4.03g Invariant points and lines

6 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \frac { 1 } { 13 } \left( \begin{array} { r r } 5 & 12 \\ 12 & - 5 \end{array} \right)\). You are given that \(\mathbf { A }\) represents the transformation T which is a reflection in a certain straight line. You are also given that this straight line, the mirror line, passes through the origin, \(O\).

1. Explain why there must be a line of invariant points for T . State the geometric significance of this line.
2. By considering the line of invariant points for T , determine the equation of the mirror line. Give your answer in the form \(y = m x + c\). The coordinates of the point \(P\) are \(( 1,5 )\).
3. By considering the image of \(P\) under the transformation T , or otherwise, determine the coordinates of the point on the mirror line which is closest to \(P\).
4. The line with equation \(y = a x + 2\) is an invariant line for T. Determine the value of \(a\).

5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - \frac { 3 } { 5 } & \frac { 4 } { 5 } \\ \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)\).

1. The diagram in the Printed Answer Booklet shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { M }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and clearly label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
2. Find the equation of the line of invariant points of this transformation.
3. (a) Find the determinant of \(\mathbf { M }\).
   (b) Describe briefly how this value relates to the transformation represented by \(\mathbf { M }\).

6 A linear transformation \(T\) of the \(x - y\) plane has an associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { c c } \lambda & k \\ 1 & \lambda - k \end{array} \right)\), and \(\lambda\) and \(k\) are real constants. and \(k\) are real constants.

1. You are given that \(\operatorname { det } \mathbf { M } > 0\) for all values of \(\lambda\).
   1. Find the range of possible values of \(k\).
   2. What is the significance of the condition \(\operatorname { det } \mathbf { M } > 0\) for the transformation T? For the remainder of this question, take \(k = - 2\).
2. Determine whether there are any lines through the origin that are invariant lines for the transformation T.
3. The transformation T is applied to a triangle with area 3 units \({ } ^ { 2 }\). The area of the resulting image triangle is 15 units \({ } ^ { 2 }\).
   Find the possible values of \(\lambda\).

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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9 A transformation T of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { c c } k + 1 & - 1 \\ 1 & k \end{array} \right)\), where \(k\) is a
constant. constant. Show that, for all values of \(k , \mathrm {~T}\) has no invariant lines through the origin.

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.

9 A linear transformation of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 1 & - 2 \\ \lambda & 3 \end{array} \right)\), where \(\lambda\) is a
constant. constant.

1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin.
2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines.

6 Given that \(y = m x\) is an invariant line of the transformation with matrix \(\left( \begin{array} { r r } 1 & 2 \\ 2 & - 2 \end{array} \right)\), determine the possible values of \(m\). Section B (113 marks)
Answer all the questions.

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

1. $$\mathbf { M } = \left( \begin{array} { l l } 4 & - 5 \\ 2 & - 7 \end{array} \right)$$

1. Show that the matrix \(\mathbf { M }\) is non-singular. The transformation \(T\) of the plane is represented by the matrix \(\mathbf { M }\).
   The triangle \(R\) is transformed to the triangle \(S\) by the transformation \(T\).
   Given that the area of \(S\) is 63 square units,
2. find the area of \(R\).
3. Show that the line \(y = 2 x\) is invariant under the transformation \(T\).

$$\mathbf { A } = \left( \begin{array} { c c } 2 & a \\ a - 4 & b \end{array} \right)$$ 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. A linear transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix

$$\mathbf { M } = \left( \begin{array} { c c } 5 & 1 \\ k & - 3 \end{array} \right)$$ where \(k\) is a constant.
Given that matrix \(\mathbf { M }\) has a repeated eigenvalue,

1. determine
   1. the value of \(k\)
   2. the eigenvalue.
2. Hence determine a Cartesian equation of the invariant line under \(T\).

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$\mathbf { A } = \left( \begin{array} { r r } 3 & k \\ - 5 & 2 \end{array} \right)$$ where \(k\) is a constant.
Given that there exists a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix where $$\mathbf { P } ^ { - 1 } \mathbf { A } \mathbf { P } = \left( \begin{array} { r r } 8 & 0 \\ 0 & - 3 \end{array} \right)$$

1. show that \(k = - 6\)
2. determine a suitable matrix \(\mathbf { P }\)

1. Given that

$$A = \left( \begin{array} { l l } 3 & 1 \\ 6 & 4 \end{array} \right)$$

1. find the characteristic equation of the matrix \(\mathbf { A }\).
2. Hence show that \(\mathbf { A } ^ { 3 } = 43 \mathbf { A } - 42 \mathbf { I }\).

3. $$\mathbf { M } = \left( \begin{array} { r r } - 2 & 5 \\ 6 & k \end{array} \right)$$ where \(k\) is a constant.
Given that $$\mathbf { M } ^ { 2 } + 11 \mathbf { M } = a \mathbf { I }$$ where \(a\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,

1. 1. determine the value of \(a\)
   2. show that \(k = - 9\)
2. Determine the equations of the invariant lines of the transformation represented by \(\mathbf { M }\).
3. State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer.

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)\).

1. Find the value of \(a\) such that \(\mathbf { A B } = \mathbf { B A }\).
2. Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
3. A triangle of area 4 square units is transformed by the matrix B. Find the area of the image of the triangle following this transformation.
4. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).

5 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & a ^ { 2 } & 0 \\ 0 & 0 & 1 \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c c } 0.6 & b & 0 \\ - b & 0.6 & 0 \\ 0 & 0 & 1 \end{array} \right)\).

1. \(\mathbf { A }\) represents a reflection. Write down the value of \(\operatorname { det } \mathbf { A }\).
2. Hence find the possible values of \(a\).
3. \(\mathbf { r }\) is the position vector of a point \(R\). Given that \(\mathbf { A r } = \mathbf { r }\) describe the location of \(R\).
4. \(\mathbf { B }\) represents a rotation. Write down the value of \(\operatorname { det } \mathbf { B }\).
5. Hence find the possible values of \(b\).

4 The point \(( 2 , - 1 )\) is invariant under the transformation represented by the matrix \(\mathbf { N }\) Which of the following matrices could be \(\mathbf { N }\) ? Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { l l } 4 & 6 \\ 2 & 5 \end{array} \right]\) \(\left[ \begin{array} { l l } 6 & 5 \\ 4 & 2 \end{array} \right]\) \(\left[ \begin{array} { l l } 5 & 2 \\ 6 & 4 \end{array} \right]\) \(\left[ \begin{array} { l l } 2 & 4 \\ 5 & 6 \end{array} \right]\)

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 \(y = m x + c\)

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.

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]

Find two invariant points under the transformation given by \(\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\) [2 marks]