IB Mathematics AI SL - Questionbank
Topic 1 All - Number & Algebra
All Questions for Topic 1 (Number & Algebra). Number Skills, Sequences & Series, Financial Mathematics, Systems of Linear Equations
Question Type
All
Paper
Difficulty
View
Question 1
[Maximum mark: 6]
After solving a problem, John has an exact answer of .
-
Write down the exact value of in the form , where .[2]
-
State the value of given correct to significant figures. [1]
-
Calculate the percentage error if is given correct to significant figures. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 2
[Maximum mark: 6]
Given that , where , and .
-
Find the exact value of . [2]
-
Write your answer to part (a)
-
correct to decimal places;
-
correct to significant figures;
-
in the form , where and .[4]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 3
[Maximum mark: 6]
Let , where , , and .
-
Find the value of . Write down your full calculator display. [2]
-
Give your answer to part (a) correct to
-
three significant figures;
-
three decimal places. [2]
-
-
Give the answer to part (b) (i) in the form , where , .[2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 4
[Maximum mark: 6]
Let , where , and .
-
Find the exact value of . [2]
-
Give your answer to part (a) correct to
-
three decimal places;
-
three significant figures. [2]
-
-
Calculate the percentage error if is given to three decimal places. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 5
[Maximum mark: 6]
The volume of a hemisphere, , is given by the formula
where is the total surface area.
The total surface area of a given hemisphere is cm.
-
Calculate the volume of this hemisphere in cm. Give your answer correct to one decimal place. [3]
-
Write down your answer to part (a) correct to the nearest integer. [1]
-
Write down your answer to part (b) in the form , where and .[2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 6
[Maximum mark: 6]
Four cement bags labelled, "5 kg", were delivered to a customer. The customer measured each bag to check their weights and recorded the following:
-
-
Find the mean of the customer's measurements.
-
Calculate the percentage error between the mean and the stated,
approximate weight of kg. [3]
-
-
Calculate , giving your answer
-
correct to the nearest integer;
-
in the form , where and . [3]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 7
[Maximum mark: 6]
The distance between two points with coordinates and is equal to .
-
Calculate the distance between points A and B. Give your answer correct to three significant figures. [3]
-
Give your answer from part (a) correct to one decimal place. [1]
-
Write the answer to part (b) in the form , where , . [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 8
[Maximum mark: 6]
The following diagram shows a rectangle with sides of length cm and cm.
- Write down the area of the rectangle in the form ,
where
and . [3]
Natalie estimates the area of the rectangle to be cm.
- Find the percentage error in Natalie's estimate.
[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 9
[Maximum mark: 6]
Let , where , and .
-
Calculate the exact value of . [2]
-
Give your answer to correct to
-
two significant figures;
-
two decimal places. [2]
-
Sasha estimates the value of to be .
- Calculate the percentage error in Sasha's estimate.
[2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 10
[Maximum mark: 6]
Given , , and ,
-
calculate the value of . [2]
Albert first writes , and correct to one significant figure and then uses these values to estimate the value of .
-
-
Write down , and each correct to one significant figure.
-
Find Albert's estimate of the value of . [2]
-
-
Calculate the percentage error in Albert's estimate of the value of . [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 11
[Maximum mark: 6]
Jeremy invests into a savings account that pays an annual interest rate of %, compounded annually.
-
Write down a formula which calculates that total value of the investment after years. [2]
-
Calculate the amount of money in the savings account after:
-
year;
-
years. [2]
-
-
Jeremy wants to use the money to put down a deposit on an apartment. Determine if Jeremy will be able to do this within a -year timeframe.[2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 12
[Maximum mark: 6]
The th term of an arithmetic sequence is and the common difference is .
-
Find the first term of the sequence. [2]
-
Find the th term of the sequence. [2]
-
Find the sum of the first terms of the sequence. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 13
[Maximum mark: 6]
Only one of the following four sequences is arithmetic and only one of them is geometric.
-
State which sequence is arithmetic and find the common difference of the sequence. [2]
-
State which sequence is geometric and find the common ratio of the sequence.[2]
-
For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 14
[Maximum mark: 6]
The Burns, Gordons and Longstaff families make meal plans for their households. The table below shows the amount of carbohydrate, fat and protein, all measured in grams, consumed by the family over a single day. The table also shows the daily calories, measured in kcal, this equates to.
Let , and represent the amount of calories, in kcal, for g of carbohydrate, fat and protein respectively.
-
Write down a system of three linear equations in terms of , and that represents the information in the table above. [2]
-
Find the values , and . [2]
The Howe family also plans meals. The table below shows the amount of carbohydrates, fat and protein consumed by the family over a single day.
- Calculate the daily calories for the Howe family. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 15
[Maximum mark: 8]
A cuboid has the following dimensions: cm, cm, and cm, measured correct to the nearest millimetre.
- Using these measurements, calculate the volume of the cuboid, in cm. Give your answer to two decimal places. [2]
The lower and upper bounds for the length of the cuboid can be expressed as .
-
Write similar expressions for
-
the width;
-
the height. [2]
-
-
Hence, calculate the minimum volume of the cuboid. Give your answer to three significant figures. [2]
-
Write your answer to part (c) in the form , where and . [2]
Formula Booklet
Mark Scheme
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Revisit
Ask Newton
Question 16
[Maximum mark: 6]
An arithmetic sequence has , , .
-
Find the common difference, . [2]
-
Find . [2]
-
Find . [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 17
[Maximum mark: 6]
Only one of the following four sequences is arithmetic and only one of them is geometric.
-
State which sequence is arithmetic and find the common difference of the sequence. [2]
-
State which sequence is geometric and find the common ratio of the sequence.[2]
-
For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 18
[Maximum mark: 6]
An arithmetic sequence has , , .
