JMAP REGENTS BY PERFORMANCE INDICATOR: TOPIC

Transcription

1 JMAP REGENTS BY PERFORMANCE INDICATOR: TOPIC NY Algebra /Trigonometry Regents Exam Questions from Fall 009 to June 0 Sorted by PI: Topic

2 TABLE OF CONTENTS TOPIC PI: SUBTOPIC QUESTION NUMBER GRAPHS AND STATISTICS PROBABILITY ABSOLUTE VALUE QUADRATICS A.S.-: Analysis of Data A.S.: Average Known with Missing Data A.S.: Dispersion...-6 A.S.6-7: Regression...7- A.S.8: Correlation Coefficient...-8 A.S.5: Normal Distributions A.S.0: Permutations...7- A.S.: Combinations A.S.9: Differentiating Permutations and Combinations A.S.: Sample Space A.S.: Geometric Probability A.S.5: Binomial Probability A.A.: Absolute Value Equations and Equalities A.A.0-: Roots of Quadratics A.A.7: Factoring Polynomials A.A.7: Factoring the Difference of Perfect Squares A.A.7: Factoring by Grouping A.A.5: Quadratic Formula A.A.: Using the Discriminant A.A.: Completing the Square A.A.: Quadratic Inequalities SYSTEMS A.A.: Quadratic-Linear Systems POWERS A.N.: Operations with Polynomials A.N., A.8-9: Negative and Fractional Exponents... - A.A.: Evaluating Exponential Expressions A.A.8: Evaluating Logarithmic Expressions A.A.5: Graphing Exponential Functions A.A.5: Graphing Logarithmic Functions A.A.9: Properties of Logarithms A.A.8: Logarithmic Equations A.A.6, 7: Exponential Equations A.A.6: Binomial Expansions A.A.6, 50: Solving Polynomial Equations i

3 RADICALS RATIONALS FUNCTIONS SEQUENCES AND SERIES TRIGONOMETRY A.N.: Operations with Irrational Expressions A.A.: Simplifying Radicals A.N., A.: Operations with Radicals A.N.5, A.5: Rationalizing Denominators A.A.: Solving Radicals A.A.0-: Exponents as Radicals A.N.6: Square Roots of Negative Numbers A.N.7: Imaginary Numbers A.N.8: Conjugates of Complex Numbers A.N.9: Multiplication and Division of Complex Numbers... - A.A.6: Multiplication and Division of Rationals A.A.6: Addition and Subtraction of Rationals... 7 A.A.: Solving Rationals and Rational Inequalities A.A.7: Complex Fractions A.A.5: Inverse Variation A.A.0-: Functional Notation A.A.5: Families of Functions... 6 A.A.6: Properties of Graphs of Functions and Relations A.A.5: Identifying the Equation of a Graph A.A.8, : Defining Functions A.A.9, 5: Domain and Range A.A.: Compositions of Functions A.A.: Inverse of Functions A.A.6: Transformations with Functions and Relations A.A.9-: Sequences A.N.0, A.: Sigma Notation A.A.5: Series A.A.55: Trigonometric Ratios A.M.-: Radian Measure A.A.60: Unit Circle A.A.60: Finding the Terminal Side of an Angle... - A.A.6, 66: Determining Trigonometric Functions A.A.6: Using Inverse Trigonometric Functions A.A.57: Reference Angles... 6 A.A.6: Arc Length A.A.58-59: Cofunction/Reciprocal Trigonometric Functions A.A.67: Proving Trigonometric Identities A.A.76: Angle Sum and Difference Identities ii

4 A.A.77: Double and Half Angle Identities A.A.68: Trigonometric Equations A.A.69: Properties of Trigonometric Functions A.A.7: Identifying the Equation of a Trigonometric Graph A.A.65, 70-7: Graphing Trigonometric Functions A.A.6: Domain and Range A.A.7: Using Trigonometry to Find Area A.A.7: Law of Sines A.A.75: Law of Sines - The Ambiguous Case A.A.7: Law of Cosines A.A.7: Vectors CONICS A.A.7-9: Equations of Circles iii

5 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic GRAPHS AND STATISTICS A.S.-: ANALYSIS OF DATA Howard collected fish eggs from a pond behind his house so he could determine whether sunlight had an effect on how many of the eggs hatched. After he collected the eggs, he divided them into two tanks. He put both tanks outside near the pond, and he covered one of the tanks with a box to block out all sunlight. State whether Howard's investigation was an example of a controlled experiment, an observation, or a survey. Justify your response. Which task is not a component of an observational study? The researcher decides who will make up the sample. The researcher analyzes the data received from the sample. The researcher gathers data from the sample, using surveys or taking measurements. The researcher divides the sample into two groups, with one group acting as a control group. A doctor wants to test the effectiveness of a new drug on her patients. She separates her sample of patients into two groups and administers the drug to only one of these groups. She then compares the results. Which type of study best describes this situation? census survey observation controlled experiment A market research firm needs to collect data on viewer preferences for local news programming in Buffalo. Which method of data collection is most appropriate? census survey observation controlled experiment 5 A school cafeteria has five different lunch periods. The cafeteria staff wants to find out which items on the menu are most popular, so they give every student in the first lunch period a list of questions to answer in order to collect data to represent the school. Which type of study does this represent? observation controlled experiment population survey sample survey 6 A survey completed at a large university asked,000 students to estimate the average number of hours they spend studying each week. Every tenth student entering the library was surveyed. The data showed that the mean number of hours that students spend studying was 5.7 per week. Which characteristic of the survey could create a bias in the results? the size of the sample the size of the population the method of analyzing the data the method of choosing the students who were surveyed 7 The yearbook staff has designed a survey to learn student opinions on how the yearbook could be improved for this year. If they want to distribute this survey to 00 students and obtain the most reliable data, they should survey every third student sent to the office every third student to enter the library every third student to enter the gym for the basketball game every third student arriving at school in the morning

6 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 Which survey is least likely to contain bias? surveying a sample of people leaving a movie theater to determine which flavor of ice cream is the most popular surveying the members of a football team to determine the most watched TV sport surveying a sample of people leaving a library to determine the average number of books a person reads in a year surveying a sample of people leaving a gym to determine the average number of hours a person exercises per week A.S.: DISPERSION The table below shows the first-quarter averages for Mr. Harper s statistics class. A.S.: AVERAGE KNOWN WITH MISSING DATA 9 The number of minutes students took to complete a quiz is summarized in the table below. If the mean number of minutes was 7, which equation could be used to calculate the value of x? 7 = 9 + x x 9 + 6x 7 = x 7 = 6 + x 6 + x 6 + 6x 7 = 6 + x What is the population variance for this set of data? The scores of one class on the Unit mathematics test are shown in the table below. 0 The table below displays the results of a survey regarding the number of pets each student in a class has. The average number of pets per student in this class is. What is the value of k for this table? 9 8 Find the population standard deviation of these scores, to the nearest tenth.

7 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic During a particular month, a local company surveyed all its employees to determine their travel times to work, in minutes. The data for all 5 employees are shown below. A.S.6-7: REGRESSION 7 Samantha constructs the scatter plot below from a set of data Determine the number of employees whose travel time is within one standard deviation of the mean. The heights, in inches, of 0 high school varsity basketball players are 78, 79, 79, 7, 75, 7, 7, 7, 8, and 7. Find the interquartile range of this data set. 5 Ten teams competed in a cheerleading competition at a local high school. Their scores were 9, 8, 9, 7, 5, 0,, 8, 7, and 8. How many scores are within one population standard deviation from the mean? For these data, what is the interquartile range? 6 The following is a list of the individual points scored by all twelve members of the Webster High School basketball team at a recent game: Find the interquartile range for this set of data. Based on her scatter plot, which regression model would be most appropriate? exponential linear logarithmic power 8 The table below shows the results of an experiment involving the growth of bacteria. Write a power regression equation for this set of data, rounding all values to three decimal places. Using this equation, predict the bacteria s growth, to the nearest integer, after 5 minutes.

