| Exam Board | Edexcel |
|---|---|
| Module | FP2 AS (Further Pure 2 AS) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring manipulation of complex loci to prove a circle equation, find its center/radius algebraically, test inequality satisfaction, and shade a region satisfying compound inequalities. While the techniques are standard for FP2 (squaring moduli, completing the square, interpreting arg inequalities), the multi-step nature and requirement to synthesize geometric understanding of both modulus and argument conditions makes this moderately challenging, though still within typical Further Maths scope. |
| 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 |
| \((x-1)^2+(y-8)^2 = 9[(x-1)^2+y^2]\); OR \(\sqrt{(x-1)^2+(y-8)^2}=3\sqrt{(x-1)^2+y^2}\) | M1 | Obtains equation in \(x\) and \(y\) using given information |
| \(8x^2-16x+8y^2+16y-56=0\) | A1 | Expands, simplifies, collects terms correctly |
| \(x^2-2x+y^2+2y-7=0\) so \((x-1)^2+(y+1)^2=9\); finds centre and radius | M1 | Completes the square |
| Circle with centre \((1,-1)\) and radius \(= 3\) | A1 | Deduces circle with centre \((1,-1)\) and radius \(3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Distance \(=\sqrt{(3-1)^2+(-3--1)^2}=\ldots\); OR \(d^2=(3-1)^2+(-3--1)^2=\ldots\) | M1 | Finds distance between \((3,-3)\) and correct centre \((1,-1)\) |
| Distance \(=\sqrt{8}=2.828\ldots < 3\ \therefore z=3-3i\) satisfies the inequality; OR \(8<9\ \therefore z=3-3i\) satisfies the inequality | A1 | Compares distance with 3 (or \(d^2\) with 9) and deduces inequality satisfied using correct centre and radius |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Circle with correct centre and radius | M1 | Circle for their centre and radius |
| Circle with centre in fourth quadrant (passing through all four quadrants) | A1 | Condone dotted circle |
| Half-line drawn from \((0,-1)\) passing through the \(x\)-axis within the circle | M1 | Condone dotted line |
| Correct region shaded | A1 | Both half-line and circle correct and not dotted; Special case: M1A1M1A0 if no coordinates stated but half-line clearly intersects axes at correct centre level |
## Question 3:
**Part (a)**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x-1)^2+(y-8)^2 = 9[(x-1)^2+y^2]$; OR $\sqrt{(x-1)^2+(y-8)^2}=3\sqrt{(x-1)^2+y^2}$ | M1 | Obtains equation in $x$ and $y$ using given information |
| $8x^2-16x+8y^2+16y-56=0$ | A1 | Expands, simplifies, collects terms correctly |
| $x^2-2x+y^2+2y-7=0$ so $(x-1)^2+(y+1)^2=9$; finds centre and radius | M1 | Completes the square |
| Circle with centre $(1,-1)$ and radius $= 3$ | A1 | Deduces circle with centre $(1,-1)$ and radius $3$ |
**Part (b)**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Distance $=\sqrt{(3-1)^2+(-3--1)^2}=\ldots$; OR $d^2=(3-1)^2+(-3--1)^2=\ldots$ | M1 | Finds distance between $(3,-3)$ and correct centre $(1,-1)$ |
| Distance $=\sqrt{8}=2.828\ldots < 3\ \therefore z=3-3i$ satisfies the inequality; OR $8<9\ \therefore z=3-3i$ satisfies the inequality | A1 | Compares distance with 3 (or $d^2$ with 9) and deduces inequality satisfied using correct centre and radius |
**Part (c)**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Circle with correct centre and radius | M1 | Circle for their centre and radius |
| Circle with centre in fourth quadrant (passing through all four quadrants) | A1 | Condone dotted circle |
| Half-line drawn from $(0,-1)$ passing through the $x$-axis within the circle | M1 | Condone dotted line |
| Correct region shaded | A1 | Both half-line and circle correct and not dotted; Special case: M1A1M1A0 if no coordinates stated but half-line clearly intersects axes at correct centre level |
---\begin{enumerate}
\item A curve $C$ in the complex plane is described by the equation
\end{enumerate}
$$| z - 1 - 8 i | = 3 | z - 1 |$$
(a) Show that $C$ is a circle, and find its centre and radius.\\
(b) Using the answer to part (a), determine whether $z = 3 - 3 \mathrm { i }$ satisfies the inequality
$$| z - 1 - 8 i | \geqslant 3 | z - 1 |$$
(c) Shade, on an Argand diagram, the set of points that satisfies both
$$| z - 1 - 8 i | \geqslant 3 | z - 1 | \quad \text { and } \quad 0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 4 }$$
\hfill \mbox{\textit{Edexcel FP2 AS 2019 Q3 [10]}}