| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Standard +0.3 This is a standard Further Maths FP2 question testing routine application of modulus/argument calculation, de Moivre's theorem for powers, and finding complex roots. While it requires multiple steps and is from Further Maths (inherently harder), the techniques are direct applications of standard methods with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \( | z | = 4\) |
| \(\arg z = \arctan\!\left(\frac{-2\sqrt{3}}{2}\right) = \arctan(-\sqrt{3}) = \frac{2\pi}{3}\) or \(120°\) | M1, A1 | M1: attempt arg using arctan, either way up, must include minus sign or other consideration of quadrant. (\(\arg = \frac{\pi}{3}\) scores M0). A1: \(\frac{2\pi}{3}\) or \(120°\); correct answer only seen, award M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z^6 = \left(4\!\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)\right)^6 = 4^6(\cos 4\pi + i\sin 4\pi)\) or \(z^6=\left(4e^{i\frac{2\pi}{3}}\right)^6\) | M1 | Apply de Moivre |
| \(= 4096\) or \(4^6\) or \(2^{12}\) | A1 cso | Must have been obtained with correct argument for \(z\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z^{\frac{3}{4}} = 4^{\frac{3}{4}}\!\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)^{\frac{3}{4}} = 4^{\frac{3}{4}}\!\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)\) | ||
| \(w = i2\sqrt{2}\) or any other correct root | B1 | For \(w=i2\sqrt{2}\) or any single correct root (0 or 0i may be included) in any form including polar |
| \(4^{\frac{3}{4}}\!\left(\cos\!\left(\frac{2\pi}{3}+2n\pi\right)+i\sin\!\left(\frac{2\pi}{3}+2n\pi\right)\right)^{\frac{3}{4}}\) | M1 | Applying de Moivre and using correct method to attempt 2 or 3 further roots |
| \(n=1:\ w=2\sqrt{2}\) oe | ||
| \(n=2:\ w=-i2\sqrt{2}\) oe | A1, A1 | For the other roots (3 correct scores A1A1; 2 correct scores A1). Accept e.g. \(2\sqrt{2},\ \sqrt{8},\ 2.83,\ 64^{\frac{1}{4}},\ 4^{\frac{3}{4}},\ 4096^{\frac{1}{8}}\). Decimals must be 3 sf min. |
| \(n=3:\ w=-2\sqrt{2}\) oe |
# Question 2:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $|z| = 4$ | B1 | Correct modulus seen. Must be 4 |
| $\arg z = \arctan\!\left(\frac{-2\sqrt{3}}{2}\right) = \arctan(-\sqrt{3}) = \frac{2\pi}{3}$ or $120°$ | M1, A1 | M1: attempt arg using arctan, either way up, must include minus sign or other consideration of quadrant. ($\arg = \frac{\pi}{3}$ scores M0). A1: $\frac{2\pi}{3}$ or $120°$; correct answer only seen, award M1A1 |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z^6 = \left(4\!\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)\right)^6 = 4^6(\cos 4\pi + i\sin 4\pi)$ or $z^6=\left(4e^{i\frac{2\pi}{3}}\right)^6$ | M1 | Apply de Moivre |
| $= 4096$ or $4^6$ or $2^{12}$ | A1 cso | Must have been obtained with correct argument for $z$ |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z^{\frac{3}{4}} = 4^{\frac{3}{4}}\!\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)^{\frac{3}{4}} = 4^{\frac{3}{4}}\!\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)$ | | |
| $w = i2\sqrt{2}$ or any other correct root | B1 | For $w=i2\sqrt{2}$ or any single correct root (0 or 0i may be included) in any form including polar |
| $4^{\frac{3}{4}}\!\left(\cos\!\left(\frac{2\pi}{3}+2n\pi\right)+i\sin\!\left(\frac{2\pi}{3}+2n\pi\right)\right)^{\frac{3}{4}}$ | M1 | Applying de Moivre and using correct method to attempt 2 or 3 further roots |
| $n=1:\ w=2\sqrt{2}$ oe | | |
| $n=2:\ w=-i2\sqrt{2}$ oe | A1, A1 | For the other roots (3 correct scores A1A1; 2 correct scores A1). Accept e.g. $2\sqrt{2},\ \sqrt{8},\ 2.83,\ 64^{\frac{1}{4}},\ 4^{\frac{3}{4}},\ 4096^{\frac{1}{8}}$. Decimals must be 3 sf min. |
| $n=3:\ w=-2\sqrt{2}$ oe | | |
---2.
$$z = - 2 + ( 2 \sqrt { 3 } ) \mathrm { i }$$
\begin{enumerate}[label=(\alph*)]
\item Find the modulus and the argument of $z$.
Using de Moivre's theorem,
\item find $z ^ { 6 }$, simplifying your answer,
\item find the values of $w$ such that $w ^ { 4 } = z ^ { 3 }$, giving your answers in the form $a + \mathrm { i } b$ where $a , b \in \mathbb { R }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2015 Q2 [9]}}