8 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 9 The table below shows the number of new stores in a coffee shop chain that opened during the years 986 through A population of single-celled organisms was grown in a Petri dish over a period of 6 hours. The number of organisms at a given time is recorded in the table below. Using x = to represent the year 986 and y to represent the number of new stores, write the exponential regression equation for these data. Round all values to the nearest thousandth. Determine the exponential regression equation model for these data, rounding all values to the nearest ten-thousandth. Using this equation, predict the number of single-celled organisms, to the nearest whole number, at the end of the 8th hour. A cup of soup is left on a countertop to cool. The table below gives the temperatures, in degrees Fahrenheit, of the soup recorded over a 0-minute period. Write an exponential regression equation for the data, rounding all values to the nearest thousandth.

9 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic The data collected by a biologist showing the growth of a colony of bacteria at the end of each hour are displayed in the table below. 5 Which calculator output shows the strongest linear relationship between x and y? Write an exponential regression equation to model these data. Round all values to the nearest thousandth. Assuming this trend continues, use this equation to estimate, to the nearest ten, the number of bacteria in the colony at the end of 7 hours. The table below shows the concentration of ozone in Earth s atmosphere at different altitudes. Write the exponential regression equation that models these data, rounding all values to the nearest thousandth. A.S.8: CORRELATION COEFFICIENT Which value of r represents data with a strong negative linear correlation between two variables?

10 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 6 As shown in the table below, a person s target heart rate during exercise changes as the person gets older. Which value represents the linear correlation coefficient, rounded to the nearest thousandth, between a person s age, in years, and that person s target heart rate, in beats per minute? The relationship between t, a student s test scores, and d, the student s success in college, is modeled by the equation d = 0.8t Based on this linear regression model, the correlation coefficient could be between and 0 between 0 and equal to equal to 0 8 Which value of r represents data with a strong positive linear correlation between two variables? A.S.5: NORMAL DISTRIBUTIONS 9 The lengths of 00 pipes have a normal distribution with a mean of 0. inches and a standard deviation of 0. inch. If one of the pipes measures exactly 0. inches, its length lies below the 6 th percentile between the 50 th and 8 th percentiles between the 6 th and 50 th percentiles above the 8 th percentile 0 An amateur bowler calculated his bowling average for the season. If the data are normally distributed, about how many of his 50 games were within one standard deviation of the mean? 7 8 Assume that the ages of first-year college students are normally distributed with a mean of 9 years and standard deviation of year. To the nearest integer, find the percentage of first-year college students who are between the ages of 8 years and 0 years, inclusive. To the nearest integer, find the percentage of first-year college students who are 0 years old or older. In a study of 8 video game players, the researchers found that the ages of these players were normally distributed, with a mean age of 7 years and a standard deviation of years. Determine if there were 5 video game players in this study over the age of 0. Justify your answer. If the amount of time students work in any given week is normally distributed with a mean of 0 hours per week and a standard deviation of hours, what is the probability a student works between 8 and hours per week?.% 8.% 5.% 68.% 6

11 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic In a certain high school, a survey revealed the mean amount of bottled water consumed by students each day was 5 bottles with a standard deviation of bottles. Assuming the survey represented a normal distribution, what is the range of the number of bottled waters that approximately 68.% of the students drink? Liz has applied to a college that requires students to score in the top 6.7% on the mathematics portion of an aptitude test. The scores on the test are approximately normally distributed with a mean score of 576 and a standard deviation of 0. What is the minimum score Liz must earn to meet this requirement? In a certain school, the heights of the population of girls are normally distributed, with a mean of 6 inches and a standard deviation of inches. If there are 50 girls in the school, determine how many of the girls are shorter than 60 inches. Round the answer to the nearest integer. PROBABILITY A.S.0: PERMUTATIONS 7 Which formula can be used to determine the total number of different eight-letter arrangements that can be formed using the letters in the word DEADLINE? 8! 8!! 8!!+! 8!!! 8 The letters of any word can be rearranged. Carol believes that the number of different 9-letter arrangements of the word TENNESSEE is greater than the number of different 7-letter arrangements of the word VERMONT. Is she correct? Justify your answer. 9 Find the total number of different twelve-letter arrangements that can be formed using the letters in the word PENNSYLVANIA. 0 A four-digit serial number is to be created from the digits 0 through 9. How many of these serial numbers can be created if 0 can not be the first digit, no digit may be repeated, and the last digit must be 5? 8 50,0,50 How many different six-letter arrangements can be made using the letters of the word TATTOO? Find the number of possible different 0-letter arrangements using the letters of the word STATISTICS. Which expression represents the total number of different -letter arrangements that can be made using the letters in the word MATHEMATICS?!!!!+!+!! 8!!!!! 7

12 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic The number of possible different -letter arrangements of the letters in the word TRIGONOMETRY is represented by!!! 6! P 8 P 6! A.S.: COMBINATIONS 5 The principal would like to assemble a committee of 8 students from the 5-member student council. How many different committees can be chosen? 0 6,5,,00 59,59,00 6 Ms. Bell's mathematics class consists of sophomores, 0 juniors, and 5 seniors. How many different ways can Ms. Bell create a four-member committee of juniors if each junior has an equal chance of being selected? 0,876 5,00 9,0 7 A blood bank needs twenty people to help with a blood drive. Twenty-five people have volunteered. Find how many different groups of twenty can be formed from the twenty-five volunteers. 8 If order does not matter, which selection of students would produce the most possible committees? 5 out of 5 5 out of 5 0 out of 5 5 out of 5 A.S.9: DIFFERENTIATING BETWEEN PERMUTATIONS AND COMBINATIONS 9 Twenty different cameras will be assigned to several boxes. Three cameras will be randomly selected and assigned to box A. Which expression can be used to calculate the number of ways that three cameras can be assigned to box A? 0! 0!! 0 C 0 P 50 Three marbles are to be drawn at random, without replacement, from a bag containing 5 red marbles, 0 blue marbles, and 5 white marbles. Which expression can be used to calculate the probability of drawing red marbles and white marble from the bag? 5 C 5 C 0 C 5 P 5 P 0 C 5 C 5 C 0 P 5 P 5 P 0 P 5 There are eight people in a tennis club. Which expression can be used to find the number of different ways they can place first, second, and third in a tournament? 8 P 8 C 8 P 5 8 C 5 8

13 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 Which problem involves evaluating 6 P? How many different four-digit ID numbers can be formed using,,,, 5, and 6 without repetition? How many different subcommittees of four can be chosen from a committee having six members? How many different outfits can be made using six shirts and four pairs of pants? How many different ways can one boy and one girl be selected from a group of four boys and six girls? 5 A math club has 0 boys and 0 girls. Which expression represents the total number of different 5-member teams, consisting of boys and girls, that can be formed? 0 P 0 P 0 C 0 C 0 P + 0 P 0 C + 0 C A.S.: SAMPLE SPACE 5 A committee of 5 members is to be randomly selected from a group of 9 teachers and 0 students. Determine how many different committees can be formed if members must be teachers and members must be students. 55 A school math team consists of three juniors and five seniors. How many different groups can be formed that consist of one junior and two seniors? A.S.: GEOMETRIC PROBABILITY 56 A dartboard is shown in the diagram below. The two lines intersect at the center of the circle, and the central angle in sector measures π. If darts thrown at this board are equally likely to land anywhere on the board, what is the probability that a dart that hits the board will land in either sector or sector? 6 A.S.5: BINOMIAL PROBABILITY 57 The members of a men s club have a choice of wearing black or red vests to their club meetings. A study done over a period of many years determined that the percentage of black vests worn is 60%. If there are 0 men at a club meeting on a given night, what is the probability, to the nearest thousandth, that at least 8 of the vests worn will be black? 9

