| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Moderate -0.3 This is a standard FP1 loci question requiring sketching a circle and a half-line, then shading a region. While it involves complex numbers (an FP1 topic), the geometric interpretation is straightforward: |z|=2 is a circle of radius 2, and arg(z-3-i)=π is a horizontal ray from (3,1) pointing left. The shading requires understanding inequalities but involves no calculation or novel insight—just routine application of loci definitions. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| B1, B1 [2] | Circle; Centre O and radius 2 |
| Answer | Marks |
|---|---|
| B1, B1, B1 [3] | Horizontal line; (3, 1) on their line; ½ line to left i.e. horizontal |
| Answer | Marks |
|---|---|
| B1, B1 [2] | Shade only inside their circle or above their horizontal line; Completely correct diagram |
### (i)(a)
| B1, B1 [2] | Circle; Centre O and radius 2
### (i)(b)
| B1, B1, B1 [3] | Horizontal line; (3, 1) on their line; ½ line to left i.e. horizontal
### (ii)
| B1, B1 [2] | Shade only inside their circle or above their horizontal line; Completely correct diagram\begin{enumerate}[label=(\roman*)]
\item Sketch on a single Argand diagram the loci given by
\begin{enumerate}[label=(\alph*)]
\item $|z| = 2$, [2]
\item $\arg(z - 3 - i) = \pi$. [3]
\end{enumerate}
\item Indicate, by shading, the region of the Argand diagram for which
$$|z| < 2 \text{ and } 0 < \arg(z - 3 - i) < \pi.$$ [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2013 Q7 [7]}}