| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Solve mixed sinh/cosh linear combinations |
| Difficulty | Standard +0.8 This is a Further Maths question involving hyperbolic functions requiring multiple techniques: solving a hyperbolic equation (converting to exponentials), differentiation of hyperbolic functions, analyzing gradients, curve sketching, and integration. While the individual steps are methodical, the multi-part nature, the need to connect parts (i) and (ii), and working with exact logarithmic forms throughout elevates this above standard A-level. The question requires sustained reasoning across 18 marks but follows established FP2 techniques without requiring novel insight. |
| Spec | 1.08d Evaluate definite integrals: between limits4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials |
| Answer | Marks |
|---|---|
| M1 | Substituting correct exponential forms |
| M1 | Obtaining quadratic in \(e^t\) |
| M1 | Solving to obtain at least one value of \(e^t\) |
| A1A1 | Condone extra values |
| A1 | These two values o.e. only. Exact form |
| Total | 6 |
| Answer | Marks |
|---|---|
| B1 | |
| M1 | Complete method to obtain an \(x\) value |
| A1 | Both \(x\) co-ordinates in any exact form |
| B1 | Both \(y\) co-ordinates |
| M1 | Any complete method |
| A1 (ag) | www |
| Total | 8 |
| Answer | Marks |
|---|---|
| M1 | Attempting integration |
| A1 | Correct result of integration |
| M1 | Using both limits and a complete method to obtain a value of \(a\) |
| A1 | Must reject \(\frac{1}{2}\ln(\frac{1}{2})\), but reason need not be given |
| Total | 4 |
## (i)
$\sinh t + 7\cosh t = 8$
$\Rightarrow \frac{1}{2}(e^t - e^{-t}) + 7 \times \frac{1}{2}(e^t + e^{-t}) = 8$
$\Rightarrow 4e^t + 3e^{-t} = 8$
$\Rightarrow 4e^{2t} - 8e^t + 3 = 0$
$\Rightarrow (2e^t - 1)(2e^t - 3) = 0$
$\Rightarrow e^t = \frac{1}{2}$ or $\frac{3}{2}$
$\Rightarrow t = \ln(\frac{1}{2})$ or $\ln(\frac{3}{2})$
| | M1 | Substituting correct exponential forms |
| | M1 | Obtaining quadratic in $e^t$ |
| | M1 | Solving to obtain at least one value of $e^t$ |
| | A1A1 | Condone extra values |
| | A1 | These two values o.e. only. Exact form |
| **Total** | **6** | |
## (ii)
$\frac{dy}{dx} = 2\sinh 2x + 14\cosh 2x$ or $8e^{2x} + 6e^{-2x}$
$2\sinh 2x + 14\cosh 2x = 16 \Rightarrow \sinh 2x + 7\cosh 2x = 8$
$\Rightarrow 2x = \ln(\frac{1}{2})$ or $\ln(\frac{3}{2}) \Rightarrow x = \frac{1}{2}\ln(\frac{1}{2})$ or $\frac{1}{2}\ln(\frac{3}{2})$
$x = \frac{1}{2}\ln(\frac{1}{2}) \Rightarrow y = -4$ $(\frac{1}{2}\ln(\frac{1}{2}), -4)$
$x = \frac{1}{2}\ln(\frac{3}{2}) \Rightarrow y = 4$ $(\frac{1}{2}\ln(\frac{3}{2}), 4)$
$\frac{dy}{dx} = 0 \Rightarrow 2\sinh 2x + 14\cosh 2x = 0$
$\Rightarrow \tanh 2x = -7$ or $e^u = -\frac{2}{7}$ etc.
No solutions because $-1 < \tanh 2x < 1$ or $e^t > 0$ etc.
| | B1 | |
| | M1 | Complete method to obtain an $x$ value |
| | A1 | Both $x$ co-ordinates in any exact form |
| | B1 | Both $y$ co-ordinates |
| | M1 | Any complete method |
| | A1 (ag) | www |
| **Total** | **8** | |
## (iii)
$\int_0^a (\cos 2x + 7\sinh 2x) dx = \frac{1}{2}$
$\Rightarrow [\frac{1}{2}\sinh 2x + \frac{7}{2}\cosh 2x]_0^a = \frac{1}{2}$
$\Rightarrow (\frac{1}{2}\sinh 2a + \frac{7}{2}\cosh 2a) - \frac{7}{2} = \frac{1}{2}$
$\Rightarrow \sinh 2a + 7\cosh 2a = 8$
$\Rightarrow 2a = \ln(\frac{1}{2})$ or $\ln(\frac{3}{2}) \Rightarrow a = \frac{1}{2}\ln(\frac{1}{2})$ or $\frac{1}{2}\ln(\frac{3}{2})$
$\Rightarrow a = \frac{1}{2}\ln(\frac{1}{2})$ $(\frac{1}{2}\ln(\frac{1}{2}) < 0)$
| | M1 | Attempting integration |
| | A1 | Correct result of integration |
| | M1 | Using both limits and a complete method to obtain a value of $a$ |
| | A1 | Must reject $\frac{1}{2}\ln(\frac{1}{2})$, but reason need not be given |
| **Total** | **4** | |
**TOTAL FOR QUESTION 4: 18**
---\begin{enumerate}[label=(\roman*)]
\item Solve the equation
$$\sinh t + 7 \cosh t = 8,$$
expressing your answer in exact logarithmic form. [6]
\end{enumerate}
A curve has equation $y = \cosh 2x + 7 \sinh 2x$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16.
Show that there is no point on the curve at which the gradient is zero.
Sketch the curve. [8]
\item Find, in an exact form, the positive value of $a$ for which the area of the region between the curve, the $x$-axis, the $y$-axis and the line $x = a$ is $\frac{1}{2}$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2011 Q4 [18]}}