| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2004 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex transformations (Möbius) |
| Difficulty | Challenging +1.2 This is a standard Further Maths FP2 Möbius transformation question requiring systematic application of known techniques: substituting parametric forms, algebraic manipulation to show circle mappings, and sketching. While it involves multiple parts and requires careful algebra with complex numbers, the methods are direct applications of syllabus content without requiring novel insight or particularly sophisticated reasoning. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction4.02m Geometrical effects: multiplication and division |
| Answer | Marks | Guidance |
|---|---|---|
| \( | (\lambda+1)+\lambda i | = |
| Answer | Marks | Guidance |
|---|---|---|
| \( | z | = 1 \Rightarrow |
(a) $\arg z = \frac{\pi}{4} \Rightarrow z = \lambda + \lambda i$ (or putting $x$ and $y$ equal at some stage) B1
$w = \frac{(\lambda+1)+\lambda i}{\lambda+(\lambda+1)i}$, and attempt modulus of numerator or denominator. M1
(Could still be in terms of $x$ and $y$)
$|(\lambda+1)+\lambda i| = |\lambda+(\lambda+1)i| = \sqrt{(\lambda+1)^2 + \lambda^2}$, $\therefore w = 1$ (*) A1 A1 (cso)
(b) $w = \frac{z+1}{z+i} \Rightarrow zw + wi = z + 1 \Rightarrow z = \frac{1-wi}{w-1}$ M1
$|z| = 1 \Rightarrow |1-wi| = |w-1|$ M1 A1
For $w = a + ib$, $(1+b) - ai = (a-1) + ib$ M1
$(1+b)^2 + a^2 =12. The transformation $T$ from the complex $z$-plane to the complex $w$-plane is given by
$$w = \frac { z + 1 } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$
\begin{enumerate}[label=(\alph*)]
\item Show that $T$ maps points on the half-line $\arg ( z ) = \frac { \pi } { 4 }$ in the $z$-plane into points on the circle $| w | = 1$ in the $w$-plane.
\item Find the image under $T$ in the $w$-plane of the circle $| Z | = 1$ in the $z$-plane.
\item Sketch on separate diagrams the circle $| \mathbf { Z } | = 1$ in the $z$-plane and its image under $T$ in the $w$-plane.
\item Mark on your sketches the point $P$, where $z = \mathrm { i }$, and its image $Q$ under $T$ in the $w$-plane.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2004 Q12 [14]}}