| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| 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 FP1 loci question requiring sketching an argument line (half-line from 2+2i at 45°) and a perpendicular bisector (vertical line Re(z)=5), then shading a region defined by inequalities. While it involves multiple steps, these are routine techniques for Further Maths students with no novel problem-solving required. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines4.02p Set notation: for loci |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Half line | B1 | Not line segment e.g \((0,0)\) to \((2,2)\); Must be half line |
| Starting at \((2,2)\) with +ve slope upwards | B1 | |
| Vertical line | B1 | |
| Clearly \(x = 5\) (must be vertical) | B1 | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Shade below sloping line and above horizontal through their \((2,2)\) | B1 | Could be for a line segment; could be earned if \(C_2\) horizontal |
| To right of their vertical line | B1 | |
| Completely correct diagram | B1 | 6/6 must be earned so far |
| [3] |
# Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Half line | B1 | Not line segment e.g $(0,0)$ to $(2,2)$; Must be half line |
| Starting at $(2,2)$ with +ve slope upwards | B1 | |
| Vertical line | B1 | |
| Clearly $x = 5$ (must be vertical) | B1 | |
| **[4]** | | |
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# Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Shade below sloping line and above horizontal through their $(2,2)$ | B1 | Could be for a line segment; could be earned if $C_2$ horizontal |
| To right of their vertical line | B1 | |
| Completely correct diagram | B1 | 6/6 must be earned so far |
| **[3]** | | |
---7 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $\arg ( z - 2 - 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi$ and $| z | = | z - 10 |$ respectively.\\
(i) Sketch on a single Argand diagram the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Indicate, by shading, the region of the Argand diagram for which
$$0 \leqslant \arg ( z - 2 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 4 } \pi \text { and } | z | \geqslant | z - 10 | .$$
\hfill \mbox{\textit{OCR FP1 2014 Q7 [7]}}