| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Solve mixed sinh/cosh linear combinations |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring standard hyperbolic identities. Part (i) is routine proof from definitions, part (ii) uses the result from (i) directly (cosh(x-y)=1 implies x=y), and part (iii) solves cosh x cosh y = 9 with x=y to get a quadratic in e^x. All steps are mechanical with clear signposting, making it slightly easier than average even for Further Maths. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use correct definition of \(\cosh\) or \(\sinh x\) | B1 | Seen anywhere in (i) |
| Attempt to mult. their \(\cosh/\sinh\) | M1 | |
| Correctly mult. out and tidy | A1√ | |
| Clearly arrive at A.G. | A1 | Accept \(e^{c-y}\) and \(e^{-x}\) |
| (ii) Get \(\cosh(x-y) = 1\) | M1 | |
| Get or imply \((x-y) = 0\) to A.G. | A1 | |
| (iii) Use \(\cosh^2 x = 9\) or \(\sinh^2 x = 8\) | B1 | |
| Attempt to solve \(\cosh x = 3\) (not \(-3\)) or \(\sinh x = \pm\sqrt{8}\) (allow \(+\sqrt{8}\) or \(-\sqrt{8}\) only) | M1 | \(x = \ln(3 + \sqrt{8})\) from formulae book or from basic \(\cosh\) definition |
| Get at least one \(x\) solution correct | A1 | |
| Get both solutions correct, \(x\) and \(y\) | A1 | \(x, y = \ln(3 \pm \sqrt{2}/2)\); AEEF |
| SR Attempt \(\tanh = \sinh/\cosh\) | B1 | |
| Get \(\tanh x = \pm\sqrt{8}/3\) (+ or -) | M1 | |
| Get at least one sol. correct | A1 | |
| Get both solutions correct | A1 | |
| SR Use exponential definition | B1 | |
| Get quadratic in \(e^x\) or \(e^{2x}\) | M1 | |
| Solve for one correct \(x\) | A1 | |
| Get both solutions, \(x\) and \(y\) | A1 |
**(i)** Use correct definition of $\cosh$ or $\sinh x$ | B1 | Seen anywhere in (i)
Attempt to mult. their $\cosh/\sinh$ | M1 |
Correctly mult. out and tidy | A1√ |
Clearly arrive at A.G. | A1 | Accept $e^{c-y}$ and $e^{-x}$
**(ii)** Get $\cosh(x-y) = 1$ | M1 |
Get or imply $(x-y) = 0$ to A.G. | A1 |
**(iii)** Use $\cosh^2 x = 9$ or $\sinh^2 x = 8$ | B1 |
Attempt to solve $\cosh x = 3$ (not $-3$) or $\sinh x = \pm\sqrt{8}$ (allow $+\sqrt{8}$ or $-\sqrt{8}$ only) | M1 | $x = \ln(3 + \sqrt{8})$ from formulae book or from basic $\cosh$ definition
Get at least one $x$ solution correct | A1 |
Get both solutions correct, $x$ and $y$ | A1 | $x, y = \ln(3 \pm \sqrt{2}/2)$; AEEF
SR Attempt $\tanh = \sinh/\cosh$ | B1 |
Get $\tanh x = \pm\sqrt{8}/3$ (+ or -) | M1 |
Get at least one sol. correct | A1 |
Get both solutions correct | A1 |
SR Use exponential definition | B1 |
Get quadratic in $e^x$ or $e^{2x}$ | M1 |
Solve for one correct $x$ | A1 |
Get both solutions, $x$ and $y$ | A1 |7 (i) Using the definitions of hyperbolic functions in terms of exponentials, prove that
$$\cosh x \cosh y - \sinh x \sinh y = \cosh ( x - y )$$
(ii) Given that $\cosh x \cosh y = 9$ and $\sinh x \sinh y = 8$, show that $x = y$.\\
(iii) Hence find the values of $x$ and $y$ which satisfy the equations given in part (ii), giving the answers in logarithmic form.
\hfill \mbox{\textit{OCR FP2 2007 Q7 [10]}}