| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| 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 Argand diagram question requiring modulus/argument calculation, sketching a circle and ray, then shading a region. While it involves multiple parts, each step follows routine procedures with no novel insight required. The geometric interpretation is straightforward, making it slightly easier than average for Further Maths content. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines4.02p Set notation: for loci |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(3\sqrt{2}, -\frac{3}{4}\) or \(-45°\) AEF | B1 B1 | 2 |
| (ii)(a) Circle, centre \((3, -3)\), through \(O\) ft for \((\pm3, \pm3)\) only | B1 B1 | |
| (ii)(b) Straight line with +ve slope, through \((3, -3)\) or their centre. Half line only starting at centre | B1 B1 B1 | 3 |
| (iii) Area above horizontal through \(a\), below (ii)(b). Outside circle | B1 ft B1 ft B1 ft | 3 |
**(i)** $3\sqrt{2}, -\frac{3}{4}$ or $-45°$ AEF | B1 B1 | 2 | State correct answers
**(ii)(a)** Circle, centre $(3, -3)$, through $O$ ft for $(\pm3, \pm3)$ only | B1 B1 | |
**(ii)(b)** Straight line with +ve slope, through $(3, -3)$ or their centre. Half line only starting at centre | B1 B1 B1 | 3 |
**(iii)** Area above horizontal through $a$, below (ii)(b). Outside circle | B1 ft B1 ft B1 ft | 3 | Total 116 The complex number $3 - 3 \mathrm { i }$ is denoted by $a$.\\
(i) Find $| a |$ and $\arg a$.\\
(ii) Sketch on a single Argand diagram the loci given by
\begin{enumerate}[label=(\alph*)]
\item $| z - a | = 3 \sqrt { 2 }$,
\item $\quad \arg ( z - a ) = \frac { 1 } { 4 } \pi$.\\
(iii) Indicate, by shading, the region of the Argand diagram for which
$$| z - a | \geqslant 3 \sqrt { 2 } \quad \text { and } \quad 0 \leqslant \arg ( z - a ) \leqslant \frac { 1 } { 4 } \pi$$
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2009 Q6 [11]}}