WS Fall Semester Final Review 2018 [198 marks]

1a.

The following table shows four different sets of numbers: , , and .

Complete the second column of the table by giving one example of a number from each set.

N Z Q R

1b. Josh states: “Every integer is a natural number”.

Write down whether Josh’s statement is correct. Justify your answer.

2. Consider the numbers and .

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

−1, 4, , √2, 0.3523

−22

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3a.

The following Venn diagram shows the relationship between the sets of numbers

The number –3 belongs to the set of

and

, but not

, and is placed in the appropriate position on the Venn diagram as an example.

Write down the following numbers in the appropriate place in the Venn diagram.

4

N, Z, Q and R.

Z, QRN

3b.13

3c. π

3d. 0.38

3e. √5

3f. −0.25

4a.

Consider c = 5200 and d = 0.0000037.

Write down the value of r = c × d.

4b. Write down your value of r in the form a × 10 , where 1 ≤ a < 10 and.

k

k ∈ Z

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4c. Consider the following statements about c, d and r. Only three of these statements are true.

Circle the true statements.

5a.

Each year the soccer team, Peterson United, plays 25 games at their home stadium. The owner of Peterson United claimed that lastyear the mean attendance per game at their home stadium was 24500.

Based on the owner’s claim, calculate the total attendance for the games at Peterson United’s home stadium last year.

5b.

The actual total attendance last year was 617700.

Calculate the percentage error in the owner’s claim.

5c. Write down your answer to part (b) in the form a × 10 where 1 ≤ a < 10, .k k ∈ Z

6a.

Abdallah owns a plot of land, near the river Nile, in the form of a quadrilateral ABCD.

The lengths of the sides are and angle .

This information is shown on the diagram.

Show that correct to the nearest metre.

AB = 40 m, BC = 115 m, CD = 60 m, AD = 84 m BAD = 90∘

BD = 93 m

6b. Calculate angle .BCD

6c. Find the area of ABCD.

6d.

The formula that the ancient Egyptians used to estimate the area of a quadrilateral ABCD is

.

Abdallah uses this formula to estimate the area of his plot of land.

Calculate Abdallah’s estimate for the area.

area =(AB+CD)(AD+BC)

4

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6e. Find the percentage error in Abdallah’s estimate.

7a.

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

Karl invests 1000 US dollars (USD) in an account that pays a nominal annual interest of 3.5%, compounded quarterly. He leaves themoney in the account for 5 years.

Calculate the amount of money he has in the account after 5 years.

7b. Write down the amount of interest he earned after 5 years.

7c. Karl decides to donate this interest to a charity in France. The charity receives 170 euros (EUR). The exchange rate is 1 USD = tEUR.

Calculate the value of t.

8a.

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 willgive Harinder a return of 17 500 USD.

Calculate the value of r.

8b.

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

The exchange rate is 1 USD = 66.91 INR.

Calculate 14 000 USD in INR.

8c.

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

Calculate the amount of this investment, in INR, in this account after five years.

8d. Harinder chose option B. At the end of five years, Harinder converted this investment back to USD. The exchange rate, at thattime, was 1 USD = 67.16 INR.

Calculate how much more money, in USD, Harinder earned by choosing option B instead of option A.

9a.

In this question, give all answers correct to 2 decimal places.

Jose travelled from Buenos Aires to Sydney. He used Argentine pesos, ARS, to buy 350 Australian dollars, AUD, at a bank. Theexchange rate was 1 ARS = 0.1559 AUD.

Use this exchange rate to calculate the amount of ARS that is equal to 350 AUD.

9b.

The bank charged Jose a commission of 2%.

Calculate the total amount of ARS Jose paid to get 350 AUD.

9c.

Jose used his credit card to pay his hotel bill in Sydney. The bill was 585 AUD. The value the credit card company charged for thispayment was 4228.38 ARS. The exchange rate used by the credit card company was 1 AUD = ARS. No commission was charged.

Find the value of .

x

x

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10a.