-
Find the common difference, . [2]
-
Find . [2]
-
Find . [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 19
[Maximum mark: 7]
Brendan is training for a long distance bike race.
In week of his training he cycled km. In week he cycled km. This pattern continues, with him cycling an extra km per week.
The distances Brendan cycled in the first weeks of training is shown in the following table.
-
Calculate how far he cycles in the th week of his training. [3]
-
In total how far has he cycled after weeks? [2]
-
Find the mean distance per week he cycled over 17 weeks. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 20
[Maximum mark: 6]
In this question give all answers correct to two decimal places.
Mia deposits Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of %, compounded semi-annually.
-
Find the amount of interest that Mia will earn over the next years. [3]
Ella also deposits AUD into a bank account. Her bank pays a nominal annual rate of %, compounded monthly. In years, the total amount in Ella's account will be AUD.
- Find the amount that Ella deposits in the bank account.
[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 21
[Maximum mark: 6]
A geometric sequence has , and .
-
Find the common ratio, . [2]
-
Find the exact value of . [2]
-
Find the exact value of . [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 22
[Maximum mark: 6]
Emily starts reading Leo Tolstoy's War and Peace on the st of February. The number of pages she reads each day increases by the same number on each successive day.
-
Calculate the number of pages Emily reads on the th of February. [3]
-
Find the exact total number of pages Emily reads in the days of February.[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 23
[Maximum mark: 6]
A geometric sequence has terms, with the first four terms given below.
-
Find , the common ratio of the sequence. Give your answer as a fraction. [1]
-
Find , the fifth term of the sequence. Give your answer as a fraction. [1]
-
Find the smallest term in the sequence that is an integer. [2]
-
Find , the sum of the first terms of the sequence. Give your answer correct to one decimal place. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 24
[Maximum mark: 6]
A tennis ball bounces on the ground times. The heights of the bounces, form a geometric sequence. The height that the ball bounces the first time, , is cm, and the second time, , is cm.
-
Find the value of the common ratio for the sequence. [2]
-
Find the height that the ball bounces the tenth time, . [2]
-
Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to decimal places. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 25
[Maximum mark: 6]
The table shows the first four terms of three sequences: , , and .
-
State which sequence is
-
arithmetic;
-
geometric. [2]
-
-
Find the sum of the first terms of the arithmetic sequence. [2]
-
Find the exact value of the th term of the geometric sequence. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 26
[Maximum mark: 6]
Hannah buys a car for . The value of the car depreciates by % each year.
-
Find the value of the car after years. [3]
Patrick buys a car for . The car depreciates by a fixed amount each year, and after years it is worth .
- Find the annual rate of depreciation of the car.
[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 27
[Maximum mark: 6]
Edward wants to buy a new car, and he decides to take out a loan of Australian dollars from a bank. The loan is for years, with a nominal annual interest rate of , compounded monthly. Edward will pay the loan in fixed monthly instalments.
-
Determine the amount Edward should pay each month. Give your answer to the nearest dollar.[3]
-
Calculate the amount Edward will still owe the bank at the end of the third year. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 28
[Maximum mark: 6]
In this question give all answers correct to two decimal places.
Elena invests in a retirement plan in which equal payments of € are made at the beginning of each year. Interest is earned on each payment at a rate of % per year, compounded annually.
-
Find the value of the investment after years. [3]
-
Find the amount of interest Elena will earn on the investment over years.[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 29
[Maximum mark: 6]
A toy rocket is fired, from a platform, vertically into the air, its height above the ground after seconds is given by , where and is measured in metres.
After second, the rocket is m above the ground; after seconds, m; after seconds, m.
-
-
Write down a system of three linear equations in terms of , and .
-
Hence find the values of , and . [3]
-
-
Find the height, above the ground, of the platform. [1]
-
Find the time it takes for the rocket to hit the ground. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 30
[Maximum mark: 6]
Maria invests into a savings account that pays a nominal annual interest rate of %, compounded monthly.
-
Calculate the amount of money in the savings account after years. [3]
-
Calculate the number of years it takes for the account to reach . [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 31
[Maximum mark: 6]
The table below shows the first four terms of three sequences: , , and .
-
State which sequence is
-
arithmetic;
-
geometric. [2]
-
-
Find the exact value of the sum of the first terms of the arithmetic
sequence. [2]
-
Find the exact value of the th term of the geometric sequence. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 32
[Maximum mark: 6]
In this question give all answers correct to two decimal places.
Charlie deposits Canadian dollars (CAD) into a bank account. The bank pays a nominal annual interest rate of %, compounded semi-monthly.
-
Find the amount of interest that Charlie will earn over the next years. [3]
Oscar also deposits CAD into a bank account. His bank pays a nominal annual interest rate of %, compounded quarterly. In years, the total amount in Oscar's account will be CAD.
- Find the amount that Oscar deposits in the bank account.
[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 33
[Maximum mark: 6]
In this question give all answers correct to the nearest whole number.
A population of goats on an island starts at . The population is
expected
to increase by % each year.
-
Find the expected population size after:
-
years;
-
years. [4]
-
-
Find the number of years it will take for the population to reach . [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 34
[Maximum mark: 6]
On the first day of September, , Gloria planted flowers in her garden. The number of flowers, which she plants at every day of the month, forms an arithmetic sequence. The number of flowers she is going to plant in the last day of September is .
-
Find the common difference of the sequence. [2]
-
Find the total number of flowers Gloria is going to plant during September.[2]
-
Gloria estimated she would plant flowers in the month of September. Calculate the percentage error in Gloria's estimate. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 35
[Maximum mark: 6]
At the beginning of each year, Jack invests in a savings account that pays annual interest, compounded quarterly
-
Find the number of years it will take until Jack has in his account. [3]
At the beginning of each year, John invests in a savings account that pays an annual interest rate, compounded semi-annually. After years John has in his account.
