Mathematical studies Standard level Paper 1 - Grade 12 - IB Math... · 2020. 5. 1. · Candidate...

Candidate session number

M16/5/MATSD/SP1/ENG/TZ2/XX

Mathematical studiesStandard levelPaper 1

© International Baccalaureate Organization 201619 pages

Instructions to candidates

yy Write your session number in the boxes above.yy Do not open this examination paper until instructed to do so.yy A graphic display calculator is required for this paper.yy A clean copy of the mathematical studies SL formula booklet is required for this paper.yy Answer all questions.yy Write your answers in the boxes provided.yy Unless otherwise stated in the question, all numerical answers should be given exactly or

correct to three significant figures.yy The maximum mark for this examination paper is [90 marks].

1 hour 30 minutes

Tuesday 10 May 2016 (afternoon)

2216 – 7405

20EP01

– 2 –

Please do not write on this page.

Answers written on this page will not be marked.

20EP02


– 3 –

Turn over

Maximum marks will be given for correct answers. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. Write your answers in the answer boxes provided. Solutions found from a graphic display calculator should be supported by suitable working, for example, if graphs are used to find a solution, you should sketch these as part of your answer.

1. Assume that the Earth is a sphere with a radius, r , of 6.38 × 103 km .

Earth

r

(a) (i) Calculate the surface area of the Earth in km2.

(ii) Write down your answer to part (a)(i) in the form a × 10k , where 1 ≤ a < 10 and k ∈ . [4]

The surface area of the Earth that is covered by water is approximately 3.61 × 108 km2 .

(b) Calculate the percentage of the surface area of the Earth that is covered by water. [2]

Working:

Answers:

(a) (i) . . . . . . . . . . . . . . . . . . . . .

(ii) . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . .

20EP03


– 4 –

2. Consider the numbers −1 , 4 , 23

, 2 , 0.35 and −22 .

Complete the following table by placing a tick () to indicate if the number is an element of the number set. The first row has been completed as an example.

−1

4

23

2

0.35

−22

[6]

20EP04


– 5 –

Turn over

3. A ladder is standing on horizontal ground and leaning against a vertical wall. The length of the ladder is 4.5 metres. The distance between the bottom of the ladder and the base of the wall is 2.2 metres.

(a) Use the above information to sketch a labelled diagram showing the ground, the ladder and the wall. [1]

(b) Calculate the distance between the top of the ladder and the base of the wall. [2]

(c) Calculate the obtuse angle made by the ladder with the ground. [3]

Working:

Answers:

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20EP05


– 6 –

4. Consider the following propositions:

p: The lesson is cancelledq: The teacher is absentr: The students are in the library.

(a) Write, in words, the compound proposition q ⇒ ( p ∧ r) . [3]

(b) Complete the following truth table. [2]

q r ¬ r q ⇒ ¬ r

T T

T F

F T

F F

(c) Hence, justify why q ⇒ ¬ r is not a tautology. [1]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20EP06


– 7 –

Turn over

5. Two friends, Sensen and Cruz, are conducting an investigation on probability.

Sensen has a fair six-sided die with faces numbered 1, 2, 2, 4, 4 and 4. Cruz has a fair disc with one red side and one blue side.

The die and the disc are thrown at the same time.

Find the probability that

(a) the number shown on the die is 1 and the colour shown on the disc is blue; [2]

(b) the number shown on the die is 1 or the colour shown on the disc is blue; [2]

(c) the number shown on the die is even given that the colour shown on the disc is red. [2]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20EP07


– 8 –

6. When Bermuda (B), Puerto Rico (P), and Miami (M) are joined on a map using straight lines, a triangle is formed. This triangle is known as the Bermuda triangle.

According to the map, the distance MB is 1650 km, the distance MP is 1500 km and angle BMP is 57˚.

diagram not to scaleB

P

M 57˚

1500 km

1650

km

(a) Calculate the distance from Bermuda to Puerto Rico, BP. [3]

(b) Calculate the area of the Bermuda triangle. [3]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20EP08


– 9 –

Turn over

7. A survey was conducted among a random sample of people about their favourite TV show. People were classified by gender and by TV show preference (Sports, Documentary, News and Reality TV).

The results are shown in the contingency table below.

