4.03d Linear transformations 2D: reflection, rotation, enlargement, shear

1. An ellipse has equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1\) and eccentricity \(e _ { 1 }\) A hyperbola has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\) and eccentricity \(e _ { 2 }\)

Given that \(e _ { 1 } \times e _ { 2 } = 1\)

1. show that \(a ^ { 2 } = 3 b ^ { 2 }\) Given also that the coordinates of the foci of the ellipse are the same as the coordinates of the foci of the hyperbola,
2. determine the equation of the hyperbola.

5

1. The transformation \(T _ { 1 }\) is defined by the matrix $$\left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right]$$ Describe this transformation geometrically.
2. The transformation \(T _ { 2 }\) is an anticlockwise rotation about the origin through an angle of \(60 ^ { \circ }\). Find the matrix of the transformation \(T _ { 2 }\). Use surds in your answer where appropriate.
   (3 marks)
3. Find the matrix of the transformation obtained by carrying out \(T _ { 1 }\) followed by \(T _ { 2 }\).
   (3 marks)

6 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \sqrt { 3 } & 3 \\ 3 & - \sqrt { 3 } \end{array} \right]$$

1. 1. Show that $$\mathbf { M } ^ { 2 } = p \mathbf { I }$$ where \(p\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   2. Show that the matrix \(\mathbf { M }\) can be written in the form $$q \left[ \begin{array} { c c } \cos 60 ^ { \circ } & \sin 60 ^ { \circ } \\ \sin 60 ^ { \circ } & - \cos 60 ^ { \circ } \end{array} \right]$$ where \(q\) is a real number. Give the value of \(q\) in surd form.
2. The matrix \(\mathbf { M }\) represents a combination of an enlargement and a reflection. Find:
   1. the scale factor of the enlargement;
   2. the equation of the mirror line of the reflection.
3. Describe fully the geometrical transformation represented by \(\mathbf { M } ^ { 4 }\).

6 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a rectangle \(R _ { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-4_652_1136_470_429}

1. The rectangle \(R _ { 1 }\) is mapped onto a second rectangle, \(R _ { 2 }\), by a transformation with matrix \(\left[ \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right]\).
   1. Calculate the coordinates of the vertices of the rectangle \(R _ { 2 }\).
   2. On Figure 1, draw the rectangle \(R _ { 2 }\).
2. The rectangle \(R _ { 2 }\) is rotated through \(90 ^ { \circ }\) clockwise about the origin to give a third rectangle, \(R _ { 3 }\).
   1. On Figure 1, draw the rectangle \(R _ { 3 }\).
   2. Write down the matrix of the rotation which maps \(R _ { 2 }\) onto \(R _ { 3 }\).
3. Find the matrix of the transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\).

7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a triangle with vertices \(A ( 1,1 ) , B ( 3,1 )\) and \(C ( 3,2 )\). \includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-04_1114_1141_552_360}

1. The triangle \(D E F\) is obtained by applying to triangle \(A B C\) the transformation T represented by the matrix $$\left[ \begin{array} { r r } 2 & 2 \\ - 2 & 2 \end{array} \right]$$
   1. Calculate the coordinates of \(D , E\) and \(F\).
   2. Draw the triangle \(D E F\) on Figure 1.
2. Given that T is a combination of an enlargement and a rotation, find the exact value of:
   1. the scale factor of the enlargement;
   2. the magnitude of the angle of the rotation.

5 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$

1. Find the matrix:
   1. \(\mathbf { M } ^ { 2 }\);
   2. \(\mathbf { M } ^ { 4 }\).
2. Describe fully the geometrical transformation represented by \(\mathbf { M }\).
3. Find the matrix \(\mathbf { M } ^ { 2006 }\).

