| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 12 |
| 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 straightforward Further Maths question testing standard loci concepts. Part (a) requires reading circles from a diagram and writing their equations/inequalities (routine recall). Part (b)(i) is a standard argument inequality sketch, and (b)(ii) is a simple calculation to check if a point satisfies the inequality. While it's Further Maths content, these are textbook exercises requiring no novel insight or complex multi-step reasoning. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\ | z-(2+6j)\ | = 4\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\ | z-(2+6j)\ | < 4\) and \(\ |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Any straight line through \(2+j\) | B1 | |
| Both correct half lines | B1 | |
| Region between their two half lines indicated | B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(43+47j-(2+j) = 41+46j\) | M1 | Attempt to calculate argument, or other valid method such as comparison with \(y=x-1\) |
| \(\arg(41+46j) = \arctan\left(\frac{46}{41}\right) = 0.843\) | A1 | Correct |
| \(\frac{\pi}{4} < 0.843 < \frac{3\pi}{4}\), so \(43+47j\) does fall within the region | E1 [3] | Justified |
# Question 8:
**Part (a)(i):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\|z-(2+6j)\| = 4$ | B1, B1, B1 **[3]** | $2+6j$ seen; expression in $z = 4$; correct equation |
**Part (a)(ii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\|z-(2+6j)\| < 4$ and $\|z-(3+7j)\| > 1$ | B1, B1, B1 **[3]** | Allow errors in inequality signs; both inequalities correct |
**Part (b)(i):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Any straight line through $2+j$ | B1 | |
| Both correct half lines | B1 | |
| Region between their two half lines indicated | B1 **[3]** | |
**Part (b)(ii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $43+47j-(2+j) = 41+46j$ | M1 | Attempt to calculate argument, or other valid method such as comparison with $y=x-1$ |
| $\arg(41+46j) = \arctan\left(\frac{46}{41}\right) = 0.843$ | A1 | Correct |
| $\frac{\pi}{4} < 0.843 < \frac{3\pi}{4}$, so $43+47j$ does fall within the region | E1 **[3]** | Justified |
---8
\begin{enumerate}[label=(\alph*)]
\item Fig. 8 shows an Argand diagram.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{df275813-15de-496f-9742-427a9e03f431-3_892_899_1048_664}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Write down the equation of the locus represented by the circumference of circle B.
\item Write down the two inequalities that define the shaded region between, but not including, circles A and B.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Draw an Argand diagram to show the region where
$$\frac { \pi } { 4 } < \arg ( z - ( 2 + \mathrm { j } ) ) < \frac { 3 \pi } { 4 }$$
\item Determine whether the point $43 + 47 \mathrm { j }$ lies within this region.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2010 Q8 [12]}}