A2 October 2021 Paper 1 Q9
9 You are given that the matrix \(\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\) represents a transformation T.
(a) You are given that the line with equation \(y = kx\) is invariant under T.
Determine the value of \(k\). [4]
Determine the value of \(k\). [4]
(b) Determine whether the line with equation \(y = kx\) in part (a) is a line of invariant points under T. [1]
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} x \\ kx \end{pmatrix} = \begin{pmatrix} 2x + kx \\ -x \end{pmatrix}\) | M1 A1 | 3.1a 1.1 |
| same line \(\Rightarrow -x = k(2x + kx)\) for all \(x\ (\neq 0)\) | M1 | 2.1 |
| \(\Rightarrow -1 = k(2 + k) \Rightarrow k^2 + 2k + 1 = 0\) \(\Rightarrow k = -1\) (i.e. \(y = -x\)) | A1 | 1.1 |
| [4] |
Notes
A1: Value of \(k\) can be implied by the correct equation
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} x \\ -x \end{pmatrix} = \begin{pmatrix} 2x - x \\ -x \end{pmatrix} = \begin{pmatrix} x \\ -x \end{pmatrix}\) so each point maps to itself and it is a line of invariant points | B1 | 2.4 |
| [1] |
Notes
B1: Must have a reason
e.g. it is sufficient to test one point other than \((0, 0)\)