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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 $U_{0}, U_{1}, U_{2}, \dots$

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

Show 111 related questions

18M.2.AHL.TZ1.H_1a:
Find the first term and the common difference of the sequence.
19M.2.AHL.TZ2.H_7:
Suppose that ${u_1}$ is the first term of a geometric series with common ratio $r$ .

Prove, by mathematical induction, that the sum of the first $n$ terms, ${s_n}$ is given by

${s_n} = \frac{{{u_1}\left( {1 - {r^n}} \right)}}{{1 - r}}$ , where $n \in {\mathbb{Z}^ + }$ .
22M.2.SL.TZ1.2c.ii:
Find the value of $w_{n}$ for this path.
22M.2.SL.TZ1.2b.ii:
$l_{n}$ .
22M.2.SL.TZ1.2c.i:
Show that Eddie needs $144$ tiles.
18M.2.AHL.TZ1.H_1b:
Calculate the number of positive terms in the sequence.
17N.2.SL.TZ0.T_2a.i:
Write down the distance Rosa runs in the third training session;
18M.1.SL.TZ1.T_7a.ii:
For that day find the weight of Sergei’s first lift.
16N.1.AHL.TZ0.H_6a:
Write down the value of ${u_1}$ .
SPM.1.SL.TZ0.2b:
Calculate the price of a ticket in the 16th row.
18M.2.SL.TZ2.T_4d:
Calculate the tea-shop’s total profit for the first 12 weeks.
22M.2.SL.TZ1.2a.ii:
$b$ .
22M.2.SL.TZ1.2b.i:
$w_{n}$ .
22M.2.SL.TZ2.2a.ii:
remaining in the bag at the end of the first day.
16N.1.SL.TZ0.S_9d:
Show that ${S_{12}} = 12{\log _2}x - 66$ .
18M.1.AHL.TZ2.H_5a:
Show that A is an arithmetic sequence, stating its common difference d in terms of r.
22M.2.SL.TZ2.2b:
Calculate the number of days that Scott can feed his dog with one bag of food.
22M.2.SL.TZ2.2c:
Determine the amount that Scott expects to spend on dog food in $2025$ . Round your answer to the nearest dollar.
17N.1.SL.TZ0.S_2b:
Find the tenth term.
21N.2.SL.TZ0.2f:
State whether, for all $n > k$ , the university will have places available for all applicants. Justify your answer.
22M.2.SL.TZ2.2a.i:
fed to the dog per day.
22M.2.SL.TZ1.2a.i:
$a$ .
21N.2.SL.TZ0.2b.i:
Write down the common ratio of the sequence.
17M.1.AHL.TZ2.H_3b:
the value of $r$ ;
16N.1.SL.TZ0.S_9b:
Show that the sum of the infinite sequence is $4{\log _2}x$ .
19M.1.AHL.TZ2.H_1:
In an arithmetic sequence, the sum of the 3rd and 8th terms is 1.

Given that the sum of the first seven terms is 35, determine the first term and the common difference.
18M.2.SL.TZ2.T_4c:
Find the tea-shop’s profit during the 11th week.
18M.2.SL.TZ2.T_4e:
In the mth week the tea-shop’s total profit exceeds the café’s total profit, for the first time since they both opened.

Find the value of m.
17M.1.AHL.TZ1.H_7b:
determine the value of $\sum\limits_{r = 1}^N {{u_r}}$ .
17M.1.SL.TZ2.T_5b:
Find the total number of sticks used by Tomás for all 24 diagrams.
18M.2.SL.TZ2.T_4b:
Calculate the café’s total profit for the first 12 weeks.
16N.2.AHL.TZ0.H_12a:
Find an expression for ${A_1}$ and show that ${A_2} = {1.004^2}x + 1.004x$ .
18M.1.SL.TZ1.T_7a.i:
For that day find how much weight was added after each lift.
19M.2.SL.TZ2.T_4d.iii:
Hence find the total amount of antibiotic, in mg, that Ted receives during the first $k$ days.
19M.2.SL.TZ2.T_4b:
Write down the value of $d$ and the value of ${u_1}$ .
17N.2.SL.TZ0.T_2a.ii:
Write down the distance Rosa runs in the $n$ th training session.
16N.1.SL.TZ0.S_9a:
Find $r$ .
18M.2.SL.TZ1.S_7a:
Given that x_k_{+ 1} = x_k + a, find a.
19M.2.SL.TZ1.T_5c:
Calculate, in years, when the bicycle value will be less than 50 USD.
17N.2.SL.TZ0.T_2b:
Find the value of $k$ .
SPM.1.SL.TZ0.2a:
Write down the value of the common difference, $d$
19M.2.SL.TZ1.T_5a:
Calculate, in CAD, the total amount John pays for the bicycle.
17N.1.SL.TZ0.S_2c:
Find the sum of the first ten terms.
16N.1.AHL.TZ0.H_6b:
Find the value of ${u_6}$ .
17N.2.SL.TZ0.T_2d:
Find the distance Carlos runs in the fifth month of training.
17M.1.AHL.TZ1.H_7a:
find the value of $d$ .
17M.1.SL.TZ1.T_5b:
The company also makes a ladder that is 1050 cm long.