Sports Documentary News Reality TV Total

Male 20 24 32 11 87

Female 18 30 20 25 93

Total 38 54 52 36 180

(a) Find the expected number of females who prefer documentary shows. [2]

A χ2 test at the 5 % significance level is used to determine whether TV show preference is independent of gender.

(b) Write down the p-value for the test. [2]

(c) State the conclusion of the test. Give a reason for your answer. [2]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20EP09


– 10 –

8. Consider the curve yx

= +1 12

, x ≠ 0 .

(a) For this curve, write down

(i) the value of the x-intercept;

(ii) the equation of the vertical asymptote. [3]

(b) Sketch the curve for −2 ≤ x ≤ 4 on the axes below. [3]

x

y

4–1 21 3–2

–1

0

–2

1

2

3

4

5

(This question continues on the following page)

20EP10


– 11 –

Turn over

(Question 8 continued)

Working:

Answers:

(a) (i) . . . . . . . . . . . . . . . . . . . . .

(ii) . . . . . . . . . . . . . . . . . . . . .

20EP11


– 12 –

9. Each day a supermarket records the midday temperature and how many cold drinks are sold on that day. The following table shows the supermarket’s data for the last 6 days. This data is also shown on a scatter diagram.

Midday temperature, C (x) 7 12 14 15 16 20

Number of cold drinks sold ( y) 280 350 380 420 400 450

y

x

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14 16 18 20 22

Num

ber o

f col

d dr

inks

sol

d

Midday temperature (C)

(a) Write down

(i) the mean temperature, x ;

(ii) the mean number of cold drinks sold, y . [2]

(b) Draw the line of best fit on the scatter diagram. [2]

(c) Use the line of best fit to estimate the number of cold drinks that are sold on a day when the midday temperature is 10C. [2]


20EP12


– 13 –

Turn over


Working:

Answers:

(a) (i) . . . . . . . . . . . . . . . . . . . . .

(ii) . . . . . . . . . . . . . . . . . . . . .

(c) . . . . . . . . . . . . . . . . . . . . . . . . .

20EP13


– 14 –

10. Obi travels from Dubai to Pretoria and changes 2000 United Arab Emirates Dirham (AED) at a bank. He receives 6160 South African Rand (ZAR).The exchange rate is 1 AED = x ZAR.

(a) Calculate the value of x . [2]

Obi decides to invest half of the money he receives, 3080 ZAR, in an account which pays a nominal interest rate of 9 %, compounded monthly.

The amount of money in the account will have doubled before the end of the nth year of the investment.

(b) Calculate the minimum value of n . [4]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20EP14


– 15 –

Turn over

11. A snack container has a cylindrical shape. The diameter of the base is 7.84 cm. The height of the container is 23.4 cm. This is shown in the following diagram.

diagram not to scale

23.4 cm

7.84 cm

(a) Write down the radius, in cm, of the base of the container. [1]

(b) Calculate the area of the base of the container. [2]

Dan is going to paint the curved surface and the base of the snack container.

(c) Calculate the area to be painted. [3]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20EP15


– 16 –

12. The equation of the straight line L1 is y = 2x − 3 .

(a) Write down the y-intercept of L1 . [1]

(b) Write down the gradient of L1 . [1]

The line L2 is parallel to L1 and passes through the point (0 , 3) .

(c) Write down the equation of L2 . [1]

The line L3 is perpendicular to L1 and passes through the point (−2 , 6) .

(d) Write down the gradient of L3 . [1]

(e) Find the equation of L3 . Give your answer in the form ax + by + d = 0 , where a , b and d are integers. [2]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20EP16


– 17 –

Turn over

13. A population of mosquitoes decreases exponentially. The size of the population, P , after t days is modelled by

P = 3200 × 2−t + 50 , where t ≥ 0 .

(a) Write down the exact size of the initial population. [1]

(b) Find the size of the population after 4 days. [2]

(c) Calculate the time it will take for the size of the population to decrease to 60. [2]

The population will stabilize when it reaches a size of k .

(d) Write down the value of k . [1]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20EP17


– 18 –

14. A group of students were asked how long they spend practising mathematics during the week. The results are shown in the following table.

Time, t (hours) Number of students

0 ≤ t < 1 35

1 ≤ t < 2 30

2 ≤ t < 3 a

3 ≤ t < 4 52

4 ≤ t < 5 43

It is known that 35 < a < 52 .

