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

6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$ Describe fully the geometrical transformation represented by each of the following matrices:

1. A ;
2. B ;
3. \(\quad \mathbf { A } ^ { 2 }\);
4. \(\quad \mathbf { B } ^ { 2 }\);
5. AB.
   □

7 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left[ \begin{array} { c c } - 1 & - \sqrt { 3 } \\ \sqrt { 3 } & - 1 \end{array} \right]$$

1. 1. Calculate the matrix \(\mathbf { A } ^ { 2 }\).
   2. Show that \(\mathbf { A } ^ { 3 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
2. Describe the single geometrical transformation, or combination of two geometrical transformations, corresponding to each of the matrices:
   1. \(\mathrm { A } ^ { 3 }\);
   2. A.

6

1. Using surd forms, find the matrix of a rotation about the origin through \(135 ^ { \circ }\) anticlockwise.
2. The matrix \(\mathbf { M }\) is defined by \(\mathbf { M } = \left[ \begin{array} { r r } - 1 & - 1 \\ 1 & - 1 \end{array} \right]\).
   1. Given that \(\mathbf { M }\) represents an enlargement followed by a rotation, find the scale factor of the enlargement and the angle of the rotation.
   2. The matrix \(\mathbf { M } ^ { 2 }\) also represents an enlargement followed by a rotation. State the scale factor of the enlargement and the angle of the rotation.
   3. Show that \(\mathbf { M } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   4. Deduce that \(\mathbf { M } ^ { 2012 } = - 2 ^ { n } \mathbf { I }\) for some positive integer \(n\).

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)

5

1. The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { c c } - 2 & c \\ d & 3 \end{array} \right]\).
   Given that the image of the point \(( 5,2 )\) under the transformation represented by \(\mathbf { A }\) is \(( - 2,1 )\), find the value of \(c\) and the value of \(d\).
   [0pt] [4 marks]
2. The matrix \(\mathbf { B }\) is defined by \(\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 2 } & \sqrt { 2 } \\ - \sqrt { 2 } & \sqrt { 2 } \end{array} \right]\).
   1. Show that \(\mathbf { B } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   2. Describe the transformation represented by the matrix \(\mathbf { B }\) as a combination of two geometrical transformations.
   3. Find the matrix \(\mathbf { B } ^ { 17 }\). \(6 \quad \mathrm {~A}\) curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$$

8 A transformation T of the plane has matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } \cos \theta & 2 \cos \theta - \sin \theta \\ \sin \theta & 2 \sin \theta + \cos \theta \end{array} \right)\).

1. Show that T leaves areas unchanged for all values of \(\theta\).
2. Find the value of \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), for which the \(y\)-axis is an invariant line of T . The matrix \(\mathbf { N }\) is \(\left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).
3. 1. Find \(\mathbf { M N } ^ { - 1 }\).
   2. Hence describe fully a sequence of two transformations of the plane that is equivalent to T . \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
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1 The transformation R of the plane is reflection in the line \(x = 0\).

1. Write down the matrix \(\mathbf { M }\) associated with R .
2. Find \(\mathbf { M } ^ { 2 }\).
3. Interpret the result of part (b) in terms of the transformation \(R\).

8

1. The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\).
   1. Find \(\mathbf { M } ^ { 2 }\).
   2. Write down the transformation represented by \(\mathbf { M }\).
   3. Hence state the geometrical significance of the result of part (i).
2. The matrix \(\mathbf { N }\) is \(\left( \begin{array} { c c } k + 1 & 0 \\ k & k + 2 \end{array} \right)\), where \(k\) is a constant. Using determinants, investigate whether \(\mathbf { N }\) can represent a reflection.

6 A transformation T of the plane has associated matrix \(\mathbf { M } = \left( \begin{array} { c c } 1 & \lambda + 1 \\ \lambda - 1 & - 1 \end{array} \right)\), where \(\lambda\) is a non-zero
constant.

1. 1. Show that T reverses orientation.
   2. State, in terms of \(\lambda\), the area scale factor of T .
2. 1. Show that \(\mathbf { M } ^ { 2 } - \lambda ^ { 2 } \mathbf { I } = \mathbf { 0 }\).
   2. Hence specify the transformation equivalent to two applications of T .
3. In the case where \(\lambda = 1 , \mathrm {~T}\) is equivalent to a transformation S followed by a reflection in the \(x\)-axis.
   1. Determine the matrix associated with S .
   2. Hence describe the transformation S .

11

1. Specify fully the transformations represented by the following matrices.

6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) respectively.

