IB Mathematics AA HL - Questionbank
Topic 1 All - Number & Algebra
All Questions for Topic 1 (Number & Algebra). Sequences & Series, Exponents & Logs, Binomial Theorem, Counting Principles, Complex Numbers, Proofs, Systems of Equations
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Question 1
[Maximum mark: 4]
Expand in descending powers of and simplify your answer.
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Question 2
[Maximum mark: 6]
An arithmetic sequence has , , .
-
Find the common difference, . [2]
-
Find . [2]
-
Find . [2]
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Question 3
[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 4
[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 5
[Maximum mark: 5]
Consider the expansion of .
-
Write down the number of terms in this expansion. [1]
-
Find the coefficient of the term in . [4]
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Question 6
[Maximum mark: 7]
Find the value of each of the following, giving your answer as an integer.
-
. [2]
-
. [2]
-
. [3]
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Question 7
[Maximum mark: 4]
Prove that the sum of three consecutive positive integers is divisible by .
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Question 8
[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 9
[Maximum mark: 6]
Find the value of each of the following, giving your answer as an integer.
-
. [2]
-
. [2]
-
. [2]
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Question 10
[Maximum mark: 4]
Consider two consecutive positive integers, and .
Show that the difference of their squares is equal to the sum of the two integers.
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Question 11
[Maximum mark: 6]
Consider an arithmetic sequence
-
Find the common difference, . [2]
-
Find the th term in the sequence. [2]
-
Find the sum of the first terms in the sequence. [2]
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Question 12
[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 13
[Maximum mark: 6]
Consider the expansion of .
-
Write down the number of terms in this expansion. [1]
-
Find the coefficient of the term in . [5]
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Question 14
[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 percentage each year, and after years it is worth .
- Find the annual rate of depreciation of the car.
[3]
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Question 15
[Maximum mark: 4]
The product of three consecutive integers is increased by the middle integer.
Prove that the result is a perfect cube.
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Question 16
[Maximum mark: 6]
Consider the infinite geometric sequence , , , , ...
-
Find the common ratio. [2]
-
Find the th term. [2]
-
Find the exact sum of the infinite sequence. [2]
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Question 17
[Maximum mark: 6]
Consider the infinite geometric sequence , , ,
-
Find the common ratio, . [2]
-
Find the th term. [2]
-
Find the exact sum of the infinite sequence. [2]
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Question 18
[Maximum mark: 6]
Let , , . Write down the following expressions in terms of , and .
-
[2]
-
[2]
-
[2]
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Question 19
[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 20
[Maximum mark: 4]
Expand in descending powers of and simplify your answer.
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Question 21
[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 22
[Maximum mark: 7]
An arithmetic sequence is given by , ,
-
Write down the value of the common difference, . [1]
-
Find
-
;
-
. [4]
-
-
Given that , find the value of . [2]
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Question 23
[Maximum mark: 5]
Consider . Given that , find the value of .
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Question 24
[Maximum mark: 7]
Let and . Write down the following expressions in terms of and .
-
[2]
-
[2]
-
[3]
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Question 25
[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 26
[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 27
[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 28
[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 29
[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 30
[Maximum mark: 7]
The first three terms of a geometric sequence are , , .
-
Find the value of the common ratio, . [2]
-
Find the sum of the first ten terms in the sequence. [2]
-
Find the greatest value of such that . [3]
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Question 31
[Maximum mark: 6]
Consider the following sequence of figures.
Figure 1 contains line segments.
-
Given that Figure contains line segments, show that .[3]
-
Find the total number of line segments in the first figures. [3]
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Question 32
[Maximum mark: 6]
On st of January , Fiona decides to take out a bank loan to purchase a new Tesla electric car. Fiona takes out a loan of with a bank that offers a nominal annual interest rate of , compounded monthly.
The size of Fiona's loan at the end of each year follows a geometric sequence with common ratio, .
-
Find the value of , giving your answer to five significant figures. [3]
The bank lets the size of Fiona's loan increase until it becomes triple the size of the original loan. Once this happens, the bank demands that Fiona pays the entire amount back to close the loan.
- Find the year during which Fiona will need to pay back the
loan. [3]
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Question 33
[Maximum mark: 4]
On the Argand diagram below, the point A represents the complex number and the point B represents the complex number . The shape ABCD is a square.
Determine the complex number represented by:
-
the point C; [2]
-
the point D. [2]
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Question 34
[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 35
[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 36
[Maximum mark: 6]
In an arithmetic sequence, , .
-
Find the common difference. [2]
-
Find the first term. [2]
-
Find the sum of the first terms in the sequence. [2]
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Question 37
[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 38
[Maximum mark: 5]
Solve the equation for .
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Question 39
[Maximum mark: 5]
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. [2]
-
Calculate the number of years it takes for the account to reach . [3]
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Question 40
[Maximum mark: 4]
Find the number of ways in which twelve different baseball cards can be given to Emily, Harry, John and Olivia, if Emily is to receive cards, Harry is to receive cards, John is to receive cards and Olivia is to receive card.
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Question 41
[Maximum mark: 6]
On Gary's th birthday, he invests in an account that pays a nominal annual interest rate of %, compounded monthly.
The amount of money in Gary's account at the end of each year follows a geometric sequence with common ratio, .
-
Find the value of , giving your answer to four significant figures. [3]
Gary makes no further deposits or withdrawals from the account.
- Find the age Gary will be when the amount of money in his
account will be double the amount he invested. [3]
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Question 42
[Maximum mark: 7]
In an arithmetic sequence, the third term is and the ninth term is .
