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Date May 2022 Marks available 4 Reference code 22M.2.AHL.TZ1.7
Level Additional Higher Level Paper Paper 2 Time zone Time zone 1
Command term Find Question number 7 Adapted from N/A

Question

A transformation, T, of a plane is represented by r=Pr+q, where P is a 2×2 matrix, q is a 2×1 vector, r is the position vector of a point in the plane and r the position vector of its image under T.

The triangle OAB has coordinates (0, 0), (0, 1) and (1, 0). Under T, these points are transformed to (0, 1)14, 1+34 and 34, 34 respectively.

P can be written as P=RS, where S and R are matrices.

S represents an enlargement with scale factor 0.5, centre (0, 0).

R represents a rotation about (0, 0).

The transformation T can also be described by an enlargement scale factor 12, centre (a, b), followed by a rotation about the same centre (a, b).

By considering the image of (0, 0), find q.

[2]
a.i.

By considering the image of (1, 0) and (0, 1), show that

P=34 14-14 34.

[4]
a.ii.

Write down the matrix S.

[1]
b.

Use P=RS to find the matrix R.

[4]
c.i.

Hence find the angle and direction of the rotation represented by R.

[3]
c.ii.

Write down an equation satisfied by ab.

[1]
d.i.

Find the value of a and the value of b.

[3]
d.ii.

Markscheme

P00+q=01        (M1)

q=01          A1

 

[2 marks]

a.i.

EITHER

P10+01=3434          M1

hence P10=34-14          A1

P01+01=141+34          M1

hence P01=1434          A1


OR

a bc d10+01=3434          M1

hence a bc d10=34-14          A1

ac=34-14

a bc d01+01=141+34          M1

a bc d01=1434          A1

bd=1434


THEN

P=34 14-14 34          AG

 

[4 marks]

a.ii.

12 00 12          A1

 

[1 mark]

b.

EITHER

S-1=2 00 2         (A1)

R=PS-1         (M1)


Note: The M1 is for an attempt at rearranging the matrix equation. Award even if the order of the product is reversed.


R=34 14-14 342 00 2         (A1)


OR

34 14-14 34=R0.5 00 0.5

let R=a bc d

attempt to solve a system of equations         M1

34=0.5a,   14=0.5b

-14=0.5c,  34=0.5d          A2


Note: Award A1 for two correct equations, A2 for all four equations correct.

 

THEN

R=32 12-12 32  OR  0.866 0.5-0.5 0.866  OR  0.866025 0.5-0.5 0.866025          A1

 

Note: The correct answer can be obtained from reversing the matrices, so do not award if incorrect product seen. If the given answer is obtained from the product R=S-1P, award (A1)(M1)(A0)A0.

 

[4 marks]

c.i.

clockwise         A1

arccosine or arcsine of value in matrix seen         (M1)

30°         A1


Note:
Both A1 marks are dependent on the answer to part (c)(i) and should only be awarded for a valid rotation matrix.

 

[3 marks]

c.ii.

METHOD 1

ab=Pab+q         A1

 

METHOD 2

x'y'=Px-ay-b+ab         A1

 

Note: Accept substitution of x and y (and x' and y') with particular points given in the question.

[1 mark]

d.i.

METHOD 1

solving ab=Pab+q using simultaneous equations or a=I-P-1q         (M1)

a=0.651  0.651084,  b=1.48  1.47662        A1A1

a=5+2313, b=14+3313

 

METHOD 2

01=P0-a0-b+ab         (M1)

 

Note: This line, with any of the points substituted, may be seen in part (d)(i) and if so the M1 can be awarded there.


01=I-Pab

a=0.651084,  b=1.47662         A1A1

a=5+2313, b=14+3313

 

[3 marks]

d.ii.

Examiners report

Part (i) proved to be straightforward for most candidates. A common error in part (ii) was for candidates to begin with the matrix P and to show it successfully transformed the points to their images. This received no marks. For a ‘show that’ question it is expected that the work moves to rather than from the given answer.

a.i.
[N/A]
a.ii.

(b), (c) These two parts dealt generally with more familiar aspects of matrix transformations and were well done.

b.

(b), (c) These two parts dealt generally with more familiar aspects of matrix transformations and were well done.

c.i.
[N/A]
c.ii.

The trick of recognizing that (a,b) was invariant was generally not seen and as such the question could not be successfully answered.

d.i.
[N/A]
d.ii.

Syllabus sections

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