M5 June 2014 Q2
2. A particle \(P\) moves in a plane so that its position vector, \(\mathbf{r}\) metres at time \(t\) seconds, satisfies the differential equation
\[\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} + \mathbf{r} = t\mathbf{i} + \mathrm{e}^{-t}\mathbf{j}\]When \(t = 0\) the particle is at the point with position vector \((\mathbf{i} + \mathbf{j})\) m.
Find \(\mathbf{r}\) in terms of \(t\). (9)
| Scheme | Marks |
|---|---|
| IF \(= \mathrm{e}^{\int \mathrm{d}t} = \mathrm{e}^t\) | M1 A1 |
| \(\dfrac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\mathrm{e}^t) = t\mathrm{e}^t\mathbf{i} + \mathbf{j}\) | |
| \(\mathbf{r}\mathrm{e}^t = \displaystyle\int t\mathrm{e}^t\mathbf{i} + \mathbf{j}\ \mathrm{d}t\) | M1 |
| \(\mathbf{r}\mathrm{e}^t = (t\mathrm{e}^t - \mathrm{e}^t)\mathbf{i} + t\mathbf{j} + \mathbf{C}\) | M1 A2 |
| \(t = 0,\ \mathbf{r} = \mathbf{i} + \mathbf{j} \Rightarrow \mathbf{C} = 2\mathbf{i} + \mathbf{j}\) | M1 |
| \(\mathbf{r}\mathrm{e}^t = (t\mathrm{e}^t - \mathrm{e}^t)\mathbf{i} + t\mathbf{j} + 2\mathbf{i} + \mathbf{j}\) | A1 |
| \(\mathbf{r} = (t - 1)\mathbf{i} + t\mathrm{e}^{-t}\mathbf{j} + (2\mathbf{i} + \mathbf{j})\mathrm{e}^{-t}\) | |
| \(= (t - 1 + 2\mathrm{e}^{-t})\mathbf{i} + (t + 1)\mathrm{e}^{-t}\mathbf{j}\) | A1 |
| (9 marks) |
Notes
First M1 for IF
First A1 for \(\mathrm{e}^t\)
Second M1 see scheme
Third M1 for attempt to integrate (must include parts)
A2 for a correct integral
Fourth M1 for use of limits
A1 for a correct \(\mathbf{C}\)
A1 for answer (any equivalent form)