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Candidate session number
M16/5/MATSD/SP1/ENG/TZ1/XX
Mathematical studiesStandard levelPaper 1
© International Baccalaureate Organization 201620 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 – 7403
20EP01
– 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. The probability that Nikita wins a tennis match depends on the surface of the tennis court on which she is playing. The probability that she plays on a grass court is 0.4.The probability that Nikita wins on a grass court is 0.35. The probability that Nikita wins when the court is not grass is 0.25.
(a) Complete the following tree diagram. [3]
win
win
lose
lose
grass
not grass
0.35
0.4
0.25
(b) Find the probability that Nikita wins a match. [3]
Working:
Answer:
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP03
M16/5/MATSD/SP1/ENG/TZ1/XX
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2. Temi’s sailing boat has a sail in the shape of a right-angled triangle, ABC. BC = 5.45 m , angle CAB = 76˚ and angle ABC = 90˚ .
(a) Calculate AC, the height of Temi’s sail. [2]
diagram not to scaleC T
A
B
R S76˚
5.45 m
2.80 m
Temi's boat William's boat
William also has a sailing boat with a sail in the shape of a right-angled triangle, TRS. RS = 2.80 m . The area of William’s sail is 10.7 m2.
(b) Calculate RT, the height of William’s sail. [2]
(c) Calculate the size of angle RST. [2]
(This question continues on the following page)
20EP04
M16/5/MATSD/SP1/ENG/TZ1/XX
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Turn over
(Question 2 continued)
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP05
M16/5/MATSD/SP1/ENG/TZ1/XX
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3. In a school 160 students sat a mathematics examination. Their scores, given as marks out of 90, are summarized on the cumulative frequency diagram.
Cum
ulat
ive
frequ
ency
0 10 20 30 40 50 60 70 80 90
20
40
60
80
100
120
140
160
Marks
(a) Write down the median score. [1]
The lower quartile of these scores is 40.
(b) Find the interquartile range. [2]
(This question continues on the following page)
20EP06
M16/5/MATSD/SP1/ENG/TZ1/XX
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Turn over
(Question 3 continued)
The lowest score was 6 marks and the highest score was 90 marks.
(c) Draw a box-and-whisker diagram on the grid below to represent the students’ examination scores. [3]
0 10 20 30 40 50 60 70 80 90 100Marks
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP07
M16/5/MATSD/SP1/ENG/TZ1/XX
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4. FreshWave brand tuna is sold in cans that are in the shape of a cuboid with length 8 cm, width 5 cm and height 3.5 cm. HappyFin brand tuna is sold in cans that are cylindrical with diameter 7 cm and height 4 cm.
diagram not to scale
FreshWave HappyFin
(a) Find the volume, in cm3, of a can of
(i) FreshWave tuna;
(ii) HappyFin tuna. [4]
The price of tuna per cm3 is the same for each brand. A can of FreshWave tuna costs 90 cents.
(b) Calculate the price, in cents, of a can of HappyFin tuna. [2]
Working:
Answers:
(a) (i) . . . . . . . . . . . . . . . . . . . . .
(ii) . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . .
20EP08
M16/5/MATSD/SP1/ENG/TZ1/XX
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Turn over
5. Consider the following statements
z : x is an integerq : x is a rational numberr : x is a real number.
(a) (i) Write down, in words, ¬ q .
(ii) Write down a value for x such that the statement ¬ q is true. [2]
(b) Write the following argument in symbolic form: “If x is a real number and x is not a rational number, then x is not an integer”. [3]
Phoebe states that the argument in part (b) can be shown to be valid, without the need of a truth table.
(c) Justify Phoebe’s statement. [1]
Working:
Answers:
(a) (i) . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
(ii) . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . .
(c) . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
20EP09
M16/5/MATSD/SP1/ENG/TZ1/XX
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6. One of the locations in the 2016 Olympic Games is an amphitheatre. The number of seats in the first row of the amphitheatre, u1 , is 240. The number of seats in each subsequent row forms an arithmetic sequence. The number of seats in the sixth row, u6 , is 270.
(a) Calculate the value of the common difference, d . [2]
There are 20 rows in the amphitheatre.
(b) Find the total number of seats in the amphitheatre. [2]
Anisha visits the amphitheatre. She estimates that the amphitheatre has 6500 seats.
