| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Convert to exponential/polar form |
| Difficulty | Standard +0.3 This is a straightforward complex numbers question requiring standard techniques: simplifying a complex fraction (multiply by conjugate), converting to modulus-argument form, and finding square roots using De Moivre's theorem. While it involves multiple steps, each is routine for A-level Further Maths students with no novel problem-solving required. Slightly above average difficulty due to the algebraic manipulation and multiple parts, but well within standard FM Pure content. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02h Square roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Either Multiply numerator and denominator by \(\sqrt{3} + i\) and use \(i^2 = -1\) | M1 | |
| Obtain correct numerator \(18 + 18\sqrt{3}i\) or correct denominator \(4\) | B1 | |
| Obtain \(\frac{9}{2} + \frac{9}{2}\sqrt{3}i\) or \(\left | 18 + 18\sqrt{3}i\right | /4\) |
| Obtain modulus or argument | M1 | |
| Obtain \(9e^{i\pi/3}\) | A1 | [5] |
| OR Obtain modulus and argument of numerator or denominator, or both | M1 | |
| Obtain moduli and argument \(18\) and \(\frac{1}{6}\pi\) or \(2\) and \(-\frac{1}{6}\pi\) | B1 | |
| Obtain moduli \(18\) and \(2\) or arguments \(\frac{1}{6}\pi\) and \(-\frac{1}{6}\pi\) (allow degrees) | B1 | |
| Obtain \(18e^{i\pi/6} + 2e^{-i\pi/6}\) or equivalent | A1 | |
| Divide moduli and subtract arguments | M1 | |
| Obtain \(9e^{i\pi/3}\) | A1 | [5] |
| (ii) State \(3e^{i\pi/6}\), following through their answer to part (i) | B1✓ | |
| State \(3e^{i(\pi+\pi/2)/6}\), following through their answer to part (i) | B1✓ | |
| Obtain \(3e^{5\pi i/6}\) | B1 | [3] |
(i) Either **Multiply numerator and denominator by $\sqrt{3} + i$ and use $i^2 = -1$** | M1 |
Obtain correct numerator $18 + 18\sqrt{3}i$ or correct denominator $4$ | B1 |
Obtain $\frac{9}{2} + \frac{9}{2}\sqrt{3}i$ or $\left|18 + 18\sqrt{3}i\right|/4$ | A1 |
Obtain modulus or argument | M1 |
Obtain $9e^{i\pi/3}$ | A1 | [5]
OR **Obtain modulus and argument of numerator or denominator, or both** | M1 |
Obtain moduli and argument $18$ and $\frac{1}{6}\pi$ or $2$ and $-\frac{1}{6}\pi$ | B1 |
Obtain moduli $18$ and $2$ or arguments $\frac{1}{6}\pi$ and $-\frac{1}{6}\pi$ (allow degrees) | B1 |
Obtain $18e^{i\pi/6} + 2e^{-i\pi/6}$ or equivalent | A1 |
Divide moduli and subtract arguments | M1 |
Obtain $9e^{i\pi/3}$ | A1 | [5]
(ii) State $3e^{i\pi/6}$, following through their answer to part (i) | B1✓ |
State $3e^{i(\pi+\pi/2)/6}$, following through their answer to part (i) | B1✓ |
Obtain $3e^{5\pi i/6}$ | B1 | [3]5 The complex number $z$ is defined by $z = \frac { 9 \sqrt { } 3 + 9 i } { \sqrt { } 3 - i }$. Find, showing all your working,\\
(i) an expression for $z$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$,\\
(ii) the two square roots of $z$, giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
\hfill \mbox{\textit{CAIE P3 2014 Q5 [8]}}