A-LevelMathematicsComplex numbers9709/31 · May/June 2022 · Q78 Marks

The complex number u is defined by u = `(√2 – a√2i) / (1 + 2i)`, where a is a positive integer.

(a)Express u in terms of a, in the form x + iy, where x and y are real and exact.[3]
(b)It is now given that a = 3. Express u in the form reiº, where r > 0 and −π < θ < π, giving the exact values of r and 9.[2]
(c)Using your answer to part (b), find the two square roots of u. Give your answers in the form reiº, where r > 0 and −π < θ < π, giving the exact values of r and 0.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[M1] Multiply numerator and denominator by 1 – 2i, or equivalent At least one multiplication completed.
[A1] Obtain correct numerator (1-2a)√2-(2+a)√2i
[A1] Obtain final answer (1-2a)√2/5 - (2+a)√2/5 i OE
Alternative method for question 7(a)
[M1] Multiply x + iy by 1+2i and compare real and imaginary parts
[A1] Obtain x-2y=√2 and 2x+y=a√2
[A1] Obtain final answer (1-2a)√2/5 - (2+a)√2/5 i OE
[B1 FT] Obtain r = 2
[B1] Obtain θ = -3π/4
[M1] Use correct method to find r or θ
[A1 FT] State answer √2e^(-3πi/8)
[A1 FT] State answer √2e^(5πi/8)

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