| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Challenging +1.2 This is a multi-part locus question requiring visualization of a circle in the complex plane, geometric reasoning about distances, and optimization of argument on a circular boundary. While it involves several steps and requires spatial reasoning about tangent lines from the origin to find the maximum argument, the techniques are standard for FP1 and the geometric setup is relatively straightforward once the circle is sketched. The optimization is guided by the diagram rather than requiring algebraic methods. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Circle, radius 10, centre \((-8,\ 15)\), all points inside but not on circumference of correctly placed circle | B4 | Circle B1; radius 10, B1; centre \((-8,15)\) B1; all points inside but not on circumference of the correctly placed circle, B1 |
| The set of points for which \(\ | z-(-8+15j)\ | < 10\) is all points inside the circle, radius 10, centre \((-8,\ 15)\), excluding the points on the circumference. |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Origin to centre of circle \(= \sqrt{(-8)^2 + 15^2} = 17\) | M1 | Allow centre at \(\pm 8 \pm 15j\) and FT |
| Origin to centre of the circle \(\pm 10\) | M1 | Use of radius of circle |
| Point A is the point on the circle furthest from the origin. Since the radius of the circle is 10, \(OA = 27\). Point B is the point on the circle closest to the origin. Since the radius of the circle is 10, \(OB = 7\). Hence for \(z\) in the circle \(7 < | z | < 27\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| P is the point where a line from the origin is a tangent to the circle giving the greatest argument \(\theta\), \(-\pi < \theta \leq \pi\) | B1 | Correctly positioned on circle |
| \( | p | = \sqrt{17^2 - 10^2} = \sqrt{189} = 13.7\) (3 s.f.) |
| \(\arg p = \dfrac{\pi}{2} + \arcsin\dfrac{8}{17} + \arcsin\dfrac{10}{17}\) | M1 | Attempt to calculate the correct angle |
| \(= 2.69\) (3 s.f.) | A1 | cao Accept \(154°\) |
| [4] |
## Question 8:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Circle, radius 10, centre $(-8,\ 15)$, all points inside but not on circumference of correctly placed circle | B4 | Circle B1; radius 10, B1; centre $(-8,15)$ B1; all points inside but not on circumference of the correctly placed circle, B1 |
| The set of points for which $\|z-(-8+15j)\| < 10$ is all points inside the circle, radius 10, centre $(-8,\ 15)$, excluding the points on the circumference. | | The circle should be reasonably circular. Radius shown to be 10 by annotation. Centre point indicated and correct. Region shown by key or description. Accept a "dotty" outline to a shaded interior. Correctly placed: circle must lie above the Re axis and intersect the Im axis twice. |
| | **[4]** | |
## Question 8:
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Origin to centre of circle $= \sqrt{(-8)^2 + 15^2} = 17$ | M1 | Allow centre at $\pm 8 \pm 15j$ and FT |
| Origin to centre of the circle $\pm 10$ | M1 | Use of radius of circle |
| Point A is the point on the circle furthest from the origin. Since the radius of the circle is 10, $OA = 27$. Point B is the point on the circle closest to the origin. Since the radius of the circle is 10, $OB = 7$. Hence for $z$ in the circle $7 < |z| < 27$ | E1 | Correct explanation for both |
| **[3]** | | |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| P is the point where a line from the origin is a tangent to the circle giving the greatest argument $\theta$, $-\pi < \theta \leq \pi$ | B1 | Correctly positioned on circle | Allow circles centred as in (ii) |
| $|p| = \sqrt{17^2 - 10^2} = \sqrt{189} = 13.7$ (3 s.f.) | B1 | Accept $\sqrt{189}$ or $3\sqrt{21}$ or 13.7 |
| $\arg p = \dfrac{\pi}{2} + \arcsin\dfrac{8}{17} + \arcsin\dfrac{10}{17}$ | M1 | Attempt to calculate the correct angle | Correct circle only |
| $= 2.69$ (3 s.f.) | A1 | cao Accept $154°$ |
| **[4]** | | |
---8 (i) Indicate on an Argand diagram the set of points $z$ for which $| z - ( - 8 + 15 \mathrm { j } ) | < 10$.\\
(ii) Using the diagram, show that $7 < | z | < 27$.\\
(iii) Mark on your Argand diagram the point, $P$, at which $| z - ( - 8 + 15 \mathrm { j } ) | = 10$ and $\arg z$ takes its maximum value. Find the modulus and argument of $z$ at $P$.
\hfill \mbox{\textit{OCR MEI FP1 2013 Q8 [11]}}