| Exam Board | Edexcel |
|---|---|
| Module | FP2 AS (Further Pure 2 AS) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Cartesian equation from argument condition |
| Difficulty | Challenging +1.2 This is a standard Further Maths FP2 locus question requiring recognition that the argument condition defines a circular arc, followed by geometric optimization. While it involves multiple steps (sketching the locus, identifying it as an arc of a circle, then finding the maximum distance from origin), these are well-practiced techniques in FP2. The optimization is straightforward once the locus is identified—finding the farthest point on the arc from the origin. More routine than the average FP2 question but requires solid understanding of argument loci. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines4.02p Set notation: for loci |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Interprets locus as a circle or arc of a circle | M1 | Correct interpretation of locus |
| Circle/arc passing through or touching at 3 and 6 on positive imaginary axis | A1 | Circle or arc touching at 3 and 6 on positive imaginary axis |
| Major arc wholly to the left of imaginary axis and wholly above real axis, with 3 and 6 marked | A1 | Correct diagram with correct region and markings |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y\)-coordinate of centre is \(4.5\) | B1 | Correct \(y\)-coordinate of centre |
| \(x\)-coordinate of centre is \(-\dfrac{1.5}{\tan\frac{\pi}{3}}=−\dfrac{\sqrt{3}}{2}\) | M1 | Correct strategy for \(x\)-coordinate of centre |
| Radius \(= \dfrac{1.5}{\sin\frac{\pi}{3}}\) or \(\sqrt{1.5^2+\left(\dfrac{1.5}{\tan\frac{\pi}{3}}\right)^2}\) | M1 | Correct strategy for radius |
| \(d=\sqrt{4.5^2+0.75+\sqrt{3}}\) | M1 | Fully correct method for maximum using their values |
| \(d=\sqrt{21}+\sqrt{3}\) | A1 | Correct value |
# Question 5:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Interprets locus as a circle or arc of a circle | M1 | Correct interpretation of locus |
| Circle/arc passing through or touching at 3 and 6 on positive imaginary axis | A1 | Circle or arc touching at 3 and 6 on positive imaginary axis |
| Major arc wholly to the left of imaginary axis and wholly above real axis, with 3 and 6 marked | A1 | Correct diagram with correct region and markings |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y$-coordinate of centre is $4.5$ | B1 | Correct $y$-coordinate of centre |
| $x$-coordinate of centre is $-\dfrac{1.5}{\tan\frac{\pi}{3}}=−\dfrac{\sqrt{3}}{2}$ | M1 | Correct strategy for $x$-coordinate of centre |
| Radius $= \dfrac{1.5}{\sin\frac{\pi}{3}}$ or $\sqrt{1.5^2+\left(\dfrac{1.5}{\tan\frac{\pi}{3}}\right)^2}$ | M1 | Correct strategy for radius |
| $d=\sqrt{4.5^2+0.75+\sqrt{3}}$ | M1 | Fully correct method for maximum using their values |
| $d=\sqrt{21}+\sqrt{3}$ | A1 | Correct value |\begin{enumerate}
\item A complex number $z$ is represented by the point $P$ on an Argand diagram.
\end{enumerate}
Given that $\arg \left( \frac { z - 6 i } { z - 3 i } \right) = \frac { \pi } { 3 }$\\
(a) sketch the locus of $P$ as $z$ varies,\\
(b) find the exact maximum possible value of $| z |$
\hfill \mbox{\textit{Edexcel FP2 AS 2018 Q5 [8]}}