| Exam Board | Edexcel |
|---|---|
| Module | FP2 AS (Further Pure 2 AS) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Circle of Apollonius locus |
| Difficulty | Standard +0.3 This is a standard Further Maths FP2 question requiring conversion of a modulus equation to Cartesian form using |z|² = x² + y², then completing the square to identify circle parameters. Part (c) adds a straightforward geometric interpretation (finding horizontal extent). While it's Further Maths content, the technique is routine and well-practiced, making it slightly easier than average overall. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x-9)^2+(y+12)^2=4[x^2+y^2]\) | M1 | Obtains equation in terms of \(x\) and \(y\) using the given information |
| \(3x^2+3y^2+18x-24y-225=0\) which is the equation of a circle | A1* | Expands and simplifies, collecting terms and obtains a circle equation correctly, deducing that this is a circle |
| As \(x^2+y^2+6x-8y-75=0\) so \((x+3)^2+(y-4)^2=10^2\) | M1 | Completes the square for their equation to find centre and radius |
| Giving centre at \((-3,4)\) and radius \(=10\) | A1ft | Both correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Circle drawn with centre and radius as given from their equation | M1 | Draws a circle with centre and radius as given from their equation |
| Correct circle drawn with centre at \(-3+4i\) passing through all four quadrants | A1 | Correct circle drawn, with centre at \(-3+4i\) and passing through all four quadrants |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Values range from their \(-3-10\) to their \(-3+10\) | M1 | Attempts to find where a line parallel to real axis, passing through the centre, meets the circle |
| So \(-13\leq\text{Re}(w)\leq7\) | A1ft | Correctly obtains the correct answer for their centre and radius |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x-9)^2+(y+12)^2=4[x^2+y^2]$ | M1 | Obtains equation in terms of $x$ and $y$ using the given information |
| $3x^2+3y^2+18x-24y-225=0$ which is the equation of a circle | A1* | Expands and simplifies, collecting terms and obtains a circle equation correctly, deducing that this is a circle |
| As $x^2+y^2+6x-8y-75=0$ so $(x+3)^2+(y-4)^2=10^2$ | M1 | Completes the square for their equation to find centre and radius |
| Giving centre at $(-3,4)$ and radius $=10$ | A1ft | Both correct |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Circle drawn with centre and radius as given from their equation | M1 | Draws a circle with centre and radius as given from their equation |
| Correct circle drawn with centre at $-3+4i$ passing through all four quadrants | A1 | Correct circle drawn, with centre at $-3+4i$ and passing through all four quadrants |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Values range from their $-3-10$ to their $-3+10$ | M1 | Attempts to find where a line parallel to real axis, passing through the centre, meets the circle |
| So $-13\leq\text{Re}(w)\leq7$ | A1ft | Correctly obtains the correct answer for their centre and radius |
---\begin{enumerate}
\item A curve $C$ is described by the equation
\end{enumerate}
$$| z - 9 + 12 i | = 2 | z |$$
(a) Show that $C$ is a circle, and find its centre and radius.\\
(b) Sketch $C$ on an Argand diagram.
Given that $w$ lies on $C$,\\
(c) find the largest value of $a$ and the smallest value of $b$ that must satisfy
$$a \leqslant \operatorname { Re } ( w ) \leqslant b$$
\hfill \mbox{\textit{Edexcel FP2 AS Q3 [8]}}