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

7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\)
2. Write down the matrix \(\mathbf { Q }\). Given that the transformation \(V\) followed by the transformation \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
3. find the matrix \(\mathbf { R }\).
4. Show that there is a value of \(k\) for which the transformation \(T\) maps each point on the straight line \(y = k x\) onto itself, and state the value of \(k\). \section*{II}

10. In your answers to this question, the elements of each matrix should be expressed in exact form in surds where necessary. The transformation \(U\), represented by the \(2 \times 2\) matrix \(\mathbf { P }\), is a rotation through \(45 ^ { \circ }\) anticlockwise about the origin.

1. Write down the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through \(60 ^ { \circ }\) anticlockwise about the origin.
2. Write down the matrix \(\mathbf { Q }\). The transformation \(U\) followed by the transformation \(V\) is the transformation \(T\). The transformation \(T\) is represented by the matrix \(\mathbf { R }\).
3. Use your matrices from parts (a) and (b) to find the matrix \(\mathbf { R }\).
4. Give a full geometric description of \(T\) as a single transformation.
5. Deduce from your answers to parts (c) and (d) that \(\sin 75 ^ { \circ } = \frac { 1 + \sqrt { 3 } } { 2 \sqrt { 2 } }\) and find the
   exact value of \(\cos 75 ^ { \circ }\), explaining your answers fully.

6. (i) $$\mathbf { A } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$

1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a rotation of \(45 ^ { \circ }\) clockwise about the origin.
2. Write down the matrix \(\mathbf { B }\), giving each element of the matrix in exact form. The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\).
3. Determine \(\mathbf { C }\).
   (ii) The trapezium \(T\) has vertices at the points \(( - 2,0 ) , ( - 2 , k ) , ( 5,8 )\) and \(( 5,0 )\), where \(k\) is a positive constant. Trapezium \(T\) is transformed onto the trapezium \(T ^ { \prime }\) by the matrix $$\left( \begin{array} { r r } 5 & 1 \\ - 2 & 3 \end{array} \right)$$ Given that the area of trapezium \(T ^ { \prime }\) is 510 square units, calculate the exact value of \(k\).

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

1. Determine the matrix \(\mathbf { A } ^ { 2 }\)
2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\)
3. Hence determine the smallest positive integer value of \(n\) for which \(\mathbf { A } ^ { n } = \mathbf { I }\) The matrix \(\mathbf { B }\) represents a stretch scale factor 4 parallel to the \(x\)-axis.
4. Write down the matrix \(\mathbf { B }\) The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\)
5. Determine the matrix \(\mathbf { C }\) The parallelogram \(P\) is transformed onto the parallelogram \(P ^ { \prime }\) by the matrix \(\mathbf { C }\)
6. Given that the area of parallelogram \(P ^ { \prime }\) is 20 square units, determine the area of parallelogram \(P\)

1. 1. The matrix \(\mathbf { A }\) is defined by

$$\mathbf { A } = \left( \begin{array} { c c } 3 k & 4 k - 1 \\ 2 & 6 \end{array} \right)$$ where \(k\) is a constant.

1. Determine the value of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
2. determine \(\mathbf { A } ^ { - 1 }\) in terms of \(k\), giving your answer in simplest form.
   (ii) The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left( \begin{array} { l l } p & 0 \\ 0 & q \end{array} \right)$$ where \(p\) and \(q\) are integers.
   State the value of \(p\) and the value of \(q\) when \(\mathbf { B }\) represents
3. an enlargement about the origin with scale factor - 2
4. a reflection in the \(y\)-axis.

7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\)
2. Write down the matrix \(\mathbf { Q }\). Given that the transformation \(V\) followed by the transformation \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
3. find the matrix \(\mathbf { R }\).
4. Show that there is a value of \(k\) for which the transformation \(T\) maps each point on the straight line \(y = k x\) onto itself, and state the value of \(k\).
   
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8. (i) The transformation \(U\) is represented by the matrix \(\mathbf { P }\) where, $$P = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\ \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$

1. Describe fully the transformation \(U\). The transformation \(V\), represented by the matrix \(\mathbf { Q }\), is a stretch scale factor 3 parallel to the \(x\)-axis.
2. Write down the matrix \(\mathbf { Q }\). Transformation \(U\) followed by transformation \(V\) is a transformation which is represented by matrix \(\mathbf { R }\).
3. Find the matrix \(\mathbf { R }\).
   (ii) $$S = \left( \begin{array} { r r } 1 & - 3 \\ 3 & 1 \end{array} \right)$$ Given that the matrix \(\mathbf { S }\) represents an enlargement, with a positive scale factor and centre \(( 0,0 )\), followed by a rotation with centre \(( 0,0 )\),
   1. find the scale factor of the enlargement,
   2. find the angle and direction of rotation, giving your answer in degrees to 1 decimal place.

