| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | May |
| Marks | 3 |
| Topic | Hyperbolic functions |
| Type | Express hyperbolic in exponential form |
| Difficulty | Moderate -0.3 This is a straightforward algebraic manipulation requiring substitution of standard exponential definitions of cosh x and sinh x, followed by expanding a sum of cubes and simplifying. While it involves hyperbolic functions (a Further Maths topic), the question is purely mechanical with no problem-solving insight needed—just recall definitions and apply algebraic identities systematically. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
3.
Show that
$$\cosh ^ { 3 } x + \sinh ^ { 3 } x = \frac { 1 } { 4 } \mathrm { e } ^ { m x } + \frac { 3 } { 4 } \mathrm { e } ^ { n x }$$
where $m$ and $n$ are integers.\\[0pt]
[3 marks]\\
\hfill \mbox{\textit{SPS SPS FM 2020 Q3 [3]}}