| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Session | Specimen |
| 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 decomposing a matrix into elementary transformations (rotation and enlargement), finding the rotation angle using arctan(2), and expressing the rotation matrix in exact form with cos(arctan(2)) = 1/√5. It goes beyond standard matrix multiplication to require geometric insight and inverse reasoning about transformation composition. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\begin{pmatrix} 1 & -2 \\ 2 & 1 \end{pmatrix}\begin{pmatrix} 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 & -2 & -1 \\ 0 & 2 & 1 & 3 \end{pmatrix}\) | M1, A1, A1 | For at least one correct image / For all vertices correct / For correct diagram |
| Answer | Marks | Guidance |
|---|---|---|
| So its area is 5 times that of the unit square | B1, M1, A1 | For identifying det as area scale factor / For calculation method relating to large sq. / For a complete explanation |
| Answer | Marks | Guidance |
|---|---|---|
| Enlargement with scale factor \(\sqrt{5}\) | B1, B1, B1 | For \(\tan^{-1}(2)\), or equivalent / For stating 'enlargement' / For correct (exact) scale factor |
| Answer | Marks | Guidance |
|---|---|---|
| (iv) \(\begin{pmatrix} \frac{1}{\sqrt{5}} & -\frac{2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{pmatrix}\) | M1, A1 | For correct \(\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\) pattern / For correct matrix in exact form |
**(i)** $\begin{pmatrix} 1 & -2 \\ 2 & 1 \end{pmatrix}\begin{pmatrix} 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 & -2 & -1 \\ 0 & 2 & 1 & 3 \end{pmatrix}$ | M1, A1, A1 | For at least one correct image / For all vertices correct / For correct diagram
**Total: 3 marks**
**(ii)** The area scale-factor is 5
The transformed square has side length $\sqrt{5}$
So its area is 5 times that of the unit square | B1, M1, A1 | For identifying det as area scale factor / For calculation method relating to large sq. / For a complete explanation
**Total: 3 marks**
**(iii)** Angle is $\tan^{-1}(2) = 63.4°$
Enlargement with scale factor $\sqrt{5}$ | B1, B1, B1 | For $\tan^{-1}(2)$, or equivalent / For stating 'enlargement' / For correct (exact) scale factor
**Total: 3 marks**
**(iv)** $\begin{pmatrix} \frac{1}{\sqrt{5}} & -\frac{2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{pmatrix}$ | M1, A1 | For correct $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ pattern / For correct matrix in exact form
**Total: 2 marks**
**Question 7 Total: 11 marks**
---7 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r } 1 & - 2 \\ 2 & 1 \end{array} \right)$.\\
(i) Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { A }$.\\
(ii) The value of $\operatorname { det } \mathbf { A }$ is 5 . Show clearly how this value relates to your diagram in part (i).
A represents a sequence of two elementary geometrical transformations, one of which is a rotation $R$.\\
(iii) Determine the angle of $R$, and describe the other transformation fully.\\
(iv) State the matrix that represents $R$, giving the elements in an exact form.
\hfill \mbox{\textit{OCR FP1 Q7 [11]}}