| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Matrix powers and repeated transformations |
| Difficulty | Standard +0.3 This is a straightforward FP1 matrices question testing standard techniques: matrix multiplication (trivial for a diagonal matrix), geometric interpretation, finding a reflection matrix using the standard formula with angle -π/6, and working backwards through composite transformations. All parts follow textbook methods with no novel problem-solving required, though the multi-step nature and exact trigonometric values elevate it slightly above the easiest FP1 questions. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03o Inverse 3x3 matrix4.03q Inverse transformations |
| Answer | Marks |
|---|---|
| (a)(i) \((A^2 =) \begin{bmatrix}4 & 0\\0 & 1\end{bmatrix}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(ii) Stretch parallel to \(x\)-axis scale factor 4 | B1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| (b) \(y = \frac{-1}{\sqrt{3}}x = \tan(-30°)x\) | M1 | Attempting to write \(x + \sqrt{3}y = 0\) in the form \(y = x\tan\theta\), PI then seeing/using \(B = \begin{bmatrix}\cos 2\theta & \sin 2\theta\\\sin 2\theta & -\cos 2\theta\end{bmatrix}\) with a value of \(\theta\) such that \(\tan\theta = \frac{-1}{\sqrt{3}}\) or \(\frac{1}{\sqrt{3}}\) |
| \(\mathbf{B} = \begin{bmatrix}\frac{1}{2} & -\frac{\sqrt{3}}{2}\\-\frac{\sqrt{3}}{2} & -\frac{1}{2}\end{bmatrix}\) | A1 | Any correct exact non-trig form |
| Answer | Marks | Guidance |
|---|---|---|
| (c) \(\mathbf{BA}^2 = \begin{bmatrix}\frac{1}{2} & -\frac{\sqrt{3}}{2}\\-\frac{\sqrt{3}}{2} & -\frac{1}{2}\end{bmatrix}\begin{bmatrix}4 & 0\\0 & 1\end{bmatrix}\) | M1 | Setting up the product of \(c\)'s \(\mathbf{B}\) and \(c\)'s \(\mathbf{A}^2\) in any order |
| \(\begin{bmatrix}2 & -\frac{\sqrt{3}}{2}\\-2\sqrt{3} & -\frac{1}{2}\end{bmatrix}\begin{bmatrix}\mathbf{x}\\\mathbf{y}\end{bmatrix} = \begin{bmatrix}0\\-4\end{bmatrix}\) | m1 | Multiply \(c\)'s \(\mathbf{B}\) by \(c\)'s \(\mathbf{A}^2\) in correct order to obtain a 2 by 2 matrix. PI by correct 2 by 1 matrix for \(\mathbf{BA}^2\begin{bmatrix}x\\y\end{bmatrix}\) in terms of \(x\) and \(y\) |
| \(2x - \frac{\sqrt{3}}{2}y = 0\) | A1 | OE |
| \(-2\sqrt{3}x - \frac{1}{2}y = -4\) | A1 | OE |
| m1 | Solving two correct equations to find a correct value for either the \(x\)-coordinate or the \(y\)-coordinate of \(P\) | |
| \(P\left(\frac{\sqrt{3}}{2},2\right)\) | A1 | Correct coordinates. Condone answer left as eg \(x = \frac{\sqrt{3}}{2}\), \(y = 2\) but do not accept answer left as a matrix |
