A2 June 2022 Paper 1 Q1
1 In this question you must show detailed reasoning.
The region \(R\) is bounded by the curve with equation \(y = \sqrt{\sinh x}\), the \(x\)-axis and the line with equation \(x = 2\ln 3\) (see diagram). The units of the axes are centimetres.
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.
| Scheme | Marks | AO |
|---|---|---|
| DR \(\cosh(2\ln 3) = \dfrac{\mathrm{e}^{2\ln 3} + \mathrm{e}^{-2\ln 3}}{2}\) | M1 | 1.1 |
| \(= \dfrac{1}{2}\left(9 + \dfrac{1}{9}\right) = \dfrac{41}{9}\) | A1 | 2.1 |
| [2] |
Notes
M1: Correct use of definition of \(\cosh x\) must be seen
A1: AG, must see either \(\mathrm{e}^{\ln 9}\) and \(\mathrm{e}^{\ln\frac{1}{9}}\) or \(3^2\) and \(3^{-2}\) or \(\dfrac{1}{2}\left(9 + \dfrac{1}{9}\right)\)
| Scheme | Marks | AO |
|---|---|---|
| DR \(\displaystyle V = \pi\int_0^{2\ln 3} \left(\sqrt{\sinh x}\right)^2\,\mathrm{d}x\) | M1 | 3.3 |
| \(= \pi\left[\cosh x\right]_0^{2\ln 3}\) | A1 | 1.1 |
| \(= \pi\left(\cosh(2\ln 3) - \cosh 0\right)\) \(= \pi\left(\dfrac{41}{9} - 1\right)\) | M1 | 3.4 |
| \(= \dfrac{32}{9}\pi\ \left(\text{cm}^3\right)\) oe | A1 | 1.1 |
| [4] |
Notes
M1: oe, intention to integrate \(y^2\).
Condone missing \(\pi\), ignore limits.
A1: For \(+\cosh x\). Ignore any reference to \(c\)
M1: Substituting correct limits and subtracting
A1: Ignore units