| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Surface area of revolution |
| Difficulty | Challenging +1.2 This is a standard Further Maths parametric question requiring differentiation of trig functions, use of double angle formulas to simplify, and application of the surface of revolution formula. Part (a) is routine algebraic manipulation with trig identities, while part (b) requires knowing and applying a standard formula with some integration technique. More mechanical than insightful, but slightly above average due to the algebraic complexity and Further Maths context. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\frac{dx}{d\theta} = -2\sin\theta + 2\sin 2\theta\), \(\frac{dy}{d\theta} = 2\cos\theta - 2\cos 2\theta\) | B1, B1 | Correct derivatives |
| \(\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2 = 4\sin^2\theta - 8\sin\theta\sin 2\theta + 4\sin^2 2\theta + 4\cos^2\theta - 8\cos\theta\cos 2\theta + 4\cos^2 2\theta\) | M1 | Squares and adds their derivatives |
| \(= 8 - 8(\cos 2\theta\cos\theta + \sin 2\theta\sin\theta)\) | ||
| \(= 8 - 8\cos(2\theta - \theta) = 8(1-\cos\theta)\) * | M1A1* | M1: Use of at least one correct trig identity; A1*: Correct proof with no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(S = 2\pi\int y\sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2}\, d\theta\) | ||
| \(S = 2\pi\int(2\sin\theta - \sin 2\theta)\sqrt{8(1-\cos\theta)}\, d\theta\) | M1 | Substitutes \(y = 2\sin\theta - \sin 2\theta\) and \(8(1-\cos\theta)\) into a correct formula |
| \(= 2\pi\int 2\sin\theta(1-\cos\theta)\sqrt{8(1-\cos\theta)}\, d\theta\) | ||
| \(= 8\pi\sqrt{2}\int\sin\theta(1-\cos\theta)^{\frac{3}{2}}\, d\theta\) | M1 | Processes to reach integrand of the form \(k\sin\theta(1-\cos\theta)^{\frac{3}{2}}\) |
| \(= 8\sqrt{2}\pi\left[\frac{2}{5}(1-\cos\theta)^{\frac{5}{2}}\right]_0^\pi\) | dM1 | Integrates to obtain expression of form \(\alpha(1-\cos\theta)^{\frac{5}{2}}\); dependent on previous M mark |
| \(= 8\pi\sqrt{2}\left(\frac{2}{5}(2)^{\frac{5}{2}} - 0\right)\) | dddM1 | Use of limits 0 and \(\pi\) and subtracts; dependent on all previous method marks |
| \(= \frac{128}{5}\pi\) | A1 | Allow equivalents e.g. \(25.6\pi\) |
## Question 6(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{dx}{d\theta} = -2\sin\theta + 2\sin 2\theta$, $\frac{dy}{d\theta} = 2\cos\theta - 2\cos 2\theta$ | B1, B1 | Correct derivatives |
| $\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2 = 4\sin^2\theta - 8\sin\theta\sin 2\theta + 4\sin^2 2\theta + 4\cos^2\theta - 8\cos\theta\cos 2\theta + 4\cos^2 2\theta$ | M1 | Squares and adds their derivatives |
| $= 8 - 8(\cos 2\theta\cos\theta + \sin 2\theta\sin\theta)$ | | |
| $= 8 - 8\cos(2\theta - \theta) = 8(1-\cos\theta)$ * | M1A1* | M1: Use of at least one correct trig identity; A1*: Correct proof with no errors |
---
## Question 6(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $S = 2\pi\int y\sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2}\, d\theta$ | | |
| $S = 2\pi\int(2\sin\theta - \sin 2\theta)\sqrt{8(1-\cos\theta)}\, d\theta$ | M1 | Substitutes $y = 2\sin\theta - \sin 2\theta$ and $8(1-\cos\theta)$ into a correct formula |
| $= 2\pi\int 2\sin\theta(1-\cos\theta)\sqrt{8(1-\cos\theta)}\, d\theta$ | | |
| $= 8\pi\sqrt{2}\int\sin\theta(1-\cos\theta)^{\frac{3}{2}}\, d\theta$ | M1 | Processes to reach integrand of the form $k\sin\theta(1-\cos\theta)^{\frac{3}{2}}$ |
| $= 8\sqrt{2}\pi\left[\frac{2}{5}(1-\cos\theta)^{\frac{5}{2}}\right]_0^\pi$ | dM1 | Integrates to obtain expression of form $\alpha(1-\cos\theta)^{\frac{5}{2}}$; dependent on previous M mark |
| $= 8\pi\sqrt{2}\left(\frac{2}{5}(2)^{\frac{5}{2}} - 0\right)$ | dddM1 | Use of limits 0 and $\pi$ and subtracts; dependent on all previous method marks |
| $= \frac{128}{5}\pi$ | A1 | Allow equivalents e.g. $25.6\pi$ |
---6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0ddee434-f7e1-4f56-91fc-f487112dbf6b-11_709_1269_292_349}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curve $C$ with parametric equations
$$x = 2 \cos \theta - \cos 2 \theta , y = 2 \sin \theta - \sin 2 \theta , \quad 0 \leqslant \theta \leqslant \pi$$
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = 8 ( 1 - \cos \theta )$$
The curve $C$ is rotated through $2 \pi$ radians about the $x$-axis.
\item Find the area of the surface generated, giving your answer in the form $k \pi$, where $k$ is a rational number.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F3 2015 Q6 [10]}}