| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring recognition of a shear transformation, matrix multiplication to find Y = ZX^(-1), and identification of a composite transformation (rotation + enlargement). While systematic, it demands fluency with matrix operations, inverse matrices, and geometric interpretation of non-standard matrix forms—significantly above average A-level difficulty but standard for FP1. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks |
|---|---|
| B1* | Shear |
| depB1 | e.g. image of \((0, 1)\) is \((2, 1)\) or parallel to the \(x\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | State \(\mathbf{Z} = \mathbf{YX}\) | |
| B1 | Obtain \(\mathbf{Y} = \mathbf{ZX}^{-1}\) | |
| \(\begin{pmatrix} 1 & -2 \\ 0 & 1 \end{pmatrix}\) | B1 | State or use correct inverse |
| M1 | Matrix multiplication, 2 elements correct | |
| \(\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}\) | A1 | Obtain completely correct simplified exact matrix |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{Z} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\) | B1 | Correct order for matrix multiplication |
| \(\begin{pmatrix} a & 2a+b \\ c & 2c+d \end{pmatrix}\) | B1 | Obtain 2 correct elements |
| B1 | Obtain other 2 correct elements | |
| M1 | Equate elements, 2 correct | |
| \(\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}\) | A1 | Obtain completely correct simplified exact matrix |
| Answer | Marks |
|---|---|
| B1* | Rotation |
| depB1 | \(60°\) clockwise |
## Question 9:
### Part (i)
| B1* | Shear
| depB1 | e.g. image of $(0, 1)$ is $(2, 1)$ or parallel to the $x$-axis
**[2]**
### Part (ii)
**Either**
| B1 | State $\mathbf{Z} = \mathbf{YX}$
| B1 | Obtain $\mathbf{Y} = \mathbf{ZX}^{-1}$
$\begin{pmatrix} 1 & -2 \\ 0 & 1 \end{pmatrix}$ | B1 | State or use correct inverse
| M1 | Matrix multiplication, 2 elements correct
$\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}$ | A1 | Obtain completely correct simplified exact matrix
**[5]**
**Or**
$\mathbf{Z} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ | B1 | Correct order for matrix multiplication
$\begin{pmatrix} a & 2a+b \\ c & 2c+d \end{pmatrix}$ | B1 | Obtain 2 correct elements
| B1 | Obtain other 2 correct elements
| M1 | Equate elements, 2 correct
$\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}$ | A1 | Obtain completely correct simplified exact matrix
### Part (iii)
| B1* | Rotation
| depB1 | $60°$ clockwise
**[2]**9 (i) The matrix $\mathbf { X }$ is given by $\mathbf { X } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)$. Describe fully the geometrical transformation represented by $\mathbf { X }$.\\
(ii) The matrix $\mathbf { Z }$ is given by $\mathbf { Z } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { 1 } { 2 } ( 2 + \sqrt { 3 } ) \\ - \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } ( 1 - 2 \sqrt { 3 } ) \end{array} \right)$. The transformation represented by $\mathbf { Z }$ is equivalent to the transformation represented by $\mathbf { X }$, followed by another transformation represented by the matrix $\mathbf { Y }$. Find $\mathbf { Y }$.\\
(iii) Describe fully the geometrical transformation represented by $\mathbf { Y }$.
\hfill \mbox{\textit{OCR FP1 2012 Q9 [9]}}