| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| 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 AS locus question requiring students to sketch a circle and a half-line, then find the maximum modulus at their intersection. The geometric interpretation is standard and the calculation involves basic trigonometry or coordinate geometry. While it's Further Maths content, it's a routine application of argand diagram techniques with no novel insight required. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Draws a circle with radius 2 or centre \(2i\) | M1 | Condone freehand circle if intention is clear |
| Draws a circle with radius 2 and centre \(2i\), with no other curves | A1 | Condone freehand circle if intention is clear |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Draws a half-line from \(O\) into the 1st quadrant at an angle of more than \(45°\) to the real axis, with no other straight lines | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Selects a method to find the maximum value of \( | w | \), e.g. identifies triangle with diameter/radius as a side and intersections as vertices; or substitutes \(y = x\tan\left(\frac{\pi}{3}\right)\) into \((x-0)^2+(y-2)^2=2^2\) |
| \(\max | w | = 4\cos\frac{\pi}{6}\) or correct value for \(x_{\max\ |
| \(\max | w | = 2\sqrt{3}\) |
## Question 12:
### Part 12(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Draws a circle with radius 2 **or** centre $2i$ | M1 | Condone freehand circle if intention is clear |
| Draws a circle with radius 2 **and** centre $2i$, with no other curves | A1 | Condone freehand circle if intention is clear |
### Part 12(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Draws a half-line from $O$ into the 1st quadrant at an angle of more than $45°$ to the real axis, with no other straight lines | B1 | |
### Part 12(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Selects a method to find the maximum value of $|w|$, e.g. identifies triangle with diameter/radius as a side and intersections as vertices; or substitutes $y = x\tan\left(\frac{\pi}{3}\right)$ into $(x-0)^2+(y-2)^2=2^2$ | M1 | |
| $\max|w| = 4\cos\frac{\pi}{6}$ or correct value for $x_{\max\|w\|}$ **and** $y_{\max\|w\|}$; note $x_{\max\|w\|}=\sqrt{3}$, $y_{\max\|w\|}=3$ | A1 | May be unsimplified; PI by 3.46 or 3 or 1.73 or better |
| $\max|w| = 2\sqrt{3}$ | A1 | Accept 3.46 or better |12
\begin{enumerate}[label=(\alph*)]
\item Sketch, on the Argand diagram below, the locus of points satisfying the equation
$$| z - 2 \mathrm { i } | = 2$$
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-18_1219_1260_477_392}
\end{center}
12
\item Sketch, also on the Argand diagram above, the locus of points satisfying the equation
$$\arg z = \frac { \pi } { 3 }$$
[1 mark]
12
\item For the complex number $w$ find the maximum value of $| w |$ such that
$$| w - 2 \mathrm { i } | \leq 2 \quad \text { and } \quad 0 \leq \arg w \leq \frac { \pi } { 3 }$$
$$y = \frac { 2 x + 7 } { 3 x + 5 }$$
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2022 Q12 [6]}}