| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Sketch graphs of hyperbolic functions |
| Difficulty | Moderate -0.8 Part (a) requires sketching a standard hyperbolic function (cosh x) with its y-intercept, which is direct recall of a basic graph. Part (b) involves solving cosh x = 2 using the standard logarithmic formula, a routine application of a known result. Both parts are straightforward textbook exercises requiring minimal problem-solving, though the hyperbolic functions topic itself is Further Maths content. |
| Spec | 4.07b Hyperbolic graphs: sketch and properties4.07e Inverse hyperbolic: definitions, domains, ranges |
| Answer | Marks | Guidance |
|---|---|---|
| Correct shape, approximately symmetrical about \(y\)-axis, above \(x\)-axis | B1 | Draws correct shape |
| \(y\)-intercept at \((0,1)\) with 1 clearly labelled on \(y\)-axis | B1 | Indicates one \(y\)-axis intercept at \((0,1)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = \pm\cosh^{-1}2 = \pm1.32\) (3sf) | M1 | Obtains at least one correct solution; accept \(\ln(2+\sqrt{3})\) or \(\ln(2-\sqrt{3})\) or equivalent \(\ln\) expression; accept AWRT 1.32 or \(-\)1.32; do not accept \(\cosh^{-1}2\) |
| \(x = \pm1.32\) | A1 | Obtains AWRT \(\pm1.32\) with no other solutions given |
## Question 6:
### Part 6(a):
| Correct shape, approximately symmetrical about $y$-axis, above $x$-axis | B1 | Draws correct shape |
| $y$-intercept at $(0,1)$ with 1 clearly labelled on $y$-axis | B1 | Indicates one $y$-axis intercept at $(0,1)$ |
### Part 6(b):
| $x = \pm\cosh^{-1}2 = \pm1.32$ (3sf) | M1 | Obtains at least one correct solution; accept $\ln(2+\sqrt{3})$ or $\ln(2-\sqrt{3})$ or equivalent $\ln$ expression; accept AWRT 1.32 or $-$1.32; do not accept $\cosh^{-1}2$ |
| $x = \pm1.32$ | A1 | Obtains AWRT $\pm1.32$ with no other solutions given |
---6
\begin{enumerate}[label=(\alph*)]
\item On the axes below, sketch the graph of
$$y = \cosh x$$
Indicate the value of any intercept of the curve with the axes.\\
\includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-05_1114_1121_552_447}
6
\item Solve the equation
$$\cosh x = 2$$
Give your answers to three significant figures.\\
\includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-06_2491_1755_173_123}
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2024 Q6 [4]}}