A-LevelMathematicsComplex numbers9709/33 · May/June 2022 · Q58 Marks

The complex number 3 – i is denoted by u.

(a)Show, on an Argand diagram with origin O, the points A, B and C representing the complex numbers u, u* and u* − u respectively. State the type of quadrilateral formed by the points O, A, B and C.[3]
(b)Express u*/u in the form x + iy, where x and y are real.[3]
(c)By considering the argument of u*/u, or otherwise, prove that tan⁻¹(3/4) = 2tan⁻¹(1/3).[2]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[B1] Show u and u* in relatively correct positions. Must have sense of scale on axes
[B1] Show u* − u in a relatively correct position. Must have sense of scale on axes
[B1] State that OABC is a parallelogram [independent of previous marks]
[M1] Multiply their numerator and the given denominator by 3 + i and attempt to evaluate either
[A1] Obtain numerator 8 + 6i or denominator 10
[A1] State final answer 4/5 + 3/5 i or 8/10 + 6/10 i or 0.8 + 0.6i
[M1] State or imply arg(u*/u) = arg u* − arg u or 2arg u*
[A1] Justify the given statement correctly

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