-
Find the common difference. [2]
-
Find the first term. [2]
-
Find the smallest value of such that . [3]
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Question 43
[Maximum mark: 6]
Given that .
-
Find the exact value of . [2]
-
Find the exact value of . [2]
-
Find the value of , giving your answer correct to significant figures. [2]
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Question 44
[Maximum mark: 6]
The first three terms of a geometric sequence are , , .
-
Find the value of the common ratio, . [2]
-
Find . [2]
-
Find . [2]
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Question 45
[Maximum mark: 6]
In an arithmetic sequence, , .
-
Find the common difference. [2]
-
Find the first term. [2]
-
Find the sum of the first terms in the sequence. [2]
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Question 46
[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 months, until Emily withdraws the
money from her bank account. [3]
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Question 47
[Maximum mark: 6]
-
Show that for . [3]
-
Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. [3]
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Question 48
[Maximum mark: 5]
Solve the equation for .
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Question 49
[Maximum mark: 6]
Let , where . Write down each of the following
expressions
in terms of .
-
[2]
-
[2]
-
[2]
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Question 50
[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 51
[Maximum mark: 6]
Using mathematical induction, prove that for all .
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Question 52
[Maximum mark: 7]
The first three terms of a geometric sequence are , , .
-
Find the value of the common ratio, . [2]
-
Find the value of . [2]
-
Find the least value of such that . [3]
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Question 53
[Maximum mark: 5]
In an arithmetic sequence, the sum of the 2nd and 6th term is .
Given that the sum of the first six terms is , determine the first
term and common difference of the sequence.
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Question 54
[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 55
[Maximum mark: 5]
-
Prove that . [3]
-
Determine the set of numbers for which the proof in part (a) is valid. [2]
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Question 56
[Maximum mark: 5]
An arithmetic sequence has first term and common difference .
-
Given that the th term of the sequence is zero, find the value of . [2]
Let denote the sum of the first terms of the sequence.
- Find the maximum value of . [3]
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Question 57
[Maximum mark: 6]
Landmarks are placed along the road from London to Edinburgh and the distance between each landmark is km. The first landmark placed on the road is km from London, and the last landmark is near Edinburgh. The length of the road from London to Edinburgh is km.
-
Find the distance between the fifth landmark and London. [3]
-
Determine how many landmarks there are along the road. [3]
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Question 58
[Maximum mark: 6]
Given that .
-
Find the exact value of . [2]
-
Find the exact value of . [2]
-
Find the value of , giving your answer correct to significant figures. [2]
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Question 59
[Maximum mark: 7]
In a geometric sequence, , .
-
Find the common ratio, . [2]
-
Find . [2]
-
Find the greatest value of such that . [3]
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Question 60
[Maximum mark: 6]
-
Write the expression in the form of , where . [3]
-
Hence, or otherwise, solve . [3]
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Question 61
[Maximum mark: 7]
Use the principle of mathematical induction to prove that
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Question 62
[Maximum mark: 6]
Solve the equation , giving your answers in Cartesian form.
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Question 63
[Maximum mark: 5]
The third term, in descending powers of , in the expansion of is . Find the possible values of .
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Question 64
[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 65
[Maximum mark: 4]
Using the method of proof by contradiction, prove that is irrational.
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Question 66
[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 67
[Maximum mark: 6]
On st of January , Grace invests in an account that pays a nominal annual interest rate of %, compounded quarterly.
The amount of money in Grace's account at the end of each year follows a geometric sequence with common ratio, .
-
Find the value of , giving your answer to four significant figures. [3]
Grace makes no further deposits or withdrawals from the account.
- Find the year in which the amount of money in Grace's
account will become triple the amount she invested. [3]
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Question 68
[Maximum mark: 4]
Tyler needs to decide the order in which to schedule exams for his school. Two of these exams are Chemistry ( SL and HL).
Find the number of different ways Tyler can schedule the exams given that the two Chemistry subjects must not be consecutive.
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Question 69
[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 70
[Maximum mark: 5]
Solve the equation .
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Question 71
[Maximum mark: 5]
Find the values of when .
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Question 72
[Maximum mark: 5]
Solve the equation .
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Question 73
[Maximum mark: 6]
-
Write down the value of
-
;
-
;
-
. [3]
-
-
Hence solve .[3]
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Question 74
[Maximum mark: 6]
-
Write down the value of
-
;
-
;
-
. [3]
-
-
Hence solve .[3]
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Question 75
[Maximum mark: 5]
Consider the expansion of . The constant term is .
Find the possible values of .
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Question 76
[Maximum mark: 5]
Solve , for .
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Question 77
[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 78
[Maximum mark: 6]
Let . Use the method of mathematical induction to prove that
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Question 79
[Maximum mark: 6]
A circle of radius 3 and centre (0,3) is drawn on an Argand diagram. 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 80
[Maximum mark: 6]
A police department has male and female officers. A special group of officers is to be assembled for an undercover operation.
-
Determine how many possible groups can be chosen. [2]
-
Determine how many groups can be formed consisting of males and [2]
-
Determine how many groups can be formed consisting of at least one male. [2]
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Question 81
[Maximum mark: 6]
A school basketball team of students is selected from boys and girls.
-
Determine how many possible teams can be chosen. [2]
-
Determine how many teams can be formed consisting of boys and girls? [2]
-
Determine how many teams can be formed consisting of at most girls? [2]
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Question 82
[Maximum mark: 6]
In this question give all angles in radians.
Let and .
