IB Maths AI HL/Question Bank/1 Number and algebra

IB Maths AI HL 1 Number and algebra Question Bank

HL102 questions10 previewsSyllabus linked

[Maximum number: 2]

A large water reservoir is built in the form of part of an upside-down right pyramid with a horizontal square base of length 80 metres. The point C is the centre of the square base and point V is the vertex of the pyramid.

diagram not to scale

The bottom of the reservoir is a square of length 60 metres that is parallel to the base of the pyramid, such that the depth of the reservoir is 6 metres as shown in the diagram.

The second diagram shows a vertical cross section, MNOPC, of the reservoir.

(a)

By finding an appropriate value, determine whether Joshua is correct.

To avoid water leaking into the ground, the five interior sides of the reservoir have been painted with a watertight material.

[ 2 ]

[Maximum number: 3]

A suitable site for the landing of a spacecraft on the planet Mars is identified at a point, A. The shortest time from sunrise to sunset at point A must be found.
Radians should be used throughout this question. All values given in the question should be treated as exact.
Mars completes a full orbit of the Sun in 669 Martian days, which is one Martian year.

On day t, where $t \in \mathbb{Z}$ , the length of time, in hours, from the start of the Martian day until sunrise at point A can be modelled by a function, R(t), where

R(t)=a \sin (b t)+c, t \in \mathbb{R} .

The graph of R is shown for one Martian year.

(a)

(i)

Write down $z_{1}$ and $z_{2}$ in exponential form, with a constant modulus.

[ 3 ]

[Maximum number: 19]

Give your answers in parts (a), (d)(i), (e) and (f) to the nearest dollar.
Daisy invested 37000 Australian dollars (AUD) in a fixed deposit account with an annual interest rate of 6.4 % compounded quarterly.

(a)

Calculate the value of Daisy's investment after 2 years.

After m months, the amount of money in the fixed deposit account has appreciated to more than 50000 AUD.

[ 3 ]

(b)

Find the minimum value of m, where $m \in \mathbb{N}$ .

Daisy is saving to purchase a new apartment. The price of the apartment is 200000 AUD.
Daisy makes an initial payment of 25 % and takes out a loan to pay the rest.

[ 4 ]

(c)

Write down the amount of the loan.

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

[ 1 ]

(d)

For this loan, find

[ 5 ]

(i)

the amount of interest paid by Daisy.

(ii)

the annual interest rate of the loan.

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

[ 5 ]

(e)

Find the amount of Daisy's final payment.

[ 3 ]

(f)

Find how much money Daisy saved by making one final payment after 5 years.

[ 3 ]

[Maximum number: 16]

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.

(a)

Find the number of cups of dog food

[ 4 ]

(i)

fed to the dog per day;

(ii)

remaining in the bag at the end of the first day.

[ 4 ]

(b)

Calculate the number of days that Scott can feed his dog with one bag of 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 %.

[ 2 ]

(c)

Determine the amount that Scott expects to spend on dog food in 2025. Round your answer to the nearest dollar.

[ 3 ]

(d)

(i)

Calculate the value of $\sum_{n=1}^{10}\left(625 \times 1.064^{(n-1)}\right)$ .

[ 3 ]

(ii)

Describe what the value in part (d)(i) represents in this context.

[ 3 ]

(e)

Comment on the appropriateness of modelling this scenario with a geometric sequence.

[ 1 ]

[Maximum number: 1]

Madhu is designing a jogging track for the campus of her school. The following diagram shows an incomplete portion of the track.
Madhu wants to design the track such that the inner edge is a smooth curve from point A to point B , and the other edge is a smooth curve from point C to point D . The distance between points A and B is 50 metres.

To create a smooth curve, Madhu first walks to M , the midpoint of $[\mathrm{AB}]$ .

(a)

Write down the length of $[\mathrm{BM}]$ .

[ 1 ]

[Maximum number: 7]

The growth of a particular type of seashell is being studied by Manon. At the end of each month Manon records the increase in the width of a seashell since the end of the previous month.
She models the monthly increase in the width of the seashell by a geometric sequence with common ratio 0.8 . In the first month, the width of the seashell increases by 4 mm .

