| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| 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 question testing basic loci notation in the Argand diagram. Parts (i)-(iii) require reading information from a diagram and writing standard forms (|z-a|=r, arg(z-a)=θ, converting to a+bj). Part (iv) combines these into inequalities, which is routine for FP1. No problem-solving or novel insight required—pure recall and application of standard notation. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \( | z-(4+2\text{j}) | = 2\) |
| B1 | \(z-(4+2\text{j})\) or \(z-4-2\text{j}\) | |
| B1 [3] | All correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\arg(z-(4+2\text{j})) = 0\) | B1 | Equation involving the argument of a complex variable |
| B1 | Argument \(= 0\) | |
| B1 [3] | All correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = 4 - 2\cos\frac{\pi}{4} = 4-\sqrt{2}\) | M1 | Valid attempt to use trigonometry involving \(\frac{\pi}{4}\), or coordinate geometry |
| \(b = 2 + 2\sin\frac{\pi}{4} = 2+\sqrt{2}\) | ||
| \(P = 4-\sqrt{2} + (2+\sqrt{2})\text{j}\) | A2 [3] | 1 mark each for \(a\) and \(b\); s.c. A1 only for \(a=2.59\), \(b=3.41\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{3}{4}\pi > \arg(z-(4+2\text{j})) > 0\) | B1 | \(\arg(z-(4+2\text{j})) > 0\) |
| and \( | z-(4+2\text{j}) | < 2\) |
| B1 [3] | \( | z-(4+2\text{j}) |
# Question 8(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $|z-(4+2\text{j})| = 2$ | B1 | Radius $= 2$ |
| | B1 | $z-(4+2\text{j})$ or $z-4-2\text{j}$ |
| | B1 **[3]** | All correct |
---
# Question 8(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\arg(z-(4+2\text{j})) = 0$ | B1 | Equation involving the argument of a complex variable |
| | B1 | Argument $= 0$ |
| | B1 **[3]** | All correct |
---
# Question 8(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 4 - 2\cos\frac{\pi}{4} = 4-\sqrt{2}$ | M1 | Valid attempt to use trigonometry involving $\frac{\pi}{4}$, or coordinate geometry |
| $b = 2 + 2\sin\frac{\pi}{4} = 2+\sqrt{2}$ | | |
| $P = 4-\sqrt{2} + (2+\sqrt{2})\text{j}$ | A2 **[3]** | 1 mark each for $a$ and $b$; s.c. A1 only for $a=2.59$, $b=3.41$ |
---
# Question 8(iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{3}{4}\pi > \arg(z-(4+2\text{j})) > 0$ | B1 | $\arg(z-(4+2\text{j})) > 0$ |
| and $|z-(4+2\text{j})| < 2$ | B1 | $\arg(z-(4+2\text{j})) < \frac{3}{4}\pi$ |
| | B1 **[3]** | $|z-(4+2\text{j})| < 2$; Deduct one mark if only error is use of inclusive inequalities |
---8 Fig. 8 shows an Argand diagram.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-3_421_586_1105_778}
\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 perimeter of the circle in the Argand diagram.
\item Write down the equation of the locus represented by the half-line $\ell$ in the Argand diagram.
\item Express the complex number represented by the point P in the form $a + b \mathrm { j }$, giving the exact values of $a$ and $b$.
\item Use inequalities to describe the set of points that fall within the shaded region (excluding its boundaries) in the Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2009 Q8 [12]}}