1. In this question you must show detailed reasoning. Determine whether \(\mathbf { M }\) and \(\mathbf { N }\) commute under matrix multiplication.
2. Specify the transformation of the plane associated with each of the following matrices.
   1. M
   2. N
3. State the significance of the result in part (a) for the transformations associated with \(\mathbf { M }\) and \(\mathbf { N }\). [1]
4. Use an algebraic method to show that all lines parallel to the \(x\)-axis are invariant lines of the transformation associated with N.

8

1. Specify fully the transformation T of the plane associated with the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } 1 & \lambda \\ 0 & 1 \end{array} \right)\) and \(\lambda\) is a non-zero constant.
2. 1. Find detM.
   2. Deduce two properties of the transformation T from the value of detM.
3. Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & n \lambda \\ 0 & 1 \end{array} \right)\), where \(n\) is a positive integer.
4. Hence specify fully a single transformation which is equivalent to \(n\) applications of the transformation T.

9 The transformation Too the plane has associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } - 1 & 0 \\ - 2 & 1 \end{array} \right)\).

1. On the grid in the Printed Answer Booklet, plot the image \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) of the unit square OABC under the transformation T.
2. 1. Calculate the value of \(\operatorname { det } \mathbf { M }\).
   2. Explain the significance of the value of \(\operatorname { det } \mathbf { M }\) in relation to the image \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\).
3. T is equivalent to a sequence of two transformations of the plane.
   1. Specify fully two transformations equivalent to T .
   2. Use matrices to verify your answer.

2. (a) The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left( \begin{array} { c c } 3 & 4 \\ - 1 & - 2 \end{array} \right) , \quad \mathbf { B } = \binom { - 11 } { 7 }$$ Given that \(\mathbf { A X } = \mathbf { B }\), find the matrix \(\mathbf { X }\).
(b) (i) Find the \(2 \times 2\) matrix, \(\mathbf { T }\), which represents a reflection in the line \(y = - 2 x\).
(ii) The images of the points \(C ( 2,7 )\) and \(D ( 3,1 )\), under \(\mathbf { T }\), are \(E\) and \(F\) respectively. Find the coordinates of the midpoint of \(E F\).

4. The transformation \(T\) in the plane consists of a translation in which the point \(( x , y )\) is transformed to the point ( \(x + 2 , y - 2\) ), followed by a reflection in the line \(y = x\).

1. Determine the \(3 \times 3\) matrix which represents \(T\).
2. Determine how many invariant points exist under the transformation \(T\).

1. A point \(P\) is reflected in the line \(y = k x\), where \(k\) is a constant. It is then rotated anticlockwise about \(O\) through an acute angle \(\theta\), where \(\cos \theta = 0 \cdot 8\). The resulting transformation matrix is given by \(T\), where

$$T = \frac { 1 } { 85 } \left[ \begin{array} { r r } - 84 & - 13 \\ - 13 & 84 \end{array} \right]$$

1. Determine the value of \(k\).
   Find the invariant points of \(T\).

5. $$\mathbf { A } = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\ \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { A }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { B }\), is a reflection in the line \(y = - x\)
2. Write down the matrix \(\mathbf { B }\). Given that \(U\) followed by \(V\) is the transformation \(T\), which is represented by the matrix \(\mathbf { C }\), (c) find the matrix \(\mathbf { C }\).
   (d) Show that there is a real number \(k\) for which the point \(( 1 , k )\) is invariant under \(T\).

   V349 SIHI NI IMIMM ION OC

   VJYV SIHIL NI LIIIM ION OO

   VJYV SIHIL NI JIIYM ION OC

$$\mathbf { A } = \left( \begin{array} { c c } 2 & a \\ a - 4 & b \end{array} \right)$$ where \(a\) and \(b\) are non-zero constants.
Given that the matrix \(\mathbf { A }\) is self-inverse,

1. determine the value of \(b\) and the possible values for \(a\). The matrix \(\mathbf { A }\) represents a linear transformation \(M\).
   Using the smaller value of \(a\) from part (a),
2. show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line.
   (ii) $$\mathbf { P } = \left( \begin{array} { c c } p & 2 p \\ - 1 & 3 p \end{array} \right)$$ where \(p\) is a positive constant.
   The matrix \(\mathbf { P }\) represents a linear transformation \(U\).
   The triangle \(T\) has vertices at the points with coordinates ( 1,2 ), ( 3,2 ) and ( 2,5 ). The area of the image of \(T\) under the linear transformation \(U\) is 15
3. Determine the value of \(p\). The transformation \(V\) consists of a stretch scale factor 3 parallel to the \(x\)-axis with the \(y\)-axis invariant followed by a stretch scale factor - 2 parallel to the \(y\)-axis with the \(x\)-axis invariant. The transformation \(V\) is represented by the matrix \(\mathbf { Q }\).
4. Write down the matrix \(\mathbf { Q }\). Given that \(U\) followed by \(V\) is the transformation \(W\), which is represented by the matrix \(\mathbf { R }\), (c) find the matrix \(\mathbf { R }\).

