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IB Math 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: 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 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

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 4

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 5

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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 6

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 7

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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 8

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 9

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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 10

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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 11

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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 12

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 13

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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 14

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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 15

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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 16

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 17

calculator

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 18

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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 19

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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 20

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easy

[Maximum mark: 7]

Let a=ln2a=\ln 2 and b=ln10b = \ln 10. Write down the following\text{following} expressions\text{expressions} in terms of aa and bb.

  1. ln20\ln 20 [2]

  2. ln5\ln 5 [2]

  3. ln160\ln 160 [3]

easy

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Question 21

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 22

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easy

[Maximum mark: 5]

Solve the equation 2lnx=ln25+62\ln x=\ln 25 +6, giving your answer in the form x=aebx=ae^b where aa, bZ+b \in \mathbb{Z}^+.

easy

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Question 23

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easy

[Maximum mark: 5]

Consider b=log8081×log7980×log7879××log34b = \log_{80}81\times\log_{79}80\times\log_{78}79\times\dots\times\log_{3}4. Given that bZb\in\mathbb{Z}, find the value of bb.

easy

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Question 24

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 25

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 26

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easy

[Maximum mark: 5]

Consider the expansion of (x+2)5(x+2)^5.

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

  2. Find the term in x3x^3. [4]

easy

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Question 27

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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 28

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 29

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 30

no calculator

easy

[Maximum mark: 5]

Consider an arithmetic sequence where u12=S12=12u_{12} = S_{12} = 12. Find the value of the first term, u1u_1, and the value of the common difference, dd.

easy

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Question 31

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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 32

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 33

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 34

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easy

[Maximum mark: 5]

In an arithmetic sequence, the sum of the 2nd and 6th term is 3232.
Given that the sum of the first six terms is 120120, determine the first term and common difference of the sequence.

easy

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Question 35

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easy

[Maximum mark: 5]

An arithmetic sequence has first term 4545 and common difference 1.5-1.5.

  1. Given that the kkth term of the sequence is zero, find the value of kk. [2]

Let SnS_n denote the sum of the first nn terms of the sequence.

  1. Find the maximum value of SnS_n. [3]

easy

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