| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Solve using substitution u = cosh x or u = sinh x |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on hyperbolic functions with clear scaffolding: part (a) guides students through proving an identity using definitions, part (b) applies it directly to solve for θ, and part (c) requires back-substitution. While it involves multiple steps and Further Maths content (inherently harder), the pathway is well-signposted and uses standard techniques throughout, making it moderately above average difficulty. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07f Inverse hyperbolic: logarithmic forms |
(a) Using the definition $\sinh y = \frac{1}{2}(e^y - e^{-y})$, prove the identity $4\sinh^3 y + 3\sinh y = \sinh 3y$
[3 marks]
(b) Given that $x = \sinh y$ and $16x^3 + 12x - 3 = 0$, find the value of $y$ in terms of a natural logarithm.
[4 marks]
(c) Hence find the real root of the equation $16x^3 + 12x - 3 = 0$, giving your answer in the form $2^p - 2^q$, where $p$ and $q$ are rational numbers.
[2 marks]5
\begin{enumerate}[label=(\alph*)]
\item Using the definition $\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right)$, prove the identity
$$4 \sinh ^ { 3 } \theta + 3 \sinh \theta = \sinh 3 \theta$$
\item Given that $x = \sinh \theta$ and $16 x ^ { 3 } + 12 x - 3 = 0$, find the value of $\theta$ in terms of a natural logarithm.
\item Hence find the real root of the equation $16 x ^ { 3 } + 12 x - 3 = 0$, giving your answer in the form $2 ^ { p } - 2 ^ { q }$, where $p$ and $q$ are rational numbers.\\[0pt]
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2014 Q5 [9]}}