June 2024 Paper 2 Q15
15 Two forces, \(\mathbf{F_1}\) and \(\mathbf{F_2}\), are acting on a particle of mass 3 kilograms.
It is given that
\[\mathbf{F_1} = \begin{bmatrix} a \\ 23 \end{bmatrix} \text{ newtons} \quad \text{and} \quad \mathbf{F_2} = \begin{bmatrix} 4 \\ b \end{bmatrix} \text{ newtons}\]where \(a\) and \(b\) are constants.
The particle has an acceleration of \(\begin{bmatrix} 4b \\ a \end{bmatrix}\) m s−2
Find the value of \(a\) and the value of \(b\) [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains \(a\) + 4 and \(b\) + 23 OE | B1 | 1.1b |
| Uses F = ma with m = 3 and a = \(\begin{bmatrix} 4b \\ a \end{bmatrix}\) | M1 | 3.3 |
| Obtains two linear simultaneous equations in \(a\) and \(b\) | M1 | 1.1a |
| Obtains \(a\) = 8 and \(b\) = 1 | A1 | 1.1b |
| (4 marks) |
Typical solution
\[\begin{bmatrix} a + 4 \\ b + 23 \end{bmatrix}\]\[\begin{bmatrix} a + 4 \\ b + 23 \end{bmatrix} = 3\begin{bmatrix} 4b \\ a \end{bmatrix}\]\[a + 4 = 12b\]\[b + 23 = 3a\]\(a\) = 8 and \(b\) = 1