solve them completely showing your steps along the way

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1 Dear IB Math Studies SL Year 2 Students, We have covered chapter 1 (number and algebra 1), chapter 2 (descriptive statistics), chapter 5 (statistical applications), chapter 7 (number and algebra 2), chapter 8 (sets and probability), and chapter 9 (logic) during your first year in IB Math Studies SL. To be ready for the first day in your second year in math studies, certain concepts taught on your first year need to be reviewed. Remember that the finals exam especially the IB exams (paper 1 and 2) will cover all topics from chapters 1 through 10. To prepare you, the attached worksheet needs to be completed over the summer. It is to be turned in on the first day of class and will count as your first quiz grade. You need to solve them completely showing your steps along the way. If just an answer is given without any work to accompany it, it will not receive credit. Any graphing needs to be done on actual graph paper (with a ruler no freehand graphs!) Graphs not done on graph paper will receive no credit. Do NOT put multiple problems in the same graph unless told to do so. Late work loses 10% per school day it is late. Please do not hesitate to me at amallari@calvertonschool.org, if you have questions or concerns. However, please know that I will have professional development workshop for the months of June to July so I will not be able to quickly respond but you will definitely receive a reply between hours. Have a great summer and good luck! Thanks, Mr. Mallari 1

2 IB Math Studies SL Year 2 - Summer Work Name: Date: Score: 1a. [2 marks] In this question give all answers correct to two decimal places. Bobby takes 5000 US dollars (USD) on a business trip to Venezuela. He exchanges 3000 USD into Venezuelan bolívars (VEF). The exchange rate is 1 USD VEF. Calculate the amount of VEF that Bobby receives. 1b. [4 marks] During his time in Venezuela, Bobby spends 1250 USD and VEF. On his return home, Bobby exchanges his remaining VEF into USD. The exchange rate is 1 USD VEF. Calculate the total amount, in USD, that Bobby has remaining from his 5000 USD after his trip to Venezuela. 2a. [2 marks] A hospital collected data from 1000 patients in four hospital wards to review the quality of its healthcare. The data, showing the number of patients who became infected during their stay in hospital, was recorded in the following table. A -test was performed at the 5% significance level. The critical value for this test is The null hypothesis for the test is : Becoming infected during a stay in the hospital is independent of the ward. Find the expected frequency of the patients who became infected whilst in Nightingale ward. 2

3 2b. [2 marks] For this test, write down the statistic. 2c. [2 marks] State, giving a reason, whether the null hypothesis should be rejected. 3a. [2 marks] A hydraulic hammer drives a metal post vertically into the ground by striking the top of the post. The distance that the post is driven into the ground, by the strike of the hammer, is. The distances form a geometric sequence. The distance that the post is driven into the ground by the first strike of the hammer,, is 64 cm. The distance that the post is driven into the ground by the second strike of the hammer,, is 48 cm. Find the value of the common ratio for this sequence. 3b. [2 marks] Find the distance that the post is driven into the ground by the eighth strike of the hammer. 3c. [2 marks] Find the total depth that the post has been driven into the ground after 10 strikes of the hammer. 4a. [2 marks] On a work day, the probability that Mr. Duterte 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. 3

4 4b. [4 marks] The probability that Mr. Duterte arrives on time for work is. Find the value of. 5a. [5 marks] A group of 66 people went on holiday to Hawaii. During their stay, three trips were arranged: a boat trip ( coach trip ( ) and a helicopter trip ( ). ), a From this group of people: 3 went on all three trips; 16 went on the coach trip only; 13 went on the boat trip only; 5 went on the helicopter trip only; x 2x 4x went on the coach trip and the helicopter trip but not the boat trip; went on the boat trip and the helicopter trip but not the coach trip; went on the boat trip and the coach trip but not the helicopter trip; 8 did not go on any of the trips. Draw a Venn diagram to represent the given information, using sets labelled, and. 5b. [2 marks] Show that. 5c. [1 mark] Write down the value of. 5d. [4 marks] One person in the group is selected at random. Find the probability that this person (i) went on at most one trip; (ii) went on the coach trip, given that this person also went on both the helicopter trip and the boat trip. 4

