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IB Mathematics AI HL - Questionbank

Topic 1 All - Number & Algebra

All Questions for Topic 1 (Number & Algebra). Number Skills, Sequences & Series, Financial Mathematics, Complex Numbers, Matrices, Systems of Linear Equations

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Question 1

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easy

[Maximum mark: 6]

After solving a problem, John has an exact answer of z=0.1475z = 0.1475.

  1. Write down the exact value of zz in the form a×10ka\times10^k, where 1a<10,kZ1 \leq a < 10, k\in \mathbb{Z}.[2]

  2. State the value of zz given correct to 22 significant figures. [1]

  3. Calculate the percentage error if zz is given correct to 22 significant figures. [3]

easy

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Question 2

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easy

[Maximum mark: 6]

The Burns, Gordons and Longstaff families make meal plans for their households. The table below shows the amount of carbohydrate, fat and protein, all measured in grams, consumed by the family over a single day. The table also shows the daily calories, measured in kcal, this equates to.

tab1

Let xx, yy and zz represent the amount of calories, in kcal, for 11 g of carbohydrate, fat and protein respectively.

  1. Write down a system of three linear equations in terms of xx, yy and zz that represents the information in the table above. [2]

  2. Find the values xx, yy and zz. [2]

The Howe family also plans meals. The table below shows the amount of carbohydrates, fat and protein consumed by the family over a single day.

tab2

  1. Calculate the daily calories for the Howe family. [2]

easy

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Question 3

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easy

[Maximum mark: 6]

Given that z=10sinα3x+yz = \dfrac{10\sin \alpha}{3x+y}, where α=30°\alpha = \ang{30}, x=6x = 6 and y=46y = 46.

  1. Find the exact value of zz. [2]

  2. Write your answer to part (a)

    1. correct to 22 decimal places;

    2. correct to 33 significant figures;

    3. in the form a×10ka\times10^k, where 1a<101 \leq a < 10 and kZk\in \mathbb{Z}.[4]

easy

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Question 4

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easy

[Maximum mark: 6]

Let A=sinαsinβx2+2yA = \sqrt{\dfrac{\sin \alpha - \sin \beta}{x^2 + 2y}}, where α=54°\alpha = \ang{54}, β=18°\beta = \ang{18}, x=24x = 24 and y=18.25y = 18.25.

  1. Find the value of AA. Write down your full calculator display. [2]

  2. Give your answer to part (a) correct to

    1. three significant figures;

    2. three decimal places. [2]

  3. Give the answer to part (b) (i) in the form a×10ka\times10^k, where 1a<101 \leq a < 10, kZk \in \mathbb{Z}.[2]

easy

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Question 5

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easy

[Maximum mark: 6]

Let Q=(sin2x+b)(2sinx1)a24tanxQ = \dfrac{(\sin 2x + b)(2\sin x - 1)}{a^2 - 4\tan x}, where x=45°x = \ang{45}, a=18a = 18 and b=2b = \sqrt{2}.

  1. Find the exact value of QQ. [2]

  2. Give your answer to part (a) correct to

    1. three decimal places;

    2. three significant figures. [2]

  3. Calculate the percentage error if QQ is given to three decimal places. [2]

easy

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Question 6

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easy

[Maximum mark: 6]

The volume of a hemisphere, VV, is given by the formula

V=4S3243π,V = \sqrt{\dfrac{4S^3}{243\pi}}\hspace{0.05em},

where SS is the total surface area.

The total surface area of a given hemisphere is 529529 cm2^2.

  1. Calculate the volume of this hemisphere in cm3^3. Give your answer correct to one decimal place. [3]

  2. Write down your answer to part (a) correct to the nearest integer. [1]

  3. Write down your answer to part (b) in the form a×10ka\times10^k, where 1a<101 \leq a < 10 and kZk \in \mathbb{Z}.[2]

easy

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Question 7

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easy

[Maximum mark: 6]

Four cement bags labelled, "5 kg", were delivered to a customer. The customer measured each bag to check their weights and recorded the following:

4.92,4.95,5.02,4.95\begin{aligned} 4.92,\hspace{0.3em} 4.95,\hspace{0.3em} 5.02,\hspace{0.3em}4.95 \\ \end{aligned}
    1. Find the mean of the customer's measurements.

