| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a standard Further Maths loci question requiring sketching a circle and half-line, shading a region, and finding a minimum argument geometrically. While it involves multiple parts and some geometric reasoning (finding tangent from origin to circle), these are routine FP1 techniques with straightforward execution. Slightly easier than average A-level difficulty overall due to its procedural nature. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines4.02p Set notation: for loci |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Circle, radius 2, centre \(3j\) | B3 | Circle B1; radius 2, B1; centre \(3j\), B1 |
| Half line from \(-1\), at angle \(\frac{\pi}{4}\) to \(x\)-axis | B3 [6] | Half line B1; from \(-1\), B1; \(\frac{\pi}{4}\) to \(x\)-axis, B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct region between circle and half line indicated | B2(ft) [2] | s.c. B1 for interior of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Tangent from origin to circle | M1 | Attempt to use right-angled triangle |
| \(\arg z = \frac{\pi}{2} - \arcsin\frac{2}{3} = 0.84\) (2d.p.) | A1 [4] | c.a.o. Accept \(48.20°\) (2d.p.) |
# Question 8(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Circle, radius 2, centre $3j$ | B3 | Circle B1; radius 2, B1; centre $3j$, B1 |
| Half line from $-1$, at angle $\frac{\pi}{4}$ to $x$-axis | B3 **[6]** | Half line B1; from $-1$, B1; $\frac{\pi}{4}$ to $x$-axis, B1 |
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# Question 8(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct region between circle and half line indicated | B2(ft) **[2]** | s.c. B1 for interior of circle |
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# Question 8(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Tangent from origin to circle | M1 | Attempt to use right-angled triangle |
| $\arg z = \frac{\pi}{2} - \arcsin\frac{2}{3} = 0.84$ (2d.p.) | A1 **[4]** | c.a.o. Accept $48.20°$ (2d.p.) |
---8
\begin{enumerate}[label=(\roman*)]
\item On a single Argand diagram, sketch the locus of points for which\\
(A) $| z - 3 \mathrm { j } | = 2$,\\
(B) $\quad \arg ( z + 1 ) = \frac { 1 } { 4 } \pi$.
\item Indicate clearly on your Argand diagram the set of points for which
$$| z - 3 \mathrm { j } | \leqslant 2 \quad \text { and } \quad \arg ( z + 1 ) \leqslant \frac { 1 } { 4 } \pi .$$
\item (A) By drawing an appropriate line through the origin, indicate on your Argand diagram the point for which $| z - 3 j | = 2$ and $\arg z$ has its minimum possible value.\\
(B) Calculate the value of $\arg z$ at this point.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2008 Q8 [12]}}