Eddie decides to construct a path across his rectangular grass lawn using pairs of tiles.

Each tile is $10 \text{cm}$ wide and $20 \text{cm}$ long. The following diagrams show the path after Eddie has laid one pair and three pairs of tiles. This pattern continues until Eddie reaches the other side of his lawn. When $n$ pairs of tiles are laid, the path has a width of $w_n$ centimetres and a length $l_n$ centimetres.

The following diagrams show this pattern for one pair of tiles and for three pairs of tiles, where the white space around each diagram represents Eddie’s lawn.

The following table shows the values of $w_n$ and $l_n$ for the first three values of $n$.

Find the value of

Write down an expression in terms of $n$ for

Eddie’s lawn has a length $740 \text{cm}$.

The tiles cost $\$24.50$ per square metre and are sold in packs of five tiles.

To allow for breakages Eddie wants to have at least $8\%$ more tiles than he needs.

There is a fixed delivery cost of $\$35$.

The marks obtained by nine Mathematical Studies SL students in their projects (x) and their final IB examination scores (y) were recorded. These data were used to determine whether the project mark is a good predictor of the examination score. The results are shown in the table.

The equation of the regression line y on x is y = mx + c.

A tenth student, Jerome, obtained a project mark of 17.

Scott purchases food for his dog in large bags and feeds the dog the same amount of dog food each day. The amount of dog food left in the bag at the end of each day can be modelled by an arithmetic sequence.

On a particular day, Scott opened a new bag of dog food and fed his dog. By the end of the third day there were $115.5$ cups of dog food remaining in the bag and at the end of the eighth day there were $108$ cups of dog food remaining in the bag.

Find the number of cups of dog food

In $2021$, Scott spent $\$625$ on dog food. Scott expects that the amount he spends on dog food will increase at an annual rate of $6.4\%$.

Sophie is planning to buy a house. She needs to take out a mortgage for $120000. She is considering two possible options.

Option 1:  Repay the mortgage over 20 years, at an annual interest rate of 5%, compounded annually.

Option 2:  Pay $1000 every month, at an annual interest rate of 6%, compounded annually, until the loan is fully repaid.

Give a reason why Sophie might choose

Sophie decides to choose option 1. At the end of 10 years, the interest rate is changed to 7%, compounded annually.

Consider the function $f\left(x\right)=\frac{27}{x^2}-<!--\; -\; -->16x,x\ne <!--\; \ne \; -->0$.

In this question, give all answers to two decimal places.

Bryan decides to purchase a new car with a price of €14 000, but cannot afford the full amount. The car dealership offers two options to finance a loan.

Finance option A:

A 6 year loan at a nominal annual interest rate of 14 % compounded quarterly. No deposit required and repayments are made each quarter.

Finance option B:

A 6 year loan at a nominal annual interest rate of $r$ % compounded monthly. Terms of the loan require a 10 % deposit and monthly repayments of €250.

Paul wants to buy a car. He needs to take out a loan for $7000. The car salesman offers him a loan with an interest rate of 8%, compounded annually. Paul considers two options to repay the loan.

Option 1: Pay $200 each month, until the loan is fully repaid

Option 2: Make 24 equal monthly payments.

Use option 1 to calculate

Use option 2 to calculate

Give a reason why Paul might choose

A new café opened and during the first week their profit was $60.

The café’s profit increases by $10 every week.

A new tea-shop opened at the same time as the café. During the first week their profit was also $60.

The tea-shop’s profit increases by 10 % every week.

The admissions team at a new university are trying to predict the number of student applications they will receive each year.

Let $n$ be the number of years that the university has been open. The admissions team collect the following data for the first two years.

It is assumed that the number of students that apply to the university each year will follow a geometric sequence, $u_n$.

In the first year there were $10 380$ places at the university available for applicants. The admissions team announce that the number of places available will increase by $600$ every year.

Let $v_n$ represent the number of places available at the university in year $n$.

For the first $10$ years that the university is open, all places are filled. Students who receive a place each pay an $\$80$ acceptance fee.

When $n=k$, the number of places available will, for the first time, exceed the number of students applying.

Boris recorded the number of daylight hours on the first day of each month in a northern hemisphere town.

This data was plotted onto a scatter diagram. The points were then joined by a smooth curve, with minimum point $(0, 8)$ and maximum point $(6, 16)$ as shown in the following diagram.

