| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Roots of unity applications |
| Difficulty | Challenging +1.2 This is a structured Further Maths question with clear scaffolding. Part (i) is a standard complex number identity proof using Euler's formula. Part (ii) applies this result to cube roots of unity—a direct application once you recognize z³=1. Part (iii) requires algebraic manipulation to clear denominators and solve a sextic, but the structure heavily guides the solution. While it requires multiple techniques and is Further Maths content, the extensive scaffolding and standard methods make it moderately above average difficulty rather than genuinely challenging. |
| Spec | 4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02r nth roots: of complex numbers |
(i) Show that if $z = \mathrm { e } ^ { \mathrm { i } \theta }$ and $z \neq - 1$ then
$$\frac { z - 1 } { z + 1 } = \mathrm { i } \tan \frac { 1 } { 2 } \theta$$
(ii) Hence, or otherwise, show that if $z$ is a cube root of unity then
$$\frac { z ^ { 3 } - 1 } { z ^ { 3 } + 1 } + \frac { z ^ { 2 } - 1 } { z ^ { 2 } + 1 } + \frac { z - 1 } { z + 1 } = 0$$
(iii) Hence write down three roots of the equation
$$\left( z ^ { 3 } - 1 \right) \left( z ^ { 2 } + 1 \right) ( z + 1 ) + \left( z ^ { 2 } - 1 \right) \left( z ^ { 3 } + 1 \right) ( z + 1 ) + ( z - 1 ) \left( z ^ { 3 } + 1 \right) \left( z ^ { 2 } + 1 \right) = 0$$
and find the other three roots. Give your answers in an exact form.\\
\hfill \mbox{\textit{CAIE FP1 2018 Q11 EITHER}}