Find line of invariant points

3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } 1 & 0 \\ 0 & k \end{array} \right) \left( \begin{array} { c c } 1 & 0 \\ k & 1 \end{array} \right)\), where \(k\) is a constant and \(k \neq 0\) or 1 .

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(\mathrm { y } = \frac { \mathrm { k } ^ { 2 } } { 1 - \mathrm { k } } \mathrm { x }\). [4]
4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Find the value of \(k\) for which the area of triangle \(D E F\) is equal to the area of triangle \(A B C\).

1 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 \(\mathbf { M } = \left( \begin{array} { c c } k & k \\ 0 & 1 \end{array} \right)\).
2. The transformation represented by \(\mathbf { M }\) has a line of invariant points. Find, in terms of \(k\), the equation of this line. \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-02_2722_43_107_2005} The unit square \(S\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto the parallelogram \(P\).
3. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\).
4. Given that the area of \(P\) is \(3 k ^ { 2 }\) units \({ } ^ { 2 }\), find the possible values of \(k\).

1 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 \(\mathbf { M } = \left( \begin{array} { c c } k & k \\ 0 & 1 \end{array} \right)\).
2. The transformation represented by \(\mathbf { M }\) has a line of invariant points. Find, in terms of \(k\), the equation of this line.
   The unit square \(S\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto the parallelogram \(P\).
3. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\).
4. Given that the area of \(P\) is \(3 k ^ { 2 }\) units \({ } ^ { 2 }\), find the possible values of \(k\).

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}

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 ( \(\mathbf { v }\) ) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?

3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 3 & - 1 \\ 2 & 0 \end{array} \right)\).

3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } - 1 & - 1 \\ 2 & 2 \end{array} \right)\).

7 Find two invariant points under the transformation given by \(\left[ \begin{array} { l l } 2 & 3 \\ 1 & 4 \end{array} \right]\) \(82 - 3 \mathrm { i }\) is one root of the equation $$z ^ { 3 } + m z + 52 = 0$$ where \(m\) is real.

9 The matrix \(\mathbf { M }\) represents the transformation T and is given by $$\mathbf { M } = \left[ \begin{array} { c c } 3 p + 1 & 12 \\ p + 2 & p ^ { 2 } - 3 \end{array} \right]$$ 9

1. In the case when \(p = 0\) show that the image of the point \(( 4,5 )\) under T is the point \(( 64 , - 7 )\) 9
2. In the case when \(p = - 2\) find the gradient of the line of invariant points under \(T\) 9
3. Show that \(p = 3\) is the only real value of \(p\) for which \(\mathbf { M }\) is singular.
   The curve \(C\) has equation $$y = \frac { 3 x ^ { 2 } + m x + p } { x ^ { 2 } + p x + m }$$ where \(m\) and \(p\) are integers.
   The vertical asymptotes of \(C\) are \(x = - 4\) and \(x = - 1\) The curve \(C\) is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-12_867_1102_733_463}

8 The \(2 \times 2\) matrix A represents a transformation T which has the following properties.

• The image of the point \(( 0,1 )\) is the point \(( 3,4 )\).
• An object shape whose area is 7 is transformed to an image shape whose area is 35 .
• T has a line of invariant points.
  1. Find a possible matrix for \(\mathbf { A }\).

The transformation S is represented by the matrix \(\mathbf { B }\) where \(\mathbf { B } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)\).

Find the equation of the line of invariant points of S .

Show that any line of the form \(y = x + c\) is an invariant line of S .

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 \(a\).

9 You are given that matrix \(\mathbf { M } = \left( \begin{array} { l l } - 3 & 8 \\ - 2 & 5 \end{array} \right)\).

1. Prove that, for all positive integers \(n , \mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 - 4 n & 8 n \\ - 2 n & 1 + 4 n \end{array} \right)\).
2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf { M }\). It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf { M } ^ { n }\), for any positive integer \(n\).
3. Explain geometrically why this claim is true.
4. Verify algebraically that this claim is true. \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

2 A surface \(S\) is defined by \(z = 4 x ^ { 2 } + 4 y ^ { 2 } - 4 x + 8 y + 11\).

1. Show that the point \(\mathrm { P } ( 0.5 , - 1,6 )\) is the only stationary point on \(S\).
2. 1. On the axes in the Printed Answer Booklet, draw a sketch of the contour of the surface corresponding to \(z = 42\).
   2. By using the sketch in part (b)(i), deduce that P must be a minimum point on \(S\).
3. In the section of \(S\) corresponding to \(y = c\), the minimum value of \(z\) occurs at the point where \(x = a\) and \(z = 22\).
   Find all possible values of \(a\) and \(c\).

