| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Applied context: real-world solid |
| Difficulty | Standard +0.8 This question requires knowledge of hyperbolic functions (Further Maths content), manipulation of exponential definitions of cosh, and volume of revolution integration. Part (a) involves non-trivial algebraic manipulation with exponentials, while part (b) requires integrating sinh x and applying hyperbolic identities. The combination of Further Maths content with multi-step reasoning elevates this above standard A-level, though the techniques themselves are relatively standard once the setup is understood. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | DR |
| Answer | Marks |
|---|---|
| 2 9 9 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 2.1 | Correct use of definition of must be seen |
| Answer | Marks |
|---|---|
| (b) | DR |
| Answer | Marks |
|---|---|
| 9 | M1 |
| Answer | Marks |
|---|---|
| [4] | 3.3 |
| Answer | Marks |
|---|---|
| 1.1 | oe, intention to integrate . |
Question 1:
1 | (a) | DR
e2ln3 +e−2ln3
cosh(2ln3)=
2
1 1 41
= 9+ =
2 9 9 | M1
A1
[2] | 1.1
2.1 | Correct use of definition of must be seen
cosh𝑥𝑥
AG, must see either and or and or
1
1 1 ln9 ln9 2 −2
9+ e e 3 3
2 9
(b) | DR
2ln3
( )2
V =π ∫ sinhx dx
0
=π[ ]2ln3
coshx
0
=π( cosh(2ln3)−cosh0 )
41
=π −1
9
32 ( )
= π cm3 oe
9 | M1
A1
M1
A1
[4] | 3.3
1.1
3.4
1.1 | oe, intention to integrate .
Condone missing , ignore2 limits.
𝑦𝑦
𝜋𝜋
For + . Ignore any reference to c
cosh𝑥𝑥
Substituting correct limits and subtracting
Ignore units1 In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item Show that $\cosh ( 2 \ln 3 ) = \frac { 41 } { 9 }$.
The region $R$ is bounded by the curve with equation $\mathrm { y } = \sqrt { \operatorname { sinhx } }$, the $x$-axis and the line with equation $x = 2 \ln 3$ (see diagram). The units of the axes are centimetres.\\
\includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-2_652_668_740_242}
A manufacturer produces bell-shaped chocolate pieces. Each piece is modelled as being the shape of the solid formed by rotating $R$ completely about the $x$-axis.
\item Determine, according to the model, the exact volume of one chocolate piece.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2022 Q1 [6]}}