M5 June 2018 Q1
1. A small bead is threaded on a smooth straight horizontal wire. The wire is modelled as a line with vector equation \(\mathbf{r} = (2 + \lambda)\mathbf{i} + (2\lambda - 1)\mathbf{j}\), where the unit of length is the metre. The bead is moved a distance of \(\sqrt{80}\) m along the wire by a force \(\mathbf{F} = (4\mathbf{i} - 3\mathbf{j})\) N. Find the magnitude of the work done by \(\mathbf{F}\). (5)
| Scheme | Marks |
|---|---|
| ALTERNATIVE 1 | |
| Direction of line is \((\mathbf{i} + 2\mathbf{j})\) | B1 |
| \(\mathbf{d} = \dfrac{\sqrt{80}}{\sqrt{1^2 + 2^2}}(\mathbf{i} + 2\mathbf{j}) = (4\mathbf{i} + 8\mathbf{j})\) | M1 A1 |
| W.D. \(= \left|(4\mathbf{i} - 3\mathbf{j})\cdot(4\mathbf{i} + 8\mathbf{j})\right| = 8\) J | M1 A1 |
| Allow column vectors throughout | |
| (5) |
Notes
See alternatives. Answer must be positive. Ignore units.
B1 For \((\mathbf{i} + 2\mathbf{j})\) or multiple seen
ALTERNATIVE 1
First M1 for attempt to find the displacement vector \(\mathbf{d}\)
First A1 for \((4\mathbf{i} + 8\mathbf{j})\)
Second M1 for WD \(= (4\mathbf{i} - 3\mathbf{j})\cdot\mathbf{d}\) (\(\mathbf{d}\) must be a multiple of \((\mathbf{i} + 2\mathbf{j})\))
Second A1 for 8 (J)
ALTERNATIVE 2
| Direction of line is \((\mathbf{i} + 2\mathbf{j})\) | B1 |
| \(\cos\theta = \dfrac{(4\mathbf{i} - 3\mathbf{j})\cdot(\mathbf{i} + 2\mathbf{j})}{5\sqrt{5}} = \dfrac{-2}{5\sqrt{5}}\) | M1 A1 |
| W.D. \(= \left|(4\mathbf{i} - 3\mathbf{j})\right|\left|\cos\theta\right| \times \sqrt{80} = 8\) J | M1 A1 |
First M1 for attempt to find the angle (or cos thereof) between their \(\mathbf{d}\) (must be a multiple of \((\mathbf{i} + 2\mathbf{j})\)) and \((4\mathbf{i} - 3\mathbf{j})\)
First A1 for \(\cos\theta = \dfrac{-2}{5\sqrt{5}}\) oe
Second M1 for W.D. \(= \left|(4\mathbf{i} - 3\mathbf{j})\right|\left|\cos\theta\right| \times \sqrt{80}\)
Second A1 for 8 (J)
ALTERNATIVE 3
| Direction of line is \((\mathbf{i} + 2\mathbf{j})\) | B1 |
| \(\hat{\mathbf{d}} = \dfrac{(\mathbf{i} + 2\mathbf{j})}{\sqrt{5}}\) | M1 A1 |
| W.D. \(= \left|(4\mathbf{i} - 3\mathbf{j})\cdot\hat{\mathbf{d}}\right| \times \sqrt{80} = 8\) J | M1 A1 |
First M1 for attempt to find a unit vector in the direction of their direction vector
First A1 for \(\hat{\mathbf{d}} = \dfrac{(\mathbf{i} + 2\mathbf{j})}{\sqrt{5}}\) oe
Second M1 for W.D. \(= \left|(4\mathbf{i} - 3\mathbf{j})\cdot\hat{\mathbf{d}}\right| \times \sqrt{80}\) (\(\hat{\mathbf{d}}\) must be a multiple of \((\mathbf{i} + 2\mathbf{j})\))
Second A1 for 8 (J)