| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Prove inverse hyperbolic logarithmic form |
| Difficulty | Standard +0.8 This is a Further Maths question requiring multiple techniques: defining hyperbolic functions, proving the inverse logarithmic form (standard but requires algebraic manipulation), solving a hyperbolic equation, and combining inverse hyperbolic and logarithmic functions. While these are bookwork results for FM students, the multi-part nature and the final part requiring substitution of the proven result makes this moderately challenging, above average difficulty but not exceptionally hard for the FM syllabus. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 14.07f Inverse hyperbolic: logarithmic forms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Get \(\frac{e^y - e^{-y}}{e^y + e^{-y}}\) | B1 | Allow \(\frac{e^{2y}-1}{e^{2y}+1}\) or if \(x\) used |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt quadratic in \(e^y\); solve for \(e^y\); clearly get AG | M1, M1, A1 | Multiply by \(e^y\) and tidy |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Rewrite as \(\tanh x = k\); use (ii) for \(x = \frac{1}{2}\ln 7\) or equivalent | M1 | SR Use hyp def\(^n\) to get quadratic in \(e^x\) |
| A1 | Solve \(e^{2x} = 7\) for \(x\) to \(\frac{1}{2}\ln 7\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use of log laws; correctly equate \(\ln A = \ln B\) to \(A = B\) | B1 | One used correctly |
| Get \(x = \pm\frac{3}{5}\) | M1, A1 | Or \(\ln(^A\!B) = 0\) |
## Question 8:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Get $\frac{e^y - e^{-y}}{e^y + e^{-y}}$ | B1 | Allow $\frac{e^{2y}-1}{e^{2y}+1}$ or if $x$ used |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt quadratic in $e^y$; solve for $e^y$; clearly get AG | M1, M1, A1 | Multiply by $e^y$ and tidy |
### Part (iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Rewrite as $\tanh x = k$; use (ii) for $x = \frac{1}{2}\ln 7$ or equivalent | M1 | SR Use hyp def$^n$ to get quadratic in $e^x$ |
| | A1 | Solve $e^{2x} = 7$ for $x$ to $\frac{1}{2}\ln 7$ |
### Part (iv)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of log laws; correctly equate $\ln A = \ln B$ to $A = B$ | B1 | One used correctly |
| Get $x = \pm\frac{3}{5}$ | M1, A1 | Or $\ln(^A\!B) = 0$ |
---8 (i) Define tanh $y$ in terms of $\mathrm { e } ^ { y }$ and $\mathrm { e } ^ { - y }$.\\
(ii) Given that $y = \tanh ^ { - 1 } x$, where $- 1 < x < 1$, prove that $y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$.\\
(iii) Find the exact solution of the equation $3 \cosh x = 4 \sinh x$, giving the answer in terms of a logarithm.\\
(iv) Solve the equation
$$\tanh ^ { - 1 } x + \ln ( 1 - x ) = \ln \left( \frac { 4 } { 5 } \right)$$
\hfill \mbox{\textit{OCR FP2 2007 Q8 [9]}}