Matrix powers and repeated transformations

Edexcel F1 2022 June Q7

9 marks Standard +0.3

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

Determine the matrix $\mathbf { A } ^ { 2 }$
Describe fully the single geometrical transformation represented by the matrix $\mathbf { A } ^ { 2 }$
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.
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 $\mathbf { C }$
Determine the matrix $\mathbf { C }$ The parallelogram $P$ is transformed onto the parallelogram $P ^ { \prime }$ by the matrix $\mathbf { C }$
Given that the area of parallelogram $P ^ { \prime }$ is 20 square units, determine the area of parallelogram $P$

Edexcel FP1 Q1

5 marks Moderate -0.8

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

Find $\mathbf { R } ^ { 2 }$.
Find $\mathbf { R S }$.
Describe the geometrical transformation represented by $\mathbf { R S }$.

Edexcel FP1 Specimen Q3

5 marks Standard +0.3

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

Find $\mathbf { R } ^ { 2 }$.
Describe the geometrical transformation represented by $\mathbf { R } ^ { 2 }$.
Describe the geometrical transformation represented by $\mathbf { R }$.

OCR FP1 2010 January Q10

11 marks Standard +0.8

10 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)$.

Find $\mathbf { M } ^ { 2 }$ and $\mathbf { M } ^ { 3 }$.
Hence suggest a suitable form for the matrix $\mathbf { M } ^ { n }$.
Use induction to prove that your answer to part (ii) is correct.
Describe fully the single geometrical transformation represented by $\mathbf { M } ^ { 10 }$.

OCR MEI FP1 2012 January Q9

12 marks Standard +0.3

9 The matrix $\mathbf { R }$ is $\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)$.

Explain in terms of transformations why $\mathbf { R } ^ { 4 } = \mathbf { I }$.
Describe the transformation represented by $\mathbf { R } ^ { - 1 }$ and write down the matrix $\mathbf { R } ^ { - 1 }$.
$\mathbf { S }$ is the matrix representing rotation through $60 ^ { \circ }$ anticlockwise about the origin. Find $\mathbf { S }$.
Write down the smallest positive integers $m$ and $n$ such that $\mathbf { S } ^ { m } = \mathbf { R } ^ { n }$, explaining your answer in terms of transformations.
Find $\mathbf { R S }$ and explain in terms of transformations why $\mathbf { R S } = \mathbf { S R }$. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

AQA FP1 2011 June Q7

9 marks Moderate -0.3

7 The matrix $\mathbf { A }$ is defined by $$\mathbf { A } = \left[ \begin{array} { c c } - 1 & - \sqrt { 3 } \\ \sqrt { 3 } & - 1 \end{array} \right]$$

1. Calculate the matrix $\mathbf { A } ^ { 2 }$.
2. Show that $\mathbf { A } ^ { 3 } = k \mathbf { I }$, where $k$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix.
Describe the single geometrical transformation, or combination of two geometrical transformations, corresponding to each of the matrices:
1. $\mathrm { A } ^ { 3 }$;
2. A.

OCR MEI Further Pure Core AS 2023 June Q1

3 marks Easy -1.2

1 The transformation R of the plane is reflection in the line $x = 0$.

Write down the matrix $\mathbf { M }$ associated with R .
Find $\mathbf { M } ^ { 2 }$.
Interpret the result of part (b) in terms of the transformation $R$.

OCR MEI Further Pure Core 2024 June Q8

10 marks Standard +0.3

8

Specify fully the transformation T of the plane associated with the matrix $\mathbf { M }$, where $\mathbf { M } = \left( \begin{array} { l l } 1 & \lambda \\ 0 & 1 \end{array} \right)$ and $\lambda$ is a non-zero constant.
1. Find detM.
2. Deduce two properties of the transformation T from the value of detM.
Prove that $\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & n \lambda \\ 0 & 1 \end{array} \right)$, where $n$ is a positive integer.
Hence specify fully a single transformation which is equivalent to $n$ applications of the transformation T.

AQA FP1 2006 June Q5

9 marks Moderate -0.3

5 The matrix $\mathbf { M }$ is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$

Find the matrix:
1. $\mathbf { M } ^ { 2 }$;
2. $\mathbf { M } ^ { 4 }$.
Describe fully the geometrical transformation represented by $\mathbf { M }$.
Find the matrix $\mathbf { M } ^ { 2006 }$.

OCR Further Pure Core AS 2023 June Q9

10 marks Standard +0.8

9 Matrix $\mathbf { R }$ is given by $\mathbf { R } = \left( \begin{array} { c c c } a & 0 & - b \\ 0 & 1 & 0 \\ b & 0 & a \end{array} \right)$ where $a$ and $b$ are constants.

Find $\mathbf { R } ^ { 2 }$ in terms of $a$ and $b$. The constants $a$ and $b$ are given by $a = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } + 1 )$ and $b = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } - 1 )$.
By determining exact expressions for $a b$ and $a ^ { 2 } - b ^ { 2 }$ and using the result from part (a), show that $\mathbf { R } ^ { 2 } = k \left( \begin{array} { c c c } \sqrt { 3 } & 0 & - 1 \\ 0 & 2 & 0 \\ 1 & 0 & \sqrt { 3 } \end{array} \right)$ where $k$ is a real number whose value is to be determined.
Find $\mathbf { R } ^ { 6 } , \mathbf { R } ^ { 12 }$ and $\mathbf { R } ^ { 24 }$.
Describe fully the transformation represented by $\mathbf { R }$. \section*{END OF QUESTION PAPER}

OCR MEI FP1 Q9

Standard +0.3

9 You are given the matrix $\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)$.

Calculate $\mathbf { M } ^ { 2 }$. You are now given that the matrix $M$ represents a reflection in a line through the origin.
Explain how your answer to part (i) relates to this information.
By investigating the invariant points of the reflection, find the equation of the mirror line.
Describe fully the transformation represented by the matrix $\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)$.
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.
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.

AQA FP1 2016 June Q8

10 marks Standard +0.3

The matrix $\mathbf{A}$ is defined by $\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$.

1. Find the matrix $\mathbf{A}^2$. [1 mark]
2. Describe fully the single geometrical transformation represented by the matrix $\mathbf{A}^2$. [1 mark]
Given that the matrix $\mathbf{B}$ represents a reflection in the line $x + \sqrt{3}y = 0$, find the matrix $\mathbf{B}$, giving the exact values of any trigonometric expressions. [2 marks]
Hence find the coordinates of the point $P$ which is mapped onto $(0, -4)$ under the transformation represented by $\mathbf{A}^2$ followed by a reflection in the line $x + \sqrt{3}y = 0$. [6 marks]