Sergei is training to be a weightlifter. Each day he trains at the local gym by lifting a metal bar that has heavy weights attached. Hecarries out successive lifts. After each lift, the same amount of weight is added to the bar to increase the weight to be lifted.

The weights of each of Sergei’s lifts form an arithmetic sequence.

Sergei’s friend, Yuri, records the weight of each lift. Unfortunately, last Monday, Yuri misplaced all but two of the recordings of Sergei’slifts.

On that day, Sergei lifted 21 kg on the third lift and 46 kg on the eighth lift.

For that day find how much weight was added after each lift.

10b. For that day find the weight of Sergei’s first lift.

10c. On that day, Sergei made 12 successive lifts. Find the total combined weight of these lifts.

11a.

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

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

Find the café’s profit during the 11th week.

11b. Calculate the café’s total profit for the first 12 weeks.

11c.

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.

Find the tea-shop’s profit during the 11th week.

11d. Calculate the tea-shop’s total profit for the first 12 weeks.

11e. In the mth week the tea-shop’s total profit exceeds the café’s total profit, for the first time since they both opened.

Find the value of m.

12a.

The first three terms of a geometric sequence are .

Find the value of , the common ratio of the sequence.

u1 = 486, u2 = 162, u3 = 54

r

12b. Find the value of for which .n un = 2

12c. Find the sum of the first 30 terms of the sequence.

13a.

Consider the geometric sequence .

Write down the common ratio of the sequence.

u1 = 18, u2 = 9, u3 = 4.5, …

13b. Find the value of .u5

13c. Find the smallest value of for which is less than .n un 10−3

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14a.

The lengths of trout in a fisherman’s catch were recorded over one month, and are represented in the following histogram.

Complete the following table.

14b. State whether length of trout is a continuous or discrete variable.

14c. Write down the modal class.

14d. Any trout with length 40 cm or less is returned to the lake.

Calculate the percentage of the fisherman’s catch that is returned to the lake.

15a.

In a high school, 160 students completed a questionnaire which asked for the number of people they are following on a social mediawebsite. The results were recorded in the following box-and-whisker diagram.

Write down the median.

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15b.

The following incomplete table shows the distribution of the responses from these 160 students.

Complete the table.

15c. Write down the mid-interval value for the 100 < x ≤ 150 group.

15d. Using the table, calculate an estimate for the mean number of people being followed on the social media website by these 160students.

16a.

A survey was conducted to determine the length of time,

, in minutes, people took to drink their coffee in a café. The information is shown in the following grouped frequency

table.

Write down the total number of people who were surveyed.

t

16b. Write down the mid-interval value for the group.10 < t ⩽ 15

16c. Find an estimate of the mean time people took to drink their coffee.

16d. The information above has been rewritten as a cumulative frequency table.

Write down the value of and the value of.

a

b

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16e. This information is shown in the following cumulative frequency graph.

For the people who were surveyed, use the graph to estimate

(i) the time taken for the first people to drink their coffee;

(ii) the number of people who take less than minutes to drink their coffee;

(iii) the number of people who take more than minutes to drink their coffee.

40

8

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17a.

A group of 20 students travelled to a gymnastics tournament together. Their ages, in years, are given in the following table.

For the students in this group find the mean age;

17b. For the students in this group write down the median age.

17c.

The lower quartile of the ages is 16 and the upper quartile is 18.5.

Draw a box-and-whisker diagram, for these students’ ages, on the following grid.

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18a. In a school 160 students sat a mathematics examination. Their scores, given as marks out of 90, are summarized on the cumulativefrequency diagram.

Write down the median score.

18b. The lower quartile of these scores is 40.

Find the interquartile range.

18c. The lowest score was 6 marks and the highest score was 90 marks.

Draw a box-and-whisker diagram on the grid below to represent the students’ examination scores.

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19a.

Each month the number of days of rain in Cardiff is recorded.The following data was collected over a period of 10 months.

11 13 8 11 8 7 8 14 x 15

For these data the median number of days of rain per month is 10.

Find the value of x.

19b. Find the standard deviation

19c. Find the interquartile range.

