| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Solve using double angle formulas |
| Difficulty | Standard +0.3 Part (i) is a straightforward proof using standard hyperbolic definitions—routine for FP2 students. Part (ii) requires substituting the identity from (i), forming a quadratic in sinh x, then using the inverse sinh formula. While this involves multiple steps, it follows a standard pattern for FP2 hyperbolic equation questions with no novel insight required. Slightly easier than average A-level due to the guided structure. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Correct definitions used. Attempt at \((e^x-e^{-x})^2/4 + 1\). Clearly derive A.G. | B1, M1, A1 | Allow \((e^x+e^{-x})^2+1\); allow \(/2\) |
| (ii) Form a quadratic in \(\sinh x\). Attempt to solve. Get \(\sinh x = -\frac{1}{2}\) or 3 | M1, M1, A1 | Factors or formula |
| Use correct ln expression. Get \(\ln(-\frac{1}{2}+\sqrt{\frac{7}{4}})\) and \(\ln(3+\sqrt{10})\) | M1, A1 | On their answer(s) seen once |
(i) Correct definitions used. Attempt at $(e^x-e^{-x})^2/4 + 1$. Clearly derive A.G. | B1, M1, A1 | Allow $(e^x+e^{-x})^2+1$; allow $/2$
(ii) Form a quadratic in $\sinh x$. Attempt to solve. Get $\sinh x = -\frac{1}{2}$ or 3 | M1, M1, A1 | Factors or formula
Use correct ln expression. Get $\ln(-\frac{1}{2}+\sqrt{\frac{7}{4}})$ and $\ln(3+\sqrt{10})$ | M1, A1 | On their answer(s) seen once\begin{enumerate}[label=(\roman*)]
\item Using the definitions of $\cosh x$ and $\sinh x$ in terms of $e^x$ and $e^{-x}$, show that
$$1 + 2\sinh^2 x = \cosh 2x.$$ [3]
\item Solve the equation
$$\cosh 2x - 5\sinh x = 4,$$
giving your answers in logarithmic form. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2009 Q6 [8]}}