| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.3 This is a standard FP1 loci question requiring sketching a circle and half-line, finding their intersection using basic trigonometry, and shading a region defined by inequalities. All techniques are routine for Further Maths students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| B1 | Circle centre \((-2, 0)\) or circle centre \((2, 0)\) | |
| B1 | Touching \(y\)-axis at origin | |
| B1 | Half line with negative slope upwards | |
| B1 | Completely correct diagram | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(-2 - \sqrt{3} + i\) | B1ft | Correct real part and correct imaginary part of a complex number, ft for their half line from centre of their circle, allow decimals (\(-3.73\) or better) or trig expressions |
| B1ft | ||
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| B1ft | Shade inside their circle | |
| B1 | Completely correct diagram and shading | |
| [2] | S.C. allow last B1 for radius or complete line |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| | B1 | Circle centre $(-2, 0)$ or circle centre $(2, 0)$ |
| | B1 | Touching $y$-axis at origin |
| | B1 | Half line with negative slope upwards |
| | B1 | Completely correct diagram |
| **[4]** | | |
---
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-2 - \sqrt{3} + i$ | B1ft | Correct real part and correct imaginary part of a complex number, ft for their half line from centre of their circle, allow decimals ($-3.73$ or better) or trig expressions |
| | B1ft | |
| **[2]** | | |
---
## Question 5(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| | B1ft | Shade inside their circle |
| | B1 | Completely correct diagram and shading |
| **[2]** | **S.C. allow last B1 for radius or complete line** | |
---5 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z + 2 | = 2$ and $\arg ( z + 2 ) = \frac { 5 } { 6 } \pi$ respectively.\\
(i) Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Find the complex number represented by the intersection of $C _ { 1 }$ and $C _ { 2 }$.\\
(iii) Indicate, by shading, the region of the Argand diagram for which
$$| z + 2 | \leqslant 2 \text { and } \frac { 5 } { 6 } \pi \leqslant \arg ( z + 2 ) \leqslant \pi .$$
\hfill \mbox{\textit{OCR FP1 2015 Q5 [8]}}