Grade 12 Prototype Examination Foundations of Mathematics 30 Course Code 8425 Barcode Number Month Date of Birth Day November 2013 Revised October 2016
TIME: Two and One-Half Hours Foundations of Mathematics 30 Calculating devices MUST meet the requirements of the Calculator Use Policy. Before an examination begins, devices must be removed from their cases and placed on the students desks for inspection by a mathematics or science teacher. Cases must be placed on the floor and left there for the duration of the examination. Students using a standard scientific or graphing calculator must clear all information stored in its memory before the examination begins. Devices such as cell phones, tablets, and ipods may be used as a calculating device if they meet the requirements of the Calculator Use Policy. The school or writing center must be able to lock and control the device using a feature such as Guided Access, or a management software that limits its functionality to permissible graphing and financial applications (apps) with similar functionality to an approved graphing calculator. It is the student s responsibility to ensure their device complies with the Calculator Use Policy in advance of the departmental examination session. Do not spend too much time on any question. Read the questions carefully. The examination consists of 38 multiple-choice questions and 7 numeric response questions of equal value which will be machine scored. Record your answers on the Student Examination Form which is provided. Each multiple-choice question has four suggested answers, one of which is better than the others. Select the best answer and record it on the Student Examination Form as shown in the example below: Student Examination Form: Multiple-Choice Questions What subject is this examination is being written in? A. Chemistry. B. Foundations of Mathematics. C. Pre-calculus. D. Workplace and Apprenticeship Mathematics. Numeric Response Questions Record your answer in the numeric response section on the answer sheet. What is 10% of $2000? (Round to the nearest dollar.) 1. A B C D i
What is 10% of $248.50? (Round to the nearest dollar.) What is 10% of 24 125? (Round to the nearest whole number.) Use an ordinary HB pencil to mark your answers on the Student Examination Form. If you change your mind about an answer, be sure to erase the first mark completely. There should be only one answer marked for each question. Be sure there are no stray pencil marks on your answer sheet. If you need space for rough work, use the space in the examination booklet beside each question. Do not fold either the Student Examination Form or the examination booklet. Check that your personal information on the Student Examination Form is correct and complete. Make any necessary changes, and fill in any missing information. Be sure to complete the Month and Day of Your Birth section. ii
Foundations of Mathematics 30 Simple Interest I Prt A P Prt OR A P I Compound Interest n r A P 1 + i OR A P 1 + n nt A nt r R 1 1 n r n S n t 1 1 r 1 r n Payment M nt r r 1 n n P nt r 1 1 n Pi OR 1 i M n i n 1 1 OR P M 1 1i i n Permutations and Combinations n P r n! ( n r)! n C r n! ( n r)! r! Probability P( A B) P( A) P( B) P( A B) P( A) P( B) P( A B) P( A) P( B) P( A B) P( A B) P( A) P( B A) iii November - Revised October 2016
iv November - Revised October 2016
GRADE 12 DEPARTMENTAL EXAMINATION FOUNDATIONS OF MATHEMATICS 30 PROTOTYPE, NOVEMBER 2013, REVISED OCTOBER 2016 VALUE 90 (45 2) Answer the following 45 questions on the computer sheet entitled Student Examination Form. MULTIPLE-CHOICE QUESTIONS 1. Toni s grandfather deposited $5000 into an education savings plan for her. If the yearly interest rate is 3.75%, which of the compounding periods will earn the most interest in 18 years? A. annually B. monthly C. quarterly D. semi-annually 2. Martha invests $2500 in an account that earns an annual interest rate of 4% interest compounded annually. How many years will it take for her money to double? A. 3 years B. 18 years C. 35 years D. 63 years - 1 -
