Write down transformation matrix - Linear transformations

Edexcel F1 2018 June Q2

7 marks Moderate -0.8

The transformation represented by the $2 \times 2$ matrix $\mathbf { P }$ is an anticlockwise rotation about the origin through 45 degrees.
1. Write down the matrix $\mathbf { P }$, giving the exact numerical value of each element.
$$\mathbf { Q } = \left( \begin{array} { c c } k \sqrt { 2 } & 0 \\ 0 & k \sqrt { 2 } \end{array} \right) \text {, where } k \text { is a constant and } k > 0$$
Describe fully the single geometrical transformation represented by the matrix $\mathbf { Q }$. The combined transformation represented by the matrix $\mathbf { P Q }$ transforms the rhombus $R _ { 1 }$ onto the rhombus $R _ { 2 }$. The area of the rhombus $R _ { 1 }$ is 6 and the area of the rhombus $R _ { 2 }$ is 147
Find the value of the constant $k$.

OCR FP1 2008 January Q1

4 marks Easy -1.2

1 The transformation S is a shear with the $y$-axis invariant (i.e. a shear parallel to the $y$-axis). It is given that the image of the point $( 1,1 )$ is the point $( 1,0 )$.

Draw a diagram showing the image of the unit square under the transformation S .
Write down the matrix that represents S .

OCR FP1 2006 June Q2

4 marks Easy -1.2

2 The transformation S is a shear parallel to the $x$-axis in which the image of the point ( 1,1 ) is the point $( 0,1 )$.

Draw a diagram showing the image of the unit square under S .
Write down the matrix that represents S .

OCR MEI FP1 2008 January Q9

13 marks Moderate -0.8

9 A transformation T acts on all points in the plane. The image of a general point P is denoted by $\mathrm { P } ^ { \prime }$. $\mathrm { P } ^ { \prime }$ always lies on the line $y = x$ and has the same $x$-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{225bff01-f2c4-421f-ac91-c6a0fcb01e6f-4_807_825_402_660} \captionsetup{labelformat=empty} \caption{Fig. 9}

\end{figure}

Write down the image of the point ( $- 3,7$ ) under transformation T .
Write down the image of the point $( x , y )$ under transformation T .
Find the $2 \times 2$ matrix which represents the transformation.
Describe the transformation M represented by the matrix $\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)$.
Find the matrix representing the composite transformation of T followed by M .
Find the image of the point $( x , y )$ under this composite transformation. State the equation of the line on which all of these images lie.

OCR FP1 2012 January Q5

6 marks Moderate -0.8

5

Find the matrix that represents a reflection in the line $y = - x$.
The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 4 \end{array} \right)$.
1. Describe fully the geometrical transformation represented by $\mathbf { C }$.
2. State the value of the determinant of $\mathbf { C }$ and describe briefly how this value relates to the transformation represented by $\mathbf { C }$.

OCR FP1 2014 June Q4

6 marks Moderate -0.3

4

Find the matrix that represents a shear with the $y$-axis invariant, the image of the point $( 1,0 )$ being the point $( 1,4 )$.
The matrix $\mathbf { X }$ is given by $\mathbf { X } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \\ - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)$.
1. Describe fully the geometrical transformation represented by $\mathbf { X }$.
2. Find the value of the determinant of $\mathbf { X }$ and describe briefly how this value relates to the transformation represented by $\mathbf { X }$.

OCR FP1 2011 January Q7

9 marks Moderate -0.8

Write down the matrix, $\mathbf { A }$, that represents a shear with $x$-axis invariant in which the image of the point $( 1,1 )$ is $( 4,1 )$.
The matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { c c } \sqrt { 3 } & 0 \\ 0 & \sqrt { 3 } \end{array} \right)$. Describe fully the geometrical transformation represented by $\mathbf { B }$.
The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { l l } 2 & 6 \\ 0 & 2 \end{array} \right)$.
1. Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { C }$.
2. Write down the determinant of $\mathbf { C }$ and explain briefly how this value relates to the transformation represented by $\mathbf { C }$. 8 The quadratic equation $2 x ^ { 2 } - x + 3 = 0$ has roots $\alpha$ and $\beta$, and the quadratic equation $x ^ { 2 } - p x + q = 0$ has roots $\alpha + \frac { 1 } { \alpha }$ and $\beta + \frac { 1 } { \beta }$.
  1. Show that $p = \frac { 5 } { 6 }$.
  2. Find the value of $q$. 9 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1 \\ 3 & a & 1 \\ 4 & 2 & 1 \end{array} \right)$.
    1. Find, in terms of $a$, the determinant of $\mathbf { M }$.
    2. Hence find the values of $a$ for which $\mathbf { M } ^ { - 1 }$ does not exist.
    3. Determine whether the simultaneous equations $$\begin{aligned} & 6 x - 6 y + z = 3 k \\ & 3 x + 6 y + z = 0 \\ & 4 x + 2 y + z = k \end{aligned}$$ where $k$ is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
    4. Show that $\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$.
    5. Hence find an expression, in terms of $n$, for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
    6. Show that $\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }$.

