| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.8 This FP1 question requires understanding matrix decomposition into component transformations, identifying a shear transformation from geometric properties, then finding the complementary transformation through matrix multiplication. While systematic, it demands conceptual understanding of transformation composition and the ability to work backwards from a given decomposition—more sophisticated than routine matrix operations. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03q Inverse transformations |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Find coordinates \((0,0)\) \((3,1)\) \((2,1)\) \((5,2)\) found. Accurate diagram sketched | B1 B1 B1 | 3 |
| (ii) \(\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\) | B1 B1 | 2 |
| (iii) Either \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\) or | B1 M1 A1 ft | |
| Or | M1 A2 ft | |
| Shear, \(x\) axis invariant or parallel to \(x\)-axis eg image of \((1,1)\) is \((3,1)\). SR allow s.f. 2 or shearing angle of correct angle to appropriate axis | B1 B1 B1 | 6 |
| 11 |
**(i)** Find coordinates $(0,0)$ $(3,1)$ $(2,1)$ $(5,2)$ found. Accurate diagram sketched | B1 B1 B1 | 3 |
**(ii)** $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ | B1 B1 | 2 | Each column correct
**(iii)** Either $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ or | B1 M1 A1 ft | | Correct inverse for their (ii) stated. Post multiply C by inverse of (ii). Correct answer found
Or | M1 A2 ft | | Set up 4 equations for elements from correct matrix multiplication. All elements correct, -1 each error
Shear, $x$ axis invariant or parallel to $x$-axis eg image of $(1,1)$ is $(3,1)$. SR allow s.f. 2 or shearing angle of correct angle to appropriate axis | B1 B1 B1 | 6 |
| | | **11** |8 The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { l l } 3 & 2 \\ 1 & 1 \end{array} \right)$.\\
(i) Draw a diagram showing the image of the unit square under the transformation represented by $\mathbf { C }$.
The transformation represented by $\mathbf { C }$ is equivalent to a transformation S followed by another transformation T.\\
(ii) Given that S is a shear with the $y$-axis invariant in which the image of the point ( 1,1 ) is ( 1,2 ), write down the matrix that represents $S$.\\
(iii) Find the matrix that represents transformation T and describe fully the transformation T .
\hfill \mbox{\textit{OCR FP1 2009 Q8 [11]}}