| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Area scale factor from determinant |
| Difficulty | Moderate -0.3 Part (i)(a) requires recognizing a standard rotation matrix with cos(45°) and sin(45°) values—straightforward pattern matching. Part (i)(b) is trivial recall. Part (ii) uses the determinant-area relationship (area scales by |det M|), requiring one calculation: |det M| = 224/16 = 14, so 9+2k = 14, giving k = 2.5. This is a standard FP1 question testing basic matrix transformation concepts with minimal problem-solving, slightly easier than average A-level due to its routine nature. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Rotation of 45 degrees anticlockwise, about the origin | B1B1 | B1: Rotation about \((0,0)\); B1: 45 degrees (anticlockwise). -45 or clockwise award B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}\) | B1 | Correct matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{224}{16} (= 14)\) | B1 | Correct area scale factor. Allow \(\pm 14\) |
| \(\det \mathbf{M} = 3 \times 3 - k \times -2 = 14\) | M1 | Attempt determinant and set equal to their area scale factor. Accept \(\det \mathbf{M} = 3 \times 3 \pm 2k\) only |
| \(k = 2.5\) | A1 | oe |
## Question 3:
### Part (i)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Rotation of 45 degrees anticlockwise, about the origin | B1B1 | B1: Rotation about $(0,0)$; B1: 45 degrees (anticlockwise). -45 or clockwise award B0 |
### Part (i)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}$ | B1 | Correct matrix |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{224}{16} (= 14)$ | B1 | Correct area scale factor. Allow $\pm 14$ |
| $\det \mathbf{M} = 3 \times 3 - k \times -2 = 14$ | M1 | Attempt determinant and set equal to their area scale factor. Accept $\det \mathbf{M} = 3 \times 3 \pm 2k$ only |
| $k = 2.5$ | A1 | oe |
---3. (i)
$$\mathbf { A } = \left( \begin{array} { c c }
\frac { 1 } { \sqrt { } 2 } & \frac { - 1 } { \sqrt { } 2 } \\
\frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 }
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Describe fully the single transformation represented by the matrix $\mathbf { A }$.
The matrix $\mathbf { B }$ represents an enlargement, scale factor - 2 , with centre the origin.
\item Write down the matrix $\mathbf { B }$.\\
(ii)
$$\mathbf { M } = \left( \begin{array} { c c }
3 & k \\
- 2 & 3
\end{array} \right) , \quad \text { where } k \text { is a positive constant. }$$
Triangle $T$ has an area of 16 square units.
Triangle $T$ is transformed onto the triangle $T ^ { \prime }$ by the transformation represented by the matrix M.
Given that the area of the triangle $T ^ { \prime }$ is 224 square units, find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2014 Q3 [6]}}