M1 June 2013 (R) Q6
6. [In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively. Position vectors are given with respect to a fixed origin \(O\).]
A ship \(S\) is moving with constant velocity \((3\mathbf{i} + 3\mathbf{j})\) km h\(^{-1}\). At time \(t = 0\), the position vector of \(S\) is \((-4\mathbf{i} + 2\mathbf{j})\) km.
A ship \(T\) is moving with constant velocity \((-2\mathbf{i} + n\mathbf{j})\) km h\(^{-1}\). At time \(t = 0\), the position vector of \(T\) is \((6\mathbf{i} + \mathbf{j})\) km. The two ships meet at the point \(P\).
| Scheme | Marks |
|---|---|
| Use of \(\mathbf{r} = \mathbf{r}_0 + \mathbf{v}t\) | M1 |
| \((-4\mathbf{i} + 2\mathbf{j}) + (3\mathbf{i} + 3\mathbf{j})t = (-4 + 3t)\mathbf{i} + (2 + 3t)\mathbf{j}\) | A1 |
| (2) |
Notes
M1 for clear attempt to use \(\mathbf{r}_0 + t\mathbf{v}\) (M0 if \(\mathbf{r}_0\) and \(\mathbf{v}\) reversed)
A1 for answer in any form.
| Scheme | Marks |
|---|---|
| \((6\mathbf{i} + \mathbf{j}) + (-2\mathbf{i} + n\mathbf{j})t = (6 - 2t)\mathbf{i} + (1 + nt)\mathbf{j}\) | B1 |
| Position vectors identical \(\Rightarrow -4 + 3t = 6 - 2t\) AND \(5t = 10\), | M1 |
| Either equation | A1 |
| \(2 + 3 \times 2 = 1 + 2n\), | DM1 |
| \(n = 3.5\) | A1 |
| (5) |
Notes
B1 for \((6\mathbf{i} + \mathbf{j}) + (-2\mathbf{i} + n\mathbf{j})t\) seen or implied
First M1 for equating their \(\mathbf{i}\)- cpts and their \(\mathbf{j}\)- cpts. (must have both equations in terms of same \(t\))
First A1 for a correct equation (either)
Second M1 dependent on first M1 for producing an equation in \(n\) only.
Second A1 for \(n = 3.5\) oe
| Scheme | Marks |
|---|---|
| Position vector of P is \((-4 + 6)\mathbf{i} + (2 + 6)\mathbf{j} = 2\mathbf{i} + 8\mathbf{j}\) | M1A1 |
| Distance OP \(= \sqrt{2^2 + 8^2} = \sqrt{68} = 8.25\) (km) | M1A1 |
| (4) | |
| (11 marks) |
Notes
First M1 for clear attempt to find pv of \(P\), using their \(t\) and/or \(n\) value(s)
First A1 for \(2\mathbf{i} + 8\mathbf{j}\)
Second M1 for attempt to find magnitude of their \(\mathbf{p}\)
Second A1 for \(\sqrt{68}\), \(2\sqrt{17}\), 8.2 or better (km)