| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Standard +0.8 This is a multi-part polar coordinates question requiring sketching a rose curve, applying the polar area formula with integration, and converting to Cartesian form using double-angle identities. While the techniques are standard for Further Maths, the cos 2θ curve requires careful handling of the restricted domain and the integration involves trigonometric manipulation beyond basic A-level, making it moderately challenging. |
| 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 | Marks | Guidance |
| Correct position including label of 1 on initial line, symmetric about initial line | B1 | Correct position including label of 1 on initial line, and symmetric about initial |
| Single correct loop | B1 | Single correct loop |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos^2 2\theta \, d\theta\) | M1 | Correct integral |
| \(= \frac{1}{4}\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos 4\theta + 1 \, d\theta\) | M1 | Correct use of double angle formula |
| \(\frac{1}{4}\left[\frac{1}{4}\sin 4\theta + \theta\right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} = \frac{\pi}{8} = 0.393\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(r = \cos^2\theta - \sin^2\theta\) OE | B1 | Uses trig identity |
| Thus \(\left(x^2 + y^2\right)^{\frac{3}{2}} = x^2 - y^2\) | M1 A1 | Uses \(x = r\cos\theta\) or \(y = r\sin\theta\) or both, AEF |
## Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct position including label of 1 on initial line, symmetric about initial line | B1 | Correct position including label of 1 on initial line, and symmetric about initial |
| Single correct loop | B1 | Single correct loop |
## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos^2 2\theta \, d\theta$ | M1 | Correct integral |
| $= \frac{1}{4}\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos 4\theta + 1 \, d\theta$ | M1 | Correct use of double angle formula |
| $\frac{1}{4}\left[\frac{1}{4}\sin 4\theta + \theta\right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} = \frac{\pi}{8} = 0.393$ | A1 | |
## Question 3(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $r = \cos^2\theta - \sin^2\theta$ OE | B1 | Uses trig identity |
| Thus $\left(x^2 + y^2\right)^{\frac{3}{2}} = x^2 - y^2$ | M1 A1 | Uses $x = r\cos\theta$ or $y = r\sin\theta$ or both, AEF |3 The curve $C$ has polar equation $r = \cos 2 \theta$, for $- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$.\\
(i) Sketch $C$.\\
(ii) Find the area of the region enclosed by $C$, showing full working.\\
(iii) Find a cartesian equation of $C$.\\
\hfill \mbox{\textit{CAIE FP1 2018 Q3 [8]}}