AS October 2021 Paper 1 Q6
6 A transformation T of the plane has associated matrix \(\mathbf{M} = \begin{pmatrix} 1 & \lambda + 1 \\ \lambda - 1 & -1 \end{pmatrix}\), where \(\lambda\) is a non-zero constant.
(a)
(i) Show that T reverses orientation. [3]
(ii) State, in terms of \(\lambda\), the area scale factor of T. [1]
(b)
(i) Show that \(\mathbf{M}^2 - \lambda^2\mathbf{I} = \mathbf{0}\). [2]
(ii) Hence specify the transformation equivalent to two applications of T. [1]
(c) In the case where \(\lambda = 1\), T is equivalent to a transformation S followed by a reflection in the \(x\)-axis.
(i) Determine the matrix associated with S. [3]
(ii) Hence describe the transformation S. [2]
| Scheme | Marks | AO |
|---|---|---|
| (i) \(\det\mathbf{M} = -1 - (\lambda + 1)(\lambda - 1)\) | M1 | 1.1a |
| \(= -\lambda^2\) | A1 | 1.1 |
| Always negative, so reverses orientation | A1 | 2.2a |
| [3] | ||
| (ii) Area scale factor \(= \lambda^2\) | B1 | 1.1 |
| [1] |
Notes
(a)(ii)
B1: not \(-\lambda^2\)
| Scheme | Marks | AO |
|---|---|---|
| (i) \(\mathbf{M}^2 = \begin{pmatrix} 1 & \lambda + 1 \\ \lambda - 1 & -1 \end{pmatrix}^2 = \begin{pmatrix} \lambda^2 & 0 \\ 0 & \lambda^2 \end{pmatrix} = \lambda^2\mathbf{I}\) | M1 | 1.2 |
| \(\Rightarrow \mathbf{M}^2 - \lambda^2\mathbf{I} = \mathbf{0}\) | A1 | 2.2a |
| [2] | ||
| (ii) enlargement [about O] scale factor \(\lambda^2\) | B1 | 1.1 |
| [1] |
Notes
(b)(i)
M1: correct matrix multiplication for \(\mathbf{M}^2\)
| Scheme | Marks | AO |
|---|---|---|
| (i) reflection in O\(x\) has matrix \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) | B1 | 1.1 |
| matrix for S \(= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}^{-1}\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix}\) | M1 | 3.1a |
| \(= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}^{-1}\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\) | A1 | 1.1 |
| [3] | ||
| (ii) S is a shear | M1 | 1.1 |
| with invariant line O\(x\) mapping \((0, 1)\) to \((2, 1)\) | A1 | 1.1 |
| [2] |
Notes
(c)(i)
M1: or \(\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)
A1: \(\Rightarrow a = 1,\ b = 2,\ c = 0,\ d = 1\)
(c)(ii)
A1: oe