1 A surface is defined in 3-D by \(z = 3 x ^ { 3 } + 6 x y + y ^ { 2 }\).
Determine the coordinates of any stationary points on the surface.

3 A surface, \(S\), is defined by \(g ( x , y , z ) = 0\) where \(g ( x , y , z ) = 2 x ^ { 3 } - x ^ { 2 } y + 2 x y ^ { 2 } + 27 z\). The normal to \(S\) at the point \(\left( 1,1 , - \frac { 1 } { 9 } \right)\) and the tangent plane to \(S\) at the point \(( 3,3 , - 3 )\) intersect at \(P\). Determine the position vector of P .

1 A surface, \(S\), is defined in 3-D by \(z = f ( x , y )\) where \(f ( x , y ) = 12 x - 30 y + 6 x y\).

1. Determine the coordinates of any stationary points on the surface.
2. The equation \(\mathrm { z } = \mathrm { f } ( \mathrm { x } , \mathrm { a } )\), where \(a\) is a constant, defines a section of S . Given that this equation is \(\mathrm { z } = 24 \mathrm { x } + \mathrm { b }\), find the value of \(a\) and the value of \(b\). The diagram shows the contour \(z = 12\) and its associated asymptotes. \includegraphics[max width=\textwidth, alt={}, center]{33c9e321-6044-45c4-bf37-0a6da3ecaf0d-2_860_1143_742_242}
3. Find the equations of the asymptotes.
4. By forming grad \(g\), where \(g ( x , y , z ) = f ( x , y ) - z\), find the equation of the tangent plane to \(S\) at the point where \(x = 3\) and \(y = 2\). Give your answer in vector form. The point \(( 0,4 , - 120 )\), which lies on S , is denoted by A .
   The plane with equation \(\mathbf { r }\). \(\left( \begin{array} { r } 3 \\ 3 \\ - 2 \end{array} \right) = 52\) is denoted by \(\Pi\).
5. Show that the normal to S at A intersects \(\Pi\) at the point \(( - 360,304 , - 110 )\).

6 A surface \(S\) is defined by \(z = \mathrm { f } ( x , y ) = 4 x ^ { 4 } + 4 y ^ { 4 } - 17 x ^ { 2 } y ^ { 2 }\).

1. 1. Show that there is only one stationary point on \(S\). The value of \(z\) at the stationary point is denoted by \(s\).
   2. State the value of \(s\).
   3. By factorising \(\mathrm { f } ( x , y )\), sketch the contour lines of the surface for \(z = s\).
   4. Hence explain whether the stationary point is a maximum point, a minimum point or a saddle point. C is a point on \(S\) with coordinates ( \(a , a , \mathrm { f } ( a , a )\) ) where \(a\) is a constant and \(a \neq 0\). \(\Pi\) is the tangent plane to \(S\) at C .
2. 1. Find the equation of \(\Pi\) in the form r.n \(= p\).
   2. The shortest distance from the origin to \(\Pi\) is denoted by \(d\). Show that \(\frac { d } { a } \rightarrow \frac { 3 \sqrt { 2 } } { 4 }\) as \(a \rightarrow \infty\).
   3. Explain whether the origin lies above or below \(\Pi\). \section*{END OF QUESTION PAPER}

1 In this question you must show detailed reasoning.
A surface \(S\) is defined by \(z = f ( x , y )\) where \(f ( x , y ) = x ^ { 3 } + x ^ { 2 } y - 2 y ^ { 2 }\).

