| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Sketch polar curve |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths polar coordinates question requiring standard techniques: finding tangents at the pole (where r=0), maximizing r using calculus, and converting to Cartesian form. While it involves several steps and Further Maths content, each part follows routine procedures without requiring novel insight or particularly complex manipulation. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(r=0 \Rightarrow \cos\theta=0,\,\sin 2\theta=0\) | M1 | For \(r=0\) (soi) and attempt to solve for \(\theta\) |
| \(\Rightarrow \theta=0,\,\frac{1}{2}\pi\) | A1 | For both values and no others (ignore values outside range) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dr}{d\theta}=-\sin\theta\sin 2\theta+2\cos 2\theta\cos\theta\) | M1 | For attempt to find \(\frac{dr}{d\theta}\) using product rule |
| \(=0\) | A1 | For correct \(\frac{dr}{d\theta}\) set \(=0\) soi |
| *Alternatively:* \(r=2\cos^2\theta\sin\theta \Rightarrow \frac{dr}{d\theta}=2\cos^3\theta-4\cos\theta\sin^2\theta\) | ||
| \(\Rightarrow 2\sin^2\theta\cos\theta=2(1-2\sin^2\theta)\cos\theta\) | ||
| \(\Rightarrow \sin\theta=\frac{1}{\sqrt{3}}\left(\cos\theta=\frac{\sqrt{2}}{\sqrt{3}},\;\tan\theta=\frac{1}{\sqrt{2}}\right)\) | A1 | For correct value of \(\sin\theta\) (OR \(\cos\theta\) OR \(\tan\theta\)) or decimal equivalent; \(\sin\theta=0.546\) or \(\cos\theta=0.816\) or \(\tan\theta=0.707\) |
| \(\Rightarrow r=\frac{4}{3\sqrt{3}}=\frac{4}{9}\sqrt{3}\) | A1 | For correct \(r\) or anything that rounds to 0.77 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x=r\cos\theta,\;y=r\sin\theta\) | M1 | For substituting \(x=r\cos\theta\) OR \(y=r\sin\theta\) |
| \(\Rightarrow r=\frac{x}{r}\cdot 2\cdot\frac{y}{r}\cdot\frac{x}{r}\) | M1 | For \(r^2=x^2+y^2\) soi |
| \(\Rightarrow (x^2+y^2)^2=2x^2y\) | A1 | For a correct Cartesian equation. Any equivalent form without fractions |
# Question 2:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $r=0 \Rightarrow \cos\theta=0,\,\sin 2\theta=0$ | M1 | For $r=0$ (soi) and attempt to solve for $\theta$ |
| $\Rightarrow \theta=0,\,\frac{1}{2}\pi$ | A1 | For both values and no others (ignore values outside range) |
**[2 marks]**
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dr}{d\theta}=-\sin\theta\sin 2\theta+2\cos 2\theta\cos\theta$ | M1 | For attempt to find $\frac{dr}{d\theta}$ using product rule |
| $=0$ | A1 | For correct $\frac{dr}{d\theta}$ set $=0$ soi |
| *Alternatively:* $r=2\cos^2\theta\sin\theta \Rightarrow \frac{dr}{d\theta}=2\cos^3\theta-4\cos\theta\sin^2\theta$ | | |
| $\Rightarrow 2\sin^2\theta\cos\theta=2(1-2\sin^2\theta)\cos\theta$ | | |
| $\Rightarrow \sin\theta=\frac{1}{\sqrt{3}}\left(\cos\theta=\frac{\sqrt{2}}{\sqrt{3}},\;\tan\theta=\frac{1}{\sqrt{2}}\right)$ | A1 | For correct value of $\sin\theta$ (OR $\cos\theta$ OR $\tan\theta$) or decimal equivalent; $\sin\theta=0.546$ or $\cos\theta=0.816$ or $\tan\theta=0.707$ |
| $\Rightarrow r=\frac{4}{3\sqrt{3}}=\frac{4}{9}\sqrt{3}$ | A1 | For correct $r$ or anything that rounds to 0.77 |
**[4 marks]**
## Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x=r\cos\theta,\;y=r\sin\theta$ | M1 | For substituting $x=r\cos\theta$ **OR** $y=r\sin\theta$ |
| $\Rightarrow r=\frac{x}{r}\cdot 2\cdot\frac{y}{r}\cdot\frac{x}{r}$ | M1 | For $r^2=x^2+y^2$ soi |
| $\Rightarrow (x^2+y^2)^2=2x^2y$ | A1 | For a correct Cartesian equation. Any equivalent form without fractions |
**[3 marks]**
---2 A curve has polar equation $r = \cos \theta \sin 2 \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. Find\\
(i) the equations of the tangents at the pole,\\
(ii) the maximum value of $r$,\\
(iii) a cartesian equation of the curve, in a form not involving fractions.
\hfill \mbox{\textit{OCR FP2 2012 Q2 [9]}}