| 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}
(i) Write down the equation of the locus represented by the perimeter of the circle in the Argand diagram.\\
(ii) Write down the equation of the locus represented by the half-line $\ell$ in the Argand diagram.\\
(iii) Express the complex number represented by the point P in the form $a + b \mathrm { j }$, giving the exact values of $a$ and $b$.\\
(iv) Use inequalities to describe the set of points that fall within the shaded region (excluding its boundaries) in the Argand diagram.
\hfill \mbox{\textit{OCR MEI FP1 2009 Q8 [12]}}