IB Mathematics AI SL - Popular Quizzes
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Question 1
[Maximum mark: 6]
Let , where , and .
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Find the exact value of . [2]
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Give your answer to part (a) correct to
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three decimal places;
-
three significant figures. [2]
-
-
Calculate the percentage error if is given to three decimal places. [2]
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Question 2
[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]
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Question 3
[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]
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Question 4
[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]
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Question 5
[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]
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Question 6
[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]
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Question 7
[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]
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Question 8
[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]
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Question 9
[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]
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Question 10
[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]
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Question 11
[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]
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Question 12
[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]
-
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Question 13
[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]
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Question 14
[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]
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Question 15
[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]
-
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Question 16
[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]
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Question 17
[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]
-
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Question 18
[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]
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Question 19
[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]
-
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Question 20
[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]
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