3 Which of the following transformations is represented by the matrix \(\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\) ?
Tick ( \(\checkmark\) ) one box.
[0pt] [1 mark] Rotation of \(180 ^ { \circ }\) about the \(x\)-axis □ Reflection in the plane \(x = 0\) □ Rotation of \(180 ^ { \circ }\) about the \(y\)-axis □ Reflection in the plane \(y = 0\) □

14 The matrix \(\mathbf { M }\) represents the transformation T , and is given by $$\mathbf { M } = \left[ \begin{array} { c c } 3 & - 1 \\ - 2 & 6 \end{array} \right]$$ 14

1. The point \(A\) has coordinates ( \(4 , - 5\) )
   Find the coordinates of the image of \(A\) under T
   14
2. Show that the only invariant point under T is the origin.
   14
3. The line \(L _ { 1 }\) has equation \(y = x + 1\) The transformation \(T\) maps the line \(L _ { 1 }\) onto the line \(L _ { 2 }\) Find the equation of \(L _ { 2 }\) in the form \(y = m x + c\)

1 A reflection is represented by the matrix \(\left[ \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right]\) State the equation of the line of invariant points. Circle your answer.
[0pt] [1 mark] $$x = 0 \quad y = 0 \quad y = x \quad y = - x$$

4

1. Find the value of \(k\) for which matrix \(\mathbf { A }\) is singular. 4
2. Describe the transformation represented by matrix \(\mathbf { B }\). 4
3. (i) Given that \(\mathbf { A }\) and \(\mathbf { B }\) are both non-singular, verify that \(\mathbf { A } ^ { \mathbf { - 1 } } \mathbf { B } ^ { \mathbf { - 1 } } = ( \mathbf { B A } ) ^ { \mathbf { - 1 } }\).
   [0pt] [4 marks]
   4 (c) (ii) Prove the result \(\mathbf { M } ^ { - \mathbf { 1 } } \mathbf { N } ^ { - \mathbf { 1 } } = ( \mathbf { N M } ) ^ { - \mathbf { 1 } }\) for all non-singular square matrices \(\mathbf { M }\) and \(\mathbf { N }\) of the same size.
   [0pt] [4 marks]

13

1. The matrix A represents a reflection in the line \(y = m x\), where \(m\) is a constant. Show that \(\mathbf { A } = \left( \frac { 1 } { m ^ { 2 } + 1 } \right) \left[ \begin{array} { c c } 1 - m ^ { 2 } & 2 m \\ 2 m & m ^ { 2 } - 1 \end{array} \right]\) You may use the result in the formulae booklet. 13
2. \(\quad\) The matrix \(\mathbf { B }\) is defined as \(\mathbf { B } = \left[ \begin{array} { l l } 3 & 0 \\ 0 & 3 \end{array} \right]\) Show that \(( \mathbf { B A } ) ^ { 2 } = k \mathbf { I }\) where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix and \(k\) is an integer.
   13
3. (i) The diagram below shows a point \(P\) and the line \(y = m x\) Draw four lines on the diagram to demonstrate the result proved in part (b).
   Label as \(P ^ { \prime }\) the image of \(P\) under the transformation represented by (BA) \({ } ^ { 2 }\) \includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-20_579_1068_584_488} 13 (c) (ii) Explain how your completed diagram shows the result proved in part (b).
   13
4. The matrix \(\mathbf { C }\) is defined as \(\mathbf { C } = \left[ \begin{array} { c c } \frac { 12 } { 5 } & \frac { 9 } { 5 } \\ \frac { 9 } { 5 } & - \frac { 12 } { 5 } \end{array} \right]\) Find the value of \(m\) such that \(\mathbf { C } = \mathbf { B A }\) Fully justify your answer.
   [0pt] [4 marks]

9 Matrix \(\mathbf { R }\) is given by \(\mathbf { R } = \left( \begin{array} { c c c } a & 0 & - b \\ 0 & 1 & 0 \\ b & 0 & a \end{array} \right)\) where \(a\) and \(b\) are constants.

