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Date	November Example question	Marks available	1	Reference code	EXN.3.AHL.TZ0.2
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	Write down	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 $U_{0}, U_{1}, U_{2}, \dots$

The first square, $U_{0}$ , has sides of length $4 cm$ .

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

$(\begin{matrix} x_{n} \\ y_{n} \end{matrix}) = A_{n} (\begin{matrix} x_{0} \\ y_{0} \end{matrix}) + b_{n}$

where $A_{n}$ is a $2 \times 2$ matrix that represents an enlargement, $b_{n}$ is a $2 \times 1$ column vector that represents a translation, $(x_{0}, y_{0})$ is a point in $U_{0}$ and $(x_{n}, y_{n})$ is its image in $U_{n}$ .

By considering the case where $(x_{0}, y_{0})$ 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 $E_{θ}$ 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 $U_{n}$ in centimetres.

[2]

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

[3]

Write down $A_{1}$ .

[1]

c.i.

Write down $A_{n}$ , in terms of $n$ .

[1]

c.ii.

state the coordinates, $(x_{1}, y_{1})$ , of its image in $U_{1}$ .

[1]

d.i.

hence find $b_{1}$ .

[2]

d.ii.

show that $b_{n} = (\begin{matrix} 8 (1 - 2^{- n}) \\ 8 (1 - 2^{- n}) \end{matrix})$ .

[3]

d.iii.

Hence or otherwise, find the coordinates of the top left-hand corner in $U_{7}$ .

[3]

Find, $s$ , in the form $s (θ) = m θ + c$ .

[4]

f.i.

Write down $E_{θ}$ .

[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]

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]

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.

$4 (\frac{1}{2^{n}})$ M1A1

[2 marks]

$\frac{4}{2^{n}} > 0.025$ (A1)

$2^{n} < 160$

$n \leq 7$ (A1)

Note: Accept equations in place of inequalities.

Hence there are $8$ squares A1

[3 marks]

$(\begin{matrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{matrix})$ A1

[1 mark]

c.i.

$A_{n} = (\begin{matrix} \frac{1}{2^{n}} & 0 \\ 0 & \frac{1}{2^{n}} \end{matrix})$ A1

[1 mark]

c.ii.

$(4, 4)$ A1

[1 mark]

d.i.

$A_{1} (\begin{matrix} 0 \\ 0 \end{matrix}) + b_{1} = (\begin{matrix} 4 \\ 4 \end{matrix})$ (M1)

$b_{1} = (\begin{matrix} 4 \\ 4 \end{matrix})$ A1

[2 marks]

d.ii.

Recognise the geometric series $b_{n} = (\begin{matrix} 4 + 2 + 1 + \dots \\ 4 + 2 + 1 + \dots \end{matrix})$ M1

Each component is equal to $\frac{4 (1 - \frac{1}{2^{n}})}{\frac{1}{2}} (= 8 (1 - \frac{1}{2^{n}}))$ M1A1

$(\begin{matrix} 8 (1 - \frac{1}{2^{n}}) \\ 8 (1 - \frac{1}{2^{n}}) \end{matrix})$ AG

[3 marks]

d.iii.

$(\begin{matrix} \frac{1}{2^{7}} & 0 \\ 0 & \frac{1}{2^{7}} \end{matrix}) (\begin{matrix} 0 \\ 4 \end{matrix}) + (\begin{matrix} 8 (1 - \frac{1}{2^{7}}) \\ 8 (1 - \frac{1}{2^{7}}) \end{matrix})$ M1A1

$(7.9375, 7.96875)$ A1

[3 marks]

$s (θ) = m θ + c$

$s (0) = 1, c = 1$ M1A1

$s (360) = \frac{1}{2}$ A1

$\frac{1}{2} = 360 m + 1 \Rightarrow m = - \frac{1}{720}$ A1

$s (θ) = - \frac{θ}{720} + 1$

[4 marks]

f.i.

$E_{θ} = (\begin{matrix} - \frac{θ}{720} + 1 & 0 \\ 0 & - \frac{θ}{720} + 1 \end{matrix})$ A1

[1 mark]

f.ii.

$(\begin{matrix} - \frac{135}{720} + 1 & 0 \\ 0 & - \frac{135}{720} + 1 \end{matrix}) (\begin{matrix} \cos 135 ° & - \sin 135 ° \\ \sin 135 ° & \cos 135 ° \end{matrix}) (\begin{matrix} 1 \\ 1 \end{matrix})$ M1A1A1

$(- 1.15, 0)$ A1

[4 marks]

f.iii.