3. Peter and Paul both acquire a car. Peter leases his car for 48 months while Paul buys his car with a bank loan over 48 months. The costs both have are shown below: Monthly payment Additional costs Lease residual Peter $328.54 $390.00 $14 700.00 Paul $618.29 $490.00 N/A After 48 months, Peter will buy out the lease when it expires. In total, how much more money will Peter have spent? A. $692 B. $792 C. $13 908 D. $14 700 4. A $1000 investment grew to $1500 after 2 years when the annual interest rate was compounded quarterly. What was the annual interest rate? A. 5.2% B. 10.4% C. 20.8% D. 50.0% - 2 -
5. When Brenna was born her parents purchased a $10 000 savings bond that earned a yearly interest rate of 3.2% compounded semi-annually. Taylor invested $35 000 on her 40th birthday, into the same savings bond program. When they are both 60 years old, whose savings bond will be worth more and by how much? A. Taylor will have about $500 more than Brenna. B. Brenna will have about $500 more than Taylor. C. Taylor will have about $1100 more than Brenna. D. Brenna will have about $1100 more than Taylor. 6. Arianna purchased a $215 000 house. She made a 15% down payment and financed the remainder at an annual interest rate of 4.8% compounded monthly for 25 years. What was her monthly payment? A. $1047 B. $1232 C. $7696 D. $8423 7. When David was born, his grandparents invested $1000 for him in an account that had an annual interest rate of 2.75% compounded quarterly. Since his birth, David s parents have deposited $100 at the end of each month in an account with an annual interest rate of 1.75% compounded monthly. After 15 years, what is the total value of David s portfolio? A. $1638 B. $3201 C. $19 028 D. $22 075-3 -
8. Linden can invest $5000 in 1 of the 2 options described below: Option A: deposit $5000 in an account that has a yearly interest rate of 2.5% compounded monthly Option B: deposit $2000 in an account that has a yearly interest rate of 3.5% compounded semi-annually and put $3000 in an account that has a yearly interest rate of 1.25% compounded monthly Which option would be worth the most at the end of 10 years and by how much? A. Option B is worth $79.57 more than Option A. B. Option A is worth $79.57 more than Option B. C. Option B is worth $189.68 more than Option A. D. Option A is worth $189.68 more than Option B. 9. Which of the following statements best describes the end behaviour of the graph of y 6x 3 5x 2 2? A. The graph extends from quadrant II to quadrant I. B. The graph extends from quadrant III to quadrant I. C. The graph extends from quadrant II to quadrant IV. D. The graph extends from quadrant III to quadrant IV. - 4 -
10. Which of the following graphs represents a cubic function with a positive leading coefficient and a constant of 1? A. B. C. D. - 5 -
11. What type of function is represented in the table of values shown below? A. sinusoidal B. polynomial C. logarithmic D. exponential x y 1 100 2 1 10 1 1 0 10 1 100 2 12. What is the horizontal translation, with respect to y sin, of the sine function shown below? A. B. 1 2 2 C. 2 D. 2-6 -
13. The location of a dolphin moving in consistent rhythmic fashion (above and below the surface of the water) is recorded over a time span of 4.0 seconds. The results are shown on the graph below: Where is the dolphin located at the 3 second mark? A. 2 metres below sea level B. 2 metres above sea level C. 3 metres below sea level D. 3 metres above sea level 14. What is the range of the sinusoidal function shown below? A. x x 0, x B. y y 0, y C. x 0.5 x 1.5, x D. y 0.5 y 1.5, y - 7 -
15. Which of the following polynomials have a constant of 4 and 2 turning points? A. B. C. D. 2 y x 3x 2 2 y x 3x 4 3 y x 3x 2 3 y x 3x 4 16. A base 10 logarithmic function has the following characteristics: the function is increasing; the x-intercept is (1,0); the domain is x x y 0, ; and, the function passes through the point (10, 5). Which of the following equations best model the logarithmic function described above? A. y 5 log5 x B. y 5 log10 x C. y10 log5 x D. y10 log10 x 17. What 2 characteristics of the graphs of f( x) 3 sin 2 x and g( x) 3 sin 2 x 5 are the same? A. period and amplitude B. period and vertical shift C. maximum value and amplitude D. maximum value and vertical shift - 8 -
18. Which of the following statements is true for y 4 x? A. The domain is x x x 0,. B. The x-intercept of the graph is (0, 4). C. The graph has a vertical asymptote at the y-axis. D. The graph passes through quadrants I and II only. - 9 -
19. Which of the following graphs represents y log( x1) 2? A. B. C. D. - 10 -
