| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Loci of complex numbers |
| Difficulty | Moderate -0.8 Part (i) is a direct verification using standard complex number definitions (z = re^(iθ), z* = re^(-iθ)), requiring only basic manipulation. Part (ii) immediately follows from recognizing |z|² = 9 means |z| = 3, which is a circle of radius 3. This is a straightforward recall question testing fundamental definitions rather than problem-solving, making it easier than average even for Further Maths. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(z \cdot z^* = re^{i\theta} \cdot re^{-i\theta} = r^2 = | z | ^2\) |
| (ii) Circle Centre \(0 + (0+i)\) OR \((0, 0)\) OR \(O\), radius \(3\) | B1 | For stating circle |
| B1 2 | For stating correct centre and radius |
**(i)** $z \cdot z^* = re^{i\theta} \cdot re^{-i\theta} = r^2 = |z|^2$ | B1 | For verifying result AG
**(ii)** Circle Centre $0 + (0+i)$ OR $(0, 0)$ OR $O$, radius $3$ | B1 | For stating circle
| B1 2 | For stating correct centre and radius1 (i) By writing $z$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, show that $z z ^ { * } = | z | ^ { 2 }$.\\
(ii) Given that $z z ^ { * } = 9$, describe the locus of $z$.
\hfill \mbox{\textit{OCR FP3 2007 Q1 [3]}}