1. Find \(\mathbf { R } ^ { 2 }\) in terms of \(a\) and \(b\). The constants \(a\) and \(b\) are given by \(a = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } + 1 )\) and \(b = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } - 1 )\).
2. By determining exact expressions for \(a b\) and \(a ^ { 2 } - b ^ { 2 }\) and using the result from part (a), show that \(\mathbf { R } ^ { 2 } = k \left( \begin{array} { c c c } \sqrt { 3 } & 0 & - 1 \\ 0 & 2 & 0 \\ 1 & 0 & \sqrt { 3 } \end{array} \right)\) where \(k\) is a real number whose value is to be determined.
3. Find \(\mathbf { R } ^ { 6 } , \mathbf { R } ^ { 12 }\) and \(\mathbf { R } ^ { 24 }\).
4. Describe fully the transformation represented by \(\mathbf { R }\). \section*{END OF QUESTION PAPER}

5 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r l } - 1 & 0 \\ 0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)\).

1. Use \(\mathbf { A }\) and \(\mathbf { B }\) to disprove the proposition: "Matrix multiplication is commutative". Matrix \(\mathbf { B }\) represents the transformation \(\mathrm { T } _ { \mathrm { B } }\).
2. Describe the transformation \(\mathrm { T } _ { \mathrm { B } }\).
3. By considering the inverse transformation of \(\mathrm { T } _ { \mathrm { B } }\), determine \(\mathbf { B } ^ { - 1 }\). Matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 3 \end{array} \right)\) and represents the transformation \(\mathrm { T } _ { \mathrm { C } }\).
   The transformation \(\mathrm { T } _ { \mathrm { BC } }\) is transformation \(\mathrm { T } _ { \mathrm { C } }\) followed by transformation \(\mathrm { T } _ { \mathrm { B } }\).
   An object shape of area 5 is transformed by \(\mathrm { T } _ { \mathrm { BC } }\) to an image shape \(N\).
4. Determine the area of \(N\).

3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)\) where \(t\) is a constant.

1. Show that \(| \mathrm { A } | = | \mathrm { B } |\).
2. Verify that \(| \mathrm { AB } | = | \mathrm { A } | | \mathrm { B } |\).
3. Given that \(| \mathbf { A B } | = - 1\) explain what this means about the constant \(t\). The \(2 \times 2\) matrix \(A\) represents a transformation \(T\) which has the following properties.
   The transformation \(S\) is represented by the matrix \(B\) where \(B = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)\).
   (b) Find the equation of the line of invariant points of S .
   (c) Show that any line of the form \(y = x + c\) is an invariant line of S .

2
You are given the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { 4 }\).
2. Describe the transformation that A represents. The matrix \(\mathbf { B }\) represents a reflection in the plane \(x = 0\).
3. Write down the matrix \(B\). The point \(P\) has coordinates \(( 2,3,4 )\). The point \(P ^ { \prime }\) is the image of \(P\) under the transformation represented by \(\mathbf { B }\).
4. Find the coordinates of \(P ^ { \prime }\).

2 A 2-D transformation \(T\) is a shear which leaves the \(y\)-axis invariant and which transforms the object point \(( 2,1 )\) to the image point \(( 2,9 )\). \(A\) is the matrix which represents the transformation \(T\).

1. Find A .
2. By considering the determinant of A , explain why the area of a shape is invariant under T .

9 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)\).

1. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(M\) represents a reflection in a line through the origin.
2. Explain how your answer to part (i) relates to this information.
3. By investigating the invariant points of the reflection, find the equation of the mirror line.
4. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)\).
5. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
6. The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} \section*{MEI STRUCTURED MATHEMATICS
   4755
   \textbackslash section*\{Further Concepts For Advanced Mathematics (FP1)\}}

   Tuesday 7 JUNE 2005	Afternoon	1 hour 30 minutes
   Additional materials: Answer booklet Graph paper MEI Examination Formulae and Tables (MF2)

   TIME 1 hour 30 minutes
   • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
   • Answer all the questions.
   • You are permitted to use a graphical calculator in this paper.
   • The number of marks is given in brackets [ ] at the end of each question or part question.
   • You are advised that an answer may receive no marks unless you show sufficient defail of the working to indicate that a correct method is being used.
   • Final answers should be given to a degree of accuracy appropriate to the context.
   • The total number of marks for this paper is 72.