-
Find . [1]
-
Find:
-
;
-
. [3]
-
-
Find , the angle shown on the diagram below. [2]
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Question 83
[Maximum mark: 6]
Prove by contradiction that the equation has no integer roots.
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Question 84
[Maximum mark: 6]
The system of equations given below represents three planes in space.
-
Show that this system of equations has an infinite number of solutions. [3]
-
Find the parametric equations of the line of intersection of the three planes. [3]
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Question 85
[Maximum mark: 5]
Find the values of when .
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Question 86
[Maximum mark: 6]
Jack rides his bike to work each morning. During the first minute, he travels metres. In each subsequent minute, he travels % of the distance travelled during the previous minute.
The distance from his home to work is metres. Jack leaves his house at : am and must be at work at : am.
Will Jack arrive to work on time? Justify your answer.
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Question 87
[Maximum mark: 6]
Consider the complex number where and .
-
Express and in modulus-argument form and write down
-
the modulus of ;
-
the argument of . [4]
-
-
Find the smallest positive integer value of such that is a real number. [2]
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Question 88
[Maximum mark: 5]
Solve the equation .
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Question 89
[Maximum mark: 6]
Find the value of
-
; [2]
-
. [4]
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Question 90
[Maximum mark: 6]
Sarah walks to school each morning. During the first minute, she travels . In each subsequent minute, she travels metres less than the distance she travelled during the previous minute. The distance from her home to school is metres. Sarah leaves her house at : am and must be at school by : am.
Will Sarah arrive to school on time? Justify your answer.
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Question 91
[Maximum mark: 6]
Find the value of
-
; [2]
-
. [4]
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Question 92
[Maximum mark: 8]
Let and .
-
Find . [2]
-
Illustrate , and on the same Argand diagram. [3]
-
Let be the angle between and . Find , giving your answer in radians.[3]
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Question 93
[Maximum mark: 6]
In an art museum, there are 8 different paintings by Picasso, 5 different paintings by Van Gogh, and 3 different paintings by Rembrandt. The curator of the museum wants to hold an exhibition in a hall that can only display a maximum of 7 paintings at a time.
The curator wants to include at least two paintings from each artist in the exhibition.
-
Given that 7 paintings will be displayed, determine how many ways they can be selected. [4]
-
Find the probability that more Rembrandt paintings will be selected than Picasso paintings or Van Gogh paintings. [2]
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Question 94
[Maximum mark: 6]
The fourth term of an arithmetic sequence is equal to and the sum of the first terms is .
-
Find the common difference and the first term. [4]
-
Determine the greatest value of such that the th term is positive. [2]
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Question 95
[Maximum mark: 6]
Let where .
-
For ,
-
express and in the form where ;
-
draw and on the following Argand diagram. [4]
-
-
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 96
[Maximum mark: 6]
Consider the expansion of .
-
Write down the number of terms in this expansion. [1]
-
Find the coefficient of the term in . [5]
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Question 97
[Maximum mark: 6]
Consider the equation , where
and , .
Find the value of and the value of .
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Question 98
[Maximum mark: 6]
The sum of an infinite geometric sequence is . The second term of the sequence is . Find the possible values of .
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Question 99
[Maximum mark: 6]
Let where .
-
For ,
-
express and in the form where ;
-
draw and on the following Argand diagram. [4]
-
-
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 100
[Maximum mark: 7]
Let .
In the expansion of the derivative, , the coefficient of the term in is . Find the possible values of .
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Question 101
[Maximum mark: 7]
The Fibonacci sequence is defined as follows:
Prove by mathematical induction that , where .
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Question 102
[Maximum mark: 6]
In the expansion of , the coefficient of the term in is . Find the value of .
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Question 103
[Maximum mark: 6]
The first term and the common ratio of a geometric series are denoted, respectively, by and , where . Given that the fourth term is and the sum to infinity is , find the value of and the value of .
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Question 104
[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 105
[Maximum mark: 7]
Points A and B represent the complex numbers and as shown on the Argand diagram below.
-
Find the angle AOB. [3]
-
Find the argument of . [1]
-
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 106
[Maximum mark: 6]
The sum of the first three terms of a geometric sequence is , and the sum of the infinite sequence is . Find the common ratio.
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Question 107
[Maximum mark: 6]
The seventh term of an arithmetic sequence is equal to and the sum of the first terms is .
Find the common difference and the first term.
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Question 108
[Maximum mark: 7]
The complex numbers and satisfy the equations
Find and in the form where .
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Question 109
[Maximum mark: 4]
Peter needs to decide the order in which to schedule exams for his school. Two of these exams are Chemistry ( SL and HL).
Find the number of different ways Peter can schedule the exams given that the two Chemistry subjects must not be consecutive.
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Question 110
[Maximum mark: 6]
An arts and crafts store is offering a special package on personalized keychains.
The store has a selection of distinct types of charms.
Customers can personalize their keychains with up to distinct charms from the selection mentioned above.
Determine how many ways a customer can personalize a keychain if
-
The order of the selections is important. [3]
-
The order of the selections is not important. [3]
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Question 111
[Maximum mark: 6]
Ten students are to be arranged in a new chemistry lab. The chemistry lab is set out in two rows of five desks as shown in the following diagram.
-
Find the number of ways the ten students may be arranged in the lab. [1]
Two of the students, Hugo and Leo, were noticed to talk to each other during previous lab sessions.
