| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2017 |
| Session | December |
| Marks | 9 |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a standard FP1 loci question requiring sketching a circle and half-line from a center point, finding their intersection using basic trigonometry (2cos(5π/6), 2sin(5π/6)), and shading a region. All techniques are routine for Further Maths students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: Circle | B1 | |
| Answer: Centre \(3 + 2i\) and radius \(2\) indicated | B1 | Could be (3, 2). Either could be inferred from axes but not just 'tick marks'. No "daylight" between curve and axis. |
| Answer: Half line starting at \(3 + 2i\) | B1 | Condone half line starting at \(±3±2i\) with \(\pi/6\) or \(5\pi/6\) labelled |
| Answer: Fully correct | B1 | Condone no dot/closed dot |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(2\cos\frac{5\pi}{6}\) or \(2\sin\frac{5\pi}{6}\) | [4] M1 | Seen or implied. Could be \(-2\cos\frac{\pi}{6}\) or \(2\sin\frac{\pi}{6}\) |
| Answer: \(3 - \sqrt{3} + 3i\) | A1 | For \(3 - \sqrt{3}\) |
| Answer: (blank) | A1 | For \(3i\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: Shading contained between their half-line and a horizontal half-line from their \(3+2i\) in negative direction. Shading inside circle. | M1ft | Ignore specification of boundaries (ie dotted or solid lines) |
| Answer: (blank) | A1 |
## (i)
Answer: Circle | B1 |
Answer: Centre $3 + 2i$ and radius $2$ indicated | B1 | Could be (3, 2). Either could be inferred from axes but not just 'tick marks'. No "daylight" between curve and axis.
Answer: Half line starting at $3 + 2i$ | B1 | Condone half line starting at $±3±2i$ with $\pi/6$ or $5\pi/6$ labelled
Answer: Fully correct | B1 | Condone no dot/closed dot
## (ii)
Answer: $2\cos\frac{5\pi}{6}$ or $2\sin\frac{5\pi}{6}$ | [4] M1 | Seen or implied. Could be $-2\cos\frac{\pi}{6}$ or $2\sin\frac{\pi}{6}$
Answer: $3 - \sqrt{3} + 3i$ | A1 | For $3 - \sqrt{3}$
Answer: (blank) | A1 | For $3i$
## (iii)
Answer: Shading contained between their half-line and a horizontal half-line from their $3+2i$ in negative direction. Shading inside circle. | M1ft | Ignore specification of boundaries (ie dotted or solid lines)
Answer: (blank) | A1 |
---The loci $C_1$ and $C_2$ are given by $|z - (3 + 2i)| = 2$ and $\arg(z - (3 + 2i)) = \frac{5\pi}{6}$ respectively.
\begin{enumerate}[label=(\roman*)]
\item Sketch $C_1$ and $C_2$ on a single Argand diagram. [4]
\item Find, in surd form, the number represented by the point of intersection of $C_1$ and $C_2$. [3]
\item Indicate, by shading, the region of the Argand diagram for which
$$|z - (3 + 2i)| \leq 2 \text{ and } \frac{5\pi}{6} \leq \arg(z - (3 + 2i)) \leq \pi.$$ [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2017 Q2 [9]}}