| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 12 |
| 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 testing standard loci concepts. Parts (a)-(c) involve routine verification that a point lies on both a circle and half-line, then showing tangency. Part (d) requires finding where arg is maximized, which is a standard technique. All steps are direct applications of known methods with no novel insight required. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| 4(a) radius \(\sqrt{2}\) | B1, B1 | 2 marks |
| centre \(-5+i\) | condone \((-5, 1)\) for centre; do not accept \((-5, i)\) | |
| 4(b) \(\arg(\bar{z}_1 + 2i) = \arg(-4 + 4i)\) | M1 | |
| \(= \frac{3\pi}{4}\) | A1 | 2 marks; clearly shown eg \(\tan^{-1}\left(\frac{-1}{-1}\right)\) |
| 4(c)(i) \( | z_1 + 5 - i | = |
| 4(c)(ii) Gradient of line from \((-5,1)\) to \((-4,2)\) is \(1\) \(\left(\frac{\pi}{4}\right)\) | M1A1 | M1 for a complete method |
| radius \(\perp\) line \(\therefore\) tangent | E1 | 3 marks |
| 4(c)(iii) [Circle diagram with tangent line shown correctly] | B1F | ft incorrect centre or radius |
| Half line correct | B1 | 2 marks; line must touch C generally above the circle |
| 4(d) \(z_2\) in correct place | B1 | B0 if \(z_2\) is directly below the centre of C |
| with tangent shown | B1 | 2 marks (12 marks for question) |
**4(a)** radius $\sqrt{2}$ | B1, B1 | 2 marks
centre $-5+i$ | | condone $(-5, 1)$ for centre; do not accept $(-5, i)$
**4(b)** $\arg(\bar{z}_1 + 2i) = \arg(-4 + 4i)$ | M1
$= \frac{3\pi}{4}$ | A1 | 2 marks; clearly shown eg $\tan^{-1}\left(\frac{-1}{-1}\right)$
**4(c)(i)** $|z_1 + 5 - i| = |1 + i| = \sqrt{2}$ | B1 | 1 mark
**4(c)(ii)** Gradient of line from $(-5,1)$ to $(-4,2)$ is $1$ $\left(\frac{\pi}{4}\right)$ | M1A1 | M1 for a complete method
radius $\perp$ line $\therefore$ tangent | E1 | 3 marks
**4(c)(iii)** [Circle diagram with tangent line shown correctly] | B1F | ft incorrect centre or radius
Half line correct | B1 | 2 marks; line must touch C generally above the circle
**4(d)** $z_2$ in correct place | B1 | B0 if $z_2$ is directly below the centre of C
with tangent shown | B1 | 2 marks (12 marks for question)4
\begin{enumerate}[label=(\alph*)]
\item A circle $C$ in the Argand diagram has equation
$$| z + 5 - \mathrm { i } | = \sqrt { 2 }$$
Write down its radius and the complex number representing its centre.
\item A half-line $L$ in the Argand diagram has equation
$$\arg ( z + 2 \mathrm { i } ) = \frac { 3 \pi } { 4 }$$
Show that $z _ { 1 } = - 4 + 2 \mathrm { i }$ lies on $L$.
\item \begin{enumerate}[label=(\roman*)]
\item Show that $z _ { 1 } = - 4 + 2 \mathrm { i }$ also lies on $C$.
\item Hence show that $L$ touches $C$.
\item Sketch $L$ and $C$ on one Argand diagram.
\end{enumerate}\item The complex number $z _ { 2 }$ lies on $C$ and is such that $\arg \left( z _ { 2 } + 2 \mathrm { i } \right)$ has as great a value as possible.
Indicate the position of $z _ { 2 }$ on your sketch.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2008 Q4 [12]}}