IB Mathematics AI HL - Questionbank
Topic 1 All - Number & Algebra
All Questions for Topic 1 (Number & Algebra). Number Skills, Sequences & Series, Financial Mathematics, Complex Numbers, Matrices, Systems of Linear Equations
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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]
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Question 2
[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]
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Question 3
[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]
-
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Question 4
[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]
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Question 5
[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]
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Question 6
[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]
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Question 7
[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]
-
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Question 8
[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]
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Question 9
[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]
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Question 10
[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]
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Question 11
[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]
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Question 12
[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]
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Question 13
[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]
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Question 14
[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]
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Question 15
[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]
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Question 16
[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]
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Question 17
[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]
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Question 18
[Maximum mark: 6]
An arithmetic sequence has , , .
-
Find the common difference, . [2]
-
Find . [2]
-
Find . [2]
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Question 19
[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]
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Question 20
[Maximum mark: 6]
An arithmetic sequence has , , .
-
Find the common difference, . [2]
-
Find . [2]
-
Find . [2]
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Question 21
[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 22
[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]
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Question 23
[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]
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Question 24
[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]
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Question 25
[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]
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Question 26
[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 27
[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]
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Question 28
[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]
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Question 29
[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]
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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]
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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]
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Question 32
[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]
-
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Question 33
[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]
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Question 34
[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]
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Question 35
[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]
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Question 36
[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]
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Question 37
[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]
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Question 38
[Maximum mark: 6]
The following transition diagram reflects the proportions of customers that Qatar Airways loses to its competitor airlines each year, and vice versa.
-
Construct a transition matrix with elements in decimal form. [2]
-
Interpret the meaning of the elements with values
-
-
[2]
-
Assume that the initial state of the market share is
.
- Determine the market share of Qatar Airways after
years. [2]
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Question 39
[Maximum mark: 5]
There are senior students studying Computer Science and senior students studying Mathematics at a university. According to an academic survey, % of these tell they will pursue a postgraduate degree, % will start a business, % will get employed, % will start freelancing and the remaining students will become assistants.
The column matrix
represents the number of senior students studying
Computer Science and Mathematics.
-
Write down a row matrix, , to represent the percentages, in decimal form, of senior students who will choose one of the five routes after graduation. [1]
-
Hence calculate the product . Give each element of the matrix correct to the nearest whole number. [1]
-
In the context of this problem, explain what the element means. [1]
The cost for textbooks per year for a computer science student is and for a mathematics student is .
-
Write down a matrix calculation that gives the total cost for textbooks paid by all the senior students studying Computer Science and Mathematics. [1]
-
Hence calculate the total cost for all the textbooks. Give your answer correct to the nearest dollar. [1]
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Question 40
[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 41
[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]
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Question 42
[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]
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Question 43
[Maximum mark: 5]
In this question give all angles in radians.
Let and .
-
Find . [1]
-
Find:
-
;
-
. [2]
-
-
Find , the angle shown on the diagram below. [2]
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Question 44
[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 45
[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]
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Question 46
[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]
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Question 47
[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]
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Question 48
[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]
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Question 49
[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 50
[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]
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Question 51
[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]
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Question 52
[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]
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Question 53
[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 54
[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]
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Question 55
[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]
-
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Question 56
[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]
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Question 57
[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]
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Question 58
[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]
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Question 59
[Maximum mark: 4]
Domino's Pizza owns two pizzerias at Asia Mall and Metrocity shopping centres. The number of Pacific Veggie, Pepperoni and Buffalo Chicken large pizzas sold during the last week at the two pizzerias is shown in the table below.
The selling price of each type of pizza is shown in the table below.
-
Write down a matrix multiplication that finds the total amount of income from sales of the three types of pizzas that each pizzeria generated during the last week. [2]
-
Hence find the total amount of income from sales of the three types of pizzas that each pizzeria generated during the last week. Give your answers correct to two decimal places. [2]
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Question 60
[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 61
[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]
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Question 62
[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]
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Question 63
[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]
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Question 64
[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]
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Question 65
[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]
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Question 66
[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]
-
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Question 67
[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]
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Question 68
[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 69
[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 70
[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]
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Question 71
[Maximum mark: 6]
Peter is playing on a swing during a school lunch break. The height of the first swing was m and every subsequent swing was % of the previous one. Peter's friend, Ronald, gives him a push whenever the height falls below m.
