| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Second derivative relations with hyperbolics |
| Difficulty | Standard +0.8 This is a Further Maths FP3 question requiring differentiation of inverse hyperbolic functions and manipulation of derivatives to verify given relations. Part (a) involves first derivative and algebraic manipulation, while part (b) requires computing the second derivative and substitution. The topic itself (inverse hyperbolics) is advanced, but the execution follows standard differentiation techniques with careful algebra—moderately challenging but within reach for well-prepared FM students. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = 2\text{arcosh}(3x) \times \frac{3}{\sqrt{9x^2-1}}\) | M1 A1 A1 A1 | Chain rule; correct derivative of arcosh |
| \(\sqrt{9x^2-1}\frac{dy}{dx} = 6\text{arcosh}(3x)\) | dM1 | Rearranging |
| \((9x^2-1)\left(\frac{dy}{dx}\right)^2 = 36\left(\text{arcosh}(3x)\right)^2\) | Squaring | |
| \((9x^2-1)\left(\frac{dy}{dx}\right)^2 = 36y\) ✽ | A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left\{18x\left(\frac{dy}{dx}\right)^2 + (9x^2-1)\times 2\frac{dy}{dx}\times\frac{d^2y}{dx^2}\right\} = 36\frac{dy}{dx}\) | M1 {A1} A1 | Differentiating implicitly; {A1} for previous result used |
| \((9x^2-1)\frac{d^2y}{dx^2} + 9x\frac{dy}{dx} = 18\) ✽ | A1 | (4) |
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 2\text{arcosh}(3x) \times \frac{3}{\sqrt{9x^2-1}}$ | M1 A1 A1 A1 | Chain rule; correct derivative of arcosh |
| $\sqrt{9x^2-1}\frac{dy}{dx} = 6\text{arcosh}(3x)$ | dM1 | Rearranging |
| $(9x^2-1)\left(\frac{dy}{dx}\right)^2 = 36\left(\text{arcosh}(3x)\right)^2$ | | Squaring |
| $(9x^2-1)\left(\frac{dy}{dx}\right)^2 = 36y$ ✽ | A1 | (5) |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left\{18x\left(\frac{dy}{dx}\right)^2 + (9x^2-1)\times 2\frac{dy}{dx}\times\frac{d^2y}{dx^2}\right\} = 36\frac{dy}{dx}$ | M1 {A1} A1 | Differentiating implicitly; {A1} for previous result used |
| $(9x^2-1)\frac{d^2y}{dx^2} + 9x\frac{dy}{dx} = 18$ ✽ | A1 | (4) |
---\begin{enumerate}
\item Given that $y = ( \operatorname { arcosh } 3 x ) ^ { 2 }$, where $3 x > 1$, show that\\
(a) $\left( 9 x ^ { 2 } - 1 \right) \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = 36 y$,\\
(b) $\left( 9 x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 9 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 18$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 2010 Q5 [9]}}