AS June 2024 Paper 1 Q5
5
(a) Find the volume scale factor of the transformation with associated matrix \(\begin{pmatrix} 1 & 2 & 0 \\ 0 & 3 & -1 \\ -1 & 0 & 2 \end{pmatrix}\). [2]
(b) The transformations S and T of the plane have associated \(2 \times 2\) matrices \(\mathbf{P}\) and \(\mathbf{Q}\) respectively.
(i) Write down an expression for the associated matrix of the combined transformation S followed by T. [1]
The determinant of \(\mathbf{P}\) is 3 and \(\mathbf{Q} = \begin{pmatrix} k & 3 \\ -1 & 2 \end{pmatrix}\), where \(k\) is a constant.
(ii) Given that this combined transformation preserves both orientation and area, determine the value of \(k\). [3]
| Scheme | Marks | AO |
|---|---|---|
| det = 8 | B1 | 1.1a |
| volume scale factor = 8 | B1ft | 1.1 |
| [2] |
Notes
B1: soi BC
B1ft: ft their determinant value
| Scheme | Marks | AO |
|---|---|---|
| (i) \(\mathbf{QP}\) | B1 | 1.1 |
| [1] | ||
| (ii) \(\det(\mathbf{Q}) = 2k + 3\) | B1 | 1.1 |
| \(3(2k + 3) = 1\) | M1 | 1.1 |
| \(\Rightarrow k = -\tfrac{4}{3}\) | A1 | 1.1 |
| [3] |
Notes
(b)(i)
B1: Allow \(\begin{pmatrix} k & 3 \\ -1 & 2 \end{pmatrix}\mathbf{P}\)
(b)(ii)
B1: soi
M1: \(3 \times\) their \((2k + 3) = 1\) oe, e.g. \(2k + 3 = 1/3\)