Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.

Find a single matrix $\mathit{P}$ that defines a transformation that represents the overall change in position.

Find ${\mathit{P}}^2$.

Hence state what the value of ${\mathit{P}}^2$ indicates for the possible movement of the drone.

Three drones are initially positioned at the points $\text{A}$, $\text{B}$ and $\text{C}$. After performing the movements listed above, the drones are positioned at points $\text{A}\prime$, $\text{B}\prime$ and $\text{C}\prime$ respectively.

Show that the area of triangle $\text{ABC}$ is equal to the area of triangle $\text{A}\prime \text{B}\prime \text{C}\prime$ .

Find a single transformation that is equivalent to the three transformations represented by matrix $\mathit{P}$.

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values $\left(\text{e.g.} \frac{\sqrt{3}}{2}=0.866\right)$.

rotation anticlockwise $\frac{\mathrm{\pi }}{6}$ is $\left(\begin{array}{cc}0.866& -0.5\\ 0.5& 0.866\end{array}\right)$  OR  $\left(\begin{array}{cc}\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ \frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right)$           (M1)A1

reflection in $y=\frac{x}{\sqrt{3}}$

$\tan \theta =\frac{1}{\sqrt{3}}$           (M1)

$\Rightarrow 2\theta =\frac{\mathrm{\pi }}{3}$           (A1)

matrix is $\left(\begin{array}{cc}0.5& 0.866\\ 0.866& -0.5\end{array}\right)$  OR  $\left(\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& -\frac{1}{2}\end{array}\right)$            A1

rotation clockwise $\frac{\mathrm{\pi }}{3}$ is $\left(\begin{array}{cc}0.5& 0.866\\ -0.866& 0.5\end{array}\right)$  OR  $\left(\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2}& \frac{1}{2}\end{array}\right)$            A1

[6 marks]

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values $\left(\text{e.g.} \frac{\sqrt{3}}{2}=0.866\right)$.

an attempt to multiply three matrices           (M1)

$\mathit{P}=\left(\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2}& \frac{1}{2}\end{array}\right)\left(\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& -\frac{1}{2}\end{array}\right)\left(\begin{array}{cc}\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ \frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right)$           (A1)

$\mathit{P}=\left(\begin{array}{cc}\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ -\frac{1}{2}& -\frac{\sqrt{3}}{2}\end{array}\right)$  OR  $\left(\begin{array}{cc}0.866& -0.5\\ -0.5& -0.866\end{array}\right)$            A1

[3 marks]

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values $\left(\text{e.g.} \frac{\sqrt{3}}{2}=0.866\right)$.

$\left({\mathit{P}}^2=\left(\begin{array}{cc}\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ -\frac{1}{2}& -\frac{\sqrt{3}}{2}\end{array}\right)\left(\begin{array}{cc}\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ -\frac{1}{2}& -\frac{\sqrt{3}}{2}\end{array}\right)=\right) \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$            A1

Note: Do not award A1 if final answer not resolved into the identity matrix $I$.

[1 mark]

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values $\left(\text{e.g.} \frac{\sqrt{3}}{2}=0.866\right)$.

if the overall movement of the drone is repeated          A1

the drone would return to its original position          A1

[2 marks]

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values $\left(\text{e.g.} \frac{\sqrt{3}}{2}=0.866\right)$.

METHOD 1

$\left|\text{det} \mathit{P}\right|=\left|\left(-\frac{3}{4}\right)-\left(\frac{1}{4}\right)\right|=1$            A1

area of triangle $\text{ABC}=$ area of triangle $\text{A}\prime \text{B}\prime \text{C}\prime$ $\times \left|\text{det} \mathit{P}\right|$            R1

area of triangle $\text{ABC}=$ area of triangle $\text{A}\prime \text{B}\prime \text{C}\prime$            AG

Note: Award at most A1R0 for responses that omit modulus sign.

METHOD 2

statement of fact that rotation leaves area unchanged            R1

statement of fact that reflection leaves area unchanged            R1

area of triangle $\text{ABC}=$ area of triangle $\text{A}\prime \text{B}\prime \text{C}\prime$            AG

[2 marks]

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values $\left(\text{e.g.} \frac{\sqrt{3}}{2}=0.866\right)$.

attempt to find angles associated with values of elements in matrix $\mathit{P}$            (M1)

$\left(\begin{array}{cc}\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ -\frac{1}{2}& -\frac{\sqrt{3}}{2}\end{array}\right)=\left(\begin{array}{cc}\cos \left(-\frac{\mathrm{\pi }}{6}\right)& \sin \left(-\frac{\mathrm{\pi }}{6}\right)\\ \sin \left(-\frac{\mathrm{\pi }}{6}\right)& -\cos \left(-\frac{\mathrm{\pi }}{6}\right)\end{array}\right)$

reflection (in $y=\left(\tan \theta \right)x$)            (M1)

where $2\theta =-\frac{\mathrm{\pi }}{6}$            A1

reflection in $y=\tan \left(-\frac{\mathrm{\pi }}{12}\right)x \left(=-0.268x\right)$            A1

[4 marks]

There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for $\mathit{P}$ then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing $0.5$ or $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ rather than, for example $\cos \left(\frac{\mathrm{\pi }}{3}\right)$. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.

There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for $\mathit{P}$ then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing $0.5$ or $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ rather than, for example $\cos \left(\frac{\mathrm{\pi }}{3}\right)$. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.

There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for $\mathit{P}$ then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing $0.5$ or $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ rather than, for example $\cos \left(\frac{\mathrm{\pi }}{3}\right)$. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.

There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for $\mathit{P}$ then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing $0.5$ or $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ rather than, for example $\cos \left(\frac{\mathrm{\pi }}{3}\right)$. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.

There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for $\mathit{P}$ then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing $0.5$ or $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ rather than, for example $\cos \left(\frac{\mathrm{\pi }}{3}\right)$. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.

There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for $\mathit{P}$ then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing $0.5$ or $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ rather than, for example $\cos \left(\frac{\mathrm{\pi }}{3}\right)$. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.