- Find the annual interest rate. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 36
[Maximum mark: 6]
The fifth term, , of a geometric sequence is . The sixth term, , of the sequence is .
-
Write down the common ratio of the sequence. [1]
-
Find . [2]
The sum of the first terms in the sequence is .
- Find the value of . [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 37
[Maximum mark: 6]
In this question give all answers correct to the nearest whole number.
Benjamin spends € buying bitcoin mining hardware for his cryptocurrency business. The hardware depreciates by % each year.
-
Find the value of the hardware after two years. [3]
-
Find the number of years it will take for the hardware to be worth less than . [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 38
[Maximum mark: 6]
Ali bought a car for . The value of the car depreciates by % each year.
-
Find the value of the car at the end of the first year. [2]
-
Find the value of the car after years. [2]
-
Calculate the number of years it will take for the car to be worth exactly half its original value. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 39
[Maximum mark: 6]
In this question give all answers correct to two decimal places.
George invests in a retirement plan in which equal payments of are made at the end of each year. Interest is earned on each payment at a rate of % per year, compounded semi-annually.
-
Find the value of the investment after years. [3]
-
Find the amount of interest George will earn on the investment over years.[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 40
[Maximum mark: 6]
An owl takes off from from a tree branch and flies higher into the sky. Its height above the ground after seconds, where , is given by , where and is measured in metres.
Initially the owl is metres above the ground.
- Write down the value of . [1]
After second, the owl is m above the ground; after seconds, m; after seconds, m.
-
-
Write down a system of three linear equations in terms of , and .
-
Hence find the values of , and . [3]
-
After some time the owl reaches a maximum height. At this time it spots some prey at ground level and then immediately swoops down to catch it.
-
-
Find the maximum height of the owl above the ground as it spots the prey.
-
Find the time it catches the prey. [2]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 41
[Maximum mark: 6]
The third term, , of an arithmetic sequence is . The common
difference of
the sequence, , is .
-
Find , the first term of the sequence. [2]
-
Find , the th term of sequence. [2]
The first and fourth terms of this arithmetic sequence are the first two
terms
of a geometric sequence.
- Calculate the sixth term of the geometric sequence.
[2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 42
[Maximum mark: 6]
Isabella and Charlotte both receives Australian dollars (AUD) on their th birthday to invest for later in their life.
Isabella deposits her AUD in a bank account that pays a nominal annual interest rate of %, compounded monthly.
-
The amount in a bank account after years will be AUD. Find the nominal annual interest rate. Give your answer correct to two decimal places.[3]
Charlotte uses her AUD to buy a house that increases in value at a rate of % per year.
- Find the house price after years. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 43
[Maximum mark: 6]
A D printer builds a set of Eifel Tower Replicas in different sizes. The height of the largest tower in this set is cm. The heights of successive smaller towers are % of the preceding larger tower, as shown in the diagram below.
-
Find the height of the smallest tower in this set. [3]
-
Find the total height if all towers were placed one on top of another. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 44
[Maximum mark: 6]
Michael buys a second hand Tesla car for . The value of the car depreciates by each year.
-
Find the total amount the car will depreciate after 5 years, giving your answer correct to the nearest dollar. [3]
The price of a different used car depreciates by each year.
- Find the value reduction of this car after years as a
percentage, when compared to the original purchase price.
[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 45
[Maximum mark: 6]
Charles plans to invest in a retirement plan for years. In this plan, he will deposit British pounds (GBP) at the end of every month and receive a interest rate per annum, compounded monthly.
-
Find the future value of the investment at the end of the years. Give your answer correct to the nearest pound.[3]
After the -year period, Charles will start receiving regular monthly payments of GBP.
- Calculate the number of years it will take Charles's
monthly retirement to match the total
amount originally invested. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 46
[Maximum mark: 6]
The fourth term, , of a geometric sequence is . The fifth term, , is .
-
Find the common ratio of the sequence. [2]
-
Find , the first term of the sequence. [2]
-
Calculate the sum of the first terms of the sequence. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 47
[Maximum mark: 6]
The fifth term, , of a geometric sequence is . The sixth term, , is .
-
Find the common ratio of the sequence. [2]
-
Find , the first term of the sequence. [2]
-
Calculate the sum of the first terms of the sequence. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 48
[Maximum mark: 6]
Mike wants to deposit part of his savings in a bank account that pays an annual interest rate of . The annual inflation rate is expected to be per year throughout the following years. Mike wants his initial deposit to have a real value of after years, compared to current values.
The bank gives Mike two proposals:
-
Find the minimum amount Mike should deposit if he accepts proposal 1. Round your answer to the nearest dollar. [3]
-
Find the minimum value of the annual payments, , if Mike accepts proposal 2. Round your answer to the nearest dollar. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 49
[Maximum mark: 6]
The fifth term, , of an arithmetic sequence is . The eighth term, , of the same sequence is .
-
Find , the common difference of the sequence. [2]
-
Find , the first term of the sequence. [2]
-
Find , the sum of the first terms of the sequence. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 50
[Maximum mark: 6]
The fifth term, , of an arithmetic sequence is . The eleventh term, , of the same sequence is .
-
Find , the common difference of the sequence. [2]
-
Find , the first term of the sequence. [2]
-
Find , the sum of the first terms of the sequence. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 51
[Maximum mark: 6]
Alex invests an amount of USD into a savings account which pays 3.3% (p.a.) interest, compounded monthly. After 5 years Alex has USD in the account.
- Find the amount of USD initially invested, rounding your answer to
two decimal places.[3]
With this money, Alex purchases a used car for dollars, and sells it 3 years later for .
- Find the rate at which the car depreciates per year over
the 3 year period.[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 52
[Maximum mark: 8]
The graph below shows the amount of money (in thousands of dollars), in the account of a contractor, where is the time in months since he opened the account.
