| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths polar coordinates question requiring symmetry verification (straightforward substitution), finding tangents at the pole (standard technique of solving r=0), and computing area using the polar area formula (routine integration). While Further Maths content is inherently harder, these are all standard textbook techniques with no novel insight required, placing it moderately above average difficulty. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\cos 3\theta = \cos(-3\theta)\) OR \(\cos\theta=\cos(-\theta)\) for all \(\theta\) | M1 | For a correct procedure for symmetry related to the equation OR to \(\cos 3\theta\) |
| \(\Rightarrow\) equation is unchanged, so symmetrical about \(\theta=0\) | A1 2 | For correct explanation relating to equation AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r=0 \Rightarrow \cos 3\theta=-1\) | M1 | For obtaining equation for tangents |
| \(\Rightarrow \theta=\pm\frac{1}{3}\pi,\,\pi\) | A1 | A1 for any 2 values |
| A1 3 | A1 for all, no extras (ignore outside range) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_{-\frac{1}{3}\pi}^{\frac{1}{3}\pi}\frac{1}{2}(1+\cos 3\theta)^2\,d\theta\) | B1 | For correct integral with limits soi (limits may be \(\left[0,\frac{1}{3}\pi\right]\) at any stage) |
| \(=\frac{1}{2}\int_{-\frac{1}{3}\pi}^{\frac{1}{3}\pi}1+2\cos 3\theta+\cos^2 3\theta\,d\theta\) | M1* | For multiplying out, giving at least 2 terms |
| \(=\frac{1}{2}\int_{-\frac{1}{3}\pi}^{\frac{1}{3}\pi}1+2\cos 3\theta+\frac{1}{2}(1+\cos 6\theta)\,d\theta\) | M1 | For integration to \(A\theta+B\sin 3\theta+C\sin 6\theta\) AEF |
| \(=\frac{1}{2}\left[\theta+\frac{2}{3}\sin 3\theta+\left(\frac{1}{2}\theta+\frac{1}{12}\sin 6\theta\right)\right]_{-\frac{1}{3}\pi}^{\frac{1}{3}\pi}\) | M1(*dep) | For completing integration and substituting their limits into terms in \(\frac{\cos}{\sin}n\theta\) |
| \(=\frac{1}{2}\pi\) | A1 5 | For correct area www |
## Question 7(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\cos 3\theta = \cos(-3\theta)$ OR $\cos\theta=\cos(-\theta)$ for all $\theta$ | M1 | For a correct procedure for symmetry related to the equation OR to $\cos 3\theta$ |
| $\Rightarrow$ equation is unchanged, so symmetrical about $\theta=0$ | A1 **2** | For correct explanation relating to equation **AG** |
## Question 7(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r=0 \Rightarrow \cos 3\theta=-1$ | M1 | For obtaining equation for tangents |
| $\Rightarrow \theta=\pm\frac{1}{3}\pi,\,\pi$ | A1 | A1 for any 2 values |
| | A1 **3** | A1 for all, no extras (ignore outside range) |
## Question 7(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_{-\frac{1}{3}\pi}^{\frac{1}{3}\pi}\frac{1}{2}(1+\cos 3\theta)^2\,d\theta$ | B1 | For correct integral with limits soi (limits may be $\left[0,\frac{1}{3}\pi\right]$ at any stage) |
| $=\frac{1}{2}\int_{-\frac{1}{3}\pi}^{\frac{1}{3}\pi}1+2\cos 3\theta+\cos^2 3\theta\,d\theta$ | M1* | For multiplying out, giving at least 2 terms |
| $=\frac{1}{2}\int_{-\frac{1}{3}\pi}^{\frac{1}{3}\pi}1+2\cos 3\theta+\frac{1}{2}(1+\cos 6\theta)\,d\theta$ | M1 | For integration to $A\theta+B\sin 3\theta+C\sin 6\theta$ **AEF** |
| $=\frac{1}{2}\left[\theta+\frac{2}{3}\sin 3\theta+\left(\frac{1}{2}\theta+\frac{1}{12}\sin 6\theta\right)\right]_{-\frac{1}{3}\pi}^{\frac{1}{3}\pi}$ | M1(*dep) | For completing integration and substituting their limits into terms in $\frac{\cos}{\sin}n\theta$ |
| $=\frac{1}{2}\pi$ | A1 **5** | For correct area **www** |
---7 A curve has polar equation $r = 1 + \cos 3 \theta$, for $- \pi < \theta \leqslant \pi$.\\
(i) Show that the line $\theta = 0$ is a line of symmetry.\\
(ii) Find the equations of the tangents at the pole.\\
(iii) Find the exact value of the area of the region enclosed by the curve between $\theta = - \frac { 1 } { 3 } \pi$ and $\theta = \frac { 1 } { 3 } \pi$.
\hfill \mbox{\textit{OCR FP2 2011 Q7 [10]}}