| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.8 This FP1 question requires geometric understanding of loci in the complex plane, visualization of tangent lines from the origin to a circle, and trigonometric calculation. While the circle sketch is routine, identifying maximum/minimum arguments requires insight about tangency, and calculating the angles involves non-trivial right-angle triangle trigonometry. It's moderately challenging for Further Maths but not exceptionally difficult. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Circle drawn | B1 | Circle, centre 4, B1; radius 3 with evidence of scale B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Tangent OA drawn; Tangent OB drawn | B1, B1 | Tangent OA; Tangent OB |
| Answer | Marks | Guidance |
|---|---|---|
| Region outside circle indicated; Correct region shown | B1, B1 | Region outside their circle indicated; Correct region shown |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha = \arcsin\dfrac{3}{4}\) | M1 | Valid method ft their tangents if circle centred on any axis |
| \(\alpha = 0.848\); \(\beta = -0.848\) | A2ft | One for each; accept \(48.6°\) and \(-48.6°\); A1 max if \(\alpha < \beta\) |
## Question 8:
### Part (i):
Circle drawn | B1 | Circle, centre 4, B1; radius 3 with evidence of scale B1 | **[3]**
### Part (ii):
Tangent OA drawn; Tangent OB drawn | B1, B1 | Tangent OA; Tangent OB | **[2]**
### Part (iii):
Region outside circle indicated; Correct region shown | B1, B1 | Region outside their circle indicated; Correct region shown | **[2]**
### Part (iv):
$\alpha = \arcsin\dfrac{3}{4}$ | M1 | Valid method ft their tangents if circle centred on any axis |
$\alpha = 0.848$; $\beta = -0.848$ | A2ft | One for each; accept $48.6°$ and $-48.6°$; A1 max if $\alpha < \beta$ | **[3]**
---8 (i) Sketch on an Argand diagram the locus, $C$, of points for which $| z - 4 | = 3$.\\
(ii) By drawing appropriate lines through the origin, indicate on your Argand diagram the point A on the locus $C$ where $\arg z$ has its maximum value. Indicate also the point B on the locus $C$ where $\arg z$ has its minimum value.\\
(iii) Given that $\arg z = \alpha$ at A and $\arg z = \beta$ at B , indicate on your Argand diagram the set of points for which $\beta \leqslant \arg z \leqslant \alpha$ and $| z - 4 | \geqslant 3$.\\
(iv) Calculate the value of $\alpha$ and the value of $\beta$.
\hfill \mbox{\textit{OCR MEI FP1 2012 Q8 [10]}}