A-LevelMathematicsComplex numbers9709/31 · May/June 2025 · Q66 Marks

It is given that z₁ = 3e^(i π/4), z₂ = (1/2)e^(i π/2) and w = 2e^(-i π/2).

(a)State the values of wz₁ and wz₂. Give your answers in the form re^(iθ), where r > 0 and -π < θ ≤ π.[2]
(b)On a sketch of an Argand diagram with origin O, show the points A, B, C and D representing the complex numbers z₁, z₂, wz₁ and wz₂, respectively.[2]
(c)State the geometric effects of multiplying z₁ and z₂ by w.[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 2025). The mark or mark type is shown in brackets.

[B1] Obtain wz₁ = 6e^(-i π/4)
[B1] Obtain wz₂ = 3e^(i0)
SC B1 for both moduli correct or both arguments correct.
[B1] A and B plotted correctly
[B1FT] C and D correct, or FT their A and B.
[B1] Rotation π/2 radians (anticlockwise), or rotation -π/2 radians clockwise Accept 90°. Not required to state the centre of the rotation, but B0 if an incorrect statement seen.
[B1] Enlargement (scale) factor 2 Not required to state the centre of the enlargement, but B0 if an incorrect statement seen. Allow 'expansion' or 'stretch'.

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