| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| 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 testing standard knowledge of transformation matrices. Part (a) requires recognizing a reflection matrix, part (b) is direct recall of the rotation matrix formula with standard angle substitution, and part (c) is routine matrix multiplication. While it's Further Maths content, all parts are textbook exercises with no problem-solving or novel insight required. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks |
|---|---|
| Transformation is a reflection in \(y = x\) | B2 (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Matrix is \(\begin{bmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix}\) | M1, A2,1 (3 marks) | M1 for matrix for a rotation; A1 for correct trig expressions |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to multiply matrices in the correct order | M1, m1 | |
| Matrix is \(\begin{bmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}\) | A1ft (3 marks) | ft wrong answer to (b) |
## Question 5:
**Part (a)**
Transformation is a reflection in $y = x$ | B2 (2 marks) |
**Part (b)**
Matrix is $\begin{bmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix}$ | M1, A2,1 (3 marks) | M1 for matrix for a rotation; A1 for correct trig expressions
**Part (c)**
Attempt to multiply matrices in the correct order | M1, m1 |
Matrix is $\begin{bmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}$ | A1ft (3 marks) | ft wrong answer to (b)
---5
\begin{enumerate}[label=(\alph*)]
\item The transformation $T _ { 1 }$ is defined by the matrix
$$\left[ \begin{array} { l l }
0 & 1 \\
1 & 0
\end{array} \right]$$
Describe this transformation geometrically.
\item The transformation $T _ { 2 }$ is an anticlockwise rotation about the origin through an angle of $60 ^ { \circ }$.
Find the matrix of the transformation $T _ { 2 }$. Use surds in your answer where appropriate.\\
(3 marks)
\item Find the matrix of the transformation obtained by carrying out $T _ { 1 }$ followed by $T _ { 2 }$.\\
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q5 [8]}}