A2 June 2024 Paper 2 Q10
10 The matrix \(\mathbf{C}\) is defined by
\[\mathbf{C} = \begin{bmatrix} 3 & 2 \\ -4 & 5 \end{bmatrix}\]Prove that the transformation represented by \(\mathbf{C}\) has no invariant lines of the form \(y = kx\) [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains \(3x + 2kx\) and \(-4x + 5kx\) Accept any letter for \(k\) Condone use of \(y = kx + c\) Or Uses \(\det(\mathbf{C} - \lambda\mathbf{I})\) | M1 | 1.1a |
| Substitutes their \(x^{\prime}\) and \(y^{\prime}\) in \(y^{\prime} = kx^{\prime}\) Accept any letter for \(k\) Condone use of \(y = kx + c\) Or Expands \(\det(\mathbf{C} - \lambda\mathbf{I})\) | M1 | 1.1a |
| Deduces \(2k^2 - 2k + 4 = 0\) OE Or \(\lambda^2 - 8\lambda + 23 = 0\) | A1 | 2.2a |
| Completes a reasoned argument justifying that the quadratic equation has no real roots to prove that the transformation represented by \(\mathbf{C}\) has no invariant lines of the form \(y = kx\) If \(y = kx + c\) is used then must state and use \(c = 0\) | R1 | 2.1 |
| (4 marks) |
Typical solution
For an invariant line \(y = kx\)
\[\begin{bmatrix} 3 & 2 \\ -4 & 5 \end{bmatrix}\begin{bmatrix} x \\ kx \end{bmatrix} = \begin{bmatrix} x^{\prime} \\ y^{\prime} \end{bmatrix}\]\[3x + 2kx = x^{\prime}\]\[-4x + 5kx = y^{\prime}\]\[y^{\prime} = kx^{\prime}\]\[-4x + 5kx = k(3x + 2kx)\]\[-4 + 5k = 2k^2 + 3k\]\[2k^2 - 2k + 4 = 0\]\[\Delta = 2^2 - 32 = -28\]\[\Delta \lt 0\]The equation has no real roots, so there is no invariant line of the form \(y = kx\).