-
Find the height of the third swing. [2]
-
Find the number of swings before Ronald gives Peter a push. [2]
-
Calculate the total height of swings if Peter is left to swing until coming
to rest. [2]
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Question 72
[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 73
[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 74
[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]
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Question 75
[Maximum mark: 6]
Data scientists, web designers and developers are paid according to an standard. The total annual salary spend for three tech startups paying to the industry standard are summarised in the table below.
Let , and represent the salaries, in thousand dollars, for data scientists, web designers and web developers respectively.
-
Write down a system of three linear equations in terms of , and
that represent the information in the table above. [2]
-
Using matrices, solve the system of linear equations from part (a)
to determine the salaries for the three roles. [2]
Data Quant is a tech startup that also pays to the industry standard and employs data scientists, web designers and web developers.
- Calculate the exact value of the total salary bill for Data
Quant. [2]
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Question 76
[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]
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Question 77
[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]
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Question 78
[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 79
[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]
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Question 80
[Maximum mark: 6]
The Brown, Miller and Taylor families pay utility bills for their houses each month. The table below shows the amount of electricity, water and gas consumed during January by each family, and the total cost of the utilities.
Let , and represent the prices, in dollars, for kWh of electricity, m of water and m of gas, respectively.
-
Write down a system of three linear equations in terms of , and
that represents the information in the table above. [2]
-
Using matrices, find the price for each of the utility. [2]
The Smith family also pay utility bills each month. The table below shows the amount of electricity, water and gas consumed during January by the Smith family.
- Calculate the total cost of the utilities for the Smiths.
[2]
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Question 81
[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]
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Question 82
[Maximum mark: 7]
Let and .
-
Find . [2]
-
Illustrate , and on the same Argand diagram. [3]
-
Let be the angle between and . Find , giving your answer in radians.[2]
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Question 83
[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]
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Question 84
[Maximum mark: 6]
Taste of Home Magazine recommends using a combination of Cheddar, Brie and Swiss when putting together cheese boards for parties. The recommended total cheese board size for a party of - people is kilogram. The table below shows the weight, in hundred of grams, of each kind of cheese required to make one kilogram of cheese combination, and the cost of making each combination.
-
By setting up a system of linear equations and using matrices, find
the price per kilogram of each type of cheese. [4]
John prepares a cheese board with proportion of each cheese type, in hundred grams, as shown in the table below.
- Calculate the amount of money John spent on cheese.
[2]
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Question 85
[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]
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Question 86
[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]
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Question 87
[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]
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Question 88
[Maximum mark: 6]
Three Internet Service Providers (ISPs) are available in a small town. During the year, ISP A is expected to retain % of its customers; % will be lost to ISP B and % to ISP C. ISP B is expected to retain % of its customers; % will be lost to each of the other two ISPs. ISP C is expected to retain % of its customers; % will be lost to ISP A and % to ISP B.
-
Write down a transition matrix that describes the exchange of market shares between the three ISPs during the year. [2]
The current market share held by ISP A is , by ISP B is and by ISP C is .
-
Find the market share held by each ISP after one year. [2]
-
Find the market share held by each ISP after five years if the same trend of market share exchanges between the three ISPs continues. [2]
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Question 89
[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]
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Question 90
[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 91
[Maximum mark: 8]
Two competing radio stations, station A and station B, each have % of the listener market at some point in time. Over each one-year period, station A to take away % of station B's share, and station B manages to take away % of station A's share.
-
Write down a transition matrix that describes the exchange of market shares between the two stations over each one-year period. [1]
-
Find the market share held by each station after one year. [2]
-
Write down the market shares of stations A and B over a five-year period. [2]
-
Find the market share held by each station in the long term if the same trend of market share exchanges between the two stations continues indefinitely. [3]
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Question 92
[Maximum mark: 6]
When a coin is thrown from the top of a skyscraper, its height above the ground after seconds is given by , where and is measured in metres. After second, the coin 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 of the skyscraper. [1]
-
Find the time it takes for the coin to hit the ground. [2]
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Question 93
[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]
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Question 94
[Maximum mark: 7]
Elon is challenged to a speed climb at a local mountain. He has to reach a height of metres above the ground within four hours.
Elon knows he can climb metres in the first hour. Due to increasing tiredness, each hour he can only climb of the height climbed in the previous hour.
-
Verify that Elon reaches his target height of metres in four hours.[2]
The mountain has a height of metres. Elon decides to attempt to climb to the summit.