-
Find the number of ways the students may be arranged if Hugo and Leo must sit so that one is directly behind the other. For example, Desk and Desk . [2]
-
Find the number of ways the students may be arranged if Hugo and Leo must not sit next to each other in the same row. [3]
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Question 112
[Maximum mark: 5]
The third term of an arithmetic sequence is equal to and the sum of the first terms is .
Find the common difference and the first term.
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Question 113
[Maximum mark: 6]
Consider the expansion of , where . The coefficient of the term
in is equal to the coefficient of the term in . Find .
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Question 114
[Maximum mark: 6]
The st, th and th terms of an arithmetic sequence, with common difference , , are the first three terms of a geometric sequence, with common ratio , . Given that the st term of both sequences is , find the value of and the value of .
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Question 115
[Maximum mark: 5]
The system of equations given below represents three planes in space.
Find the set of values of
and such that the three planes have no points of
intersection.
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Question 116
[Maximum mark: 7]
A professor and five of his students attend a talk given in a lecture series. They have a row of 8 seats to themselves.
Find the number of ways the professor and his students can sit if
-
the professor and his students sit together. [3]
-
the students decide to sit at least one seat apart from their professor. [4]
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Question 117
[Maximum mark: 6]
Let where . Find the modulus and argument of , expressing your answers in terms of .
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Question 118
[Maximum mark: 5]
Solve the equation for . Express your answer in terms of and .
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Question 119
[Maximum mark: 6]
Consider the expansion of , where . The coefficient of the term in is , and the coefficient of the term in is .
Find the value of and the value of .
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Question 120
[Maximum mark: 7]
Consider , for , where .
The equation has exactly one solution. Find the value of .
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Question 121
[Maximum mark: 8]
Let .
-
-
Write , and in the form where .
-
Draw , and on an Argand diagram. [6]
-
-
Find the smallest integer such that is a real number. [2]
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Question 122
[Maximum mark: 6]
The system of equations given below represents three planes in space.
Find the set of values of and such that the three planes have exactly one intersection point.
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Question 123
[Maximum mark: 6]
Solve , for .
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Question 124
[Maximum mark: 7]
Consider the equation where and .
Three of the roots of the equation are , and , where .
-
Find the value of .[4]
-
Hence, or otherwise, find the value of .[3]
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Question 125
[Maximum mark: 9]
-
Find three distinct roots of the equation , , giving your answers in modulus-argument form. [6]
The roots are represented by the vertices of a triangle in an Argand diagram.
- Show that the area of the triangle is .
[3]
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Question 126
[Maximum mark: 7]
Consider the complex numbers and .
-
Given that , express in the form where . [4]
-
Find and express it in the form . [3]
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Question 127
[Maximum mark: 5]
Solve the equation for . Express your answer in terms of and .
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Question 128
[Maximum mark: 7]
Consider the expansion of , . The coefficient of the term
in is twelve times the coefficient of the term in . Find .
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Question 129
[Maximum mark: 7]
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. [4]
-
Find the total area of the shaded region if the process is repeated indefinitely.[3]
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Question 130
[Maximum mark: 9]
Consider the following system of equations:
where .
-
Find conditions on and for which
-
the system has no solutions;
-
the system has only one solution;
-
the system has an infinite number of solutions. [6]
-
-
In the case where the number of solutions is infinite, find the general
solution of the system of equations in Cartesian form. [3]
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Question 131
[Maximum mark: 6]
-
Write down and simplify the expansion of in descending order of powers of . [3]
-
Hence find the exact value of . [3]
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Question 132
[Maximum mark: 15]
The first three terms of an infinite geometric sequence are , where .
-
-
Write down an expression for the common ratio, .
-
Hence show that satisfies the equation .[5]
-
-
-
Find the possible values for .
-
Find the possible values for . [5]
-
-
The geometric sequence has an infinite sum.
-
Which value of leads to this sum. Justify your answer.
-
Find the sum of the sequence. [5]
-
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Question 133
[Maximum mark: 8]
Let , for .
-
Find . [1]
-
Prove by mathematical induction that
[7]
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Question 134
[Maximum mark: 15]
The equation has two solutions, and .
- Find the value of and the value of .[5]
A second equation, , also has two solutions, and .
-
-
Show that this second equation can be expressed as
-
Hence find the value of and the value of . [7]
-
-
Given that , find the value of . Give your answer in the form , where .[3]
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Question 135
[Maximum mark: 14]
Alex and Julie each have a goal of saving to put towards a house deposit. They each have to invest.
-
Alex chooses his local bank and invests his in a savings account that offers an interest rate of per annum compounded annually.
-
Find the value of Alex's investment after years, to the nearest hundred dollars.
-
Alex reaches his goal after n years, where n is an integer. Determine the value of n. [4]
-
-
Julie chooses a different bank and invests her in a savings account that offers an interest rate of per annum compounded monthly, where is set to two decimal places.
Find the minimum value of needed for Julie to reach her goal after years. [3]
-
Xavier also wants to reach a savings goal of . He doesn't trust his local bank so he decides to put his money into a safety deposit box where it does not earn any interest. His system is to add more money into the safety deposit box each year. Each year he will add one third of the amount he added in the previous year.
-
Show that Xavier will never reach his goal if his initial deposit into the safety deposit box is .
-
Find the amount Xavier needs to initially deposit in order to reach his goal after years. Give your answer to the nearest dollar. [7]
-
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Question 136
[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.
-
Show that the total distance travelled by the ball until coming to rest can be expressed by
[2]
-
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 137
[Maximum mark: 6]
The sum of the first three terms of a geometric sequence is , and the sum of the infinite sequence is . Find the common ratio.
