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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

no calculator

easy

[Maximum mark: 4]

Expand (2x+1)4(2x + 1)^4 in descending powers of xx and simplify your answer.

easy

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Question 2

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easy

[Maximum mark: 6]

An arithmetic sequence has u1=40u_1= 40, u2=32u_2 = 32, u3=24u_3 = 24.

  1. Find the common difference, dd. [2]

  2. Find u8u_8. [2]

  3. Find S8S_8. [2]

easy

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Question 3

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easy

[Maximum mark: 6]

Jeremy invests $8000\$8000 into a savings account that pays an annual interest rate of 5.55.5 %, compounded annually.

  1. Write down a formula which calculates that total value of the investment after nn years. [2]

  2. Calculate the amount of money in the savings account after:

    1. 11 year;

    2. 33 years. [2]

  3. Jeremy wants to use the money to put down a $10000\$10\hspace{0.15em}000 deposit on an apartment. Determine if Jeremy will be able to do this within a 55-year timeframe.[2]

easy

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Question 4

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easy

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

an=13,14,15,16,cn=3,1,13,19,bn=2.5,5,7.5,10,dn=1,3,6,10,\begin{array}{rcccccl} a_n &=& \dfrac{1}{3},\,\dfrac{1}{4},\,\dfrac{1}{5},\,\dfrac{1}{6},\,\dots &\,\hspace{4em}\,& c_n &=& 3,\,1,\,\dfrac{1}{3},\,\dfrac{1}{9},\,\dots \\[12pt] b_n &=& 2.5,\,5,\,7.5,\,10,\,\dots &\,\hspace{4em}\,& d_n &=& 1,\,3,\,6,\,10,\,\dots \end{array}
  1. State which sequence is arithmetic and find the common difference of the sequence. [2]

  2. State which sequence is geometric and find the common ratio of the sequence.[2]

  3. For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction. [2]

easy

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Question 5

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easy

[Maximum mark: 5]

Consider the expansion of (x3)8(x-3)^8.

  1. Write down the number of terms in this expansion. [1]

  2. Find the coefficient of the term in x6x^6. [4]

easy

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Question 6

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easy

[Maximum mark: 7]

Find the value of each of the following, giving your answer as an integer.

  1. log10100\log_{10} 100. [2]

  2. log1050+log102\log_{10} 50 + \log_{10} 2. [2]

  3. log104log1040\log_{10} 4 - \log_{10} 40. [3]

easy

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Question 7

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easy

[Maximum mark: 4]

Prove that the sum of three consecutive positive integers is divisible by 33.

easy

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Question 8

calculator

easy

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

an=1,5,10,15,cn=1.5,3,4.5,6,bn=12,23,34,45,dn=2,1,12,14,\begin{array}{rcccccl} a_n &=& 1,\,5,\,10,\,15,\,\dots &\,\hspace{4em}\,& c_n &=& 1.5,\,3,\,4.5,\,6,\,\dots \\[12pt] b_n &=& \dfrac{1}{2},\,\dfrac{2}{3},\,\dfrac{3}{4},\,\dfrac{4}{5},\,\dots &\,\hspace{4em}\,& d_n &=& 2,\,1,\,\dfrac{1}{2},\,\dfrac{1}{4},\,\dots \end{array}
  1. State which sequence is arithmetic and find the common difference of the sequence. [2]

  2. State which sequence is geometric and find the common ratio of the sequence.[2]

  3. For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction. [2]

easy

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Question 9

no calculator

easy

[Maximum mark: 6]

Find the value of each of the following, giving your answer as an integer.

  1. log66\log_6 6. [2]

  2. log69+log64\log_6 9 + \log_6 4. [2]

  3. log672log62\log_6 72 - \log_6 2. [2]

easy

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Question 10

no calculator

easy

[Maximum mark: 4]

Consider two consecutive positive integers, kk and k+1k+1.

Show that the difference of their squares is equal to the sum of the two integers.

easy

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Question 11

no calculator

easy

[Maximum mark: 6]

Consider an arithmetic sequence 2,6,10,14,2,6,10,14,\dots

  1. Find the common difference, dd. [2]

  2. Find the 1010th term in the sequence. [2]

  3. Find the sum of the first 1010 terms in the sequence. [2]

easy

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Question 12

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easy

[Maximum mark: 6]

The fifth term, u5u_5, of a geometric sequence is 125125. The sixth term, u6u_6, is 156.25156.25.

  1. Find the common ratio of the sequence. [2]

  2. Find u1u_1, the first term of the sequence. [2]

  3. Calculate the sum of the first 1212 terms of the sequence. [2]

easy

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Question 13

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easy

[Maximum mark: 6]

Consider the expansion of (2x1)9(2x-1)^9.