(a)

Find by how much the width of the seashell will increase during the third month, according to her model.

[ 2 ]

(b)

Find the total increase in the width of the seashell, predicted by Manon's model, during the first year.

Manon's seashell had a width of 25 mm at the beginning of the first month.

[ 2 ]

(c)

Find the maximum possible width of the seashell, predicted by Manon's model.

[ 3 ]

[Maximum number: 9]

Imani invests $ 3000 in a bank that pays a nominal annual interest rate of 1.25 % compounded monthly.

(a)

Calculate the amount of money Imani will have in the bank at the end of 6 years. Give your answer correct to two decimal places.

[ 3 ]

(b)

Calculate the number of months it takes until Imani has at least $ 3550 in the bank.

Imani uses the $ 3550 as a partial payment for a used car costing $ 22000. For the remainder she takes out a loan from a bank.

[ 2 ]

(c)

Write down the amount of money that Imani takes out as a loan.

The loan is for 8 years and the nominal annual interest rate is 12.6 % compounded monthly. Imani will pay the loan in fixed monthly instalments at the end of each month.

[ 1 ]

(d)

Calculate the amount, correct to the nearest dollar, that Imani will have to pay the bank each month.

[ 3 ]

[Maximum number: 6]

Maan deposited $ 100000 into a savings account with a nominal annual interest rate of I % compounded monthly. At the end of the eighth year, the amount in the account had increased to $ 150000.

(a)

Find the value of I.

Maan withdraws the $ 150000 and places it in an annuity, earning a nominal annual interest rate of 6.1 % compounded monthly. At the end of each month, Maan will receive a payment of $ 1000.

[ 3 ]

(b)

Find the amount of money remaining in the annuity at the end of 10 years. Express your answer to the nearest dollar.

[ 3 ]

[Maximum number: 3]

This question considers whether it is reasonable to go on all the rides in a theme park and get back to the entrance in two and a half hours.
Martin is visiting a theme park. He will enter the park at 09:00 and must leave the park by 11:30. He uses information available on the internet to calculate whether he will be able to go on all of the rides in the two and a half hours.
He begins by constructing a graph which shows the main paths between the rides and the route of the cable car between the entrance/exit A and ride D.
His graph and the names of the rides are shown in the following diagram.

The weights on the edges of the graph represent the times, in minutes, to walk between the rides and the time to travel by cable car between A and D.

Let T be the shortest possible time, in minutes, taken to visit all the rides, beginning and ending at A .

Martin notices that the graph contains a Hamiltonian cycle. He decides to use the weight of the Hamiltonian cycle as an upper bound for T.

(a)

Find the expected time he will leave the park.

[ 3 ]

[Maximum number: 13]

The following question explores how sequences, series and Markov chains may be used in modelling the number of customers in a commercial setting.
In a town, there are three stores: Aroma, Bodega and Clover.
Ashley is the manager of Aroma. She gathers data to determine whether there is significant movement of customers between the three stores over the course of one year.
She found that:
- 91 % of Aroma customers stayed with Aroma, 5 % moved to Bodega, and 4 % moved to Clover.
- 95 % of Bodega customers stayed with Bodega, 4 % moved to Aroma, and 1 % moved to Clover.
- 92 % of Clover customers stayed with Clover, 6 % moved to Aroma, and 2 % moved to Bodega.
This information is used to form a transition matrix, T.

(a)

Use Model 1 to find the number of customers Ashley can expect Dusk to have in her 10th week.

Ashley wants to increase the number of customers at Dusk, so she introduces a loyalty scheme after her 5th week. She anticipates that the number of new customers she can expect each week should be 12 % greater than the number of new customers the previous week (Model 2).

[ 2 ]

(b)

Show that the number of customers in Ashley's 6th week, according to Model 2, is 753.6 .

[ 2 ]

(c)

Use Model 2 to find how many customers Ashley should expect in her 10th week.

[ 3 ]

(d)

By comparing the two models, determine the first week in which there would be an expected difference of at least 500 customers.

[ 6 ]