| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2018 |
| Session | September |
| Marks | 6 |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a standard Further Maths locus question requiring students to sketch a circle centered at (1,0) with radius 5 and a half-line from (-4,-4) at angle π/4, then shade their intersection. While it involves multiple loci and set notation, the techniques are routine for FM students with no novel problem-solving required. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(C_1\): Circle, centre \((1,0)\) radius 5 | B1, B1 [4] | Soi eg by a circle through points \((6,0)\) and \((-4,0)\) |
| \(C_2\): Half line starting at (but not including) \((-4, -4)\) Gradient 1 (i.e. through \((0,0)\)) | B1, B1 [4] | |
| (ii) Shading: Inside circle, Under line but above horizontal through \((-4,-4)\). But nothing extra | B1, B1 [2] | Withhold one mark for anything extra or omissions |
**(i)** $C_1$: Circle, centre $(1,0)$ radius 5 | B1, B1 [4] | Soi eg by a circle through points $(6,0)$ and $(-4,0)$
$C_2$: Half line starting at (but not including) $(-4, -4)$ Gradient 1 (i.e. through $(0,0)$) | B1, B1 [4]
**(ii)** Shading: Inside circle, Under line but above horizontal through $(-4,-4)$. But nothing extra | B1, B1 [2] | Withhold one mark for anything extra or omissions
---2 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z - 1 | = 5$ and $\arg ( z + 4 + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi$ respectively.\\
(i) Sketch on a single Argand diagram the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Indicate by shading on your Argand diagram the following set of points.
$$\{ z : | z - 1 | \leqslant 5 \} \cap \left\{ z : 0 \leqslant \arg ( z + 4 + 4 i ) \leqslant \frac { 1 } { 4 } \pi \right\}$$
\hfill \mbox{\textit{OCR Further Pure Core 1 2018 Q2 [6]}}