| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Basic roots of unity properties |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on roots of unity with standard techniques: writing fifth roots of unity, transforming an equation to find roots in exponential form, applying geometric series formula, and finding minimum modulus. All parts follow routine procedures with no novel insight required, making it slightly easier than average even for FM content. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\exp(2\pi ki/5), k = 0, 1, 2, 3, 4\) (AEF) | M1A1 | |
| [2] | ||
| M1 for 1 correct fifth root of unity; A1 for exactly 5 distinct, correct roots | ||
| (ii) \(z^5 = 32 \exp(-2\pi i/3)\) | M1 | |
| \(z_k = 2 \exp(-2\pi i/15 + 2\pi ki/5)\) | A1 | |
| Roots equally spaced on circle \( | z | = 2\); correctly placed |
| [4] | ||
| (iii) \(\sum_{k=0}^{4} (w/2)^k = \frac{[1-(w/2)^5]}{[1-w/2]}\) | M1 | |
| \(= \frac{1-(1/32)(-16-16\sqrt{3}i)}{[1-w/2]}\) | A1 | |
| \(= ... = (3 + \sqrt{3}i)/(2-w)\) (AG) | A1 | |
| [3] | ||
| (iv) Deduces from diagram in (ii) that minimum of \( | 2-w | \) occurs when \(w = 2e^{2\pi i/15}\) or \(2e^{-\frac{2\pi i}{15}}\) |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Evaluates 5 possible values of \( | 2-w | \); Identifies minimum of \( |
**(i)** $\exp(2\pi ki/5), k = 0, 1, 2, 3, 4$ (AEF) | M1A1 |
| [2] |
M1 for 1 correct fifth root of unity; A1 for exactly 5 distinct, correct roots |
**(ii)** $z^5 = 32 \exp(-2\pi i/3)$ | M1 |
$z_k = 2 \exp(-2\pi i/15 + 2\pi ki/5)$ | A1 |
Roots equally spaced on circle $|z| = 2$; correctly placed | M1A1 |
| [4] |
**(iii)** $\sum_{k=0}^{4} (w/2)^k = \frac{[1-(w/2)^5]}{[1-w/2]}$ | M1 |
$= \frac{1-(1/32)(-16-16\sqrt{3}i)}{[1-w/2]}$ | A1 |
$= ... = (3 + \sqrt{3}i)/(2-w)$ (AG) | A1 |
| [3] |
**(iv)** Deduces from diagram in (ii) that minimum of $|2-w|$ occurs when $w = 2e^{2\pi i/15}$ or $2e^{-\frac{2\pi i}{15}}$ | M1A1 |
| [2] |
**OR**
Evaluates 5 possible values of $|2-w|$; Identifies minimum of $|2-w|$ correctly | M1 A1 |9 (i) Write down the five fifth roots of unity.\\
(ii) Hence find all the roots of the equation
$$z ^ { 5 } + 16 + ( 16 \sqrt { } 3 ) i = 0$$
giving answers in the form $r \mathrm { e } ^ { \mathrm { i } q \pi }$, where $r > 0$ and $q$ is a rational number. Show these roots on an Argand diagram.
Let $w$ be a root of the equation in part (ii).\\
(iii) Show that
$$\sum _ { k = 0 } ^ { 4 } \left( \frac { w } { 2 } \right) ^ { k } = \frac { 3 + \mathrm { i } \sqrt { } 3 } { 2 - w }$$
(iv) Identify the root for which $| 2 - w |$ is least.
\hfill \mbox{\textit{CAIE FP1 2010 Q9 [11]}}