| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Moderate -0.5 This is a straightforward Further Maths question requiring basic understanding of modulus-argument form and conversion to Cartesian form. Students sketch a circle and a ray, identify their intersection geometrically, then use standard trigonometry (cos π/3, sin π/3) to find the Cartesian coordinates. While it's Further Maths content, it's a routine textbook exercise with no problem-solving insight required. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| - In 1st quadrant only | B1, B1, B1, B1, 5 | |
| Part (ii) \(1 + \sqrt{3}\) | M1 | Attempt to find intersections by trig, solving equations or from graph |
| A1, 2 | Correct answer stated as complex number |
**Part (i)** Sketch showing correct features:
- Circle, Centre $O$, radius 2
- One straight line
- Through $O$ with $+ve$ slope
- In 1st quadrant only | B1, B1, B1, B1, 5 |
**Part (ii)** $1 + \sqrt{3}$ | M1 | Attempt to find intersections by trig, solving equations or from graph
| A1, 2 | Correct answer stated as complex number6 In an Argand diagram the loci $C _ { 1 }$ and $C _ { 2 }$ are given by
$$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$
respectively.\\
(i) Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Hence find, in the form $x + \mathrm { i } y$, the complex number representing the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.
\hfill \mbox{\textit{OCR FP1 2006 Q6 [7]}}