| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Prove inverse hyperbolic logarithmic form |
| Difficulty | Standard +0.8 This FP2 question tests hyperbolic functions at a moderate-to-challenging level. Part (i) is a standard bookwork proof requiring algebraic manipulation of the inverse tanh definition. Part (ii)(a) requires differentiation and inequality reasoning. Part (ii)(b) is the most demanding, requiring finding stationary points by solving a transcendental equation involving hyperbolic functions, then verifying it's a minimum and computing its value—this involves multiple sophisticated steps including using cosh²x - sinh²x = 1. The 12-mark allocation and multi-layered reasoning place this above average difficulty, though it remains within standard FP2 scope without requiring exceptional insight. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07f Inverse hyperbolic: logarithmic forms |
| Answer | Marks | Guidance |
|---|---|---|
| Rewrite tanh y as \(\frac{e^t - e^{-t}}{e^t + e^{-t}}\) | B1 | Or equivalent |
| Attempt to write as quadratic in \(e^{2x}\) | M1 | |
| Clearly get A.G. | A1 |
| Answer | Marks |
|---|---|
| Attempt to diff. and solve \(= 0\) | M1 |
| Get \(\tanh x = b/a\) | A1 |
| Use \((-1) < \tanh x < 1\) to show \(b < a\) | B1 |
| SC Use exponentials | M1 |
| Get \(e^{2x} = \frac{(a+b)}{(a-b)}\) | A1 |
| Use \(e^{2x} > 0\) to show \(b < a\) | B1 |
| SC Write \(x = \tanh^{-1}(b/a)\) | M1 |
| \(= \frac{1}{2}\ln\left(\frac{1+b/a}{1-b/a}\right)\) | A1 |
| Use \(( ) > 0\) to show \(b < a\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Get tanh \(x = -1/a\) from part (ii)(a) | B1 | |
| Replace as ln from their answer | M1 | |
| Get \(x = \frac{1}{2}\ln\left(\frac{(a+1)}{(a-1)}\right)\) | A1 | |
| Use \(e^{\sqrt{\ln((a+1)/(a-1))}} = \sqrt{((a+1)/(a-1))}\) | M1 | At least once |
| Clearly get A.G. | A1 | |
| Test for minimum correctly | B1 | |
| SC Use of \(y = \cosh(x(a-\tanh x))\) and \(\cosh x = 1/\text{sech} x = 1/\sqrt{(1-\tanh^2 x)}\) |
## (i)
Rewrite tanh y as $\frac{e^t - e^{-t}}{e^t + e^{-t}}$ | B1 | Or equivalent
Attempt to write as quadratic in $e^{2x}$ | M1 |
Clearly get A.G. | A1 |
## (ii)
### (a)
Attempt to diff. and solve $= 0$ | M1 |
Get $\tanh x = b/a$ | A1 |
Use $(-1) < \tanh x < 1$ to show $b < a$ | B1 |
**SC** Use exponentials | M1 |
Get $e^{2x} = \frac{(a+b)}{(a-b)}$ | A1 |
Use $e^{2x} > 0$ to show $b < a$ | B1 |
**SC** Write $x = \tanh^{-1}(b/a)$ | M1 |
$= \frac{1}{2}\ln\left(\frac{1+b/a}{1-b/a}\right)$ | A1 |
Use $( ) > 0$ to show $b < a$ | B1 |
### (b)
Get tanh $x = -1/a$ from part (ii)(a) | B1 |
Replace as ln from their answer | M1 |
Get $x = \frac{1}{2}\ln\left(\frac{(a+1)}{(a-1)}\right)$ | A1 |
Use $e^{\sqrt{\ln((a+1)/(a-1))}} = \sqrt{((a+1)/(a-1))}$ | M1 | At least once
Clearly get A.G. | A1 |
Test for minimum correctly | B1 |
**SC** Use of $y = \cosh(x(a-\tanh x))$ and $\cosh x = 1/\text{sech} x = 1/\sqrt{(1-\tanh^2 x)}$ |\begin{enumerate}[label=(\roman*)]
\item Given that $y = \tanh^{-1} x$, for $-1 < x < 1$, prove that $y = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$. [3]
\item It is given that $f(x) = a\cosh x - b\sinh x$, where $a$ and $b$ are positive constants.
\begin{enumerate}[label=(\alph*)]
\item Given that $b \geq a$, show that the curve with equation $y = f(x)$ has no stationary points. [3]
\item In the case where $a > 1$ and $b = 1$, show that $f(x)$ has a minimum value of $\sqrt{a^2 - 1}$. [6]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2010 Q9 [12]}}