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Christ the Redeemer Catholic Schools Mathematics 30-2 Course Review Package Mrs. Sample 2014 Fun Facts about the Diploma Exam How long is the exam? The exam is designed to be completed in 2.5 hours; you are allowed an addition half hour if you need it. How many questions are there on the diploma exam? There are 28 multiple choice questions and 12 numerical response questions. What is the breakdown of the exam in terms of topics learned in class? 50% of the exam (approximately 20 questions) will cover topics such as rational expressions, rational equations, polynomials, exponential functions, logarithmic functions, and sinusoidal functions. 33% of the exam (approximately 13 questions) will cover topics such as the fundamental counting principle, permutations, combinations, and probability. 17% of the exam (approximately 7 questions) will cover topic such as logical reasoning, problem solving strategies, and set theory. Numerical Response Questions You should be prepared to see three different styles of numerical response questions on the diploma. It is important to note that for all types, responses should be aligned to the left and numerical bubbles must be filled in. Question and Solution When the answer to be recorded cannot be a decimal, students are asked to determine a whole number value. If the answer can be a decimal value, then students are asked to record their answer to the nearest tenth or nearest hundredth, as specified in the question. Students should retain all decimals throughout the question and rounding should only occur in the final answer. Correct-Order Question and Solution Any-Order Question and Solution Rational Expressions A rational expression is the ratio (think fraction) of two polynomials. They are written in the form 1. An expression that is equivalent to A. π₯ + 1, π₯ ≠ 0 C. 2π₯, π₯ ≠ 0 π(π₯) π(π₯) π₯ 2 +π₯ ,π₯ π₯ ≠ 0, is B. π₯ 2 + 1, π₯ ≠ 0 D. π₯ 2 , π₯ ≠ 0 2. Numerical Response The non-permissible value of π₯ in the expression 2π₯+1 is _______. 3π₯−9 Non-Permissible Values: 2π₯+4 Since division by zero is impossible within the laws of mathematics, the denominator of a rational expression may never be equal to zero. Therefore, any values of the variable that make the denominator equal to zero are nonpermissible. *When working with division, you must also look in the numerator of the second expression! 3. When the rational expression π₯ 2 −4 is simplified, the equivalent expression is A. C. 2 ,π₯ π₯−2 2 ,π₯ ≠ π₯ ≠ −2, 2 B. −2, 0, 2 D. 2 ,π₯ ≠ π₯+2 2 ,π₯ ≠ 0 π₯ −2 To simplify rational expressions: Fully factor both the numerator and denominator, and then cancel out any common factors. To multiply rational expressions: Simplify the expressions as much as possible. Then, multiply together the numerators and denominators of both expressions. Simplify further, if possible. 4. Numerical Response One possible selection to form a correct simplification is (π₯ − 3), (π₯ + 4), and (π₯ 2 − 9), so Henry records the code 159. To form another correct simplification, a code for another possibility for a is _______________ (Record in the first column) b is _______________ (Record in the second column) c is _______________ (Record in the third column) To divide rational expressions: Simplify the expressions as much as possible. Then, multiply the first expression by the reciprocal of the second. 5. The simplified form of A. C. To add rational expressions: Simplify both expressions as much as possible. Then, multiply the top and bottom of the expressions to create a common denominator, and add the expressions together. 6. The simplified form of A. C. To subtract rational expressions: The same process as for adding, but don’t forget to distribute the negative! 1. A 2. 3 π₯ π₯−2 π₯ π₯+3 D. π₯ 2 +3π₯ π₯ 2 −4 A. 3. A ≠ −2, 0, 2 π₯+3 D. 5π₯+3 π₯+2 π₯+2 π₯+3 π₯+3 π₯+2 is ≠ −2 B. ≠ −2 D. 4. 