| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 6 |
| 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 function definitions and algebraic manipulation. Part (a) is direct substitution of sinh x = (e^x - e^{-x})/2 and cosh x = (e^x + e^{-x})/2. Part (b) involves solving a quadratic in e^x, which is routine for FP2 students. While it's Further Maths content (inherently harder than Core), it's a standard textbook exercise with no novel insight required, placing it slightly above average overall. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials |
| Answer | Marks | Guidance |
|---|---|---|
| 1(a) \(5\left(\frac{e^x - e^{-x}}{2}\right) + \left(\frac{e^x + e^{-x}}{2}\right) = 3e^x - 2e^{-x}\) | M1 | M0 if no 2s in denominator |
| \(3e^x - 2e^{-x}\) | A1 | 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \((3e^x - 1)(e^x + 2) = 0\) | M1, A1F, E1 | ft if 2s missing in (a); any indication of rejection |
| \(e^x \neq -2\) | E1 | |
| \(e^x = \frac{1}{3}\) | A1F | |
| \(x = \ln \frac{1}{3}\) | A1F | 4 marks total (6 marks for question) |
**1(a)** $5\left(\frac{e^x - e^{-x}}{2}\right) + \left(\frac{e^x + e^{-x}}{2}\right) = 3e^x - 2e^{-x}$ | M1 | M0 if no 2s in denominator
$3e^x - 2e^{-x}$ | A1 | 2 marks total
**1(b)** $3e^x - 2e^{-x} + 5 = 0$
$3e^{2x} + 5e^x - 2 = 0$
$(3e^x - 1)(e^x + 2) = 0$ | M1, A1F, E1 | ft if 2s missing in (a); any indication of rejection
$e^x \neq -2$ | E1
$e^x = \frac{1}{3}$ | A1F
$x = \ln \frac{1}{3}$ | A1F | 4 marks total (6 marks for question)1
\begin{enumerate}[label=(\alph*)]
\item Express
$$5 \sinh x + \cosh x$$
in the form $A \mathrm { e } ^ { x } + B \mathrm { e } ^ { - x }$, where $A$ and $B$ are integers.
\item Solve the equation
$$5 \sinh x + \cosh x + 5 = 0$$
giving your answer in the form $\ln a$, where $a$ is a rational number.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2008 Q1 [6]}}