-
Determine whether he can reach the summit of the mountain if he continues climbing, given his increasing tiredness. Justify your answer.[2]
On a different day, Elon climbs with energy snacks, which help to reduce his tiredness as he climbs. On this day, Elon again climbs 150 metres in the first hour, but then of the height he climbed in the previous hour, where .
- Calculate the minimum value of , given that on this day
Elon is able to reach the summit. Give your answer as a percentage,
to the nearest whole number.[3]
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Question 95
[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]
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Question 96
[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]
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Question 97
[Maximum mark: 8]
Austin allocates a portion of his employment salary each month to investing and invests this money into two stock funds: A and B. He adjusts his investment portfolio each month according to the following transition diagram.
-
Construct a transition matrix with elements in decimal form. [2]
-
Interpret the meaning of the elements with values
-
-
[2]
-
The initial state of his investment portfolio is in stock fund B.
-
-
Find the investment proportion in stock fund A after months.
-
Determine the long term steady state proportion of his investment between the two stock funds. [4]
-
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Question 98
[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]
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Question 99
[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]
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Question 100
[Maximum mark: 6]
The sides of a square are cm long. A new square is formed by joining the midpoints of the adjacent sides and two of the resulting triangles are shaded as shown. This process is repeated more times to form the right hand diagram below.
-
Find the total area of the shaded region in the right hand diagram above. [3]
-
Find the total area of the shaded region if the process is repeated indefinitely.[3]
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Question 101
[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]
-
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Question 102
[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 103
[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]
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Question 104
[Maximum mark: 6]
When a ball is thrown from the top of a tall building, its height above
the ground after seconds is given by
where
and is measured in metres. After
second, the ball is m above the ground;
after seconds, m; after seconds,
m.
-
-
Write down a system of three linear equations in terms of , and .
-
Use matrices to find the values of , and . [3]
-
-
Find the height of the building. [1]
-
Find the time it takes for the ball to hit the ground. [2]
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Question 105
[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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Question 106
[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]
-
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Question 107
[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 108
[Maximum mark: 6]
The vegetables sold at supermarkets in a town are supplied by three major retail suppliers: A, B and C. According to an analysis report, supplier A retains % of their customers each year and lose % to supplier B and % to supplier C. Meanwhile, supplier B retains % of their customers each year and lose % to supplier A and % to supplier C. Supplier C retains % of their customers each year and lose % to supplier A and % to supplier B.
The report also shows that suppliers A, B and C currently hold a market share of %, % and %, respectively.
-
Find the market share held by each supplier after three years. [4]
-
Determine the steady state market share held by each supplier if the same trend remains unchanged. [2]
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Question 109
[Maximum mark: 5]
Laura creates a list of her favorite songs that includes three genres: Jazz, Slow Rock and Country. After her current song ends she randomly selects the next song and the probabilities of genre of the next song are outlined in the following table.
Laura starts her day with a Slow Rock song and is now listening to her fourth song.
-
Determine the genre of music she is currently most likely listening to. [3]
-
Determine which genre of music she listens to most over the long term. [2]
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Question 110
[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 111
[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]
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Question 112
[Maximum mark: 6]
The complex numbers and correspond to the points A and B as shown on the diagram below.
-
Find the exact value of . [2]
-
-
Find the exact perimeter of triangle AOB.
-
Find the exact area of triangle AOB. [4]
-
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Question 113
[Maximum mark: 6]
Let , , and be represented by the points
A, B and C on an Argand diagram as shown below.
The shape OABC is a rectangle.
-
Show that . [1]
-
Find . [1]
-
Express in modulus-argument form. [4]
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Question 114
[Maximum mark: 6]
A circle is drawn on an Argand diagram as shown below. The tangent to the circle from the point B meets the circle at the point A as shown. Let .
-
Show that . [2]
-
Find . [2]
-
Hence write in the form where . [2]
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Question 115
[Maximum mark: 6]
A discrete dynamical system is described by the following transition matrix, ,
The state of the system is defined by the proportions of population with a particular characteristic.
-
Use the characteristic polynomial of to find its eigenvalues. [2]
-
Find the corresponding eigenvectors of . [2]
-
Hence find the steady state matrix of the system. [2]
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Question 116
[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 117
[Maximum mark: 6]
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.