AQA FP1 2014 June Q7

10 marks Moderate -0.3

Write down the $2 \times 2$ matrix corresponding to each of the following transformations:
1. a reflection in the line $y = -x$; [1 mark]
2. a stretch parallel to the $y$-axis of scale factor $7$. [1 mark]
Hence find the matrix corresponding to the combined transformation of a reflection in the line $y = -x$ followed by a stretch parallel to the $y$-axis of scale factor $7$. [2 marks]
The matrix $\mathbf{A}$ is defined by $\mathbf{A} = \begin{bmatrix} -3 & -\sqrt{3} \\ -\sqrt{3} & 3 \end{bmatrix}$.
1. Show that $\mathbf{A}^2 = k\mathbf{I}$, where $k$ is a constant and $\mathbf{I}$ is the $2 \times 2$ identity matrix. [1 mark]
2. Show that the matrix $\mathbf{A}$ corresponds to a combination of an enlargement and a reflection. State the scale factor of the enlargement and state the equation of the line of reflection in the form $y = (\tan \theta)x$. [5 marks]

OCR FP1 Q9

12 marks Standard +0.3

Write down the matrix $\mathbf{C}$ which represents a stretch, scale factor $2$, in the $x$-direction. [2]
The matrix $\mathbf{D}$ is given by $\mathbf{D} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}$. Describe fully the geometrical transformation represented by $\mathbf{D}$. [2]
The matrix $\mathbf{M}$ represents the combined effect of the transformation represented by $\mathbf{C}$ followed by the transformation represented by $\mathbf{D}$. Show that $$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ [2]
Prove by induction that $\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}$, for all positive integers $n$. [6]

OCR FP1 2005 June Q9

12 marks Standard +0.3

Write down the matrix $\mathbf{C}$ which represents a stretch, scale factor 2, in the $x$-direction. [2]
The matrix $\mathbf{D}$ is given by $\mathbf{D} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}$. Describe fully the geometrical transformation represented by $\mathbf{D}$. [2]
The matrix $\mathbf{M}$ represents the combined effect of the transformation represented by $\mathbf{C}$ followed by the transformation represented by $\mathbf{D}$. Show that $$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ [2]
Prove by induction that $\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}$, for all positive integers $n$. [6]

OCR FP1 2010 June Q5

6 marks Easy -1.2

Write down the matrix that represents a reflection in the line $y = x$. [2]
Describe fully the geometrical transformation represented by each of the following matrices:
1. $\begin{pmatrix} 5 & 0 \\ 0 & 1 \end{pmatrix}$, [2]
2. $\begin{pmatrix} \frac{1}{2} & \frac{1}{2}\sqrt{3} \\ -\frac{1}{2}\sqrt{3} & \frac{1}{2} \end{pmatrix}$. [2]

OCR Further Pure Core AS 2020 November Q2

10 marks Moderate -0.8

P, Q and T are three transformations in 2-D. P is a reflection in the $x$-axis. A is the matrix that represents P.

Write down the matrix A. [1]

Q is a shear in which the $y$-axis is invariant and the point $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is transformed to the point $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$. B is the matrix that represents Q.

Find the matrix B. [2]

T is P followed by Q. C is the matrix that represents T.

Determine the matrix C. [2]

$L$ is the line whose equation is $y = x$.

Explain whether or not $L$ is a line of invariant points under T. [2]

An object parallelogram, $M$, is transformed under T to an image parallelogram, $N$.

Explain what the value of the determinant of C means about
[3]

SPS SPS FM 2020 December Q5

4 marks Moderate -0.8

The $2 \times 2$ matrix A represents a rotation by $90°$ anticlockwise about the origin. The $2 \times 2$ matrix B represents a reflection in the line $y = -x$. The matrix B is given by $$\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$

Write down the matrix representing A. [1]
The $2 \times 2$ matrix C represents a rotation by $90°$ anticlockwise about the origin, followed by a reflection in the line $y = -x$. Compute the matrix C and describe geometrically the single transformation represented by C. [3]

SPS SPS FM 2022 February Q4

4 marks Moderate -0.8

The transformation $S$ is a shear parallel to the $x$-axis in which the image of the point $(1, 1)$ is the point $(0, 1)$.

Draw a diagram showing the image of the unit square under $S$. [2]
Write down the matrix that represents $S$. [2]

SPS SPS FM 2025 February Q5

10 marks Moderate -0.8

$P$, $Q$ and $T$ are three transformations in 2-D. $P$ is a reflection in the $x$-axis. $\mathbf{A}$ is the matrix that represents $P$. Write down the matrix $\mathbf{A}$. [1]
$Q$ is a shear in which the $y$-axis is invariant and the point $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is transformed to the point $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$. $\mathbf{B}$ is the matrix that represents $Q$. Find the matrix $\mathbf{B}$. [2]
$T$ is $P$ followed by $Q$. $\mathbf{C}$ is the matrix that represents $T$. Determine the matrix $\mathbf{C}$. [2]
$L$ is the line whose equation is $y = x$. Explain whether or not $L$ is a line of invariant points under $T$. [2]
An object parallelogram, $M$, is transformed under $T$ to an image parallelogram, $N$. Explain what the value of the determinant of $\mathbf{C}$ means about • the area of $N$ compared to the area of $M$. • the orientation of $N$ compared to the orientation of $M$. [3]

OCR Further Pure Core 2 2021 June Q1

8 marks Moderate -0.8

In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the x-axis. A is the matrix which represents S.

Write down A. [1]
By considering the transformation represented by $\mathbf{A}^{-1}$, determine the matrix $\mathbf{A}^{-1}$. [2]

Matrix B is given by $\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$. T is the transformation represented by B.

Describe T. [1]
Determine the matrix which represents the transformation S followed by T. [2]
Demonstrate, by direct calculation, that $(\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}$. [2]

OCR Further Pure Core 2 2018 December Q2

8 marks Moderate -0.8

In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the $x$-axis. A is the matrix which represents S.

Write down A. [1]
By considering the transformation represented by $\mathbf{A}^{-1}$, determine the matrix $\mathbf{A}^{-1}$. [2]

Matrix B is given by $\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$. T is the transformation represented by B.

Describe T. [1]
Determine the matrix which represents the transformation S followed by T. [2]
Demonstrate, by direct calculation, that $(\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}$. [2]