Throughout this question the use of a calculator is not permitted. The complex numbers −3√3 + i and √3 + 2i are denoted by u and v respectively.

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

1. [M1] Substitute in uv, expand the product and use i² = –1
2. [A1] Obtain answer uv = –11–5√3i
3. [M1] EITHER: Substitute in u/v and multiply numerator and denominator by the conjugate of v, or equivalent
4. [A1] Obtain numerator –7+7√3i or denominator 7
5. [A1] Obtain final answer –1+√3i
6. OR: Substitute in u/v, equate to x + iy and solve for x or for y
7. [M1] {-3√3 = √3x - 2y; 1 = 2x + √3y}
8. [A1] Obtain x = -1 or y = √3
9. [A1] Obtain final answer -1+√3 i
10. [B1] Show the points A and B representing u and v in relatively correct positions
11. [M1] Carry out a complete method for finding angle AOB, e.g. calculate arg(u/v)
12. [A1] Prove the given statement