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Find the height the ball reaches after the th bounce. [3]
-
Find the total distance travelled by the ball until it comes to rest. [3]
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Question 118
[Maximum mark: 6]
A bouncy ball is dropped from a height of metres onto a concrete floor. After hitting the floor, the ball rebounds back up to % of it's previous height, and this pattern continues on repeatedly, until coming to rest.
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Show that the total distance travelled by the ball until coming to rest can be expressed by
[2]
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Find an expression for the total distance travelled by the ball, in terms of the number of bounces, . [2]
-
Find the total distance travelled by the ball. [2]
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Question 119
[Maximum mark: 6]
On an Argand diagram, the complex numbers , and are represented by the vertices of a triangle.
Find the area of the triangle.
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Question 120
[Maximum mark: 8]
Consider two power sources with voltages and respectively, where is the frequency of the power source in Hertz (Hz) and is time in seconds.
It is given that
When the two power sources are combined, the resultant voltage is given by
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Find . Give your answer in the form , where .[6]
-
Hence, determine the difference in phase shift between and .[2]
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Question 121
[Maximum mark: 6]
Points A and B represent the complex numbers and as shown on an Argand diagram below.
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Find the angle AOB. [2]
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Find the argument of . [1]
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Given that the real powers of , for , all lie on a unit circle centred at the origin, find the exact value of . [3]
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Question 122
[Maximum mark: 8]
Let and , where .
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Find the value of . Give your answer in the form , where , . [2]
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Find the value of for . Give your answer in the form , where , . [3]
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Find the smallest integer such that . [3]
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Question 123
[Maximum mark: 14]
A biologist conducts an experiment to study the pollination preference of bumblebees' on different floral types. In a flight cage, bumblebees are free to choose between two species of floral: A. majus striatum or A. majus pseudomajus. The changes of pollination behaviors of these bumblebees after every minute are reflected in the following table.
Initially, bumblebees choose A. majus striatum and bumblebees choose A. majus pseudomajus.
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Write down the initial state and the transition matrix . [2]
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Determine and interpret the result. [2]
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Find the eigenvalues and corresponding eigenvectors of . [4]
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Write an expression for the number of bumblebees choosing to pollinate on A. majus pseudomajus after minutes, .
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Hence find the number of bumblebees choose to pollinate on A. majus pseudomajus in the long term. [6]
-
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Question 124
[Maximum mark: 6]
Let where .
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For ,
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express and in the form where ;
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draw and on the following Argand diagram. [4]
-
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Given that the integer powers of lie on a unit circle centred at the origin, find the value of . [2]
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Question 125
[Maximum mark: 6]
Let where .
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For ,
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express and in the form where ;
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draw and on the following Argand diagram. [4]
-
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Given that the integer powers of lie on a unit circle centred
at the origin, find the value of . [2]
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Question 126
[Maximum mark: 14]
An information technology (IT) company offers paid travelling vacation to its employees every year. The employees can choose between travelling domestically or internationally. It is observed that % of the employees who choose to travel domestically one year, choose internationally the next year. Conversely, % of those who choose to travel internationally one year change to travel domestically the following year. For this year, employees chose travelling domestically and employees chose travelling internationally.
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Write down the initial state and the transition matrix . [2]
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Determine and interpret the result. [2]
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Find the eigenvalues and corresponding eigenvectors of . [4]
-
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Write an expression for the number of employees who choose travelling internationally after years, .
-
Hence find the long term steady state number of employees to choose to travel internationally. [6]
-
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Question 127
[Maximum mark: 28]
Steve watches a TV show that has a small section dedicated to trivia quiz questions, where a randomly selected person from the audience is asked to answer 5 general knowledge questions in a row. The following table records the number of correct answers Steve obtains from watching 100 episodes of the program.
- Find Steve's mean number of correct answers per show. [2]
Steve suspects that his number of correct answers can be modelled by a binomial distribution with , and decides to carry out a goodness of fit test.
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Using the data from the table, find the estimated value of for the binomial model.[1]
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Use the binomial model to find the probability that Steve gets all 5 questions incorrect. Express your answer to two significant figures.
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Find the expected frequency for zero correct answers.[3]
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As some expected frequencies are less than , Steve combines two rows in the table to obtain the following updated table.
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Steve uses this table to carry out a goodness of fit test to test the hypothesis that the data follow a binomial distribution with at the significance level. For this test, state:
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the null hypothesis;
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the number of degrees of freedom;
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the value;
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the conclusion, justifying your answer.[6]
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Using the binomial model, find the probability that during the next show, Steve answers all five questions correctly.[2]
Steve's friend Tony considers it might be better to model the problem by using states, where R means a correct answer, and W is an incorrect answer. To simplify the model, Tony proposes a transition of only two states with frequencies shown in the following table.
