OCR MEI FP1 2009 June — Question 9 12 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2009
SessionJune
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeCombined transformation matrix product
DifficultyModerate -0.8 This is a straightforward Further Maths question testing basic matrix multiplication and recognition of standard transformations (stretch, reflection, rotation). Part (i) tests associativity (a fundamental property), part (ii) requires identifying three standard 2×2 transformation matrices, and part (iii) involves applying the composite transformation to plot points. While it's Further Maths content, these are routine exercises with no problem-solving or novel insight required—essentially testing recall and mechanical application of techniques.
Spec4.03a 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, shear
9 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right) , \mathbf { N } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).
  1. The matrix products \(\mathbf { Q } ( \mathbf { M N } )\) and \(( \mathbf { Q M } ) \mathbf { N }\) are identical. What property of matrix multiplication does this illustrate? Find QMN. \(\mathbf { M } , \mathbf { N }\) and \(\mathbf { Q }\) represent the transformations \(\mathrm { M } , \mathrm { N }\) and Q respectively.
  2. Describe the transformations \(\mathrm { M } , \mathrm { N }\) and Q . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-4_668_908_788_621} \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{figure}
  3. The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 9 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively by the composite transformation N followed by M followed by Q . Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of each point clearly.
Question 9(i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Matrix multiplication is associativeB1 [1]
\(\mathbf{MN} = \begin{pmatrix}3&0\\0&2\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}\)M1 Attempt to find MN or QM
\(\Rightarrow \mathbf{MN} = \begin{pmatrix}0&3\\2&0\end{pmatrix}\)A1 or \(\mathbf{QM} = \begin{pmatrix}0&-2\\3&0\end{pmatrix}\)
\(\mathbf{QMN} = \begin{pmatrix}-2&0\\0&3\end{pmatrix}\)A1(ft) [3]
Question 9(ii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
M is a stretch, factor 3 in the \(x\) direction, factor 2 in the \(y\) directionB1, B1 Stretch factor 3 in \(x\); Stretch factor 2 in \(y\)
N is a reflection in the line \(y=x\)B1
Q is an anticlockwise rotation through \(90°\) about the originB1 [4]
Question 9(iii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\begin{pmatrix}-2&0\\0&3\end{pmatrix}\begin{pmatrix}1&1&2\\2&0&2\end{pmatrix} = \begin{pmatrix}-2&-2&-4\\6&0&6\end{pmatrix}\)M1, A1(ft) Applying their QMN to points; Minus 1 each error to a minimum of 0
Correct labelled image points on diagramB2 [4] Correct labelled image points, minus 1 each error to minimum of 0; Give B4 for correct diagram with no workings 
# Question 9(i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Matrix multiplication is associative | B1 **[1]** | |
| $\mathbf{MN} = \begin{pmatrix}3&0\\0&2\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}$ | M1 | Attempt to find **MN** or **QM** |
| $\Rightarrow \mathbf{MN} = \begin{pmatrix}0&3\\2&0\end{pmatrix}$ | A1 | or $\mathbf{QM} = \begin{pmatrix}0&-2\\3&0\end{pmatrix}$ |
| $\mathbf{QMN} = \begin{pmatrix}-2&0\\0&3\end{pmatrix}$ | A1(ft) **[3]** | |

---

# Question 9(ii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| **M** is a stretch, factor 3 in the $x$ direction, factor 2 in the $y$ direction | B1, B1 | Stretch factor 3 in $x$; Stretch factor 2 in $y$ |
| **N** is a reflection in the line $y=x$ | B1 | |
| **Q** is an anticlockwise rotation through $90°$ about the origin | B1 **[4]** | |

---

# Question 9(iii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix}-2&0\\0&3\end{pmatrix}\begin{pmatrix}1&1&2\\2&0&2\end{pmatrix} = \begin{pmatrix}-2&-2&-4\\6&0&6\end{pmatrix}$ | M1, A1(ft) | Applying their **QMN** to points; Minus 1 each error to a minimum of 0 |
| Correct labelled image points on diagram | B2 **[4]** | Correct labelled image points, minus 1 each error to minimum of 0; Give B4 for correct diagram with no workings |
9 You are given that $\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right) , \mathbf { N } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)$ and $\mathbf { Q } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)$.\\
(i) The matrix products $\mathbf { Q } ( \mathbf { M N } )$ and $( \mathbf { Q M } ) \mathbf { N }$ are identical. What property of matrix multiplication does this illustrate?

Find QMN.\\
$\mathbf { M } , \mathbf { N }$ and $\mathbf { Q }$ represent the transformations $\mathrm { M } , \mathrm { N }$ and Q respectively.\\
(ii) Describe the transformations $\mathrm { M } , \mathrm { N }$ and Q .

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-4_668_908_788_621}
\captionsetup{labelformat=empty}
\caption{Fig. 9}
\end{center}
\end{figure}

(iii) The points $\mathrm { A } , \mathrm { B }$ and C in the triangle in Fig. 9 are mapped to the points $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ and $\mathrm { C } ^ { \prime }$ respectively by the composite transformation N followed by M followed by Q . Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of each point clearly.

\hfill \mbox{\textit{OCR MEI FP1 2009 Q9 [12]}}
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