**(a)(i)** $(A^2 =) \begin{bmatrix}4 & 0\\0 & 1\end{bmatrix}$ | B1 |
**Total for (a)(i): 1 mark**
**(a)(ii)** Stretch parallel to $x$-axis scale factor 4 | B1 | OE
**Total for (a)(ii): 1 mark**
**(b)** $y = \frac{-1}{\sqrt{3}}x = \tan(-30°)x$ | M1 | Attempting to write $x + \sqrt{3}y = 0$ in the form $y = x\tan\theta$, PI then seeing/using $B = \begin{bmatrix}\cos 2\theta & \sin 2\theta\\\sin 2\theta & -\cos 2\theta\end{bmatrix}$ with a value of $\theta$ such that $\tan\theta = \frac{-1}{\sqrt{3}}$ or $\frac{1}{\sqrt{3}}$
$\mathbf{B} = \begin{bmatrix}\frac{1}{2} & -\frac{\sqrt{3}}{2}\\-\frac{\sqrt{3}}{2} & -\frac{1}{2}\end{bmatrix}$ | A1 | Any correct exact non-trig form
**Total for (b): 2 marks**
**(c)** $\mathbf{BA}^2 = \begin{bmatrix}\frac{1}{2} & -\frac{\sqrt{3}}{2}\\-\frac{\sqrt{3}}{2} & -\frac{1}{2}\end{bmatrix}\begin{bmatrix}4 & 0\\0 & 1\end{bmatrix}$ | M1 | Setting up the product of $c$'s $\mathbf{B}$ and $c$'s $\mathbf{A}^2$ in any order
$\begin{bmatrix}2 & -\frac{\sqrt{3}}{2}\\-2\sqrt{3} & -\frac{1}{2}\end{bmatrix}\begin{bmatrix}\mathbf{x}\\\mathbf{y}\end{bmatrix} = \begin{bmatrix}0\\-4\end{bmatrix}$ | m1 | Multiply $c$'s $\mathbf{B}$ by $c$'s $\mathbf{A}^2$ in correct order to obtain a 2 by 2 matrix. PI by correct 2 by 1 matrix for $\mathbf{BA}^2\begin{bmatrix}x\\y\end{bmatrix}$ in terms of $x$ and $y$
$2x - \frac{\sqrt{3}}{2}y = 0$ | A1 | OE
$-2\sqrt{3}x - \frac{1}{2}y = -4$ | A1 | OE
m1 | Solving two correct equations to find a correct value for either the $x$-coordinate or the $y$-coordinate of $P$
$P\left(\frac{\sqrt{3}}{2},2\right)$ | A1 | Correct coordinates. Condone answer left as eg $x = \frac{\sqrt{3}}{2}$, $y = 2$ but do not accept answer left as a matrix
**Total for (c): 6 marks**
**Overall Total: 10 marks**
**Alternative (c):**
$\mathbf{BA}^2\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}0\\-4\end{bmatrix}$ (M1) PI; $\mathbf{B}^2\mathbf{A}^2\begin{bmatrix}x\\y\end{bmatrix} = \mathbf{A}^2\begin{bmatrix}x\\y\end{bmatrix}$ (m1 with both products attempted); $(\mathbf{B}^2 = \mathbf{I}$ seen/used (m1)) $\mathbf{A}^2\begin{bmatrix}x\\y\end{bmatrix} = \mathbf{B}\begin{bmatrix}0\\-4\end{bmatrix}$ (m1 with both products attempted); $\begin{bmatrix}4x\\y\end{bmatrix} = \begin{bmatrix}2\sqrt{3}\\2\end{bmatrix}$ (A1 LHS) (A1 RHS); $P\left(\frac{\sqrt{3}}{2},2\right)$ A1
---The matrix $\mathbf{A}$ is defined by $\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the matrix $\mathbf{A}^2$.
[1 mark]
\item Describe fully the single geometrical transformation represented by the matrix $\mathbf{A}^2$.
[1 mark]
\end{enumerate}
\item Given that the matrix $\mathbf{B}$ represents a reflection in the line $x + \sqrt{3}y = 0$, find the matrix $\mathbf{B}$, giving the exact values of any trigonometric expressions.
[2 marks]
\item Hence find the coordinates of the point $P$ which is mapped onto $(0, -4)$ under the transformation represented by $\mathbf{A}^2$ followed by a reflection in the line $x + \sqrt{3}y = 0$.
[6 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2016 Q8 [10]}}