(a) Write down

(i) the modal class;

(ii) the mid-interval value of the modal class;

(iii) the class in which the median lies. [3]

For this group of students, the estimated mean number of hours spent practising mathematics is 2.69.

(b) Calculate the value of a . [3]

Working:

Answers:

(a) (i) . . . . . . . . . . . . . . . . . . . . .

(ii) . . . . . . . . . . . . . . . . . . . . .

(iii) . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . .

20EP18


– 19 –

15. Consider the function f (x) = x3 − 3x2 + 2x + 2 . Part of the graph of f is shown below.

5

– 2

20

25

10

– 1– 5

1 2 3 4 x

y

15

0

– 10

– 15

(a) Find f ′(x) . [3]

(b) There are two points at which the gradient of the graph of f is 11. Find the x-coordinates of these points. [3]

Working:

Answers:

(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20EP19


Please do not write on this page.

Answers written on this page will not be marked.

20EP20


Mathematical studiesStandard levelPaper 2

© International Baccalaureate Organization 20169 pages

Instructions to candidates

yy Do not open this examination paper until instructed to do so.yy A graphic display calculator is required for this paper.yy A clean copy of the mathematical studies SL formula booklet is required for this paper.yy Answer all the questions in the answer booklet provided.yy Unless otherwise stated in the question, all numerical answers should be given exactly or

correct to three significant figures.yy The maximum mark for this examination paper is [90 marks].

1 hour 30 minutes

Wednesday 11 May 2016 (morning)

2216 – 7406

– 2 –

Answer all questions in the answer booklet provided. Please start each question on a new page. You are advised to show all working, where possible. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. Solutions found from a graphic display calculator should be supported by suitable working, for example, if graphs are used to find a solution, you should sketch these as part of your answer.

1. [Maximum mark: 14]

180 people were interviewed and asked what types of transport they had used in the last year from a choice of airplane (A), train (T ) or bus (B). The following information was obtained.

47 had travelled by airplane68 had travelled by train

122 had travelled by bus25 had travelled by airplane and train32 had travelled by airplane and bus35 had travelled by train and bus20 had travelled by all three types of transport

(a) Draw a Venn diagram to show the above information. [4]

(b) Find the number of people who, in the last year, had travelled by

(i) bus only;

(ii) both airplane and bus but not by train;

(iii) at least two types of transport;

(iv) none of the three types of transport. [6]

A person is selected at random from those who were interviewed.

(c) Find the probability that the person had used only one type of transport in the last year. [2]

(d) Given that the person had used only one type of transport in the last year, find the probability that the person had travelled by airplane. [2]


– 3 –

Turn over


Prachi is on vacation in the United States. She is visiting the Grand Canyon.

When she reaches the top, she drops a coin down a cliff. The coin falls down a distance of 5 metres during the first second, 15 metres during the next second, 25 metres during the third second and continues in this way. The distances that the coin falls during each second forms an arithmetic sequence.

(a) (i) Write down the common difference, d , of this arithmetic sequence.

(ii) Write down the distance the coin falls during the fourth second. [2]

(b) Calculate the distance the coin falls during the 15th second. [2]

(c) Calculate the total distance the coin falls in the first 15 seconds. Give your answer in kilometres. [3]

Prachi drops the coin from a height of 1800 metres above the ground.

(d) Calculate the time, to the nearest second, the coin will take to reach the ground. [3]

Prachi visits a tourist centre nearby. It opened at the start of 2015 and in the first year there were 17 000 visitors. The number of people who visit the tourist centre is expected to increase by 10 % each year.

(e) Calculate the number of people expected to visit the tourist centre in 2016. [2]

(f) Calculate the total number of people expected to visit the tourist centre during the first 10 years since it opened. [3]


– 4 –


A speed camera on Peterson Road records the speed of each passing vehicle. The speeds are found to be normally distributed with a mean of 67 km h−1 and a standard deviation of 3.4 km h−1.

(a) Sketch a diagram of this normal distribution and shade the region representing the probability that the speed of a vehicle is between 60 and 70 km h−1. [2]

A vehicle on Peterson Road is chosen at random.

(b) Find the probability that the speed of this vehicle is

(i) more than 60 km h−1;

(ii) less than 70 km h−1;

(iii) between 60 and 70 km h−1. [3]

It is found that 19 % of the vehicles are exceeding the speed limit of s km h−1.