258 or 148 or 269 or 357 2(2π₯+3) (π₯+2)(π₯−2) 2(2π₯−3) ,π₯ (π₯+2)(π₯−2) ÷ π₯+2 , π₯ ≠ −3, −2, 2 is B. 4π₯ −π₯−3 ,π₯ 2π₯+4 −π₯−3 ,π₯ π₯+2 is B. 7. The simplified form of π₯+2 − C. Answers: π₯+3 (π₯+2)(π₯−2) 4π₯ (π₯+2)(π₯−2) π₯ 3π₯ + π₯ 2 +2π₯ π₯ 2 −4 5. D −π₯+3 ,π₯ 2π₯+4 −π₯+3 ,π₯ π₯+2 6. A ≠ −2 ≠ −2 7. C Rational Equations 1. The solution to A. π₯ = 3 2 5π₯−1 4π₯+11 3 4 = is B. π₯ = 3 37 8 C. π₯ = 37 32 D. π₯ = 29 8 C. π₯ = 9 35 D. π₯ = 9 25 5 2. The solution to x + 3 = 10 is 1 A. π₯ = 5 B. π₯ = 4 3 5 6x 3. The solution to + = 6 is x π₯+1 1 A. π₯ = 2 B. π₯ = 3 C. π₯ = −1 2 D. π₯ = 3 You can solve a rational equation algebraically by multiplying each term in the equation by the lowest common denominator (LCD) to eliminate fractions from the equation. Then solve the resulting linear or quadratic equation. If a root of a rational equation is a non-permissible value of the rational expressions in the original equation, it is an extraneous root and must be dismissed as a valid solution. Each solution to a rational equation can be verified by substituting it into the original equation. If the left side equals the right side, then it is a valid solution to a rational equation in a non-contextual problem. 4. In which step did the student make their first error? A. Step 1 B. Step 2 C. Step 3 2π₯ π₯ D. Step 4 18 5. The solution to + = 2 is π₯+3 π₯−3 π₯ −9 A. π₯ = 3 B. π₯ = −2, 3 C. π₯ = −3, 2 D. π₯ = −2 6. What is the correct equation to model this situation? 160 160 160 160 A. π₯−40 + π₯ = 1.6 B. π₯−40 − π₯ = 1.6 C. Answers: 160 π₯−40 = 1.6 1. B D. 2. D 3. A 160 π₯ 4. A = 1.6 5. D 6. B When solving a contextual problem, it is important to check for solutions that are valid solutions to the equation but are inadmissible in the context of the problem (for example, a rectangle cannot have a negative side length). Exponential Functions and Equations From your formula sheet: π¦ = π π₯ ↔ π₯ = log π π¦ 1. Which of the following exponential functions could be used to model the value of the car, π£(π‘), after π‘ years? A. π£(π‘) = 5274(1.18)π‘ B. π£(π‘) = 29300(1.18)π‘ C. π£(π‘) = 5274(0.82)π‘ D. π£(π‘) = 29300(0.82)π‘ Exponential Functions: π¦ = π β ππ₯ Properties of Exponential Functions: ο· no x-intercepts ο· y-intercept of a ο· curve extends from quadrant II to quadrant I ο· domain {π₯|π₯ ∈ π } ο· range {π¦|π¦ > 0, π¦ ∈ π } 2. How many times as intense is an earthquake of magnitude 8.5 than an earthquake with a magnitude of 6.9? A. 10 B. 20 C. 30 D. 40 3. A painting was purchased in 2012 for $10 000. If the painting appreciates in value at 5%/a, then which of the following graphs best models the value of this painting for the next 40 years? A. B. To estimate the solution to an exponential equation, you can enter the exponential equation as a system of equations on a graphing calculator and graph the system. To determine an exact solution to an exponential equation algebraically, write both sides of the equation as powers with the same base (if possible). Then set the exponents equal to each other, and solve the resulting equation. C. Financial Applications The exponential function that models compound interest is π΄(π) = π(1 + π)π . Remember, π= D. 4. Solve 2π₯−1 = 3π₯−2 . A. π₯ = 1.0 B. π₯ = 1.3 ππππ’ππ πππ‘π # π‘ππππ πππ‘ππππ π‘ ππ ππππ πππ π¦πππ C. π₯ = 3.7 D. π₯ = 5.4 5. Determine how long it will take for the investment to be worth at least $150 at 2.4%/a, compounded monthly. A. 202 months B. 203 months C. 204 months D. 205 months Answers: 1. D 2. D 3. A 4. C 5. B Logarithmic Functions and Equations Change of Base Formula (to evaluate logs of bases other than 10 in your calculator!) log π π log π π = log π π Laws of Logarithms log π (ππ) = log π π + log π π π log π ( ) = log π π − log π π π log π (ππ ) = π log π π Logarithmic Regression Functions π¦ = π + π ln π₯ The common logarithm has base 10, and is written as π¦ = log π₯ 1. Use logarithmic regression to determine the Minimum Perceivable Difference for a 2100 g object, to the nearest whole gram. A. 17 g B. 22 g C. 27 g D. 32 g 2. The expression log 6 3 + log 6 12 in simplified form is A. log 6 15 B. log 6 36 C. log 6 (−9) The natural logarithm has base e, and is written as π¦ = ln π₯ 1 D. log 6 (4) 3. Numerical Response If the concentration of hydrogen ions in the solution is doubled, the new pH of the solution, to the nearest tenth is _______. Sometimes, to solve an exponential equation, it is useful to apply the common logarithm to both sides of the equation. 