A-LevelMathematicsComplex numbers9709/33 · Oct/Nov 2024 · Q45 Marks

Find the complex number z satisfying the equation (z-3i)/(z+3i) = (2-9i)/5. Give your answer in the form x+ iy, where x and y are real.

Find the complex number z satisfying the equation (z-3i)/(z+3i) = (2-9i)/5. Give your answer in the form x+ iy, where x and y are real.[5]

Mark scheme

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

[*M1] Substitute z = x + iy and obtain a horizontal equation. E.g. 5(x+(y-3)i)=(2−9i)(x+(y+3)i) OR Obtain a horizontal equation in z. E.g. 5z – 15i = 2z + 6i – 27i² – 9iz. Do not allow if this would lead to an equation containing xy terms which do not cancel.
[M1] Use i² = -1 anywhere.
[A1] Obtain e.g. 5x +5(y - 3)i = (2x + 9y +27)+i(2y+6-9x) OR z = (9+7i)/(1+3i).
[DM1] Obtain simultaneous equations by equating real and imaginary parts OR Multiply top and bottom by 1-3i or equivalent for their z.
[A1] Obtain [z =] 3-2i only.

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Topic: Complex numbers, A-Level Mathematics (syllabus 9709). Structured question, 5 marks.
Source: A-Level Mathematics, Oct/Nov 2024 Paper 3 Variant 3, question 4.

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