| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.3 This is a straightforward FP1 loci question requiring standard techniques: recognizing |z| = |z - 4i| as a perpendicular bisector (the line y = 2), sketching arg z = π/6 as a half-line from the origin, then finding their intersection using basic trigonometry (2/tan(π/6) + 2i). While it involves multiple steps, each is routine for Further Maths students and requires no novel insight. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Horizontal straight line in 2 quadrants | B1 | |
| Through \((0, 2)\) | B1 | |
| Straight line | B1 | |
| Through \(O\) with positive slope | B1 | |
| In 1st quadrant only | B1 | 5 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or obtain algebraically that \(y = 2\) | B1 | |
| Use suitable trigonometry | M1 | |
| \(2\sqrt{3} + 2\text{i}\) | A1 | 3 marks Obtain correct answer a.e.f. decimals OK, must be a complex number |
# Question 6:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Horizontal straight line in 2 quadrants | B1 | |
| Through $(0, 2)$ | B1 | |
| Straight line | B1 | |
| Through $O$ with positive slope | B1 | |
| In 1st quadrant only | B1 | **5 marks** |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| State or obtain algebraically that $y = 2$ | B1 | |
| Use suitable trigonometry | M1 | |
| $2\sqrt{3} + 2\text{i}$ | A1 | **3 marks** Obtain correct answer a.e.f. decimals OK, must be a complex number |
---6 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by
$$| z | = | z - 4 \mathbf { i } | \quad \text { and } \quad \arg z = \frac { 1 } { 6 } \pi$$
respectively.\\
(i) Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.\\
(ii) Hence find, in the form $x +$ i $y$, the complex number represented by the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.
\hfill \mbox{\textit{OCR FP1 2008 Q6 [8]}}