Challenging +1.2 This question requires finding the matrix for reflection in a non-standard line (y = x√3), then solving MN·P = Q by finding the inverse of a 2×2 matrix product. While it involves multiple steps (deriving reflection matrix, matrix multiplication, matrix inversion, solving system), each component uses standard Further Maths techniques with no novel insight required. The 5 marks and calculator-permitted nature (3 d.p.) indicate computational rather than conceptual challenge.
The transformation \(S\) is represented by the matrix \(\mathbf{M} = \begin{bmatrix} 1 & -6 \\ 2 & 7 \end{bmatrix}\)
The transformation \(T\) is a reflection in the line \(y = x\sqrt{3}\) and is represented by the matrix \(\mathbf{N}\)
The point \(P(x, y)\) is transformed first by \(S\), then by \(T\)
The result of these transformations is the point \(Q(3, 8)\)
Find the coordinates of \(P\)
Give your answers to three decimal places.
[5 marks]
The transformation $S$ is represented by the matrix $\mathbf{M} = \begin{bmatrix} 1 & -6 \\ 2 & 7 \end{bmatrix}$
The transformation $T$ is a reflection in the line $y = x\sqrt{3}$ and is represented by the matrix $\mathbf{N}$
The point $P(x, y)$ is transformed first by $S$, then by $T$
The result of these transformations is the point $Q(3, 8)$
Find the coordinates of $P$
Give your answers to three decimal places.
[5 marks]
\hfill \mbox{\textit{AQA Further Paper 2 2024 Q12 [5]}}