| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | June |
| Marks | 11 |
| Topic | Hyperbolic functions |
| Type | Intersection points of hyperbolic curves |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths question involving hyperbolic functions and curve fitting. While it requires knowledge of cosh properties and solving simultaneous equations with hyperbolic functions, the steps are fairly guided: using the minimum point to find k, then substituting two known points to solve for a and b. The comparison with a quadratic model is straightforward evaluation. The question is more computational than conceptually demanding, making it moderately above average difficulty but not requiring exceptional insight. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07b Hyperbolic graphs: sketch and properties |
Two thin poles, $OA$ and $BC$, are fixed vertically on horizontal ground. A chain is fixed at $A$ and $C$ such that it touches the ground at point $D$ as shown in the diagram.
On a coordinate system the coordinates of $A$, $B$ and $D$ are $(0, 3)$, $(5, 0)$ and $(2, 0)$.
\includegraphics{figure_5}
It is required to find the height of pole $BC$ by modelling the shape of the curve that the chain forms.
Jofra models the curve using the equation $y = k \cosh(ax - b) - 1$ where $k$, $a$ and $b$ are positive constants.
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $k$. [2]
\item Find the exact value of $a$ and the exact value of $b$, giving your answers in logarithmic form. [5]
\end{enumerate}
Holly models the curve using the equation $y = \frac{1}{4}x^2 - 3x + 3$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Write down the coordinates of the point, $(u, v)$ where $u$ and $v$ are both non-zero, at which the two models will agree. [1]
\item Show that Jofra's model and Holly's model disagree in their predictions of the height of pole $BC$ by $3.32$m to 3 significant figures. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q5 [11]}}