A2 June 2024 Q1
1.
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
A particle \(P\) moves along a straight line.
Initially \(P\) is at rest at the point \(O\) on the line.
At time \(t\) seconds, where \(t \geqslant 0\)
- the displacement of \(P\) from \(O\) is \(x\) metres
- the velocity of \(P\) is \(v\ \text{m s}^{-1}\) in the positive \(x\) direction
- the acceleration of \(P\) is \(\dfrac{96}{(3t + 5)^3}\ \text{m s}^{-2}\) in the positive \(x\) direction
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle\int \dfrac{96}{(3t + 5)^3}\,\mathrm{d}t = \int 1\,\mathrm{d}v \quad \Rightarrow v = \ldots\) | M1 | 2.1 |
| \(\Rightarrow -\dfrac{96}{2 \times 3 \times (3t + 5)^2}\ (+C) = v\) | A1 | 1.1b |
| Use limits \(v = 0,\ t = 0\) | M1 | 1.1b |
| \(\Rightarrow v = \dfrac{96}{6 \times (5)^2} - \dfrac{96}{6 \times (3t + 5)^2} = \dfrac{16}{25} - \dfrac{16}{(3t + 5)^2}\) * | A1* | 2.2a |
| (4) |
Notes
M1: Form a differential equation in \(v\) and \(t\) and integrate.
Must attempt integration of \(\dfrac{k}{(3t + 5)^3}\). RHS can be implied.
A1: Correct integration. Ignore any limits. Accept without constant of integration.
M1: Use \(v = 0,\ t = 0\) as limits in a definite integral or to find the constant of integration
A1*: Obtain given answer in the form \(v = p - \dfrac{q}{(3t + 5)^2}\) from correct working. Accept if correct form given and values of \(p\) and \(q\) stated separately. Must have “\(v =\)”.
| Scheme | Marks | AO |
|---|---|---|
| \(t \to \infty \Rightarrow v \to \dfrac{16}{25}\ (= 0.64)\) | B1ft | 2.2a |
| (1) |
Notes
B1ft: Follow through their \(p\)
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle\int 1\,\mathrm{d}x = \int \dfrac{16}{25} - \dfrac{16}{(3t + 5)^2}\,\mathrm{d}t \Rightarrow x = rt + s\dfrac{1}{3t + 5}\) | M1 | 2.1 |
| \(x = \dfrac{16}{25}t + \dfrac{16}{3(3t + 5)}\ (+D)\) | A1ft | 1.1b |
| \(x = \left[\dfrac{16}{25}t + \dfrac{16}{3(3t + 5)}\right]_0^2\) | M1 | 1.1b |
| \(x = \left(\dfrac{32}{25} + \dfrac{16}{3(11)}\right) - \left(\dfrac{16}{3(5)}\right) = \left(\dfrac{192}{275}\right) = 0.70\) or better | A1 | 2.2a |
| (4) | ||
| (9 marks) |
Notes
M1: Form a differential equation in \(x\) and \(t\) and integrate to obtain \(rt + s\dfrac{1}{3t + 5}\) where \(r\) and \(s\) are rational
A1ft: Correct integration. Ignore limits and condone no constant of integration.
Follow through their \(p\) and their \(-\dfrac{q}{3}\)
M1: Use \(x = 0,\ t = 0\) as limits in a definite integral or substituted to find the constant of integration and find \(x\) when \(t = 2\)
A1: 0.70 or better. (0.698181…)