| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of modulus on loci |
| Difficulty | Standard +0.8 This is a Further Maths question requiring understanding of complex loci (circle and half-line), geometric visualization, and optimization. Parts (a) and (b) are routine sketching, but part (c) requires recognizing that the expression represents the squared distance between points on the two loci and finding the minimum distance from a circle to a ray—a non-trivial geometric optimization problem requiring coordinate geometry or perpendicular distance calculations. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| 18(a) | Draws a circle with centre and radius 2. |
| Answer | Marks | Guidance |
|---|---|---|
| (0,0) | 1.1b | B1 |
| Answer | Marks |
|---|---|
| 18(b) | Draws a straight line from at to the real axis. |
| Answer | Marks | Guidance |
|---|---|---|
| Accept a reasonably accur(a−te4 ,u0n)rule4d line. | 1.1b | B1 |
| Answer | Marks |
|---|---|
| 18(c) | Selects a method to find the required expression by |
| Answer | Marks | Guidance |
|---|---|---|
| or an indication of the use of the point . | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| or the distance from to( −2,2) . | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 12−8√2 | 3.2a | A1 |
| Total | 5 | = �√8−2� |
| Answer | Marks |
|---|---|
| Paper total | 80 |
Question 18:
--- 18(a) ---
18(a) | Draws a circle with centre and radius 2.
Accept a reasonably accurate freehand circle.
(0,0) | 1.1b | B1 | 2 2
�(−2−0) +(2−0)
shortest distance
least possible value
= √8−2
2
--- 18(b) ---
18(b) | Draws a straight line from at to the real axis.
𝜋𝜋
Accept a reasonably accur(a−te4 ,u0n)rule4d line. | 1.1b | B1
--- 18(c) ---
18(c) | Selects a method to find the required expression by
relating it to the shortest distance between the circle and
the line.
e.g. a perpendicular drawn from the line to the origin (or to
the circle).
or an indication of the use of the point . | 3.1a | M1
Calculates the distance from to( t−h2e, o2r)igin,
or the distance from to( −2,2) . | 1.1a | M1
(−2,2) �−√2,√2�
Obtains the correct value =
ACF, need not be simplified, exact value not required.
12−8√2 | 3.2a | A1
Total | 5 | = �√8−2�
= 12−8√2
Paper total | 80The locus of points $L_1$ satisfies the equation $|z| = 2$
The locus of points $L_2$ satisfies the equation $\arg(z + 4) = \frac{\pi}{4}$
\begin{enumerate}[label=(\alph*)]
\item Sketch $L_1$ on the Argand diagram below.
\includegraphics{figure_18}
[1 mark]
\item Sketch $L_2$ on the Argand diagram above.
[1 mark]
\item The complex number $a + ib$, where $a$ and $b$ are real, lies on $L_1$
The complex number $c + id$, where $c$ and $d$ are real, lies on $L_2$
Calculate the least possible value of the expression
$$(c - a)^2 + (d - b)^2$$
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2020 Q18 [5]}}