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Date November 2021 Marks available 6 Reference code 21N.2.AHL.TZ0.4
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Write down Question number 4 Adapted from N/A

Question

A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin O and a set of x-y-axes.

In each case, the drone moves to a new position represented by the following transformations:

All the movements are performed in the listed order.

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

[6]
a.i.

Find a single matrix P that defines a transformation that represents the overall change in position.

[3]
a.ii.

Find P2.

[1]
a.iii.

Hence state what the value of P2 indicates for the possible movement of the drone.

[2]
a.iv.

Three drones are initially positioned at the points A, B and C. After performing the movements listed above, the drones are positioned at points A, B and C respectively.

Show that the area of triangle ABC is equal to the area of triangle ABC .

[2]
b.

Find a single transformation that is equivalent to the three transformations represented by matrix P.

[4]
c.

Markscheme

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values e.g. 32=0.866.

 

rotation anticlockwise π6 is 0.866-0.50.50.866  OR  32-121232           (M1)A1

reflection in y=x3

tanθ=13           (M1)

2θ=π3           (A1)

matrix is 0.50.8660.866-0.5  OR  123232-12            A1

rotation clockwise π3 is 0.50.866-0.8660.5  OR  1232-3212            A1

  

[6 marks]

a.i.

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values e.g. 32=0.866.

 

an attempt to multiply three matrices           (M1)

P=1232-3212123232-1232-121232           (A1)

P=32-12-12-32  OR  0.866-0.5-0.5-0.866            A1

   

[3 marks]

a.ii.

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values e.g. 32=0.866.

 

P2=32-12-12-3232-12-12-32= 1001            A1


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

   

[1 mark]

a.iii.

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values e.g. 32=0.866.

 

if the overall movement of the drone is repeated          A1

the drone would return to its original position          A1

   

[2 marks]

a.iv.

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values e.g. 32=0.866.

 

METHOD 1

detP=-34-14=1            A1

area of triangle ABC= area of triangle ABC ×detP            R1

area of triangle ABC= area of triangle ABC            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 ABC= area of triangle ABC            AG

 

[2 marks]

b.

Note: For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values e.g. 32=0.866.

 

attempt to find angles associated with values of elements in matrix P            (M1)

32-12-12-32=cos-π6sin-π6sin-π6-cos-π6

reflection (in y=tanθx)            (M1)

where 2θ=-π6            A1

reflection in y=tan-π12x  =-0.268x            A1

 

[4 marks]

c.

Examiners report

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 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 12 rather than, for example cosπ3. 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.

a.i.

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 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 12 rather than, for example cosπ3. 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.

a.ii.

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 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 12 rather than, for example cosπ3. 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.

a.iii.

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 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 12 rather than, for example cosπ3. 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.

a.iv.

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 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 12 rather than, for example cosπ3. 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.

b.

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 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 12 rather than, for example cosπ3. 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.

c.

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.9—Matrix transformations
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