| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Maclaurin series for inverse hyperbolics |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring differentiation of inverse hyperbolic functions using the chain rule, then computing a Maclaurin series. While it involves Further Maths content (making it inherently harder than standard A-level), the execution is methodical: apply standard derivative formula for tanh^(-1), differentiate again, then evaluate at x=0. The 'show that' structure provides guidance, and the techniques are standard for FP2 students. More challenging than typical C3/C4 calculus but routine within Further Maths. |
| Spec | 4.07f Inverse hyperbolic: logarithmic forms4.08a Maclaurin series: find series for function4.08g Derivatives: inverse trig and hyperbolic functions |
| Answer | Marks |
|---|---|
| (i) State \(A = 42\) | B1 |
| State \(k = \frac{1}{5}\) | B1 |
| Attempt correct process for finding \(m\) | M1 |
| Obtain \(\frac{1}{3}\ln 2\) or \(0.077\) | A1 |
| (ii) Attempt solution for \(t\) using either formula | M1 |
| Obtain \(11.3\) | A1 |
| (iii) Differentiate to obtain form \(Be^{mt}\) | M1 |
| Obtain \(3.235e^{0.077t}\) | A1√ |
| Obtain \(47.9\) | A1 |
| | |
|---|---|
| (i) State $A = 42$ | B1 |
| State $k = \frac{1}{5}$ | B1 |
| Attempt correct process for finding $m$ | M1 |
| Obtain $\frac{1}{3}\ln 2$ or $0.077$ | A1 |
| (ii) Attempt solution for $t$ using either formula | M1 |
| Obtain $11.3$ | A1 |
| (iii) Differentiate to obtain form $Be^{mt}$ | M1 |
| Obtain $3.235e^{0.077t}$ | A1√ |
| Obtain $47.9$ | A1 |7 It is given that $\mathrm { f } ( x ) = \tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)$, for $x > - \frac { 1 } { 2 }$.\\
(i) Show that $\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { 1 + 2 x }$, and find $\mathrm { f } ^ { \prime \prime } ( x )$.\\
(ii) Show that the first three terms of the Maclaurin series for $\mathrm { f } ( x )$ can be written as $\ln a + b x + c x ^ { 2 }$, for constants $a , b$ and $c$ to be found.
\hfill \mbox{\textit{OCR FP2 2008 Q7 [10]}}