| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Moderate -0.3 This is a straightforward FP1 loci question requiring standard techniques: recognizing C₁ as a circle (center 2i, radius 2) and C₂ as a perpendicular bisector (the line y = -x). Finding intersections involves substituting the line equation into the circle equation, which is routine. While it requires multiple steps and understanding of complex number geometry, it's a textbook exercise with no novel insight needed, making it slightly easier than average. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Circle | B1 | Sketch(s) showing correct features, each mark independent |
| Centre \((0, 2)\) | B1 | |
| Radius \(2\) | B1 | |
| Straight line | B1 | |
| Through origin with positive slope | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(0\) or \(0 + 0i\) and \(2 + 2i\) | B1 ftB1 ft | Obtain intersections as complex numbers |
### (i)
Circle | B1 | Sketch(s) showing correct features, each mark independent
Centre $(0, 2)$ | B1 |
Radius $2$ | B1 |
Straight line | B1 |
Through origin with positive slope | B1 |
**Subtotal: 5 marks**
### (ii)
$0$ or $0 + 0i$ and $2 + 2i$ | B1 ftB1 ft | Obtain intersections as complex numbers
**Subtotal: 2 marks**
**Total: 7 marks**The loci $C_1$ and $C_2$ are given by
$$|z - 2\text{i}| = 2 \quad \text{and} \quad |z + 1| = |z + \text{i}|$$
respectively.
\begin{enumerate}[label=(\roman*)]
\item Sketch, on a single Argand diagram, the loci $C_1$ and $C_2$. [5]
\item Hence write down the complex numbers represented by the points of intersection of $C_1$ and $C_2$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2005 Q6 [7]}}