A2 June 2023 Q5
5.
A uniform lamina \(OAB\) is modelled by the finite region bounded by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = 9 - x^2\), for \(x \geqslant 0\), as shown shaded in Figure 3.
The unit of length on both axes is 1 m.
The area of the lamina is \(18\ \text{m}^2\)
[ Solutions relying on calculator technology are not acceptable.]
(4)A light string has one end attached to the lamina at \(O\) and the other end attached to the ceiling. A second light string has one end attached to the lamina at \(A\) and the other end attached to the ceiling.
The lamina hangs in equilibrium with the strings vertical and \(OA\) horizontal.
The weight of the lamina is \(W\)
The tension in the string attached to the lamina at \(A\) is \(\lambda W\)
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle\int \dfrac{1}{2}y^2\,\mathrm{d}x = \dfrac{1}{2}\int \left(9 - x^2\right)^2\mathrm{d}x\ \left(= \dfrac{1}{2}\int \left(81 - 18x^2 + x^4\right)\mathrm{d}x\right)\) | M1 | 2.1 |
| \(= \dfrac{1}{2}\left[81x - 6x^3 + \dfrac{1}{5}x^5\right]_0^3\) | DM1 | 1.1b |
| \(= \dfrac{1}{2}\left(3 \times 81 - 6 \times 27 + \dfrac{243}{5}\right) \left(= \dfrac{324}{5}\right)\) | A1 | 1.1b |
| \(\Rightarrow \text{distance} = \dfrac{64.8}{18} = 3.6\) * | A1* | 2.2a |
| (4) |
Notes
M1: Use of \(\displaystyle\int \dfrac{1}{2}y^2\,\mathrm{d}x\) or \(\displaystyle\int xy\,\mathrm{d}y\). Ignore any limits.
DM1: Integrate and use correct limits (0 and 3 for \(x\), 0 and 9 for \(y\)).
Usual rules for integration: powers increasing by 1
A1: Correct unsimplified expression.
A1*: Obtain given answer from correct working.
Alternative (a)
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle\int xy\,\mathrm{d}y = \int y\sqrt{9 - y}\,\mathrm{d}y\) | M1 | 2.1 |
| \(= \left[-\dfrac{2}{3}y(9 - y)^{\frac{3}{2}} - \dfrac{4}{15}(9 - y)^{\frac{5}{2}}\right]_0^9\) | DM1 | 1.1b |
| \(= \dfrac{4}{15} \times 9^{\frac{5}{2}} \quad \left(= \dfrac{324}{5}\right)\) | A1 | 1.1b |
| \(\Rightarrow \text{distance} = \dfrac{64.8}{18} = 3.6\ \text{(m)}\) * | A1* | 2.2a |
| (4) |
| Scheme | Marks | AO |
|---|---|---|
| Moments about \(O\): | M1 | 3.1a |
| \(3.6W = 9\lambda W\) | A1 | 1.1b |
| \(\lambda = 0.4\) | A1 | 1.1b |
| (3) | ||
| (7 marks) |
Notes
M1: Complete method to obtain \(\lambda\), e.g. by taking moments about \(O\) or by resolving and taking moments about a different point.
Allow for an equation in \(T_A\) or \(\lambda\)
A1: Correct unsimplified equation in \(\lambda\)
A1: Correct only.