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

1. Describe fully the transformations described by each of the matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\). It is given that the matrix \(\mathbf { D } = \mathbf { C A }\), and that the matrix \(\mathbf { E } = \mathbf { D B }\).
2. Find \(\mathbf { D }\).
3. Show that \(\mathbf { E } = \left( \begin{array} { c c } - 3 & 3 \\ 3 & 3 \end{array} \right)\). The triangle \(O R S\) has vertices at the points with coordinates \(( 0,0 ) , ( - 15,15 )\) and \(( 4,21 )\). This triangle is transformed onto the triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation described by \(\mathbf { E }\).
4. Find the coordinates of the vertices of triangle \(O R ^ { \prime } S ^ { \prime }\).
5. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\) and deduce the area of triangle \(O R S\).

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

1. Describe fully the geometrical transformation represented by the matrix \(\mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(A\) with coordinates \(( p , q )\) onto the point \(B\) with coordinates \(( 3 \sqrt { } 2,4 \sqrt { } 2 )\).
2. Find the value of \(p\) and the value of \(q\).
3. Find, in its simplest surd form, the length \(O A\), where \(O\) is the origin.
4. Find \(\mathbf { M } ^ { 2 }\). The point \(B\) is mapped onto the point \(C\) by the transformation represented by \(\mathbf { M } ^ { 2 }\).
5. Find the coordinates of \(C\).

2. $$\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 5 & 3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 3 & - 1 \\ 5 & 2 \end{array} \right)$$

1. Find \(\mathbf { A B }\). Given that $$\mathbf { C } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)$$
2. describe fully the geometrical transformation represented by \(\mathbf { C }\),
3. write down \(\mathbf { C } ^ { 100 }\). \includegraphics[max width=\textwidth, alt={}, center]{d20fa710-2d91-4ac2-adbc-46ccdcb93380-03_99_97_2631_1784}

4. A right angled triangle \(T\) has vertices \(A ( 1,1 ) , B ( 2,1 )\) and \(C ( 2,4 )\). When \(T\) is transformed by the matrix \(\mathbf { P } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\), the image is \(T ^ { \prime }\).

1. Find the coordinates of the vertices of \(T ^ { \prime }\).
2. Describe fully the transformation represented by \(\mathbf { P }\). The matrices \(\mathbf { Q } = \left( \begin{array} { c c } 4 & - 2 \\ 3 & - 1 \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)\) represent two transformations. When \(T\) is transformed by the matrix \(\mathbf { Q R }\), the image is \(T ^ { \prime \prime }\).
3. Find \(\mathbf { Q R }\).
4. Find the determinant of \(\mathbf { Q R }\).
5. Using your answer to part (d), find the area of \(T ^ { \prime \prime }\).

4. The transformation \(U\), represented by the \(2 \times 2\) matrix \(\mathbf { P }\), is a rotation through \(90 ^ { \circ }\) anticlockwise about the origin.

1. Write down the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line \(y = - x\).
2. Write down the matrix \(\mathbf { Q }\). Given that \(U\) followed by \(V\) is transformation \(T\), which is represented by the matrix \(\mathbf { R }\), (c) express \(\mathbf { R }\) in terms of \(\mathbf { P }\) and \(\mathbf { Q }\),
   (d) find the matrix \(\mathbf { R }\),
   (e) give a full geometrical description of \(T\) as a single transformation.

6. \(\mathbf { X } = \left( \begin{array} { l l } 1 & a \\ 3 & 2 \end{array} \right)\), where \(a\) is a constant.

1. Find the value of \(a\) for which the matrix \(\mathbf { X }\) is singular. $$\mathbf { Y } = \left( \begin{array} { r r } 1 & - 1 \\ 3 & 2 \end{array} \right)$$
2. Find \(\mathbf { Y } ^ { - 1 }\). The transformation represented by \(\mathbf { Y }\) maps the point \(A\) onto the point \(B\).
   Given that \(B\) has coordinates ( \(1 - \lambda , 7 \lambda - 2\) ), where \(\lambda\) is a constant,
3. find, in terms of \(\lambda\), the coordinates of point \(A\).

5. \(\mathbf { R } = \left( \begin{array} { l l } a & 2 \\ a & b \end{array} \right)\), where \(a\) and \(b\) are constants and \(a > 0\).

1. Find \(\mathbf { R } ^ { 2 }\) in terms of \(a\) and \(b\). Given that \(\mathbf { R } ^ { 2 }\) represents an enlargement with centre ( 0,0 ) and scale factor 15 ,
2. find the value of \(a\) and the value of \(b\).

