| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| 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 multi-part question on circle loci in the complex plane. Part (a) requires simple recognition that a circle intersecting the imaginary axis at 2i and 8i has center 5i and radius 3. Part (b) involves finding the tangent point from the origin to this circle using basic geometry (Pythagoras) and trigonometry. All steps are standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Deduces \(a = 5\); accept \( | z - 5\text{i} | = b\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Deduces \(b = 3\); accept \( | z - a\text{i} | = 3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Forms a correct equation in \(OP\); FT their \(a\) and \(b\): \(OP^2 = 5^2 - 3^2\) | M1 (3.1a) | |
| Obtains \(OP = 4\); FT their \(a\) and \(b\) | A1F (1.1b) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Obtains a correct expression or equation for an acute angle in the 3,4,5 triangle; FT their \(a\), \(b\) and \(OP\) | M1 (3.1a) | |
| Deduces \(k = \frac{4}{3}\); \(\tan\theta = \frac{4}{3}\) | A1 (2.2a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Forms a correct equation in the real and/or imaginary part of \(P\); e.g. \(x^2 + (y-5)^2 = 3^2\), \(y = \frac{4}{3}x\); FT their \(a\), \(b\) and \(OP\) | M1 (3.1a) | |
| Obtains \(2.4\) or \(3.2\); \(x = \frac{3}{5} \times 4 = 2.4\) | A1 (1.1b) | |
| Obtains \(z = 2.4 + 3.2\text{i}\); \(y = \frac{4}{5} \times 4 = 3.2\) | A1 (1.1b) |
## Question 17(a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Deduces $a = 5$; accept $|z - 5\text{i}| = b$ | B1 (2.2a) | |
---
## Question 17(a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Deduces $b = 3$; accept $|z - a\text{i}| = 3$ | B1 (2.2a) | |
---
## Question 17(b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Forms a correct equation in $OP$; FT their $a$ and $b$: $OP^2 = 5^2 - 3^2$ | M1 (3.1a) | |
| Obtains $OP = 4$; FT their $a$ and $b$ | A1F (1.1b) | |
---
## Question 17(b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Obtains a correct expression or equation for an acute angle in the 3,4,5 triangle; FT their $a$, $b$ and $OP$ | M1 (3.1a) | |
| Deduces $k = \frac{4}{3}$; $\tan\theta = \frac{4}{3}$ | A1 (2.2a) | |
---
## Question 17(b)(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Forms a correct equation in the real and/or imaginary part of $P$; e.g. $x^2 + (y-5)^2 = 3^2$, $y = \frac{4}{3}x$; FT their $a$, $b$ and $OP$ | M1 (3.1a) | |
| Obtains $2.4$ or $3.2$; $x = \frac{3}{5} \times 4 = 2.4$ | A1 (1.1b) | |
| Obtains $z = 2.4 + 3.2\text{i}$; $y = \frac{4}{5} \times 4 = 3.2$ | A1 (1.1b) | |17 The circle $C$ represents the locus of points satisfying the equation
$$| z - a \mathrm { i } | = b$$
where $a$ and $b$ are real constants.
The circle $C$ intersects the imaginary axis at 2 i and 8 i\\
The circle $C$ is shown on the Argand diagram in Figure 2
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-24_764_770_778_699}
\end{center}
\end{figure}
17
\begin{enumerate}[label=(\alph*)]
\item (i) Write down the value of $a$
17 (a) (ii) Write down the value of $b$\\
17
\item The half-line $L$ represents the locus of points satisfying the equation
$$\arg ( z ) = \tan ^ { - 1 } ( k )$$
where $k$ is a positive constant.\\
The point $P$ is the only point which lies on both $C$ and $L$, as shown in Figure 3
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-25_766_770_685_699}
\end{center}
\end{figure}
17 (b) (i) The point $O$ represents the number $0 + 0 \mathrm { i }$\\
Calculate the length $O P$\\
17 (b) (ii) Calculate the exact value of $k$\\
17 (b) (iii) Find the complex number represented by point $P$\\
Give your answer in the form $x + y i$ where $x$ and $y$ are real.
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2024 Q17 [8]}}