| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Standard +0.3 This is a structured Further Maths question requiring standard techniques: sketching complex numbers in polar form, recognizing an equilateral triangle, converting to polar form using geometry, and applying De Moivre's theorem. While it involves multiple steps, each part follows directly from the previous with clear guidance, making it slightly easier than average for FP3 material. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation4.02q De Moivre's theorem: multiple angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch | B1 | Must have axes, \(A\) shown 3 across and either scale (or co-ordinates) with \(B\) in rough position, or angle and distance on argand diagram. No inconsistencies |
| \(OA = | 3 | = 3,\ OB = \left |
| Hence \(\triangle OAB\) equilateral | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3e^{-\frac{1}{3}\pi i}\) | M1A1 | Or \(3e^{\frac{5}{3}\pi i}\), isw. M1 for evidence they are considering \(BA\), or for \(\frac{3}{2} - \frac{3}{2}\sqrt{3}\,i\). For full marks can use CiS form, or clear polar co-ordinates, in radians. Not C-iS |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(3 - 3e^{\frac{1}{3}\pi i}\right)^5 = 3^5 e^{-\frac{5}{3}\pi i}\) | M1 | For \(\text{mod}^5\) and \(\arg \times 5\). "Hence" so must use their \(3e^{-\frac{1}{3}\pi i}\) |
| \(= 243\!\left(\cos\tfrac{5}{3}\pi - i\sin\tfrac{5}{3}\pi\right)\) | A1ft | aef. Condone use of "121.5" |
| \(= \frac{243}{2} + \frac{243}{2}\sqrt{3}\,i\) | B1 |
# Question 4:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch | B1 | Must have axes, $A$ shown 3 across and either scale (or co-ordinates) with $B$ in rough position, or angle and distance on argand diagram. No inconsistencies |
| $OA = |3| = 3,\ OB = \left|3e^{\frac{1}{3}\pi i}\right| = 3$ and $\angle BOA = \frac{1}{3}\pi$ | M1 | Can be seen on diagram. Alt. Attempts $AB$ or second angle |
| Hence $\triangle OAB$ equilateral | A1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3e^{-\frac{1}{3}\pi i}$ | M1A1 | Or $3e^{\frac{5}{3}\pi i}$, isw. M1 for evidence they are considering $BA$, or for $\frac{3}{2} - \frac{3}{2}\sqrt{3}\,i$. For full marks can use CiS form, or clear polar co-ordinates, in radians. Not C-iS |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(3 - 3e^{\frac{1}{3}\pi i}\right)^5 = 3^5 e^{-\frac{5}{3}\pi i}$ | M1 | For $\text{mod}^5$ and $\arg \times 5$. "Hence" so must use their $3e^{-\frac{1}{3}\pi i}$ |
| $= 243\!\left(\cos\tfrac{5}{3}\pi - i\sin\tfrac{5}{3}\pi\right)$ | A1ft | aef. Condone use of "121.5" |
| $= \frac{243}{2} + \frac{243}{2}\sqrt{3}\,i$ | B1 | |4 The complex numbers 0,3 and $3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }$ are represented in an Argand diagram by the points $O , A$ and $B$ respectively.\\
(i) Sketch the triangle $O A B$ and show that it is equilateral.\\
(ii) Hence express $3 - 3 e ^ { \frac { 1 } { 3 } \pi i }$ in polar form.\\
(iii) Hence find $\left( 3 - 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } } \right) ^ { 5 }$, giving your answer in the form $a + b \sqrt { 3 } \mathrm { i }$ where $a$ and $b$ are rational numbers.
\hfill \mbox{\textit{OCR FP3 2013 Q4 [8]}}