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Question 138
[Maximum mark: 5]
In the expansion of , the coefficient of the term in is , where . Find .
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Question 139
[Maximum mark: 5]
Consider a geometric sequence with common ratio such that .
- Show that .[2]
A geometric sequence has a first term of 150 and a second term of 120.
- Find the smallest value of such that .[3]
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Question 140
[Maximum mark: 5]
Find the integer values of and for which
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Question 141
[Maximum mark: 13]
Grant wants to save over 5 years to help his son pay for his college tuition. He deposits into a savings account that has an interest rate of per annum compounded monthly for years.
-
Show that Grant will not be able to reach his target. [2]
-
Find the minimum amount, to the nearest dollar, that Grant would need to deposit initially for him to reach his target. [3]
Grant only has to invest, so he asks his sister, Caroline, to help him accelerate the saving process. Caroline is happy to help and offers to contribute part of her income each year. Her annual income is per year. She starts by contributing one fifth of her annual income, and then decreases her contributions by half each year until the target is reached. Caroline's contributions do not yield any interest.
-
Show that Grant and Caroline together can reach the target in 5 years. [4]
Grant and Caroline agree that Caroline should stop contributing once she contributes enough to complement the deficit of Grant's investment.
- Find the whole number of years after which Caroline
will will stop contributing. [4]
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Question 142
[Maximum mark: 8]
Let , for .
The th maximum point on the graph of has -coordinate , where .
-
Given that , find . [4]
-
Hence find the value of such that . [4]
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Question 143
[Maximum mark: 8]
It is known that the number of trees in a small forest will decrease by % each year unless some new trees are planted. At the end of each year, new trees are planted to the forest. At the start of , there are trees in the forest.
-
Show that there will be roughly trees in the forest at the start of . [4]
-
Find the approximate number of trees in the forest at the start of . [4]
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Question 144
[Maximum mark: 5]
In the expansion of , the coefficient of the term in is , where . Find .
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Question 145
[Maximum mark: 18]
The first three terms of an infinite sequence, in order, are
First consider the case in which the series is geometric.
-
-
Find the possible values of .
-
Hence or otherwise, show that the series is convergent. [3]
-
-
Given that and , find the value of . [3]
Now suppose that the series is arithmetic.
-
-
Show that .
-
Write down the common difference in the form , where . [4]
-
-
Given that the sum of the first terms of the sequence is , find the value of . [8]
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Question 146
[Maximum mark: 13]
-
The following diagram shows [PQ], with length cm. The line is divided into an infinite number of line segments. The diagram shows the first four segments.
The length of the line segments are cm, cm, cm, , where .
Show that . [5]
-
The following diagram shows [RS], with length cm, where . Squares with side lengths cm, cm, cm, , where , are drawn along [RS]. This process is carried on indefinitely. The diagram shows the first four squares.
The total sum of the areas of all the squares is . Find the value of . [8]
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Question 147
[Maximum mark: 8]
-
Show that . [3]
-
Hence, or otherwise, solve , for .[5]
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Question 148
[Maximum mark: 6]
Julie works at a book store and has nine books to display on the main shelf of the store. Four of the books are non-fiction and five are fiction. Each book is different. Determine the number of possible ways Julie can line up the nine books on the main shelf, given that
-
the non-fiction books should stand together; [2]
-
the non-fiction books should stand together on either end; [2]
-
the non-fiction books should stand together and do not stand on either end. [2]
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Question 149
[Maximum mark: 7]
Solve the simultaneous equations:
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Question 150
[Maximum mark: 5]
Use the extension of the binomial theorem for to show that , .
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Question 151
[Maximum mark: 7]
Consider the equation , where and . Two of the roots of the equation are and and the sum of all the roots is .
Show that .
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Question 152
[Maximum mark: 9]
Let , .
-
Find . [2]
-
Prove by induction that for all .[7]
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Question 153
[Maximum mark: 9]
Consider the following system of equations:
where .
-
Find conditions on and for which
-
the system has no solutions;
-
the system has only one solution;
-
the system has an infinite number of solutions. [6]
-
-
In the case where the number of solutions is infinite, find the general
solution of the system of equations in Cartesian form. [3]
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Question 154
[Maximum mark: 6]
Using the principle of mathematical induction, prove that is divisible by for all integers .
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Question 155
[Maximum mark: 14]
The first two terms of an infinite geometric sequence, in order, are
-
Find the common ratio, . [2]
-
Show that the sum of the infinite sequence is . [3]
The first three terms of an arithmetic sequence, in order, are
-
Find the common difference , giving your answer as an integer. [3]
Let be the sum of the first terms of the arithmetic sequence.
-
Show that . [3]
-
Given that is equal to one third of the sum of the infinite geometric
sequence, find , giving your answer in the form where . [3]
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Question 156
[Maximum mark: 8]
-
Show that . [3]
-
Hence, or otherwise, solve , for .[5]
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Question 157
[Maximum mark: 8]
Consider the following system of equations:
where .
-
Show that this system does not have a unique solution for any value of . [4]
-
-
Determine the value of for which the system is consistent.
-
For this value of , find the general solution of the system. [4]
-
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Question 158
[Maximum mark: 18]
-
Express in the form , where and . [5]
Let the roots of the equation be , and .
-
Find , and expressing your answers in the form , where and . [5]
On an Argand diagram, , and are represented by the points A, B and C, respectively.
-
Find the area of the triangle ABC. [4]
-
By considering the sum of the roots , and , show that
[4]
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Question 159
[Maximum mark: 6]
Given a sequence of integers, between and , which are divisible by .
