| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 8 |
| 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 students to sketch two basic loci (perpendicular bisector and half-line from a point) and shade a region satisfying inequalities. While it involves multiple steps, each component uses routine techniques taught in Further Maths, making it slightly easier than average for an A-level question overall but typical for FP1. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (a) | B1* | Vertical line |
| depB1 2 | Clearly through \((4, 0)\) | |
| (b) | B1 | Sloping line with +ve slope |
| B1 | Through \((0, -2)\) | |
| B1ft 3 | Half line starting on \(y\)-axis at \(45°\) shown convincingly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| B1ft | Shaded to left of their (i)(a) | |
| B1ft | Shaded below their (i)(b); must be +ve slope | |
| B1ft 3 | Shaded above horizontal through their \((0,-2)\); NB these 3 marks are independent, but 3/3 only for fully correct answer |
## Question 6:
**Part (i)(a) and (b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| (a) | B1* | Vertical line |
| | depB1 **2** | Clearly through $(4, 0)$ |
| (b) | B1 | Sloping line with +ve slope |
| | B1 | Through $(0, -2)$ |
| | B1ft **3** | Half line starting on $y$-axis at $45°$ shown convincingly |
**Part (ii)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| | B1ft | Shaded to left of their (i)(a) |
| | B1ft | Shaded below their (i)(b); must be +ve slope |
| | B1ft **3** | Shaded above horizontal through their $(0,-2)$; NB these 3 marks are independent, but 3/3 only for fully correct answer |
---6 (i) Sketch on a single Argand diagram the loci given by
\begin{enumerate}[label=(\alph*)]
\item $\quad | z | = | z - 8 |$,
\item $\quad \arg ( z + 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi$.\\
(ii) Indicate by shading the region of the Argand diagram for which
$$| z | \leqslant | z - 8 | \quad \text { and } \quad 0 \leqslant \arg ( z + 2 i ) \leqslant \frac { 1 } { 4 } \pi$$
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2011 Q6 [8]}}