| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| 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 loci question requiring students to sketch three familiar geometric objects: a half-line from arg condition, a circle from modulus equation, and a shaded region combining inequalities. While it requires understanding of complex number geometry, these are textbook loci types with straightforward interpretation and no novel problem-solving required. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Accept un-numbered evenly spaced marks on axes to show scale | B1 | Line at acute angle, all or part in Im \(z > 0\) |
| B1 | Half line from \(-1-j\) through 0 [don't penalise if point \(-1-j\) is included]. Allow near miss to 0 if \(\pi/4\) marked | |
| [2] | SC correct diagram, no annotations seen B1 B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| B1 | Circle centre \(1 + 2j\) | |
| B1 | Radius 2. Must touch real axis | |
| [2] | SC correct diagram, no annotations seen B1 B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Shaded region as shown in diagram | B1 | The shaded region must be outside their circle and have a border with the circumference |
| B1 | Fully correct | |
| [2] | SC correct diagram, no annotations seen allow B1 B1 |
## Question 4:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Accept un-numbered evenly spaced marks on axes to show scale | B1 | Line at acute angle, all or part in Im $z > 0$ |
| | B1 | Half line from $-1-j$ through 0 [don't penalise if point $-1-j$ is included]. Allow near miss to 0 if $\pi/4$ marked |
| | **[2]** | SC correct diagram, no annotations seen B1 B0 |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| | B1 | Circle centre $1 + 2j$ |
| | B1 | Radius 2. Must touch real axis |
| | **[2]** | SC correct diagram, no annotations seen B1 B0 |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Shaded region as shown in diagram | B1 | The shaded region must be outside their circle and have a border with the circumference |
| | B1 | Fully correct |
| | **[2]** | SC correct diagram, no annotations seen allow B1 B1 |
---4 Indicate, on a single Argand diagram\\
(i) the set of points for which $\arg ( z - ( - 1 - \mathrm { j } ) ) = \frac { \pi } { 4 }$,\\
(ii) the set of points for which $| z - ( 1 + 2 j ) | = 2$,\\
(iii) the set of points for which $| z - ( 1 + 2 j ) | \geqslant 2$ and $0 \leqslant \arg ( z - ( - 1 - j ) ) \leqslant \frac { \pi } { 4 }$.
\hfill \mbox{\textit{OCR MEI FP1 2015 Q4 [6]}}