A-LevelMathematicsComplex numbers9709/33 · May/June 2017 · Q1110 Marks

Throughout this question the use of a calculator is not permitted.

(a)The complex numbers z and w satisfy the equations z + (1 + i)w = i and (1 – i)z + iw = 1. Solve the equations for z and w, giving your answers in the form x + iy, where x and y are real.[6]
(b)The complex numbers u and v are given by u = 1 + (2√3)i and v = 3 + 2i. In an Argand diagram, u and v are represented by the points A and B. A third point C lies in the first quadrant and is such that BC = 2AB and angle ABC = 90°. Find the complex number z represented by C, giving your answer in the form x + iy, where x and y are real and exact.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

[M1] Solve for z or for w
[M1] Use i² = -1
[A1] Obtain w = i/(2-i) or z = (2+i)/(2-i)
[M1] Multiply numerator and denominator by the conjugate of the denominator
[A1] Obtain w = -1/5 + 2/5 i
[A1] Obtain z = 3/5 + 1/5 i
[B1] EITHER: Find ±[2+(2-2√3)i]. OR: State (z-v)/(v-u) = ki, or equivalent.
[M1*] Multiply by 2i (or -2i) (if first method). OR: State k = 2 (if second method).
[DM1] Add result to v (if first method). OR: Substitute and solve for z even if i omitted (if second method).
[A1] Obtain answer 4√3 - 1 + 6i

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