A-LevelMathematicsComplex numbers9709/32 · May/June 2020 · Q810 Marks

Solve the equation (1 + 2i)w + iw* = 3 + 5i. Give your answer in the form x + iy, where x and y are real.

(a)Solve the equation (1 + 2i)w + iw* = 3 + 5i. Give your answer in the form x + iy, where x and y are real.[4]
(b)(i)On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z – 2 – 2i| ≤ 1 and arg(z – 4i) ≥ −¼π.[4]
(b)(ii)Find the least value of Im z for points in this region, giving your answer in an exact form.[2]

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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.

[B1] Substitute and obtain a correct equation in x and y
[M1] Use i² = –1 and equate real and imaginary parts
[A1] Obtain two correct equations in x and y, e.g. x – y = 3 and 3x + y = 5
[A1] Solve and obtain answer z = 2 – i
[B1] Show a point representing 2 + 2i
[B1 FT] Show a circle with radius 1 and centre not at the origin (FT is on the point representing the centre)
[B1] Show the correct half line from 4i
[B1] Shade the correct region
[M1] Carry out a complete method for finding the least value of Im z
[A1] Obtain answer 2 – 1/√2, or exact equivalent

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