| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Extract enlargement and rotation parameters |
| Difficulty | Moderate -0.5 Standard Core Pure question on matrix transformations: finding determinant (trivial), using det for area scale factor, then identifying enlargement scale factor k=√(det M)=2 and rotation angle θ=60° by comparing to standard rotation-enlargement matrix form. All parts are routine with no novel insight required. |
| Spec | 4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\det(\mathbf{M})=(1)(1)-(\sqrt{3})(-\sqrt{3})\) | M1 | An attempt to find \(\det(\mathbf{M})\) |
| \(\mathbf{M}\) is non-singular because \(\det(\mathbf{M})=4\) and so \(\det(\mathbf{M})\neq 0\) | A1 | \(\det(\mathbf{M})=4\) and reference to zero, e.g. \(4\neq 0\) and conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Area}(S)=4(5)=20\) | B1ft | 20 or a correct ft based on their answer to part (a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(k=\sqrt{(1)(1)-(\sqrt{3})(-\sqrt{3})}\) | M1 | \(\sqrt{\text{their }\det(\mathbf{M})}\) |
| \(k=2\) | A1ft |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\cos\theta=\dfrac{1}{2}\) or \(\sin\theta=\dfrac{\sqrt{3}}{2}\) or \(\tan\theta=\sqrt{3}\) | M1 | Either \(\cos\theta=\dfrac{1}{\text{their }k}\) or \(\sin\theta=\dfrac{\sqrt{3}}{\text{their }k}\) or \(\tan\theta=\sqrt{3}\) |
| \(\theta=60°\) or \(\dfrac{\pi}{3}\) | A1 | Also accept any value satisfying \(360n+60°\), \(n\in\mathbb{Z}\) |
# Question 5:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\det(\mathbf{M})=(1)(1)-(\sqrt{3})(-\sqrt{3})$ | M1 | An attempt to find $\det(\mathbf{M})$ |
| $\mathbf{M}$ is non-singular because $\det(\mathbf{M})=4$ and so $\det(\mathbf{M})\neq 0$ | A1 | $\det(\mathbf{M})=4$ **and** reference to zero, e.g. $4\neq 0$ **and** conclusion |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Area}(S)=4(5)=20$ | B1ft | 20 or a correct ft based on their answer to part (a) |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $k=\sqrt{(1)(1)-(\sqrt{3})(-\sqrt{3})}$ | M1 | $\sqrt{\text{their }\det(\mathbf{M})}$ |
| $k=2$ | A1ft | |
## Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\cos\theta=\dfrac{1}{2}$ or $\sin\theta=\dfrac{\sqrt{3}}{2}$ or $\tan\theta=\sqrt{3}$ | M1 | Either $\cos\theta=\dfrac{1}{\text{their }k}$ or $\sin\theta=\dfrac{\sqrt{3}}{\text{their }k}$ or $\tan\theta=\sqrt{3}$ |
| $\theta=60°$ or $\dfrac{\pi}{3}$ | A1 | Also accept any value satisfying $360n+60°$, $n\in\mathbb{Z}$ |5.
$$\mathbf { M } = \left( \begin{array} { c c }
1 & - \sqrt { 3 } \\
\sqrt { 3 } & 1
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathbf { M }$ is non-singular.
The hexagon $R$ is transformed to the hexagon $S$ by the transformation represented by the matrix $\mathbf { M }$.
Given that the area of hexagon $R$ is 5 square units,
\item find the area of hexagon $S$.
The matrix $\mathbf { M }$ represents an enlargement, with centre $( 0,0 )$ and scale factor $k$, where $k > 0$, followed by a rotation anti-clockwise through an angle $\theta$ about $( 0,0 )$.
\item Find the value of $k$.
\item Find the value of $\theta$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS Q5 [7]}}