| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 5 |
| Topic | Hyperbolic functions |
| Type | Solve mixed sinh/cosh linear combinations |
| Difficulty | Standard +0.3 Part (i) is a standard bookwork proof using exponential definitions that most students memorize. Part (ii) requires applying the identity to rewrite the equation, then solving a quadratic in e^x, which is a routine technique for hyperbolic equations. The final form requires some algebraic manipulation but follows a well-practiced method. This is slightly easier than average due to being a standard textbook exercise with clear steps. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
**(i)**
$\cosh A\cosh B + \sinh A\sinh B \equiv \frac{1}{2}(e^A+e^{-A})\frac{1}{2}(e^B+e^{-B}) + \frac{1}{2}(e^A-e^{-A})\frac{1}{2}(e^B-e^{-B})$
$\equiv \frac{1}{4}(e^{A+B}+e^{B-A}+e^{A-B}+e^{-A-B}) + \frac{1}{4}(e^{A+B}-e^{B-A}-e^{A-B}+e^{-A-B})$
$\equiv \frac{1}{2}(e^{A+B}+e^{-(A+B)})$
$\equiv \cosh(A+B)$ **B1** Shown legitimately [1]
**(ii)** METHOD I $5\cdot\frac{1}{2}(e^x+e^{-x})+3\cdot\frac{1}{2}(e^x-e^{-x}) = 4e^x+e^{-x}$ **M1** Use of exponential forms
$4(e^x)^2 - 12(e^x) + 1 = 0$ **M1** Writing as quadratic in $e^x$; **A1** correct
$e^x = \frac{12\pm\sqrt{144-16}}{8}$ **M1** $e^x = \frac{3}{2}\pm\sqrt{2}$
**A1** for $\ln(1.5\pm\sqrt{2})$
METHOD II $4\cosh\left(x+\tanh^{-1}\frac{3}{5}\right)=12$ **M1A1**
**M1** $x = (\pm)\cosh^{-1}3 - \tanh^{-1}\frac{3}{5}$
**M1** Use of $\cosh^{-1}x = \ln\left(x\pm\sqrt{x^2-1}\right)$, $\tanh^{-1}x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$ ignore $\pm$
**A1** $\ln(1.5\pm\sqrt{2})$ **[5]**4 (i) Using the definitions of sinh and cosh in terms of exponentials, prove that
$$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$
(ii) Solve the equation $5 \cosh x + 3 \sinh x = 12$, giving your answers in the form $\ln ( p \pm q \sqrt { 2 } )$ for rational numbers $p$ and $q$ to be determined.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2010 Q4 [5]}}