| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Surface area of revolution with hyperbolics |
| Difficulty | Challenging +1.8 This is a Further Maths question requiring differentiation of hyperbolic functions (straightforward using standard identities), followed by a surface area of revolution integral. The parametric setup and hyperbolic identities make the integral tractable but require careful manipulation. The multi-step nature, Further Maths content, and integration challenge place it well above average difficulty but below the most demanding proof-based questions. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07d Differentiate/integrate: hyperbolic functions8.06b Arc length and surface area: of revolution, cartesian or parametric |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{dx}{d\theta} = 1 - \text{sech}^2\theta\) | B1 | Correct derivative |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{dy}{d\theta} = -\text{sech}\,\theta\tanh\theta\) | B1 | Correct derivative. If both derivatives in different variable but otherwise correct, allow B1B1. If one or both incorrect award B0B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(S = (2\pi)\int \text{sech}\,\theta\sqrt{(1-\text{sech}^2\theta)^2 + (-\text{sech}\,\theta\tanh\theta)^2}\,d\theta\) | M1 | Uses correct formula with their derivatives; \(2\pi\) not needed |
| \(S = 2\pi\int \text{sech}\,\theta\tanh\theta\,d\theta\) | A1 | Correct integral after full simplification; \(2\pi\) and limits not needed |
| \(S = 2\pi[-\text{sech}\,\theta]\) | A1 | Correct integration; limits not needed |
| \(S = -2\pi(\text{sech}(\ln 3) - \text{sech}(0)) = 0.8\pi\) | dM1, A1 (cao and cso) | Include \(2\pi\) and use limits (0 to \(\ln 3\)) correctly in a multiple of \(\text{sech}\,\theta\); A1: cao and cso |
## Question 6:
**Given:** $x = \theta - \tanh\theta$, $y = \text{sech}\,\theta$, $0 \leq \theta \leq \ln 3$
### Part (a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{dx}{d\theta} = 1 - \text{sech}^2\theta$ | B1 | Correct derivative |
### Part (a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{dy}{d\theta} = -\text{sech}\,\theta\tanh\theta$ | B1 | Correct derivative. If both derivatives in different variable but otherwise correct, allow B1B1. If one or both incorrect award B0B0 |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $S = (2\pi)\int \text{sech}\,\theta\sqrt{(1-\text{sech}^2\theta)^2 + (-\text{sech}\,\theta\tanh\theta)^2}\,d\theta$ | M1 | Uses correct formula with their derivatives; $2\pi$ not needed |
| $S = 2\pi\int \text{sech}\,\theta\tanh\theta\,d\theta$ | A1 | Correct integral after full simplification; $2\pi$ and limits not needed |
| $S = 2\pi[-\text{sech}\,\theta]$ | A1 | Correct integration; limits not needed |
| $S = -2\pi(\text{sech}(\ln 3) - \text{sech}(0)) = 0.8\pi$ | dM1, A1 (cao and cso) | Include $2\pi$ and use limits (0 to $\ln 3$) correctly in a multiple of $\text{sech}\,\theta$; A1: cao and cso |
---6. The curve $C$ has parametric equations
$$x = \theta - \tanh \theta , \quad y = \operatorname { sech } \theta , \quad 0 \leqslant \theta \leqslant \ln 3$$
\begin{enumerate}[label=(\alph*)]
\item Find
\begin{enumerate}[label=(\roman*)]
\item $\frac { \mathrm { d } x } { \mathrm {~d} \theta }$
\item $\frac { \mathrm { d } y } { \mathrm {~d} \theta }$
The curve $C$ is rotated through $2 \pi$ radians about the $x$-axis.
\end{enumerate}\item Find the exact area of the curved surface formed, giving your answer as a multiple of $\pi$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F3 2018 Q6 [7]}}