| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.8 This FP1 question requires students to decompose a matrix into a sequence of transformations (rotation followed by enlargement), which demands geometric insight beyond routine matrix multiplication. While the individual transformations are standard, recognizing the decomposition and describing both transformations fully requires conceptual understanding rather than algorithmic application. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}\), \(\begin{pmatrix} 0 \\ -1 \end{pmatrix}\), \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\), \(\begin{pmatrix} 3 \\ -1 \end{pmatrix}\) | M1 | For at least two correct images |
| A1 | For correct diagram, co-ords. clearly written down | |
| (ii) \(90°\) clockwise, centre origin | B1 B1 | Or equivalent correct description |
| \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) | B1 | Correct matrix, not in trig form |
| (iii) Stretch parallel to \(x\)-axis, s.f. 3 | B1 B1 | Or equivalent correct description, but must be a stretch for 2nd B1 |
| \(\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}\) | B1 B1 | Each correct column |
(i) $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$, $\begin{pmatrix} 0 \\ -1 \end{pmatrix}$, $\begin{pmatrix} 3 \\ 0 \end{pmatrix}$, $\begin{pmatrix} 3 \\ -1 \end{pmatrix}$ | M1 | For at least two correct images
| A1 | For correct diagram, co-ords. clearly written down
(ii) $90°$ clockwise, centre origin | B1 B1 | Or equivalent correct description
$\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ | B1 | Correct matrix, not in trig form
(iii) Stretch parallel to $x$-axis, s.f. 3 | B1 B1 | Or equivalent correct description, but must be a stretch for 2nd B1
$\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$ | B1 B1 | Each correct column | (4 marks, 9 total)9 The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { r r } 0 & 3 \\ - 1 & 0 \end{array} \right)$.\\
(i) Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { C }$.
The transformation represented by $\mathbf { C }$ is equivalent to a rotation, R , followed by another transformation, S.\\
(ii) Describe fully the rotation R and write down the matrix that represents R .\\
(iii) Describe fully the transformation S and write down the matrix that represents S .
\hfill \mbox{\textit{OCR FP1 2007 Q9 [9]}}