| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring verification of two given facts (substitution and checking a point lies on a line through the centre), sketching standard loci (circle and half-line), and shading a region. All techniques are routine for FP2 students with no novel problem-solving required, making it slightly easier than average even for Further Maths. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \(z = -i\) | M1 | \( |
| \( | -2\sqrt{3} - 2i | \) |
| \(4\) | ||
| (ii) Centre of circle is \(2\sqrt{3} + i\) | B1 | Do not accept \((2\sqrt{3}, 1)\) unless attempt to solve using trig |
| Substitute into line | M1 | |
| \(\arg(2\sqrt{3} + 2i) = \frac{\pi}{6}\) shown | A1 | 3 marks |
| (b) Circle: centre correct through \((0, -1)\) | B1 | |
| B1 | ||
| Half line: through \((0, -1)\) | B1 | |
| through centre of circle | B1 | 4 marks |
| Shading inside circle and below line | B1F | |
| Bounded by \(y = -1\) | B1 | 2 marks |
**(a)(i)** $z = -i$ | M1 | $|-2\sqrt{3} - 2i| = \sqrt{12 + 4} = 4$
$|-2\sqrt{3} - 2i|$ | A1 | 2 marks
$4$ | |
**(ii)** Centre of circle is $2\sqrt{3} + i$ | B1 | Do not accept $(2\sqrt{3}, 1)$ unless attempt to solve using trig
Substitute into line | M1 |
$\arg(2\sqrt{3} + 2i) = \frac{\pi}{6}$ shown | A1 | 3 marks
**(b)** Circle: centre correct through $(0, -1)$ | B1 |
| B1 |
Half line: through $(0, -1)$ | B1 |
| through centre of circle | B1 | 4 marks
Shading inside circle and below line | B1F |
Bounded by $y = -1$ | B1 | 2 marks
**Question 3 Total: 11 marks**
---3 A circle $C$ and a half-line $L$ have equations
$$| z - 2 \sqrt { 3 } - \mathrm { i } | = 4$$
and
$$\arg ( z + i ) = \frac { \pi } { 6 }$$
respectively.
\begin{enumerate}[label=(\alph*)]
\item Show that:
\begin{enumerate}[label=(\roman*)]
\item the circle $C$ passes through the point where $z = - \mathrm { i }$;
\item the half-line $L$ passes through the centre of $C$.
\end{enumerate}\item On one Argand diagram, sketch $C$ and $L$.
\item Shade on your sketch the set of points satisfying both
$$| z - 2 \sqrt { 3 } - \mathrm { i } | \leqslant 4$$
and
$$0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 6 }$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2008 Q3 [11]}}