| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Challenging +1.2 This is a Further Maths question combining loci in the complex plane with geometric reasoning. Part (a) is routine (circle sketching). Part (b) requires finding where a circle and ray intersect uniquely, involving geometric insight about tangency and then coordinate geometry/trigonometry to find the exact value. While requiring multiple techniques and some spatial reasoning, it follows standard Further Maths patterns without requiring exceptional creativity. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| 14(a) | Draws a circle with centre and radius 2. |
| Answer | Marks | Guidance |
|---|---|---|
| Ignore any straight lines drawn on the diagram. | 1.1b | B1 |
| Answer | Marks |
|---|---|
| 14(b)(i) | Uses fully correct method for or or |
| Answer | Marks | Guidance |
|---|---|---|
| sinπΌπΌ = 4 cosπΌπΌ = 4 tanπΌπΌ = β4 2 β2 2 | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Condone 30 | 1.1b | A1 |
| Answer | Marks |
|---|---|
| 14(b)(ii) | Forms an equation in using cosine rule or equivalent. |
| Answer | Marks | Guidance |
|---|---|---|
| π₯π₯ π¦π¦ | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Or forms a second coπΌπΌrrect equation in and . | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Or obtains correct values ππfor ππ and . | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| β1 β21 β1 3 | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Total | 7 | |
| Q | Marking instructions | AO |
Question 14:
--- 14(a) ---
14(a) | Draws a circle with centre and radius 2.
Accept freehand circle.
(3,0)
Ignore any straight lines drawn on the diagram. | 1.1b | B1
--- 14(b)(i) ---
14(b)(i) | Uses fully correct method for or or
π π πππππΌπΌ πππππ π πΌπΌ π‘π‘πππππΌπΌ
2 2
2 β4 β2 2
sinπΌπΌ = 4 cosπΌπΌ = 4 tanπΌπΌ = β4 2 β2 2 | 3.1a | M1 | 2
sinπΌπΌ = 4
β1 2
πΌπΌ = si n οΏ½4οΏ½
Obtains correct value for
Accept 0.52(35987756)
πΌπΌ
Condone 30 | 1.1b | A1
--- 14(b)(ii) ---
14(b)(ii) | Forms an equation in using cosine rule or equivalent.
Follow through their .
ππ
Or forms a correct eqπΌπΌuation in and .
π₯π₯ π¦π¦ | 3.1a | M1 | 2 2 2 ππ
ππ = 2 +3 β2Γ2Γ3Γcos3
ππ = β7
ππ
π π π π πππ π π π π π ππ3
2 = β7
β3 β21
π π ππππππ = 2Γ Γ·β7 =
2 7
Forms an equation in using sine rule or equivalent.
Follow through their .
ππ
Or forms a second coπΌπΌrrect equation in and . | 1.1a | M1
π₯π₯ π¦π¦
Obtains correct value for or .
Or obtains correct values ππfor ππ and . | 1.1b | A1
π₯π₯ π¦π¦
Expresses in the required form.
Accept or for
π€π€
Accept 20..67 1[4[357725413371819]] or β 7 ππ or for
β1 β21 β1 3 | 1.1b | A1 | ππ = 0.71
π€π€ =2.6(πππππ π 0.71+πππ π ππππ0.71)
sin οΏ½ 7 οΏ½ sin οΏ½οΏ½7οΏ½ ππ
Total | 7
Q | Marking instructions | AO | Mark | Typical solution\begin{enumerate}[label=(\alph*)]
\item Sketch, on the Argand diagram below, the locus of points satisfying the equation
$$|z - 3| = 2$$
[1 mark]
\includegraphics{figure_14a}
\item There is a unique complex number $w$ that satisfies both
$$|w - 3| = 2 \quad \text{and} \quad \arg(w + 1) = \alpha$$
where $\alpha$ is a constant such that $0 < \alpha < \pi$
\begin{enumerate}[label=(\roman*)]
\item Find the value of $\alpha$.
[2 marks]
\item Express $w$ in the form $r(\cos \theta + i \sin \theta)$.
Give each of $r$ and $\theta$ to two significant figures.
[4 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2018 Q14 [7]}}