| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Moderate -0.3 This is a straightforward Further Pure 1 question requiring recognition of a 90° rotation matrix and decomposition into reflection + reflection. While it involves matrix transformations (a Further Maths topic), the geometric interpretations are standard and the matrix calculation is routine, making it slightly easier than average overall. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks |
|---|---|
| Rotation 90° (about origin) anticlockwise | B1, B1, 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) | M1 | Show image of unit square after reflection in \(y = -x\) |
| A1 | Deduce reflection in x-axis | |
| B1ft, B1ft, 4, M1 | Each column correct; fit for matrix of their transformation; Post multiply by correct reflection matrix | |
| A1 | Obtain correct answer | |
| B1B1 | State reflection, in x-axis |
| Answer | Marks |
|---|---|
| M1 | Post multiply by correct reflection matrix |
| A1 | Obtain correct answer |
| B1B1 | State reflection, in x-axis |
**Part (i)**
Rotation 90° (about origin) anticlockwise | B1, B1, 2 |
---
**Part (ii)**
Either:
$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ | M1 | Show image of unit square after reflection in $y = -x$
| A1 | Deduce reflection in x-axis
| B1ft, B1ft, 4, M1 | Each column correct; fit for matrix of their transformation; Post multiply by correct reflection matrix
| A1 | Obtain correct answer
| B1B1 | State reflection, in x-axis
Or:
| M1 | Post multiply by correct reflection matrix
| A1 | Obtain correct answer
| B1B1 | State reflection, in x-axis
S.C. If pre-multiplication, M0 but B1 B1 available for correct description of their matrix
---5 (i) The transformation T is represented by the matrix $\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)$. Give a geometrical description of T .\\
(ii) The transformation T is equivalent to a reflection in the line $y = - x$ followed by another transformation S . Give a geometrical description of S and find the matrix that represents S .
\hfill \mbox{\textit{OCR FP1 2010 Q5 [6]}}