AS June 2018 Paper 1 Q6
6 The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} t & 6 \\ t & -2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 2t & 4 \\ t & -2 \end{pmatrix}\) where \(t\) is a constant.
| Scheme | Marks | AO |
|---|---|---|
| \(|\mathbf{A}| = -2t - 6t\) or \(|\mathbf{B}| = -4t - 4t\) | M1 | 1.1a |
| \(|\mathbf{B}| = -8t = |\mathbf{A}|\) | A1 | 2.2a |
| [2] |
Notes
M1: Correct expression for either seen or implied
A1: Both correct and statement of equality
Need to have an indication that candidate understands that they have shown that these are equal. Could be done by re-writing \(|\mathbf{A}| = -8t\) immediately next to \(|\mathbf{B}| = -8t\). \(|\mathbf{A}| = |\mathbf{B}|\) is fine after having shown both are equal to \(-8t\), but \(-8t = -8t\) is not ok for the A mark.
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} t & 6 \\ t & -2 \end{pmatrix}\begin{pmatrix} 2t & 4 \\ t & -2 \end{pmatrix} = \begin{pmatrix} 2t^2 + 6t & 4t - 12 \\ 2t^2 - 2t & 4t + 4 \end{pmatrix}\) | M1 | 3.1a |
| \(|\mathbf{AB}| = \begin{vmatrix} 2t^2 + 6t & 4t - 12 \\ 2t^2 - 2t & 4t + 4 \end{vmatrix} =\) \((2t^2 + 6t)(4t + 4) - (2t^2 - 2t)(4t - 12)\) | M1 | 2.1 |
| \(= 8t^3 + 8t^2 + 24t^2 + 24t - (8t^3 - 24t^2 - 8t^2 + 24t)\) \(= 64t^2 = (-8t)(-8t) = |\mathbf{A}||\mathbf{B}|\) | A1 | 2.1 |
| [3] |
Notes
M1: (1st) Must be attempt at proper matrix multiplication (i.e. columns into rows). Condone one error
M1: (2nd) Correct expression of determinant of their matrix.
Condone one error
A1: Convincing expansion, correct answer and conclusion
Similar to question above. Need candidate to conclude that \(|\mathbf{AB}| = |\mathbf{A}|\,|\mathbf{B}|\)
Condone not seeing \((-8t)(-8t)\) explicitly
| Scheme | Marks | AO |
|---|---|---|
| Set their \(|\mathbf{AB}| = -1\) or \(|\mathbf{A}||\mathbf{A}| = |\mathbf{A}|^2 = -1\) | M1 | 3.1a |
| \(\Rightarrow 64t^2 = -1\) so \(t\) must be complex/imaginary/not real | A1ft | 3.2a |
| [2] |
Notes
M1: Seen or implied
\(64t^2 = -1\) or \((-8t)^2 = -1\)
A1ft: Accept \(t = -\mathrm{i}/8\) or \(t = \mathrm{i}/8\)
Allow follow through if their \(|\mathbf{AB}|\) is of the form \(kt^2\).