DP IB Maths: AI HL

Topic Questions

1.1 Number Toolkit

1a
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2 marks

Let P space equals space fraction numerator open parentheses 4 space sin space 2 q minus 2 close parentheses open parentheses 6 space tan space q plus 2 close parentheses over denominator 10 open parentheses r plus s close parentheses squared end fraction, where q equals 30 degree, r equals 6 and s equals 2.

Calculate the exact value of P.

1b
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2 marks

Give your answer from part (a) correct to

(i)
two decimal places.

(ii)
two significant figures.
1c
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2 marks

Michael estimates the value of P to be 0.015

Calculate the percentage error of Michael’s estimate.

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2a
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2 marks

Let W space equals space fraction numerator open parentheses 2 cos space 2 x plus y close parentheses open parentheses tan x over 2 minus z close parentheses over denominator 10 open parentheses 5 sin x plus z squared close parentheses end fraction, where x equals 90 degree comma y equals negative 1 and z equals 2.

Find the value of W. Give your answer as a fraction.

2b
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2 marks

Give your answer from part (a) to

(i)
three decimal places.

(ii)
three significant figures.
2c
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2 marks

Louis estimates the value of W to be 0.03.

Calculate the percentage error of Louis’ estimate.

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3a
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2 marks

A prism has a cross sectional area of 5.83 cross times 10 to the power of 7 space cm squared and volume of 4.22 cross times 10 to the power of 8 space cm cubed.

Calculate the height of the prism.

3b
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3 marks

Lily estimates the height of the prism to be 7 cm.

Calculate the percentage error in Lily’s estimate.

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4a
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1 mark

Mary has found the exact answer for R is  45 over 16.

Write down the exact answer of as a decimal.

4b
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4 marks

Mary rounds her exact answer so that the percentage error is 6.67%.

State the number of significant figures Mary rounded the exact answer to and write down the approximate value of R used in calculating the percentage error.

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5a
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2 marks

It is given that sin space a space equals fraction numerator square root of 3 over denominator 2 end fraction and sin space b space equals space 1 half, where a less or equal than 180 degree space a n d space b less or equal than 180 degree.

Find the size of the angles a and b.

5b
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2 marks

A circle has radius  equal to square root of fraction numerator sin space a over denominator sin space b end fraction end root space cm.

Find the area of the circle, giving your answer in terms of straight pi.

5c
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3 marks

James rounds the answer from part (b) to the nearest integer.

Calculate the percentage error of James’ estimate.

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6a
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1 mark

A medium rare steak should have an internal temperature of 55 degree straight C to 56 degree straight C. Max decides to go to 10 different steak houses, he measures the internal temperature of a medium rare steak at each establishment and records the following:

51.0,   52.1,  62.9,  49.0,  59.8,  50.2,  54.3,  47.7,  48.6,   65.4

Find the mean internal temperature of Max’s recordings.

6b
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3 marks

Max goes to 5 more steak houses and calculates the mean of all 15 restaurants to be 55.2 degree straight C.

Calculate the mean internal temperature from the 5 additional steak houses Max went to.

6c
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3 marks

Max records one last steak that has an internal temperature of T degree straight C.

Calculate the interval of T such that the mean internal temperature for all 16 steaks is within the temperature range for a medium rare steak.

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7
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5 marks

The diameter of Earth is 1.274 cross times 10 to the power of 7 m, correct to four significant figures.  The circumference of Mars is 2.134 cross times 10 to the power of 7 m, correct to four significant figures. 

Modelling Earth and Mars as perfect spheres, find the difference between the volume of Earth and the volume of Mars, giving your answer in the form a cross times 10 to the power of k, where 1 less or equal than a less than 10 comma k element of straight integer numbers.

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8
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6 marks

Solve the following systems of linear equations using technology.

(i)
5 x plus 3 y minus 2 z equals negative 12
3 x minus 4 y minus z equals 17
10 x minus 10 y plus z equals 65

(ii)
4 x minus 5 y plus z equals 50
3 x plus y plus 3 z equals negative 16
6 x minus 2 z equals 61 plus y

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1a
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2 marks

Let Q fraction numerator 30 space sin space 2 a over denominator 8 b end fraction, where a equals 45 degree and b equals 2.

Calculate the exact value of Q.

1b
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2 marks

Give your answer from part (a) correct to

(i)
two decimal places

(ii)
two significant figures.
1c
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2 marks

Nina estimates the value of Q to be 2.

Calculate the percentage error in Nina’s estimate.

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2a
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2 marks

Let  R equals fraction numerator 4 x over denominator 6 space cos 5 y end fraction, where x equals 1.25 and y equals 36 degree.

Find the value of R. Give your answer as a fraction.

2b
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2 marks

Give your answer from part (a) to

(i)
one decimal place

(ii)
three significant figures.
2c
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2 marks

Kieran estimates the value of R to be -1.

Calculate the percentage error in Kieran’s estimate.

