| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Standard +0.8 This is a standard cardioid problem in Further Maths polar coordinates. Part (i) requires finding where r=0 and using the tangent formula, part (ii) is routine sketching, and part (iii) applies the standard polar area formula ½∫r²dθ. While it requires multiple techniques and is Further Maths content (inherently harder), it's a textbook example with no novel insight needed—straightforward application of learned methods across multiple parts. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow \theta = \pi\) | M1, A1 | soi; Only this answer: A0 if anything else |
| [2] | ||
| (ii) | B1, B1 | Correct shape, correct orientation, roughly symmetric; All 3 intersections on axes indicated, cusp at pole dep on 1st B. |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \frac{a^2}{2} \cdot 3\pi = \frac{3\pi a^2}{2}\) | M1, A1, M1, A1, M1, A1 | Use of formula with limits; Condone omission of \(a^2\); Dealing with \(\cos^2\). Condone omission of \(a^2\); Substitute limits dep on 2nd M |
| [6] |
**(i)** $a 1+\cos\theta = 0$ when $\cos\theta = -1$
$\Rightarrow \theta = \pi$ | M1, A1 | soi; Only this answer: A0 if anything else
| [2] |
**(ii)** | B1, B1 | Correct shape, correct orientation, roughly symmetric; All 3 intersections on axes indicated, cusp at pole dep on 1st B.
| [2] |
**(iii)** $r = a(1+\cos\theta)$
$A = \frac{1}{2}\int_0^{2\pi} r^2 \, d\theta = \frac{1}{2}\int_0^{2\pi} a^2(1+\cos\theta)^2 \, d\theta$
$= \frac{a^2}{2}\int_0^{2\pi} (1+2\cos\theta + \cos^2\theta) \, d\theta$
$= \frac{a^2}{2}\int_0^{2\pi} \left(1+2\cos\theta + \frac{1}{2}\cos 2\theta + 1\right) d\theta$
$= \frac{a^2}{2}\left[\theta + 2\sin\theta + \frac{1}{2}\left(\frac{1}{2}\sin 2\theta + \theta\right)\right]_0^{2\pi} = \frac{a^2}{2}\left(2\pi + 0 + \frac{1}{2} \cdot 0 + 2\pi\right)$
$= \frac{a^2}{2} \cdot 3\pi = \frac{3\pi a^2}{2}$ | M1, A1, M1, A1, M1, A1 | Use of formula with limits; Condone omission of $a^2$; Dealing with $\cos^2$. Condone omission of $a^2$; Substitute limits dep on 2nd M
| [6] |
---8 A curve has polar equation $r = a ( 1 + \cos \theta )$, where $a$ is a positive constant and $0 \leqslant \theta < 2 \pi$.\\
(i) Find the equation of the tangent at the pole.\\
(ii) Sketch the curve.\\
(iii) Find the area enclosed by the curve.
\hfill \mbox{\textit{OCR FP2 2014 Q8 [10]}}