June 2023 Paper 3 Q10
10 A particle \(P\) of mass \(m\,\mathrm{kg}\) is moving on a smooth horizontal surface under the action of two constant horizontal forces \((-4\mathbf{i} + 2\mathbf{j})\,\mathrm{N}\) and \((a\mathbf{i} + b\mathbf{j})\,\mathrm{N}\). The resultant of these two forces is \(\mathbf{R}\,\mathrm{N}\). It is given that \(\mathbf{R}\) acts in a direction which is parallel to the vector \(-\mathbf{i} + 3\mathbf{j}\).
It is given that \(a = 6\) and that \(P\) moves with an acceleration of magnitude \(5\sqrt{10}\,\mathrm{m\,s^{-2}}\).
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{R} = (-4 + a)\mathbf{i} + (2 + b)\mathbf{j}\) | B1 | 1.1 |
| \((-4 + a)\mathbf{i} + (2 + b)\mathbf{j} = k(-\mathbf{i} + 3\mathbf{j})\) | M1 | 3.1b |
| \(k = 4 - a\) therefore \(2 + b = 3(4 - a)\) so \(3a + b = 10\) | A1 | 2.2a |
| [3] |
Notes
B1: oe e.g. \(\begin{pmatrix} -4 \\ 2 \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix}\)
M1: Sets their \(\mathbf{R}\) equal to \(k\) times \(-\mathbf{i} + 3\mathbf{j}\) (or \(k\) times \(\mathbf{R}\)) where \(k\) is non-numerical/unknown
Implied by a correct equation in \(a\) and \(b\) e.g. \(\frac{a - 4}{2 + b} = -\frac{1}{3}\) but not from stating \(-4 + a = -1\) and \(2 + b = 3\) (so must have come from a ‘gradient’ approach if \(k\) not seen)
A1: AG Eliminate \(k\) and derive given result
Using an assumed value of \(k\) is A0
| Scheme | Marks | AO |
|---|---|---|
| \(a = 6 \Rightarrow b = -8 \quad \therefore \mathbf{R} = 2\mathbf{i} - 6\mathbf{j}\) | B1 | 1.1 |
| \(|\mathbf{R}| = \sqrt{2^2 + (-6)^2}\) | M1* | 3.3 |
| \(\sqrt{2^2 + (-6)^2} = m(5\sqrt{10})\) | M1dep* | 3.4 |
| \(m = \dfrac{\sqrt{40}}{5\sqrt{10}} = 0.4\) | A1 | 2.2a |
| [4] |
Notes
M1*: Calculate \(|\mathbf{R}|\) or \(|\mathbf{R}|^2\) for their \(\mathbf{R}\)
M1dep*: N2L applied to the magnitude of their \(\mathbf{R}\) with \(5\sqrt{10}\)
A1: www 0.4 (oe) but square roots cancelled