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1 Math2UU3*TEST1 Day Class Duration of Test: 60 minutes McMaster University, 26 September 2017 Name (PLEASE PRINT): Dr M. Lovrić This test has 6 pages. Calculators allowed: Casio fx991ms or fx991ms PLUS, or similar (i.e., two lines of display and no graphing capabilities). If some piece of information is missing, you must decide on its value. 1. Multiple choice questions. Circle ONE answer, no justification is needed. (a)[2] Assume that an implication is true. Its converse is (A) Always true (B) Never true (C) Sometimes true (b)[2] The contrapositive of Cats are not birds is (A) Cats are birds (B) Birds are cats (C) Birds are not cats (D) Cats are not birds (E) None of the above (c)[2] Consider the theorem If the last digit of a positive number is 3, 7 or 8, then it cannot be the square of some number. What can you conclude from this theorem? (A) The square of a number does not have 3, 7 or 8 as its last digit (B) The number is not the square of some number (C) The number is the square of some number (D) The number is not the square of some number (E) None of the above (d)[2] A mathematical statement that has been rigorously shown to be true through logical reasoning from a set of axioms, or from previously established (proven) theorems is called (A) Theorem (B) Axiom (C) Definition (D) Algorithm 1

2 (e)[2] Nanotechnology (i.e., nano scale) refers to objects, tools and structures whose size is approximately (A) 10 3 metres (B) 10 6 metres (C) 10 9 metres (D) metres (f)[2] The order of magnitude of the number is (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 (g)[2] In September, businesses in Westdale do better than in summer months. As well, in September, there is more garbage on streets in Westdale. A separate study shows an increase in noise levels. In September, students come back to McMaster, which increases the population of Westdale. Based on these statements only (and nothing else), what is the correct conclusion? (A) Increased business activity in Westdale generates more garbage (B) There is more garbage due to an increased number of students in Westdale (C) Population of students in Westdale, garbage and noise are correlated (D) Westdale is noisier because there are more students (E) Larger numbers of students cause business activity in Westdale to grow (h)[2] Identify a major difference between the cause and effect structure (i.e., an implication) when you compare mathematics with real life. (A) In real life there could be hundreds of causes for some event to occur, which is not possible in mathematics. (B) In mathematics you can have only one cause and only one effect, whereas in real life we usually have multiple causes and multiple effects. (C) In real life no cause and effect statement is certain; i.e., a cause, with some chance (probability), is responsible for the effect. (D) In mathematics you can switch cause and effect (thus getting a converse), but in real life that never works. 2

3 2. (a)[2] Consider the statement Multiplying a positive number by 5 makes it larger. Rewrite this statement as an implication, by identifying assumption(s) and conclusion(s). (b)[2] Consider the statement Low energy, chest pain, rapid heartbeat, and frequent infections are symptoms of stress. Rewrite this statement as an implication, by identifying cause(s) and effect(s). 3. Consider the statement There is a woman in Windsor, Nova Scotia who is tall or has dark hair. (Note: or is inclusive or; i.e., if you order coffee or tea, that means that you might have ordered coffee only, tea only, or both coffee and tea.) (a)[2] State clearly what you need to do in order to prove that the given statement is true. (b)[2] State clearly what you need to do in order to prove that the given statement is false. 3

4 4. A light year is an astronomical unit of distance, equal to the distance that light travels in one year. (a)[2] A good approximation of the speed of light is 299,792 kilometres per second. Use it to express one light year in kilometres (recall that distance = speed times time). Write your answer in scientific notation (round off to three decimal places). Make sure to explicitly state assumption(s) that you are using. (b)[2] We often take the speed of light to be 300,000 kilometres per second. Use it to express one light year in kilometres. Write your answer in scientific notation (round off to three decimal places). Use the same assumption(s) that you used in (a). (c)[1] The answers in (a) and (b) are not equal, but approximately equal. However, they have one quantity in common. What does this quantity describe, i.e., what is it called? 4

5 5. (a) [3] The area code for North America consists of three digits. The first digit cannot be 0 or 1; the second digit can be any number except 9; the third digit can be any number. How many different area codes are there? (b)[2] Ontario automobile license plates are of the format aaaa bbb, where a represents any letter of the alphabet A, B,..., Z, and b represents any digit 0, 1,...9. What is the largest number of different license plates that can be made this way? Justify your answer. (c)[no credit; if you finished your test early, you can think of this one] The actual number of plates is smaller than your answer in (b). Name one reason why it s so. (Obvious reason is that we don t need that many plates). 5

6 6. Recall that 1 trillion is 1 thousand billion. (a)[3] In the article Humans Have Bogged Down the Earth with 30 Trillion Metric Tons of Stuff, Study Finds published by Smithsonian, we read that the amount of stuff humans have produced is around 30 trillion metric tons, or 5 million times the mass of the Great Pyramid of Giza. And you thought you owned a lot of crap. Metric ton is equal to 1000 kg. Based on the data from this excerpt, calculate the mass of the Great Pyramid of Giza in kilograms. Write your answer in scientific notation. (b)[3] How much is one trillion dollars? (The Canadian government debt is of the order of magnitude of a trillion dollars.) Someone said that if you spend $1 million per day starting in year 0, you would still have not spent one trillion by the end of Is that true? Justify your reasoning. State all assumptions that you re making. 6 THE END