4. Which pair of equations are not equivalent? A. log 100 = 2 and 102 = 100 B. log 2 8 = 3 and 23 = 8 C. ln π₯ = 2 and 102 = π₯ D. log π 5 = 2 and π2 = 5 5. What is 2 ln π₯ − ln π¦ expressed as a single logarithm? π₯2 B. ln(2π₯ − π¦) C. ln(π₯ 2 − π¦) A. ln π¦ π₯ 2 π¦ D. ln ( ) 6. What is log π − 4 log π − log π expressed as a single logarithm? π ππ π ππ A. log (π4 π ) B. log (4π) C. log (4ππ ) D. log ( π4 ) Answers: 1. C 2. B 3. 6.3 4. C Characteristics of Logarithmic Functions: ο· x-intercept of 1 ο· no y-intercept ο· curve extends from quadrant IV to I or from quadrant I to IV ο· domain {π₯|π₯ > 0, π₯ ∈ π } ο· range {π¦|π¦ ∈ π } 5. A 6. D Polynomial Functions Characteristics of Degree 0 (constant) Functions No x-intercept Domain {π₯|π₯ ∈ π } Range {π¦|π¦ = constant, π¦ ∈ π } No turning points 1. Consider the graph of π(π₯) = −(π₯ + 1)(π₯ − 2)2 . Which of the following statements are false? A. π(π₯) is a cubic function B. π(π₯) has two x-intercepts C. π(π₯) extends from quadrant II to quadrant IV D. π(π₯) has a y-intercept of 4 Characteristics of Degree 1 (linear) Functions 1 x-intercept Domain {π₯|π₯ ∈ π } Range {π¦|π¦ ∈ π } No turning points 2. The rate at which snow fell on a driveway on a particular day can be modelled by π¦ = −3π₯ 2 + 6π₯, where y represents the rate of snowfall in ft 2 /hr, and x represents the time in hours. To estimate the length of time that snow fell on this particular day, a student should determine the A. y-intercept B. x-coordinate of the vertex C. y-coordinate of the vertex D. difference between the x-intercepts 3. Which of the following graphs would most likely represent the graph of a cubic function? A. B. C. D. Characteristics of Degree 2 (quadratic) Functions Number of x-intercepts Domain {π₯|π₯ ∈ π } Range π¦|π¦ ≥ ππππππ’π, π¦ ∈ π } or {π¦|π¦ ≤ πππ₯πππ’π, π¦ ∈ π } 1 turning point Characteristics of Degree 3 (cubic) Functions May be 1, 2 or 3 x-intercepts Domain {π₯|π₯ ∈ π } Range π¦|π¦ ∈ π } May have 0 or 2 turning points Inflection points: a curve in a cubic function that is not a turning point 4. The new box must hold 1000 mL of juice. Determine the largest amount by which each dimension of the juice box can be increased, to the nearest tenth of a centimeter. A. 2.7 cm B. 3.5 cm C. 4.2 cm D. 5.3 cm Answers: 1. D 2. D 3. B 4. B Sinusoidal Functions From your formula sheet: π¦ = π β sin(ππ₯ + π) + π Period: 1. Numerical Response The height of the pendulum at the moment of release, to the nearest tenth of an inch, is _________ in. 2π π Angles can be measured in both degrees and radians. π radians = 180° Periodic function: a function whose graph repeats in regular intervals or cycles 2. According to the sinusoidal regression function, the maximum height of the rider above the ground is A. 2 m B. 6 m C. 8 m D. 14 m 3. Numerical Response When the rider is at least 11.5 m above the ground, she can see the rodeo grounds. During each rotation of the Ferris wheel, the length of time that the rider can see the rodeo grounds, to the nearest tenth of a minute, is _________ min. The height of the Ferris wheel can be represented by a sinusoidal function of the form β = π sin(ππ‘ − 1.57) + π. 4. What is the value of a in this function? A. 0.13 B. 8 C. 10 D. 48 5. What is the value of b in this function? A. 0.13 B. 8 C. 10 D. 48 6. What is the value of d in this function? A. 0.13 B. 8 C. 10 D. 48 Answers: 1. 