| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | December |
| Marks | 4 |
| Topic | Linear transformations |
| Type | Write down transformation matrix |
| Difficulty | Moderate -0.8 This question tests standard 2×2 transformation matrices with straightforward recall and computation. Part (i) requires writing down a well-known rotation matrix (pure recall). Part (ii) involves matrix multiplication of two simple matrices and recognizing the result as a reflection in y=x—all routine procedures for Further Maths students with no problem-solving or novel insight required. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
The $2 \times 2$ matrix A represents a rotation by $90°$ anticlockwise about the origin. The $2 \times 2$ matrix B represents a reflection in the line $y = -x$.
The matrix B is given by
$$\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$
\begin{enumerate}[label=(\roman*)]
\item Write down the matrix representing A. [1]
\item The $2 \times 2$ matrix C represents a rotation by $90°$ anticlockwise about the origin, followed by a reflection in the line $y = -x$. Compute the matrix C and describe geometrically the single transformation represented by C. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q5 [4]}}