mathematical studies standard level paper 2 - grade 12 math studies...2018/05/03 · standard level...

Mathematical studiesStandard levelPaper 2

10 pages

Thursday 3 May 2018 (morning)

1 hour 30 minutes

Instructions to candidates

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

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

© International Baccalaureate Organization 2018

M18/5/MATSD/SP2/ENG/TZ2/XX

2218 – 7406

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: 16]

In a company it is found that 25 % of the employees encountered traffic on their way to work. From those who encountered traffic the probability of being late for work is 80 %.

From those who did not encounter traffic, the probability of being late for work is 15 %.

The tree diagram illustrates the information.

traffic

notraffic

0.75

0.80

a

b

0.25

late forwork

late forwork

not latefor work

not latefor work

0.15

(a) Write down the value of

(i) a ;

(ii) b . [2]

(b) Use the tree diagram to find the probability that an employee

(i) encountered traffic and was late for work;

(ii) was late for work;

(iii) encountered traffic given that they were late for work. [8]

(This question continues on the following page)

– 2 – M18/5/MATSD/SP2/ENG/TZ2/XX

(Question 1 continued)

The company investigates the different means of transport used by their employees in the past year to travel to work. It was found that the three most common means of transport used to travel to work were public transportation (P ), car (C ) and bicycle (B ).

The company finds that 20 employees travelled by car, 28 travelled by bicycle and 19 travelled by public transportation in the last year.

Some of the information is shown in the Venn diagram.

UCP

y

x

13

4

2 6

8

B

(c) Find the value of

(i) x ;

(ii) y . [2]

There are 54 employees in the company.

(d) Find the number of employees who, in the last year, did not travel to work by car, bicycle or public transportation. [2]

(e) Find n C B P( )∪ ∩ ′( ) . [2]

Turn over



A transportation company owns 30 buses. The distance that each bus has travelled since being purchased by the company is recorded. The cumulative frequency curve for these data is shown.

0

5

10

15

20

25

30

5000 10 000 15 000 20 000 25 000Distance (km)

Cum

ulat

ive

frequ

ency

(a) Find the number of buses that travelled a distance between 15 000 and 20 000 kilometres. [2]

(b) Use the cumulative frequency curve to find the

(i) median distance;

(ii) lower quartile;

(iii) upper quartile. [4]




(c) Hence write down the interquartile range. [1]

(d) Write down the percentage of buses that travelled a distance greater than the upper quartile. [1]

(e) Find the number of buses that travelled a distance less than or equal to 12 000 km. [1]

It is known that 8 buses travelled more than m kilometres.

(f) Find the value of m . [2]

The smallest distance travelled by one of the buses was 2500 km. The longest distance travelled by one of the buses was 23 000 km.

(g) On graph paper, draw a box-and-whisker diagram for these data. Use a scale of 2 cm to represent 5000 km. [4]

Turn over



The weight, W , of basketball players in a tournament is found to be normally distributed with a mean of 65 kg and a standard deviation of 5 kg.

(a) (i) Find the probability that a basketball player has a weight that is less than 61 kg.

In a training session there are 40 basketball players.

(ii) Find the expected number of players with a weight less than 61 kg in this training session. [4]

(b) The probability that a basketball player has a weight that is within 1.5 standard deviations of the mean is q .

(i) Sketch a normal curve to represent this probability.

(ii) Find the value of q . [3]

(c) Given that P (W > k) = 0.225 , find the value of k . [2]

A basketball coach observed 60 of her players to determine whether their performance and their weight were independent of each other. Her observations were recorded as shown in the table.

Performance

Satisfactory Excellent

Weight

Below average 6 10

Average 7 15

Above average 12 10

She decided to conduct a χ 2 test for independence at the 5 % significance level.

(d) For this test,

(i) state the null hypothesis; [1]

(ii) find the p-value. [2]

(e) State a conclusion for this test. Justify your answer. [2]



A new café opened and during the first week their profit was $60.

The café’s profit increases by $10 every week.

(a) Find the café’s profit during the 11th week. [3]

(b) Calculate the café’s total profit for the first 12 weeks. [3]

A new tea-shop opened at the same time as the café. During the first week their profit was also $60.

The tea-shop’s profit increases by 10 % every week.

