 Marks

Find the complex numbers z for which z+5i / z-5 is real and |z|= √17. Give your answers in the form z = x+iy, where x and y are real.

  1. Find the complex numbers z for which z+5i / z-5 is real and |z|= √17. Give your answers in the form z = x+iy, where x and y are real.[6]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

  1. [*M1] Multiply numerator (and denominator) by the conjugate of the denominator.
  2. [DM1] Numerator = x(x-5)+y(y+5)+i((y+5)(x-5)-xy).
  3. [A1] Obtain x-y=5
  4. [M1] Correct use of modulus and solve for x and y x²+(x-5)² =17 => x²−5x+4=0 Or x - y = 5 and xy = -4.
  5. [A1] ⇒z=4-i
  6. [A1] ⇒z=1-4i
  7. Alternative Method for the first two marks of Question 3
  8. [M1] Quotient real => x/(x+5) = (y+5)/y
  9. [M1] Rearrange to linear form and simplify

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About this question

Topic
Complex numbers, A-Level Mathematics (syllabus 9709). Structured question, 6 marks.
Source
A-Level Mathematics, May/June 2025 Paper 3 Variant 1, question 3.

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