20. In the Venn diagram shown, what is the n A B B C ( ) \ ( )? A. 2 B. 6 C. 8 D. 9 21. In the Venn diagram shown below, what are the elements of A B \ C? A. {pizza} B. {chef salad} C. {chef salad, pizza} D. {tomatoes, chef salad, cheddar, mozzarella} - 11 -
22. Jaimie surveyed 100 students in her school to see how many students own cellphones or tablets. She collected the following results: 47 students own a tablet 76 students own a cellphone 33 students own both a cellphone and a tablet Which of the following Venn diagrams best represents the data Jaimie collected? A. B. C. D. - 12 -
23. What is the inverse of the following conditional statement? If it rains, then Jane will need an umbrella. A. If it rains, Jane will not need an umbrella. B. If Jane will need an umbrella, then it will rain. C. If it does not rain, then Jane will not need an umbrella. D. If Jane does not need an umbrella, then it will not rain. 24. What is the converse of the following conditional statement? If Jennie eats peanuts, then she will have an allergic reaction. A. If Jennie has an allergic reaction, then she ate peanuts. B. If Jennie does not eat peanuts, then she will not have an allergic reaction. C. If Jennie does not have an allergic reaction, then she did not eat peanuts. D. If Jennie does not eat peanuts, then she will have an allergic reaction. 25. The conditional statement If Karlee is eligible to vote in Saskatchewan, then she is at least 18 years old was rewritten as If Karlee is not at least 18 years old, then she is not eligible to vote. What type of statement is this? A. inverse B. converse C. biconditional D. contrapositive - 13 -
26. Given the Venn diagram shown below, which of the following statements is TRUE? Real Numbers ( ) Rational Numbers (Q) Integers (I) Whole Numbers (W) Irrational Numbers ( ) Natural Numbers (N) A. N I B. QQ C. I W N D. N W I - 14 -
27. The odds of NOT selecting an even number from a given set of numbers is 2 : 5. What is the probability of selecting an even number? A. B. C. D. 2 7 2 5 3 5 5 7 28. The table below shows the residential status of Canadians based on a sampling of 100 people who recently applied for a mortgage. Residential status Number Rent 43 Own 45 Live free with family 5 Other 7 What are the odds the mortgage applicant was renting? A. 43 : 57 B. 43 : 100 C. 57 : 43 D. 57 : 100-15 -
29. Which of the following is an example of 2 dependent events? A. Samuel selects a marble from bag 1, keeps it, then selects a second marble from bag 1. B. Joshua selects a marble from bag 1, replaces it, then selects a marble from bag 2. C. Arland selects a marble from bag 1, replaces it, then selects a second marble from bag 1. D. Brayden selects a marble from bag 1, does not replace it, then selects a marble from bag 2 30. Which statement, when combined with It is midnight in Yorkton, will be an example of mutually exclusive events? A. It is cool outside. B. It is dark outside. C. It is windy outside. D. It is sunny outside. 31. How many permutations of all the letters from the word CIRCUIT are possible? A. 1260 B. 2520 C. 5040 D. 20 160-16 -
32. Mr. Brown flips a fair coin to determine what games will be played in his gym classes. If the flip result is a head, the class will play badminton. If the flip result is a tail, the class will play handball. What is the probability that the 4 different gym classes Mr. Brown teaches that day will all play badminton OR all play handball? A. B. C. D. 1 16 1 8 1 7 1 4 33. A bag contains 6 ping-pong balls numbered 1 through 6. An experiment is conducted where the balls are randomly taken out of the bag, one at a time. The number is recorded and once the ball is chosen it is NOT returned to the bag. The first 10 trials of this experiment shown in the table below: Trial 1st ball 2nd ball 3rd ball 4th ball 5th ball 6th ball 1 4 6 2 3 1 5 2 4 3 1 5 2 6 3 6 5 3 2 1 4 4 3 4 2 6 1 5 5 5 3 1 4 6 2 6 2 4 1 5 3 6 7 1 2 6 5 4 3 8 3 1 4 6 5 2 9 5 3 2 6 1 4 10 5 4 6 3 2 1 Based on this experiment, what are the odds that the 2nd ball taken from the bag is a 4? A. 3 : 7 B. 3 :10 C. 7 : 3 D. 7 :10-17 -
34. On Saturday, the probability of rain is 0.47. On Sunday, the probability of rain is 0.42. What is the probability that it will NOT rain on either day? A. 0.20 B. 0.31 C. 0.45 D. 0.56 35. John rolls 2 regular 6-sided dice. What is the probability that John will roll a sum of 8 OR a sum of 10? A. B. C. D. 1 9 2 9 5 9 7 9 36. A bag contains 5 green marbles and 3 blue marbles. If 2 marbles are drawn at the same time, what is the probability that both marbles are green? A. B. C. D. 9 56 5 14 9 15 5 8-18 -
