| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 6 |
| Topic | 3x3 Matrices |
| Type | Transformation mapping problems |
| Difficulty | Standard +0.3 This is a straightforward Further Maths linear transformations question testing standard properties of reflection matrices. Part (a) requires recognizing that a diagonal matrix with entries 1, k, 1 reflects in the xz-plane (routine observation). Part (b) uses the key property that reflection matrices have eigenvalue -1, so (2a-a²)/3 = -1, giving a simple quadratic to solve. Part (c) asks for a conceptual explanation that reflecting twice returns to the original position. All parts are standard bookwork applications with minimal problem-solving required. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03f Linear transformations 3D: reflections and rotations about axes |
You are given that the matrix $\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{2a-a^2}{3} & 0 \\ 0 & 0 & 1 \end{pmatrix}$, where $a$ is a positive constant, represents the transformation $R$ which is a reflection in 3-D.
\begin{enumerate}[label=(\alph*)]
\item State the plane of reflection of $R$. [1]
\item Determine the value of $a$. [3]
\item With reference to $R$ explain why $\mathbf{A}^2 = \mathbf{I}$, the $3 \times 3$ identity matrix. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q4 [6]}}