1. $$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$

1. 1. Describe fully the single geometrical transformation \(P\) represented by the matrix \(\mathbf { P }\).
   2. Describe fully the single geometrical transformation \(Q\) represented by the matrix \(\mathbf { Q }\). The transformation \(P\) followed by the transformation \(Q\) is the transformation \(R\), which is represented by the matrix \(\mathbf { R }\).
2. Determine \(\mathbf { R }\).
3. 1. Evaluate the determinant of \(\mathbf { R }\).
   2. Explain how the value obtained in (c)(i) relates to the transformation \(R\).

3. $$\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ 0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$

1. Describe fully the single geometric transformation \(A\) represented by the matrix \(\mathbf { A }\). $$\mathbf { B } = \left( \begin{array} { c c c } 1 & 3 & 0 \\ \sqrt { 3 } & 0 & 5 \sqrt { 3 } \\ 1 & 2 & 0 \end{array} \right)$$ The transformation \(B\) is represented by the matrix \(\mathbf { B }\).
   The transformation \(A\) followed by the transformation \(B\) is the transformation \(C\), which is represented by the matrix \(\mathbf { C }\). To determine matrix \(\mathbf { C }\), a student attempts the following matrix multiplication. $$\left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ 0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right) \left( \begin{array} { c c c } 1 & 3 & 0 \\ \sqrt { 3 } & 0 & 5 \sqrt { 3 } \\ 1 & 2 & 0 \end{array} \right)$$
2. State the error made by the student.
3. Determine the correct matrix \(\mathbf { C }\).

1. (i)

$$\mathbf { P } = \left( \begin{array} { r r r } k & - 2 & 7 \\ - 3 & - 5 & 2 \\ k & k & 4 \end{array} \right)$$ where \(k\) is a constant Show that \(\mathbf { P }\) is non-singular for all real values of \(k\).
(ii) $$\mathbf { Q } = \left( \begin{array} { r r } 2 & - 1 \\ - 3 & 0 \end{array} \right)$$ The matrix \(\mathbf { Q }\) represents a linear transformation \(T\) Under \(T\), the point \(A ( a , 2 )\) and the point \(B ( 4 , - a )\), where \(a\) is a constant, are transformed to the points \(A ^ { \prime }\) and \(B ^ { \prime }\) respectively. Given that the distance \(A ^ { \prime } B ^ { \prime }\) is \(\sqrt { 58 }\), determine the possible values of \(a\).

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 16 } = 1$$ The points \(S\) and \(S ^ { \prime }\) are the foci of \(E\).

1. Find the coordinates of \(S\) and \(S ^ { \prime }\)
2. Show that for any point \(P\) on \(E\), the triangle \(P S S ^ { \prime }\) has constant perimeter and determine its value.

1. The points \(P \left( 9 p ^ { 2 } , 18 p \right)\) and \(Q \left( 9 q ^ { 2 } , 18 q \right) , p \neq q\), lie on the parabola \(C\) with equation

$$y ^ { 2 } = 36 x$$ The line \(l\) passes through the points \(P\) and \(Q\)

1. Show that an equation for the line \(l\) is $$( p + q ) y = 2 ( x + 9 p q )$$ The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(A\).
2. Show that the coordinates of \(A\) are $$\left( 9 \left( p ^ { 2 } + q ^ { 2 } + p q + 2 \right) , - 9 p q ( p + q ) \right)$$ Given that the points \(P\) and \(Q\) vary such that \(l\) always passes through the point \(( 12,0 )\)
3. find, in the form \(y ^ { 2 } = \mathrm { f } ( x )\), an equation for the locus of \(A\), giving \(\mathrm { f } ( x )\) in simplest form.

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$$ Find

1. the coordinates of the foci of \(E\),
2. the equations of the directrices of \(E\).

1. The parabola \(C\) has equation

$$y ^ { 2 } = 32 x$$ and the hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 9 } = 1$$

1. Write down the equations of the asymptotes of \(H\). The line \(l _ { 1 }\) is normal to \(C\) and parallel to the asymptote of \(H\) with positive gradient. The line \(l _ { 2 }\) is normal to \(C\) and parallel to the asymptote of \(H\) with negative gradient.
2. Determine
   1. an equation for \(l _ { 1 }\)
   2. an equation for \(l _ { 2 }\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet \(H\) at the points \(P\) and \(Q\) respectively.
3. Find the area of the triangle \(O P Q\), where \(O\) is the origin.