6. Write down the \(2 \times 2\) matrix that represents

1. an enlargement with centre \(( 0,0 )\) and scale factor 8 ,
2. a reflection in the \(x\)-axis. Hence, or otherwise,
3. find the matrix \(\mathbf { T }\) that represents an enlargement with centre ( 0,0 ) and scale factor 8, followed by a reflection in the \(x\)-axis. $$\mathbf { A } = \left( \begin{array} { l l } 6 & 1 \\ 4 & 2 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } k & 1 \\ c & - 6 \end{array} \right) , \text { where } k \text { and } c \text { are constants. }$$
4. Find \(\mathbf { A B }\). Given that \(\mathbf { A B }\) represents the same transformation as \(\mathbf { T }\),
5. find the value of \(k\) and the value of \(c\).

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

1. find \(\mathbf { A } ^ { 2 }\),
2. describe fully the geometrical transformation represented by \(\mathbf { A } ^ { 2 }\).
   (b) Given that $$\mathbf { B } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$ describe fully the geometrical transformation represented by \(\mathbf { B }\).
   (c) Given that $$\mathbf { C } = \left( \begin{array} { c c } k + 1 & 12 \\ k & 9 \end{array} \right)$$ where \(k\) is a constant, find the value of \(k\) for which the matrix \(\mathbf { C }\) is singular.

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

1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents an enlargement, scale factor - 2 , with centre the origin.
2. Write down the matrix \(\mathbf { B }\).
   (ii) $$\mathbf { M } = \left( \begin{array} { c c } 3 & k \\ - 2 & 3 \end{array} \right) , \quad \text { where } k \text { is a positive constant. }$$ Triangle \(T\) has an area of 16 square units. Triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the transformation represented by the matrix M. Given that the area of the triangle \(T ^ { \prime }\) is 224 square units, find the value of \(k\).

7. (i) In each of the following cases, find a \(2 \times 2\) matrix that represents

1. a reflection in the line \(y = - x\),
2. a rotation of \(135 ^ { \circ }\) anticlockwise about \(( 0,0 )\),
3. a reflection in the line \(y = - x\) followed by a rotation of \(135 ^ { \circ }\) anticlockwise about \(( 0,0 )\).
   (ii) The triangle \(T\) has vertices at the points \(( 1 , k ) , ( 3,0 )\) and \(( 11,0 )\), where \(k\) is a constant. Triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the matrix $$\left( \begin{array} { r r } 6 & - 2 \\ 1 & 2 \end{array} \right)$$ Given that the area of triangle \(T ^ { \prime }\) is 364 square units, find the value of \(k\).

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

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(U\) maps the point \(A\), with coordinates \(( p , q )\), onto the point \(B\), with coordinates \(( 6 \sqrt { } 2,3 \sqrt { } 2 )\).
2. Find the value of \(p\) and the value of \(q\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\).
3. Write down the matrix \(\mathbf { Q }\). The transformation \(U\) followed by the transformation \(V\) is the transformation \(T\). The transformation \(T\) is represented by the matrix \(\mathbf { R }\).
4. Find the matrix \(\mathbf { R }\).
5. Deduce that the transformation \(T\) is self-inverse.

3. (i) Given that $$\mathbf { A } = \left( \begin{array} { r r } - 2 & 3 \\ 1 & 1 \end{array} \right) , \quad \mathbf { A } \mathbf { B } = \left( \begin{array} { r r r } - 1 & 5 & 12 \\ 3 & - 5 & - 1 \end{array} \right)$$

1. find \(\mathbf { A } ^ { - 1 }\)
2. Hence, or otherwise, find the matrix \(\mathbf { B }\), giving your answer in its simplest form.
   (ii) Given that $$\mathbf { C } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$
   1. describe fully the single geometrical transformation represented by the matrix \(\mathbf { C }\).
   2. Hence find the matrix \(\mathbf { C } ^ { 39 }\)

1. $$\mathbf { R } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right) \text { and } \mathbf { S } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$

1. Find \(\mathbf { R } ^ { 2 }\).
2. Find \(\mathbf { R S }\).
3. Describe the geometrical transformation represented by \(\mathbf { R S }\).

2. The rectangle \(R\) has vertices at the points \(( 0,0 ) , ( 1,0 ) , ( 1,2 )\) and \(( 0,2 )\).

1. Find the coordinates of the vertices of the image of \(R\) under the transformation given by the matrix \(\mathbf { A } = \left( \begin{array} { c c } a & 4 \\ - 1 & 1 \end{array} \right)\), where \(a\) is a constant.
2. Find det \(\mathbf { A }\), giving your answer in terms of \(a\). Given that the area of the image of \(R\) is 18 ,
3. find the value of \(a\).

3. The matrix \(\mathbf { R }\) is given by \(\mathbf { R } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right)\)

1. Find \(\mathbf { R } ^ { 2 }\).
2. Describe the geometrical transformation represented by \(\mathbf { R } ^ { 2 }\).
3. Describe the geometrical transformation represented by \(\mathbf { R }\).