A-LevelMathematicsComplex numbers9709/33 · May/June 2020 · Q910 Marks

The complex numbers u and w are such that u - w = 2i and uw = 6. Find u and w, giving your answers in the form x + iy, where x and y are real and exact.

(a)The complex numbers u and w are such that u - w = 2i and uw = 6. Find u and w, giving your answers in the form x + iy, where x and y are real and exact.[5]
(b)On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z − 2 − 2i| ≤ 2, 0 ≤ arg z ≤ ¼π and Re z ≤ 3.[5]

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Mark scheme

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

[M1] Eliminate u or w and obtain an equation in w or u
[A1] Obtain a quadratic in u or w, e.g. u² – 2iu – 6 = 0 or w² + 2iw – 6 = 0
[M1] Solve a 3-term quadratic for u or for w
[A1] Obtain answer u = √5 + i, w = √5 − i
[A1] Obtain answer u = −√5 + i, w = −√5 − i
[B1] Show the point representing 2 + 2i
[B1 FT] Show a circle with centre 2 + 2i and radius 2 (FT is on the position of 2 + 2i)
[B1] Show half-line from origin at 45° to the positive x-axis
[B1] Show line for Re z = 3
[B1] Shade the correct region

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