| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.8 This question requires converting to modulus-argument form (routine), sketching a locus region defined by argument and modulus inequalities (standard FP1), but then demands geometric optimization to find the maximum distance from w to points in the region. The optimization step requires spatial reasoning about which point in the constrained region maximizes |z-w|, going beyond routine loci sketching to problem-solving with geometric insight. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | w | = \sqrt{2^2 + (2\sqrt{3})^2} = 4\) |
| \(\arg w = \arctan\dfrac{2\sqrt{3}}{2} = \dfrac{\pi}{3}\) | M1 | |
| \(w = 4\left(\cos\dfrac{\pi}{3} + j\sin\dfrac{\pi}{3}\right)\) | A1 | Accept \(\left(4, \dfrac{\pi}{3}\right)\), 1.05 rad, 60° in place of \(\dfrac{\pi}{3}\), or \(4e^{j\frac{\pi}{3}}\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Circle or arc of circle, centre the origin | B1 | |
| Radius 4 | B1 | |
| Half line from origin \(\dfrac{\pi}{4} < \text{angle} < \dfrac{\pi}{2}\) with positive real axis, or acute angle labelled as \(\pi/3\) | B1 | |
| Use of negative Im axis clearly indicated | B1 | |
| Correct region indicated (dependent on first 4 B marks). Ignore placing of \(w\) | B1 | |
| \(w\) at intersection of \(\dfrac{\pi}{3}\) line and circle | B1 | Dependent on 1st 3 B marks |
| Maximum \( | z - w | = \sqrt{2^2 + (4 + 2\sqrt{3})^2} = 7.73\) (3 s.f.) or \(2 \times 4\cos15° = 2\sqrt{6} + 2\sqrt{2}\) |
| M1 | Valid attempt to calculate maximum \( | z - w |
| A1 | Allow \(\sqrt{32 + 16\sqrt{3}}\) (accept 2 s.f. or better) | |
| [9] |
## Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|w| = \sqrt{2^2 + (2\sqrt{3})^2} = 4$ | B1 | |
| $\arg w = \arctan\dfrac{2\sqrt{3}}{2} = \dfrac{\pi}{3}$ | M1 | |
| $w = 4\left(\cos\dfrac{\pi}{3} + j\sin\dfrac{\pi}{3}\right)$ | A1 | Accept $\left(4, \dfrac{\pi}{3}\right)$, 1.05 rad, 60° in place of $\dfrac{\pi}{3}$, or $4e^{j\frac{\pi}{3}}$ |
| | **[3]** | |
---
## Question 8(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Circle or arc of circle, centre the origin | B1 | |
| Radius 4 | B1 | |
| Half line from origin $\dfrac{\pi}{4} < \text{angle} < \dfrac{\pi}{2}$ with positive real axis, or acute angle labelled as $\pi/3$ | B1 | |
| Use of negative Im axis clearly indicated | B1 | |
| Correct region indicated (dependent on first 4 B marks). Ignore placing of $w$ | B1 | |
| $w$ at intersection of $\dfrac{\pi}{3}$ line and circle | B1 | Dependent on 1st 3 B marks |
| Maximum $|z - w| = \sqrt{2^2 + (4 + 2\sqrt{3})^2} = 7.73$ (3 s.f.) or $2 \times 4\cos15° = 2\sqrt{6} + 2\sqrt{2}$ | B1 | Maximum $|z-w|$ indicated by chord on diagram or sight of $-4j - (2 + 2\sqrt{3}j)$ |
| | M1 | Valid attempt to calculate maximum $|z - w|$ |
| | A1 | Allow $\sqrt{32 + 16\sqrt{3}}$ (accept 2 s.f. or better) |
| | **[9]** | |
---8 You are given the complex number $w = 2 + 2 \sqrt { 3 } \mathrm { j }$.\\
(i) Express $w$ in modulus-argument form.\\
(ii) Indicate on an Argand diagram the set of points, $z$, which satisfy both of the following inequalities.
$$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$
Mark $w$ on your Argand diagram and find the greatest value of $| z - w |$.
\hfill \mbox{\textit{OCR MEI FP1 2014 Q8 [12]}}