| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question requiring standard matrix operations: finding a rotation matrix (standard result for 90° clockwise), applying transformations to vertices, and multiplying two 2×2 matrices. While it's Further Maths content, the techniques are routine and well-practiced, making it slightly easier than average overall but typical for FP1. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Mark |
| (a)(i) | Rectangle with vertices \((0, 0)\), \((0, -3)\), \((2, -3)\), \((2, 0)\) | B1 |
| (a)(ii) | Rectangle with vertices either whose x-coords are c's (a)(i) x-coord vertices multiplied by 4 or whose y-coords are c's (a)(i) y-coord vertices multiplied by 2 | M1 |
| A2,1 | 3 marks; A2 if rectangle with vertices \((0, 0)\), \((0, -6)\), \((8, -6)\), \((8, 0)\) (A1 if either the four x-coords are correct or the four y-coords are correct) | |
| (b)(i) | Matrix is \(\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\) | B1 |
| (b)(ii) | Attempt to multiply \(\begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\) with c's (b)(i) matrix in either order | M1 |
| Multiplication in correct order with at least two of the four ft multiplications carried out correctly | m1 | |
| \(\begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix}\) | A1 | 3 marks; For \(\begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix}\); NMS \(\begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix}\) scores B3; \(\begin{bmatrix} 0 & 2 \\ -4 & 0 \end{bmatrix}\) scores B1 |
| Part | Answer/Working | Mark | Guidance |
|------|---|---|---|
| (a)(i) | Rectangle with vertices $(0, 0)$, $(0, -3)$, $(2, -3)$, $(2, 0)$ | B1 | 1 mark |
| (a)(ii) | Rectangle with vertices either whose x-coords are c's (a)(i) x-coord vertices multiplied by 4 or whose y-coords are c's (a)(i) y-coord vertices multiplied by 2 | M1 | |
| | | A2,1 | 3 marks; A2 if rectangle with vertices $(0, 0)$, $(0, -6)$, $(8, -6)$, $(8, 0)$ (A1 if either the four x-coords are correct or the four y-coords are correct) |
| (b)(i) | Matrix is $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ | B1 | 1 mark |
| (b)(ii) | Attempt to multiply $\begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ with c's (b)(i) matrix in either order | M1 | |
| | Multiplication in correct order with at least two of the four ft multiplications carried out correctly | m1 | |
| | $\begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix}$ | A1 | 3 marks; For $\begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix}$; NMS $\begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix}$ scores B3; $\begin{bmatrix} 0 & 2 \\ -4 & 0 \end{bmatrix}$ scores B1 |8 The diagram below shows a rectangle $R _ { 1 }$ which has vertices $( 0,0 ) , ( 3,0 ) , ( 3,2 )$ and $( 0,2 )$.
\begin{enumerate}[label=(\alph*)]
\item On the diagram, draw:
\begin{enumerate}[label=(\roman*)]
\item the image $R _ { 2 }$ of $R _ { 1 }$ under a rotation through $90 ^ { \circ }$ clockwise about the origin;
\item the image $R _ { 3 }$ of $R _ { 2 }$ under the transformation which has matrix
$$\left[ \begin{array} { l l }
4 & 0 \\
0 & 2
\end{array} \right]$$
\end{enumerate}\item Find the matrix of:
\begin{enumerate}[label=(\roman*)]
\item the rotation which maps $R _ { 1 }$ onto $R _ { 2 }$;
\item the combined transformation which maps $R _ { 1 }$ onto $R _ { 3 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-5_913_910_1228_598}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2012 Q8 [8]}}