A2 October 2020 Paper 1 Q9
9 A linear transformation of the plane is represented by the matrix \(\mathbf{M} = \begin{pmatrix} 1 & -2 \\ \lambda & 3 \end{pmatrix}\), where \(\lambda\) is a constant.
(a) Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin. [5]
(b) Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines. [3]
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} 1 & -2 \\ \lambda & 3 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x - 2y \\ \lambda x + 3y \end{pmatrix}\) | B1 | 2.1 |
| suppose \(y = mx\) is invariant \(\lambda x + 3y = m(x - 2y)\) | M1 | 2.1 |
| \(\Rightarrow \lambda + 3m = m(1 - 2m)\) \(\Rightarrow 2m^2 + 2m + \lambda = 0\) | A1 | 1.1 |
| no solutions if discriminant \(\lt 0\) \(\Rightarrow 4 - 8\lambda \lt 0,\ \lambda \gt \frac{1}{2}\) | M1 A1 | 3.1a 3.2a |
| [5] |
Notes
B1: or \(y = mx + c\)
M1: or \(\lambda x + 3y = m(x - 2y) + c\)
| Scheme | Marks | AO |
|---|---|---|
| \(\det\mathbf{M} = 3 + 2\lambda\) or \(\det\mathbf{M} = -5\) | B1 | 1.1 |
| \(\lambda = -4\) | B1 | 2.1 |
| \(m^2 + m - 2 = 0,\ m = -2\) or \(1\) so lines are \(y = x\) and \(y = -2x\) | B1 | 2.2a |
| [3] |