A-LevelMathematicsComplex numbers9709/31 · Oct/Nov 2023 · Q45 Marks

The complex number u is defined by u = (3 + 2i) / (a - 5i), where a is real.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[M1] (a) Multiply numerator and denominator by a + 5i OE
[M1] (a) Use i² = -1 At least once.
[A1] (a) Obtain answer (3a-10)/(a²+25) + (2a+15)/(a²+25) i
[M1] (a) Alternative Method: Multiply x + iy by a – 5i and use i² = –1
[M1] (a) Alternative Method: Compare real and imaginary parts 3 = ax + 5y, 2 = ay – 5x.
[A1] (a) Alternative Method: Obtain answer (3a-10)/(a²+25) + (2a+15)/(a²+25) i
[M1] (b) State or imply Im(a) ÷ Re(a) = 1 Or Im(a) = Re(a) or equivalent for their u.
[A1] (b) Obtain answer a = 25

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Topic: Complex numbers, A-Level Mathematics (syllabus 9709). Structured question, 5 marks.
Source: A-Level Mathematics, Oct/Nov 2023 Paper 3 Variant 1, question 4.

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