  1. Write down the number of terms in this expansion. [1]

  2. Find the coefficient of the term in x2x^2. [5]

easy

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Question 14

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easy

[Maximum mark: 6]

Hannah buys a car for $24900\$24\hspace{0.15em}900. The value of the car depreciates by 1616 % each year.

  1. Find the value of the car after 1010 years. [3]

Patrick buys a car for $12000\$12\hspace{0.15em}000. The car depreciates by a fixed percentage each year, and after 66 years it is worth $6200\$6200.

  1. Find the annual rate of depreciation of the car. [3]

easy

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Question 15

no calculator

easy

[Maximum mark: 4]

The product of three consecutive integers is increased by the middle integer.

Prove that the result is a perfect cube.

easy

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Question 16

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easy

[Maximum mark: 6]

Consider the infinite geometric sequence 90009000, 7200-7200, 57605760, 4608-4608, ...

  1. Find the common ratio. [2]

  2. Find the 2525th term. [2]

  3. Find the exact sum of the infinite sequence. [2]

easy

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Question 17

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easy

[Maximum mark: 6]

Consider the infinite geometric sequence 44804480, 3360-3360, 25202520, 1890,-1890,\dots

  1. Find the common ratio, rr. [2]

  2. Find the 2020th term. [2]

  3. Find the exact sum of the infinite sequence. [2]

easy

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Question 18

no calculator

easy

[Maximum mark: 6]

Let log2a=p\log_2 a = p, log2b=q\log_2 b = q, log2c=r\log_2 c = r. Write down the following expressions in terms of pp, qq and rr.

  1. log2(abc)\log_2\Big(\dfrac{ab}{c}\Big) [2]

  2. log2(a2cb3)\log_2\Big(\dfrac{a^2c}{b^3}\Big) [2]

  3. logab\log_a b [2]

easy

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Question 19

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easy

[Maximum mark: 6]

A 33D printer builds a set of 4949 Eif\text{f}fel Tower Replicas in different sizes. The height of the largest tower in this set is 6464 cm. The heights of successive smaller towers are 9595 % of the preceding larger tower, as shown in the diagram below.

AA724a

  1. Find the height of the smallest tower in this set. [3]

  2. Find the total height if all 4949 towers were placed one on top of another. [3]

easy

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Question 20

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easy

[Maximum mark: 4]

Expand (2x3)4(2x - 3)^4 in descending powers of xx and simplify your answer.

easy

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Question 21

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easy

[Maximum mark: 6]

A tennis ball bounces on the ground nn times. The heights of the bounces, h1,h2,h3,,hn,h_1, h_2, h_3, \dots,h_n, form a geometric sequence. The height that the ball bounces the first time, h1h_1, is 8080 cm, and the second time, h2h_2, is 6060 cm.

  1. Find the value of the common ratio for the sequence. [2]

  2. Find the height that the ball bounces the tenth time, h10h_{10}. [2]

  3. Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to 22 decimal places. [2]

easy

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Question 22

no calculator

easy

[Maximum mark: 7]

An arithmetic sequence is given by 33, 55, 7,7,\dots

  1. Write down the value of the common difference, dd. [1]

  2. Find

    1. u10u_{10};

    2. S10S_{10}. [4]

  3. Given that un=253u_n = 253, find the value of nn. [2]

easy

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Question 23

no calculator

easy

[Maximum mark: 5]

Consider a=log6364×log6263×log6162××log23a = \log_{63}64\times\log_{62}63\times\log_{61}62\times\dots\times\log_{2}3. Given that aZa\in\mathbb{Z}, find the value of aa.

easy

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Question 24

no calculator

easy

[Maximum mark: 7]

Let p=ln2p=\ln 2 and q=ln6q = \ln 6. Write down the following expressions in terms of pp and qq.

  1. ln12\ln 12 [2]

  2. ln3\ln 3 [2]

  3. ln48\ln 48 [3]

easy

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Question 25

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easy

[Maximum mark: 6]

The fourth term, u4u_4, of a geometric sequence is 135135. The fifth term, u5u_5, is 8181.

  1. Find the common ratio of the sequence. [2]

  2. Find u1u_1, the first term of the sequence. [2]

  3. Calculate the sum of the first 2020 terms of the sequence. [2]

easy

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Question 26

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easy

[Maximum mark: 6]

The fifth term, u5u_5, of an arithmetic sequence is 2525. The eleventh term, u11u_{11}, of the same sequence is 4949.

