| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reduction Formulae |
| Type | Trigonometric power reduction |
| Difficulty | Challenging +1.2 This is a standard Further Maths reduction formula question with typical polar coordinates application. Part (i) requires integration by parts with a well-known technique (differentiating sin^n θ), while part (ii) involves routine polar curve sketching and area calculation using the derived formula. The question is harder than average A-level due to being Further Maths content, but it's a textbook example of this topic with no novel insights required. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve8.06a Reduction formulae: establish, use, and evaluate recursively |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use \(\sin\theta \cdot \sin^{n-1}\theta\) and parts | M1 | Reasonable attempt with 2 parts, one yet to be integrated |
| Get \(-\cos\theta\cdot\sin^{n-1}\theta + (n-1)\int \sin^{n-2}\theta\cdot\cos^2\theta\, d\theta\) | A1 | Signs need to be carefully considered |
| Replace \(\cos^2 = 1 - \sin^2\) | M1 | |
| Clearly use limits and get AG | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Solve for \(r=0\) for at least one \(\theta\) | M1 | \(\theta\) need not be correct |
| Get \(\theta = 0\) and \(\pi\) | A1 | Ignore extra answers out of range |
| General shape (symmetry stated or approximately seen) | B1 | |
| Tangents at \(\theta=0\), \(\pi\) and max \(r\) seen | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct formula used; correct \(r\) | M1 | May be \(\int r^2\, d\theta\) with correct limits |
| Use \(6I_6 = 5I_4\), \(4I_4 = 3I_2\) | M1 | At least one |
| Attempt \(I_0\) (or \(I_2\)) | M1 | \((I_0 = \frac{1}{2}\pi)\) |
| Replace their values to get \(I_6\) | M1 | |
| Get \(\frac{5\pi}{32}\) | A1 | |
| Use symmetry to get \(\frac{5\pi}{32}\) | A1 | May be implied but correct use of limits must be given somewhere in answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct formula used; correct \(r\) | M1 | |
| Reasonable attempt at formula \((2i\sin\theta)^6 = (z - \frac{1}{z})^6\) | M1 | |
| Attempt to multiply out both sides (7 terms) | M1 | |
| Get correct expansion | A1 | |
| Convert to trig. equivalent and integrate their expression | M1 | cwo |
| Get \(\frac{5\pi}{32}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct formula used; correct \(r\) | M1 | |
| Use double-angle formula and attempt to cube (4 terms) | M1 | |
| Get correct expression | A1 | |
| Reasonable attempt to put \(\cos^2 2\theta\) into integrable form and integrate | M1 | |
| Reasonable attempt to integrate \(\cos^3 2\theta\) as e.g. \(\cos^2 2\theta \cdot \cos 2\theta\) | M1 | cwo |
| Get \(\frac{5\pi}{32}\) | A1 |
## Question 9(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use $\sin\theta \cdot \sin^{n-1}\theta$ and parts | M1 | Reasonable attempt with 2 parts, one yet to be integrated |
| Get $-\cos\theta\cdot\sin^{n-1}\theta + (n-1)\int \sin^{n-2}\theta\cdot\cos^2\theta\, d\theta$ | A1 | Signs need to be carefully considered |
| Replace $\cos^2 = 1 - \sin^2$ | M1 | |
| Clearly use limits and get AG | A1 | |
---
## Question 9(ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Solve for $r=0$ for at least one $\theta$ | M1 | $\theta$ need not be correct |
| Get $\theta = 0$ and $\pi$ | A1 | Ignore extra answers out of range |
| General shape (symmetry stated or approximately seen) | B1 | |
| Tangents at $\theta=0$, $\pi$ and max $r$ seen | B1 | |
---
## Question 9(ii)(b):
**Method 1:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct formula used; correct $r$ | M1 | May be $\int r^2\, d\theta$ with correct limits |
| Use $6I_6 = 5I_4$, $4I_4 = 3I_2$ | M1 | At least one |
| Attempt $I_0$ (or $I_2$) | M1 | $(I_0 = \frac{1}{2}\pi)$ |
| Replace their values to get $I_6$ | M1 | |
| Get $\frac{5\pi}{32}$ | A1 | |
| Use symmetry to get $\frac{5\pi}{32}$ | A1 | May be implied but correct use of limits must be given somewhere in answer |
**Method 2:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct formula used; correct $r$ | M1 | |
| Reasonable attempt at formula $(2i\sin\theta)^6 = (z - \frac{1}{z})^6$ | M1 | |
| Attempt to multiply out both sides (7 terms) | M1 | |
| Get correct expansion | A1 | |
| Convert to trig. equivalent and integrate their expression | M1 | cwo |
| Get $\frac{5\pi}{32}$ | A1 | |
**Method 3:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct formula used; correct $r$ | M1 | |
| Use double-angle formula and attempt to cube (4 terms) | M1 | |
| Get correct expression | A1 | |
| Reasonable attempt to put $\cos^2 2\theta$ into integrable form and integrate | M1 | |
| Reasonable attempt to integrate $\cos^3 2\theta$ as e.g. $\cos^2 2\theta \cdot \cos 2\theta$ | M1 | cwo |
| Get $\frac{5\pi}{32}$ | A1 | |9 (i) It is given that, for non-negative integers $n$,
$$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$
Show that, for $n \geqslant 2$,
$$n I _ { n } = ( n - 1 ) I _ { n - 2 } .$$
(ii) The equation of a curve, in polar coordinates, is
$$r = \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \pi$$
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the tangents at the pole and sketch the curve.
\item Find the exact area of the region enclosed by the curve.
RECOGNISING ACHIEVEMENT
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2009 Q9 [14]}}