14 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 58 A study shows that 5% of the fish caught in a local lake had high levels of mercury. Suppose that 0 fish were caught from this lake. Find, to the nearest tenth of a percent, the probability that at least 8 of the 0 fish caught did not contain high levels of mercury. 59 The probability that the Stormville Sluggers will win a baseball game is. Determine the probability, to the nearest thousandth, that the Stormville Sluggers will win at least 6 of their next 8 games. 60 The probability that a professional baseball player will get a hit is. Calculate the exact probability that he will get at least hits in 5 attempts. 6 A spinner is divided into eight equal sections. Five sections are red and three are green. If the spinner is spun three times, what is the probability that it lands on red exactly twice? A study finds that 80% of the local high school students text while doing homework. Ten students are selected at random from the local high school. Which expression would be part of the process used to determine the probability that, at most, 7 of the 0 students text while doing homework? 0 C 6 0 C 7 0 C 8 0 C On a multiple-choice test, Abby randomly guesses on all seven questions. Each question has four choices. Find the probability, to the nearest thousandth, that Abby gets exactly three questions correct. 6 Because Sam s backyard gets very little sunlight, the probability that a geranium planted there will flower is 0.8. Sam planted five geraniums. Determine the probability, to the nearest thousandth, that at least four geraniums will flower. 65 Whenever Sara rents a movie, the probability that it is a horror movie is Of the next five movies she rents, determine the probability, to the nearest hundredth, that no more than two of these rentals are horror movies. ABSOLUTE VALUE A.A.: ABSOLUTE VALUE EQUATIONS AND INEQUALITIES 66 What is the solution set of the equation a + 6 a = 0? 0 0, 67 Which graph represents the solution set of 6x 7 5? 0

15 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 68 Graph the inequality 6 x < 5 for x. Graph the solution on the line below. 69 Which graph represents the solution set of x 5 >? 70 Determine the solution of the inequality x 7. [The use of the grid below is optional.] 7 What is the graph of the solution set of x >5? 7 Solve x + 5 < algebraically for x. 7 Solve x >5 algebraically. QUADRATICS A.A.0-: ROOTS OF QUADRATICS 7 Find the sum and product of the roots of the equation 5x + x = What are the sum and product of the roots of the equation 6x x = 0? sum = ; product = sum = ; product = sum = ; product = sum = ; product = 76 Determine the sum and the product of the roots of x = x Determine the sum and the product of the roots of the equation x + x 6 = What is the product of the roots of the quadratic equation x 7x = 5?

16 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 79 For which equation does the sum of the roots equal and the product of the roots equal? x 8x + = 0 x + 8x + = 0 x x 8 = 0 x + x = 0 80 For which equation does the sum of the roots equal and the product of the roots equal? x + x = 0 x x + = 0 x + 6x + = 0 x 6x + = 0 8 Write a quadratic equation such that the sum of its roots is 6 and the product of its roots is 7. 8 Which equation has roots with the sum equal to 9 and the product equal to? x + 9x + = 0 x + 9x = 0 x 9x + = 0 x 9x = 0 8 What is the product of the roots of x x + k = 0 if one of the roots is 7? 77 A.A.7: FACTORING POLYNOMIALS 8 Factored completely, the expression 6x x x is equivalent to x(x + )(x ) x(x )(x + ) x(x )(x + ) x(x + )(x ) 85 Factored completely, the expression x + 0x x is equivalent to x (x + 6)(x ) (x + x)(x x) x (x )(x + ) x (x + )(x ) 86 Factor completely: 0ax ax 5a A.A.7: FACTORING THE DIFFERENCE OF PERFECT SQUARES 87 Factor the expression t 8 75t completely. A.A.7: FACTORING BY GROUPING 88 When factored completely, x + x x equals (x + )(x )(x ) (x + )(x )(x + ) (x )(x + ) (x )(x ) 89 When factored completely, the expression x 5x 8x + 80 is equivalent to (x 6)(x 5) (x + 6)(x 5)(x + 5) (x + )(x )(x 5) (x + )(x )(x 5)(x 5) 90 The expression x (x + ) (x + ) is equivalent to x x x + x x + (x + )(x )(x + )

17 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.5: QUADRATIC FORMULA 9 The solutions of the equation y y = 9 are ± i ± i 5 ± 5 ± 5 9 The roots of the equation x + 7x = 0 are and and 7 ± 7 7 ± 7 9 Solve the equation 6x x = 0 and express the answer in simplest radical form. 9 A cliff diver on a Caribbean island jumps from a height of 05 feet, with an initial upward velocity of 5 feet per second. An equation that models the height, h(t), above the water, in feet, of the diver in time elapsed, t, in seconds, is h(t) = 6t + 5t How many seconds, to the nearest hundredth, does it take the diver to fall 5 feet below his starting point? A.A.: USING THE DISCRIMINANT 95 Use the discriminant to determine all values of k that would result in the equation x kx + = 0 having equal roots. 96 The roots of the equation 9x + x = 0 are imaginary real, rational, and equal real, rational, and unequal real, irrational, and unequal 97 The roots of the equation x 0x + 5 = 0 are imaginary real and irrational real, rational, and equal real, rational, and unequal 98 The discriminant of a quadratic equation is. The roots are imaginary real, rational, and equal real, rational, and unequal real, irrational, and unequal 99 The roots of the equation x + = 9x are real, rational, and equal real, rational, and unequal real, irrational, and unequal imaginary 00 For which value of k will the roots of the equation x 5x + k = 0 be real and rational numbers? 5 0 A.A.: COMPLETING THE SQUARE 0 Solve x x + = 0 by completing the square, expressing the result in simplest radical form. 0 If x + = 6x is solved by completing the square, an intermediate step would be (x + ) = 7 (x ) = 7 (x ) = (x 6) =

18 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 0 Brian correctly used a method of completing the square to solve the equation x + 7x = 0. Brian s first step was to rewrite the equation as x + 7x =. He then added a number to both sides of the equation. Which number did he add? Max solves a quadratic equation by completing the square. He shows a correct step: (x + ) = 9 What are the solutions to his equation? ± i ± i ± i ± i 05 Which step can be used when solving x 6x 5 = 0 by completing the square? x 6x + 9 = x 6x 9 = 5 9 x 6x + 6 = x 6x 6 = 5 6 A.A.: QUADRATIC INEQUALITIES 06 Which graph best represents the inequality y + 6 x x?

19 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 07 The solution set of the inequality x x > 0 is {x < x < 5} {x 0 < x < } {x x < or x > 5} {x x < 5 or x > } 08 Find the solution of the inequality x x > 5, algebraically. SYSTEMS A.A.: QUADRATIC-LINEAR SYSTEMS 09 Which values of x are in the solution set of the following system of equations? y = x 6 0, 0, 6, 6, y = x x 6 0 Solve the following systems of equations algebraically: 5 = y x x = 7x + y + Which ordered pair is a solution of the system of equations shown below? x + y = 5 (, ) (5, 0) ( 5, 0) (, 9) (x + ) + (y ) = 5 Which ordered pair is in the solution set of the system of equations shown below? (, 6) (, ) (, ) ( 6, ) y x + = 0 y x = 0 Determine algebraically the x-coordinate of all points where the graphs of xy = 0 and y = x + intersect. POWERS A.N.: OPERATIONS WITH POLYNOMIALS Express x as a trinomial. 5 When x x is subtracted from 5 x x +, the difference is x + x 5 x x + 5 x x x x 6 Express the product of y y and y + 5 as a trinomial. 7 What is the product of x 8 9 x 6 9 x 8 x 6 9 x 6 x 6 9 x and x +? 5

20 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 What is the product of 5 x y and 5 x + y? 5 x 9 6 y 5 x 9 6 y 5 x y 5 x 9 When x + x is subtracted from x + x x, the difference is x + x 5x + x + x + x x + x + x x x + 5x + 0 The expression x is equivalent to 9x x x + 9x x + 6x A.N., A.8-9: NEGATIVE AND FRACTIONAL EXPONENTS If a = and b =, what is the value of the expression a b? If n is a negative integer, then which statement is always true? 6n < n n > 6n 6n < n n > (6n) When simplified, the expression equivalent to w 7 w w 7 w w 5 w 9 Which expression is equivalent to 9x y 6? xy xy xy xy 5 Which expression is equivalent to x? x x 9x 9x is 6

21 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 6 The expression (a) is equivalent to 8a 6 a a 6a 7 The expression a b a 6 b 5 b 5 a 6 a b a b a b is equivalent to 8 When x is divided by x, the quotient is x x (x ) x y 5 9 Simplify the expression and write the (x y 7 ) answer using only positive exponents. 0 When x + is divided by x +, the quotient equals x x Which expression is equivalent to x y 5 x 5 y x y 5 y x 5 x y x 5 y? Which expression is equivalent to x y? y 5 y x y x x y x y A.A.: EVALUATING EXPONENTIAL EXPRESSIONS Matt places $,00 in an investment account earning an annual rate of 6.5%, compounded continuously. Using the formula V = Pe rt, where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, that Matt will have in the account after 0 years. Evaluate e x ln y when x = and y =. x 7

