A-LevelMathematicsComplex numbers9709/32 · Oct/Nov 2020 · Q68 Marks

The complex number u is defined by u = 7+ i 1-i

(a)Express u in the form x + iy, where x and y are real.[3]
(b)Show on a sketch of an Argand diagram the points A, B and C representing u, 7 + i and 1 – i respectively.[2]
(c)By considering the arguments of 7 + i and 1 – i, show that tan⁻¹(4/3) = tan⁻¹(4) + π/4.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

[M1] Multiply numerator and denominator by 1 + i, or equivalent
[A1] Obtain numerator 6 + 8i or denominator 2
[A1] Obtain final answer u = 3 + 4i
[B1] Show the point A representing u in a relatively correct position
[B1] Show the other two points B and C in relatively correct positions: approximately equal distance above / below real axis
[B1] State or imply arg(1 – i) = −π/4
[M1] Substitute exact arguments in arg(7 + i) – arg(1 – i) = arg u
[A1] Obtain given answer correctly from their u = k(3+4i)

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Topic: Complex numbers, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source: A-Level Mathematics, Oct/Nov 2020 Paper 3 Variant 2, question 6.

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