| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a standard Further Maths FP1 loci question requiring sketching a circle and half-line, finding their intersection, and shading a region. While it involves multiple parts and requires understanding of modulus/argument geometry, these are routine techniques for FP1 students with no novel problem-solving required. The intersection point can be read directly from the sketch, making this slightly easier than average even for Further Maths. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| \(C_1\): circle centre \((-3, 4)\), radius \(5\) | B1 B1 |
| \(C_2\): ray from \((-3, 6)\) at angle \(\frac{\pi}{2}\) (vertical ray upward) | B1 B1 B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(z = -3 + 6\text{j}\) ... checking: \( | -3+3-4(6-6)\text{j} | \)... intersection at \(z = -3+6\text{j}\)... verify on circle: \( |
| Answer | Marks |
|---|---|
| Outside/on circle shaded, between rays \(\frac{\pi}{2} \leq \arg \leq \frac{3\pi}{4}\) | B1 B1 |
## Question 5:
**(i)**
$C_1$: circle centre $(-3, 4)$, radius $5$ | B1 B1 |
$C_2$: ray from $(-3, 6)$ at angle $\frac{\pi}{2}$ (vertical ray upward) | B1 B1 B1 |
**(ii)**
$z = -3 + 6\text{j}$ ... checking: $|-3+3-4(6-6)\text{j}|$... intersection at $z = -3+6\text{j}$... verify on circle: $|0+2\text{j}|=2\neq5$; intersection: $(-3, 9)$, so $z = -3+9\text{j}$ | B1 |
**(iii)**
Outside/on circle shaded, between rays $\frac{\pi}{2} \leq \arg \leq \frac{3\pi}{4}$ | B1 B1 |
---5 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z + 3 - 4 \mathrm { j } | = 5$ and arg $( z + 3 - 6 \mathrm { j } ) = \frac { 1 } { 2 } \pi$ respectively.\\
(i) Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Write down the complex number represented by the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.\\
(iii) Indicate, by shading on your sketch, the region satisfying
$$| z + 3 - 4 \mathrm { j } | \geqslant 5 \text { and } \frac { 1 } { 2 } \pi \leqslant \arg ( z + 3 - 6 \mathrm { j } ) \leqslant \frac { 3 } { 4 } \pi .$$
\hfill \mbox{\textit{OCR MEI FP1 2016 Q5 [8]}}