| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Solve using sech/tanh identities |
| Difficulty | Standard +0.3 Part (i) is a standard FP2 proof using definitions of hyperbolic functions—straightforward algebraic manipulation with no conceptual difficulty. Part (ii) requires substitution using the identity from (i), solving a quadratic in sech x, then inverting to find x in logarithmic form. While this involves multiple steps, it's a routine FP2 exercise following a predictable pattern. The question is slightly above average difficulty due to the multi-step nature and FP2 content, but remains a standard textbook-style problem. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Use correct definitions of cosh and sinh | B1 | |
| Attempt to square and subtract | M1 | On their definitions |
| Clearly get A.G. | A1 | |
| Show division by cosh² | B1 | Or clear use of first result |
| Answer | Marks | Guidance |
|---|---|---|
| Rewrite as quadratic in sech and attempt to solve | M1 | Or quadratic in cosh |
| Eliminate values outside \(0 < \text{sech} \leq 1\) | B1 | Or eliminate values outside cosh \(\geq 1\) (allow positive) |
| Get \(x = \ln(2+\sqrt{3})\) | A1 | |
| Get \(x = -\ln(2+\sqrt{3})\) or \(\ln(2-\sqrt{3})\) | A1 |
## (i)
Use correct definitions of cosh and sinh | B1 |
Attempt to square and subtract | M1 | On their definitions
Clearly get A.G. | A1 |
Show division by cosh² | B1 | Or clear use of first result
## (ii)
Rewrite as quadratic in sech and attempt to solve | M1 | Or quadratic in cosh
Eliminate values outside $0 < \text{sech} \leq 1$ | B1 | Or eliminate values outside cosh $\geq 1$ (allow positive)
Get $x = \ln(2+\sqrt{3})$ | A1 |
Get $x = -\ln(2+\sqrt{3})$ or $\ln(2-\sqrt{3})$ | A1 |
---\begin{enumerate}[label=(\roman*)]
\item Using the definitions of $\sinh x$ and $\cosh x$ in terms of $e^x$ and $e^{-x}$, show that
$$\cosh^2 x - \sinh^2 x \equiv 1.$$
Deduce that $1 - \tanh^2 x \equiv \operatorname{sech}^2 x$. [4]
\item Solve the equation $2\tanh^2 x - \operatorname{sech} x = 1$, giving your answer(s) in logarithmic form. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2010 Q5 [8]}}