| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Challenging +1.2 This is a standard Further Maths polar coordinates question requiring routine techniques: sketching from the polar equation, applying the standard area formula ∫½r²dθ with straightforward trigonometric integration, and converting to Cartesian form using standard substitutions. While it requires multiple steps and Further Maths content (making it harder than typical A-level), the techniques are all textbook-standard with no novel insight required. |
| 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/Working | Mark | Guidance |
| Correct shape for correct \(\theta\); start at \((r, 0)\) | B1 | Ignore other \(\theta\); used |
| \(\theta=0\), \(r=1\) and increasing \(r\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use correct formula with correct \(r\); \(\int \sec^2 x\,dx = \tan x\) used | B1 | |
| Quote \(\int 2\sec x \tan x\,dx = 2\sec x\) | B1 | Or substitute correctly |
| Replace \(\tan^2 x\) by \(\sec^2 x - 1\) to integrate | M1 | |
| Reasonable attempt to integrate 3 terms and use limits correctly | M1 | |
| Get \(\sqrt{3} + 1 - \frac{1}{6}\pi\) | A1 | Exact only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use \(x = r\cos\theta\), \(y = r\sin\theta\), \(r = (x^2+y^2)^{1/2}\) | M1 | |
| Reasonable attempt to eliminate \(r, \theta\) | M1 | |
| Get \(y = (x-1)\sqrt{x^2+y^2}\) | A1 | Or equivalent |
## Question 9:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct shape for correct $\theta$; start at $(r, 0)$ | B1 | Ignore other $\theta$; used |
| $\theta=0$, $r=1$ and increasing $r$ | B1 | |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct formula with correct $r$; $\int \sec^2 x\,dx = \tan x$ used | B1 | |
| Quote $\int 2\sec x \tan x\,dx = 2\sec x$ | B1 | Or substitute correctly |
| Replace $\tan^2 x$ by $\sec^2 x - 1$ to integrate | M1 | |
| Reasonable attempt to integrate 3 terms and use limits correctly | M1 | |
| Get $\sqrt{3} + 1 - \frac{1}{6}\pi$ | A1 | Exact only |
### Part (iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use $x = r\cos\theta$, $y = r\sin\theta$, $r = (x^2+y^2)^{1/2}$ | M1 | |
| Reasonable attempt to eliminate $r, \theta$ | M1 | |
| Get $y = (x-1)\sqrt{x^2+y^2}$ | A1 | Or equivalent |9 The equation of a curve, in polar coordinates, is
$$r = \sec \theta + \tan \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$$
(i) Sketch the curve.\\
(ii) Find the exact area of the region bounded by the curve and the lines $\theta = 0$ and $\theta = \frac { 1 } { 3 } \pi$.\\
(iii) Find a cartesian equation of the curve.
\hfill \mbox{\textit{OCR FP2 2007 Q9 [11]}}