-
Find their sum. [2]
-
Express this sum using sigma notation. [2]
An arithmetic sequence has first term and common difference of . The sum of the first terms of this sequence is negative.
- Find the greatest value of . [2]
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Question 160
[Maximum mark: 7]
Given that , find the value of .
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Question 161
[Maximum mark: 5]
Sophia and Zoe compete in a freestyle swimming race where there are no tied finishes and there is a total of competitors.
Find the total number of possible ways in which the ten swimmers can finish if Zoe finishes
-
in the position immediately after Sophia;[2]
-
in any position after Sophia.[3]
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Question 162
[Maximum mark: 8]
Use the extension of the binomial theorem for to show that , .
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Question 163
[Maximum mark: 12]
Consider the complex numbers and .
-
Calculate giving your answer both in modulus-argument form and
Cartesian form. [7]
-
Use your results from part (a) to find the exact value of ,
giving your answer in the form where . [5]
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Question 164
[Maximum mark: 18]
Consider the family of polynomials of the form where coefficients , , and belong to the set .
-
Find the number of possible polynomials if
-
each coefficient value can be repeated;
-
each coefficient must be different.[4]
-
Consider the case where has as a factor, two purely imaginary roots, and all the coefficients are different.
-
-
By considering the sum of the roots, find the two possible combinations for coefficients and .
-
Show that there is only one way to assign the values , , , and if .[7]
-
Now, consider the polynomial with the coefficients found in part (b) (ii).
-
-
Express as a product of one linear and one quadratic factor.
-
Determine the three roots of .[7]
-
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Question 165
[Maximum mark: 8]
-
Solve the inequality . [2]
-
Use mathematical induction to prove that for all , .[6]
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Question 166
[Maximum mark: 15]
The first two terms of an infinite geometric sequence are and , where , and .
-
-
Find an expression for in terms of .
-
Find the possible values of . [5]
-
-
Show that the sum of the infinite sequence is . [4]
-
Find the values of which give the greatest value of the sum. [6]
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Question 167
[Maximum mark: 6]
There are players on a football team who are asked to line up in one straight line for a team photo. Three of the team members named Adam, Brad and Chris refuse to stand next to each other. There is no restriction on the order in which the other team members position themselves.
Find the number of different orders in which the team members can be positioned for the photo.
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Question 168
[Maximum mark: 7]
Given that , find the possible values of and .
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Question 169
[Maximum mark: 15]
Consider where and .
-
If ,
-
write in the form ;
-
find the value of . [5]
-
-
Show that in general,
[4]
-
Find condition under which . [2]
-
State condition under which is:
-
real;
-
purely imaginary. [2]
-
-
Find the modulus of given that . [2]
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Question 170
[Maximum mark: 7]
There are six office cubicles arranged in a grid with two rows and three columns as shown in the following diagram. Aria, Bella, Charlotte, Danna, and Emma are to be stationed inside the cubicles to work on various company projects.
Find the number of ways of placing the team members in the cubicles in each of the following cases.
-
Each cubicle is large enough to contain the five team members, but Danna and Emma must not be placed in the same cubicle.[2]
-
Each cubicle may only contain one team member. But Aria and Bella must not be placed in cubicles which share a boundary, as they tend to get distracted by each other.[5]
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Question 171
[Maximum mark: 7]
-
Write down the quadratic expression in the form .[2]
-
Hence, or otherwise, find the coefficient of the term in in the expansion
of . [5]
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Question 172
[Maximum mark: 6]
On an Argand diagram, the complex numbers , and are represented by the vertices of a triangle.
The exact area of the triangle can be expressed in the form . Find the value of and of .
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Question 173
[Maximum mark: 8]
The first three terms of a geometric sequence are , , , for .
-
Find the common ratio. [3]
-
Solve . [5]
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Question 174
[Maximum mark: 11]
Sophie and Ella play a game. They each have five cards showing roman numerals I, V, X, L, C. Sophie lays her cards face up on the table in order I, V, X, L, C as shown in the following diagram.
Ella shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Sophie's 4 card directly above. Sophie wins if no matches occur; otherwise Ella wins.
-
Show that the probability that Sophie wins the game is .[6]
Sophie and Ella repeat their game so that they play a total of times. Let the discrete random variable represent the number of times Sophie wins.
- Determine:
-
the mean of ;
-
the variance of . [5]
-
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Question 175
[Maximum mark: 6]
The barcode strings of a new product are created from four letters A, B, C, D and ten digits . No three of the letters may be written consecutively in a barcode string. There is no restriction on the order in which the numbers can be written.
Find the number of different barcode strings that can be created.
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Question 176
[Maximum mark: 19]
Let , for .
-
-
Find using the binomial theorem.
-
Use de Moivre's theorem to show that and . [8]
-
-
Hence show that . [6]
-
Given that , find the exact value of . [5]
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Question 177
[Maximum mark: 14]
-
Show that , where . [3]
-
Hence show that . [2]
-
Prove by mathematical induction that
[9]
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Question 178
[Maximum mark: 19]
-
-
Expand by using the binomial theorem.
-
Hence use de Moivre's theorem to prove that
-
State a similar expression for in terms of and . [6]
-
Let
,
where is measured in degrees, be the solution
of which has the
smallest positive argument.
-
Find the modulus and argument of . [4]
-
Use (a) (ii) and your answer from (b) to show that . [4]
-
Hence express in the form where . [5]
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Question 179
[Maximum mark: 22]
-
Solve , for . [5]
-
Show that . [3]
-
Let , for , .