(c) Find the tea-shop’s profit during the 11th week. [3]

(d) Calculate the tea-shop’s total profit for the first 12 weeks. [3]

In the mth week the tea-shop’s total profit exceeds the café’s total profit, for the first time since they both opened.

(e) Find the value of m . [4]

Turn over



The Tower of Pisa is well known worldwide for how it leans.

Giovanni visits the Tower and wants to investigate how much it is leaning. He draws a diagram showing a non-right triangle, ABC.

On Giovanni’s diagram the length of AB is 56 m, the length of BC is 37 m, and angle ACB is 60. AX is the perpendicular height from A to BC.

diagram not to scale

A

B CX

56

37

60°

(a) Use Giovanni’s diagram to

(i) show that angle ABC, the angle at which the Tower is leaning relative to the horizontal, is 85 to the nearest degree.

(ii) calculate the length of AX.

(iii) find the length of BX, the horizontal displacement of the Tower. [9]




Giovanni’s tourist guidebook says that the actual horizontal displacement of the Tower, BX, is 3.9 metres.

(b) Find the percentage error on Giovanni’s diagram. [2]

Giovanni adds a point D to his diagram, such that BD = 45 m , and another triangle is formed.


A

B C D45X

56

(c) Find the angle of elevation of A from D. [3]

Turn over



Consider the curve y = 2x3 -9x2 + 12x + 2 , for -1

Mathematical studiesStandard levelPaper 2

11 pages

Thursday 3 May 2018 (morning)

1 hour 30 minutes

Instructions to candidates

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

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

© International Baccalaureate Organization 2018

M18/5/MATSD/SP2/ENG/TZ1/XX

2218 – 7404

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.


Farmer Brown has built a new barn, on horizontal ground, on his farm. The barn has a cuboid base and a triangular prism roof, as shown in the diagram.


A

E

F

G

H

B C

K

L

D

10

16

7

5

15˚

The cuboid has a width of 10 m, a length of 16 m and a height of 5 m. The roof has two sloping faces and two vertical and identical sides, ADE and GLF. The face DEFL slopes at an angle of 15 to the horizontal and ED = 7 m .

(a) Calculate the area of triangle EAD. [3]

(b) Calculate the total volume of the barn. [3]


M18/5/MATSD/SP2/ENG/TZ1/XX– 2 –


The roof was built using metal supports. Each support is made from five lengths of metal AE, ED, AD, EM and MN, and the design is shown in the following diagram.

diagram not to scaleE

N

DM

A

7

10

15

ED = 7 m , AD = 10 m and angle ADE = 15 . M is the midpoint of AD. N is the point on ED such that MN is at right angles to ED.

(c) Calculate the length of MN. [2]

(d) Calculate the length of AE. [3]

Farmer Brown believes that N is the midpoint of ED.

(e) Show that Farmer Brown is incorrect. [3]

(f) Calculate the total length of metal required for one support. [4]

Turn over



On one day 180 flights arrived at a particular airport. The distance travelled and the arrival status for each incoming flight was recorded. The flight was then classified as on time, slightly delayed, or heavily delayed.

The results are shown in the following table.

Distance travelled

At most 500 kmBetween 500 km

and 5000 kmAt least 5000 km TOTAL

Arr

ival

Sta

tus On time 19 17 16 52

Slightly delayed 13 18 14 45

Heavily delayed 28 15 40 83

TOTAL 60 50 70 180

A χ2 test is carried out at the 10 % significance level to determine whether the arrival status of incoming flights is independent of the distance travelled.

(a) State the alternative hypothesis. [1]

(b) Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed. [2]

(c) Write down the number of degrees of freedom. [1]

(d) Write down

(i) the χ2 statistic;

(ii) the associated p-value. [3]




The critical value for this test is 7.779.

(e) State, with a reason, whether you would reject the null hypothesis. [2]

A flight is chosen at random from the 180 recorded flights.

(f) Write down the probability that this flight arrived on time. [2]

(g) Given that this flight was not heavily delayed, find the probability that it travelled between 500 km and 5000 km. [2]

Two flights are chosen at random from those which were slightly delayed.

(h) Find the probability that each of these flights travelled at least 5000 km. [3]

Turn over



Give your answers to parts (b), (c) and (d) to the nearest whole number.