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3a
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2 marks

Consider the numbers a equals 4.14 cross times 10 to the power of 6 and b equals 2.54 cross times 10 to the power of negative 7 end exponent.

Calculate C space equals space root index 10 of open parentheses a over b close parentheses cubed end root. Give your answer correct to the nearest integer.

3b
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2 marks

Give your answer to part (a) in the form a cross times 10 to the power of k, where 1 less or equal than a less or equal than 10 and k element of straight integer numbers.

3c
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2 marks

Calculate the percentage error if C was approximated to be 9000.

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4a
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3 marks

A cylinder has radius of 12.7 cm and height of 14.4 cm.

Calculate the volume of the cylinder correct to

(i)
one decimal place

(ii)
three significant figures

(iii)

the nearest integer.
4b
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2 marks

Write your answer to part (a) (ii) in the form a cross times 10 to the power of k, where 1 less or equal than a less or equal than 10 and k element of straight integer numbers.

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5a
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2 marks

A rectangular field has length, L, of 25.2 m and width, W, of 21.4 m, each correct to 1 decimal place.

Calculate the lower and upper bound for

(i)
L

(ii)

W
5b
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4 marks

Calculate the lower and upper bound for the

(i)
perimeter, P

(ii)
area, A comma of the field.

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6
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6 marks

Calculate the following, giving your answer in the form a cross times 10 to the power of k, where 1 less or equal than a less or equal than 10 and k element of straight integer numbers

(i)
4 cross times open parentheses 6.2 space cross times 10 to the power of negative 5 end exponent close parentheses

(ii)

open parentheses 4 space cross times 10 to the power of 5 close parentheses minus open parentheses 5 cross times 10 to the power of 4 close parentheses

(iii)

open parentheses 4321 to the power of negative 1 end exponent close parentheses open parentheses 1.2 cross times 10 to the power of negative 1 end exponent close parentheses

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7a
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2 marks

Consider the following four numbers.

a = 0.272 b = 0.0272 cross times105 c = e(10e)-1 d = 2.72 cross times102

Write down

(i)
the number that is in the form a cross times 10 to the power of k, where 1 less or equal than a less or equal than 10 and k element of straight integer numbers 

(ii)

the largest of these numbers.
7b
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4 marks
(i)
Find the value of a plus b minus c plus d.

(ii)
Give your answer to part (b)(i) in the form a cross times 10 to the power of k, where 1 less or equal than a less or equal than 10 and k element of straight integer numbers

                      .

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8a
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3 marks

Five Olympic barbells labelled, “2.2 m in length”, were delivered to an Olympic weightlifting team. The coach measured each barbell to check its length, in metres, and recorded the following:

2.18 2.21 2.23 2.19 2.24

(i)
Find the mean of the coach’s recorded measurements.

(ii)
Calculate the percentage error between the mean and the stated length of 2.2 m.
8b
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3 marks

The weights of the barbells are labelled 20 kg. The coach also weighed each barbell, in kg, and recorded the following:

20.3 19.9 20.3 20.4 20.1

(i)
Find the mean of the coach’s recorded weights.

(ii)
Calculate the percentage error between the mean and the stated weight of 20 kg.

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9a
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2 marks

In a game show, there is a transparent box filled with identical cubes. Contestants must estimate the number of cubes in the box. The box is 60 cm wide, 80 cm long and 20 cm tall.

Find the volume of the box.

9b
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2 marks

Monica estimates the volume of one cube is 300 cm3. She uses this value to estimate the number of cubes in the box.

Find Monica’s estimated number of cubes in the box.

9c
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2 marks

The actual number of cubes in the box is 280.

Find the percentage error in Monica’s estimated number of cubes in the box.

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10
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6 marks

Solve the following systems of linear equations using technology.

(i)
5 x plus 3 y minus 2 z equals negative 12
3 x minus 4 y minus z equals 17
10 x minus 10 y plus z equals 65

(ii)
4 x minus 5 y plus z equals 50
3 x plus y plus 3 z equals negative 16
6 x minus 2 z equals 61 plus y

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11
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6 marks

Solve the following systems of linear equations using technology.

(i)
2 x minus 5 y minus 7 z equals negative 21
3 z plus x minus 4 y equals 44
x plus z minus y equals 12

(ii)
z minus x minus y equals negative 11
5 x plus 11 z minus 2 y equals negative 28
3 y minus 4 z plus x equals 30

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1a
Sme Calculator
2 marks

Consider the numbers  a equals 11 square root of 2 comma space b equals left parenthesis 5 plus 6 pi right parenthesis comma space c equals square root of 2 comma space d equals 6 left parenthesis pi minus 1 right parenthesis.

Giving your answer to 1 decimal place, calculate the value of

(i)
a

(ii)
b

(iii)
c

(iv)
d
1b
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2 marks

Points P and Q have coordinates open parentheses a comma space b close parentheses and open parentheses c comma space d close parentheses respectively.