22 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 The formula for continuously compounded interest is A = Pe rt, where A is the amount of money in the account, P is the initial investment, r is the interest rate, and t is the time in years. Using the formula, determine, to the nearest dollar, the amount in the account after 8 years if $750 is invested at an annual rate of %. 6 If $5000 is invested at a rate of % interest compounded quarterly, what is the value of the investment in 5 years? (Use the formula A = P + r nt n, where A is the amount accrued, P is the principal, r is the interest rate, n is the number of times per year the money is compounded, and t is the length of time, in years.) $590. $ $ $ The expression log 5 5 is equivalent to A.A.5: GRAPHING EXPONENTIAL FUNCTIONS x 0 The graph of the equation y = has an asymptote. On the grid below, sketch the graph of x y = and write the equation of this asymptote. 7 The formula to determine continuously compounded interest is A = Pe rt, where A is the amount of money in the account, P is the initial investment, r is the interest rate, and t is the time, in years. Which equation could be used to determine the value of an account with an $8,000 initial investment, at an interest rate of.5% for months? A = 8, 000e.5.5 A = 8, 000e A = 8, 000e A = 8, 000e A.A.8: EVALUATING LOGARITHMIC EXPRESSIONS 8 The expression log 8 6 is equivalent to 8 8 8

23 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic On the axes below, for x, graph y = x +. A.A.5: GRAPHING LOGARITHMIC FUNCTIONS If a function is defined by the equation f(x) = x, which graph represents the inverse of this function? What is the equation of the graph shown below? y = x y = x x = y x = y 9

24 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Which graph represents the function log x = y? 5 Which sketch shows the inverse of y = a x, where a >? A.A.9: PROPERTIES OF LOGARITHMS 6 The expression logx ( logy + logz) is equivalent to log x y z log x z y log x yz log xz y 0

25 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A 7 If r = B, then logr can be represented by C 6 loga + logb logc (loga + logb logc) log(a + B) C loga + logb logc 8 If logx loga = loga, then logx expressed in terms of loga is equivalent to log5a log6 + loga log6 + loga log6 + loga 9 If log b x = log b p log b t + log b r, then the value of x is p t r p t r p t r p t r 50 If log = a and log = b, the expression log 9 0 is equivalent to b a + b a b a + 0 b a + 5 The expression logm is equivalent to (log + logm) log + logm log + logm log6 + logm 5 If x = y, then logy equals log(x) + log log(x) log + logx log + logx A.A.8: LOGARITHMIC EQUATIONS 5 What is the solution of the equation log (5x) =? Solve algebraically for x: log x + x + x x = 55 The temperature, T, of a given cup of hot chocolate after it has been cooling for t minutes can best be modeled by the function below, where T 0 is the temperature of the room and k is a constant. ln(t T 0 ) = kt +.78 A cup of hot chocolate is placed in a room that has a temperature of 68. After minutes, the temperature of the hot chocolate is 50. Compute the value of k to the nearest thousandth. [Only an algebraic solution can receive full credit.] Using this value of k, find the temperature, T, of this cup of hot chocolate if it has been sitting in this room for a total of 0 minutes. Express your answer to the nearest degree. [Only an algebraic solution can receive full credit.] 56 What is the value of x in the equation log 5 x =? ,0

26 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 57 If log x =.5 and log y 5 =, find the numerical value of x, in simplest form. y 58 Solve algebraically for all values of x: log (x + ) (7x ) = 59 Solve algebraically for x: log 7 (x ) = 60 Solve algebraically for all values of x: log (x + ) (x + ) + log (x + ) (x + 5) = 6 Solve algebraically for x: log 5x = A.A.6, 7: EXPONENTIAL EQUATIONS 6 Akeem invests $5,000 in an account that pays.75% annual interest compounded continuously. Using the formula A = Pe rt, where A = the amount in the account after t years, P = principal invested, and r = the annual interest rate, how many years, to the nearest tenth, will it take for Akeem s investment to triple? A population of rabbits doubles every 60 days t according to the formula P = 0() 60, where P is the population of rabbits on day t. What is the value of t when the population is 0? Susie invests $500 in an account that is compounded continuously at an annual interest rate of 5%, according to the formula A = Pe rt, where A is the amount accrued, P is the principal, r is the rate of interest, and t is the time, in years. Approximately how many years will it take for Susie s money to double? The solution set of x + x = 6 is {, } {, } {, } {, } 67 What is the value of x in the equation 9 x + = 7 x +? 68 Solve algebraically for x: 6 x + = 6 x + 69 The value of x in the equation x + 5 = 8 x is Solve algebraically for all values of x: 8 x + x = 7 5x 6 The number of bacteria present in a Petri dish can be modeled by the function N = 50e t, where N is the number of bacteria present in the Petri dish after t hours. Using this model, determine, to the nearest hundredth, the number of hours it will take for N to reach 0,700.

27 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 Which value of k satisfies the equation 8 k + = k? 9 5 A.A.6: BINOMIAL EXPANSIONS 7 What is the fourth term in the expansion of (x ) 5? 70x 0x 70x, 080x 7 Write the binomial expansion of (x ) 5 as a polynomial in simplest form. 7 What is the coefficient of the fourth term in the expansion of (a b) 9? 5, ,76 75 Which expression represents the third term in the expansion of (x y)? y 6x y 6x y x y 77 What is the fourth term in the binomial expansion (x ) 8? 8x 5 8x 8x 5 8x A.A.6, 50: SOLVING POLYNOMIAL EQUATIONS 78 Solve the equation 8x + x 8x 9 = 0 algebraically for all values of x. 79 Which values of x are solutions of the equation x + x x = 0? 0,, 0,, 0,, 0,, 80 What is the solution set of the equation x 5 8x = 0? {0, ±} {0, ±, } {0, ±, ±i} {±, ±i} 8 Solve algebraically for all values of x: x + x + x = 6x 8 Solve x + 5x = x + 0 algebraically. 76 What is the middle term in the expansion of 6 x y? 0x y 5 x y 0x y 5 x y

28 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 The graph of y = f(x) is shown below. 8 The graph of y = x x + x + 6 is shown below. Which set lists all the real solutions of f(x) = 0? {, } {, } {, 0, } {, 0, } What is the product of the roots of the equation x x + x + 6 = 0? How many negative solutions to the equation x x + x = 0 exist? 0 RADICALS A.N.: OPERATIONS WITH IRRATIONAL EXPRESSIONS 86 The product of ( + 5 ) and ( 5 ) is

29 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.: SIMPLIFYING RADICALS 87 Express in simplest form: 88 The expression 6a 6 8a 8a 8 a 5 a a a 5 a 6 b 9 6 is equivalent to A.N., A.: OPERATIONS WITH RADICALS 89 Express 5 x 7x in simplest radical form. 90 The sum of 6a b and 6a b, expressed in simplest radical form, is 6 68a 8 b a b a b a 6ab 0a b 8 9 The expression 7x to x x x 6x x 6x 6x is equivalent 9 Express 08x 5 y 8 6xy 5 in simplest radical form. A.N.5, A.5: RATIONALIZING DENOMINATORS 5 9 Express with a rational denominator, in simplest radical form. 95 Which expression is equivalent to The expression 5 5 (5 ) (5 + ) ? is equivalent to 9 The expression ab b a 8b + 7ab 6b is equivalent to ab 6b 6ab b 5ab + 7ab 6b 5ab b + 7ab 6b 5

30 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 97 The expression The fraction a b b ab b ab a 99 The expression 7 a b (x + ) x x (x + ) x x x x + x + x + is equivalent to is equivalent to is equivalent to 00 Expressed with a rational denominator and in x simplest form, x x is x + x x x x x + x x x + x x x A.A.: SOLVING RADICALS 0 The solution set of the equation x + = x is {} {0} {, 6} {, } 0 The solution set of x + 6 = x + is {, } {, } {} { } 0 Solve algebraically for x: x 5 = 0 What is the solution set for the equation 5x + 9 = x +? {} { 5} {, 5} { 5, } 05 Solve algebraically for x: x + x + x = 7x + 06 The solution set of the equation x = x is {, } {, } {} { } A.A.0-: EXPONENTS AS RADICALS 07 The expression (x ) (x ) (x ) (x ) (x ) is equivalent to 6

