| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Modulus-argument form conversion |
| Difficulty | Moderate -0.5 This is a straightforward Further Maths question testing standard conversions to modulus-argument form and basic complex number operations. While it's FP1 material (inherently harder than C1-C4), the question requires only routine application of formulas: finding r using Pythagoras, θ using standard angles, and applying |z₁z₂| = |z₁||z₂|. The exact values involve recognizable angles (π/3, 5π/6) making this easier than average even for Further Maths. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z_1 = \frac{1}{2}+i\frac{\sqrt{3}}{2}\), \(r=\sqrt{\frac{1}{4}+\frac{3}{4}}=1\), \(\tan\theta=\sqrt{3}\) so \(\theta=\frac{\pi}{3}\) | M1A1 | First M for use of Pythagoras, A1 for \(r=1\) and \(2\) |
| \(z_2 = -\sqrt{3}+i\), \(r=\sqrt{3+1}=2\), \(\tan\theta=\frac{-1}{\sqrt{3}}\) so \(\theta=\frac{5\pi}{6}\) | M1A1 | Second M for use of \(\tan\) or \(\tan^{-1}\), A1 for \(\theta=\frac{\pi}{3}\) and \(\frac{5\pi}{6}\) |
| (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\ | z_1 z_2\ | = \ |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Argand diagram with \(z_1\) plotted near \((1,0)\) in first quadrant and \(z_2\) plotted near \((-2,0)\) in second quadrant | M1, A1ft | M for either number plotted correctly; A for both plotted correctly on a single diagram |
| (2) [8] |
# Question 3:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z_1 = \frac{1}{2}+i\frac{\sqrt{3}}{2}$, $r=\sqrt{\frac{1}{4}+\frac{3}{4}}=1$, $\tan\theta=\sqrt{3}$ so $\theta=\frac{\pi}{3}$ | M1A1 | First M for use of Pythagoras, A1 for $r=1$ and $2$ |
| $z_2 = -\sqrt{3}+i$, $r=\sqrt{3+1}=2$, $\tan\theta=\frac{-1}{\sqrt{3}}$ so $\theta=\frac{5\pi}{6}$ | M1A1 | Second M for use of $\tan$ or $\tan^{-1}$, A1 for $\theta=\frac{\pi}{3}$ and $\frac{5\pi}{6}$ |
| | **(4)** | |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\|z_1 z_2\| = \|z_1\|\|z_2\| = 2$ | M1A1 | M for their $r_1 r_2$ |
| | **(2)** | |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Argand diagram with $z_1$ plotted near $(1,0)$ in first quadrant and $z_2$ plotted near $(-2,0)$ in second quadrant | M1, A1ft | M for either number plotted correctly; A for both plotted correctly on a single diagram |
| | **(2) [8]** | |3.
$$z _ { 1 } = \frac { 1 } { 2 } ( 1 + \mathrm { i } \sqrt { } 3 ) , z _ { 2 } = - \sqrt { } 3 + \mathrm { i }$$
\begin{enumerate}[label=(\alph*)]
\item Express $z _ { 1 }$ and $z _ { 2 }$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$ giving exact values of $r$ and $\theta$.\\
(4)
\item Find $\left| z _ { 1 } z _ { 2 } \right|$.
\item Show and label $z _ { 1 }$ and $z _ { 2 }$ on a single Argand diagram.\\
(2)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2013 Q3 [8]}}