37. A student was asked to colour the following figure so that 2 of the squares are coloured red, 2 are coloured blue, and 1 is coloured green. What is the probability that the student colours the figure in a left to right order of red, blue, green, blue, then red? A. B. C. D. 1 5 1 30 1 60 1 120 38. A graduating class consists of 5 girls and 11 boys. A committee of 1 girl and 2 boys is to be formed from this class. What is one correct way to calculate the number of ways this committee can be formed? A. 16C 3 B. 16P 3 C C. 5 1 11 2 P C P D. 5 1 11 2-19 -
NUMERIC RESPONSE QUESTIONS Record your answer in the Numeric Response section of the Student Examination Form. 39. At the end of each year for 3 consecutive years, Kurri puts $2000 into an account that has an annual interest rate of 4.2% compounded annually. How much would his investment be worth at the end of the third year? (Round to the nearest dollar.) 40. Two model rockets were launched at the same time. Their altitudes, h, in metres, t seconds after being launched, are shown below. h t What was the altitude, in metres, of rocket 1 when rocket 2 reached its maximum height? (Round to the nearest metre.) - 20 -
41. Sets U, A, and B are defined as follows: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} A = {1, 3, 6, 9, 12, 15} B = {2, 4, 6, 8, 10, 12, 14} ( ) What is n A B? 42. A game is played with 11 coins. The object of the game is to leave your opponent with the last coin. Each player must take either 1 or 2 coins on their turn. If you want to ensure you win in the last play, how many coins do you want to leave your opponent with after your 2nd last play of the game? - 21 -
43. Sam is ordering a 2-topping pizza. The toppings he has to choose from are: ham pepperoni salami mushrooms pineapple green peppers How many different 2-topping pizzas would be available for Sam to order? 44. Liz can take 3 CDs to a party. If she has 8 different CDs to choose from, how many groups of 3 CDs could she make? 45. A teacher can form 45 different groups by selecting exactly 2 students from her class. How many students are in her class? - 22 -
(See Explanation of Answers) GRADE 12 DEPARTMENTAL EXAMINATION FOUNDATIONS OF MATHEMATICS 30 PROTOTYPE EXAM Answer Key 1. B 11. C 21. B 31. A 41. 4 2. B 12. B 22. B 32. B 42. 4 3. A 13. A 23. C 33. A 43. 15 4. C 14. D 24. A 34. B 44. 56 5. D 15. D 25. D 35. B 45. 10 6. A 16. B 26. A 36. B 7. D 17. A 27. D 37. B 8. D 18. D 28. A 38. C 9. C 19. D 29. A 39. 6256 10. C 20. B 30. D 40. 120 Explanation of Answers 1. B. The number of compounding periods has an exponential impact on the future value of an investment. If you want to increase the amount of interest earned the exponent needs to increase. Hence, monthly will offer 12 pay periods per year (the greatest payout), quarterly will offer 4 pay periods per year, semi-annually will offer 2 pay periods and annually will offer 1 pay period. OR by calculation: 0.0375 Annually: A 5000 1 1 A=$9699.65 0.0375 Monthly: A 5000 1 12 A= $9809.83 181 1812 0.0375 Quarterly: A 5000 1 4 A=$9789.33 184 Semi- Annually: A = $9758.97 0.0375 A 5000 1 2 - i - 182 Foundations of Mathematics, Prototype Exam Answer Key
2. B. Using the Rule of 72 72 18 4 years 3. A. Peter: ($328.54 48) $390 $14 700 $30 859.92 Paul: ($618.29 48) $490 $30167.92 $30 859.92 $30167.92 $692.00 Peter spent $692.00 more than Paul. 4. C. i 1500 1000 1 4 i 1.5 1 4 i 8 1.5 1 4 i 1.051 9 1 4 i 0.0519 4 0.2079 i or 20.8% 8 2(4) 5. D. Brenna: 0.032 A 10 000 1 2 A 67180.77 602 Taylor: 0.032 A 35 000 1 2 A 66 041.41 202 $67180.77 $66 041.41 $1139.36 Brenna will have $1139.36 more than Taylor. - ii - Foundations of Mathematics, Prototype Exam Answer Key
6. A. ($215 000)(0.85) $182 750 (amount financed) 0.048 0.048 1 12 12 M =182 750 2512 0.048 1 1 12 $1047.16 2512 7. D. Grandparents Invest 1: 415 0.027 5 A $1000 1 $1508.46 4 Parents Invest 2: 1215 0.017 5 $100 1 1 12 A 0.017 5 12 $20 566.48 Total portfolio is $1508.46 $20566.48 $22 074.94 - iii - Foundations of Mathematics, Prototype Exam Answer Key
