| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| 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 identification of a circle and perpendicular bisector, sketching them, finding intersections algebraically, and shading a region. While it involves multiple parts and some algebraic manipulation, these are routine techniques for Further Maths students with no novel insight required—slightly easier than average A-level difficulty overall. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| B1B1 | Circle, centre \((3, 4)\) |
| B1ft | Touching \(x\)-axis, ft for \((3, -4)\) etc as centre |
| B1ft | Crossing \(y\)-axis twice |
| B1B1 | Horizontal line, \(y\) intercept \(4\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(-1 + 4i \quad 7 + 4i\) | B1B1 | State correct answers |
| Answer | Marks |
|---|---|
| B1ft | Inside circle or above line |
| B1 | Completely correct diagram |
## Question 7:
### Part (i)
| B1B1 | Circle, centre $(3, 4)$
| B1ft | Touching $x$-axis, ft for $(3, -4)$ etc as centre
| B1ft | Crossing $y$-axis twice
| B1B1 | Horizontal line, $y$ intercept $4$
**[6]**
### Part (ii)
$-1 + 4i \quad 7 + 4i$ | B1B1 | State correct answers
**[2]**
### Part (iii)
| B1ft | Inside circle or above line
| B1 | Completely correct diagram
**[2]**
---7 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z - 3 - 4 \mathrm { i } | = 4$ and $| z | = | z - 8 \mathrm { i } |$ respectively.\\
(i) Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Hence find the complex numbers represented by the points of intersection of $C _ { 1 }$ and $C _ { 2 }$.\\
(iii) Indicate, by shading, the region of the Argand diagram for which
$$| z - 3 - 4 i | \leqslant 4 \text { and } | z | \geqslant | z - 8 i | .$$
\hfill \mbox{\textit{OCR FP1 2012 Q7 [10]}}