| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Session | Specimen |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Derive equation from calculus condition |
| Difficulty | Standard +0.8 This is a multi-part FP2 question requiring differentiation of hyperbolic functions, quotient rule, graphical analysis, Newton-Raphson iteration, and error estimation. While each individual technique is standard for Further Maths, the combination of skills and the error estimation in part (iv) elevates it above routine exercises. The hyperbolic function context and multi-step reasoning place it moderately above average difficulty. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.09d Newton-Raphson method4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07d Differentiate/integrate: hyperbolic functions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{dy}{dx} = \frac{\cosh x - x\sinh x}{\cosh^2 x}\) | M1 | For differentiating and equating to zero |
| Max occurs when \(\cosh x = x \sinh x\), i.e. \(x \tanh x = 1\) | A1 | For showing given result correctly |
| \(-\) | 2 | |
| (ii) [Correct sketch of \(y = \tanh x\)] | B1 | For correct sketch of \(y = \tanh x\) |
| [Correct identification of asymptote \(y=1\)] | B1 | For identification of asymptote \(y=1\) |
| [Correct explanation of \(a > 1\) based on intersection \((1,1)\) of \(y = 1/x\) with \(y=1\)] | B1 | For correct explanation of \(a > 1\) based on intersection \((1, 1)\) of \(y = 1/x\) with \(y = 1\) |
| \(-\) | 3 | |
| (iii) \(x_{n+1} = x_n - \frac{x_n \tanh x_n - 1}{\tanh x_n + x_n \operatorname{sech}^2 x_n}\) | M1 | For correct Newton-Raphson structure |
| \(-\) | A1 | For all details in \(x - \frac{f(x)}{f'(x)}\) correct |
| \(x_1 = 1 \Rightarrow x_2 = 1.20177\ldots\) | M1 | For using Newton-Raphson at least once |
| \(-\) | A1 | For \(x_2\) correct to at least 3sf |
| \(x_3 = 1.1996785\ldots\) | A1 | For \(x_3\) correct to at least 4sf |
| \(-\) | 5 | |
| (iv) \(e_1 = 0.2\), \(e_2 = -0.002\) | B1√ | For both magnitudes correct |
| \(\frac{e_3}{e_2^2} = \frac{e_2}{e_1^2} \Rightarrow e_3 = -2 \times 10^{-7}\) | M1 | For use of quadratic convergence property |
| \(-\) | A1 | For answer of correct magnitude |
| \(-\) | 3 |
(i) $\frac{dy}{dx} = \frac{\cosh x - x\sinh x}{\cosh^2 x}$ | M1 | For differentiating and equating to zero |
Max occurs when $\cosh x = x \sinh x$, i.e. $x \tanh x = 1$ | A1 | For showing given result correctly |
$-$ | **2** | |
(ii) [Correct sketch of $y = \tanh x$] | B1 | For correct sketch of $y = \tanh x$ |
[Correct identification of asymptote $y=1$] | B1 | For identification of asymptote $y=1$ |
[Correct explanation of $a > 1$ based on intersection $(1,1)$ of $y = 1/x$ with $y=1$] | B1 | For correct explanation of $a > 1$ based on intersection $(1, 1)$ of $y = 1/x$ with $y = 1$ |
$-$ | **3** | |
(iii) $x_{n+1} = x_n - \frac{x_n \tanh x_n - 1}{\tanh x_n + x_n \operatorname{sech}^2 x_n}$ | M1 | For correct Newton-Raphson structure |
$-$ | A1 | For all details in $x - \frac{f(x)}{f'(x)}$ correct |
$x_1 = 1 \Rightarrow x_2 = 1.20177\ldots$ | M1 | For using Newton-Raphson at least once |
$-$ | A1 | For $x_2$ correct to at least 3sf |
$x_3 = 1.1996785\ldots$ | A1 | For $x_3$ correct to at least 4sf |
$-$ | **5** | |
(iv) $e_1 = 0.2$, $e_2 = -0.002$ | B1√ | For both magnitudes correct |
$\frac{e_3}{e_2^2} = \frac{e_2}{e_1^2} \Rightarrow e_3 = -2 \times 10^{-7}$ | M1 | For use of quadratic convergence property |
$-$ | A1 | For answer of correct magnitude |
$-$ | **3** | |
---7 The curve with equation
$$y = \frac { x } { \cosh x }$$
has one stationary point for $x > 0$.\\
(i) Show that the $x$-coordinate of this stationary point satisfies the equation $x \tanh x - 1 = 0$.
The positive root of the equation $x \tanh x - 1 = 0$ is denoted by $\alpha$.\\
(ii) Draw a sketch showing (for positive values of $x$ ) the graph of $y = \tanh x$ and its asymptote, and the graph of $y = \frac { 1 } { x }$. Explain how you can deduce from your sketch that $\alpha > 1$.\\
(iii) Use the Newton-Raphson method, taking first approximation $x _ { 1 } = 1$, to find further approximations $x _ { 2 }$ and $x _ { 3 }$ for $\alpha$.\\
(iv) By considering the approximate errors in $x _ { 1 }$ and $x _ { 2 }$, estimate the error in $x _ { 3 }$.
\hfill \mbox{\textit{OCR FP2 Q7 [13]}}