1. On the coordinate axes in the Printed Answer Booklet, sketch the section \(z = f ( 2 , y )\) giving the coordinates of any turning points and any points of intersection with the axes.
2. Find the stationary points on \(S\). \(2 G\) is a group of order 8.
3. Explain why there is no subgroup of \(G\) of order 6 . You are now given that \(G\) is a cyclic group with the following features:
   • \(e\) is the identity element of \(G\),
   • \(g\) is a generator of \(G\),
   • \(H\) is the subgroup of \(G\) of order 4.
   • Write down the possible generators of \(H\). \(M\) is the group ( \(\{ 0,1,2,3,4,5,6,7 \} , + _ { 8 }\) ) where \(+ _ { 8 }\) denotes the binary operation of addition modulo 8. You are given that \(M\) is isomorphic to \(G\).
   • Specify all possible isomorphisms between \(M\) and \(G\).

18. $$\mathbf { M } = \left( \begin{array} { l l } 4 & - 5 \\ 6 & - 9 \end{array} \right)$$

1. Find the eigenvalues of \(\mathbf { M }\). A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { M }\). There is a line through the origin for which every point on the line is mapped onto itself under \(T\).
2. Find a cartesian equation of this line.
   [0pt] [P6 June 2003 Qn 3]

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 image of the point \(( 0,1 )\) is the point \(( 3,4 )\).
   • An object shape whose area is 7 is transformed to an image shape whose area is 35 .
   • T has a line of invariant points.
   • Find a possible matrix for \(\mathbf { A }\).
   The transformation \(S\) is represented by the matrix \(B\) where \(B = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)\).
4. Find the equation of the line of invariant points of S .
5. Show that any line of the form \(y = x + c\) is an invariant line of S . \section*{Total Marks for Question Set 4: 30} \section*{Mark scheme} \section*{Marking Instructions} a An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not always be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed.
   b The following types of marks are available. \section*{M} A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified.
   A method mark may usually be implied by a correct answer unless the question includes the DR statement, the command words "Determine" or "Show that", or some other indication that the method must be given explicitly. \section*{A} Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. \section*{B} Mark for a correct result or statement independent of Method marks. \section*{E} A given result is to be established or a result has to be explained. This usually requires more working or explanation than the establishment of an unknown result. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.
   c When a part of a question has two or more 'method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation 'dep*' is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.
   d The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be 'follow through'.
   e We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so.
   • When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value.
   • When a value is not given in the paper accept any answer that agrees with the correct value to \(\mathbf { 3 ~ s } . \mathbf { f }\). unless a different level of accuracy has been asked for in the question, or the mark scheme specifies an acceptable range.
   Follow through should be used so that only one mark in any question is lost for each distinct accuracy error.
   Candidates using a value of \(9.80,9.81\) or 10 for \(g\) should usually be penalised for any final accuracy marks which do not agree to the value found with 9.8 which is given in the rubric.
   f Rules for replaced work and multiple attempts:
   • If one attempt is clearly indicated as the one to mark, or only one is left uncrossed out, then mark that attempt and ignore the others.
   • If more than one attempt is left not crossed out, then mark the last attempt unless it only repeats part of the first attempt or is substantially less complete.
   • if a candidate crosses out all of their attempts, the assessor should attempt to mark the crossed out answer(s) as above and award marks appropriately.
   For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate's data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A or B mark in the question. Marks designated as cao may be awarded as long as there are no other errors.
   If a candidate corrects the misread in a later part, do not continue to follow through. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
   h If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers, provided that there is nothing in the wording of the question specifying that analytical methods are required such as the bold "In this question you must show detailed reasoning", or the command words "Show" or "Determine". Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Abbreviations}

   Abbreviations used in the mark scheme	Meaning
   dep*	Mark dependent on a previous mark, indicated by *. The * may be omitted if only one previous M mark
   cao	Correct answer only
   ое	Or equivalent
   rot	Rounded or truncated
   soi	Seen or implied
   www	Without wrong working
   AG	Answer given
   awrt	Anything which rounds to
   BC	By Calculator
   DR	This question included the instruction: In this question you must show detailed reasoning.

   \end{table}