Find the maximum number of rungs in this 1050 cm long ladder.
16N.1.AHL.TZ0.H_6c:
Prove that $\{ {u_n}\}$ is an arithmetic sequence, stating clearly its common difference.
18N.2.SL.TZ0.S_5:
The sum of an infinite geometric sequence is 33.25. The second term of the sequence is 7.98. Find the possible values of $r$ .
19N.2.SL.TZ0.T_3b:
Show that the total number of triangular panes, $S_{n}$ , in the first $n$ levels is given by:

$S_{n} = n^{2} + 4 n$ .
17N.1.SL.TZ0.S_2a:
Find the common difference.
19M.2.SL.TZ2.T_4a.i:
Write down an equation, in terms of ${u_1}$ and $d$ , for the amount of the drug that she receives on the seventh day.
18N.1.SL.TZ0.S_3a:
Find u₈.
16N.1.SL.TZ0.S_9e:
Given that ${S_{12}}$ is equal to half the sum of the infinite geometric sequence, find $x$ , giving your answer in the form ${2^p}$ , where $p \in \mathbb{Q}$ .
16N.2.AHL.TZ0.H_12e:
As soon as Mary was 18 she decided to invest $\$ 15\,000$ of this money in an account of the same type earning 0.4% interest per month. She withdraws $\$ 1000$ every year on her birthday to buy herself a present. Determine how long it will take until there is no money in the account.
18M.1.SL.TZ1.T_7b:
On that day, Sergei made 12 successive lifts. Find the total combined weight of these lifts.
16N.2.AHL.TZ0.H_12d:
Mary’s grandparents wished for the amount in her account to be at least $\$ 20\,000$ the day before she was 18. Determine the minimum value of the monthly deposit $\$ x$ required to achieve this. Give your answer correct to the nearest dollar.
20N.2.SL.TZ0.T_5c:
Write down the common difference, $d$ .
20N.2.SL.TZ0.T_5f:
Find the total amount of fuel pumped into the tank in the first $8$ hours.
20N.2.SL.TZ0.T_5g:
Show that the tank will never be completely filled using this pump.
19M.2.SL.TZ2.T_4d.ii:
The daily amount of antibiotic Ted receives will first be less than 0.06 mg on the $k$ th day. Find the value of $k$ .
19N.2.SL.TZ0.T_3e:
Each triangular pane has an area of $1.84 m^{2}$ .

Find the total area of the decorative glass face, if the maximum number of complete levels were built. Express your area to the nearest $m^{2}$ .
20N.2.SL.TZ0.T_5e.ii:
Write down the number of hours that the pump was pumping fuel into the tank.
EXN.1.SL.TZ0.12a:
Write down the value of $Σ_{i = 1}^{9} u_{i}$ .
EXN.1.SL.TZ0.12b:
Find the value of $u_{1}$ .
19M.2.SL.TZ2.T_4d.i:
Find the amount of antibiotic, in mg, that Ted receives on the fifth day.
18M.2.SL.TZ2.T_4a:
Find the café’s profit during the 11th week.
19M.2.SL.TZ2.S_10b:
Find the exact value of ${S_k}$ .
17N.2.SL.TZ0.T_2e:
Calculate the total distance Carlos runs in the first year.
18M.2.AHL.TZ1.H_7a:
Show that there will be approximately 2645 fish in the lake at the start of 2020.
16N.1.SL.TZ0.S_9c:
Find $d$ , giving your answer as an integer.
19N.2.SL.TZ0.T_3c:
Hence, find the total number of panes in a glass face with $18$ levels.
16N.2.AHL.TZ0.H_12b:
(i) Write down a similar expression for ${A_3}$ and ${A_4}$ .