    2. Calculate the percentage error between the mean and the stated,
      approximate weight of 55 kg. [3]

  1. Calculate 2.1585.120.8\sqrt{2.15^8-5.12^{-0.8}}, giving your answer

    1. correct to the nearest integer;

    2. in the form a×10ka\times10^k, where 1a<101 \leq a < 10 and kZk\in \mathbb{Z}. [3]

easy

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Question 8

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easy

[Maximum mark: 6]

The distance between two points with coordinates (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) is equal to (x2x1)2+(y2y1)2\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.

  1. Calculate the distance between points A(40,100)(40,-100) and B(1,2)(1,-2). Give your answer correct to three significant figures. [3]

  2. Give your answer from part (a) correct to one decimal place. [1]

  3. Write the answer to part (b) in the form a×10ka\times10^k, where 1a<101 \leq a < 10, kZk \in \mathbb{Z}. [2]

easy

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Question 9

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easy

[Maximum mark: 6]

The following diagram shows a rectangle with sides of length 7.6×1027.6\times10^2 cm and 1.5×1031.5\times10^3 cm.

95dfeaca3907795a90048551ee5f6c3d7e41c444.svg

  1. Write down the area of the rectangle in the form a×10ka\times10^k, where
    1a<101 \leq a < 10 and kZk \in \mathbb{Z}. [3]

Natalie estimates the area of the rectangle to be 12000001\hspace{0.1em}200\hspace{0.15em}000 cm2^2.

  1. Find the percentage error in Natalie's estimate. [3]

easy

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Question 10

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easy

[Maximum mark: 8]

A cuboid has the following dimensions: length=9.6\text{length} = 9.6\hspace{0.25em}cm, width=7.4\text{width} = 7.4\hspace{0.25em}cm, and height=5.2\text{height} = 5.2\hspace{0.25em}cm, measured correct to the nearest millimetre.

  1. Using these measurements, calculate the volume of the cuboid, in cm3^3. Give your answer to two decimal places. [2]

The lower and upper bounds for the length of the cuboid can be expressed as 9.55l<9.659.55 \leq l < 9.65.

  1. Write similar expressions for

    1. the width;

    2. the height. [2]

  2. Hence, calculate the minimum volume of the cuboid. Give your answer to three significant figures. [2]

  3. Write your answer to part (c) in the form a×10ka\times10^k, where 1a<101 \leq a < 10 and kZk \in \mathbb{Z}. [2]

easy

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Question 11

calculator

easy

[Maximum mark: 6]

Let F=(4sin2z1)(2tan3z+1)x2y2F = \dfrac{(4\sin 2z-1)(2\tan 3z+1)}{x^2-y^2}, where x=12x = 12, y=8y = 8 and z=15°z = \ang{15}.

  1. Calculate the exact value of FF. [2]

  2. Give your answer to FF correct to

    1. two significant figures;

    2. two decimal places. [2]

Sasha estimates the value of FF to be 0.030.03.

  1. Calculate the percentage error in Sasha's estimate. [2]

easy

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Question 12

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easy

[Maximum mark: 7]

Brendan is training for a long distance bike race.

In week 11 of his training he cycled 2222\,km. In week 22 he cycled 3434\,km. This pattern continues, with him cycling an extra 1212\,km per week.

The distances Brendan cycled in the first 55 weeks of training is shown in the following table.

Screenshot 2023-08-31 at 2.15.24 PM

  1. Calculate how far he cycles in the 1717th week of his training. [3]

  2. In total how far has he cycled after 1717 weeks? [2]

  3. Find the mean distance per week he cycled over 17 weeks. [2]

easy

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Question 13

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easy

[Maximum mark: 6]

The 1515th term of an arithmetic sequence is 2121 and the common difference is 4-4.

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

  2. Find the 2929th term of the sequence. [2]

  3. Find the sum of the first 4040 terms of the sequence. [2]

easy

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Question 14

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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=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 15

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easy

[Maximum mark: 6]

Given r=2abcr = 2a - \dfrac{\sqrt{b}}{c}, a=0.975a = 0.975, b=4.41b = 4.41 and c=35c = 35,

  1. calculate the value of rr. [2]

Albert first writes aa, bb and cc correct to one significant figure and then uses these values to estimate the value of rr.

