| Exam Board | WJEC |
|---|---|
| Module | Further Unit 4 (Further Unit 4) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Solve using substitution u = cosh x or u = sinh x |
| Difficulty | Standard +0.3 This is a straightforward composition of functions question from Further Maths. Part (a) requires basic understanding of function composition domains/ranges (routine for FM students). Part (b) involves solving cosh(x²-1)=3, which requires knowing cosh inverse or using the exponential definition—standard technique with calculator work. The hyperbolic function context adds slight difficulty above core A-level, but the problem-solving is mechanical with no novel insight required. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials |
| Answer | Marks | Guidance |
|---|---|---|
| a) | Domain of \(fg\): \((0, \infty)\) | B1 |
| Range of \(fg\): \([1, \infty)\) | B1 | |
| (2) | ||
| b) | \(\cosh(x^2 - 1) = 3\) | B1 |
| \(x^2 - 1 = \cosh^{-1} 3\) | M1 | |
| \(x^2 = 1 + \cosh^{-1} 3\) | ||
| \(x = \pm 1.662\) | A1 | A0 for \(x = \pm 1.662\) as final answer |
| \(x = 1.662\) since \(x = -1.662\) not in domain | ||
| (3) | ||
| [5] |
a) | Domain of $fg$: $(0, \infty)$ | B1 | |
| Range of $fg$: $[1, \infty)$ | B1 | |
| | (2) | |
b) | $\cosh(x^2 - 1) = 3$ | B1 | |
| $x^2 - 1 = \cosh^{-1} 3$ | M1 | |
| $x^2 = 1 + \cosh^{-1} 3$ | | |
| $x = \pm 1.662$ | A1 | A0 for $x = \pm 1.662$ as final answer |
| $x = 1.662$ since $x = -1.662$ not in domain | | |
| | (3) | |
| | [5] | |The functions $f$ and $g$ have domains $(-1, \infty)$ and $(0, \infty)$ respectively and are defined by
$$f(x) = \cosh x, \qquad g(x) = x^2 - 1.$$
\begin{enumerate}[label=(\alph*)]
\item State the domain and range of $fg$. [2]
\item Solve the equation $fg(x) = 3$. Give your answer correct to three decimal places. [3]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 4 2023 Q1 [5]}}