(ii) Hence show that the amount in Mary’s account the day before she turned 10 years old is given by $251({1.004^{120}} - 1)x$ .
18M.2.SL.TZ1.S_7b:
Hence find the value of n such that $\sum\limits_{k = 1}^n {{x_k} = 861}$ .
19M.2.SL.TZ2.T_4c:
Calculate the total amount of the drug, in mg, that she receives.
20N.2.SL.TZ0.T_5b:
Show that the volume of the tank is $624 000 m^{3}$ , correct to three significant figures.
19M.2.SL.TZ2.S_10d:
An infinite geometric series is given as ${S_\infty } = a + \frac{a}{{\sqrt 2 }} + \frac{a}{2} + \ldots$ , $a \in {\mathbb{Z}^ + }$ .

Find the largest value of $a$ such that ${S_\infty } < F$ .
18M.2.AHL.TZ1.H_7b:
Find the approximate number of fish in the lake at the start of 2042.
19M.2.SL.TZ1.S_7a:
Write down the first three non-zero terms of ${w_n}$ .
20N.2.SL.TZ0.T_5a:
Find $h$ , the height of the tank.
17M.1.SL.TZ2.T_5a:
Diagram $n$ is formed with 52 sticks. Find the value of $n$ .
21M.1.SL.TZ1.8a.i:
Charlie ran on day $20$ of his fitness programme.
20N.2.SL.TZ0.T_5d:
Find the amount of fuel pumped into the tank in the $13th$ hour.
19N.2.SL.TZ0.T_3a:
Find the number of triangular panes in the $12 th$ level.
SPM.1.SL.TZ0.2c:
Find the total cost of buying 2 tickets in each of the first 16 rows.
17M.1.SL.TZ1.T_5a:
Find the distance from the base of this ladder to the top rung.
19N.2.SL.TZ0.T_3d:
Find the maximum number of complete levels that Maegan can build.
18M.1.SL.TZ2.S_7b:
Let $p = {c^2}$ and $q = {c^3}$ . Find the value of $\sum\limits_{n = 1}^{20} {{u_n}}$ .
21M.2.SL.TZ2.3a.ii:
the total number of seats in the concert hall.
21M.2.SL.TZ2.3a.i:
the number of seats in the last row.
17M.1.AHL.TZ2.H_3a:
the value of $d$ ;
19M.2.SL.TZ2.T_4a.ii:
Write down an equation, in terms of ${u_1}$ and $d$ , for the amount of the drug that she receives on the eleventh day.
18M.1.AHL.TZ2.H_5b:
A particular geometric sequence has u₁ = 3 and a sum to infinity of 4.

Find the value of d.
19M.2.SL.TZ1.S_7b.ii:
Find the value of $m$ .
19M.2.SL.TZ2.S_10c:
Show that $F = 3240$ .
19M.2.SL.TZ1.T_5d:
Find the total amount John has paid to insure his bicycle for the first 5 years.
20N.2.SL.TZ0.T_5e.i:
Find the value of $n$ such that $u_{n} = 0$ .
16N.2.AHL.TZ0.H_12c:
Write down an expression for ${A_n}$ in terms of $x$ on the day before Mary turned 18 years old showing clearly the value of $n$ .
17N.2.SL.TZ0.T_2c:
Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.
19M.2.SL.TZ1.T_5b:
Find the value of the bicycle during the 5th year. Give your answer to two decimal places.
EXN.1.SL.TZ0.12c:
A game is played in which the arrow attached to the centre of the disc is spun and the sector in which the arrow stops is noted. If the arrow stops in sector $1$ the player wins $10$ points, otherwise they lose $2$ points.

Let $X$ be the number of points won

Find $E (X)$ .
19M.2.SL.TZ2.S_10a:
Find the value of $k$ .
19M.2.SL.TZ1.S_7b.i:
Find the value of $r$ .
21M.1.SL.TZ1.8a.ii:
Daniella ran on day $20$ of her fitness programme.
EXN.3.AHL.TZ0.2b:
Given the width of a pixel is approximately $0.025 cm$ , find the number of squares in the final image.
21N.2.SL.TZ0.2a:
Calculate the percentage increase in applications from the first year to the second year.
21N.2.SL.TZ0.2b.ii:
Find an expression for $u_{n}$ .
21N.2.SL.TZ0.2b.iii:
Find the number of student applications the university expects to receive when $n = 11$ . Express your answer to the nearest integer.
21N.2.SL.TZ0.2c:
Write down an expression for $v_{n}$ .
21N.2.SL.TZ0.2d:
Calculate the total amount of acceptance fees paid to the university in the first $10$ years.
21N.2.SL.TZ0.2e:
Find $k$ .

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Topic 1—Number and algebra

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