The formula for the distance, d, between two points with coordinates open parentheses x subscript 1 comma space y subscript 1 close parentheses and open parentheses x subscript 2 comma space end subscript y subscript 2 close parentheses is given in your formula booklet.

d equals square root of left parenthesis x subscript 1 minus x subscript 2 right parenthesis squared plus left parenthesis y subscript 1 minus y subscript 2 right parenthesis squared end root

Using your answers from part (a), calculate the distance, d, between points P and Q. Give your answer correct to 1 decimal place.

1c
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4 marks

Find the percentage error between the distance, correct to 1 decimal place, found in part (b) and the exact distance between points P and Q.

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2a
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2 marks

Let Y equals space open parentheses p q close parentheses to the power of negative 1 end exponent r and T space equals space p q r to the power of negative 1 end exponent, wherespace p equals sin space 60 degreeq equals square root of 3r equals 2

Giving your answer to 1 decimal place, calculate the value of

(i)
Y.

(ii)
T.
2b
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1 mark

Using your answers to part (a), estimate the value of Y T. Give your answer as a fraction.

2c
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4 marks

Calculate the percentage error between your estimated value of Y T found in part (b) and the exact value of Y T.

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3a
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2 marks

A cuboid has length, l = 0.102 m, width, w = 9.4 cm and height, h = 0.25 m.

Calculate the exact volume of the cuboid

(i)
in cm cubed.

(ii)
in straight m cubed.
3b
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2 marks

Write your answers to part (a) (i) and (ii) in the form a cross times 10 to the power of k, where 1 less or equal than a less than 10 comma k element of straight integer numbers.

3c
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4 marks

William estimates the volume of the cuboid as being Q space cm cubed and the percentage error in his estimate is 5%.

Calculate the exact possible values of Q.

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4a
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2 marks

Let S space equals space open parentheses a space sin to the power of 2 space end exponent 4 b close parentheses open parentheses c squared space tan squared space 12 d close parentheses to the power of negative 1 end exponent open parentheses square root of a space plus c minus cos space 48 b close parentheses comma where a = 16b = 7.5 degreec = 3 and d = 5 degree

Note: sin2 theta = (sin theta)2

Find the value of S, giving your answer as a fraction.

4b
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2 marks

Let Xfraction numerator square root of a plus c squared minus 2 s i n space 54 space d over denominator square root of left parenthesis a cubed right parenthesis end root minus a minus c end fraction

Find the value of X comma giving your answer as a fraction.

4c
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2 marks

Calculate the value of S X comma giving your answer as a fraction.

4d
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2 marks

John estimates the value of S X to be 0.3.

Calculate the percentage error in John’s estimate.

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5a
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4 marks

Consider the numbersspace p equals 2.41 cross times 10 to the power of 4 and q equals 4.12 cross times 10 to the power of 5.

Giving your answers in the form a cross times 10 to the power of k comma where 1 less or equal than a less than 10 comma space k element of straight integer numbers comma calculate

(i)
space p plus q

(ii)
space p minus q

(iii)
q minus p

(iv)
p over q
5b
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2 marks

The formula for the distance, d comma between two points with coordinates left parenthesis x subscript 1 comma space y subscript 1 right parenthesis and open parentheses x subscript 2 comma space y subscript 2 close parentheses is given in your formula booklet.

d equals square root of left parenthesis x subscript 1 minus x subscript 2 right parenthesis squared plus left parenthesis y subscript 1 minus y subscript 2 right parenthesis squared end root

Using your answers to part (a), estimate the distance between points Aleft parenthesis p plus q comma space p minus q right parenthesis  and Bopen parentheses q minus p comma space p over q close parentheses.

5c
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4 marks

Calculate the percentage error between the estimate of the distance between points P and Q found in part (b) and the exact distance.

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6a
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2 marks

A shop sells bags of potatoes labelled as “5 kg”. The shop owner weighs five bags, in kilograms, at random and recorded the following:

4.96,   4.89,   5.07,   5.11,   5.02

(i)
Find the mean of the shop owner’s recorded weights.

(ii)
Calculate the percentage error between the mean and the stated weight of 5 kg.
6b
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3 marks

The shop owner shares his findings with his potato supplier, who weighs another five bags and recorded the following:

5.05,   5.01,   4.97,   5.09,   X

The supplier allows a maximum percentage error of 1%.

Find the interval for the values of X such that the percentage error from the five bags that the supplier weighed is less than 1%.

6c
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3 marks

Find the interval for the values of X such that the percentage error from all the ten bags weighed is less than 1%.

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7
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6 marks

Solve the following systems of linear equations using technology.

(i)
2 x minus 5 y minus 7 z equals negative 21
3 z plus x minus 4 y equals 44
x plus z minus y equals 12

(ii)
z minus x minus y equals negative 11
5 x plus 11 z minus 2 y equals negative 28
3 y minus 4 z plus x equals 30

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