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IB Mathematics AA SL - Questionbank

Topic 1 All - Number & Algebra

All Questions for Topic 1 (Number & Algebra). Sequences & Series, Exponents & Logs, Binomial Theorem, Proofs

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Paper 2

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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

calculator

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

no calculator

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 4

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 5

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 6

calculator

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 7

calculator

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 8

calculator

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 9

no calculator

easy

[Maximum mark: 4]

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

easy

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Question 10

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 11

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 12

calculator

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 13

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 14

calculator

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 15

calculator

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 16

calculator

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 17

calculator

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 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

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 20

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 21

no calculator

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 22

calculator

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 23

calculator

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 24

no calculator

easy

[Maximum mark: 6]

Let a=log5ba = \log_5b, where b>0b > 0. Write down each of the following expressions
in terms of aa.

  1. log5b4\log_5b^4 [2]

  2. log5(25b)\log_5 (25b) [2]

  3. log25b\log_{25}b [2]

easy

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Question 25

calculator

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 26

calculator

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 27

calculator

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 28

calculator

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 %, compounded semi-annually.

  1. Find the amount of interest that Mia will earn over the next 2.52.5 years. [3]

Ella also deposits AUD into a bank account. Her bank pays a nominal annual interest\text{interest} rate of 44 %, compounded monthly. In 2.52.5 years, the total amount in Ella's account will be 40004000 AUD.

  1. Find the amount that Ella deposits in the bank account. [3]

easy

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Question 29

calculator

easy

[Maximum mark: 5]

Maria invests $25000\$25\hspace{0.15em}000 into a savings account that pays a nominal annual interest rate of 4.254.25%, compounded monthly.

  1. Calculate the amount of money in the savings account after 33 years. [2]

  2. Calculate the number of years it takes for the account to reach $40000\$40\hspace{0.15em}000. [3]

easy

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Question 30

no calculator

easy

[Maximum mark: 6]

The first three terms of a geometric sequence are u1=32u_1 = 32, u2=16u_2 = -16, u3=8u_3 = 8.

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

  2. Find u6u_6. [2]

  3. Find SS_{\infty}. [2]

easy

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Question 31

no calculator

easy

[Maximum mark: 6]

In an arithmetic sequence, u4=12u_4 = 12, u11=9u_{11} = -9.

  1. Find the common difference. [2]

  2. Find the first term. [2]

  3. Find the sum of the first 1111 terms in the sequence. [2]

easy

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Question 32

calculator

easy

[Maximum mark: 6]

Emily deposits 20002000 Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of 44 %, compounded monthly.

  1. Find the amount of money that Emily will have in her bank account after 55 years. Give your answer correct to two decimal places. [3]

Emily will withdraw the money back from her bank account when the amount reaches 30003000 AUD.

  1. Find the time, in months, until Emily withdraws the money from her bank account. [3]

easy

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Question 33

no calculator

easy

[Maximum mark: 6]

  1. Show that (2n1)3+(