(c) Calculate the percentage error in Anisha’s estimate. [2]
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP10
M16/5/MATSD/SP1/ENG/TZ1/XX
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Turn over
7. The equation of line L1 is y = 2.5x + k . Point A (3 , −2) lies on L1 .
(a) Find the value of k . [2]
The line L2 is perpendicular to L1 and intersects L1 at point A.
(b) Write down the gradient of L2 . [1]
(c) Find the equation of L2 . Give your answer in the form y = mx + c . [2]
(d) Write your answer to part (c) in the form ax + by + d = 0 where a , b and d ∈ . [1]
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP11
M16/5/MATSD/SP1/ENG/TZ1/XX
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8. The lifetime, L , of light bulbs made by a company follows a normal distribution. L is measured in hours. The normal distribution curve of L is shown below.
580052004600 6400 7000hours
(a) Write down the mean lifetime of the light bulbs. [1]
The standard deviation of the lifetime of the light bulbs is 850 hours.
(b) Find the probability that 5000 ≤ L ≤ 6000 , for a randomly chosen light bulb. [2]
The company states that 90 % of the light bulbs have a lifetime of at least k hours.
(c) Find the value of k . Give your answer correct to the nearest hundred. [3]
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP12
M16/5/MATSD/SP1/ENG/TZ1/XX
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Turn over
9. In this question give all answers correct to the nearest whole number.
Loic travelled from China to Brazil. At the airport he exchanged 3100 Chinese Yuan, CNY, to Brazilian Real, BRL, at an exchange rate of 1 CNY = 0.3871 BRL.No commission was charged.
(a) Calculate the amount of BRL he received. [2]
When he returned to China, Loic changed his remaining BRL at a bank. The exchange rate at the bank was 1 CNY = 0.3756 BRL and a commission of 5 % was charged. He received 285 CNY.
(b) (i) Calculate the amount of CNY Loic would have received if no commission was charged.
(ii) Calculate the amount of BRL Loic exchanged when he returned to China. [4]
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) (i) . . . . . . . . . . . . . . . . . . . . .
(ii) . . . . . . . . . . . . . . . . . . . . .
20EP13
M16/5/MATSD/SP1/ENG/TZ1/XX
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10. The manager of a travel agency surveyed 1200 travellers. She wanted to find out whether there was a relationship between a traveller’s age and their preferred destination. The travellers were asked to complete the following survey.
Traveller survey
My age is:
25 or younger 26–40 41–60 61 or older
My preferred destination is:
New York Tokyo Melbourne Dubai Marrakech
A χ 2 test was carried out, at the 5 % significance level, on the data collected.
(a) Write down the null hypothesis. [1]
(b) Find the number of degrees of freedom. [2]
The critical value of this χ 2 test is 21.026.
(c) Use this information to write down the values of the χ 2 statistic for which the null hypothesis is rejected. [1]
From the travellers taking part in the survey, 285 were 61 years or older and 420 preferred Tokyo.
(d) Calculate the expected number of travellers who preferred Tokyo and were 61 years or older. [2]
(This question continues on the following page)
20EP14
M16/5/MATSD/SP1/ENG/TZ1/XX
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Turn over
(Question 10 continued)
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP15
M16/5/MATSD/SP1/ENG/TZ1/XX
– 16 –
11. Consider the function f (x) = ax2 + c .
(a) Find f ′(x). [1]
Point A (−2 , 5) lies on the graph of y = f (x) . The gradient of the tangent to this graph at A is −6.
(b) Find the value of a . [3]
(c) Find the value of c . [2]
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP16
M16/5/MATSD/SP1/ENG/TZ1/XX
– 17 –
Turn over
12. In this question give all answers correct to two decimal places.
Diogo deposited 8000 Argentine pesos, ARS, in a bank account which pays a nominal annual interest rate of 15 %, compounded monthly.
(a) Find how much interest Diogo has earned after 2 years. [3]
Carmen also deposited ARS in a bank account. Her account pays a nominal annual interest rate of 17 %, compounded yearly. After three years, the total amount in Carmen’s account is 10 000 ARS.
(b) Find the amount that Carmen deposited in the bank account. [3]
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP17
M16/5/MATSD/SP1/ENG/TZ1/XX
– 18 –
13. The golden ratio, r , was considered by the Ancient Greeks to be the perfect ratio between
the lengths of two adjacent sides of a rectangle. The exact value of r is 1 52+ .
(a) Write down the value of r
(i) correct to 5 significant figures;
(ii) correct to 2 decimal places. [2]
Phidias is designing rectangular windows with adjacent sides of length x metres and y metres. The area of each window is 1 m2 .
