AS June 2019 Paper 1 Q6
6 A linear transformation T of the \(x\)-\(y\) plane has an associated matrix \(\mathbf{M}\), where \(\mathbf{M} = \begin{pmatrix} \lambda & k \\ 1 & \lambda - k \end{pmatrix}\), and \(\lambda\) and \(k\) are real constants.
(a) You are given that \(\det\mathbf{M} \gt 0\) for all values of \(\lambda\).
(i) Find the range of possible values of \(k\). [3]
(ii) What is the significance of the condition \(\det\mathbf{M} \gt 0\) for the transformation T? [1]
For the remainder of this question, take \(k = -2\).
(b) Determine whether there are any lines through the origin that are invariant lines for the transformation T. [4]
(c) The transformation T is applied to a triangle with area 3 units2. The area of the resulting image triangle is 15 units2.
Find the possible values of \(\lambda\). [3]
Find the possible values of \(\lambda\). [3]
| Scheme | Marks | AO |
|---|---|---|
| (i) \(\det\mathbf{M} = \lambda(\lambda - k) - k\) | B1 | 1.1 |
| \(\det\mathbf{M} \gt 0 \Rightarrow \lambda^2 - k\lambda - k \gt 0\) for all \(\lambda \Rightarrow k^2 + 4k \lt 0\) | M1 | 3.1a |
| \(\Rightarrow -4 \lt k \lt 0\) | A1cao | 1.1 |
| [3] | ||
| (ii) The transformation represented by \(\mathbf{M}\) always preserves the orientation of shapes | B1 | 1.2 |
| [1] |
Notes
(a)(i)
M1: attempt to find discriminant
or \((\lambda - k/2)^2 \gt k^2/4 + k\)
(a)(ii)
B1: condone ‘doesn’t reflect’
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} \lambda & -2 \\ 1 & \lambda + 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \lambda x - 2y \\ x + (\lambda + 2)y \end{pmatrix}\) | B1 | 2.1 |
| invariant line if \(x + \lambda mx + 2mx = m(\lambda x - 2mx)\) | M1 | 2.1 |
| \(\Rightarrow 2m^2 + 2m + 1 = 0\) | A1 | 1.1 |
| discriminant is \(2^2 - 2 \times 4 = -4 \lt 0\) so no real roots for \(m\), i.e. there are no invariant lines | A1 | 2.3 |
| [4] |
Notes
B1: oe (e.g. with \(y = mx\) [\(+ c\)])
M1: subst \(y = mx[+c]\) into \(x + (\lambda + 2)y = m(\lambda x - 2y)[+c]\)
\(x + (\lambda + 2)(mx + c) = m(\lambda x - 2mx - 2c) + c\)
| Scheme | Marks | AO |
|---|---|---|
| \(\det\mathbf{M} = \lambda^2 + 2\lambda + 2 = 5\) | M1 | 3.1a |
| \(\Rightarrow \lambda^2 + 2\lambda - 3 = 0\) | A1 | 1.1 |
| \(\Rightarrow \lambda = -3\) or 1 | A1 | 1.1 |
| [3] |
Notes
M1: det × area by 5 soi
A1: (1st) correct equation in any form
A1: (2nd) BC