| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Basic roots of unity properties |
| Difficulty | Moderate -0.3 This is a straightforward application of roots of unity with a standard factorization technique. Part (i) is direct recall, and part (ii) uses the well-known identity that z^10 + z^5 + 1 factors via multiplying by (z^5 - 1), leading to primitive 15th roots excluding certain cases. While it requires some algebraic manipulation, this is a standard Further Maths exercise with no novel insight needed. |
| Spec | 4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| \(\exp\left(i\frac{2\pi k}{5}\right)\), \(k = 0, \pm1, \pm2\) | B2 | B1 for 1 correct fifth root; B1 for all 5 distinct correct roots (AEF); SCB1 if only arguments given |
| Answer | Marks | Guidance |
|---|---|---|
| \(z^5 = -\frac{1}{2} \pm i\frac{\sqrt{3}}{2}\) | M1 | Correctly solves quadratic in \(z^5\) |
| \(z^5 = \exp\left(i2\pi\left(\frac{1}{3}+k\right)\right)\) or \(\exp\left(i2\pi\left(-\frac{1}{3}+k\right)\right)\) | M1 A1 | Writes in polar/exponential form and adds multiples of \(2\pi\) |
| \(\exp\left(\pm i\frac{2\pi}{15}\right)\), \(\exp\left(\pm i\frac{4\pi}{15}\right)\), \(\exp\left(\pm i\frac{8\pi}{15}\right)\), \(\exp\left(\pm i\frac{2\pi}{3}\right)\), \(\exp\left(\pm i\frac{14\pi}{15}\right)\) | A2 | A1 for 5 distinct correct roots; A1 for exactly 10 distinct correct roots. Allow \(\theta = \frac{16\pi}{15}, \frac{4\pi}{3}, \frac{22\pi}{15}, \frac{26\pi}{15}, \frac{28\pi}{15}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(z^5 + z^{-5} = -1\) | M1 | Divides through by \(z^5\) |
| \(2\cos 5\theta = -1\) | M1 A1 | Applies de Moivre's theorem |
| Same 10 roots as above | A2 | Same criteria as above |
## Question 3(i):
| $\exp\left(i\frac{2\pi k}{5}\right)$, $k = 0, \pm1, \pm2$ | B2 | B1 for 1 correct fifth root; B1 for all 5 distinct correct roots (AEF); SCB1 if only arguments given |
## Question 3(ii):
| $z^5 = -\frac{1}{2} \pm i\frac{\sqrt{3}}{2}$ | M1 | Correctly solves quadratic in $z^5$ |
| $z^5 = \exp\left(i2\pi\left(\frac{1}{3}+k\right)\right)$ or $\exp\left(i2\pi\left(-\frac{1}{3}+k\right)\right)$ | M1 A1 | Writes in polar/exponential form and adds multiples of $2\pi$ |
| $\exp\left(\pm i\frac{2\pi}{15}\right)$, $\exp\left(\pm i\frac{4\pi}{15}\right)$, $\exp\left(\pm i\frac{8\pi}{15}\right)$, $\exp\left(\pm i\frac{2\pi}{3}\right)$, $\exp\left(\pm i\frac{14\pi}{15}\right)$ | A2 | A1 for 5 distinct correct roots; A1 for exactly 10 distinct correct roots. Allow $\theta = \frac{16\pi}{15}, \frac{4\pi}{3}, \frac{22\pi}{15}, \frac{26\pi}{15}, \frac{28\pi}{15}$ |
**Alternative method for 3(ii):**
| $z^5 + z^{-5} = -1$ | M1 | Divides through by $z^5$ |
| $2\cos 5\theta = -1$ | M1 A1 | Applies de Moivre's theorem |
| Same 10 roots as above | A2 | Same criteria as above |
---3 (i) Write down the fifth roots of unity.\\
(ii) Find all the roots of the equation
$$z ^ { 10 } + z ^ { 5 } + 1 = 0$$
giving each root in the form $\mathrm { e } ^ { \mathrm { i } \theta }$.\\
\hfill \mbox{\textit{CAIE FP1 2019 Q3 [7]}}