M5 June 2018 Q3
3. A particle \(P\) moves in the \(xy\)-plane in such a way that its position vector \(\mathbf{r}\) metres at time \(t\) seconds, where \(0 \leqslant t \lt \pi\), satisfies the differential equation
\[\sec^2\left(\frac{1}{2}t\right)\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} + \sec^3\left(\frac{1}{2}t\right)\sin\left(\frac{1}{2}t\right)\mathbf{r} = \sin\left(\frac{1}{2}t\right)\mathbf{i} + \sec^2\left(\frac{1}{2}t\right)\mathbf{j}\]When \(t = 0\), the particle is at the point with position vector \((-\mathbf{i} + \mathbf{j})\) m.
Find \(\mathbf{r}\) in terms of \(t\). (8)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} + \tan\left(\tfrac{1}{2}t\right)\mathbf{r} = \sin\left(\tfrac{1}{2}t\right)\cos^2\left(\tfrac{1}{2}t\right)\mathbf{i} + \mathbf{j}\) | M1 |
| \(\mathrm{R} = \mathrm{e}^{\int\tan\left(\tfrac{1}{2}t\right)\mathrm{d}t} = \sec^2\left(\tfrac{1}{2}t\right)\) | M1 A1 |
| \(\mathbf{r}\sec^2\left(\dfrac{1}{2}t\right) = \displaystyle\int \sin\left(\tfrac{1}{2}t\right)\mathbf{i} + \sec^2\left(\tfrac{1}{2}t\right)\mathbf{j}\,\mathrm{d}t\) | M1 A1 |
| \(= -2\cos\left(\tfrac{1}{2}t\right)\mathbf{i} + 2\tan\left(\tfrac{1}{2}t\right)\mathbf{j}\ (+\mathbf{C})\) | A1 |
| \(t = 0,\ \mathbf{r} = -\mathbf{i} + \mathbf{j} \Rightarrow \mathbf{C} = \mathbf{i} + \mathbf{j}\) | M1 |
| \(\mathbf{r} = \left(\cos^2\left(\tfrac{1}{2}t\right) - 2\cos^3\left(\tfrac{1}{2}t\right)\right)\mathbf{i} + \left(2\sin\left(\tfrac{1}{2}t\right)\cos\left(\tfrac{1}{2}t\right) + \cos^2\left(\tfrac{1}{2}t\right)\right)\mathbf{j}\) | A1 |
| (8) |
Notes
First M1 for dividing by \(\sec^2\left(\tfrac{1}{2}t\right)\)
Second M1 for using a correct formula for the IF
First A1 for a correct integrating factor in terms of \(t\)
Third M1 for multiplying through by IF and attempting to integrate both sides
Second A1 for a correct equation with LHS integrated correctly
Third A1 for a fully correct equation, \(\mathbf{C}\) not needed
Fourth M1 for use of limits to find \(\mathbf{C}\)
Fourth A1 for the answer in any form
N.B. If components used, marks can only be scored once the components have been put together