| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Sketch graphs of hyperbolic functions |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question combining routine sketching of a standard hyperbolic function (coth x with known asymptotes), mechanical application of Newton-Raphson iteration requiring only differentiation of tanh x, and simple algebraic manipulation to show an equivalence. All components are standard textbook exercises with no novel insight required, though the Further Maths context places it slightly above average A-level difficulty. |
| Spec | 1.09d Newton-Raphson method4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07b Hyperbolic graphs: sketch and properties |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sketch: \(y\)-axis asymptote | B1 | Equation may be implied if clear |
| Shape | B1 | |
| \(y = \pm 1\) asymptotes | B1 | May be implied if seen on graph |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Reasonable attempt at product rule, giving two terms | M1 | |
| Use correct Newton-Raphson at least once with their \(f'(x)\) to produce \(x_2\) | M1 | May be implied |
| Get \(x_2 = 2.0651\) | A1\(\sqrt{}\) | One correct at any stage if reasonable |
| Get \(x_3 = 2.0653\), \(x_4 = 2.0653\) | A1 | cao; or greater accuracy which rounds |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Clearly derive \(\coth x = \frac{1}{2}x\) | B1 | AG; allow derivation from AG. Two roots only |
| Attempt to find second root e.g. symmetry | M1 | |
| Get \(\pm 2.0653\) | A1\(\sqrt{}\) | \(\pm\) their iteration in part (ii) |
## Question 7(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sketch: $y$-axis asymptote | B1 | Equation may be implied if clear |
| Shape | B1 | |
| $y = \pm 1$ asymptotes | B1 | May be implied if seen on graph |
---
## Question 7(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Reasonable attempt at product rule, giving two terms | M1 | |
| Use correct Newton-Raphson at least once with their $f'(x)$ to produce $x_2$ | M1 | May be implied |
| Get $x_2 = 2.0651$ | A1$\sqrt{}$ | One correct at any stage if reasonable |
| Get $x_3 = 2.0653$, $x_4 = 2.0653$ | A1 | cao; or greater accuracy which rounds |
---
## Question 7(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Clearly derive $\coth x = \frac{1}{2}x$ | B1 | AG; allow derivation from AG. Two roots only |
| Attempt to find second root e.g. symmetry | M1 | |
| Get $\pm 2.0653$ | A1$\sqrt{}$ | $\pm$ their iteration in part (ii) |
---7 (i) Sketch the graph of $y = \operatorname { coth } x$, and give the equations of any asymptotes.\\
(ii) It is given that $\mathrm { f } ( x ) = x \tanh x - 2$. Use the Newton-Raphson method, with a first approximation $x _ { 1 } = 2$, to find the next three approximations $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$ to a root of $\mathrm { f } ( x ) = 0$. Give the answers correct to 4 decimal places.\\
(iii) If $\mathrm { f } ( x ) = 0$, show that $\operatorname { coth } x = \frac { 1 } { 2 } x$. Hence write down the roots of $\mathrm { f } ( x ) = 0$, correct to 4 decimal places.
\hfill \mbox{\textit{OCR FP2 2009 Q7 [10]}}