8. D. Option A 0.025 A $5000 1 12 A $6418.46 1012 $6418.46 $6228.78 $189.68 Option A is worth $189.68 more Option B 0.035 0.012 5 A $2000 1 $3000 1 2 12 A $2829.56 $3399.22 A $6228.78 20 120 9. C. 3 2 For y 6 x 5 x 2, the degree is odd (3), therefore it will behave differently on the two sides of the y-axis. For example, the end behaviour will extend from quadrant III to quadrant I or extend from quadrant II to quadrant IV. Since the leading coefficient is negative, the graph will drop to the right (quadrant IV). Therefore, the end behavior is the graph will extend from quadrant II to quadrant IV. 10. C. A positive leading coefficient will cause the graph of the cubic function to rise to the right (extend into quadrant I) and since the constant or y-intercept is 1, the graph must also pass through (0,1). 11. C. By sketching the points in the table, the graph looks like Therefore, by its shape, it is a logarithmic function. - iv - Foundations of Mathematics, Prototype Exam Answer Key
12. B. A normal y sin has an x-intercept at (0, 0) and has a maximum point when. Since this graph has an x-intercept of and a maximum 2 2 occurring when, it has been shifted to the right. 2 13. A. By reading the sine graph at the 3 second mark (x-axis), the height is 2 metres or 2 metres below sea level (y-axis). 14. D. Range is defined as the possible values of a function on the vertical, or the y-axis. This graph goes to a height of 1.5 and a low of -0.5. Therefore, the range is all the values between 1.5 and -0.5 and since the graph is a solid line, it encompasses all real numbers between. The range is written as y 0.5 y 1.5, y 15. D. A polynomial with 2 turning points has to have a degree of 3 or higher. Turning points are determined by finding the degree and subtracting 1. A constant of 4 represents the d value in a general polynomial. Therefore the 3 2 3 general polynomial must be y ax bx cx d and y x 3x 4 best fits this description. 16. B. The general form of a logarithmic function is y a log b x, where b is the base and a represents how quickly the function increases or decreases. The characteristics tell us that the base is 10 and the function is increasing, so a is positive and b is 10. With the information about the domain and the x-intercept being (1, 0) we know the graph is regular log graph, not one that has been translated. If the function was y log 10 x, substituting an x value of 10 would give a y value of 1. Because the point is instead (10,5) it tells us the a value is 5. Therefore, the correct equation is y 5 log10 x. - v - Foundations of Mathematics, Prototype Exam Answer Key
17. A. f( x) 3 sin 2 x g( x) 3 sin 2 x 5 amplitude 3 3 same maximum 3 2 minimum 3 8 period same horizontal shift none none same vertical shift none down 5 18. D. The graph of y 4 x, as shown to the right, has the following properties: domain x y-intercept at (0, 1); no x-intercept horizontal asymptote at the x-axis extends from quadrant II to quadrant I 19. D. The graph of y log x always looks as shown to the right However, the equation y log( x1) 2 has been modified so that the y values of the graph move up 2 units and the x values move 1 unit to the right. Substituting a few key points into both equations will confirm the behaviour. y log x y log( x1) 2 x y x y 1 0 1 undefined 10 1 2 2 10 2.9542 11 3 y log x is written in the form y log( x1) 2 - vi - Foundations of Mathematics, Prototype Exam Answer Key
20. B. n ( A B) \ ( B C ) translates to the number of elements in Set A and Set B (union symbol), excluding all the elements in the intersection of Set B and Set C (intersection symbol). The elements described are: 10, 11, 6, 7, 1, 2, 4, 9 and 3, excluding 4, 7 and 9. Hence there are 6 elements. 21. B. A B translates to elements that are common to A and B. These are chef salad and pizza. The \C in A B \C means not including elements in C, which would not include the pizza. Hence {chef salad}. 22. B. There are 33 students are in the intersection set (both cellphone and tablet). If there are 47 with a tablet and 33 in the intersection with a tablet and cellphone, that leaves 47 33 = 14 with just a tablet. If 76 have a cellphone and 33 in the intersection with a cellphone and tablet that leaves 76 33 =43 with just a cellphone. Total student students with either or both = students without either 100 (43 33 14) 100 90 10 without either 23. C. The inverse statement negates both parts of the if p, then q statement to if not p, then not q. Hence: If it does not rain; then Jane will not need an umbrella is the inverse statement. - vii - Foundations of Mathematics, Prototype Exam Answer Key
