| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Solve mixed sinh/cosh linear combinations |
| Difficulty | Standard +0.8 This is a Further Maths FP3 question requiring knowledge of hyperbolic functions and their exponential definitions. Part (a) requires sketching with asymptotes and intercepts (routine but multi-step). Part (b) requires converting sinh to exponential form, leading to a quadratic in e^{2x}, which is a standard technique but requires careful algebraic manipulation across 5 marks. Slightly above average difficulty due to the Further Maths content and multi-step algebraic reasoning required. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06d Natural logarithm: ln(x) function and properties4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07b Hyperbolic graphs: sketch and properties |
| Answer | Marks |
|---|---|
| - Graph of \(y = 3\sinh 2x\): Shape of \(-e^{2x}\) graph | B1, B1 |
| - Asymptote: \(y = 13\) | B1 |
| - Value 10 on y axis and value 0.7 or \(\frac{1}{2}\ln\left(\frac{1}{3}\right)\) on x axis | B1 |
| Answer | Marks |
|---|---|
| \(\therefore e^{2x} = -\frac{1}{9}\) or 3 | M1 A1, DM1 A1 |
| \(\therefore x = \frac{1}{2}\ln(3)\) | B1 |
## Part (a)
**Answer/Working:**
- Graph of $y = 3\sinh 2x$: Shape of $-e^{2x}$ graph | **B1, B1**
- Asymptote: $y = 13$ | **B1**
- Value 10 on y axis and value 0.7 or $\frac{1}{2}\ln\left(\frac{1}{3}\right)$ on x axis | **B1**
(4 marks)
## Part (b)
**Answer/Working:** Use definition $\frac{3}{2}(e^{2x}-e^{-2x}) = 13 - 3e^{2x} \to 9e^{4x} - 26e^{2x} - 3 = 0$ to form quadratic
$\therefore e^{2x} = -\frac{1}{9}$ or 3 | **M1 A1, DM1 A1**
$\therefore x = \frac{1}{2}\ln(3)$ | **B1**
(5 marks)
**Notes:**
- (a) 1B1: $y = 3\sinh 2x$ first and third quadrant
- 2B1: Shape of $y = -e^{2x}$ correct intersects on positive axes
- 3B1: Equation of asymptote, $y = 13$, given. Penlise 'extra' asymptotes here
- 4B1: Intercepts correct both
- (b) 1M1: Getting a three terms quadratic in $e^{2x}$
- 1A1: Correct three term quadratic
- 2DM1: Solving for $e^{2x}$
- 2A1: CAO for $e^{2x}$ condone omission of negative value
- B1: CAO one answer only
---The curve $C_1$ has equation $y = 3\sinh 2x$, and the curve $C_2$ has equation $y = 13 - 3e^{2x}$.
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of the curves $C_1$ and $C_2$ on one set of axes, giving the equation of any asymptote and the coordinates of points where the curves cross the axes.
[4]
\item Solve the equation $3\sinh 2x = 13 - 3e^{2x}$, giving your answer in the form $\frac{1}{2}\ln k$, where $k$ is an integer.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 2011 Q5 [9]}}