20a.

Consider the following Venn diagrams.

Write down an expression, in set notation, for the shaded region represented by Diagram 1.

20b. Write down an expression, in set notation, for the shaded region represented by Diagram 2.

20c. Write down an expression, in set notation, for the shaded region represented by Diagram 3.

20d. Shade, on the Venn diagram, the region represented by the set .

(H ∪ I)′

20e. Shade, on the Venn diagram, the region represented by the set .J ∩ K

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21a.

In a company it is found that 25 % of the employees encountered traffic on their way to work. From those who encountered traffic theprobability 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.

Write down the value of a.

21b. Write down the value of b.

21c. Use the tree diagram to find the probability that an employee encountered traffic and was late for work.

21d. Use the tree diagram to find the probability that an employee was late for work.

21e. Use the tree diagram to find the probability that an employee encountered traffic given that they were late for work.

21f.

The company investigates the different means of transport used by their employees in the past year to travel to work. It was found thatthe 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.

Find the value of x.

21g. Find the value of y.

21h.

There are 54 employees in the company.

Find the number of employees who, in the last year, did not travel to work by car, bicycle or public transportation.

21i. Find .n ((C ∪ B) ∩ P ′)

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22a.

Dune Canyon High School organizes its school year into three trimesters: fall/autumn ( ), winter ( ) and spring ( ). The school offersa variety of sporting activities during and outside the school year.

The activities offered by the school are summarized in the following Venn diagram.

Write down the number of sporting activities offered by the school during its school year.

F W S

22b. Determine whether rock-climbing is offered by the school in the fall/autumn trimester.

22c. Write down the elements of the set ;F ∩ W ′

22d. Write down .n(W ∩ S)

22e. Write down, in terms of , and , an expression for the set which contains only archery, baseball, kayaking and surfing.F W S

23a.

Rosewood College has 120 students. The students can join the sports club ( ) and the music club ( ).

For a student chosen at random from these 120, the probability that they joined both clubs is and the probability that they joined the

music club is .

There are 20 students that did not join either club.

Complete the Venn diagram for these students.

S M

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23b. One of the students who joined the sports club is chosen at random. Find the probability that this student joined both clubs.

23c. Determine whether the events and are independent.S M

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24a.

In an international competition, participants can answer questions in only one of the three following languages: Portuguese, Mandarinor Hindi. 80 participants took part in the competition. The number of participants answering in Portuguese, Mandarin or Hindi is shown inthe table.

State the number of boys who answered questions in Portuguese.

24b.

A boy is chosen at random.

Find the probability that the boy answered questions in Hindi.

24c. Two girls are selected at random.

Calculate the probability that one girl answered questions in Mandarin and the other answered questions in Hindi.

25a.

Sara regularly flies from Geneva to London. She takes either a direct flight or a non-directflight that goes via Amsterdam.

If she takes a direct flight, the probability that her baggage does not arrive in London is 0.01.If she takes a non-direct flight the probability that her baggage arrives in London is 0.95.

The probability that she takes a non-direct flight is 0.2.

Complete the tree diagram.

25b. Find the probability that Sara’s baggage arrives in London.

26a.

All the children in a summer camp play at least one sport, from a choice of football ( ) or basketball ( ). 15 children play both sports.

The number of children who play only football is double the number of children who play only basketball.

Let be the number of children who play only football.

Write down an expression, in terms of , for the number of children who play only basketball.

F B

x

x

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Printed for Raymore-Peculiar High School

© International Baccalaureate Organization 2018

International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®

26b. Complete the Venn diagram using the above information.

26c.

There are 120 children in the summer camp.

Find the number of children who play only football.

26d. Write down the value of .n(F)

27a.

On a work day, the probability that Mr Van Winkel wakes up early is .

If he wakes up early, the probability that he is on time for work is .

If he wakes up late, the probability that he is on time for work is .

Complete the tree diagram below.

45

p

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27b.

The probability that Mr Van Winkel arrives on time for work is .

Find the value of .

35

p

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