| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Topic | Complex Numbers Argand & Loci |
| Type | Circle of Apollonius locus |
| Difficulty | Challenging +1.8 This is a Further Maths question requiring students to manipulate a complex logarithm equation, convert to Cartesian form, and recognize the resulting locus. It demands understanding that ln(w) being real implies w is positive real, leading to an argument condition, then algebraic manipulation to identify a circle. The multi-step nature, conceptual depth (complex logarithms, locus identification), and requirement for geometric insight place it well above average difficulty. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02o Loci in Argand diagram: circles, half-lines |
$C$ is the locus of numbers, $z$, for which $\ln\left(\frac{z + 7i}{z - 24}\right) = \frac{1}{4}$.
By writing $z = x + iy$ give a complete description of the shape of $C$ on an Argand diagram. [7]
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q5 [7]}}