- Write down one characteristic of the graph which suggests that a cubic function might be an appropriate model for the amount of money in the account. [1]
The equation of the model can be expressed as , where , , and . It is given that the graph of the model passes through the following points.
-
-
State the value of .
-
Using the values in the table, write down three equations in , , and .
-
By solving the system of equations from part (ii), find the values of , and . [4]
-
If has a negative value, the contractor is in debt.
- Use the model from part (b) to find the number of months the contractor expects to be in debt. Give your answer to the nearest month. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 53
[Maximum mark: 15]
Charles has a New Years Resolution that he wants to be able to complete pushups in one go without a break. He sets out a training regime whereby he completes pushups on the first day, then adds pushups each day thereafter.
- Write down the number of pushups Charles completes
-
on the th training day;
-
on the th training day. [3]
-
On the th training day Charles completes pushups for the first time.
-
Find the value of . [2]
-
Calculate the total number of pushups Charles completes on the first training days. [4]
Charles is also working on improving his long distance swimming in preparation for an Iron Man event in weeks time. He swims a total of metres in the first week, and plans to increase this by % each week up until the event.
-
Find the distance Charles swims in the th week of training. [3]
-
Calculate the total distance Charles swims until the event. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 54
[Maximum mark: 6]
The second and the third terms of a geometric sequence are and .
-
Find the value of , the common ratio of the sequence. [2]
-
Find the value of . [2]
-
Find the largest value of for which is less than .[2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 55
[Maximum mark: 12]
The sum of the first terms of an arithmetic sequence, , is given by .
-
Write down the values of and . [2]
-
Write down the values of and . [2]
-
Find , the common difference of the sequence. [1]
-
Find , the tenth term of the sequence. [2]
-
Find the greatest value of , for which is less than . [3]
-
Find the value of , for which is equal to . [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Video (f)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 56
[Maximum mark: 12]
Coral is a wildlife expert who tags flying fish and records their movement using an electronic device.
The tagging device tells her the height of a fish relative to the water level, at any given time.
She knows that the fish swim mostly in the water, but occasionally they fly (jump!) out of the water.
The height is measured in metres and the time in seconds. If the height is negative the fish is under the water, if the height is positive the fish is flying.
Coral notices one particular fish as it flies out of the water. The moment it re-enters the water the device begins tracking its height.
At seconds the fish is at a height of m; at seconds the fish is at a height of m and at seconds the fish is also at a height of m.
The height of the fish can be expressed as , where , , and .
-
-
Write down the value of .
-
Using the information given write down three equations involving , and .
-
Solve the system of equations to find the values of , and . [4]
-
From previous research, Coral knows that if a fish is flying for more than second then a seagull will attempt to catch it.
- Use a justification to explain why a seagull will attempt to catch this fish. [4]
At s a seagull begins swooping down to catch the fish.
Its height is given by .
-
-
Find the height at which the bird catches the fish.
-
Interpret the answer in the context of the problem. [4]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 57
[Maximum mark: 7]
Consider the quadratic function . The graph of is shown in the diagram below. The vertex of the graph has coordinates .
The graph intersects the -axis at two points; and .
-
Find the value of . [1]
-
Find the values of , , and .[5]
-
Write down the equation of the axis of symmetry of the graph. [1]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 58
[Maximum mark: 6]
The Australian Koala Foundation estimates that there are about koalas left in the wild in . A year before, in , the population of koalas was estimated as . Assuming the population of koalas continues to decrease by the same percentage each year, find:
-
the exact population of koalas in ; [3]
-
the number of years it will take for the koala population to reduce to half of its number in . [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 59
[Maximum mark: 6]
A battalion is arranged, per row, according to an arithmetic sequence. There are troops in the third row and troops in the sixth row.
-
Find the first term and the common difference of this arithmetic sequence. [3]
There are rows in the battalion.
- Find the total number of troops in the battalion.
[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 60
[Maximum mark: 12]
The graph below shows the profit (in thousands of dollars), that business A makes, where is the time in months since January 1st.
- Write down one characteristic of the graph which suggests that a cubic function might be an appropriate model for the business profit. [1]
The model can be expressed as , where , , and . It is given that the graph of the model passes through the following points.
-
-
State the value of .
-
Using the values in the table, write down three equations in , , and .
-
By solving the system of equations from part (ii), find the values of , and . [4]
-
If has a negative value, business A is losing money. The owner has decided they will not open during the corresponding time next year.
- Use the model from part (b) to find the approximate dates during which business A will not open next year. [4]
Business B has a profit function given by , for .
- Determine the total amount of time for which business B is more profitable than business A. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 61
[Maximum mark: 6]
Smith has saved from working a part-time job and wants to invest this money so that it grows over time. His bank offers a savings account that pays an annual interest rate of , compounded quarterly.
- Find how many years it will take for Smith's investment amount to
double in value, rounding your answer to the nearest integer.
[3]
Smith wants his money to grow faster than this first option. His wants to invest the money so that it will double in value in years. He considers an high-growth, higher-risk option, which pays an annual interest of , compounding half-yearly.
- Determine the value of required in this option, rounding your
answer to two decimal places. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 62
[Maximum mark: 6]
Greg has saved British pounds (GBP) over the last six months. He decided to deposit his savings in a bank which offers a nominal annual interest rate of , compounded monthly, for two years.
-
Calculate the total amount of interest Greg would earn over the two years. Give your answer correct to two decimal places. [3]
Greg would earn the same amount of interest, compounded semi-annually, for two years if he deposits his savings in a second bank.
- Calculate the nominal annual interest rate the second bank
offers. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 63
[Maximum mark: 7]
On January 1st 2023, Virgil deposits 1500 Canadian dollars (CAD) into a savings account which pays a nominal annual interest rate of compounded monthly. At the end of each month, Virgil deposits an extra CAD into the savings account.