31 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 08 The expression x 5 is equivalent to x 5 5 x 5 x 5 x 09 The expression 6x y 7 is equivalent to 7 y x x 8 y 8 7 y x x 8 y 8 A.N.6: SQUARE ROOTS OF NEGATIVE NUMBERS 0 In simplest form, 00 is equivalent to i 0 5i 0i i 5 Expressed in simplest form, 8 is 7 i 7i A.N.7: IMAGINARY NUMBERS The product of i 7 and i 5 is equivalent to i i The expression i + i is equivalent to i i + i + i Determine the value of n in simplest form: i + i 8 + i + n = 0 5 Express xi + 5yi 8 + 6xi + yi in simplest a + bi form. A.N.8: CONJUGATES OF COMPLEX NUMBERS 6 What is the conjugate of + i? + i i i + i 7 The conjugate of 7 5i is 7 5i 7 + 5i 7 5i 7 + 5i 8 What is the conjugate of + i? + i i + i i 9 The conjugate of the complex expression 5x + i is 5x i 5x + i 5x i 5x + i 0 Multiply x + yi by its conjugate, and express the product in simplest form. 7

32 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.N.9: MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS The expression ( 7i) is equivalent to 0 + 0i 0 i i 58 i The expression (x + i) (x i) is equivalent to 0 + xi xi If x = i, y = i, and z = m + i, the expression xy z equals mi 6 6mi mi 6 6mi RATIONALS A.A.6: MULTIPLICATION AND DIVISION OF RATIONALS Perform the indicated operations and simplify completely: x x + 6x 8 x x 5 Express in simplest form: x x x x + x 8 6 x x x + 7x + x x + 6 The expression x + 9x ( x) is equivalent x to x x x x A.A.6: ADDITION AND SUBTRACTION OF RATIONALS 7 Expressed in simplest form, equivalent to 6y + 6y 5 (y 6)(6 y) y 9 y 6 y y y is A.A.: SOLVING RATIONALS AND RATIONAL INEQUALITIES 8 Solve for x: x x = + x 9 Solve algebraically for x: x + x = x 9 0 Solve the equation below algebraically, and express the result in simplest radical form: x = 0 x What is the solution set of the equation 0 x 9 + = 5 x? {, } {} {} { } Which graph represents the solution set of x + 6 x 7? 8

33 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.7: COMPLEX FRACTIONS Written in simplest form, the expression is equivalent to x x x x x + Express in simplest form: 5 The simplest form of x x + x x x 6 The expression c + d a + b d b ac + b cd b ac + cd a + b c d b c d d + d x x 8 x is is equivalent to x x x + 7 Express in simplest terms: + x 5 x x A.A.5: INVERSE VARIATION 8 For a given set of rectangles, the length is inversely proportional to the width. In one of these rectangles, the length is and the width is 6. For this set of rectangles, calculate the width of a rectangle whose length is 9. 9 If p varies inversely as q, and p = 0 when q =, what is the value of p when q = 5? The quantities p and q vary inversely. If p = 0 when q =, and p = x when q = x +, then x equals and and The points (, ),,, and (6, d) lie on the graph of a function. If y is inversely proportional to the square of x, what is the value of d? 7 9

34 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic If d varies inversely as t, and d = 0 when t =, what is the value of t when d = 5? 8 8 FUNCTIONS A.A.0-: FUNCTIONAL NOTATION The equation y sin θ = may be rewritten as f(y) = sinx + f(y) = sinθ + f(x) = sinθ + f(θ) = sinθ + A.A.5: FAMILIES OF FUNCTIONS 6 On January, a share of a certain stock cost $80. Each month thereafter, the cost of a share of this stock decreased by one-third. If x represents the time, in months, and y represents the cost of the stock, in dollars, which graph best represents the cost of a share over the following 5 months? If fx = 5 x, what is the value of f( 0)? x If g(x) = ax x, express g(0) in simplest form. 0

35 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.5: PROPERTIES OF GRAPHS OF FUNCTIONS AND RELATIONS 50 Which equation is represented by the graph below? 7 Which statement about the graph of the equation y = e x is not true? It is asymptotic to the x-axis. The domain is the set of all real numbers. It lies in Quadrants I and II. It passes through the point (e, ). 8 Theresa is comparing the graphs of y = x and y = 5 x. Which statement is true? The y-intercept of y = x is (0, ), and the y-intercept of y = 5 x is (0, 5). Both graphs have a y-intercept of (0, ), and y = x is steeper for x > 0. Both graphs have a y-intercept of (0, ), and y = 5 x is steeper for x > 0. Neither graph has a y-intercept. A.A.5: IDENTIFYING THE EQUATION OF A GRAPH y = 5 x y = 0.5 x y = 5 x y = 0.5 x 9 Four points on the graph of the function f(x) are shown below. {(0, ), (, ), (, ), (, 8)} Which equation represents f(x)? f(x) = x f(x) = x f(x) = x + f(x) = log x

36 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.8, : DEFINING FUNCTIONS 5 Which graph does not represent a function? 5 Which graph does not represent a function? 5 Which relation is not a function? (x ) + y = x + x + y = x + y = xy =

37 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 Which graph represents a relation that is not a function? 56 Which graph represents a function? 57 Which function is not one-to-one? {(0, ), (, ), (, ), (, )} {(0, 0), (, ), (, ), (, )} {(0, ), (, 0), (, ), (, )} {(0, ), (, 0), (, 0), (, )} 55 Given the relation {(8,),(,6),(7,5),(k, )}, which value of k will result in the relation not being a function?

38 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 58 Which graph represents a one-to-one function? 6 Which diagram represents a relation that is both one-to-one and onto? 6 Which relation is both one-to-one and onto? 59 Which function is one-to-one? f(x) = x f(x) = x f(x) = x f(x) = sinx 60 Which function is one-to-one? k(x) = x + g(x) = x + f(x) = x + j(x) = x +

39 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.9, 5: DOMAIN AND RANGE 6 What is the domain of the function f(x) = x +? (, ) (, ) [, ) [, ) 68 What are the domain and the range of the function shown in the graph below? 6 What is the range of f(x) = (x + ) + 7? y y y = 7 y 7 65 What is the range of f(x) = x +? {x x } {y y } {x x real numbers} {y y real numbers} 66 If f(x) = 9 x, what are its domain and range? domain: {x x }; range: {y 0 y } domain: {x x ±}; range: {y 0 y } domain: {x x or x }; range: {y y 0} domain: {x x }; range: {y y 0} {x x > }; {y y > } {x x }; {y y } {x x > }; {y y > } {x x }; {y y } 69 The graph below represents the function y = f(x). 67 For y =, what are the domain and range? x {x x > } and {y y > 0} {x x } and {y y > 0} {x x > } and {y y 0} {x x } and {y y 0} State the domain and range of this function. 5

40 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 70 What is the domain of the function shown below? 7 The graph below shows the average price of gasoline, in dollars, for the years 997 to 007. x 6 y 6 x 5 y 5 7 What is the range of the function shown below? x 0 x 0 y 0 y 0 What is the approximate range of this graph? 997 x x y.8.7 y.8 A.A.: COMPOSITIONS OF FUNCTIONS 7 If f(x) = x and g(x) = x + 5, what is the value of (g f)()? If f(x) = x 5 and g(x) = 6x, then g(f(x)) is equal to 6x 0x 6x 0 6x 5 x + 6x 5 75 If f(x) = x 6 and g(x) = x, determine the value of (g f)( ). 6

