| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2018 |
| Session | March |
| Marks | 8 |
| Topic | Hyperbolic functions |
| Type | Solve using substitution u = cosh x or u = sinh x |
| Difficulty | Challenging +1.2 This is a structured two-part question where part (i) guides students through proving an identity using standard definitions, and part (ii) applies this result with a clearly signposted substitution. While it requires connecting the cubic equation to hyperbolic functions and manipulating the result into the required form, the pathway is well-scaffolded and uses routine Further Maths techniques. Moderately above average difficulty due to the multi-step reasoning and Further Maths content, but not exceptionally challenging. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
7 (i) Using the definition of $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, show that
$$4 \sinh ^ { 3 } x = \sinh 3 x - 3 \sinh x$$
\section*{(ii) In this question you must show detailed reasoning.}
By making a suitable substitution, find the real root of the equation
$$16 u ^ { 3 } + 12 u = 3 .$$
Give your answer in the form $\frac { \left( a ^ { \frac { 1 } { b } } - a ^ { - \frac { 1 } { b } } \right) } { c }$ where $a , b$ and $c$ are integers.
\hfill \mbox{\textit{OCR Further Pure Core 1 2018 Q7 [8]}}