$θ = 720 °$ A1

[1 mark]

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]

Examiners report

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

d.iii.

[N/A]

f.i.

[N/A]

f.ii.

[N/A]

f.iii.

[N/A]

Syllabus sections

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

Show 36 related questions

22M.2.AHL.TZ1.6c.i:
Find an expression for $h (x)$ .
22M.2.AHL.TZ1.6c.ii:
Find the value of $a$ .
22M.2.AHL.TZ1.7a.i:
By considering the image of $(0, 0)$ , find $q$ .
22M.2.AHL.TZ1.7c.ii:
Hence find the angle and direction of the rotation represented by $R$ .
22M.2.AHL.TZ1.7b:
Write down the matrix $S$ .
22M.1.AHL.TZ2.15c:
Show that the equation of the resulting line does not depend on $m$ or $c$ .
EXM.2.AHL.TZ0.15d:
Describe fully the geometrical transformation represented by X.
EXM.2.AHL.TZ0.16c.i:
Find the area of triangle X.
22M.2.AHL.TZ1.6c.iii:
Find the value of $b$ .
22M.2.AHL.TZ1.7a.ii:
By considering the image of $(1, 0)$ and $(0, 1)$ , show that

$P = (\begin{matrix} \frac{\sqrt{3}}{4} & \frac{1}{4} \\ - \frac{1}{4} & \frac{\sqrt{3}}{4} \end{matrix})$ .
22M.2.AHL.TZ1.7d.ii:
Find the value of $a$ and the value of $b$ .
21N.2.AHL.TZ0.4b:
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 $A' B' C'$ .
21N.2.AHL.TZ0.4a.iii:
Find $P^{2}$ .
21N.2.AHL.TZ0.4c:
Find a single transformation that is equivalent to the three transformations represented by matrix $P$ .
22M.2.AHL.TZ1.7d.i:
Write down an equation satisfied by $(\begin{matrix} a \\ b \end{matrix})$ .
22M.2.AHL.TZ1.7c.i:
Use $P = R S$ to find the matrix $R$ .
21N.2.AHL.TZ0.4a.i:
Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.
21N.2.AHL.TZ0.4a.ii:
Find a single matrix $P$ that defines a transformation that represents the overall change in position.
21N.2.AHL.TZ0.4a.iv:
Hence state what the value of $P^{2}$ indicates for the possible movement of the drone.
EXM.2.AHL.TZ0.16b:
Find the coordinates of triangle X.
EXM.2.AHL.TZ0.15a:
Describe fully the geometrical transformation represented by A.
EXM.2.AHL.TZ0.16a:
Describe fully the geometrical transformation represented by B.
EXM.2.AHL.TZ0.15c:
Find the matrix X.
20N.1.AHL.TZ0.F_7b.i:
Show that the points $A (1, 4), B (4, 8), C (8, 5), D (5, 1)$ form a square.
EXM.2.AHL.TZ0.15b:
Find the area of the image of P.
EXM.2.AHL.TZ0.16d:
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.
EXN.3.AHL.TZ0.2c.i:
Write down $A_{1}$ .
20N.1.AHL.TZ0.F_7a:
Describe, in words, the effect of the transformation $T$ .
20N.1.AHL.TZ0.F_7b.v:
Determine the area of this parallelogram.
20N.1.AHL.TZ0.F_7b.ii:
Determine the area of this square.
20N.1.AHL.TZ0.F_7b.iv:
Show that $A' B' C' D'$ is a parallelogram.
EXM.2.AHL.TZ0.16c.ii:
Hence find the area of triangle Y.
20N.1.AHL.TZ0.F_7b.iii:
Find the coordinates of $A', B', C', D'$ , the points to which $A, B, C, D$ are transformed under $T$ .
21M.1.AHL.TZ2.14a:
Find the coordinates of the image of the point $(6, - 2)$ .
21M.1.AHL.TZ2.14b:
Given that $T : (\begin{matrix} p \\ q \end{matrix}) \mapsto 2 (\begin{matrix} p \\ q \end{matrix})$ , find the value of $p$ and the value of $q$ .
21M.1.AHL.TZ2.14c:
A triangle $L$ with vertices lying on the $x y$ plane is transformed by $T$ .

Explain why both $L$ and its image will have exactly the same area.

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Topic 3—Geometry and trigonometry

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