| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Maximum/minimum distance from pole or line |
| Difficulty | Challenging +1.2 This is a multi-part polar coordinates question requiring area integration, distance calculations, and optimization. While it involves several techniques (polar area formula, converting to Cartesian distance, finding maxima), each part follows standard procedures without requiring novel insight. The integration is straightforward, and the optimization is routine calculus. Slightly above average difficulty due to the multi-step nature and Further Maths content, but not exceptionally challenging. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{25}{2}\int_{0.01}^{\frac{\pi}{2}} \cot\theta \, d\theta\) | M1 | Uses \(\frac{1}{2}\int r^2 \, d\theta\) |
| \(= \frac{25}{2}[\ln\sin\theta]_{0.01}^{\frac{\pi}{2}}\) | A1 | |
| \(= -\frac{25}{2}\ln\sin 0.01 \approx 57.6\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = 5\cos^{\frac{1}{2}}\theta\sin^{\frac{1}{2}}\theta = \frac{5}{\sqrt{2}}\sin^{\frac{1}{2}} 2\theta\) | M1 | Uses \(y = r\sin\theta\) |
| \(\theta = 0.01 \Rightarrow y \approx 0.5\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{d\theta} = \frac{5}{\sqrt{2}}\sin^{-\frac{1}{2}} 2\theta \cos 2\theta = 0\) or \(\max(\sin 2\theta) = 1\) | M1 A1 | Sets \(\frac{dy}{d\theta} = 0\) or considers max (AEF) |
| \(\Rightarrow y = \frac{5\sqrt{2}}{2}\) \((= 3.54)\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Curve intersecting initial line only when \(x=0\) and \(y=0\)] | B1 | Intersecting the initial line only when \(x = 0\) and \(y = 0\) |
| [Correct shape shown] | B1 | Correct shape |
| 2 |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{25}{2}\int_{0.01}^{\frac{\pi}{2}} \cot\theta \, d\theta$ | M1 | Uses $\frac{1}{2}\int r^2 \, d\theta$ |
| $= \frac{25}{2}[\ln\sin\theta]_{0.01}^{\frac{\pi}{2}}$ | A1 | |
| $= -\frac{25}{2}\ln\sin 0.01 \approx 57.6$ | A1 | |
| | **3** | |
## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = 5\cos^{\frac{1}{2}}\theta\sin^{\frac{1}{2}}\theta = \frac{5}{\sqrt{2}}\sin^{\frac{1}{2}} 2\theta$ | M1 | Uses $y = r\sin\theta$ |
| $\theta = 0.01 \Rightarrow y \approx 0.5$ | A1 | |
| | **2** | |
## Question 9(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{d\theta} = \frac{5}{\sqrt{2}}\sin^{-\frac{1}{2}} 2\theta \cos 2\theta = 0$ or $\max(\sin 2\theta) = 1$ | M1 A1 | Sets $\frac{dy}{d\theta} = 0$ or considers max (AEF) |
| $\Rightarrow y = \frac{5\sqrt{2}}{2}$ $(= 3.54)$ | A1 | |
| | **3** | |
## Question 9(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Curve intersecting initial line only when $x=0$ and $y=0$] | B1 | Intersecting the initial line only when $x = 0$ and $y = 0$ |
| [Correct shape shown] | B1 | Correct shape |
| | **2** | |9 The curve $C$ has polar equation
$$r = 5 \sqrt { } ( \cot \theta ) ,$$
where $0.01 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.\\
(i) Find the area of the finite region bounded by $C$ and the line $\theta = 0.01$, showing full working. Give your answer correct to 1 decimal place.\\
Let $P$ be the point on $C$ where $\theta = 0.01$.\\
(ii) Find the distance of $P$ from the initial line, giving your answer correct to 1 decimal place.\\
(iii) Find the maximum distance of $C$ from the initial line.\\
(iv) Sketch $C$.
\hfill \mbox{\textit{CAIE FP1 2018 Q9}}