41 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 76 If f(x) = x x and g(x) = x, then (f g) equal to 7 7 is A.A.6: TRANSFORMATIONS WITH FUNCTIONS AND RELATIONS 8 The graph below shows the function f(x). 77 Which expression is equivalent to (n m p)(x), given m(x) = sin x, n(x) = x, and p(x) = x? sin(x) sin x sin (x) sin x 78 If g(x) = x + 8 and h(x) = x, what is the value of g(h( 8))? If f(x) = x x + and g(x) = x + 5, what is f(g(x))? x + 7x + 6 x + 7x + 66 x x + 6 x x + 6 Which graph represents the function f(x + )? A.A.: INVERSE OF FUNCTIONS 80 Which two functions are inverse functions of each other? f(x) = sin x and g(x) = cos(x) f(x) = + 8x and g(x) = 8x f(x) = e x and g(x) = ln x f(x) = x and g(x) = x + 8 If f(x) = x 6, find f (x). 7

42 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 The minimum point on the graph of the equation y = f(x) is (, ). What is the minimum point on the graph of the equation y = f(x) + 5? (, ) (, 8) (, ) ( 6, ) 8 The function f(x) is graphed on the set of axes below. On the same set of axes, graph f(x + ) What is a formula for the nth term of sequence B shown below? B = 0,,, 6,... b n = 8 + n b n = 0 + n b n = 0() n b n = 0() n 87 A sequence has the following terms: a =, a = 0, a = 5, a = 6.5. Which formula represents the nth term in the sequence? a n = +.5n a n = +.5(n ) a n = (.5) n a n = (.5) n 88 In an arithmetic sequence, a = 9 and a 7 =. Determine a formula for a n, the n th term of this sequence. SEQUENCES AND SERIES A.A.9-: SEQUENCES 85 What is the formula for the nth term of the sequence 5, 8, 6,...? a n = 6 n a n = 6 n a n = 5 n a n = 5 n 89 What is the common difference of the arithmetic sequence 5, 8,,? Which arithmetic sequence has a common difference of? {0, n, 8n, n,... } {n, n, 6n, 6n,... } {n +, n + 5, n + 9, n +,... } {n +, n + 6, n + 6, n + 56,... } 9 What is the common difference in the sequence a +, a +, 6a + 7, 8a + 0,...? a + a a + 5 a + 5 8

43 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 9 What is the common difference of the arithmetic sequence below? 7x, x, x, x, 5x,... x x 9 What is the common ratio of the geometric sequence whose first term is 7 and fourth term is 6? What is the common ratio of the geometric sequence shown below?,, 8,6, What is the common ratio of the sequence 6 a5 b, a b 9, 6 ab5,...? b a 6b a a b 6a b 96 What is the fifteenth term of the sequence 5, 0, 0, 0, 80,...? 6, 80 8, 90 8,90 7, What is the fifteenth term of the geometric sequence 5, 0, 5,...? Find the first four terms of the recursive sequence defined below. a = a n = a (n ) n 99 Find the third term in the recursive sequence a k + = a k, where a =. A.N.0, A.: SIGMA NOTATION 00 The value of the expression (n + n ) is 6 0 Evaluate: 0 + (n ) 5 n = n = 0 0 The value of the expression ( r + r) is Evaluate: ( n n) n = 5 r = 9

44 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 0 The expression + (k x) is equal to 58 x 6 x 58 x 6 x 5 k = 05 Which expression is equivalent to (a n)? a + 7 a + 0 a 0a + 7 a 0a + 0 n = 06 Mrs. Hill asked her students to express the sum using sigma notation. Four different student answers were given. Which student answer is correct? 0 (k ) k = 0 (k ) k = 7 (k + ) k = 9 (k ) k = 07 Express the sum using sigma notation. 08 Which summation represents ? n n = 5 0 (n + ) n = (n ) n = (n ) n = 09 A jogger ran mile on day, and mile on day, and miles on day, and miles on day, and this pattern continued for more days. Which expression represents the total distance the jogger ran? 7 d = 7 d = 7 ()d ()d d = 7 d = d A.A.5: SERIES d 0 An auditorium has rows of seats. The first row has 8 seats, and each succeeding row has two more seats than the previous row. How many seats are in the auditorium?

45 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic What is the sum of the first 9 terms of the sequence, 0, 7,,,...? Determine the sum of the first twenty terms of the sequence whose first five terms are 5,,,,. The sum of the first eight terms of the series is, 07, 85 9, 65, 55 TRIGONOMETRY A.A.55: TRIGONOMETRIC RATIOS 5 Which ratio represents csca in the diagram below? In the diagram below of right triangle JTM, JT =, JM = 6, and m JMT = 90. In the diagram below of right triangle KTW, KW = 6, KT = 5, and m KTW = 90. What is the measure of K, to the nearest minute? ' ' 55' 56' What is the value of cotj?

46 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 In the diagram below, the length of which line segment is equal to the exact value of sin θ? 0 Find, to the nearest minute, the angle whose measure is.5 radians. What is the number of degrees in an angle whose radian measure is π? What is the radian measure of an angle whose measure is 0? 7π TO TS OR OS 8 In the right triangle shown below, what is the measure of angle S, to the nearest minute? 8 ' 8 ' 6 56' 6 9' A.M.-: RADIAN MEASURE 9 What is the radian measure of the smaller angle formed by the hands of a clock at 7 o clock? π π 5π 6 7π 6 7π 6 7π 6 7π Find, to the nearest tenth of a degree, the angle whose measure is.5 radians. What is the number of degrees in an angle whose measure is radians? 60 π π Find, to the nearest tenth, the radian measure of 6º. 6 Convert radians to degrees and express the answer to the nearest minute. 7 What is the number of degrees in an angle whose radian measure is 8π 5?

47 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 Approximately how many degrees does five radians equal? π 6 5π 9 Convert.5 radians to degrees, and express the answer to the nearest minute. In which graph is θ coterminal with an angle of 70? A.A.60: UNIT CIRCLE 0 On the unit circle shown in the diagram below, sketch an angle, in standard position, whose degree measure is 0 and find the exact value of sin 0.

48 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic If m θ = 50, which diagram represents θ drawn in standard position? If sin θ < 0 and cotθ > 0, in which quadrant does the terminal side of angle θ lie? I II III IV A.A.56, 6, 66: DETERMINING TRIGONOMETRIC FUNCTIONS 5 In the interval 0 x < 60, tanx is undefined when x equals 0º and 90º 90º and 80º 80º and 70º 90º and 70º 6 Express the product of cos 0 and sin 5 in simplest radical form. 7 If θ is an angle in standard position and its terminal side passes through the point (, ), find the exact value of cscθ. A.A.60: FINDING THE TERMINAL SIDE OF AN ANGLE An angle, P, drawn in standard position, terminates in Quadrant II if cosp < 0 and cscp < 0 sin P > 0 and cosp > 0 cscp > 0 and cotp < 0 tanp < 0 and secp > 0 8 The value of tan6 to the nearest ten-thousandth is Which expression, when rounded to three decimal places, is equal to.55? sec 5π 6 tan(9 0 ) sin π 5 csc( 8 ) 0 The value of csc8 rounded to four decimal places is

49 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.6: USING INVERSE TRIGONOMETRIC FUNCTIONS What is the principal value of cos ? If sin 5 8 = A, then sin A = 5 8 sin A = 8 5 cosa = 5 8 cosa = 8 5 In the diagram below of a unit circle, the ordered pair, represents the point where the terminal side of θ intersects the unit circle. If tan Arccos k =, then k is 5 If sin A = 7 5 tana equals and A terminates in Quadrant IV, What is m θ? A.A.57: REFERENCE ANGLES 6 Expressed as a function of a positive acute angle, cos( 05 ) is equal to cos 55 cos 55 sin 55 sin 55 5

50 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.6: ARC LENGTH 7 A circle has a radius of inches. In inches, what is the length of the arc intercepted by a central angle of radians? π 8π 8 8 A circle is drawn to represent a pizza with a inch diameter. The circle is cut into eight congruent pieces. What is the length of the outer edge of any one piece of this circle? π π π π 9 Circle O shown below has a radius of centimeters. To the nearest tenth of a centimeter, determine the length of the arc, x, subtended by an angle of 8 50'. A.A.58-59: COFUNCTION AND RECIPROCAL TRIGONOMETRIC FUNCTIONS 50 If A is acute and tana =, then cota = cota = cot(90 A) = cot(90 A) = 5 The expression sin θ + cos θ is equivalent to sin θ cos θ sin θ sec θ csc θ 5 Express cos θ(sec θ cos θ), in terms of sin θ. 5 If sec(a + 5) =csc(a), find the smallest positive value of a, in degrees. 5 Express cotxsin x as a single trigonometric secx function, in simplest form, for all values of x for which it is defined. 55 Show that secθ sin θ cotθ = is an identity. 56 The expression cotx cscx sinx cosx tanx secx is equivalent to 57 Express the exact value of csc60, with a rational denominator. 6

