OCR MEI FP1 2007 January — Question 3 7 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2007
SessionJanuary
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeDescribe enlargement or stretch from matrix
DifficultyEasy -1.2 This is a straightforward FP1 question requiring recognition that a diagonal matrix represents a stretch. Students need to identify two perpendicular stretches (scale factor 2 in x-direction, scale factor 1/2 in y-direction) and plot the transformed triangle. This is standard textbook material with no problem-solving required, making it easier than average even for Further Maths.
Spec4.03a Matrix language: terminology and notation4.03d Linear transformations 2D: reflection, rotation, enlargement, shear
3 The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 3 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively under the transformation represented by the matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 0 \\ 0 & \frac { 1 } { 2 } \end{array} \right)\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4a339746-195f-477a-952e-02fbdfd9cce5-2_446_444_1046_808} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly.
  2. Describe fully the transformation represented by the matrix \(\mathbf { M }\).
The points A, B and C in the triangle in Fig. 3 are mapped to the points \(A'\), \(B'\) and \(C'\) respectively under the transformation represented by the matrix \(M = \begin{pmatrix} 2 & 0 \\ 0 & \frac{1}{2} \end{pmatrix}\).
y
4
3
A C
2
1
0 B 1 2 3 4 x
Fig. 3
(i) Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly. [4]
(ii) Describe fully the transformation represented by the matrix M. [3]
The points A, B and C in the triangle in Fig. 3 are mapped to the points $A'$, $B'$ and $C'$ respectively under the transformation represented by the matrix $M = \begin{pmatrix} 2 & 0 \\ 0 & \frac{1}{2} \end{pmatrix}$.

y
4
3
A C
2
1
0 B 1 2 3 4 x
Fig. 3

(i) Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly. [4]

(ii) Describe fully the transformation represented by the matrix M. [3]
3 The points $\mathrm { A } , \mathrm { B }$ and C in the triangle in Fig. 3 are mapped to the points $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ and $\mathrm { C } ^ { \prime }$ respectively under the transformation represented by the matrix $\mathbf { M } = \left( \begin{array} { l l } 2 & 0 \\ 0 & \frac { 1 } { 2 } \end{array} \right)$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{4a339746-195f-477a-952e-02fbdfd9cce5-2_446_444_1046_808}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{center}
\end{figure}

(i) Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly.\\
(ii) Describe fully the transformation represented by the matrix $\mathbf { M }$.

\hfill \mbox{\textit{OCR MEI FP1 2007 Q3 [7]}}
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