June 2024 Paper 1 Q4
4 The vectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are defined by \(\mathbf{v}_1 = 2a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{v}_2 = b\mathbf{i} - 3\mathbf{j}\) where \(a\) and \(b\) are constants.
Given that \(3\mathbf{v}_1 + \mathbf{v}_2 = 22\mathbf{i} - 9\mathbf{j}\), find the values of \(a\) and \(b\). [4]
| Scheme | Marks | AO |
|---|---|---|
| \(3(2a\mathbf{i} + b\mathbf{j}) + (b\mathbf{i} - 3\mathbf{j})\ \ [= 22\mathbf{i} - 9\mathbf{j}]\) | M1 | 1.1 |
| \(6a + b = 22\) and \(3b - 3 = -9\) | M1 | 3.1a |
| \(a = 4\) | A1 | 1.1 |
| \(b = -2\) | A1 | 1.1 |
| [4] |
Notes
M1: Attempt to scalar multiply \(\mathbf{v}_1\) and add to \(\mathbf{v}_2\)
Allow vector expression or 2 separate components
M1: Equate coefficients of \(\mathbf{i}\) and \(\mathbf{j}\) to form two equations. Allow if \(\mathbf{i}\) and \(\mathbf{j}\) still seen in every term of these equations
A1: cao
A1: cao