| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex transformations and mappings |
| Difficulty | Challenging +1.8 This Further Maths question requires understanding of complex transformations and the mapping of loci. Students must manipulate the transformation algebraically, substitute the circle equation, separate real and imaginary parts, and complete the square—a multi-step process requiring solid technique but following a standard approach for this topic. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(w(z+4i)=z \Rightarrow z(1-w)=4iw\) or \(z=\frac{4iw}{1-w}\) | M1A1 | Re-arrange to \(z=\ldots\) or \(z(\alpha w+\beta)=\gamma w+\delta\) |
| \( | z | =3 \Rightarrow \left |
| \( | 4iw | =3 |
| \(w=u+iv\): \(16(u^2+v^2)=9\left((1-u)^2+v^2\right)\) | ddM1A1 | Dep on both M marks; squares of moduli attempted correctly. |
| \(7u^2+7v^2+18u-9=0\) | ||
| \(\left(u+\frac{9}{7}\right)^2+v^2=\frac{144}{49}\) | dddM1 | Dep on all previous M marks; complete the square. |
| Centre \(\left(-\frac{9}{7},0\right)\), Radius \(\frac{12}{7}\) | A1A1 (8) | Either correct; both correct and exact. |
# Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $w(z+4i)=z \Rightarrow z(1-w)=4iw$ or $z=\frac{4iw}{1-w}$ | M1A1 | Re-arrange to $z=\ldots$ or $z(\alpha w+\beta)=\gamma w+\delta$ |
| $|z|=3 \Rightarrow \left|\frac{4iw}{1-w}\right|=3$ | dM1 | Dep on first M1, using $|z|=3$ |
| $|4iw|=3|1-w|$ | | |
| $w=u+iv$: $16(u^2+v^2)=9\left((1-u)^2+v^2\right)$ | ddM1A1 | Dep on both M marks; squares of moduli attempted correctly. |
| $7u^2+7v^2+18u-9=0$ | | |
| $\left(u+\frac{9}{7}\right)^2+v^2=\frac{144}{49}$ | dddM1 | Dep on all previous M marks; complete the square. |
| Centre $\left(-\frac{9}{7},0\right)$, Radius $\frac{12}{7}$ | A1A1 (8) | Either correct; both correct and exact. |
---\begin{enumerate}
\item The transformation $T$ from the $z$-plane to the $w$-plane is given by
\end{enumerate}
$$w = \frac { z } { z + 4 \mathrm { i } } \quad z \neq - 4 \mathrm { i }$$
The circle with equation $| z | = 3$ is mapped by $T$ onto the circle $C$ Determine\\
(i) a Cartesian equation of $C$\\
(ii) the centre and radius of $C$
\hfill \mbox{\textit{Edexcel F2 2022 Q3 [8]}}