| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.8 This FP1 question requires understanding two distinct loci (perpendicular bisector and half-line), finding their intersection algebraically by solving simultaneous equations involving complex numbers, and interpreting a degenerate case. While systematic, it demands solid technique across multiple complex number concepts and careful algebraic manipulation, placing it moderately above average difficulty. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
4 Two loci, $C _ { 1 }$ and $C _ { 2 }$, are defined by
$$\begin{aligned}
& C _ { 1 } = \left\{ z : | z | = \left| z - 4 d ^ { 2 } - 36 \right| \right\} \\
& C _ { 2 } = \left\{ z : \arg ( z - 12 d - 3 \mathrm { i } ) = \frac { 1 } { 4 } \pi \right\}
\end{aligned}$$
where $d$ is a real number.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $d$, the complex number which is represented on an Argand diagram by the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.\\[0pt]
[You may assume that $C _ { 1 } \cap C _ { 2 } \neq \varnothing$.]
\item Explain why the solution found in part (a) is not valid when $d = 3$.
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2021 Q4 [8]}}