51 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.67: PROVING TRIGONOMETRIC IDENTITIES 58 Starting with sin A + cos A =, derive the formula tan A + = sec A. 59 Which expression always equals? cos x sin x cos x + sin x cosx sinx cosx + sinx A.A.76: ANGLE SUM AND DIFFERENCE IDENTITIES 60 The expression cosxcosx + sin xsin x is equivalent to sinx sin 7x cosx cos 7x 6 If tana = and sin B = 5 and angles A and B are in Quadrant I, find the value of tan(a + B). 6 Express as a single fraction the exact value of sin Given angle A in Quadrant I with sin A = and angle B in Quadrant II with cosb =, what is the 5 value of cos(a B)? The value of sin(80 + x) is equivalent to sin x sin(90 x) sinx sin(90 x) 65 The expression sin(θ + 90) is equivalent to sin θ cos θ sinθ cosθ 66 If sin x = sin y = a and cosx = cosy = b, then cos(x y) is b a b + a b a b + a A.A.77: DOUBLE AND HALF ANGLE IDENTITIES 67 The expression cos θ cosθ is equivalent to sin θ sin θ cos θ + cos θ 68 If sin A = where 0 <A < 90, what is the value of sin A?

52 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 69 What is a positive value of tan x, when sin x = 0.8? If sin A =, what is the value of cosa? A.A.68: TRIGONOMETRIC EQUATIONS 7 What are the values of θ in the interval 0 θ < 60 that satisfy the equation tanθ = 0? 60º, 0º 7º, 5º 7º, 08º, 5º, 88º 60º, 0º, 0º, 00º 7 Find all values of θ in the interval 0 θ < 60 that satisfy the equation sin θ = sin θ. 7 Solve the equation tanc = tanc algebraically for all values of C in the interval 0 C < What is the solution set for cosθ = 0 in the interval 0 θ < 60? {0, 50 } {60, 0 } {0, 0 } {60, 00 } 75 What is the solution set of the equation secx = when 0 x < 60? {5, 5, 5, 5 } {5, 5 } {5, 5 } {5, 5 } 76 Find, algebraically, the measure of the obtuse angle, to the nearest degree, that satisfies the equation 5cscθ = Solve algebraically for all exact values of x in the interval 0 x < π: sin x + 5 sin x = 78 Solve secx = 0 algebraically for all values of x in 0 x < 60. A.A.69: PROPERTIES OF TRIGONOMETRIC FUNCTIONS 79 What is the period of the function y = sin x π? π 6π 80 What is the period of the function f(θ) = cosθ? π π π π 8 Which equation represents a graph that has a period of π? y = sin x y = sinx y = sin x y = sinx 8

53 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 What is the period of the graph y = sin 6x? π 6 π π 6π 8 Write an equation for the graph of the trigonometric function shown below. A.A.7: IDENTIFYING THE EQUATION OF A TRIGONOMETRIC GRAPH 8 Which equation is graphed in the diagram below? 85 Which equation is represented by the graph below? π y = cos 0 x + 8 π y = cos 5 x + 5 π y = cos 0 x + 8 π y = cos 5 x + 5 y = cosx y = sinx y = cos π x y = sin π x 9

54 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 86 Which equation represents the graph below? A.A.65, 70-7: GRAPHING TRIGONOMETRIC FUNCTIONS 87 Which graph represents the equation y = cos x? y = sinx y = sin x y = cosx y = cos x 50

55 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 88 Which graph shows y = cos x? 89 Which graph represents one complete cycle of the equation y = sin πx? 5

56 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 90 Which equation is represented by the graph below? 9 Which is a graph of y = cotx? y = cotx y = cscx y = secx y = tanx 9 Which equation is sketched in the diagram below? A.A.6: DOMAIN AND RANGE y = cscx y = secx y = cotx y = tanx 9 The function f(x) = tanx is defined in such a way that f (x) is a function. What can be the domain of f(x)? {x 0 x π} {x 0 x π} x π < x < π x π < x < π 5

57 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 9 In which interval of f(x) = cos(x) is the inverse also a function? π < x < π π x π 0 x π π x π 95 Which statement regarding the inverse function is true? A domain of y = sin x is [0, π]. The range of y = sin x is [, ]. A domain of y = cos x is (, ). The range of y = cos x is [0, π]. A.A.7: USING TRIGONOMETRY TO FIND AREA 96 In ABC, m A = 0, b = 0, and c = 8. What is the area of ABC to the nearest square inch? Two sides of a parallelogram are feet and 0 feet. The measure of the angle between these sides is 57. Find the area of the parallelogram, to the nearest square foot. 98 The sides of a parallelogram measure 0 cm and 8 cm. One angle of the parallelogram measures 6 degrees. What is the area of the parallelogram, to the nearest square centimeter? In parallelogram BFLO, OL =.8, LF = 7., and m O = 6. If diagonal BL is drawn, what is the area of BLF? The two sides and included angle of a parallelogram are 8,, and 60. Find its exact area in simplest form. 0 The area of triangle ABC is. If AB = 8 and m B = 6, the length of BC is approximately A ranch in the Australian Outback is shaped like triangle ACE, with m A =, m E = 0, and AC = 5 miles. Find the area of the ranch, to the nearest square mile. 0 Find, to the nearest tenth of a square foot, the area of a rhombus that has a side of 6 feet and an angle of Two sides of a triangular-shaped sandbox measure feet and feet. If the angle between these two sides measures 55, what is the area of the sandbox, to the nearest square foot? 8 7 A.A.7: LAW OF SINES 05 In ABC, m A =, a =, and b = 0. Find the measures of the missing angles and side of ABC. Round each measure to the nearest tenth. 5

58 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 06 The diagram below shows the plans for a cell phone tower. A guy wire attached to the top of the tower makes an angle of 65 degrees with the ground. From a point on the ground 00 feet from the end of the guy wire, the angle of elevation to the top of the tower is degrees. Find the height of the tower, to the nearest foot. 07 As shown in the diagram below, fire-tracking station A is 00 miles due west of fire-tracking station B. A forest fire is spotted at F, on a bearing 7 northeast of station A and 5 northeast of station B. Determine, to the nearest tenth of a mile, the distance the fire is from both station A and station B. [N represents due north.] A.A.75: LAW OF SINES-THE AMBIGUOUS CASE 09 In ABC, m A = 7, a = 59., and c = 60.. What are the two possible values for m C, to the nearest tenth? 7.7 and and and and How many distinct triangles can be formed if m A = 5, a = 0, and b =? 0 Given ABC with a = 9, b = 0, and m B = 70, what type of triangle can be drawn? an acute triangle, only an obtuse triangle, only both an acute triangle and an obtuse triangle neither an acute triangle nor an obtuse triangle In MNP, m = 6 and n = 0. Two distinct triangles can be constructed if the measure of angle M is In PQR, p equals rsin P sin Q rsin P sin R rsin R sin P qsin R sin Q In KLM, KL = 0, LM =, and m K = 0. The measure of M? must be between 0 and 90 must equal 90 must be between 90 and 80 is ambiguous In DEF, d = 5, e = 8, and m D =. How many distinct triangles can be drawn given these measurements? 0 5

