Edexcel A-Level Mathematics AS Fp1 6 Transformations Using Matrices Questions

Practise using 2 x 2 matrices for transformations, compositions, inverse transformations, and area scale factors in FP1 geometry.

Exam points

describe transformations from matrices, including rotations, reflections, stretches and enlargements
build composite matrices in the correct order when one transformation is followed by another
use determinants to convert areas under matrix transformations or inverse transformations

Covered syllabus points

FP1.6.1 - Linear transformations of column FP1.6.2 - Applications of 2 × 2 matrices FP1.6.3 - Combinations of transformations FP1.6.4 - Inverse transformations and determinant scale factor

[Maximum number: 11]

(a)

\mathbf{P}=\left(\begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array}\right)

The matrix P represents a geometrical transformation U

[ 6 ]

(i)

Describe U fully as a single geometrical transformation.

The transformation V, represented by the $2 \times 2$ matrix Q, is a rotation through $240^{\circ}$ anticlockwise about the origin followed by an enlargement about ( 0,0 ) with scale factor 6

[ 2 ]

(ii)

Determine the matrix Q, giving each entry in exact numerical form.

Given that U followed by V is the transformation T, which is represented by the matrix R

[ 2 ]

$\begin{pmatrix}-\frac12&\frac{\sqrt3}{2}\\-\frac{\sqrt3}{2}&-\frac12\end{pmatrix}$
or 6 times the corresponding rotation matrix for $240^\circ$ $\begin{pmatrix}-3&3\sqrt3\\-3\sqrt3&-3\end{pmatrix}$

(iii)

determine the matrix R

[ 2 ]

$\mathbf R=\mathbf Q\mathbf P=\begin{pmatrix}-3&3\sqrt3\\-3\sqrt3&-3\end{pmatrix}\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$ $\mathbf R=\begin{pmatrix}-3\sqrt3&3\\3&3\sqrt3\end{pmatrix}$

(b)

The transformation W is represented by the matrix

\left(\begin{array}{cc} -2 & 2 \sqrt{3} \\ 2 \sqrt{3} & 2 \end{array}\right)

Show that there is a real number $\lambda$ for which W maps the point $(\lambda, 1)$ onto the point ( $4 \lambda, 4$ ), giving the exact value of $\lambda$

[ 5 ]

$\begin{pmatrix}-2&2\sqrt3\\2\sqrt3&2\end{pmatrix}\binom{\lambda}{1}=\binom{4\lambda}{4}$ $\binom{-2\lambda+2\sqrt3}{2\sqrt3\lambda+2}=\binom{4\lambda}{4}$ $-2\lambda+2\sqrt3=4\lambda\quad\text{or}\quad2\sqrt3\lambda+2=4$ $6\lambda=2\sqrt3\Rightarrow\lambda=\frac{\sqrt3}{3}$
or $2\sqrt3\lambda=2\Rightarrow\lambda=\frac1{\sqrt3}$
Both equations solved leading to $\lambda=\frac{\sqrt3}{3}$ or $\frac1{\sqrt3}$

[Maximum number: 8]

\mathbf{A}=\left(\begin{array}{cc} -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{array}\right)

(a)

Describe fully the single geometrical transformation represented by the matrix $\mathbf{A}^{2}$

[ 2 ]

Rotation $-60^{\circ}$ (anticlockwise) about the origin

Rotation
$-60^{\circ}$ (anticlockwise) (Or $60^{\circ}$ clockwise
or $300^{\circ}$ (anticlockwise)) about (0,0)
(2)

(b)

Hence determine the smallest positive integer value of n for which $\mathbf{A}^{n}=\mathbf{I}$

The matrix B represents a stretch scale factor 4 parallel to the x-axis.

[ 1 ]

n=12

Cao but can be embedded
ie $A^{12}=I$
(1)

(c)

Write down the matrix B

The transformation represented by matrix A followed by the transformation represented by matrix B is represented by the matrix C

[ 1 ]

$B= \left(\begin{smallmatrix} 4 & 0\\ 0 & 1 \end{smallmatrix}\right)$

(d)

Determine the matrix C

The parallelogram P is transformed onto the parallelogram $P^{\prime}$ by the matrix C

[ 2 ]

$C=BA= \left(\begin{smallmatrix} 4 & 0\\ 0 & 1 \end{smallmatrix}\right) \left(\begin{smallmatrix} -\frac{\sqrt3}{2} & -\frac12\\ \frac12 & -\frac{\sqrt3}{2} \end{smallmatrix}\right)$ $C= \left(\begin{smallmatrix} -2\sqrt3 & -2\\ \frac12 & -\frac{\sqrt3}{2} \end{smallmatrix}\right)$

(e)

Given that the area of parallelogram $P^{\prime}$ is 20 square units, determine the area of parallelogram P

[ 2 ]

$\det C=(-2\sqrt3)\left(-\frac{\sqrt3}{2}\right)-\frac12(-2)=4$ $\text{Area of }P=\frac{20}{\det C}=5$

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Covered syllabus points

Question 1

Question (a)

Question (i)

Question (ii)

Question (iii)

Question (b)

Question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)