-
Find the modulus and argument of in terms of .
-
Hence find the fourth roots of in modulus-argument form. [14]
-
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Question 180
[Maximum mark: 16]
-
Find the roots of which satisfy the condition ,
expressing your answer in the form , where . [5]
-
Let be the sum of the roots found in part (a).
-
Show that .
-
By writing as , find the value of in the form ,
where and are integers to be determined.
-
Hence, or otherwise, show that . [11]
-
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Question 181
[Maximum mark: 17]
-
Solve the equation , , giving your answer in the form
and in the form where . [6]
-
Consider the complex numbers .
-
Write in the form .
-
Calculate and write in the form where .
-
Hence find the value of in the form where .
-
Find the smallest such that is a positive real number. [11]
-
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Question 182
[Maximum mark: 15]
Bill takes out a bank loan of to buy a premium electric car, at an annual interest rate of %. The interest is calculated at the end of each year and added to the amount outstanding.
-
Find the amount of money Bill would owe the bank after years. Give your answer to the nearest dollar. [3]
To pay off the loan, Bill makes quarterly deposits of at the end of every quarter in a savings account, paying a nominal annual interest rate of %. He makes his first deposit at the end of the first quarter after taking out the loan.
-
Show that the total value of Bill's savings after years is . [3]
-
Given that Bill's aim is to own the electric car after years, find the value for to the nearest dollar. [3]
Melinda visits a different bank and makes a single deposit of , the annual rate being %.
-
-
Melinda wishes to withdraw at the end of each year for a period of years. Show that an expression for the minimum value of is
-
Hence, or otherwise, find the minimum value of that would permit Melinda to withdraw annual amounts of indefinitely. Give your answer to the nearest dollar. [6]
-
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Question 183
[Maximum mark: 28]
This question asks you to explore the sequence defined by
where and are the roots of the quadratic equation and .
-
Find the value of and the value of . Give your answers in the form , where .[3]
-
Hence find the values of and . [4]
-
Show that and . [1]
-
Hence show that .[4]
-
Suppose that and are integers. Show that is also an integer.[2]
-
Hence show that is an integer for all .[2]
Now consider the sequence defined by
-
Find the exact values of and .[4]
-
Express in terms of and .[4]
-
Hence show that is a multiple of for all .[4]
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Question 184
[Maximum mark: 11]
Consider three planes represented by the following system of equations:
Where .
-
State the values of , and for which
-
The system has infinite solutions.
-
The system is inconsistent. [7]
-
-
In the case where the system has infinite solutions, describe the geometric relationship between the three planes. [2]
-
In the case where the system is inconsistent, identify one of the geometric relationships that could exist between the three planes. [2]
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Question 185
[Maximum mark: 24]
This question asks you to investigate some properties of hexagonal numbers.
Hexagonal numbers can be represented by dots as shown below where denotes the th hexagonal number, .
Note that points are required to create the regular hexagon with side of length , while points are required to create the next hexagon with side of length , and so on.
-
Write down the value of .[1]
-
By examining the pattern, show that , . [3]
-
By expressing as a series, show that , .[3]
-
Hence, determine whether is a hexagonal number.[3]
-
Find the least hexagonal number which is greater than .[5]
-
Consider the statement:
is the only hexagonal number which is divisible by .
Show that this statement is false.[2]
Matt claims that given and , , then
- Show, by mathematical induction, that Matt's claim is true
for all .[7]
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Question 186
[Maximum mark: 14]
The cubic polynomial equation has three roots and . By expanding the product , show that
-
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;
-
;
-
. [3]
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It is given that and for parts (b) and (c) below.
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In the case that the three roots and form an arithmetic
sequence, show that one of the roots is . -
Hence determine the value of . [5]
-
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In another case the three roots form a geometric sequence. Determine
the value of . [6]
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Question 187
[Maximum mark: 30]
This question will investigate power series, as an extension to the
Binomial Theorem for negative and fractional indices.
A power series in is defined as a function of the form
where the .
- Expand using the Binomial Theorem.[3]
Consider the geometric series
-
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State for which values the geometric series is convergent.
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Show that, for this set of values, the sum of the series is .[4]
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By differentiating the series , show that
[2]
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By differentiating the equation obtained in part (c), show that
[2]
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Hence by recognising the pattern, deduce that for ,
[4]
Now, we will determine how to generalize the expansion of for .
Suppose with can be written as the power series
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By substituting , find the value of .[1]
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By differentiating the series of and evaluating at find the value of .[2]
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By repeating the procedure of part (g) find the value of and .[4]
-
Hence, write down the first four terms of the series expansion for called the Extended Binomial Theorem.[1]
-
Write down the power series for , including the first four terms.[3]
-
Hence, integrating the series found in part (j), find the power series for , including the first four terms.[4]
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Question 188
[Maximum mark: 21]
-
Use de Moivre's theorem to find the value of . [2]
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Use mathematical induction to prove that
[6]
Let .
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Find an expression in terms of for , , where is the complex conjugate of . [2]
-
-
Show that .
-
Write down and simplify the binomial expansion of in terms of and .
-
Hence show that . [5]
-
-
Hence solve for . [6]
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Question 189
[Maximum mark: 23]
Let , for .
