Write down the values of θ, in the interval 0 ≤ θ < 2π, for which cos θ + i sin θ is a fifth root of unity. By writing the equation (z + 1)⁵ = z⁵ in the form ((z + 1)/z)⁵ = 1, show that its roots are ½{1 + i cot(kπ/5)}, k = 1, 2, 3, 4.

Mark scheme

Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.

1. [M1] Obtains fifth roots of unity by de M's Thm. 1 = cos(2kπ) + i sin(2kπ) k = 0, 1, 2, 3, 4.
2. [A1] θ = 2kπ/5 k = 0, 1, 2, 3, 4.
3. [M1] Rewrites ((z+1)/z)⁵ = 1 ⇒ z+1 = z cis(2kπ/5)
4. [A1] and factorises. ⇒ z(1-cis(2kπ/5)) = -1
5. [M1A1] Isolates z. z = -1/(1-cis(2kπ/5))
6. [A1] Obtains purely imaginary denominator -(cos(kπ/5)+isin(kπ/5)) / (-2i sin(kπ/5))
7. [A1] and obtains result. =-½{1+icot(kπ/5)} k=1,2,3,4. (AG)
8. [B1] Observes that original equation is a quartic with real coefficients, so roots occur in conjugate pairs and k = 0 must be rejected.