| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Sketch graphs of hyperbolic functions |
| Difficulty | Standard +0.3 Part (a) requires understanding that x = cosh(y+b) is a horizontal hyperbola with vertex at (1, -b), which is a straightforward transformation of the standard cosh graph. Part (b) is immediate once the sketch is done: the minimum distance is 1 since cosh has minimum value 1. This is a routine Further Maths question testing basic hyperbolic function properties with minimal problem-solving required. |
| Spec | 4.07b Hyperbolic graphs: sketch and properties |
| Answer | Marks |
|---|---|
| 6(a) | Draws a shape or or or in any position. |
| Answer | Marks | Guidance |
|---|---|---|
| cusp. | 1.2 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| cusp. | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| cusp. | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| NMS can score 4/4 | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 6(b) | Deduces the correct minimum distance between the graph of | |
| and the y-axis as 1 | 2.2a | B1 |
| 𝑥𝑥 = cosh(𝑦𝑦+𝑏𝑏) Total | 5 | |
| Q | Marking instructions | AO |
Question 6:
--- 6(a) ---
6(a) | Draws a shape or or or in any position.
Condone a graph which appears to have an asymptote and/or a
∪ ⊂ ∩ ⊃
cusp. | 1.2 | B1
Draws a shape in any orientation or position with the vertex
not positioned on an axis.
∪
Condone a half graph drawn.
Condone a graph which appears to have an asymptote and/or a
cusp. | 3.1a | M1
Draws a shape.
The line of symmetry must be parallel to the -axis.
⊂
Condone a half graph drawn, i.e. or
𝑥𝑥
Condone a graph which appears to have an asymptote and/or a
cusp. | 3.1a | M1
Draws a correct sketch of with x-intercept
identified as
𝑥𝑥 = cosh(𝑦𝑦+𝑏𝑏)
Ignore incorrect vertex coordinates.
(cosh𝑏𝑏,0)
Do not condone a half graph, or a graph with asymptotes or a
cusp.
NMS can score 4/4 | 1.1b | A1
--- 6(b) ---
6(b) | Deduces the correct minimum distance between the graph of
and the y-axis as 1 | 2.2a | B1 | 1
𝑥𝑥 = cosh(𝑦𝑦+𝑏𝑏) Total | 5
Q | Marking instructions | AO | Marks | Typical solution\begin{enumerate}[label=(\alph*)]
\item On the axes provided, sketch the graph of
$$x = \cosh(y + b)$$
where $b$ is a positive constant. [4 marks]
\item Determine the minimum distance between the graph of $x = \cosh(y + b)$ and the $y$-axis. [1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2019 Q6 [5]}}