| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Challenging +1.2 This is a multi-part polar coordinates question requiring standard techniques: verification by substitution (trivial), sketching a limaçon curve (routine), computing area using the standard polar integral formula, and finding a triangle area using polar coordinates. Part (d) requires recognizing that Q is diametrically opposite P and applying the triangle area formula, which is moderately challenging but follows established methods. Overall, this is above average difficulty due to the multi-step nature and part (d)'s geometric reasoning, but all techniques are standard for Further Maths students. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| \(\theta = 0\), \(r = 5 + 2\cos 0 = 7\) \(\{A \text{ lies on } C\}\) | B1 | |
| \(\theta = \pi\), \(r = 5 + 2\cos\pi = 3\) \(\{B \text{ lies on } C\}\) | B1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Closed single loop curve with (indication of) symmetry | B1 | |
| Critical values 3, 5, 7 indicated | B1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Area \(= \frac{1}{2}\int(5+2\cos\theta)^2\,d\theta\) | M1 | Use of \(\frac{1}{2}\int r^2\,d\theta\) |
| \(= \frac{1}{2}\int_{-\pi}^{\pi}\left(25 + 20\cos\theta + 4\cos^2\theta\right)d\theta\) | B1, B1 | OE for correct expansion; correct limits |
| \(= \frac{1}{2}\int_{-\pi}^{\pi}\left(25 + 20\cos\theta + 2(\cos 2\theta+1)\right)d\theta\) | M1 | Attempt to write \(\cos^2\theta\) in terms of \(\cos 2\theta\) |
| \(= \frac{1}{2}\left[27\theta + 20\sin\theta + \sin 2\theta\right]_{-\pi}^{\pi}\) | A1F | Correct integration ft wrong non-zero coefficients in \(a + b\cos\theta + c\cos 2\theta\) |
| \(= 27\pi\) | A1 | Total: 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Triangle \(OBQ\) with \(OB = 3\) and angle \(BOQ = \alpha\) | B1 | PI |
| \(OQ = 5 + 2\cos(-\pi + \alpha)\) | M1 | OE |
| Area of triangle \(OQB = \frac{1}{2}OB \times OQ\sin\alpha\) | m1 | Dep. on correct method to find \(OQ\) |
| \(= \frac{3}{2}(5 - 2\cos\alpha)\sin\alpha\) | A1 | Total: 4 |
# Question 8:
## Part (a):
$\theta = 0$, $r = 5 + 2\cos 0 = 7$ $\{A \text{ lies on } C\}$ | B1 | |
$\theta = \pi$, $r = 5 + 2\cos\pi = 3$ $\{B \text{ lies on } C\}$ | B1 | Total: 2 | |
## Part (b):
Closed single loop curve with (indication of) symmetry | B1 | | |
Critical values 3, 5, 7 indicated | B1 | Total: 2 | |
## Part (c):
Area $= \frac{1}{2}\int(5+2\cos\theta)^2\,d\theta$ | M1 | Use of $\frac{1}{2}\int r^2\,d\theta$ |
$= \frac{1}{2}\int_{-\pi}^{\pi}\left(25 + 20\cos\theta + 4\cos^2\theta\right)d\theta$ | B1, B1 | OE for correct expansion; correct limits |
$= \frac{1}{2}\int_{-\pi}^{\pi}\left(25 + 20\cos\theta + 2(\cos 2\theta+1)\right)d\theta$ | M1 | Attempt to write $\cos^2\theta$ in terms of $\cos 2\theta$ |
$= \frac{1}{2}\left[27\theta + 20\sin\theta + \sin 2\theta\right]_{-\pi}^{\pi}$ | A1F | Correct integration ft wrong non-zero coefficients in $a + b\cos\theta + c\cos 2\theta$ |
$= 27\pi$ | A1 | Total: 6 | CSO |
## Part (d):
Triangle $OBQ$ with $OB = 3$ and angle $BOQ = \alpha$ | B1 | PI |
$OQ = 5 + 2\cos(-\pi + \alpha)$ | M1 | OE |
Area of triangle $OQB = \frac{1}{2}OB \times OQ\sin\alpha$ | m1 | Dep. on correct method to find $OQ$ |
$= \frac{3}{2}(5 - 2\cos\alpha)\sin\alpha$ | A1 | Total: 4 | CSO |8 The polar equation of a curve $C$ is
$$r = 5 + 2 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$
\begin{enumerate}[label=(\alph*)]
\item Verify that the points $A$ and $B$, with polar coordinates ( 7,0 ) and ( $3 , \pi$ ) respectively, lie on the curve $C$.
\item Sketch the curve $C$.
\item Find the area of the region bounded by the curve $C$.
\item The point $P$ is the point on the curve $C$ for which $\theta = \alpha$, where $0 < \alpha \leqslant \frac { \pi } { 2 }$. The point $Q$ lies on the curve such that $P O Q$ is a straight line, where the point $O$ is the pole. Find, in terms of $\alpha$, the area of triangle $O Q B$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2008 Q8 [14]}}