AQA FP1 2013 June — Question 8 6 marks

Exam BoardAQA
ModuleFP1 (Further Pure Mathematics 1)
Year2013
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeCombined transformation matrix product
DifficultyStandard +0.3 This is a standard FP1 transformation question requiring finding a stretch matrix from a diagram, deriving a reflection matrix in a line through the origin (using the standard formula with angle), and computing a matrix product. While it involves multiple parts and surds, these are routine techniques for Further Maths students with no novel problem-solving required, making it slightly easier than average.
Spec4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products
8 The diagram shows two triangles, \(T _ { 1 }\) and \(T _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{d74d6295-d5b8-46da-8812-c5bf7c7a35f1-09_972_967_358_589}
  1. Find the matrix which represents the stretch that maps triangle \(T _ { 1 }\) onto triangle \(T _ { 2 }\).
  2. The triangle \(T _ { 2 }\) is reflected in the line \(y = \sqrt { 3 } x\) to give a third triangle, \(T _ { 3 }\). Find, using surd forms where appropriate:
    1. the matrix which represents the reflection that maps triangle \(T _ { 2 }\) onto triangle \(T _ { 3 }\);
    2. the matrix which represents the combined transformation that maps triangle \(T _ { 1 }\) onto triangle \(T _ { 3 }\).
      (2 marks)
The diagram shows two triangles, \(T_1\) and \(T_2\).
(a) Find the matrix which represents the stretch that maps triangle \(T_1\) onto triangle \(T_2\). (2 marks)
(b) The triangle \(T_2\) is reflected in the line \(y = \sqrt{3}x\) to give a third triangle, \(T_3\). Find, using surd forms where appropriate:
(i) the matrix which represents the reflection that maps triangle \(T_2\) onto triangle \(T_3\); (2 marks)
(ii) the matrix which represents the combined transformation that maps triangle \(T_1\) onto triangle \(T_3\). (2 marks)
The diagram shows two triangles, $T_1$ and $T_2$.

**(a) Find the matrix which represents the stretch that maps triangle $T_1$ onto triangle $T_2$. (2 marks)**

**(b) The triangle $T_2$ is reflected in the line $y = \sqrt{3}x$ to give a third triangle, $T_3$. Find, using surd forms where appropriate:**

(i) the matrix which represents the reflection that maps triangle $T_2$ onto triangle $T_3$; **(2 marks)**

(ii) the matrix which represents the combined transformation that maps triangle $T_1$ onto triangle $T_3$. **(2 marks)**
8 The diagram shows two triangles, $T _ { 1 }$ and $T _ { 2 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{d74d6295-d5b8-46da-8812-c5bf7c7a35f1-09_972_967_358_589}
\begin{enumerate}[label=(\alph*)]
\item Find the matrix which represents the stretch that maps triangle $T _ { 1 }$ onto triangle $T _ { 2 }$.
\item The triangle $T _ { 2 }$ is reflected in the line $y = \sqrt { 3 } x$ to give a third triangle, $T _ { 3 }$. Find, using surd forms where appropriate:
\begin{enumerate}[label=(\roman*)]
\item the matrix which represents the reflection that maps triangle $T _ { 2 }$ onto triangle $T _ { 3 }$;
\item the matrix which represents the combined transformation that maps triangle $T _ { 1 }$ onto triangle $T _ { 3 }$.\\
(2 marks)
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA FP1 2013 Q8 [6]}}
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