| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 8 |
| 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 straightforward Further Maths question requiring students to sketch standard loci (a circle and a half-line) and shade a region defined by simple inequalities. While it's from FP1, the techniques are routine: recognizing |z-a|=r as a circle and arg(z-a)=θ as a half-line, then correctly interpreting inequalities. No problem-solving insight or complex reasoning required, just careful application of standard methods. |
| Spec | 4.02o Loci in Argand diagram: circles, half-lines4.02p Set notation: for loci |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Circle, centre \((3, 0)\), y-axis a tangent at origin. Straight line, through \((1, 0)\) with +ve slope. In 1st quadrant only. | B1B1 B1 B1 B1 B2ft | Sketch showing correct features. N.B. treat 2 diagrams as MR. |
| (ii) Inside circle, below line, above x-axis. |
(i) Circle, centre $(3, 0)$, y-axis a tangent at origin. Straight line, through $(1, 0)$ with +ve slope. In 1st quadrant only. | B1B1 B1 B1 B1 B2ft | Sketch showing correct features. N.B. treat 2 diagrams as MR. |
(ii) Inside circle, below line, above x-axis. | | |
**Total: 6 + 2 = 8 marks**8 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z - 3 | = 3$ and arg $( z - 1 ) = \frac { 1 } { 4 } \pi$ respectively.\\
(i) Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Indicate, by shading, the region of the Argand diagram for which
$$| z - 3 | \leqslant 3 \text { and } 0 \leqslant \arg ( z - 1 ) \leqslant \frac { 1 } { 4 } \pi$$
\hfill \mbox{\textit{OCR FP1 2007 Q8 [8]}}