| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Standard +0.8 This is a multi-part polar coordinates question requiring: (i) solving |cos θ| = 1/2 for intersections, (ii) sketching a circle and figure-eight curve with symmetry, and (iii) computing area using polar integration with careful attention to which curve bounds which region. The absolute value and composite region make this moderately challenging, requiring solid understanding of polar area formulas and geometric visualization, but the techniques are standard for Further Maths. |
| Spec | 1.05g Exact trigonometric values: for standard angles4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = 2a | \cos\theta | \Rightarrow \cos\theta = \pm\frac{1}{2}\) |
| \(\left(a, \frac{\pi}{3}\right)\) and \(\left(a, \frac{2\pi}{3}\right)\) | A1 | Both points needed for A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Semicircle for C1 including \(r = a\) | B1 | Semicircle for C1 including \(r = a\) |
| Half of C2 including \(r = 2a\) | B1 | Half of C2 including \(r = 2a\) |
| Other half of C2 and line of symmetry | B1 | Other half of C2 and line of symmetry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4a^2\int_{\pi/3}^{\pi/2}\cos^2\theta\, d\theta\) | M1 | Finds area of segment \(OP_1\) and \(OP_2\) of \(C_2\) |
| \(= 2a^2\int_{\pi/3}^{\pi/2}\cos 2\theta + 1\, d\theta\) | M1 | Uses \(\cos^2\theta = \frac{1}{2}(\cos 2\theta + 1)\) |
| \(= 2a^2\left[\frac{1}{2}\sin 2\theta + \theta\right]_{\pi/3}^{\pi/2} = 2a^2\left(\frac{\pi}{2} - \left(\frac{\sqrt{3}}{4} + \frac{\pi}{3}\right)\right) = a^2\left(\frac{\pi}{3} - \frac{\sqrt{3}}{2}\right)\) | A1 | Integrates correctly |
| Area \(= \frac{\pi a^2}{6} - a^2\left(\frac{\pi}{3} - \frac{\sqrt{3}}{2}\right) = -\frac{\pi a^2}{6} + \frac{a^2\sqrt{3}}{2}\) | M1 A1FT | M1 for subtracting 'their' \(OP_1P_2\) from \(\frac{\pi a^2}{6}\) |
## Question 8:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = 2a|\cos\theta| \Rightarrow \cos\theta = \pm\frac{1}{2}$ | M1 | Eliminates $r$ |
| $\left(a, \frac{\pi}{3}\right)$ and $\left(a, \frac{2\pi}{3}\right)$ | A1 | Both points needed for A1 |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Semicircle for C1 including $r = a$ | B1 | Semicircle for C1 including $r = a$ |
| Half of C2 including $r = 2a$ | B1 | Half of C2 including $r = 2a$ |
| Other half of C2 and line of symmetry | B1 | Other half of C2 and line of symmetry |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4a^2\int_{\pi/3}^{\pi/2}\cos^2\theta\, d\theta$ | M1 | Finds area of segment $OP_1$ and $OP_2$ of $C_2$ |
| $= 2a^2\int_{\pi/3}^{\pi/2}\cos 2\theta + 1\, d\theta$ | M1 | Uses $\cos^2\theta = \frac{1}{2}(\cos 2\theta + 1)$ |
| $= 2a^2\left[\frac{1}{2}\sin 2\theta + \theta\right]_{\pi/3}^{\pi/2} = 2a^2\left(\frac{\pi}{2} - \left(\frac{\sqrt{3}}{4} + \frac{\pi}{3}\right)\right) = a^2\left(\frac{\pi}{3} - \frac{\sqrt{3}}{2}\right)$ | A1 | Integrates correctly |
| Area $= \frac{\pi a^2}{6} - a^2\left(\frac{\pi}{3} - \frac{\sqrt{3}}{2}\right) = -\frac{\pi a^2}{6} + \frac{a^2\sqrt{3}}{2}$ | M1 A1FT | M1 for subtracting 'their' $OP_1P_2$ from $\frac{\pi a^2}{6}$ |
---8 The curves $C _ { 1 }$ and $C _ { 2 }$ have polar equations, for $0 \leqslant \theta \leqslant \pi$, as follows:
$$\begin{aligned}
& C _ { 1 } : r = a \\
& C _ { 2 } : r = 2 a | \cos \theta |
\end{aligned}$$
where $a$ is a positive constant. The curves intersect at the points $P _ { 1 }$ and $P _ { 2 }$.\\
(i) Find the polar coordinates of $P _ { 1 }$ and $P _ { 2 }$.\\
(ii) In a single diagram, sketch $C _ { 1 } , C _ { 2 }$ and their line of symmetry.\\
(iii) The region $R$ enclosed by $C _ { 1 }$ and $C _ { 2 }$ is bounded by the $\operatorname { arcs } O P _ { 1 } , P _ { 1 } P _ { 2 }$ and $P _ { 2 } O$, where $O$ is the pole. Find the area of $R$, giving your answer in exact form.\\
\hfill \mbox{\textit{CAIE FP1 2018 Q8 [10]}}