Standard +0.8 This FP2 question requires understanding of complex loci (circle and half-lines), geometric visualization, and a non-trivial transformation mapping. Part (ii)(a) demands algebraic manipulation to prove the mapping property, which requires insight into how |z-1|=1 transforms. While systematic, this goes beyond routine exercises and requires connecting multiple concepts, placing it moderately above average difficulty for A-level.
On the same Argand diagram sketch the loci given by the following equations.
$$|z - 1| = 1,$$
$$\arg(z + 1) = \frac{\pi}{12},$$
$$\arg(z + 1) = \frac{\pi}{2}.$$ [4]
Shade on your diagram the region for which
$$|z - 1| \leq 1 \quad \text{and} \quad \frac{\pi}{12} \leq \arg(z + 1) \leq \frac{\pi}{2}.$$ [1]
Show that the transformation
$$w = \frac{z - 1}{z}, \quad z \neq 0,$$
maps \(|z - 1| = 1\) in the \(z\)-plane onto \(|w| = |w - 1|\) in the \(w\)-plane. [3]
The region \(|z - 1| \leq 1\) in the \(z\)-plane is mapped onto the region \(T\) in the \(w\)-plane.
\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\alph*)]
\item On the same Argand diagram sketch the loci given by the following equations.
$$|z - 1| = 1,$$
$$\arg(z + 1) = \frac{\pi}{12},$$
$$\arg(z + 1) = \frac{\pi}{2}.$$ [4]
\item Shade on your diagram the region for which
$$|z - 1| \leq 1 \quad \text{and} \quad \frac{\pi}{12} \leq \arg(z + 1) \leq \frac{\pi}{2}.$$ [1]
\end{enumerate}
\item \begin{enumerate}[label=(\alph*)]
\item Show that the transformation
$$w = \frac{z - 1}{z}, \quad z \neq 0,$$
maps $|z - 1| = 1$ in the $z$-plane onto $|w| = |w - 1|$ in the $w$-plane. [3]
The region $|z - 1| \leq 1$ in the $z$-plane is mapped onto the region $T$ in the $w$-plane.
\item Shade the region $T$ on an Argand diagram. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q22 [10]}}