AS June 2024 Paper 1 Q1
1 Use a matrix method to determine the solution of the following simultaneous equations. [4]
\[\begin{aligned} 2x - 3y + z &= 1 \\ x - 2y - 4z &= 40 \\ 5x + 6y - z &= 61 \end{aligned}\]| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & -4 \\ 5 & 6 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 40 \\ 61 \end{pmatrix}\) | M1 | 1.1 |
| \(\begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & -4 \\ 5 & 6 & -1 \end{pmatrix}^{-1} = \dfrac{1}{125}\begin{pmatrix} 26 & 3 & 14 \\ -19 & -7 & 9 \\ 16 & -27 & -1 \end{pmatrix}\) | B1* | 1.1 |
| \(\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \dfrac{1}{125}\begin{pmatrix} 26 & 3 & 14 \\ -19 & -7 & 9 \\ 16 & -27 & -1 \end{pmatrix}\begin{pmatrix} 1 \\ 40 \\ 61 \end{pmatrix}\) | M1 | 1.1 |
| \(x = 8\), \(y = 2\), \(z = -9\) | A1*dep | 1.1 |
| [4] |
Notes
M1: Reduction of system to matrix form soi
B1*: BC. Could be embedded
Inverse matrix correctly evaluated
\(\begin{pmatrix} 0.208 & 0.024 & 0.112 \\ -0.152 & -0.056 & 0.072 \\ 0.128 & -0.216 & -0.008 \end{pmatrix}\)
M1: Correctly using the inverse matrix: \((\mathbf{r} =)\ \mathbf{A}^{-1}\mathbf{b}\). It must be clear that a matrix method is being used.
Can be incorrect inverse for M1
Allow M1 for expressions of the form \(\mathbf{A}^{-1}\mathbf{b}\)
A1*dep: Could be seen in vector form but \(x\), \(y\) and \(z\) must be appropriately seen.
Correct answer with no matrix forms shown (with or without other working) is 0/4.
Need to have earned the B1