| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Find constant from given area |
| Difficulty | Standard +0.8 This is a Further Maths FP2 polar coordinates question requiring knowledge of the area formula (½∫r²dθ), expansion of (a+3cosθ)², integration of cos²θ using double angle formulas, and solving for the constant a. While the integration is standard for FP2, the multi-step algebraic manipulation and the need to recall/apply the polar area formula makes this moderately challenging, above average difficulty. |
| Spec | 1.08d Evaluate definite integrals: between limits4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| \(A = \frac{1}{2}\int_0^{2\pi}(a+3\cos\theta)^2\,\mathrm{d}\theta\) | B1 | Applies \(\frac{1}{2}\int_0^{2\pi} r^2\,\mathrm{d}\theta\) with correct limits (ignore \(\mathrm{d}\theta\)) |
| \((a+3\cos\theta)^2 = a^2+6a\cos\theta+9\cos^2\theta = a^2+6a\cos\theta+9\!\left(\frac{1+\cos 2\theta}{2}\right)\) | M1 | \(\cos^2\theta = \frac{\pm 1 \pm \cos 2\theta}{2}\) |
| A1 | Correct underlined expression | |
| \(= \frac{1}{2}\!\left[a^2\theta+6a\sin\theta+\frac{9}{2}\theta+\frac{9}{4}\sin 2\theta\right]_0^{2\pi}\) | M1* | Integrated with at least 3 of 4 terms of form \(\pm A\theta \pm B\sin\theta \pm C\theta \pm D\sin 2\theta\) |
| A1 ft | \(a^2\theta+6a\sin\theta+\) correct further integration (ignore \(\frac{1}{2}\), ignore limits) | |
| \(= \frac{1}{2}\!\left[(2\pi a^2+0+9\pi+0)-(0)\right] = \pi a^2+\frac{9\pi}{2}\) | A1 | \(\pi a^2+\frac{9\pi}{2}\) |
| \(\pi a^2+\frac{9\pi}{2} = \frac{107}{2}\pi \Rightarrow a^2+\frac{9}{2}=\frac{107}{2} \Rightarrow a^2=49\) | dM1* | Integrated expression equal to \(\frac{107}{2}\pi\) |
| \(a = 7\) (since \(a>0\)) | A1 cso | \(a=7\) (8 marks) |
# Question 4:
| $A = \frac{1}{2}\int_0^{2\pi}(a+3\cos\theta)^2\,\mathrm{d}\theta$ | B1 | Applies $\frac{1}{2}\int_0^{2\pi} r^2\,\mathrm{d}\theta$ with correct limits (ignore $\mathrm{d}\theta$) |
|---|---|---|
| $(a+3\cos\theta)^2 = a^2+6a\cos\theta+9\cos^2\theta = a^2+6a\cos\theta+9\!\left(\frac{1+\cos 2\theta}{2}\right)$ | M1 | $\cos^2\theta = \frac{\pm 1 \pm \cos 2\theta}{2}$ |
| | A1 | Correct underlined expression |
| $= \frac{1}{2}\!\left[a^2\theta+6a\sin\theta+\frac{9}{2}\theta+\frac{9}{4}\sin 2\theta\right]_0^{2\pi}$ | M1* | Integrated with at least 3 of 4 terms of form $\pm A\theta \pm B\sin\theta \pm C\theta \pm D\sin 2\theta$ |
| | A1 ft | $a^2\theta+6a\sin\theta+$ correct further integration (ignore $\frac{1}{2}$, ignore limits) |
| $= \frac{1}{2}\!\left[(2\pi a^2+0+9\pi+0)-(0)\right] = \pi a^2+\frac{9\pi}{2}$ | A1 | $\pi a^2+\frac{9\pi}{2}$ |
| $\pi a^2+\frac{9\pi}{2} = \frac{107}{2}\pi \Rightarrow a^2+\frac{9}{2}=\frac{107}{2} \Rightarrow a^2=49$ | dM1* | Integrated expression equal to $\frac{107}{2}\pi$ |
| $a = 7$ (since $a>0$) | A1 **cso** | $a=7$ (8 marks) |
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0626e500-8ae5-4c98-82bb-a4536de11bf9-05_428_803_233_577}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve with polar equation
$$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leqslant \theta < 2 \pi$$
The area enclosed by the curve is $\frac { 107 } { 2 } \pi$.\\
Find the value of $a$.\\
\hfill \mbox{\textit{Edexcel FP2 2009 Q4 [8]}}