Let the curve in the diagram be $y=f\left(t\right)$, where $t$ is the time, measured in months, since Boris first recorded these values.

Boris thinks that $f\left(t\right)$ might be modelled by a quadratic function.

Paula thinks that a better model is $f\left(t\right)=a \cos \left(bt\right)+d$, $t\ge 0$, for specific values of $a, b$ and $d$.

For Paula’s model, use the diagram to write down

The true maximum number of daylight hours was $16$ hours and $14$ minutes.

The following table shows values of ln x and ln y.

The relationship between ln x and ln y can be modelled by the regression equation ln y = a ln x + b.

Abdallah owns a plot of land, near the river Nile, in the form of a quadrilateral ABCD.

The lengths of the sides are $\text{AB}=\text{40 m, BC}=\text{115 m, CD}=\text{60 m, AD}=\text{84 m}$ and angle $\mathrm{B}\stackrel{^<!--\; ^\; -->}{\mathrm{A}}\mathrm{D}={90}^{\circ <!--\; \circ \; -->}$.

This information is shown on the diagram.

The formula that the ancient Egyptians used to estimate the area of a quadrilateral ABCD is

$\text{area}=\frac{(\text{AB}+\text{CD})(\text{AD}+\text{BC})}{4}$.

Abdallah uses this formula to estimate the area of his plot of land.

On her first day in a hospital, Kiri receives $u_1$ milligrams (mg) of a therapeutic drug. The amount of the drug Kiri receives increases by the same amount, $d$, each day. On the seventh day, she receives 21 mg of the drug, and on the eleventh day she receives 29 mg.

Kiri receives the drug for 30 days.

Ted is also in a hospital and on his first day he receives a 20 mg antibiotic injection. The amount of the antibiotic Ted receives decreases by 50 % each day. On the second day, Ted receives a 10 mg antibiotic injection, on the third day he receives 5 mg, and so on.

A water container is made in the shape of a cylinder with internal height $h$ cm and internal base radius $r$ cm.

The water container has no top. The inner surfaces of the container are to be coated with a water-resistant material.

The volume of the water container is $0.5{\text{m}}^3$.

The water container is designed so that the area to be coated is minimized.

One can of water-resistant material coats a surface area of $2000\text{ c}{\text{m}}^2$.

The Tower of Pisa is well known worldwide for how it leans.

Giovanni visits the Tower and wants to investigate how much it is leaning. He draws a diagram showing a non-right triangle, ABC.

On Giovanni’s diagram the length of AB is 56 m, the length of BC is 37 m, and angle ACB is 60°. AX is the perpendicular height from A to BC.

Giovanni’s tourist guidebook says that the actual horizontal displacement of the Tower, BX, is 3.9 metres.

The following table shows the average body weight, $x$, and the average weight of the brain, $y$, of seven species of mammal. Both measured in kilograms (kg).

The average body weight of grey wolves is 36 kg.

In fact, the average weight of the brain of grey wolves is 0.120 kg.

Let $f\left(x\right)={\text{e}}^{2\text{sin}\left(\frac{\mathrm{\pi <!--\; \pi \; -->}x}{2}\right)}$, for x > 0.

The k th maximum point on the graph of f has x-coordinate x_k where $k\in <!--\; \in \; -->{\mathbb{Z}}^{+}$.

Give your answers to parts (b), (c) and (d) to the nearest whole number.

Harinder has 14 000 US Dollars (USD) to invest for a period of five years. He has two options of how to invest the money.

Option A: Invest the full amount, in USD, in a fixed deposit account in an American bank.

The account pays a nominal annual interest rate of r % , compounded yearly, for the five years. The bank manager says that this will give Harinder a return of 17 500 USD.

Option B: Invest the full amount, in Indian Rupees (INR), in a fixed deposit account in an Indian bank. The money must be converted from USD to INR before it is invested.

The exchange rate is 1 USD = 66.91 INR.

The account in the Indian bank pays a nominal annual interest rate of 5.2 % compounded monthly.

Give your answers in parts (a), (d)(i), (e) and (f) to the nearest dollar.

Daisy invested $37 000$ Australian dollars ($\text{AUD}$) in a fixed deposit account with an annual interest rate of $6.4\%$ compounded quarterly.

