AS June 2023 Q2
2. A particle \(P\) is moving along the \(x\)-axis.
At time \(t\) seconds, \(t \geqslant 0\), \(P\) has acceleration \(a\ \text{m s}^{-2}\) and velocity \(v\ \text{m s}^{-1}\) in the direction of \(x\) increasing, where
and \(k\) is a positive constant.
When \(t = \ln 2\), \(a = 0\)
When \(t = 0\), the particle passes through the fixed point \(A\).
When \(t = \ln 2\), the particle is \(d\) metres from \(A\).
[Solutions relying entirely on calculator technology are not acceptable.] (4)
| Scheme | Marks | AO |
|---|---|---|
| Use of \(a = \dfrac{\mathrm{d}v}{\mathrm{d}t}\) | M1 | 3.1a |
| \(a = 2\mathrm{e}^{2t} + 6\mathrm{e}^{t} - k\) | A1 | 1.1b |
| Substitute \(t = \ln 2\) into their acceleration and solve for \(k\) | M1 | 1.1b |
| \(k = 20\) | A1 | 2.2a |
| (4) |
Notes
M1: Use the model and differentiate \(v\) to obtain \(a\).
Obtain form \(p\mathrm{e}^{2t} + q\mathrm{e}^{t} - k\)
A1: Correct only
M1: \((2 \times 4 + 6 \times 2 - k = 0)\) Their acceleration must come from an attempt to differentiate.
A1: Correct only
| Scheme | Marks | AO |
|---|---|---|
| Use of \(v = \dfrac{\mathrm{d}x}{\mathrm{d}t}\) | M1 | 2.1 |
| \(x = \dfrac{1}{2}\mathrm{e}^{2t} + 6\mathrm{e}^{t} - \dfrac{k}{2}t^2\ (+C)\) | A1ft | 1.1b |
| Correct use of boundary conditions in their \(x\) | M1 | 3.1a |
| \(x = 2 + 12 - 10(\ln 2)^2 - \left(\dfrac{1}{2} + 6\right)\ \left(= 7.5 - 10(\ln 2)^2\right) = 2.7\) to 2 s.f. | A1 | 1.1b |
| (4) | ||
| (8 marks) |
Notes
M1: Integrate \(v\) to obtain \(x\). Obtain form \(a\mathrm{e}^{2t} + b\mathrm{e}^{t} + ct^2\)
A1ft: Allow in \(k\) or their \(k\)
M1: Use the model to evaluate constant of integration or use boundary conditions as limits in a definite integral. Their \(x\) must come from an attempt to integrate.
A1: 2sf only