(b) Write down an equation to describe this information. [1]
Phidias designs the windows so that the ratio between the longer side, y , and the shorter side, x , is the golden ratio, r .
(c) Write down an equation in y , x and r to describe this information. [1]
(d) Find the value of x . [2]
Working:
Answers:
(a) (i) . . . . . . . . . . . . . . . . . . . . .
(ii) . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . .
(c) . . . . . . . . . . . . . . . . . . . . . . . . .
(d) . . . . . . . . . . . . . . . . . . . . . . . . .
20EP18
M16/5/MATSD/SP1/ENG/TZ1/XX
– 19 –
Turn over
14. A population of 200 rabbits was introduced to an island. One week later the number of rabbits was 210. The number of rabbits, N , can be modelled by the function
N (t) = 200 × b t , t ≥ 0 ,
where t is the time, in weeks, since the rabbits were introduced to the island.
(a) Find the value of b . [2]
(b) Calculate the number of rabbits on the island after 10 weeks. [2]
An ecologist estimates that the island has enough food to support a maximum population of 1000 rabbits.
(c) Calculate the number of weeks it takes for the rabbit population to reach this maximum. [2]
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP19
M16/5/MATSD/SP1/ENG/TZ1/XX
– 20 –
15. A company sells fruit juices in cylindrical cans, each of which has a volume of 340 cm3. The surface area of a can is A cm2 and is given by the formula
2 6802= π +A rr
,
where r is the radius of the can, in cm.
To reduce the cost of a can, its surface area must be minimized.
(a) Find ddAr
. [3]
(b) Calculate the value of r that minimizes the surface area of a can. [3]
Working:
Answers:
(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20EP20
M16/5/MATSD/SP1/ENG/TZ1/XX
M16/5/MATSD/SP2/ENG/TZ1/XX
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 – 7404
– 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: 12]
For an ecological study, Ernesto measured the average concentration ( y) of the fine dust, PM10, in the air at different distances (x) from a power plant. His data are represented on the following scatter diagram. The concentration of PM10 is measured in micrograms per cubic metre and the distance is measured in kilometres.
2 4 6 8 10 12
20
40
0
60
80
100
120
140
x
y
distance (km)
Con
cent
ratio
n of
PM
10 (µ
g m–3
)
(This question continues on the following page)
M16/5/MATSD/SP2/ENG/TZ1/XX
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Turn over
(Question 1 continued)
His data are also listed in the following table.
Distance (x) 0.6 1.2 2.6 a 5.5 6.2 7.5 8.6 10.5 12.2
Concentration of PM10 ( y) 128 115 103 89 92 80 72 b 65 62
(a) Use the scatter diagram to find the value of a and of b in the table. [2]
(b) Calculate
(i) x , the mean distance from the power plant;
(ii) y , the mean concentration of PM10;
(iii) r , the Pearson’s product–moment correlation coefficient. [4]
(c) Write down the equation of the regression line y on x . [2]
Ernesto’s school is located 14 km from the power plant. He uses the equation of the regression line to estimate the concentration of PM10 in the air at his school.
(d) (i) Calculate the value of Ernesto’s estimate.
(ii) State whether Ernesto’s estimate is reliable. Justify your answer. [4]
M16/5/MATSD/SP2/ENG/TZ1/XX
– 4 –
2. [Maximum mark: 18]
A group of students at Dune Canyon High School were surveyed. They were asked which of the following products: books (B), music (M) or films (F), they downloaded from the internet.
The following results were obtained:
100 students downloaded music;95 students downloaded films;68 students downloaded films and music;52 students downloaded books and music;50 students downloaded films and books;40 students downloaded all three products;8 students downloaded books only;25 students downloaded none of the three products.
(a) Use the above information to complete a Venn diagram. [5]
(b) Calculate the number of students who were surveyed. [2]
(c) (i) On your Venn diagram, shade the set (F ∪ M ) ∩ B′ . Do not shade any labels or values on the diagram.
(ii) Find n F M B( )∪ ∩ ′( ) . [3]
(d) A student who was surveyed is chosen at random.Find the probability that
(i) the student downloaded music;
(ii) the student downloaded books, given that they had not downloaded films;
(iii) the student downloaded at least two of the products. [6]
Dune Canyon High School has 850 students.
