| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex transformations and mappings |
| Difficulty | Standard +0.3 This is a standard FP2 question covering routine loci manipulation and basic complex transformations. Part (i) requires algebraic manipulation of modulus equations (textbook technique), part (ii)(a) tests recall of the geometric effect of z^3 (scaling modulus, tripling argument), and part (ii)(b) applies this to sketch a transformed region. All techniques are standard with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction4.02m Geometrical effects: multiplication and division |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\ | x+(y-3)i\ | =2\ |
| \(\Rightarrow x^2+y^2-6y+9=4x^2+4y^2 \Rightarrow x^2+y^2+2y-3=0\) in any order | M1, A1 | 1.1b, 1.1b |
| (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Points are twice as far from \(3i\) as from \(0\) so \(i\) and \(-3i\) are diametrically opposite points | M1 | 3.1a |
| So radius is \(\frac{\ | i-(-3i)\ | }{2}=2\) and centre is \(\frac{i-3i}{2}=-i\) |
| Hence equation is \(x^2+(y+1)^2=4\) | A1 | 1.1b |
| (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Circle drawn with the inside shaded | M1 | 2.2a |
| Correct circle drawn for their equation. Implied by the position of the centre and radius. Inside shaded. | A1ft | 3.1a |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| A point \(z\) is mapped to a point with 3 times the argument... Rotate every point by 2 times the argument | B1 | 2.4 |
| ...and with modulus as the modulus of the cube of \(z\) | B1 | 2.5 |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Sector from \(O\) along the real axis, indicated by some shading, in an anticlockwise direction | M1 | 1.1b |
| Correct sector, angle \(\frac{3\pi}{4}\) must be stated or implied by the diagram | A1 | 2.2a |
| (2) | ||
| (9 marks) |
# Question 5:
## Part (i)(a) - Locus:
| Working | Mark | Guidance |
|---------|------|----------|
| $\|x+(y-3)i\|=2\|x+yi\| \Rightarrow x^2+(y-3)^2=4(x^2+y^2)$ | **M1** | 3.1a |
| $\Rightarrow x^2+y^2-6y+9=4x^2+4y^2 \Rightarrow x^2+y^2+2y-3=0$ in any order | **M1, A1** | 1.1b, 1.1b |
| | **(3)** | |
**Alternative (a):**
| Working | Mark | Guidance |
|---------|------|----------|
| Points are twice as far from $3i$ as from $0$ so $i$ and $-3i$ are diametrically opposite points | **M1** | 3.1a |
| So radius is $\frac{\|i-(-3i)\|}{2}=2$ and centre is $\frac{i-3i}{2}=-i$ | **M1** | 1.1b |
| Hence equation is $x^2+(y+1)^2=4$ | **A1** | 1.1b |
| | **(3)** | |
## Part (i)(b) - Argand Diagram:
| Working | Mark | Guidance |
|---------|------|----------|
| Circle drawn with the inside shaded | **M1** | 2.2a |
| Correct circle drawn for their equation. Implied by the position of the centre and radius. Inside shaded. | **A1ft** | 3.1a |
| | **(2)** | |
**Notes:**
- **M1:** Circle drawn anywhere and the inside shaded
- **A1ft:** Correct circle drawn for their equation, implied by position of centre and radius. Inside shaded.
## Part (ii)(a) - Transformation Description:
| Working | Mark | Guidance |
|---------|------|----------|
| A point $z$ is mapped to a point with 3 times the argument... Rotate every point by 2 times the argument | **B1** | 2.4 |
| ...and with modulus as the modulus of the cube of $z$ | **B1** | 2.5 |
| | **(2)** | |
**Notes:**
- **B1:** Identifies that the argument triples in size
- **B1:** Identifies that the modulus scales according to the modulus of $z$ cubed
## Part (ii)(b) - Region:
| Working | Mark | Guidance |
|---------|------|----------|
| Sector from $O$ along the real axis, indicated by some shading, in an anticlockwise direction | **M1** | 1.1b |
| Correct sector, angle $\frac{3\pi}{4}$ must be stated or implied by the diagram | **A1** | 2.2a |
| | **(2)** | |
| | **(9 marks)** | |
**Notes:**
- **M1:** A sector centre $O$ and starting along the real axis and in an anticlockwise direction. Must be some shading to represent the region
- **A1:** Correct sector shaded and the angle stated or implied
---\begin{enumerate}
\item (i) A circle $C$ in the complex plane is defined by the locus of points satisfying
\end{enumerate}
$$| z - 3 i | = 2 | z |$$
(a) Determine a Cartesian equation for $C$, giving your answer in simplest form.\\
(b) On an Argand diagram, shade the region defined by
$$\{ z \in \mathbb { C } : | z - 3 \mathrm { i } | > 2 | z | \}$$
(ii) The transformation $T$ from the $z$-plane to the $w$-plane is given by
$$w = z ^ { 3 }$$
(a) Describe the geometric effect of $T$.
The region $R$ in the $z$-plane is given by
$$\left\{ z \in \mathbb { C } : 0 < \arg z < \frac { \pi } { 4 } \right\}$$
(b) On a different Argand diagram, sketch the image of $R$ under $T$.
\hfill \mbox{\textit{Edexcel FP2 2024 Q5 [9]}}