| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 5 |
| Topic | Linear transformations |
| Type | 3D transformation matrices |
| Difficulty | Moderate -0.3 This is a Further Maths question on 3D transformations that requires routine matrix multiplication and geometric understanding. Parts (a) and (c) are straightforward recall (computing A^4 and writing down a reflection matrix), part (b) requires recognizing a rotation from the matrix form, and part (d) is trivial application. While 3D transformations are FM content, the individual steps are mechanical with no problem-solving required, making it slightly easier than average overall. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03f Linear transformations 3D: reflections and rotations about axes |
You are given the matrix $\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}$.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathbf{A}^4$. [1]
\item Describe the transformation that $\mathbf{A}$ represents. [2]
\end{enumerate}
The matrix $\mathbf{B}$ represents a reflection in the plane $x = 0$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Write down the matrix $\mathbf{B}$. [1]
\end{enumerate}
The point $P$ has coordinates $(2, 3, 4)$. The point $P'$ is the image of $P$ under the transformation represented by $\mathbf{B}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the coordinates of $P'$. [1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q3 [5]}}