5 6a. [1 mark] In a school 160 students sat a mathematics examination. Their scores, given as marks out of 90, are summarized on the cumulative frequency diagram. Write down the median score. 6b. [2 marks] The lower quartile of these scores is 40. Find the interquartile range. 6c. [3 marks] 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. 5

6 7a. [2 marks] One of the locations in the Olympic Games is an amphitheatre. The number of seats in the first row of the amphitheatre,, is. The number of seats in each subsequent row forms an arithmetic sequence. The number of seats in the sixth row,, is. Calculate the value of the common difference,. 7b. [2 marks] There are rows in the amphitheatre. Find the total number of seats in the amphitheatre. 7c. [2 marks] Caroline visits the amphitheatre. She estimates that the amphitheatre has seats. Calculate the percentage error in Caroline s estimate. 8a. [2 marks] In this question give all answers correct to the nearest whole number. Steven travelled from China to Brazil. At the airport he exchanged 3100 Chinese Yuan,, at an exchange rate of., to Brazilian Real, No commission was charged. Calculate the amount of he received. 6

7 8b. [4 marks] When he returned to China, Steven changed his remaining at a bank. The exchange rate at the bank was and a commission of was charged. He received. i) Calculate the amount of Steven would have received if no commission was charged. ii) Calculate the amount of Steven exchanged when he returned to China. 9a. [1 mark] 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. A test was carried out, at the significance level, on the data collected. Write down the null hypothesis. 9b. [2 marks] Find the number of degrees of freedom. 9c. [1 mark] The critical value of this test is. Use this information to write down the values of the statistic for which the null hypothesis is rejected. 9d. [2 marks] From the travellers taking part in the survey, 285 were 61 years or older and 420 preferred Tokyo. Calculate the expected number of travellers who preferred Tokyo and were 61 years or older. 7

8 10. [6 marks] Consider the numbers and. Complete the following table by placing a tick ( first row has been completed as an example. ) to indicate if the number is an element of the number set. The 11a. [2 marks] The IB grades attained by a group of students are listed as follows. Find the median grade. 11b. [2 marks] Calculate the interquartile range. 11c. [2 marks] Find the probability that a student chosen at random from the group scored at least a grade. 8

9 12a. [3 marks] The second term of an arithmetic sequence is 30. The fifth term is 90. Calculate (i) the common difference of the sequence; (ii) the first term of the sequence. 12b. [3 marks] The first, second and fifth terms of this arithmetic sequence are the first three terms of a geometric sequence. Calculate the seventh term of the geometric sequence. 13a. [2 marks] Adam has an unbiased cubical (six faced) die on which are written the numbers 1, 2, 3, 4, 5 and 6. Eileen has an unbiased tetrahedral (four faced) die on which are written the numbers 2, 3, 5 and 7. Complete the Venn diagram with the numbers written on Adam s die ( ) and Eileen s die ( ). 13b. [2 marks] Find. 13c. [2 marks] Adam and Eileen are each going to roll their die once only. Shin says the probability that each die will show the same number is. Determine whether Shin is correct. Give a reason. 9

10 14a. [1 mark] Jessica either walks or cycles to work. The probability that she walks is If Jessica walks to work, the probability that she is late is 0.1. If she cycles to work, the probability that she is late is The tree diagram for this information is shown. On a day chosen at random, Jessica walked to work. Write down the probability that she was on time. 14b. [2 marks] For a different day, also chosen at random, find the probability that Jessica cycled to work and was late. 14c. [3 marks] For a different day, also chosen at random, find the probability that, given Jessica was late, she cycled to work. 15a. [2 marks] The weight,, of bags of rice follows a normal distribution with mean 1000 g and standard deviation 4 g. Find the probability that a bag of rice chosen at random weighs between 990 g and 1004 g. 15b. [2 marks] 95% of the bags of rice weigh less than grams. Find the value of. 10

11 15c. [2 marks] For a bag of rice chosen at random,. Find the value of. 16a. [1 mark] Two groups of 40 students were asked how many books they have read in the last two months. The results for the first group are shown in the following table. The quartiles for these results are 3 and 5. Write down the value of the median for these results. 16b. [3 marks] Draw a box-and-whisker diagram for these results on the following grid. 11

