| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Challenging +1.8 This question requires constructing four individual transformation matrices (reflection in a non-standard line, two shears with specific conditions, and a rotation), multiplying them in the correct order, and interpreting the resulting matrix geometrically. While each individual matrix is standard Further Maths content, the combination of four transformations, the non-trivial reflection angle (π/8), and the requirement to give a complete geometric description of the composite transformation demands sustained multi-step reasoning and matrix manipulation skills beyond typical A-level questions. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
Reflection in $y = x\tan\frac{1}{8}\pi$: $\begin{pmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\end{pmatrix}$ Allow $\cos(\frac{1}{4}\pi)$'s, etc. **B1**
Shear $//$ $y$-axis, mapping $(1,0)$ to $(1,2)$: $\begin{pmatrix}1 & 0\\ 2 & 1\end{pmatrix}$ **B1**
Rotation through $\frac{1}{4}\pi$ clockwise about $O$: $\begin{pmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{pmatrix}$ **B1**
Shear $//$ $x$-axis, mapping $(0,1)$ to $(-2,1)$: $\begin{pmatrix}1 & -2\\ 0 & 1\end{pmatrix}$ **B1**
Multiplying them together in this order (from right-to-left) $= \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}$ **M1 A1**
Reflection in $y = x$ **M1 A1**
**[8]**
**NB 1** Multiplying the matrices in the reverse order scores max. $4\times$**B1** + **M0**; then **B1** for correct $\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}$ and **M1** for "Reflection" and **A1** for "in $x$-axis"
**NB 2** Incorrect final matrices automatically lose the last 2 marks9 The plane transformation $T$ is the composition (in this order) of
\begin{itemize}
\item a reflection in the line $y = x \tan \frac { 1 } { 8 } \pi$; followed by
\item a shear parallel to the $y$-axis, mapping $( 1,0 )$ to $( 1,2 )$; followed by
\item a clockwise rotation through $\frac { 1 } { 4 } \pi$ radians about the origin; followed by
\item a shear parallel to the $x$-axis, mapping $( 0,1 )$ to $( - 2,1 )$.
\end{itemize}
Determine the matrix $\mathbf { M }$ which represents $T$, and hence give a full geometrical description of $T$ as a single plane transformation.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q9 [8]}}