After months, Virgil will have enough money to withdraw CAD.
- Find the smallest possible value for , given that is a whole number.[4]
At this time, months, annual interest rates have improved. Virgil withdraws CAD and re-invests the remaining money in the same account with the new nominal annual interest rate for 24 months, making no further deposits. After 24 months, Virgil has CAD in the account.
- Determine the new nominal annual interest rate.[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 64
[Maximum mark: 6]
Melinda has in a private foundation. Each year she donates of the money remaining in her private foundation to charity.
-
Find the maximum number of years Melinda can donate to charity while keeping at least in the private foundation. [3]
Bill invests in a bank account that pays a nominal interest rate of %, compounded quarterly, for ten years.
- Calculate the value of Bill's investment at the end of this
time. Give your answer correct to the nearest dollar.
[3]
Formula Booklet
Mark Scheme
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Revisit
Ask Newton
Question 65
[Maximum mark: 15]
Towards the end of 2004, a theatre company upgraded their auditorium and installed new comfortable ergonomic chairs for the audience.
After the redesign, there were seats in the first row and each subsequent row had three more seats than the previous row.
- If the auditorium had a total of rows, find
-
the total number of seats in the last row.
-
the total number of seats in the auditorium. [5]
-
The auditorium reopened for performances at the start of 2005. The average number of visitors per show during that year was . In 2006, the average number of visitors per show increased by .
- Find the average number of visitors per show in 2006. [1]
The average number of visitors per show continued to increase by each year.
- Determine the first year in which the total number of visitors to a
show exceeded the seating capacity of the auditorium. [5]
The theatre company hosts shows per year.
- Determine the total number of visitors that attended the auditorium
from when it opened in 2005 until the end of 2011. Round your answer
correct to the nearest integer. [4]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 66
[Maximum mark: 6]
The first term of an arithmetic sequence is and the common difference is .
-
Find the value of the nd term of the sequence. [2]
The first term of a geometric sequence is . The th term of the geometric sequence is equal to the th term of the arithmetic sequence given above.
-
Write down an equation using this information. [2]
-
Calculate the common ratio of the geometric sequence. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 67
[Maximum mark: 6]
Emily deposits Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of %, compounded monthly.
-
Find the amount of money that Emily will have in her bank account after years. Give your answer correct to two decimal places. [3]
Emily will withdraw the money back from her bank account when the amount reaches AUD.
- Find the time, in full years, until Emily withdraws the
money from her bank account. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 68
[Maximum mark: 6]
In this question give all answers correct to two decimal places.
Stella receives a loan of € for her flower shop business at an interest rate % per year, compounded monthly. She agrees to pay back the loan in equal installments, made at the end of each month over the next five years.
-
Calculate the amount of monthly installment. [3]
Four years after she starts repaying the loan, Stella decides to repay the rest in a final single installment.
- Calculate the amount owing at the end of the four years.
[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 69
[Maximum mark: 6]
In this question give all answers correct to the nearest whole number.
Michelle takes out a loan of . The unpaid balance on the loan has an interest rate of % per year, compounded annually.
-
The loan is to be repaid in payments of made at the end of each year.
-
Find the number of years it will take to repay the loan.
-
Calculate the total amount that has been paid in amortising the loan.[3]
-
-
The loan is to be amortised over years.
-
Find the annual payment made at the end of each year.
-
Calculate the total amount that has been paid in amortising the loan.[3]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 70
[Maximum mark: 15]
A ball is dropped from the top of the Eiffel Tower, metres from the ground. The ball falls a distance of metres during the first second, metres during the next second, metres during the third second, and so on. The distances that the ball falls each second form an arithmetic sequence.
-
-
Find , the common difference of the sequence.
-
Find , the fifth term of the sequence. [2]
-
-
Find , the sum of the first terms of the sequence. [2]
-
Find the time the ball will take to reach the ground. Give your answer in seconds correct to one decimal place. [3]
Assuming the ball is dropped another time from a much higher height than of the Eiffel Tower,
-
find the distance the ball travels from the start of the th second to the end of the th second. [3]
The Eiffel Tower in Paris, France was opened in , and million visitors ascended it during that first year. The number of people who visited the tower the following year () was million.
-
Calculate the percentage increase in the number of visitors from to . Give your answer correct to one decimal place. [2]
-
Use your answer to part (e) to estimate the number of visitors in , assuming that the number of visitors continues to increase at the same percentage rate. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Video (f)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 71
[Maximum mark: 13]
On September 1st, an orchard commences the process of harvesting hectares of apple trees. At the end of September 4th, there were hectares remaining to be harvested, and at the end of September 8th, there were hectares remaining. Assuming that the number of hectares harvested each day is constant, the total number of hectares remaining to be harvested can be described by an arithmetic sequence.
-
Find the number of hectares of apple trees that are harvested each day. [3]
-
Determine the number of hectares remaining to be harvested at the end of September 1st. [1]
-
Determine the date on which the harvest will be complete. [2]
In 2021 the orchard sold their apple crop for . It is expected that the selling price will then increase by annually for the next years.
-
Determine the amount of money the orchard will earn for their crop in 2026. Round your answer to the nearest dollar. [3]
-
-
Find the value of . Round your answer to the nearest integer.
-
Describe, in context, what the value in part (e) (i) represents. [3]
-
-
Comment on whether it is appropriate to model this situation in terms of a geometric sequence. [1]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Video (f)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 72
[Maximum mark: 16]
The number of seats in an auditorium follows a regular pattern where the first row has seats, and the amount increases by the same amount, , each row. In the fifth row, there are seats and in the thirteenth row there are seats.
-
Write down an equation, in terms of and , for the amount of seats
-
in the fifth row.
