| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Sketch graphs of hyperbolic functions |
| Difficulty | Standard +0.3 Part (i) is straightforward recall of standard hyperbolic function graphs. Part (ii) involves routine algebraic manipulation of the cosech definition, followed by a guided substitution to integrate—the substitution is explicitly given, making this a standard Further Maths exercise rather than requiring problem-solving insight. Slightly above average difficulty due to being Further Maths content, but mechanical in execution. |
| Spec | 1.08h Integration by substitution4.07b Hyperbolic graphs: sketch and properties |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct shape for \(\sinh x\) | B1 | |
| Correct shape for \(\text{cosech}\, x\) | B1 | |
| Obvious point \((dy/dx \neq O)\)/asymptotes clear | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct definition of \(\sinh x\) | B1 | May be implied |
| Invert and multiply by \(e^x\) to AG | B1 | Must be clear; allow \(2/(e^x - e^{-x})\) as minimum simplification |
| Sub. \(u = e^x\) and \(du = e^x\,dx\) | M1 | Or equivalent, all \(x\) eliminated and not \(dx = du\) |
| Replace to \(\frac{2}{u^2-1}\,du\) | ||
| Integrate to \(a\ln\!\left(\frac{u-1}{u+1}\right)\) | A1 | |
| Replace \(u\) | A1\(\sqrt{}\) | Use formulae book, PT, or \(\text{atanh}^{-1}u\) |
| A1 | No need for \(c\) |
## Question 4:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct shape for $\sinh x$ | B1 | |
| Correct shape for $\text{cosech}\, x$ | B1 | |
| Obvious point $(dy/dx \neq O)$/asymptotes clear | B1 | |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct definition of $\sinh x$ | B1 | May be implied |
| Invert and multiply by $e^x$ to AG | B1 | Must be clear; allow $2/(e^x - e^{-x})$ as minimum simplification |
| Sub. $u = e^x$ and $du = e^x\,dx$ | M1 | Or equivalent, all $x$ eliminated and not $dx = du$ |
| Replace to $\frac{2}{u^2-1}\,du$ | | |
| Integrate to $a\ln\!\left(\frac{u-1}{u+1}\right)$ | A1 | |
| Replace $u$ | A1$\sqrt{}$ | Use formulae book, PT, or $\text{atanh}^{-1}u$ |
| | A1 | No need for $c$ |
---4 (i) On separate diagrams, sketch the graphs of $y = \sinh x$ and $y = \operatorname { cosech } x$.\\
(ii) Show that $\operatorname { cosech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }$, and hence, using the substitution $u = \mathrm { e } ^ { x }$, find $\int \operatorname { cosech } x \mathrm {~d} x$.
\hfill \mbox{\textit{OCR FP2 2007 Q4 [9]}}