24. A The converse statement reverses the if p, then q statement to if q, then p. Hence, If Jennie has an allergic reaction, then she ate peanuts is the converse statement. 25. D. 26. A The statement is reversed and negated. This is the definition of contrapositive. N W I Q N I. Natural Numbers are a subset of Whole Numbers which is a subset of Integers which is a subset of Rational Numbers which is a subset of Real Numbers. 27. D. number of odd numbers : number of even numbers = 2 : 5 If there are 2 odd numbers, 5 even numbers and total of 7 numbers, then desired P (even number) 5 total 7 28. A. Odds of renting are represented by number of people renting : number of people not renting 43 : 57 29. A. Dependent events will affect one another. Not replacing the first marble before choosing the second one from the same bag will affect the sample space of marbles remaining to choose from. 30. D. Mutually exclusive events are disjoint events with no possibility of overlap. Even in Saskatchewan, when it is midnight in the summer months, it can t be sunny. - viii - Foundations of Mathematics, Prototype Exam Answer Key
31. A This is a permutation with repeated elements of the letter C and I. 7! 2! 2! 5040 4 1260 32. B. P(T) P(T) P(T) P( T ) or P(H) P(H) P(H) P(H) 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 16 16 2 1 or 16 8 33. A. In the 2nd ball column, there are 3 balls with the number 4 on them and 7 balls without the number 4 on them. number of balls with 4 on them: number of balls without 4 on them 3 : 7 34. B. P(Saturday rain) = 0.47 P (Saturday not rain) = 1 0.47 = 0.53 P(Sunday rain) = 0.42 P(Sunday not rain) = 1 0.42 = 0.58 P (Saturday not rain) P(Sunday not rain) (0.53) (0.58) 0.3074 or 0.31 - ix - Foundations of Mathematics, Prototype Exam Answer Key
35. B. Sample space of rolling an 8 (2, 6), (6, 2), (5, 3), (3, 5), (4, 4) Sample space of rolling a 10 (4, 6), (6, 4), (5, 5) Total possible outcomes: 36 P(8) or P(10) 5 3 36 36 8 36 2 9 36. B Because the marbles are drawn at the same time (there is no replacement) these are dependent events. For the first marble there are 8 to choose from, but only 7 marbles to chose from for the second one drawn. P(first green marble)p(second green marble) 5 4 8 7 20 56 5 14 37. B. P(R) & P(G) & P(B) & P(G) & P(R) OR 2 2 1 1 1 The number of different arrangements is 5 4 3 2 1 5! 120 30. 4 2! 2! 4 120 With only one possible arrangement desired, 1 1 30 the P(R, G, B, G, R). 30 - x - Foundations of Mathematics, Prototype Exam Answer Key
38. C. This is a combination from more than one set, since the order does not matter. There are five 5 girls and we need one 1 this can be found by 5C1. There are 11 boys and we need 2 students from that group so that would be 11C2 Therefore: 5 C 1 11 C 2 would be the solution to the number of ways the set could be chosen. 39. Numeric Response: 6256 A nt r R 1 1 n r n 3 0.042 1 1 1 A $2000 0.042 1 $6255.53 $6256 40. Numeric Response: 120 By reading the graph, rocket 2 reached its maximum height (80 m) at 4 seconds. Therefore, at 4 seconds rocket 1 is at an altitude of 120 m. - xi - Foundations of Mathematics, Prototype Exam Answer Key
41. Numeric Response: 4 AB {1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15} ( AB) {5, 7, 11, 13} Therefore there are 4 elements in the complement set. 42. Numeric Response: 4 If you leave your opponent 4 chips after your 2nd last play, then they can either: take 1 of the 4 remaining, leaving you with 3 and you can take 2 on your turn and they take the last one; or, take 2 of the 4 remaining, leaving you with 2 then you can take 1 on your turn and they still take the last one. 43. Numeric Response: 15 This is a combination since it does not matter what order the pizza toppings are chosen. C 6 2 6! 4! 2! 15 - xii - Foundations of Mathematics, Prototype Exam Answer Key
44. Numeric Response: 56 This is a combination since it does not matter what order the CDs are chosen. 8 3 n! ( n r)! r! C 8! (5)!(3)! 56 45. Numeric Response: 10 C n 2 45 n! 45 ( n 2)! 2! n! 90 ( n 2)! n( n 1) ( n 2)!... 90 ( n 2)!... ( n)( n1) 90 n 2 n 90 0 ( n10)( n 9) 0 n 10, n 9 (extraneous) n 10 - xiii - Foundations of Mathematics, Prototype Exam Answer Key