12 16c. [2 marks] The results for the second group of 40 students are shown in the following box-and-whisker diagram. Estimate the number of students in the second group who have read at least 6 books. 17. [6 marks] Consider the following Venn diagrams. Each diagram is shaded differently. 12

13 In the following table there are six sets. Each of these sets corresponds to the shaded region of one of the Venn diagrams. In the correct space, write the number of the diagram that corresponds to that set. 18a. [1 mark] The producer of a TV dancing show asked a group of 150 viewers their age and the type of Latin dance they preferred. The types of Latin dances in the show were Argentine tango, Samba, Rumba and Cha-cha-cha. The data obtained were organized in the following table. A test was carried out, at the 5% significance level. Write down the null hypothesis for this test. 18b. [1 mark] Write down the observed number of viewers who preferred Rumba and were older than 20 years old. 18c. [2 marks] Use your graphic display calculator to find the -value for this test. 13

14 18d. [2 marks] The producer claims that the type of Latin dance a viewer preferred is independent of their age. Decide whether this claim is justified. Give a reason for your decision. 19a. [2 marks] Only one of the following four sequences is arithmetic and only one of them is geometric. State which sequence is (i) arithmetic; (ii) geometric. 19b. [1 mark] For another geometric sequence write down the common ratio; 19c. [3 marks] For another geometric sequence find the exact value of the tenth term. Give your answer as a fraction. 20a. [2 marks] The sum of the first terms of an arithmetic sequence is given by. Write down the value of (i) ; (ii). 14

15 20b. [1 mark] The term of the arithmetic sequence is given by. Show that. 20c. [2 marks] The term of the arithmetic sequence is given by. Find the common difference of the sequence. 20d. [2 marks] The term of the arithmetic sequence is given by. Find. 20e. [3 marks] The term of the arithmetic sequence is given by. Find the lowest value of for which is greater than. 20f. [2 marks] The term of the arithmetic sequence is given by. There is a value of for which Find the value of. 21a. [1 mark] A class of 13 Mathematics students received the following grades in their final IB examination For these grades, find the mode; 21b. [2 marks] For these grades, find the median; 15

16 21c. [1 mark] For these grades, find the upper quartile; 21d. [2 marks] For these grades, find the interquartile range. 22a. [4 marks] A group of tourists went on safari to a game reserve. The game warden wanted to know how many of the tourists saw Leopard ( ), Cheetah ( ) or Rhino ( ). The results are given as follows. 5 of the tourists saw all three 7 saw Leopard and Rhino 1 saw Cheetah and Leopard but not Rhino 4 saw Leopard only 3 saw Cheetah only 9 saw Rhino only Draw a Venn diagram to show this information. 22b. [2 marks] There were 25 tourists in the group and every tourist saw at least one of the three types of animal. Find the number of tourists that saw Cheetah and Rhino but not Leopard. 22c. [6 marks] There were 25 tourists in the group and every tourist saw at least one of the three types of animal. Calculate the probability that a tourist chosen at random from the group (i) saw Leopard; (ii) saw only one of the three types of animal; (iii) saw only Leopard, given that he saw only one of the three types of animal. 16

17 22d. [2 marks] There were 25 tourists in the group and every tourist saw at least one of the three types of animal. If a tourist chosen at random from the group saw Leopard, find the probability that he also saw Cheetah. 23a. [1 mark] is the set of positive integers less than or equal to., and are subsets of. List the elements of. 23b. [1 mark] List the elements of. 23c. [4 marks] Complete the Venn diagram with all the elements of. 17

18 24a. [2 marks] Let and represent the propositions : food may be taken into the cinema : drinks may be taken into the cinema Complete the truth table below for the symbolic statement. 24b. [2 marks] Write down in words the meaning of the symbolic statement. 24c. [2 marks] Write in symbolic form the compound statement: no food and no drinks may be taken into the cinema. 18

19 25a. [4 marks] (i) Complete the truth table below. (ii) State whether the compound propositions and are equivalent. 25b. [2 marks] Consider the following propositions. p: Jamie eats sweets. q: Jamie goes swimming. Write, in symbolic form, the following proposition: Jamie either eats sweets or goes swimming, but not both. Printed for The Calverton School International Baccalaureate Organization 2017 International Baccalaureate - Baccalauréat International - Bachillerato Internacional 19