6

1. The set \(S\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n \in \mathbb { Z }\).
   1. Show that \(S\), under the operation of matrix multiplication, forms a group \(G\). [You may assume that matrix multiplication is associative.]
   2. State, giving a reason, whether \(G\) is abelian.
   3. The group \(H\) is the set \(\mathbb { Z }\) together with the operation of addition. Explain why \(G\) is isomorphic to \(H\).
   4. The plane transformation \(T\) is given by the matrix \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n\) is a non-zero integer. Describe \(T\) fully.

5

1. Write down the \(2 \times 2\) matrices which represent the following plane transformations:
   1. an anticlockwise rotation about the origin through an angle \(\alpha\);
   2. a reflection in the line \(y = x \tan \left( \frac { 1 } { 2 } \beta \right)\).
   3. A reflection in the \(x - y\) plane in the line \(y = x \tan \left( \frac { 1 } { 2 } \theta \right)\) is followed by a reflection in the line \(y = x \tan \left( \frac { 1 } { 2 } \phi \right)\). Show that the composition of these two reflections (in this order) is a rotation and describe this rotation fully.

9 The plane transformation \(T\) is the composition (in this order) of

• a reflection in the line \(y = x \tan \frac { 1 } { 8 } \pi\); followed by
• a shear parallel to the \(y\)-axis, mapping \(( 1,0 )\) to \(( 1,2 )\); followed by
• a clockwise rotation through \(\frac { 1 } { 4 } \pi\) radians about the origin; followed by
• a shear parallel to the \(x\)-axis, mapping \(( 0,1 )\) to \(( - 2,1 )\).

Determine the matrix \(\mathbf { M }\) which represents \(T\), and hence give a full geometrical description of \(T\) as a single plane transformation.

11

1. 1. Write down the matrix which represents a rotation through an angle \(\alpha\) anticlockwise about the origin.
   2. Show that the plane transformation given by the matrix $$\left( \begin{array} { c c } \cos \theta + \sin \theta & - ( \sin \theta - \cos \theta ) \\ \sin \theta - \cos \theta & \cos \theta + \sin \theta \end{array} \right)$$ is the composition of a rotation, \(R\), and a second transformation, \(S\). Describe both \(R\) and \(S\) fully.
2. 1. Write down the matrix which represents a reflection in the line \(y = x \tan \frac { 1 } { 2 } \beta\). For \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\), the plane transformation \(T\) is given by the matrix $$\left( \begin{array} { c c } 1 + \cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & - 1 - \cos 2 \theta \end{array} \right)$$
   2. Show that \(T\) is the composition of a reflection and an enlargement, and describe these transformations in full.
   3. Find also the values of \(\theta\) for which \(T\) is an area-preserving transformation.

The matrix \(\mathbf{M}\) represents the sequence of two transformations in the \(x\)-\(y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k\) (\(k \neq 0\)), followed by a shear, \(x\)-axis fixed, with \((0, 1)\) mapped to \((k, 1)\).

1. Show that \(\mathbf{M} = \begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}\). [4]
2. The transformation represented by \(\mathbf{M}\) has a line of invariant points. Find, in terms of \(k\), the equation of this line. [3]

The unit square \(S\) in the \(x\)-\(y\) plane is transformed by \(\mathbf{M}\) onto the parallelogram \(P\).

1. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\). [1]
2. Given that the area of \(P\) is \(3k^2\) units\(^2\), find the possible values of \(k\). [2]