| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2023 |
| Session | January |
| Marks | 6 |
| Topic | Linear transformations |
| Type | Extract enlargement and rotation parameters |
| Difficulty | Standard +0.3 This is a straightforward matrix transformation question requiring recognition that enlargement scale factor equals the determinant's square root and rotation angle from matrix entries. Part (a) involves standard calculations with trigonometric values, while part (b) applies the determinant-area relationship. All techniques are routine for Further Maths students with no novel problem-solving required. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03i Determinant: area scale factor and orientation |
The transformation $P$ is an enlargement, centre the origin, with scale factor $k$, where $k > 0$
The transformation $Q$ is a rotation through angle $\theta$ degrees anticlockwise about the origin.
The transformation $P$ followed by the transformation $Q$ is represented by the matrix
$$\mathbf{M} = \begin{pmatrix} -4 & -4\sqrt{3} \\ 4\sqrt{3} & -4 \end{pmatrix}$$
\begin{enumerate}[label=(\alph*)]
\item Determine
\begin{enumerate}[label=(\roman*)]
\item the value of $k$,
\item the smallest value of $\theta$ [4]
\end{enumerate}
\end{enumerate}
A square $S$ has vertices at the points with coordinates $(0, 0)$, $(a, -a)$, $(2a, 0)$ and $(a, a)$ where $a$ is a constant.
The square $S$ is transformed to the square $S'$ by the transformation represented by $\mathbf{M}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Determine, in terms of $a$, the area of $S'$ [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2023 Q10 [6]}}