A-LevelMathematicsComplex numbers9709/32 · Oct/Nov 2019 · Q79 Marks

The context for the parts involves complex numbers.

(a)Find the complex number z satisfying the equation z + iz/z* - 2 = 0, where z* denotes the complex conjugate of z. Give your answer in the form x + iy, where x and y are real.[5]
(b)(i)On a single Argand diagram sketch the loci given by the equations |z – 2i| = 2 and Im z = 3, where Im z denotes the imaginary part of z.[2]
(b)(ii)In the first quadrant the two loci intersect at the point P. Find the exact argument of the complex number represented by P.[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 (Oct/Nov 2019). The mark or mark type is shown in brackets.

[B1] Substitute and obtain a correct horizontal equation in x and y in any form
[*M1] Use i² = -1 and equate real and imaginary parts to zero OE
[A1] Obtain two correct equations e.g. x² + y² - y -2x = 0 and x + 2y = 0
[DM1] Solve for x or for y
[A1] Obtain answer (6/5) - (3/5)i and no other
[B1] Show a circle with centre 2i and radius 2
[B1] Show horizontal line y = 3 – in first and second quadrant
[M1] Carry out a complete method for finding the argument. (Not by measuring the sketch)
[A1] Obtain answer (1/3)π (or 60°)

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