-
in the thirteenth row.[2]
-
-
Find the value of and .[2]
-
Calculate the total number of seats if the auditorium has 20 rows.[3]
The cost of the ticket for a musical held at the auditorium is inversely proportional to the seat's row. The price for a seat in the first row is $120 dollars, and the price decreases each row. Thus, the value of the ticket for seats in the second row is $116.40 and $112.91 in the third one, etc.
-
-
Find the price of the ticket for a seat in the fifth row, rounding your answer to two decimal places.
-
Find the row of the seat at which the price of a ticket first falls below $70.
-
Find the total revenue the auditorium generates by tickets sales if 40 seats in each of the 20 rows are sold. Give your answer rounded to the nearest dollar.[9]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 73
[Maximum mark: 7]
Two college students, David and Lisa, decide to invest money they have saved from working part-time jobs. David's investment strategy results in an increase of his investment amount by each year. Lisa's investment strategy results in her investment amount increasing by each year.
At the start of the second year of investing, David's total investment amount is and Lisa's is .
- Calculate
-
the original amount David invested.
-
the original amount Lisa invested.[4]
-
During a certain year, , Lisa's investment amount becomes larger than David's amount for the first time.
- Find the value of . [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 74
[Maximum mark: 14]
Georgia is on vacation in Costa Rica. She is in a hot air balloon over a lush jungle in Muelle.
When she leans forward to see the treetops, she accidentally drops her purse. The purse falls down a distance of metres during the first second, metres during the next second, metres during the third second and continues in this way. The distances that the purse falls during each second forms an arithmetic sequence.
-
-
Write down the common difference, , of this arithmetic sequence.
-
Write down the distance the purse falls during the fourth second. [2]
-
-
Calculate the distance the purse falls during the th second. [2]
-
Calculate the total distance the purse falls in the first seconds. [2]
Georgia drops the purse from a height of metres above the ground.
-
Calculate the time, to the nearest second, the purse will take to reach
the ground. [3]
Georgia visits a national park in Muelle. It is opened at the start of and in the first year there were visitors. The number of people who visit the national park is expected to increase by each year.
-
Calculate the number of people expected to visit the national park in . [2]
-
Calculate the total number of people expected to visit the national park by the end of . [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Video (f)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 75
[Maximum mark: 5]
A bouncy ball is dropped out of a second story classroom window, m off the ground. Every time the ball hits the ground it bounces % of its previous height.
-
Find the height the ball reaches after the nd bounce. [2]
-
Find the total distance the ball has travelled when it hits the ground for the th time. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 76
[Maximum mark: 6]
Julia wants to buy a house that requires a deposit of Australian dollars (AUD).
Julia is going to invest an amount of AUD in an account that pays a nominal annual interest rate of %, compounded monthly.
-
Find the amount of AUD Julia needs to invest to reach AUD after years. Give your answer correct to the nearest dollar. [3]
Julia's parents offer to add AUD to her initial investment from part (a), however, only if she invests her money in a more reliable bank that pays a nominal annual interest rate only of %, compounded quarterly.
- Find the number of years it would take Julia to save the
AUD if she accepts her parents money and
follows their advice. Give your answer correct to the nearest
year. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 77
[Maximum mark: 6]
Olivia takes a mortgage (loan) of to buy an apartment in Sydney. on the loan accumulates at the rate of % per year, compounded semi-annually. Olivia agrees with the bank to amortise the loan in monthly payments, made at the beginning of each month.
-
Given that the loan is to be amortised over years, find:
-
the monthly payment amount;
-
the total amount paid in amortising the loan. [4]
-
-
Olivia has the capacity to increase her monthly payments by . Justify to Olivia why this may be a smart financial choice. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 78
[Maximum mark: 5]
Phil's phone shop sells Azura smartphones for and Bellson smartphones for . It is expected that a Bellson smartphone will depreciate at a rate of per year.
After 2 years, an Azura smartphone is worth approximately .
- Show that the expected annual depreciation rate of an Azura
smartphone is 30%. [2]
An Azura smartphone and a Bellson smartphone will have the same value years after they were purchased.
-
Find the value of . [2]
-
Comment on the validity of your answer to part (b). [1]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 79
[Maximum mark: 6]
Let , for .
-
-
Using sigma notation, write down an expression for .
-
Find the value of the sum from part (a) (i). [4]
-
A geometric sequence is defined by , for .
- Find the value of the sum of the geometric series
.[2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 80
[Maximum mark: 14]
Bruce goes into a car dealership to purchase a new vehicle. The one he wants to buy costs , however he doesn't have that much money in his bank. The salesman offers him a financing option of a % deposit followed by monthly payments of .
-
Find the amount of the deposit. [1]
-
Calculate the total cost of the loan under this financing option. [2]
Bruce's father generously offers him an interest free loan of to buy the car to avoid the expensive loan repayments. They agree that Bruce will repay the loan by paying his father in the first month and every following month until the is repaid.
The total amount Bruce's father receives after months is . This can be expressed by the equation . The total amount that Bruce's father receives after months is .
-
Write down a second equation involving and . [1]
-
Determine the value of and the value of . [2]
-
Calculate the number of months it will take Bruce's father to receive
the . [3]
Bruce decides to buy a cheaper car for and invest the remaining . He is considering two investment options over four years.
Option A: Compound interest at an annual rate of %.
Option B: Compound interest at a nominal annual rate of %, compounded monthly.
Express each answer in part (f) to the nearest dollar.
-
Calculate the value of each investment option after four years.
-
Option A.
-
Option B. [5]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Video (f)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 81
[Maximum mark: 6]
Landmarks are placed along the road from London to Edinburgh and the distance between each landmark is km. The first milestone placed on the road is km from London, and the last milestone is near Edinburgh. The length of the road from London to Edinburgh is km.
