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Date November Example question Marks available 2 Reference code EXN.3.AHL.TZ0.2
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term Find Question number 2 Adapted from N/A

Question

A graphic designer, Ben, wants to create an animation in which a sequence of squares is created by a composition of successive enlargements and translations and then rotated about the origin and reduced in size.

Ben outlines his plan with the following storyboards.

The first four frames of the animation are shown below in greater detail.

The sides of each successive square are one half the size of the adjacent larger square. Let the sequence of squares be U0, U1, U2, 

The first square, U0, has sides of length 4cm.

Ben decides the animation will continue as long as the width of the square is greater than the width of one pixel.

Ben decides to generate the squares using the transformation

xnyn=Anx0y0+bn

where An is a 2×2 matrix that represents an enlargement, bn is a 2×1 column vector that represents a translation, x0,y0 is a point in U0 and xn,yn is its image in Un.

By considering the case where x0,y0 is 0,0,

Once the image of squares has been produced, Ben wants to continue the animation by rotating the image counter clockwise about the origin and having it reduce in size during the rotation.

Let Eθ be the enlargement matrix used when the original sequence of squares has been rotated through θ degrees.

Ben decides the enlargement scale factor, s, should be a linear function of the angle, θ, and after a rotation of 360° the sequence of squares should be half of its original length.

Find an expression for the width of Un in centimetres.

[2]
a.

Given the width of a pixel is approximately 0.025cm, find the number of squares in the final image.

[3]
b.

Write down A1.

[1]
c.i.

Write down An, in terms of n.

[1]
c.ii.

state the coordinates, x1,y1, of its image in U1.

[1]
d.i.

hence find b1.

[2]
d.ii.

show that bn=81-2-n81-2-n.

[3]
d.iii.

Hence or otherwise, find the coordinates of the top left-hand corner in U7.

[3]
e.

Find, s, in the form sθ=mθ+c.

[4]
f.i.

Write down Eθ.

[1]
f.ii.

Hence find the image of (1, 1) after it is rotated 135° and enlarged.

[4]
f.iii.

Find the value of θ at which the enlargement scale factor equals zero.

[1]
g.

After the enlargement scale factor equals zero, Ben continues to rotate the image for another two revolutions.

Describe the animation for these two revolutions, stating the final position of the sequence of squares.

[3]
h.

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

412n       M1A1

 

[2 marks]

a.

42n>0.025        (A1)

2n<160

n7        (A1)

 

Note: Accept equations in place of inequalities.  

 

Hence there are 8 squares        A1

 

[3 marks]

b.

120012        A1

 

[1 mark]

c.i.

An=12n0012n        A1

 

[1 mark]

c.ii.

4,4        A1

 

[1 mark]

d.i.

A100+b1=44        (M1)

 b1=44        A1

 

[2 marks]

d.ii.

Recognise the geometric series bn=4+2+1+4+2+1+        M1

Each component is equal to 41-12n12 =81-12n        M1A1

81-12n81-12n        AG

  

[3 marks]

d.iii.

1270012704+81-12781-127        M1A1

7.9375, 7.96875        A1

  

[3 marks]

e.

sθ=mθ+c

s0=1, c=1        M1A1

s360=12        A1

12=360m+1m=-1720        A1

sθ=-θ720+1

  

[4 marks]

f.i.

Eθ=-θ720+100-θ720+1        A1

 

[1 mark]

f.ii.

-135720+100-135720+1cos135°-sin135°sin135°cos135°11        M1A1A1

 -1.15, 0        A1

 

[4 marks]

f.iii.

θ=720°      A1

 

[1 mark]

g.

The image will expand from zero (accept equivalent answers)

It will rotate counter clockwise

The design will (re)appear in the opposite (third) quadrant         A1A1

 

Note: Accept any two of the above

 

Its final position will be in the opposite (third) quadrant or 180˚ from its original position or equivalent statement.         A1

 

[3 marks]

h.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.i.
[N/A]
c.ii.
[N/A]
d.i.
[N/A]
d.ii.
[N/A]
d.iii.
[N/A]
e.
[N/A]
f.i.
[N/A]
f.ii.
[N/A]
f.iii.
[N/A]
g.
[N/A]
h.

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series
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Topic 1—Number and algebra

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