3.3 2. D 3. 1.8 4. B 5. A 6. C Midline: the horizontal line halfway between the maximum and minimum values of a periodic function πππ₯ + πππ πππππππ = 2 Amplitude: the distance from the midline to either the maximum or minimum value of a periodic function; the amplitude is always expressed as a positive number. πππ₯ − πππ ππππππ‘π’ππ = 2 Period: the length of the interval of the domain to complete one cycle. 2π ππππππ = π Logical Reasoning Good problem solving strategies include: ο· Guess and check ο· Look for a pattern ο· Make a systematic list ο· Draw or model ο· Eliminate possibilities ο· Simplify the original problem ο· Work backward ο· Develop alternative approaches Examples 1. The fifth row of the pattern will be A. 1 234 × 8 + 4 = 9 876 B. 1 234 × 8 + 5 = 9 876 C. 12 345 × 8 + 4 = 9 876 D. 12 345 × 8 + 5 = 98 765 2. NUMERICAL RESPONSE It is Margaret’s turn, and she determines that she can guarantee a win by placing the letter M in Row __________ Column __________ 3. Which of the following pictures is next in the pattern? A. B. C. Answers: 1. D 2. 42 D. 3. A Set Theory Some important terms, symbols and definitions: 1. The number of students in the universal set is A. 61 B. 64 C. 78 D. 98 π Universal set: a set of all the elements under consideration for a particular context (also called sample space). π΄′ Complement: all the elements of a universal set that do not belong to a subset of it. ∅ Empty set: a set with no elements; for example, the set of odd numbers divisible by 2 is the empty set. ∩ Intersection: the set of elements that are common to two or more sets. ⊂ Subset: a set whose elements all belong to another set. ∪ Union: the set of all the elements in two or more sets. 2. Numerical Response The number of students taking Art is _______. 3. Numerical Response The number of students not taking Math is _______. 4. The number of students taking Math or Art is A. 17 B. 61 C. 78 D. 98 5. The number of people who saw “The Princely Groom” is A. 17 B. 61 C. 78 D. 98 6. Numerical Response The number of people who saw “Metal Man” and “The Princely Groom” but not “Quick and Angry” is _______. 7. The number of people who saw “Metal Man” only is A. 17 B. 61 C. 78 D. 98 8. The number of people who saw “Metal Man” or “Quick and Angry” is A. 17 B. 61 C. 78 D. 98 9. Which of the following statements is true for sets π , π and π? A. π ⊂ π B. π ⊂ π C. π ⊂ π Disjoint sets: two or more sets having no elements in common. n(A) Notation, describing the number of elements in set A. D. π ⊂ π 10. Which of the following statements is not true for sets π , π and π? A. π ⊂ (π ∩ π) B. T⊂ (π ∩ π) C. (π ∩ π) ⊂ π D. (π ∩ π) ⊂ π Answers: 1. D 2. 50 3. 53 4. C 5. C 6. 10 7. B 8. C 9. D 10. C Fundamental Counting Principle 1. Determine the number of six-digit odd numbers that can be created using the digits 0 to 9 without repetition. A. 60480 B. 67200 C. 75600 D. 151200 π! = π(π − 1)(π − 2) … 3 β 2 β 1 The Fundamental Counting Principle: if there are a ways to perform one task and b ways to perform another, then there are π β π ways of performing both. 2. Determine the number of distinct arrangements of all the letters in the word TATTOO. A. 30 B. 60 C. 120 D. 720 The Fundamental Counting Principle can be extended to more than two tasks. The Fundamental Counting Principle does not apply when tasks are related by the word OR. 3. Determine the number of 3-letter arrangements of the letters of the word DIPLOMA. 6. 7πΆ3 7. 7π3 8. 7! 3! 9. 7! 4. Only six people have tickets for 2 prizes in a school draw, and each person has only one ticket. Once a ticket is drawn for a prize, it is not reentered in the draw. What is the probability that Bill wins the first prize and Mary wins the second prize? 1 A. B. C. D. 15 1 18 1 30 1 36 5. Numerical Response How many different flight options can Julian select from if he flies with this airline? ________ Answers: 1. B 2. B 3. B 4. C 5. 