-
Find the distance between the fifth milestone and London. [3]
-
Determine how many milestones there are along the road. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 82
[Maximum mark: 6]
Let , for .
-
-
Using sigma notation, write down an expression for .
-
Find the value of the sum from part (a) (i). [4]
-
A geometric sequence is defined by , for .
- Find the value of the sum of the geometric series
.[2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 83
[Maximum mark: 6]
Consider the sum , where is a positive integer greater than .
-
Write down the first three terms of the series. [2]
-
Write down the number of terms in the series. [1]
-
Given that , find the value of . [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 84
[Maximum mark: 5]
Tom takes out a loan of to purchase some new machinery for his farming business. He agrees to pay the bank at the end of every month to amortise the loan. Interest accumulates on the balance at a rate of % per year, compounded monthly.
-
Find the number of years and months it takes to pay back the loan. [2]
-
Calculate the total amount that Tom pays in amortising the loan. [1]
-
Tom decides to increase the monthly payment to . How much interest will Tom save in comparison to the former payment schedule.[2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 85
[Maximum mark: 19]
Nathan receives a lump sum inheritance of and invests the money into a savings account with an annual interest rate of , compounded quarterly.
- Calculate the value of Nathan's investment after 5 years, rounding
your answer to the nearest dollar. [3]
After months, the amount in the savings account has increased to more than .
- Find the minimum value of , where .[4]
Nathan is saving to purchase a property. The price of the property is . Nathan puts down a deposit and takes out a loan for the remaining amount.
- Write down the loan amount.[1]
The loan duration is for eight years, compounded monthly, with equal monthly payments of made by Nathan at the end of each month.
- For this loan, find
-
the amount of interest paid by Nathan over the life of the loan.
-
the annual interest rate of the loan, correct to two decimal places. [5]
-
After years of paying this loan, Nathan decides to pay the outstanding loan amount in one final payment.
-
Find the amount of the final payment after years, rounding your answer to the nearest dollar. [3]
-
Find the amount Nathan saved by making this final payment after years, rounding your answer to the nearest dollar.[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Video (f)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 86
[Maximum mark: 7]
Ray takes out a loan of to purchase a house. He agrees to pay the bank at the end of every month to amortise the loan, and interest accumulates on the balance at a rate of % per year, compounded monthly.
-
Find the number of years and months it takes to pay back the loan. [2]
-
Calculate the total amount that Ray has paid in amortising the loan. [2]
-
Ray decides to increase the monthly payment to . Justify this decision.[3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 87
[Maximum mark: 15]
Consider the sequence where
The sequence continues in the same manner.
-
Find the value of . [3]
-
Find the sum of the first terms of the sequence. [3]
Now consider the sequence where
This sequence continues in the same manner.
-
Find the exact value of . [3]
-
Find the sum of the first terms of this sequence. [3]
is the smallest value of for which is greater than .
- Calculate the value of . [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 88
[Maximum mark: 14]
In this question, give all answers correct to the nearest whole number.
Ann is considering investing into a term deposit in one of two banks. The first bank offers an annual interest rate of %, compounding monthly. The second bank offers a fixed payment amount of per month.
-
Calculate:
-
the amount that would be in the account in the first bank at the end of the first year;
-
the amount that would be in the account in the second bank at the end of the first year. [4]
-
-
Write down an expression for:
-
the amount in the account in the first bank at the end of the th year;
-
the amount in the account in the second bank at the end of the th year. [4]
-
-
Calculate the year in which the amount in the first bank account becomes
greater than the amount in the second bank. [2]
-
Calculate:
-
the interest that Ann would earn if she invested in the first bank for years;
-
the interest that Ann would earn if she invested in the second bank for years. [4]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 89
[Maximum mark: 19]
On Wednesday Eddy goes to a velodrome to train. He cycles the first lap of the track in seconds. Each lap Eddy cycles takes him seconds longer than the previous lap.
-
Find the time, in seconds, Eddy takes to cycle his tenth lap. [3]
Eddy cycles his last lap in seconds.
-
Find how many laps he has cycled on Wednesday. [3]
-
Find the total time, in minutes, cycled by Eddy on Wednesday. [4]
On Friday Eddy brings his friend Mario to train. They both cycled the first lap of the track in seconds. Each lap Mario cycles takes him times as long as his previous lap.
-
Find the time, in seconds, Mario takes to cycle his fifth lap. [3]
-
Find the total time, in minutes, Mario takes to cycle his first ten laps. [3]
Each lap Eddy cycles again takes him seconds longer that his
previous lap.
After a certain number of laps Eddy takes less time per lap than Mario.
- Find the number of the lap when this happens. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Video (f)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 90
[Maximum mark: 12]
Lily and Eva both receive Australian dollars (AUD) on their th birthday. Lily deposits her AUD into a bank account. The bank pays an annual interest rate of %, compounded yearly. Eva invests her AUD into a high-yield mutual fund that returns a fixed amount of AUD per year.
-
Calculate:
-
the amount in Lily's bank account at the end of the first year;
-
the total amount of Eva's funds at the end of the first year. [2]
-
-
Write down an expression for:
-
the amount in Lily's bank account at the end of the th year;
-
the total amount of Eva's funds at the end of the th year. [4]
-
-
Calculate the year in which the amount in Lily's bank account becomes
greater than the amount in Eva's fund. [2]
-
Calculate:
-
the interest amount that Lily earns if invested for years, giving your answer correct to two decimal places;
-
the amount of funds that Eva earns for her investment if invested for years. [4]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 91
[Maximum mark: 7]
The half-life, , in years, of a radioactive isotope can be modelled by the function
where is the decay rate, in percent, per year of the isotope.
