| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Find stationary points of hyperbolic curves |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on hyperbolic functions. Part (i) is routine algebraic manipulation using standard definitions. Part (ii) requires differentiation, solving a quadratic equation, and using the second derivative test—all standard techniques with no novel insight required. While it's Further Maths content, the execution is mechanical and follows textbook methods. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07b Hyperbolic graphs: sketch and properties4.07d Differentiate/integrate: hyperbolic functions |
9 (i) Using the definitions of $\cosh x$ and $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, prove that
$$\sinh 2 x = 2 \sinh x \cosh x$$
(ii) Show that the curve with equation
$$y = \cosh 2 x - 6 \sinh x$$
has just one stationary point, and find its $x$-coordinate in logarithmic form. Determine the nature of the stationary point.
\hfill \mbox{\textit{OCR FP2 Q9 [12]}}