| Exam Board | Edexcel |
|---|---|
| Module | FP2 AS (Further Pure 2 AS) |
| Year | 2022 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question requiring students to interpret and sketch the intersection of an argument inequality (forming a wedge) with vertical line constraints. While the individual components are standard FP2 material, correctly handling the intersection, determining which boundaries are included/excluded, and accurately representing this with solid/dashed lines requires careful geometric reasoning and attention to detail—making it moderately challenging but within expected FP2 scope. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02p Set notation: for loci |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| Sector drawn or identified, spanning from \(\pm 2\) or \(\pm 2i\) | M1 | Draws or identifies a sector, or relevant part thereof, spanning from either \(\pm 2\) or \(\pm 2i\) |
| Strip about either axis drawn or identified, with lines either imaginary or real axis | M1 | Either a vertical or horizontal strip drawn or identified |
| Correct strip and sector — sector starting from \(-2\) on negative real axis, with vertical strip about imaginary axis approximately halfway on real axis between start of sector and \(O\); sector at \(\pm 45°\) | A1 | Strip must be roughly evenly spaced either side of imaginary axis; sector roughly symmetric in real axis (allow small tolerance, but A0 if clearly not symmetric) |
| Fully correct — inside strip and sector shaded, with correct boundary lines | A1 | Correct boundary lines required for this mark |
| (4) | (4 marks) |
## Question 1:
**Locus/Region on Argand Diagram**
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| Sector drawn or identified, spanning from $\pm 2$ or $\pm 2i$ | **M1** | Draws or identifies a sector, or relevant part thereof, spanning from either $\pm 2$ or $\pm 2i$ |
| Strip about either axis drawn or identified, with lines either imaginary or real axis | **M1** | Either a vertical or horizontal strip drawn or identified |
| Correct strip and sector — sector starting from $-2$ on negative real axis, with vertical strip about imaginary axis approximately halfway on real axis between start of sector and $O$; sector at $\pm 45°$ | **A1** | Strip must be roughly evenly spaced either side of imaginary axis; sector roughly symmetric in real axis (allow small tolerance, but A0 if clearly not symmetric) |
| Fully correct — inside strip and sector shaded, with correct boundary lines | **A1** | Correct boundary lines required for this mark |
| | **(4)** | **(4 marks)** |\begin{enumerate}
\item Sketch on an Argand diagram the region defined by
\end{enumerate}
$$z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg ( z + 2 ) < \frac { \pi } { 4 } \cap \{ z \in \mathbb { C } : - 1 < \operatorname { Re } ( z ) \leqslant 1 \}$$
On your sketch
\begin{itemize}
\item shade the part of the diagram that is included in the region
\item use solid lines to show the parts of the boundary that are included in the region
\item use dashed lines to show the parts of the boundary that are not included in the region
\end{itemize}
\hfill \mbox{\textit{Edexcel FP2 AS 2022 Q1 [4]}}