-
The decay rate of Hydrogen- is % per year. Find its half-life.[2]
The half-life of Uranium- (U-) is years. A sample containing grams of U- is obtained and stored as a side product of a nuclear fuel cycle.
-
Find the decay rate per year of U-. [2]
-
Find the amount of U- left in the sample after:
-
years;
-
years. [3]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 92
[Maximum mark: 7]
The half-life, , in days, of a radioactive isotope can be modelled by the function
where is the decay rate, in percent, per day of the isotope.
-
The decay rate of Gold- is % per day. Find its half-life.[2]
The half-life of Phosphorus- (P-) is days. A sample containing grams of P- is produced and stored in a biochemistry laboratory.
-
Find the decay rate per day of P-. [2]
-
Find the amount of P- left in the sample after:
-
days;
-
days. [3]
-
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 93
[Maximum mark: 17]
The Burj Khalifa, located in Dubai, is the tallest building in the world. It has a height of and has a square base that covers a floor area of . A tourism shop located near the building sells souvenirs of the tower, which sit inside glass pyramids, as illustrated by the diagram below. The souvenir tower is an accurate scale replica of the actual tower.
The scaled model of Burj Khalifa has a base area of . The height and base area dimensions of the glass pyramid are 10% larger than the model.
-
-
Find the height of the souvenir tower, in cm, correct to the nearest mm.
-
Find the volume of the glass pyramid, rounding your answer correct to the nearest cubic centimetre. [5]
-
The shop owner aims to maximise profits from selling the souvenirs. The more the owner orders from the manufacturer, the cheaper the souvenirs are to buy. However, if too many are ordered, profits may decrease due to surplus stock unsold.
The number of souvenirs ordered from previous years and the resulting profits are shown in the following table.
Quantity | Profit($) |
---|---|
The shop owner decides to fit a cubic model of the form
to model the profit, , for thousand souvenirs ordered.
-
Explain why .[1]
-
Construct three linear equations to solve for , and , and hence write down the completed function . [5]
-
Find .[2]
-
Find, to the nearest hundred souvenirs, the optimal number of souvenirs the owner should buy to maximise profit, and the resulting profit from this number. [4]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 94
[Maximum mark: 7]
Jenni is conducting an experiment with a spring and has attached a mass so that it will oscillate up and down.
She is measuring the -coordinate of the centre of the mass.
At the start of the experiment the mass is at rest with its centre being at the point .
She gives the mass a nudge upwards in the positive -direction. She makes her first measurement of when the centre of the mass is at the first maximum point (). The units of the -coordinate are in millimetres.
The mass then moves downwards passing the -axis and reaching its first minimum point (). Jenni makes her second measurement of the -coordinate of the centre of a the mass as .
The mass then moves up past the -axis to the next maximum point () and Jenni makes her third measurement of .
The diagram below shows how the mass moves up and down until Jenni makes her rd measurement.
Jenni notices that the -coordinates of the three measurements form a geometric sequence.
- Find . [2]
The spring continues to oscillate up and down with Jenni measuring the -coordinate in the same way as described.
The results continue to form a geometric sequence.
-
Find the th term in the sequence. Give your answer to 3 decimal places. [2]
-
Show that the total distance travelled in the -direction by the mass when the th measurement is made is mm. [3]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Thank you Revision Village Members
#1 IB Math Resource
Revision Village is ranked the #1 IB Math Resources by IB Students & Teachers.
34% Grade Increase
Revision Village students scored 34% greater than the IB Global Average in their exams (2021).
80% of IB Students
More and more IB students are using Revision Village to prepare for their IB Math Exams.
More IB Math Applications & Interpretation SL Resources
Frequently Asked Questions
What is the IB Math AI SL Questionbank?
The IB Math Applications and Interpretation (AI) SL Questionbank is a comprehensive set of IB Mathematics exam style questions, categorised into syllabus topic and concept, and sorted by difficulty of question. The bank of exam style questions are accompanied by high quality step-by-step markschemes and video tutorials, taught by experienced IB Mathematics teachers. The IB Mathematics AI SL Question bank is the perfect exam revision resource for IB students looking to practice IB Math exam style questions in a particular topic or concept in their AI Standard Level course.
Where should I start in the AI SL Questionbank?
The AI SL Questionbank is designed to help IB students practice AI SL exam style questions in a specific topic or concept. Therefore, a good place to start is by identifying a concept that you would like to practice and improve in and go to that area of the AI SL Question bank. For example, if you want to practice AI SL exam style questions involving Compound Interest & Depreciation, you can go to AI SL Topic 1 (Number & Algebra) and go to the Financial Mathematics area of the question bank. On this page there is a carefully designed set of IB Math AI SL exam style questions, progressing in order of difficulty from easiest to hardest. If you’re just getting started with your revision, you could start at the top of the page with Question 1, or if you already have some confidence, you could start at the medium difficulty questions and progress down.
How should I use the AI SL Questionbank?
The AI SL Questionbank is perfect for revising a particular topic or concept, in-depth. For example, if you wanted to improve your knowledge of Sequences & Series, there is a designed set of full length IB Math AI SL exam style questions focused specifically on this concept. Alternatively, Revision Village also has an extensive library of AI SL Practice Exams, where students can simulate the length and difficulty of an IB exam with the Mock Exam sets, as well as AI SL Key Concepts, where students can learn and revise the underlying theory, if missed or misunderstood in class.
What if I finish the AI SL Questionbank?
With an extensive and growing library of full length IB Math Applications and Interpretation (AI) SL exam style questions in the AI SL Question bank, finishing all of the questions would be a fantastic effort, and you will be in a great position for your final exams. If you were able to complete all the questions in the AI SL Question bank, then a popular option would be to go to the AI SL Practice Exams section on Revision Village and test yourself with the Mock Exam Papers, to simulate the length and difficulty of an actual IB Mathematics AI SL exam.