  1. Find dd, the common difference of the sequence. [2]

  2. Find u1u_1, the first term of the sequence. [2]

  3. Find S100S_{100}, the sum of the first 100100 terms of the sequence. [2]

easy

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Question 27

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easy

[Maximum mark: 6]

The table shows the first four terms of three sequences: unu_n, vnv_n, and wnw_n.

c39694c1cf7513ffce115791e6b0f1c54c230963.svg

  1. State which sequence is

    1. arithmetic;

    2. geometric. [2]

  2. Find the sum of the first 5050 terms of the arithmetic sequence. [2]

  3. Find the exact value of the 1313th term of the geometric sequence. [2]

easy

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Question 28

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easy

[Maximum mark: 6]

The third term, u3u_3, of an arithmetic sequence is 77. The common difference of
the sequence, dd, is 33.

  1. Find u1u_1, the first term of the sequence. [2]

  2. Find u60u_{60}, the 6060th term of sequence. [2]

The first and fourth terms of this arithmetic sequence are the first two terms
of a geometric sequence.

  1. Calculate the sixth term of the geometric sequence. [2]

easy

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Question 29

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easy

[Maximum mark: 6]

Julia wants to buy a house that requires a deposit of 7400074\hspace{0.15em}000 Australian dollars (AUD).

Julia is going to invest an amount of AUD in an account that pays a nominal annual interest rate of 5.55.5 %, compounded monthly.

  1. Find the amount of AUD Julia needs to invest to reach 7400074\hspace{0.15em}000 AUD after 88 years. Give your answer correct to the nearest dollar. [3]

Julia's parents offer to add 50005000 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 3.53.5 %, compounded quarterly.

  1. Find the number of years it would take Julia to save the 7400074\hspace{0.15em}000 AUD if she accepts her parents money and follows their advice. Give your answer correct to the nearest year. [3]

easy

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Question 30

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easy

[Maximum mark: 7]

The first three terms of a geometric sequence are u1=0.4u_1 = 0.4, u2=0.6u_2 = 0.6, u3=0.9u_3 = 0.9.

  1. Find the value of the common ratio, rr. [2]

  2. Find the sum of the first ten terms in the sequence. [2]

  3. Find the greatest value of nn such that Sn<650S_n < 650. [3]

easy

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Question 31

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easy

[Maximum mark: 6]

Consider the following sequence of figures.

AA008

Figure 1 contains 66 line segments.

  1. Given that Figure nn contains 101101 line segments, show that n=20n = 20.[3]

  2. Find the total number of line segments in the first 2020 figures. [3]

easy

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Question 32

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easy

[Maximum mark: 6]

On 11st of January 20212021, Fiona decides to take out a bank loan to purchase a new Tesla electric car. Fiona takes out a loan of $P\$P with a bank that offers a nominal annual interest rate of 2.6%2.6\hspace{0.05em}\%, compounded monthly.

The size of Fiona's loan at the end of each year follows a geometric sequence with common ratio, α\alpha.

  1. Find the value of α\alpha, 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.

  1. Find the year during which Fiona will need to pay back the loan. [3]

easy

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Question 33

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easy

[Maximum mark: 4]

On the Argand diagram below, the point A represents the complex number 4i4{\mathrm{\hspace{0.05em}i}\mkern 1mu} and the point B represents the complex number 5+i-5+{\mathrm{\hspace{0.05em}i}\mkern 1mu}. The shape ABCD is a square.

fed41c2307d3c10825aed04db2b46a9169006d2c.svg

Determine the complex number represented by:

  1. the point C; [2]

  2. the point D. [2]

easy

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Question 34

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easy

[Maximum mark: 6]

In this question give all answers correct to the nearest whole number.

A population of goats on an island starts at 232232. The population is expected to increase by 1515 % each year.

  1. Find the expected population size after:

    1. 1010 years;

    2. 2020 years. [4]

  2. Find the number of years it will take for the population to reach 1500015\hspace{0.15em}000. [2]

easy

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Question 35

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easy

[Maximum mark: 6]

The first term of an arithmetic sequence is 2424 and the common difference is 1616.

  1. Find the value of the 6262nd term of the sequence. [2]

The first term of a geometric sequence is 88. The 44th term of the geometric sequence is equal to the 1313th term of the arithmetic sequence given above.

  1. Write down an equation using this information. [2]

  2. Calculate the common ratio of the geometric sequence. [2]

easy

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Question 36

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easy

[Maximum mark: 6]

In an arithmetic sequence, u5=24u_5 = 24, u13=80u_{13} = 80.

  1. Find the common difference. [2]

  2. Find the first term. [2]

  3. Find the sum of the first 2020 terms in the sequence. [2]

easy

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Question 37

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easy

[Maximum mark: 6]

In this question give all answers correct to two decimal places.

Mia deposits 40004000 Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of 66