| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Topic | Hyperbolic functions |
| Type | Sketch graphs of hyperbolic functions |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring sketching hyperbolic functions (less familiar than trig), analyzing intersections, verifying bounds, and applying Newton-Raphson with derivatives of hyperbolic functions. While the individual techniques are standard, the combination of hyperbolic function manipulation and numerical methods places it moderately above average difficulty. |
| Spec | 1.09d Newton-Raphson method4.07b Hyperbolic graphs: sketch and properties4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
| Answer | Marks |
|---|---|
| (i) [Graph showing \(y = \tanh x\) and \(y = \cosh x - 1\) with curves crossing twice] | B1 |
| Grad. of \(\tanh x\) should be 1 | B1 |
| Curves cross twice (at \(x = 0\) and \(x = \alpha\)) so there are 2 roots to \(\tanh x = \cosh x - 1\) i.e. \(1 + \tanh x - \cosh x = 0\) | B1 |
| [3] | |
| (ii) (a) \(f(1)f(1.5) = 0.22... \times (-0.45...) < 0\) \(\Rightarrow 1 < \alpha < 1.5\) by the "Change-of-Sign" Rule | B1 |
| [1] | |
| (b) \(f'(x) = \text{sech}^2 x - \sinh x\) | B1, B1 |
| Use of \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) at least once | M1 |
| \(x_1 = 1.25\), \(x_2 = 1.219\ 625\ 3\), \(x_3 = 1.218\ 76\) to 5 d.p. | A1 |
| [4] |
**(i)** [Graph showing $y = \tanh x$ and $y = \cosh x - 1$ with curves crossing twice] | B1 |
Grad. of $\tanh x$ should be 1 | B1 |
Curves cross twice (at $x = 0$ and $x = \alpha$) so there are 2 roots to $\tanh x = \cosh x - 1$ i.e. $1 + \tanh x - \cosh x = 0$ | B1 |
| [3] |
**(ii)** **(a)** $f(1)f(1.5) = 0.22... \times (-0.45...) < 0$ $\Rightarrow 1 < \alpha < 1.5$ by the "Change-of-Sign" Rule | B1 |
| [1] |
**(b)** $f'(x) = \text{sech}^2 x - \sinh x$ | B1, B1 |
Use of $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ at least once | M1 |
$x_1 = 1.25$, $x_2 = 1.219\ 625\ 3$, $x_3 = 1.218\ 76$ to 5 d.p. | A1 |
| [4] |
---\begin{enumerate}[label=(\roman*)]
\item On a single diagram, sketch the graphs of $y = \tanh x$ and $y = \cosh x - 1$, and use your diagram to explain why the equation $\text{f}(x) = 0$ has exactly two roots, where
$$\text{f}(x) = 1 + \tanh x - \cosh x.$$ [3]
\item The non-zero root of $\text{f}(x) = 0$ is $\alpha$.
\begin{enumerate}[label=(\alph*)]
\item Verify that $1 < \alpha < 1.5$. [1]
\item Taking $x_1 = 1.25$ as an initial approximation to $\alpha$, use the Newton-Raphson iterative method to find $x_3$, giving your answer to 5 decimal places. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2011 Q4 [8]}}