| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Standard +0.8 This question requires recall of standard transformation matrices (part i) followed by matrix multiplication and interpretation of the result (part ii). The key challenge is recognizing that the product of two reflection matrices yields a rotation matrix and determining the rotation angle from the matrix product, which requires trigonometric manipulation using double-angle formulas. While systematic, this goes beyond routine application and requires connecting geometric transformations to their algebraic representations. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
**(i)(a)** $\begin{pmatrix}\cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha\end{pmatrix}$ **B1**
**(b)** $\begin{pmatrix}\cos\beta & \sin\beta \\ \sin\beta & -\cos\beta\end{pmatrix}$ **B1**
**[2]**
**(ii)** $\begin{pmatrix}\cos\phi & \sin\phi \\ \sin\phi & -\cos\phi\end{pmatrix}\begin{pmatrix}\cos\theta & \sin\theta \\ \sin\theta & -\cos\theta\end{pmatrix}$ Multn. of 2 reflection matrices. Correct order. **M1 M1**
$= \begin{pmatrix}\cos\phi\cos\theta+\sin\phi\sin\theta & \cos\phi\sin\theta-\sin\phi\cos\theta \\ \sin\phi\cos\theta-\cos\phi\sin\theta & \cos\phi\cos\theta+\sin\phi\sin\theta\end{pmatrix}$
$= \begin{pmatrix}\cos(\phi-\theta) & -\sin(\phi-\theta) \\ \sin(\phi-\theta) & \cos(\phi-\theta)\end{pmatrix}$ Use of the addition formulae; correctly done **M1 A1**
$\ldots$ giving a Rotation (about $O$) through $(\phi - \theta)$ acw [or $(\theta-\phi)$ cw] **M1 A1**
Those who get the initial matrices in the wrong order, can get 5/6, losing only that **M** mark
**[6]**5 (i) Write down the $2 \times 2$ matrices which represent the following plane transformations:
\begin{enumerate}[label=(\alph*)]
\item an anticlockwise rotation about the origin through an angle $\alpha$;
\item a reflection in the line $y = x \tan \left( \frac { 1 } { 2 } \beta \right)$.\\
(ii) A reflection in the $x - y$ plane in the line $y = x \tan \left( \frac { 1 } { 2 } \theta \right)$ is followed by a reflection in the line $y = x \tan \left( \frac { 1 } { 2 } \phi \right)$. Show that the composition of these two reflections (in this order) is a rotation and describe this rotation fully.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2012 Q5 [6]}}