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Date May Example question Marks available 2 Reference code EXM.2.AHL.TZ0.16
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Describe Question number 16 Adapted from N/A

Question

The matrices A and B are defined by  A = ( 3 2 2 4 ) and  B = ( 1 0 0 1 ) .

Triangle X is mapped onto triangle Y by the transformation represented by AB. The coordinates of triangle Y are (0, 0), (−30, −20) and (−16, 32).

Describe fully the geometrical transformation represented by B.

[2]
a.

Find the coordinates of triangle X.

[5]
b.

Find the area of triangle X.

[2]
c.i.

Hence find the area of triangle Y.

[3]
c.ii.

Matrix A represents a combination of transformations:            

A stretch, with scale factor 3 and y-axis invariant;
Followed by a stretch, with scale factor 4 and x-axis invariant;
Followed by a transformation represented by matrix C.

Find matrix C.

[4]
d.

Markscheme

reflection in the y-axis     A1A1

[2 marks]

a.

X = ( A B ) 1 Y          M1

EITHER

A B = ( 3 2 2 4 ) , so  ( A B ) 1 = ( 1 4 1 8 1 8 3 16 )          M1A1

OR

X = B 1 A 1 Y          M1A1

THEN

X = ( 0 10 0 0 0 8 )          (A1)

So the coordinates are (0, 0), (10, 0) and (0, 8).        A1

[5 marks]

b.

10 × 8 2 = 40 units2       M1A1

[2 marks]

c.i.

det ( A B ) = 16        M1A1

Area  = 40 × 16 = 640  units2       A1

[3 marks]

c.ii.

A stretch, with scale factor 3 and y-axis invariant is given by  ( 3 0 0 1 )        A1

A stretch, with scale factor 4 and x-axis invariant is given by  ( 1 0 0 4 )        A1

So  C = A ( 3 0 0 1 ) 1 ( 1 0 0 4 ) 1 = ( 1 1 2 2 3 1 )        M1A1

[4 marks]

d.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.i.
[N/A]
c.ii.
[N/A]
d.

Syllabus sections

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

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