    1. Write down aa, bb and cc each correct to one significant figure.

    2. Find Albert's estimate of the value of rr. [2]

  1. Calculate the percentage error in Albert's estimate of the value of rr. [2]

easy

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Question 16

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 17

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easy

[Maximum mark: 6]

A toy rocket is fired, from a platform, vertically into the air, its height above the ground after tt seconds is given by s(t)=at2+bt+cs(t) = at^2 + bt + c, where a,b,cRa,b,c \in \mathbb{R} and s(t)s(t) is measured in metres.

rocket

After 22 second, the rocket is 28.328.3 m above the ground; after 44 seconds, 25.625.6 m; after 55 seconds, 14.714.7 m.

    1. Write down a system of three linear equations in terms of aa, bb and cc.

    2. Hence find the values of aa, bb and cc. [3]

  1. Find the height, above the ground, of the platform. [1]

  2. Find the time it takes for the rocket to hit the ground. [2]

easy

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Question 18

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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 19

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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 20

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easy

[Maximum mark: 6]

An arithmetic sequence has u1=12u_1 = 12, u2=21u_2 = 21, u3=30u_3 = 30.

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

  2. Find u10u_{10}. [2]

  3. Find S10S_{10}. [2]

easy

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Question 21

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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 %, 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 22

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easy

[Maximum mark: 6]

A geometric sequence has u1=5u_1 =5, u2=1u_2 = -1 and u3=15u_3 = \dfrac{1}{5}.

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

  2. Find the exact value of u7u_{7}. [2]

  3. Find the exact value of S7S_{7}. [2]

easy

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Question 23

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easy

[Maximum mark: 6]

Emily starts reading Leo Tolstoy's War and Peace on the 11st of February. The number of pages she reads each day increases by the same number on each successive day.

c94a768fb53af8987d3e1115bdd47ee0b1976776.svg

  1. Calculate the number of pages Emily reads on the 1414th of February. [3]

  2. Find the exact total number of pages Emily reads in the 2828 days of February.[3]

easy

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Question 24

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easy

[Maximum mark: 6]

A geometric sequence has 2020 terms, with the first four terms given below.

418.5,279,186,124,\begin{aligned} 418.5,\hspace{0.25em} 279,\hspace{0.25em} 186,\hspace{0.25em} 124,\hspace{0.25em}\dots \\ \end{aligned}
  1. Find rr, the common ratio of the sequence. Give your answer as a fraction. [1]

  2. Find u5u_5, the fifth term of the sequence. Give your answer as a fraction. [1]

  3. Find the smallest term in the sequence that is an integer. [2]

  4. Find S10S_{10}, the sum of the first 1010 terms of the sequence. Give your answer correct to one decimal place. [2]

easy

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Question 25

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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 26

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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 27

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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 amount 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 28

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easy

[Maximum mark: 6]

Edward wants to buy a new car, and he decides to take out a loan of 7000070\hspace{0.15em}000 Australian dollars from a bank. The loan is for 66 years, with a nominal annual interest rate of 7.2%7.2\%, compounded monthly. Edward will pay the loan in fixed monthly instalments.

  1. Determine the amount Edward should pay each month. Give your answer to the nearest dollar.[3]

  2. Calculate the amount Edward will still owe the bank at the end of the third year. [3]

easy

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Question 29

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easy

[Maximum mark: 6]

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

Elena invests in a retirement plan in which equal payments of €15001500 are made at the beginning of each year. Interest is earned on each payment at a rate of 2.492.49 % per year, compounded annually.

  1. Find the value of the investment after 2525 years. [3]

  2. Find the amount of interest Elena will earn on the investment over 2525 years.[3]

easy

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Question 30

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easy

[Maximum mark: 6]

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. [3]

  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 31

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easy

[Maximum mark: 6]

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

6896afb03e54861ed9a71ba4f129a85ea32667d8.svg

  1. State which sequence is

    1. arithmetic;

    2. geometric. [2]

  2. Find the exact value of the sum of the first 3535 terms of the arithmetic
    sequence. [2]

  3. Find the exact value of the 1010th term of the geometric sequence. [2]

easy

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Question 32

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easy

[Maximum mark: 6]

An owl takes off from from a tree branch and flies higher into the sky. Its height above the ground after tt seconds, where t0t\geq 0, is given by s(t)=at3+bt2+ct+ds(t) = at^3 + bt^2 + ct+d