| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Challenging +1.2 This is a multi-part polar coordinates question requiring conversion between Cartesian and polar forms, area calculation using the standard polar area formula, and finding intersection points. While it involves several steps and techniques (completing the square, polar area integration, solving trigonometric equations), these are all standard procedures for Further Maths students. Part (c) requires some geometric insight to identify the quadrilateral, but the overall question follows predictable patterns for FP3 polar coordinate problems without requiring novel problem-solving approaches. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x^2 + y^2 - 12y + 36 = 36\) | M1 | Use of \(y = r\sin\theta\) (\(x = r\cos\theta\) PI) |
| M1 | Use of \(x^2 + y^2 = r^2\) | |
| \(r^2 - 12r\sin\theta + 36 = 36\) | m1 | |
| \(\Rightarrow r = 12\sin\theta\) | A1 | CSO AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Area \(= \frac{1}{2}\int(2\sin\theta+5)^2\,d\theta\) | M1 | Use of \(\frac{1}{2}\int r^2\,d\theta\) |
| \(= \frac{1}{2}\int_0^{2\pi}(4\sin^2\theta + 20\sin\theta + 25)\,d\theta\) | B1 | Correct expn. of \((2\sin\theta+5)^2\) |
| B1 | Correct limits | |
| \(= \frac{1}{2}\int_0^{2\pi}(2(1-\cos 2\theta) + 20\sin\theta + 25)\,d\theta\) | M1 | Attempt to write \(\sin^2\theta\) in terms of \(\cos 2\theta\) |
| \(= \frac{1}{2}\left[27\theta - \sin 2\theta - 20\cos\theta\right]_0^{2\pi}\) | A1\(\checkmark\) | Correct integration ft wrong coeffs |
| \(= 27\pi\) | A1 | CSO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| At intersection \(12\sin\theta = 2\sin\theta + 5\) | M1 | OE e.g. \(r = 6(r-5)\) |
| \(\Rightarrow \sin\theta = \frac{5}{10}\) | A1 | OE e.g. \(r = 6\) |
| Points \(\left(6, \frac{\pi}{6}\right)\) and \(\left(6, \frac{5\pi}{6}\right)\) | A1 | OE |
| \(OPMQ\) is a rhombus of side 6 | Or two equilateral triangles of side 6 | |
| Area \(= 6\times 6\times\sin\frac{2\pi}{3}\) oe | M1 | Any valid complete method to find the area (or half area) of quadrilateral |
| A1 | ||
| \(= 18\sqrt{3}\) | A1 | Accept unsimplified surd |
## Question 6:
---
**Question 6(a):**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x^2 + y^2 - 12y + 36 = 36$ | M1 | Use of $y = r\sin\theta$ ($x = r\cos\theta$ PI) |
| | M1 | Use of $x^2 + y^2 = r^2$ |
| $r^2 - 12r\sin\theta + 36 = 36$ | m1 | |
| $\Rightarrow r = 12\sin\theta$ | A1 | **CSO AG** |
---
**Question 6(b):**
| Answer/Working | Marks | Guidance |
|---|---|---|
| Area $= \frac{1}{2}\int(2\sin\theta+5)^2\,d\theta$ | M1 | Use of $\frac{1}{2}\int r^2\,d\theta$ |
| $= \frac{1}{2}\int_0^{2\pi}(4\sin^2\theta + 20\sin\theta + 25)\,d\theta$ | B1 | Correct expn. of $(2\sin\theta+5)^2$ |
| | B1 | Correct limits |
| $= \frac{1}{2}\int_0^{2\pi}(2(1-\cos 2\theta) + 20\sin\theta + 25)\,d\theta$ | M1 | Attempt to write $\sin^2\theta$ in terms of $\cos 2\theta$ |
| $= \frac{1}{2}\left[27\theta - \sin 2\theta - 20\cos\theta\right]_0^{2\pi}$ | A1$\checkmark$ | Correct integration ft wrong coeffs |
| $= 27\pi$ | A1 | CSO |
---
**Question 6(c):**
| Answer/Working | Marks | Guidance |
|---|---|---|
| At intersection $12\sin\theta = 2\sin\theta + 5$ | M1 | OE e.g. $r = 6(r-5)$ |
| $\Rightarrow \sin\theta = \frac{5}{10}$ | A1 | OE e.g. $r = 6$ |
| Points $\left(6, \frac{\pi}{6}\right)$ and $\left(6, \frac{5\pi}{6}\right)$ | A1 | OE |
| $OPMQ$ is a rhombus of side 6 | | Or two equilateral triangles of side 6 |
| Area $= 6\times 6\times\sin\frac{2\pi}{3}$ oe | M1 | Any valid complete method to find the area (or half area) of quadrilateral |
| | A1 | |
| $= 18\sqrt{3}$ | A1 | Accept unsimplified surd |
---6
\begin{enumerate}[label=(\alph*)]
\item A circle $C _ { 1 }$ has cartesian equation $x ^ { 2 } + ( y - 6 ) ^ { 2 } = 36$. Show that the polar equation of $C _ { 1 }$ is $r = 12 \sin \theta$.
\item A curve $C _ { 2 }$ with polar equation $r = 2 \sin \theta + 5,0 \leqslant \theta \leqslant 2 \pi$ is shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{b572aeb5-bcbb-4d50-964c-7f37e223f51d-5_545_837_559_651}
Calculate the area bounded by $C _ { 2 }$.
\item The circle $C _ { 1 }$ intersects the curve $C _ { 2 }$ at the points $P$ and $Q$. Find, in surd form, the area of the quadrilateral $O P M Q$, where $M$ is the centre of the circle and $O$ is the pole.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2006 Q6 [16]}}