It is given that z₁ = r₁e^(iθ₁) and z₂ = r₂e^(iθ₂). Show that (z₁z₂)* = z₁*z₂*.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [B1] State z₁z₂ = r₁r₂e^(i(θ₁+θ₂)). Allow equivalent forms.
2. [B1] State correct conjugate of z₁z₂, z₁ or z₂. Allow equivalent forms.
3. [B1] Obtain given result from correct working. Clear demonstration that the product of the conjugates is identical to the conjugate of the product. Need to see a conclusion.
4. [B1] State (z* =) 3e^(-πi/4). Allow equivalent forms. Allow 3e^(7πi/4).
5. [M1] Complete method to find both b and c. E.g. find the product and sum of the roots, or expand (z−3e^(πi/4))(z−3e^(-πi/4)).
6. [A1] Obtain b = -3√2, c = 9. Allow z²-3√2z+9=0.