| Exam Board | Edexcel |
|---|---|
| Module | FP2 AS (Further Pure 2 AS) |
| Year | 2020 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Challenging +1.2 This is a multi-part FP2 question requiring reading an Argand diagram, identifying loci parameters, then finding a maximum distance. Part (a) is straightforward diagram reading. Part (b) requires converting w to modulus-argument form, recognizing that |w-z|² is maximized when z is at the boundary point furthest from w, and computing the distance—standard optimization on loci with clear geometric insight, but requires several coordinated steps and careful calculation. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(p = \dfrac{\pi}{4}\) and \(q = \pi\), or \(\dfrac{\pi}{4} \leq \arg z \leq \pi\) | B1 | Correct values or correct loci |
| \(r = 2\) or \( | z | \leq 2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Position for \(z\) is at intersection in quadrant 1 | B1 | Realises position for \(z\) is at intersection in quadrant 1 and makes progress in finding the distance |
| Angle between \(y = x\) and "\(OW\)" is \(\dfrac{\pi}{3} + \dfrac{\pi}{4}\) | B1 | Deduces the angle between \(y=x\) and \(OW\) is \(\pi/4 + \pi/3\) |
| \(d^2 = 4^2 + 2^2 - 2 \times 4 \times 2\cos\!\left(\dfrac{\pi}{3} + \dfrac{\pi}{4}\right)\) | M1 | Correct use of the cosine rule to find the required length\(^2\) |
| \(= 20 - 4\sqrt{2} + 4\sqrt{6}\) | A1 | Deduces the required value in simplified and exact form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Position for \(z\) is at intersection in quadrant 1 | B1 | Realises position for \(z\) is at intersection in quadrant 1 |
| \(y = x\) intersects \(x^2 + y^2 = 4\) at \((\sqrt{2},\, \sqrt{2})\) | B1 | Deduces correct coordinates for \(z\) |
| \(d^2 = \bigl(\text{``}\sqrt{2}\text{''} + 2\sqrt{3}\bigr)^2 + \bigl(\text{``}\sqrt{2}\text{''} - 2\bigr)^2\) | M1 | Correct use of Pythagoras to find the required length\(^2\) |
| \(= 20 - 4\sqrt{2} + 4\sqrt{6}\) | A1 | Deduces the required value in simplified and exact form |
## Question 5(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $p = \dfrac{\pi}{4}$ and $q = \pi$, or $\dfrac{\pi}{4} \leq \arg z \leq \pi$ | B1 | Correct values or correct loci |
| $r = 2$ or $|z| \leq 2$ | B1 | Correct value or correct loci |
## Question 5(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Position for $z$ is at intersection in quadrant 1 | B1 | Realises position for $z$ is at intersection in quadrant 1 and makes progress in finding the distance |
| Angle between $y = x$ and "$OW$" is $\dfrac{\pi}{3} + \dfrac{\pi}{4}$ | B1 | Deduces the angle between $y=x$ and $OW$ is $\pi/4 + \pi/3$ |
| $d^2 = 4^2 + 2^2 - 2 \times 4 \times 2\cos\!\left(\dfrac{\pi}{3} + \dfrac{\pi}{4}\right)$ | M1 | Correct use of the cosine rule to find the required length$^2$ |
| $= 20 - 4\sqrt{2} + 4\sqrt{6}$ | A1 | Deduces the required value in simplified and exact form |
**Alternative:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Position for $z$ is at intersection in quadrant 1 | B1 | Realises position for $z$ is at intersection in quadrant 1 |
| $y = x$ intersects $x^2 + y^2 = 4$ at $(\sqrt{2},\, \sqrt{2})$ | B1 | Deduces correct coordinates for $z$ |
| $d^2 = \bigl(\text{``}\sqrt{2}\text{''} + 2\sqrt{3}\bigr)^2 + \bigl(\text{``}\sqrt{2}\text{''} - 2\bigr)^2$ | M1 | Correct use of Pythagoras to find the required length$^2$ |
| $= 20 - 4\sqrt{2} + 4\sqrt{6}$ | A1 | Deduces the required value in simplified and exact form |5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8d0194d2-7958-4699-9c5c-02e815ac433c-18_510_714_251_689}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows an Argand diagram.\\
The set of points, $A$, that lies within the shaded region, including its boundaries, is defined by
$$A = \{ z : p \leqslant \arg ( z ) \leqslant q \} \cap \{ z : | z | \leqslant r \}$$
where $p , q$ and $r$ are positive constants.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $p , q$ and $r$.
Given that $w = - 2 \sqrt { 3 } + 2 \mathrm { i }$ and $\mathrm { z } \in A$,
\item find the maximum value of $| w - z | ^ { 2 }$ giving your answer in an exact simplified form.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 AS 2020 Q5 [6]}}