| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Session | Specimen |
| Marks | 17 |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Challenging +1.2 This question requires systematic matrix decomposition using standard techniques (factoring out scalars, recognizing rotation/reflection forms) but follows predictable patterns. Part (a) is routine; part (b) requires recognizing that det=-4cos²θ and factoring the matrix, which is methodical rather than insightful. The area-preserving condition is a direct application of determinant = ±1. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03f Linear transformations 3D: reflections and rotations about axes4.03h Determinant 2x2: calculation |
**(a)(i)** $\begin{pmatrix}\cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha\end{pmatrix}$: B1 **[1]**
**(a)(ii)** *EITHER:* $\cos\theta + \sin\theta \equiv \sqrt{2}\cos\left(\theta - \frac{\pi}{4}\right)$ and $\sin\theta - \cos\theta \equiv \sqrt{2}\sin\left(\theta - \frac{\pi}{4}\right)$: M1, A1, A1
$\begin{pmatrix}\cos\theta+\sin\theta & \cos\theta-\sin\theta \\ \sin\theta-\cos\theta & \cos\theta+\sin\theta\end{pmatrix} = \sqrt{2}\begin{pmatrix}\cos(\theta-\frac{\pi}{4}) & -\sin(\theta-\frac{\pi}{4}) \\ \sin(\theta-\frac{\pi}{4}) & \cos(\theta-\frac{\pi}{4})\end{pmatrix}$: B1
The rotation $R$ is (anticlockwise about $O$) through an angle $\left(\theta - \frac{\pi}{4}\right)$: B1
Second transformation is an enlargement: M1; (about $O$) s.f. $\sqrt{2}$ **ft**: A1 **[7]**
**(b)(i)** $\begin{pmatrix}\cos\beta & \sin\beta \\ \sin\beta & -\cos\beta\end{pmatrix}$: B1 **[1]**
**(b)(ii)** $\begin{pmatrix}2\cos^2\theta & 2\sin\theta\cos\theta \\ 2\sin\theta\cos\theta & -2\cos^2\theta\end{pmatrix}$, for use of double-angle formulae: M1
$= 2\cos\theta\begin{pmatrix}\cos\theta & \sin\theta \\ \sin\theta & -\cos\theta\end{pmatrix}$: M1, A1
Enlargement has s.f. $2\cos\theta$: B1
Reflection is in the line $y = x\tan\frac{1}{2}\theta$: B1 **[5]**
**(b)(iii)** Det $= -4\cos^2\theta$: B1
Setting $= \pm 1$ and solving for $\theta$, for at least one of $+1/-1$: M1
$\theta = \pm\frac{1}{3}\pi$, both and no extras: A1 **[3]**11
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the matrix which represents a rotation through an angle $\alpha$ anticlockwise about the origin.
\item Show that the plane transformation given by the matrix
$$\left( \begin{array} { c c }
\cos \theta + \sin \theta & - ( \sin \theta - \cos \theta ) \\
\sin \theta - \cos \theta & \cos \theta + \sin \theta
\end{array} \right)$$
is the composition of a rotation, $R$, and a second transformation, $S$. Describe both $R$ and $S$ fully.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Write down the matrix which represents a reflection in the line $y = x \tan \frac { 1 } { 2 } \beta$.
For $- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi$, the plane transformation $T$ is given by the matrix
$$\left( \begin{array} { c c }
1 + \cos 2 \theta & \sin 2 \theta \\
\sin 2 \theta & - 1 - \cos 2 \theta
\end{array} \right)$$
\item Show that $T$ is the composition of a reflection and an enlargement, and describe these transformations in full.
\item Find also the values of $\theta$ for which $T$ is an area-preserving transformation.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 Q11 [17]}}