After $m$ months, the amount of money in the fixed deposit account has appreciated to more than $50 000 \text{AUD}$.

Daisy is saving to purchase a new apartment. The price of the apartment is $200 000 \text{AUD}$.

Daisy makes an initial payment of $25\%$ and takes out a loan to pay the rest.

The loan is for $10$ years, compounded monthly, with equal monthly payments of $1700 \text{AUD}$ made by Daisy at the end of each month.

For this loan, find

After $5$ years of paying off this loan, Daisy decides to pay the remainder in one final payment.

The first term of an infinite geometric sequence is 4. The sum of the infinite sequence is 200.

Rosa joins a club to prepare to run a marathon. During the first training session Rosa runs a distance of 3000 metres. Each training session she increases the distance she runs by 400 metres.

A marathon is 42.195 kilometres.

In the $k$th training session Rosa will run further than a marathon for the first time.

Carlos joins the club to lose weight. He runs 7500 metres during the first month. The distance he runs increases by 20% each month.

John purchases a new bicycle for 880 US dollars (USD) and pays for it with a Canadian credit card. There is a transaction fee of 4.2 % charged to John by the credit card company to convert this purchase into Canadian dollars (CAD).

The exchange rate is 1 USD = 1.25 CAD.

John insures his bicycle with a US company. The insurance company produces the following table for the bicycle’s value during each year.

The values of the bicycle form a geometric sequence.

During the 1st year John pays 120 USD to insure his bicycle. Each year the amount he pays to insure his bicycle is reduced by 3.50 USD.

A large underground tank is constructed at Mills Airport to store fuel. The tank is in the shape of an isosceles trapezoidal prism, $\text{ABCDEFGH}$.

$\text{AB}=70 \text{m}$ , $\text{AF}=200 \text{m}$, $\text{AD}=40 \text{m}$, $\text{BC}=40 \text{m}$ and $\text{CD}=110 \text{m}$. Angle $\text{ADC}=60°$ and angle $\text{BCD}=60°$. The tank is illustrated below.

Once construction was complete, a fuel pump was used to pump fuel into the empty tank. The amount of fuel pumped into the tank by this pump each hour decreases as an arithmetic sequence with terms $u_1, u_2, u_3, \dots , u_n$.

Part of this sequence is shown in the table.

At the end of the $2\text{nd}$ hour, the total volume of fuel in the tank was $88 200 {\text{m}}^3$.

Maegan designs a decorative glass face for a new Fine Arts Centre. The glass face is made up of small triangular panes. The first three levels of the glass face are illustrated in the following diagram.

The $1\text{st}$ level, at the bottom of the glass face, has $5$ triangular panes. The $2\text{nd}$ level has $7$ triangular panes, and the $3\text{rd}$ level has $9$ triangular panes. Each additional level has $2$ more triangular panes than the level below it.

Maegan has $1000$ triangular panes to build the decorative glass face and does not want it to have any incomplete levels.

An infinite geometric series has first term $u_1=a$ and second term $u_2=\frac{1}{4}{a}^2-3a$, where $a>0$.

In an arithmetic sequence, $u_1=1.3$ , $u_2=1.4$ and $u_k=31.2$.

Consider the terms, $u_n$, of this sequence such that $n$ ≤ $k$.

Let $F$ be the sum of the terms for which $n$ is not a multiple of 3.

A new concert hall was built with $14$ seats in the first row. Each subsequent row of the hall has two more seats than the previous row. The hall has a total of $20$ rows.

Find:

The concert hall opened in $2019$. The average number of visitors per concert during that year was $584$. In $2020$, the average number of visitors per concert increased by $1.2\%$.

The concert organizers use this data to model future numbers of visitors. It is assumed that the average number of visitors per concert will continue to increase each year by $1.2\%$.

The first terms of an infinite geometric sequence, $u_n$, are 2, 6, 18, 54, …

The first terms of a second infinite geometric sequence, $v_n$, are 2, −6, 18, −54, …

The terms of a third sequence, $w_n$, are defined as $w_n=u_n+v_n$.

The finite series, $\underset{k=1}{\overset{225}{\sum <!--\; \sum \; -->}}{w}_k$ , can also be written in the form $\underset{k=0}{\overset{m}{\sum <!--\; \sum \; -->}}4r^k$.