(e) Find the expected number of students at Dune Canyon High School that downloaded music. [2]
M16/5/MATSD/SP2/ENG/TZ1/XX
– 5 –
Turn over
3. [Maximum mark: 13]
A distress flare is fired into the air from a ship at sea. The height, h , in metres, of the flare above sea level is modelled by the quadratic function
h (t) = −0.2t2 + 16t + 12 , t ≥ 0 ,
where t is the time, in seconds, and t = 0 at the moment the flare was fired.
(a) Write down the height from which the flare was fired. [1]
(b) Find the height of the flare 15 seconds after it was fired. [2]
The flare fell into the sea k seconds after it was fired.
(c) Find the value of k . [2]
(d) Find h′ (t) . [2]
(e) (i) Show that the flare reached its maximum height 40 seconds after being fired.
(ii) Calculate the maximum height reached by the flare. [3]
The nearest coastguard can see the flare when its height is more than 40 metres above sea level.
(f) Determine the total length of time the flare can be seen by the coastguard. [3]
M16/5/MATSD/SP2/ENG/TZ1/XX
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4. [Maximum mark: 19]
The Great Pyramid of Giza in Egypt is a right pyramid with a square base. The pyramid is made of solid stone. The sides of the base are 230 m long. The diagram below represents this pyramid, labelled VABCD.
V is the vertex of the pyramid. O is the centre of the base, ABCD. M is the midpoint of AB. Angle ABV = 58.3˚ .
diagram not to scaleV
A
D C
BM
O 230 m
(a) Show that the length of VM is 186 metres, correct to three significant figures. [3]
(b) Calculate the height of the pyramid, VO. [2]
(c) Find the volume of the pyramid. [2]
(d) Write down your answer to part (c) in the form a × 10k where 1 ≤ a < 10 and k ∈ . [2]
Ahmad is a tour guide at the Great Pyramid of Giza. He claims that the amount of stone used to build the pyramid could build a wall 5 metres high and 1 metre wide stretching from Paris to Amsterdam, which are 430 km apart.
(e) Determine whether Ahmad’s claim is correct. Give a reason. [4]
(This question continues on the following page)
M16/5/MATSD/SP2/ENG/TZ1/XX
– 7 –
Turn over
(Question 4 continued)
Ahmad and his friends like to sit in the pyramid’s shadow, ABW, to cool down.At mid-afternoon, BW = 160 m and angle ABW = 15˚ .
diagram not to scale
W
A B
160 m
D C
V
230 m
(f) (i) Calculate the length of AW at mid-afternoon.
(ii) Calculate the area of the shadow, ABW, at mid-afternoon. [6]
M16/5/MATSD/SP2/ENG/TZ1/XX
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5. [Maximum mark: 16]
Antonio and Barbara start work at the same company on the same day. They each earn an annual salary of 8000 euros during the first year of employment. The company gives them a salary increase following the completion of each year of employment. Antonio is paid using plan A and Barbara is paid using plan B.
Plan A: The annual salary increases by 450 euros each year.
Plan B: The annual salary increases by 5 % each year.
(a) Calculate
(i) Antonio’s annual salary during his second year of employment;
(ii) Barbara’s annual salary during her second year of employment. [3]
(b) Write down an expression for
(i) Antonio’s annual salary during his n th year of employment;
(ii) Barbara’s annual salary during her n th year of employment. [4]
(c) Determine the number of years for which Antonio’s annual salary is greater than or equal to Barbara’s annual salary. [2]
Both Antonio and Barbara plan to work at the company for a total of 15 years.
(d) (i) Calculate the total amount that Barbara will be paid during these 15 years.
(ii) Determine whether Antonio earns more than Barbara during these 15 years. [7]
M16/5/MATSD/SP2/ENG/TZ1/XX
– 9 –
6. [Maximum mark: 12]
A function, f , is given by
f (x) = 4 × 2−x + 1.5x − 5 .
(a) Calculate f (0) . [2]
(b) Use your graphic display calculator to solve f (x) = 0 . [2]
(c) Sketch the graph of y = f (x) for −2 ≤ x ≤ 6 and −4 ≤ y ≤ 10 , showing the x and y intercepts. Use a scale of 2 cm to represent 2 units on both the horizontal axis, x , and the vertical axis, y . [4]
The function f is the derivative of a function g . It is known that g (1) = 3 .
(d) (i) Calculate g′ (1) .
(ii) Find the equation of the tangent to the graph of y = g (x) at x = 1 . Give your answer in the form y = mx + c . [4]
M16/5/MATSD/SP2/ENG/TZ1/XX
