Throughout this question the use of a calculator is not permitted. It is given that the complex number −1 + (√3)i is a root of the equation kx³ + 5x² + 10x + 4 = 0, where k is a real constant.

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

1. [B1] State answer -1-√3i (If -1-√3i given as well at this point, still just B1)
2. [M1] Substitute x = -1 + √3i in the equation and attempt expansions of x² and x³ (Need to see sufficient working to be convinced that a calculator has not been used.)
3. [M1] Use i² = -1 correctly at least once (Allow for relevant use at any point in the solution)
4. [A1] Obtain k = 2
5. [M1] Carry out a complete method for finding a quadratic factor with zeros -1+√3i and -1-√3i (Could use factor theorem from this point. Need to see working. M1 for correct testing of correct root or allow M1 for three unsuccessful valid attempts.)
6. [A1] Obtain x² + 2x + 4 (Using factor theorem, obtain f(1/2) = 0)
7. [A1] Obtain root x = -1/2, or equivalent, via division or inspection (Final answer)