| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Moderate -0.3 This is a standard FP1 matrix transformations question requiring recognition of basic transformation types (enlargement, reflection, rotation) and routine matrix multiplication. Part (a) tests recall of standard matrices, while parts (b)-(e) involve straightforward computation with no novel problem-solving required. Slightly easier than average A-level due to the mechanical nature of the tasks. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| A represents enlargement scale factor \(3\sqrt{2}\) (centre \(O\)) | M1 A1 | Enlargement for M1; \(3\sqrt{2}\) for A1 |
| B represents reflection in the line \(y = x\) | B1 | |
| C represents rotation of \(\dfrac{\pi}{4}\), i.e. \(45°\) anticlockwise about \(O\) | B1 (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\begin{pmatrix}3 & -3 \\ 3 & 3\end{pmatrix}\) | M1 A1 (2) | Answer incorrect: require CD for M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\begin{pmatrix}3 & -3 \\ 3 & 3\end{pmatrix}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} = \begin{pmatrix}-3 & 3 \\ 3 & 3\end{pmatrix}\) | B1 (1) | Answer given so require DB as shown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\begin{pmatrix}-3 & 3 \\ 3 & 3\end{pmatrix}\begin{pmatrix}0 & -15 & 4 \\ 0 & 15 & 21\end{pmatrix} = \begin{pmatrix}0 & 90 & 51 \\ 0 & 0 & 75\end{pmatrix}\) giving \((0,0)\), \((90,0)\), \((51,75)\) | M1 A1 A1 A1 (4) | Coordinates as column vectors \(\begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}90\\0\end{pmatrix}, \begin{pmatrix}51\\75\end{pmatrix}\) for each A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Area of \(\triangle OR'S' = \dfrac{1}{2} \times 90 \times 75 = 3375\) | B1 | 3375 B1 |
| Determinant of E is \(-18\); area of \(\triangle ORS = 3375 \div 18 = 187.5\) | M1 A1 (3) [14] | Divide by theirs for M1 |
## Question 10:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| **A** represents enlargement scale factor $3\sqrt{2}$ (centre $O$) | M1 A1 | Enlargement for M1; $3\sqrt{2}$ for A1 |
| **B** represents reflection in the line $y = x$ | B1 | |
| **C** represents rotation of $\dfrac{\pi}{4}$, i.e. $45°$ anticlockwise about $O$ | B1 (4) | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix}3 & -3 \\ 3 & 3\end{pmatrix}$ | M1 A1 (2) | Answer incorrect: require **CD** for M1 |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix}3 & -3 \\ 3 & 3\end{pmatrix}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} = \begin{pmatrix}-3 & 3 \\ 3 & 3\end{pmatrix}$ | B1 (1) | Answer given so require **DB** as shown |
### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix}-3 & 3 \\ 3 & 3\end{pmatrix}\begin{pmatrix}0 & -15 & 4 \\ 0 & 15 & 21\end{pmatrix} = \begin{pmatrix}0 & 90 & 51 \\ 0 & 0 & 75\end{pmatrix}$ giving $(0,0)$, $(90,0)$, $(51,75)$ | M1 A1 A1 A1 (4) | Coordinates as column vectors $\begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}90\\0\end{pmatrix}, \begin{pmatrix}51\\75\end{pmatrix}$ for each A1 |
### Part (e):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Area of $\triangle OR'S' = \dfrac{1}{2} \times 90 \times 75 = 3375$ | B1 | 3375 B1 |
| Determinant of **E** is $-18$; area of $\triangle ORS = 3375 \div 18 = 187.5$ | M1 A1 (3) [14] | Divide by theirs for M1 |10.
$$\mathbf { A } = \left( \begin{array} { c c }
3 \sqrt { } 2 & 0 \\
0 & 3 \sqrt { } 2
\end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c }
0 & 1 \\
1 & 0
\end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { c c }
\frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 } \\
\frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 }
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Describe fully the transformations described by each of the matrices $\mathbf { A } , \mathbf { B }$ and $\mathbf { C }$.
It is given that the matrix $\mathbf { D } = \mathbf { C A }$, and that the matrix $\mathbf { E } = \mathbf { D B }$.
\item Find $\mathbf { D }$.
\item Show that $\mathbf { E } = \left( \begin{array} { c c } - 3 & 3 \\ 3 & 3 \end{array} \right)$.
The triangle $O R S$ has vertices at the points with coordinates $( 0,0 ) , ( - 15,15 )$ and $( 4,21 )$. This triangle is transformed onto the triangle $O R ^ { \prime } S ^ { \prime }$ by the transformation described by $\mathbf { E }$.
\item Find the coordinates of the vertices of triangle $O R ^ { \prime } S ^ { \prime }$.
\item Find the area of triangle $O R ^ { \prime } S ^ { \prime }$ and deduce the area of triangle $O R S$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2009 Q10 [14]}}