| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Marks | 10 |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.8 Part (i) requires solving simultaneous geometric conditions (circle and ray) in the complex plane, involving coordinate geometry and trigonometry. Part (ii) requires understanding and accurately sketching a region bounded by two circles and two rays, which demands spatial visualization and precision. This is more challenging than routine complex number exercises but doesn't require exceptional insight—it's a solid Further Maths question testing multiple techniques. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
\begin{enumerate}[label=(\roman*)]
\item Determine $x$ and $y$ given that the complex number $z = x + \text{i}y$ simultaneously satisfies
$$|z - 1| = 1 \quad \text{and} \quad \arg(z + 1) = \frac{1}{6}\pi.$$ [4]
\item On an Argand diagram, shade the region whose points satisfy
$$1 \leqslant |z - 1| \leqslant 2 \quad \text{and} \quad \frac{1}{6}\pi \leqslant \arg(z + 1) \leqslant \frac{1}{4}\pi.$$ [6]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q8 [10]}}