-
Find . [2]
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Prove by induction that for all . [7]
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Find the coordinates of any local maximum and minimum points on the graph of . Justify whether such point is a maximum or a minimum. [5]
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Find the coordinates of any points of inflexion on the graph of . Justify whether such point is a point of inflexion. [5]
-
Hence sketch the graph of , indicating clearly the points found in parts (c) and (d) and any intercepts with the axes. [4]
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Question 190
[Maximum mark: 20]
-
Solve the equation , . [5]
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Show that . [4]
-
Let , for , .
-
Find the modulus and argument of . Express each answer
in its simplest form. -
Hence find the fourth roots of in modulus-argument form. [11]
-
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Question 191
[Maximum mark: 25]
Jack and John have decided to play a game. They will be rolling a die seven times. One roll of a die is considered as one round of the game. On each round, John agrees to pay Jack $4 if or is rolled, Jack agrees to pay John $2 if or is rolled, and who is paid wins the round. In the end, who earns money wins the game.
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Show that the probability that Jack wins exactly two rounds is . [3]
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Explain why the total number of outcomes for the results of the seven rounds is .
-
Expand and choose a suitable value of to prove that
-
Give a meaning of the equality above in the context of the seven
rounds.[4]
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Find the expected amount of money earned by each player in the game.
-
Who is expected to win the game?
-
Is this game fair? Justify your answer. [3]
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Jack and John have decided to play the game again.
-
Find an expression for the probability that John wins five rounds on the first game and two rounds on the second game. Give your answer in the form
where the values of and are to be found.
-
Use your answer to (d) (i) and seven similar expressions to write down the probability that John wins a total of seven rounds over two games as the sum of eight probabilities.
-
Hence prove that
[9]
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Now Jack and John roll a die times. Let denote the number of rounds Jack wins. The expected value of can be written as
-
Find the value of and .
-
Differentiate the expansion of to prove that the expected
number of rolls Jack wins is . [6]
-
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Question 192
[Maximum mark: 28]
This questions will investigate a method to obtain Cardano's formula using elementary properties of complex numbers and algebraic expressions.
- The roots, of magnitude , described above can be written polar form such that . Where, the argument , is .
-
Show that and find the remaining roots, and , in polar form.
-
Show that .
-
Show that .
-
Hence, show that in Cartesian form. [7]
-
Consider the cubic equation for , .
Assume we can write it in a factorised form such that .
-
- Identify and sum the roots of the factorised form above and then use some of the relations relations found in part (a) to show that
Another formula for the roots , and of the cubic equation states that .
-
Using the roots identified from (i), the formula above and some of the relations found in part (a) show that .
-
Using parts (i) and (ii), write and in terms of and .
-
Hence, show that is a root of the cubic equation . [12]
Now, consider a more general cubic equation , where , .
-
-
By substituting and prove Cardano's formula, which states that
is a root of the cubic equation .
-
Use Cardano's formula to find the exact solution to .
-
Using the substitution and Cardano's formula, find an exact solution to .[9]
-
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Question 193
[Maximum mark: 21]
Let ,.
-
Show that . [3]
-
Use mathematical induction to prove that[9]
Let , .
Consider the function defined by for .
The term in the Maclaurin series for has a coefficient of .
- Find the possible values of .[9]
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Question 194
[Maximum mark: 17]
The following diagram shows the graph of
for ,
with asymptotes at and .
-
Describe a sequence of transformations that transforms the graph of
to the graph of for .[3]
-
Show that .[3]
-
Verify that .[3]
-
Using mathematical induction and the results from part (b) and (c), prove that[8]
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Frequently Asked Questions
What is the IB Math AA HL Questionbank?
The IB Math Analysis and Approaches (AA) HL Questionbank is a comprehensive set of IB Mathematics exam style questions, categorised into syllabus topic and concept, and sorted by difficulty of question. The bank of exam style questions are accompanied by high quality step-by-step markschemes and video tutorials, taught by experienced IB Mathematics teachers. The IB Mathematics AA HL Question bank is the perfect exam revision resource for IB students looking to practice IB Math exam style questions in a particular topic or concept in their AA Higher Level course.
Where should I start in the AA HL Questionbank?
The AA HL Questionbank is designed to help IB students practice AA HL exam style questions in a specific topic or concept. Therefore, a good place to start is by identifying a concept that you would like to practice and improve in and go to that area of the AA HL Question bank. For example, if you want to practice AA HL exam style questions that involve Complex Numbers, you can go to AA SL Topic 1 (Number & Algebra) and go to the Complex Numbers area of the question bank. On this page there is a carefully designed set of IB Math AA HL exam style questions, progressing in order of difficulty from easiest to hardest. If you’re just getting started with your revision, you could start at the top of the page with Question 1, or if you already have some confidence, you could start at the medium difficulty questions and progress down.
How should I use the AA HL Questionbank?
The AA HL Questionbank is perfect for revising a particular topic or concept, in-depth. For example, if you wanted to improve your knowledge of Counting Principles (Combinations & Permutations), there is a set of full length IB Math AA HL exam style questions focused specifically on this concept. Alternatively, Revision Village also has an extensive library of AA HL Practice Exams, where students can simulate the length and difficulty of an IB exam with the Mock Exam sets, as well as AA HL Key Concepts, where students can learn and revise the underlying theory, if missed or misunderstood in class.
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With an extensive and growing library of full length IB Math Analysis and Approaches (AA) HL exam style questions in the AA HL Question bank, finishing all of the questions would be a fantastic effort, and you will be in a great position for your final exams. If you were able to complete all the questions in the AA HL Question bank, then a popular option would be to go to the AA HL Practice Exams section on Revision Village and test yourself with the Mock Exam Papers, to simulate the length and difficulty of an actual IB Mathematics AA HL exam.