| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Standard +0.3 This is a standard Further Maths question testing routine matrix transformations and the area scale factor property. Part (i) requires recall of standard transformation matrices and matrix multiplication. Part (ii) applies the determinant-area relationship, which is a core FP1 technique. All steps are algorithmic with no novel problem-solving required, making it slightly easier than average even for Further Maths. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix} -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix} -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix}\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix}\) | M1A1 | M1: multiplies their (b) \(\times\) their (a) in correct order; A1: correct matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Area triangle \(T = \frac{1}{2} \times (11-3) \times k = 4k\) | M1A1 | M1: correct method for area; A1: \(4k\) |
| \(\det\begin{pmatrix} 6 & -2 \\ 1 & 2 \end{pmatrix} = 6\times2 - 1\times(-2) (= 14)\) | M1A1 | M1: correct method for determinant; A1: 14 |
| Area triangle \(T' = \frac{364}{\text{"14"}} (= 26) \Rightarrow 4k = 26\) | M1 | Uses 364 and their determinant correctly to form equation in \(k\) |
| \(k = \frac{26}{4} \left(= \frac{13}{2}\right)\) | A1 | Accept \(k = \pm\frac{13}{2}\) or \(k = -\frac{13}{2}\) |
# Question 7:
## Part (i)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ | B1 | |
## Part (i)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix} -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix}$ | B1 | |
## Part (i)(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix} -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix}\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix}$ | M1A1 | M1: multiplies their (b) $\times$ their (a) in correct order; A1: correct matrix |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Area triangle $T = \frac{1}{2} \times (11-3) \times k = 4k$ | M1A1 | M1: correct method for area; A1: $4k$ |
| $\det\begin{pmatrix} 6 & -2 \\ 1 & 2 \end{pmatrix} = 6\times2 - 1\times(-2) (= 14)$ | M1A1 | M1: correct method for determinant; A1: 14 |
| Area triangle $T' = \frac{364}{\text{"14"}} (= 26) \Rightarrow 4k = 26$ | M1 | Uses 364 and their determinant correctly to form equation in $k$ |
| $k = \frac{26}{4} \left(= \frac{13}{2}\right)$ | A1 | Accept $k = \pm\frac{13}{2}$ or $k = -\frac{13}{2}$ |
---7. (i) In each of the following cases, find a $2 \times 2$ matrix that represents
\begin{enumerate}[label=(\alph*)]
\item a reflection in the line $y = - x$,
\item a rotation of $135 ^ { \circ }$ anticlockwise about $( 0,0 )$,
\item a reflection in the line $y = - x$ followed by a rotation of $135 ^ { \circ }$ anticlockwise about $( 0,0 )$.\\
(ii) The triangle $T$ has vertices at the points $( 1 , k ) , ( 3,0 )$ and $( 11,0 )$, where $k$ is a constant. Triangle $T$ is transformed onto the triangle $T ^ { \prime }$ by the matrix
$$\left( \begin{array} { r r }
6 & - 2 \\
1 & 2
\end{array} \right)$$
Given that the area of triangle $T ^ { \prime }$ is 364 square units, find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2014 Q7 [10]}}