| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Find stationary points of hyperbolic curves |
| Difficulty | Standard +0.8 Part (i) is routine manipulation of hyperbolic identities (using cosh²t - sinh²t = 1). Part (ii) requires setting up a perimeter function P(x) = 2(x + 1/x), finding the minimum via calculus, and justifying it's a minimum—this involves several steps and optimization reasoning, making it moderately challenging but still a standard Further Maths calculus problem. |
| Spec | 1.08i Integration by parts4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (i) | xy(cid:32)cosh2t(cid:16)sinh2t |
| Answer | Marks |
|---|---|
| So xy(cid:32)1 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (ii) | Perimeter(cid:32)2x(cid:14)2y |
| Answer | Marks |
|---|---|
| So the minimum value of the perimeter is 4 | M1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
| Answer | Marks |
|---|---|
| 3.2a | n |
Question 6:
6 | (i) | xy(cid:32)cosh2t(cid:16)sinh2t
Substituting cosh2t(cid:16)sinh2t(cid:32)1
So xy(cid:32)1 | M1
A1
[2] | 1.1
2.1
6 | (ii) | Perimeter(cid:32)2x(cid:14)2y
(cid:32)2(cosht(cid:14)sinht)(cid:14)2(cosht(cid:16)sinht)
(cid:32)4cosht
The minimum value of cosht is 1
So the minimum value of the perimeter is 4 | M1
A1
B1
A1
[4] | 1.1
1.1
3.1a
3.2a | n
e\begin{enumerate}[label=(\roman*)]
\item A curve is in the first quadrant. It has parametric equations $x = \cosh t + \sinh t$, $y = \cosh t - \sinh t$ where $t \in \mathbb{R}$. Show that the cartesian equation of the curve is $xy = 1$. [2]
\end{enumerate}
Fig. 6 shows the curve from part (i). P is a point on the curve. O is the origin. Point A lies on the $x$-axis, point B lies on the $y$-axis and OAPB is a rectangle.
\includegraphics{figure_6}
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the smallest possible value of the perimeter of rectangle OAPB. Justify your answer. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core Q6 [6]}}