June 2019 Paper 2 Q1
1
(a) \(a\begin{pmatrix} 3 \\ 5 \end{pmatrix} = 4\begin{pmatrix} 2a + 3 \\ b \end{pmatrix}\)
Work out the values of \(a\) and \(b\). [3 marks]
(b) \(\begin{pmatrix} m & -1 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} 2 & 2 \\ -2 & -1 \end{pmatrix} = \mathbf{I} \qquad\) where \(\mathbf{I}\) is the identity matrix.
Work out the value of \(m\). [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(3a = 4(2a + 3)\) or \(3a = 8a + 12\) or \(5a = 4b\) | M1 | oe equation |
| \((a =)\ -\dfrac{12}{5}\) or \(-2\dfrac{2}{5}\) or \(-2.4\) | A1 | |
| \((b =)\ -3\) | A1ft | ft their \(a \times 1.25\) evaluated or \(\dfrac{\text{their } 5a}{4}\) evaluated \(a \ne 0\) if awarding ft M1 is implied |
| Alternative method 2 | ||
| \(3a + 5a = 4(2a + 3) + 4b\) or \(8a = 8a + 12 + 4b\) or \(4b = -12\) | M1 | oe equation |
| \((b =)\ -3\) | A1 | |
| \((a =)\ -\dfrac{12}{5}\) or \(-2\dfrac{2}{5}\) or \(-2.4\) | A1ft | ft their \(b \times 0.8\) evaluated or \(\dfrac{\text{their } 4b}{5}\) evaluated \(b \ne 0\) if awarding ft M1 is implied |
Additional guidance
| Alt 1 \(a = 2 \quad b = 2.5 \quad (5a = 4b\) is implied) | M1A0A1ft |
| \(\begin{pmatrix} 3a \\ 5a \end{pmatrix} = \begin{pmatrix} 8a + 12 \\ 4b \end{pmatrix}\) | M0 |
| Accept \(\dfrac{-12}{5}\) or \(\dfrac{12}{-5}\) for \(-\dfrac{12}{5}\) (apply throughout scheme for values) | |
| Only solutions seen with one correct and the other incorrect (or missing) | 2 marks |
| Answer | Mark | Comments |
|---|---|---|
| \(2m + 2 = 1\) or \(2m + 1 = 0\) or \(\dfrac{1 - 2}{2}\) or \(\begin{pmatrix} 2m + 2 & 2m + 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) | M1 | oe equation or calculation |
| \(-\dfrac{1}{2}\) or \(-0.5\) | A1 |
Additional guidance
| Condone missing brackets in \(\begin{pmatrix} 2m + 2 & 2m + 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) | |
| Allow \(\begin{pmatrix} 2m + 2 & 2m + 1 \\ 2 - 2 & 2 - 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) | |
| Mark positively eg error in matrix multiplication but \(2m + 2 = 1\) and answer \(-0.5\) | M1A1 |
| More than one answer given is A0 eg \(\ m + 2 = 1\) and \(2m + 1 = 0\) (mark positively) Answer \(-1\) and \(-0.5\) | M1 A0 |