| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Find image coordinates under transformation |
| Difficulty | Moderate -0.3 This is a standard Further Pure 1 matrix transformations question covering routine operations: describing geometric transformations from matrices, matrix multiplication for composite transformations, applying transformations to coordinates, and finding reflection matrices. While it has multiple parts (5 sub-questions), each part uses well-practiced techniques with no novel problem-solving required. The operations are mechanical and follow standard FP1 procedures, making it slightly easier than average for an A-level question overall, though appropriately pitched for Further Maths students learning this topic. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative)4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03n Inverse 2x2 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| \(P\) is a rotation through 90 degrees about the origin in a clockwise direction. | B1, B1 | Rotation about origin; 90 degrees clockwise, or equivalent |
| \(Q\) is a stretch factor 2 parallel to the \(x\)-axis | B1, B1 [4] | Stretch factor 2; Parallel to the \(x\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{QP} = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 2 \\ -1 & 0 \end{pmatrix}\) | M1, A1 [2] | Correct order; c.a.o. |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{pmatrix} 0 & 2 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 2 & 1 & 3 \\ 0 & 2 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 4 & 2 \\ -2 & -1 & -3 \end{pmatrix}\) | M1 | Pre-multiply by their \(\mathbf{QP}\) - may be implied |
| \(A' = (0,\ -2),\ B' = (4,\ -1),\ C' = (2,\ -3)\) | A1(ft) [2] | For all three points |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{R} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) | B1, B1 [2] | One for each correct column |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{RQP} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 2 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -2 \end{pmatrix}\) | M1, A1(ft) | Multiplication of their matrices in correct order |
| \(\left(\mathbf{RQP}\right)^{-1} = \dfrac{-1}{2}\begin{pmatrix} -2 & 0 \\ 0 & 1 \end{pmatrix}\) | M1, A1 [4] | Attempt to calculate inverse of their \(\mathbf{RQP}\); c.a.o. |
# Question 9:
## Part 9(i):
$P$ is a rotation through 90 degrees about the origin in a clockwise direction. | B1, B1 | Rotation about origin; 90 degrees clockwise, or equivalent
$Q$ is a stretch factor 2 parallel to the $x$-axis | B1, B1 [4] | Stretch factor 2; Parallel to the $x$-axis
---
## Part 9(ii):
$\mathbf{QP} = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 2 \\ -1 & 0 \end{pmatrix}$ | M1, A1 [2] | Correct order; c.a.o.
---
## Part 9(iii):
$\begin{pmatrix} 0 & 2 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 2 & 1 & 3 \\ 0 & 2 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 4 & 2 \\ -2 & -1 & -3 \end{pmatrix}$ | M1 | Pre-multiply by their $\mathbf{QP}$ - may be implied
$A' = (0,\ -2),\ B' = (4,\ -1),\ C' = (2,\ -3)$ | A1(ft) [2] | For all three points
---
## Part 9(iv):
$\mathbf{R} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ | B1, B1 [2] | One for each correct column
---
## Part 9(v):
$\mathbf{RQP} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 2 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -2 \end{pmatrix}$ | M1, A1(ft) | Multiplication of their matrices in correct order
$\left(\mathbf{RQP}\right)^{-1} = \dfrac{-1}{2}\begin{pmatrix} -2 & 0 \\ 0 & 1 \end{pmatrix}$ | M1, A1 [4] | Attempt to calculate inverse of their $\mathbf{RQP}$; c.a.o.9 The matrices $\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$ and $\mathbf { Q } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)$ represent transformations $P$ and $Q$ respectively.\\
(i) Describe fully the transformations P and Q .
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-4_625_849_470_648}
\captionsetup{labelformat=empty}
\caption{Fig. 9}
\end{center}
\end{figure}
Fig. 9 shows triangle T with vertices $\mathrm { A } ( 2,0 ) , \mathrm { B } ( 1,2 )$ and $\mathrm { C } ( 3,1 )$.\\
Triangle T is transformed first by transformation P , then by transformation Q .\\
(ii) Find the single matrix that represents this composite transformation.\\
(iii) This composite transformation maps triangle T onto triangle $\mathrm { T } ^ { \prime }$, with vertices $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ and $\mathrm { C } ^ { \prime }$. Calculate the coordinates of $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ and $\mathrm { C } ^ { \prime }$.
T' is reflected in the line $y = - x$ to give a new triangle, T".\\
(iv) Find the matrix $\mathbf { R }$ that represents reflection in the line $y = - x$.\\
(v) A single transformation maps $\mathrm { T } ^ { \prime \prime }$ onto the original triangle, T . Find the matrix representing this transformation.
\hfill \mbox{\textit{OCR MEI FP1 2010 Q9 [14]}}