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Find the probability, in the form , that Steve gets a question incorrect given that he correctly answered the previous question.
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Write down the transition matrix, , for this model.[3]
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-
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Show that the characteristic polynomial for the transition matrix can be expressed as .
-
Hence, or otherwise, find the eigenvalues of .
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Find the eigenvectors of .[7]
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Steve is leaving his home country for a working vacation and won't see the show for at least 6 months. When he returns, he will watch the first episode of the show and attempt to get all five questions correct. Tony claims that, according to the transition model, the probability of this occurring is approximately 10%.
- Determine whether Tony's claim is correct.[4]
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Question 128
[Maximum mark: 5]
Two voltage sources are connected to a circuit. At time milliseconds (ms), the voltage from the first source is and the voltage from the second source is , where both and are measured in volts.
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Write, in the form , an expression for the total voltage in the circuit at time ms. [4]
-
Hence write down the highest voltage in the circuit. [1]
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Question 129
[Maximum mark: 17]
Two grocery stores, store A and store B, serve in a small city. Each year, store A keeps % of its customers while % of them switch to store B. Store B keeps of its customers while % of them switch to store A.
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Write down a transition matrix representing the proportions of the moving between the two stores. [2]
At the end of , store A had customers while store B had customers.
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Find the distribution of the customers between the two stores after two years.[2]
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Show that the eigenvalues of are and .
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Find a corresponding eigenvector for each eigenvalue from part (c) (i).
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Hence express in the form . [6]
-
-
Show that
, where . [2]
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Hence find an expression for the number of customers buying groceries from store A after years, where [3]
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Verify your formula by finding the number of customers buying groceries from store A after two years and comparing with the value found in part (b). [1]
-
Write down the long-term number of customers buying groceries from store A.[1]
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Question 130
[Maximum mark: 14]
Zoologists have been collecting data about the migration habits of a particular species of mammals in two regions; region X and region Y. Each year of the mammals move from region X to region Y and % of the mammals move from region Y to region X. Assume that there are no mammal movements to or from any other neighboring regions.
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Write down a transition matrix representing the movements between the two regions in a particular year. [2]
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Find the eigenvalues of .
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Find a corresponding eigenvector for each eigenvalue of .
-
Hence write down matrices and such that . [6]
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Initially region X had and region Y had of these mammals.
-
Find an expression for the number of mammals living in region Y after
years, where . [5]
-
Hence write down the long-term number of mammals living in region Y. [1]
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Question 131
[Maximum mark: 6]
Ali is swimming in a public pool with some of his friends. At time seconds, he some waves with height from big Bobby jumping into the pool, and waves of height from small Suzie jumping into the pool. Both and are measured in metres.
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Write, in the form , an expression for the total height of the waves Ali encounters at time seconds. [3]
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Find the times in the first seconds when Ali isn't affected by any waves. [2]
-
Find the first time when the waves reaching Ali has maximum height. [1]
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Question 132
[Maximum mark: 14]
A city has two major security guard companies, company A and company B. Each year, % of customers using company A move to company B and % of the customers using company B move to company A. All additional losses and gains of customers by the companies can be ignored.
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Write down a transition matrix representing the movements between the two companies in a particular year. [2]
-
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Find the eigenvalues and corresponding eigenvectors of .
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Hence write down matrices and such that . [6]
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Initially company A and company B both have customers.
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Find an expression for the number of customers company A has after years, where . [5]
-
Hence write down the number of customers that company A
can expect to have in the long term. [1]
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Question 133
[Maximum mark: 6]
The revenues of a four seasons hotel can be modelled by the function
,
where is the number of days after midnight on December.
In a similar way, the operating costs of the hotel can be modelled by the function
.
Both and are measured in thousand dollars.
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Show that the profits of the hotel can be modelled by the function . [3]
-
According to the model, find:
-
the highest profit the hotel will make;
-
the date on which the highest profit will occur. [3]
-
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Question 134
[Maximum mark: 6]
In an unbalanced three-phase electrical circuit, the current at time ms is given by
,
where is measured in milliamperes (mA).
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Write in the form . [4]
-
Hence find the highest current flowing through the circuit, and the time it first occurs. [2]
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