8 Permutations From your formula sheet: πππ 1. Numerical Response For each context, use a 1 to indicate the problem should be solved using a permutation and use a 2 to indicate the problem should be solved using a combination. = π! (π − π)! Permutation: an arrangement of distinguishable objects in a definite order, where each object appears only once in each arrangement. In the case of repeated identical objects: π! π! π! π! 2. Numerical Response A 7-player volleyball team must stand in a straight line for a picture. How many different arrangements can be made for the picture? 3. Numerical Response Determine the number of arrangements for the volleyball team picture that can be made for the picture if the tallest player must stand in the middle. Answers: 1. 1221 2. 5040 3. 720 Combinations From your formula sheet: π! ππΆπ = (π − π)! π! ππΆπ 1. In a group of 9 students, there are 4 females and 5 males. How many different 4-member committees have 2 females and 2 males? A. 24 B. 60 C. 126 D. 3024 π =( ) π When order does not matter in a counting problem, you are determining combinations. 2. In a group of 9 students there are 4 females and 5 males. How many different committees consist of 2 or 3 students? A. 120 B. 156 C. 540 D. 576 From a set of n different objects, there are always fewer combinations than permutations when selecting r of these objects. When all n objects are being used in each combination, there is only one possible combination. 3. Ralph knows that there are 15 distinguishable possibilities when 2 people are chosen to form a committee from a particular group of n people. How many people were in the group? A. π = 4 B. π = 5 C. π = 6 D. π = 7 Solving Counting Problems When solving counting problems, you need to determine if order plays a role in the situation. Once you have determined whether a problem involves permutations or combinations you can also use these strategies: ο· Look for conditions. Consider these first as you develop your solution. ο· If there is a repetition of r of the n objects to be eliminated, it is usually done by dividing by r! ο· If a problem involves multiple tasks that are connected by the word AND, then the Fundamental Counting Principle can be applied; multiply the number of ways that each task can occur ο· If a problem involves multiple tasks that are connected by the word OR, add the number of ways that each task can occur. Answers: 1. B 2. A 4. Numerical Response The values of w, x, y, and z are _______, _______, _______, and _______, respectively. 5. Numerical Response Triangles can be formed in an octagon by connecting any 3 of its vertices. Determine the number of different triangles that can be formed in an octagon. 3. C 4. 9372 5. 56 Odds and Probability Practice Problems 1, 2, 3, 4, 5 Probability: π(π΄) = favourable outcomes 1. The odds in favour of the Renegades winning the season final in the football league are listed at 10:7. The odds against the Renegades winning the season total possible outcomes final are Odds express a level of confidence A. 3:7 about the occurrence of an event. B. 3:10 C. 7:10 Odds in favour of event A: D. 10:3 π(π΄): π(π΄′ ) 2. Statistics show that 6 out of 25 car accidents are weather-related. The odds that a car accident is weather-related can be expressed in the form a:b. The values of a and b are, respectively, _______ and _______. 3. A class of 35 students has 17 males. One student will be selected at random from the class. What are the odds against the randomly selected student being a male? A. 17:18 B. 18:17 C. 17:25 D. 18:25 4. What is the probability of winning the Flim’em game? A. 0.25 B. 0.33 C. 0.67 D. 0.75 Answers: 1. C 2. 