Harinder has 14 000 US Dollars (USD) to invest for a period of five years. He has two options of how to invest the money.

Option A: Invest the full amount, in USD, in a fixed deposit account in an American bank.

The account pays a nominal annual interest rate of r % , compounded yearly, for the five years. The bank manager says that this will give Harinder a return of 17 500 USD.

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

Option B: Invest the full amount, in Indian Rupees (INR), in a fixed deposit account in an Indian bank. The money must be converted from USD to INR before it is invested.

The exchange rate is 1 USD = 66.91 INR.

(b) Calculate 14 000 USD in INR. [2]

The account in the Indian bank pays a nominal annual interest rate of 5.2 % compounded monthly.

(c) Calculate the amount of this investment, in INR, in this account after five years. [3]

Harinder chose option B. At the end of five years, Harinder converted this investment back to USD. The exchange rate, at that time, was 1 USD = 67.16 INR.

(d) Calculate how much more money, in USD, Harinder earned by choosing option B instead of option A. [3]



Consider the function f xx

kx( ) = + −48 582 , where x > 0 and k is a constant.

The graph of the function passes through the point with coordinates (4 , 2) .

(a) Find the value of k . [2]

(b) Using your value of k , find f ′(x) . [3]

P is the minimum point of the graph of f (x) .

(c) Use your answer to part (b) to show that the minimum value of f (x) is -22 . [3]

(d) Write down the two values of x which satisfy f (x) = 0 . [2]

(e) Sketch the graph of y = f (x) for 0 < x ≤ 6 and -30 ≤ y ≤ 60 . Clearly indicate the minimum point P and the x-intercepts on your graph. [4]

Turn over



Contestants in a TV gameshow try to get through three walls by passing through doors without falling into a trap. Contestants choose doors at random.

If they avoid a trap they progress to the next wall. If a contestant falls into a trap they exit the game before the next contestant plays. Contestants are not allowed to watch each other attempt the game.

Second Wall Third WallFirst Wall

Door

Trap

Key:

The first wall has four doors with a trap behind one door.

Ayako is a contestant.

(a) Write down the probability that Ayako avoids the trap in this wall. [1]

Natsuko is the second contestant.

(b) Find the probability that only one of Ayako and Natsuko falls into a trap while attempting to pass through a door in the first wall. [3]

The second wall has five doors with a trap behind two of the doors.

The third wall has six doors with a trap behind three of the doors.




The following diagram shows the branches of a probability tree diagram for a contestant in the game.

Falls intotrap

Avoidstrap

Falls intotrap

Avoidstrap

Falls intotrap

Avoidstrap

First Wall Second Wall Third Wall

(c) Copy the probability tree diagram and write down the relevant probabilities along the branches. [3]

(d) A contestant is chosen at random. Find the probability that this contestant

(i) fell into a trap while attempting to pass through a door in the second wall;

(ii) fell into a trap. [5]

120 contestants attempted this game.

(e) Find the expected number of contestants who fell into a trap while attempting to pass through a door in the third wall. [3]

Turn over



A manufacturer makes trash cans in the form of a cylinder with a hemispherical top. The trash can has a height of 70 cm. The base radius of both the cylinder and the hemispherical top is 20 cm.


70

TRASH

20

(a) Write down the height of the cylinder. [1]

(b) Find the total volume of the trash can. [4]

A designer is asked to produce a new trash can. The new trash can will also be in the form of a cylinder with a hemispherical top. This trash can will have a height of H cm and a base radius of r cm.


H

TRASH

r




There is a design constraint such that H + 2r = 110 cm . The designer has to maximize the volume of the trash can.

(c) Find the height of the cylinder, h , of the new trash can, in terms of r . [2]

(d) Show that the volume, V cm3 , of the new trash can is given by

V r r= −110 73

2 3π π . [3]

(e) Using your graphic display calculator, find the value of r which maximizes the value of V . [2]

The designer claims that the new trash can has a capacity that is at least 40 % greater than the capacity of the original trash can.

(f) State whether the designer’s claim is correct. Justify your answer. [4]


mathematical studies standard level paper 2 - grade 12 math studies...2018/05/03 · standard level...

Documents