The complex numbers z and ω are defined by z = 1-i and ω =-3+3√3i.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [B1] State zω = (-3+3√3) + (3+3√3)i. Or exact equivalent with real and imaginary parts collected. Need brackets around the coefficient of i. Allow for a=, b= stated correctly.
2. [B1] Obtain |z| = √2.
3. [B1] Obtain arg z = -π/4 final answer.
4. [B1] Obtain |ω| = 6.
5. [B1] Obtain arg ω = 2π/3 final answer.
6. [B1] Show |OA|=|AB| = 6, hence isosceles. One mark for ‘isosceles’ and one mark for 'right angle'.
7. [B1] ∠AOB = arg ω - arg zω = -arg z = π/4 hence third angle is a right angle. There will be alternatives e.g. use of Pythagoras (ratio of lengths is 1:1:√2), expressing each number in “vector” form and using scalar product or explaining the effect of multiplying by 1 – i.
8. [M1] Using your answers to part (b), prove that tan(π/12) = (√3+1)/(√3-1). For showing correct use of their angles from part (b). Must demonstrate where 5π/12 comes from.
9. [M1] Correct method for their zω from part (a). Must link to point B on diagram or to arg zω. Need to see tan⁻¹((3+3√3)/(-3+3√3)) or tan θ = (3+3√3)/(-3+3√3) and not just tan⁻¹((1+√3)/(-1+√3)).
10. [A1] Obtain given answer from full and correct working.