June 2025 Paper 3 Q7
7 A particle \(P\) of mass 3 kg is moving on a smooth horizontal surface under the action of two constant horizontal forces \((5\mathbf{i} + 3\mathbf{j})\) N and \((a\mathbf{i} + 3b\mathbf{j})\) N. The acceleration of \(P\) is \((2\mathbf{i} - 3\mathbf{j})\) m s−2.
At time \(t = 0\) seconds the velocity of \(P\) is \(\mathbf{u}\) m s−1 and at time \(t = 5\) seconds the velocity of \(P\) is \((7\mathbf{i} - 6\mathbf{j})\) m s−1.
| Scheme | Marks | AO |
|---|---|---|
| \((5\mathbf{i} + 3\mathbf{j}) + (a\mathbf{i} + 3b\mathbf{j}) = 3(2\mathbf{i} - 3\mathbf{j})\) | M1 | 3.3 |
| \(5 + a = 6\) or \(3 + 3b = -9\) | B1 | 1.1 |
| \(a = 1\), \(b = -4\) | A1 | 1.1 |
| [3] |
Notes
M1: Applying \(\mathbf{F} = m\mathbf{a}\) for both components – no misreads of vectors but allow sign errors only
Must be \(m = 3\) for both components. M0 if including the weight as a force in \(\mathbf{F}\)
B1: One correct explicit linear equation not in vector form
This mark can be implied by one correct value for \(a\) or \(b\)
A1: ISW if correct \(a\) and \(b\) stated in working and then miscopied onto the answer lines at the bottom of the PAB
Correct answers with no working scores full marks. One correct answer with no working scores M0 B1 A0
| Scheme | Marks | AO |
|---|---|---|
| \((7\mathbf{i} - 6\mathbf{j}) = \mathbf{u} + 5(2\mathbf{i} - 3\mathbf{j})\) | B1 | 3.3 |
| \((\mathbf{u} =)\ -3\mathbf{i} + 9\mathbf{j}\) | B1 | 1.1 |
| [2] |
Notes
B1: Applying \(\mathbf{v} = \mathbf{u} + \mathbf{a}t\) correctly - no misreads of vectors. Or consider two correct linear equations i.e. \(7 = u_1 + 5 \times 2\) and \(-6 = u_2 - 5 \times 3\) (condone \(u\) for both \(u_1, u_2\))
oe correct expression/equation for \(\mathbf{u}\) if using calculus
B1: Condone given as a column vector
ISW if magnitude of correct \(\mathbf{u}\) is then considered
Correct answer with no working scores full marks