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 \(A B C\).

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 \(\mathbf { M } = \left( \begin{array} { r r } - 1 & - 1 \\ 2 & 2 \end{array} \right)\).

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 \(a\).

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}

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 .

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

The matrix \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\) represents a transformation. Which one of the points below is an invariant point under this transformation? Circle your answer. [1 mark] \((1, 1) \quad (0, 2) \quad (3, 0) \quad (2, 1)\)

Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\). [4]

$$\mathbf{A} = \begin{pmatrix} 2 & a \\ a-4 & b \end{pmatrix}$$ 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 \(a\). [5] 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. [3]

Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\) [4]

The \(2 \times 2\) matrix \(\mathbf{A}\) 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 \(\mathbf{A}\). [8]

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

1. Find the equation of the line of invariant points of \(S\). [2]
2. Show that any line of the form \(y = x + c\) is an invariant line of \(S\). [3]

Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\). [4]