619 3. B 4. A Odds against event A: π(π΄′ ): π(π΄) Mutually Exclusive Events Mutually exclusive events: two or more events that cannot occur at the same time. For example, rolling a 3 and rolling a 4 on a die are mutually exclusive events. You can represent the favourable outcomes of two mutually exclusive events, A and B, as two disjoint sets. You can represent the probability that either A or B will occur by the following formula: π(π΄ ∪ π΅) = π(π΄) + π(π΅) You can represent the favourable outcomes of two non-mutually exclusive events, A and B, as two intersecting sets. You can represent the probability that either A or B will occur by this formula: π(π΄ ∪ π΅) = π(π΄) + π(π΅) − π(π΄ ∩ π΅) 1. Numerical Response If one ball is randomly selected from the box, what is the probability that the number written on it is divisible by 3 or is an even number? 2. A particular traffic light at the outskirts of a town is red for 30 s, green for 25 s, and yellow for 5 s in every minute. When a vehicle approaches the traffic light, the probability that the light will be red or yellow is 7 1 1 1 A. 12 B. 2 C. 12 D. 24 3. Numerical Response In a non-leap year of 365 days, the average number of days of the year that a tourist could expect to experience weather other than sunshine, to the nearest whole number, is _________ days. 4. Numerical Response From the list above, the two events that are mutually exclusive are numbered _______ and _______. 5. What is the probability that a person chosen at random from the population watches TV at least once a day or uses a computer at least once a day? A. 0.65 B. 0.75 C. 0.85 D. 0.95 6. Numerical Response Determine the probability of Deborah getting hit in a game. Answers: 1. 0.65 2. A 3. 40 4. 23 or 32 5. D 6. 0.50 Conditional Probability Dependent events: events whose outcomes are affected by each other. For example, if two cards are drawn from a deck without replacement, the outcome of the second event depends on the outcome of the first event. 1. In the first experiment, Event X and Event Y are ___i___, and in the second experiment, Event X and event Y are ___ii___. The statement above is completed by the information in row Row i ii A. dependent independent B. dependent dependent C. independent independent D. independent dependent 2. Numerical Response A box contains 6 blue balls and 4 red balls. Two balls are drawn from the box, one after the other, without replacement. The probability, to the nearest hundredth, that the first ball drawn is blue and the second ball drawn is red is _________. 3. Based on previous performance, the probability of a particular baseball team 4 winning any game is . The probability that this team will win their next two 5 games is 1 4 1 16 A. B. C. D. 25 5 5 25 4. What is the probability that the team will win 1 game and lose 1 game during the next 2 games? A. 0.16 B. 0.32 C. 0.64 D. 0.80 If event B depends on event A occurring, then the conditional probability that event B will occur, given that event A has occurred can be represented as π(π΄ ∩ π΅) π(π΅|π΄) = π(π΄) or π(π΄ ∩ π΅) = π(π΄)π(π΅|π΄) A tree diagram is often useful for modelling problems that involve dependent events. Conditional probability: the probability of an event occurring given that another event has already occurred. Independent events: If the probability of event B does not depend on the probability of event A occurring, then tehse events are called independent events. For example, tossing tails with a coin and drawing the ace of spades from a standard deck of 52 playing cards are independent events. The probability that two independent events, A and B, will both occur is the product of their individual probabilities: π(π΄ ∩ π΅) = π(π΄)π(π΅) 5. Numerical Response Assuming that having a home video game system and having a TV in their bedroom are independent, determine the probability, to the nearest hundredth, that a student in the class has a TV in their bedroom. Answers: 1. D 2. 0.27 3. D 4. B 5. 0.74 A tree diagram is often useful for modelling problems that involve independent events.