(c) Find the value of s , correct to the nearest integer. [2]

There is a fine of US$65 for exceeding the speed limit on Peterson Road. On a particular day the total value of fines issued was US$14 820.

(d) (i) Calculate the number of fines that were issued on this day.

(ii) Estimate the total number of vehicles that passed the speed camera on Peterson Road on this day. [4]


– 5 –

Turn over


A playground, when viewed from above, is shaped like a quadrilateral, ABCD, where AB = 21.8 m and CD = 11 m . Three of the internal angles have been measured and angle ABC = 47˚ , angle ACB = 63˚ and angle CAD = 30˚ . This information is represented in the following diagram.


21.8 m

11 mD

A

C

B

63

30

47

(a) Calculate the distance AC. [3]

(b) Calculate angle ADC. [3]

There is a tree at C, perpendicular to the ground. The angle of elevation to the top of the tree from D is 35˚.

(c) Calculate the height of the tree. [2]

Chavi estimates that the height of the tree is 6 m.

(d) Calculate the percentage error in Chavi’s estimate. [2]

Chavi is celebrating her birthday with her friends on the playground. Her mother brings a 2 litre bottle of orange juice to share among them. She also brings cone-shaped paper cups.

Each cup has a vertical height of 10 cm and the top of the cup has a diameter of 6 cm.

(e) Calculate the volume of one paper cup. [3]

(f) Calculate the maximum number of cups that can be completely filled with the 2 litre bottle of orange juice. [3]


– 6 –


Hugo is given a rectangular piece of thin cardboard, 16 cm by 10 cm. He decides to design a tray with it.

He removes from each corner the shaded squares of side x cm, as shown in the following diagram.

diagram not to scale16 cm

10 cm

x

The remainder of the cardboard is folded up to form the tray as shown in the following diagram.


(a) Write down, in terms of x , the length and the width of the tray. [2]

(b) (i) State whether x can have a value of 5. Give a reason for your answer.

(ii) Write down the interval for the possible values of x . [4]

(c) Show that the volume, V cm3, of this tray is given by

V = 4x3 − 52x2 + 160x . [2]



– 7 –

Turn over


(d) Find ddVx

. [3]

(e) Using your answer from part (d), find the value of x that maximizes the volume of the tray. [2]

(f) Calculate the maximum volume of the tray. [2]

(g) Sketch the graph of V = 4x3 − 52x2 + 160x , for the possible values of x found in part (b)(ii), and 0 ≤ V ≤ 200 . Clearly label the maximum point. [4]


– 8 –


The table below shows the number and height (h), in metres, of plants grown for a school project.

Height, h (metres) Frequency Cumulative frequency

0.40 ≤ h < 0.60 12 12

0.60 ≤ h < 0.80 37 49

0.80 ≤ h < 1.00 46 p

1.00 ≤ h < 1.20 17 112

1.20 ≤ h < 1.40 8 120

(a) Write down the value of p . [1]

This information is shown in the following cumulative frequency curve.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

120

110

100

90

80

70

60

50

40

30

20

10

0

Height of plants (metres)

Cum

ulat

ive

frequ

ency



– 9 –


(b) Use the graph to find the median height of the plants. [1]

At the end of the project, the school will offer some of the plants to a local charity, Greentrust, and will replant some others in the school garden.

All plants whose heights are above 1.14 metres will be replanted in the school garden.

(c) Use the graph to find the number of plants that will be replanted in the school garden. [3]

All plants whose heights are greater than the lower quartile and less than the upper quartile will be offered to Greentrust.

(d) Write down the number of plants that will be offered to Greentrust. [1]

The range of heights of the plants offered to Greentrust is a < h < 0.96 .

(e) Write down the value of a . [1]

The shortest plant is 0.45 metres and the tallest plant is 1.35 metres.

(f) Draw a box-and-whisker diagram for this data, on graph paper, using a scale of 1 cm to represent 0.1 metres. [4]

Greentrust received a total of 180 plants from local schools and decided to sell them at a market. Greentrust paid 12 euros for a market stall from which to sell the plants. At the end of the day, Greentrust made a profit of 420 euros.

(g) Calculate the selling price of one plant, in euros, if 34

of them were sold and all plants

had the same selling price. [4]


Mathematical studies Standard level Paper 1 - Grade 12 - IB Math... · 2020. 5. 1. · Candidate...

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