59 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.7: LAW OF COSINES 5 In a triangle, two sides that measure 6 cm and 0 cm form an angle that measures 80. Find, to the nearest degree, the measure of the smallest angle in the triangle. 6 In ABC, a =, b = 5, and c = 7. What is m C? In ABC, a = 5, b =, and c =, as shown in the diagram below. What is the m C, to the nearest degree? 0 The measures of the angles between the resultant and two applied forces are 60 and 5, and the magnitude of the resultant is 7 pounds. Find, to the nearest pound, the magnitude of each applied force. Two forces of 0 pounds and 8 pounds act on an object. The angle between the two forces is 65. Find the magnitude of the resultant force, to the nearest pound. Using this answer, find the measure of the angle formed between the resultant and the smaller force, to the nearest degree. CONICS A.A.7-9: EQUATIONS OF CIRCLES The equation x + y x + 6y + = 0 is equivalent to (x ) + (y + ) = (x ) + (y + ) = 7 (x + ) + (y + ) = 7 (x + ) + (y + ) = Two sides of a parallelogram measure 7 cm and cm. The included angle measures 8. Find the length of the longer diagonal of the parallelogram, to the nearest centimeter. A.A.7: VECTORS What are the coordinates of the center of a circle whose equation is x + y 6x + 6y + 5 = 0? ( 8, ) ( 8, ) (8, ) (8, ) What is the equation of the circle passing through the point (6, 5) and centered at (, )? (x 6) + (y 5) = 8 (x 6) + (y 5) = 90 (x ) + (y + ) = 8 (x ) + (y + ) = 90 9 Two forces of 5 newtons and 85 newtons acting on a body form an angle of 55. Find the magnitude of the resultant force, to the nearest hundredth of a newton. Find the measure, to the nearest degree, of the angle formed between the resultant and the larger force. 55

60 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 5 Write an equation of the circle shown in the graph below. 7 Which equation represents the circle shown in the graph below that passes through the point (0, )? 6 A circle shown in the diagram below has a center of ( 5, ) and passes through point (, 7). (x ) + (y + ) = 6 (x ) + (y + ) = 8 (x + ) + (y ) = 6 (x + ) + (y ) = 8 8 Write an equation of the circle shown in the diagram below. Write an equation that represents the circle. 56

61 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 9 Which equation is represented by the graph below? (x ) + (y + ) = 5 (x + ) + (y ) = 5 (x ) + (y + ) = (x + ) + (y ) = 57

62 ID: A Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Answer Section ANS: Controlled experiment because Howard is comparing the results obtained from an experimental sample against a control sample. PTS: REF: 0800a STA: A.S. TOP: Analysis of Data ANS: PTS: REF: 07a STA: A.S. TOP: Analysis of Data ANS: PTS: REF: 060a STA: A.S. TOP: Analysis of Data ANS: PTS: REF: 060a STA: A.S. TOP: Analysis of Data 5 ANS: PTS: REF: 006a STA: A.S. TOP: Analysis of Data 6 ANS: Students entering the library are more likely to spend more time studying, creating bias. PTS: REF: fall090a STA: A.S. TOP: Analysis of Data 7 ANS: PTS: REF: 00a STA: A.S. TOP: Analysis of Data 8 ANS: PTS: REF: 060a STA: A.S. TOP: Analysis of Data 9 ANS: PTS: REF: 06a STA: A.S. TOP: Average Known with Missing Data 0 ANS: k k + = k + 6 k + = k + 6 = k + k = 8 k = PTS: REF: 06a STA: A.S. TOP: Average Known with Missing Data ANS: PTS: REF: fall09a STA: A.S. TOP: Dispersion KEY: range, quartiles, interquartile range, variance

63 ID: A ANS: 7. PTS: REF: 0609a STA: A.S. TOP: Dispersion KEY: basic, group frequency distributions ANS: σ x =.9. x = 0. There are 8 scores between 5. and 5.9. PTS: REF: 067a STA: A.S. TOP: Dispersion KEY: advanced ANS: Ordered, the heights are 7, 7, 7, 7, 7, 75, 78, 79, 79, 8. Q = 7 and Q = = 7. PTS: REF: 0a STA: A.S. TOP: Dispersion KEY: range, quartiles, interquartile range, variance 5 ANS: σ x scores are within a population standard deviation of the mean. Q Q = 7 = x 8. PTS: REF: 068a STA: A.S. TOP: Dispersion KEY: advanced 6 ANS: Q =.5 and Q = = 7. PTS: REF: 00a STA: A.S. TOP: Dispersion KEY: range, quartiles, interquartile range, variance 7 ANS: PTS: REF: 067a STA: A.S.6 TOP: Regression 8 ANS: y =.00x.98,,009. y =.00(5) PTS: REF: fall098a STA: A.S.7 TOP: Power Regression 9 ANS: y = 0.596(.586) x PTS: REF: 080a STA: A.S.7 TOP: Exponential Regression 0 ANS: y = 7.05(.509) x. y = 7.05(.509) 8 PTS: REF: 08a STA: A.S.7 TOP: Exponential Regression ANS: y = 80.77(0.95) x PTS: REF: 06a STA: A.S.7 TOP: Exponential Regression

64 ID: A ANS: y = 5.98(.65) x. 5.98(.65) PTS: REF: 07a STA: A.S.7 TOP: Exponential Regression ANS: y = 0.88(.6) x PTS: REF: 069a STA: A.S.7 TOP: Exponential Regression ANS: PTS: REF: 060a STA: A.S.8 TOP: Correlation Coefficient 5 ANS: () shows the strongest linear relationship, but if r < 0, b < 0. The Regents announced that a correct solution was not provided for this question and all students should be awarded credit. PTS: REF: 0a STA: A.S.8 TOP: Correlation Coefficient 6 ANS:. PTS: REF: 065a STA: A.S.8 TOP: Correlation Coefficient 7 ANS: Since the coefficient of t is greater than 0, r > 0. PTS: REF: 00a STA: A.S.8 TOP: Correlation Coefficient 8 ANS: PTS: REF: 066a STA: A.S.8 TOP: Correlation Coefficient 9 ANS: PTS: REF: fall095a STA: A.S.5 TOP: Normal Distributions KEY: interval 0 ANS: 68% 50 = PTS: REF: 080a STA: A.S.5 TOP: Normal Distributions KEY: predict

65 ID: A ANS: 68% of the students are within one standard deviation of the mean. 6% of the students are more than one standard deviation above the mean. PTS: REF: 0a STA: A.S.5 TOP: Normal Distributions KEY: percent ANS: no. over 0 is more than standard deviation above the mean PTS: REF: 069a STA: A.S.5 TOP: Normal Distributions KEY: predict ANS:.% + 9.% = 5.% PTS: REF: 0a STA: A.S.5 TOP: Normal Distributions KEY: probability ANS: x ± σ 5 ± 75 PTS: REF: 007a STA: A.S.5 TOP: Normal Distributions KEY: interval 5 ANS: Top 6.7% =.5 s.d. + σ =.5(0) = 7 PTS: REF: 00a STA: A.S.5 TOP: Normal Distributions KEY: predict 6 ANS: Less than 60 inches is below.5 standard deviations from the mean PTS: REF: 068a STA: A.S.5 TOP: Normal Distributions KEY: predict 7 ANS: PTS: REF: fall095a STA: A.S.0 TOP: Permutations 8 ANS: No. TENNESSEE: 9 P 9!!! = 6, =, 780. VERMONT: 7 P 7 = 5, 00 PTS: REF: 0608a STA: A.S.0 TOP: Permutations 9 ANS: 9,96,800. P!! = 79, 00, 600 = 9, 96, 800 PTS: REF: 0805a STA: A.S.0 TOP: Permutations

66 ID: A 0 ANS: = 8. The first digit cannot be 0 or 5. The second digit cannot be 5 or the same as the first digit. The third digit cannot be 5 or the same as the first or second digit. PTS: REF: 05a STA: A.S.0 TOP: Permutations ANS: 6 P 6!! = 70 = 60 PTS: REF: 0a STA: A.S.0 TOP: Permutations ANS: 0 P 0!!! =, 68, = 50, 00 PTS: REF: 060a STA: A.S.0 TOP: Permutations ANS: PTS: REF: 009a STA: A.S.0 TOP: Permutations ANS:!!!= 8 PTS: REF: 065a STA: A.S.0 TOP: Permutations 5 ANS: = 6, 5 5 C 8 PTS: REF: 080a STA: A.S. TOP: Combinations 6 ANS: 0 C = 0 PTS: REF: 06a STA: A.S. TOP: Combinations 7 ANS: = 5, 0 5 C 0 PTS: REF: 0a STA: A.S. TOP: Combinations 8 ANS: 5 C 5 =, C 5 = 5 C 0 = 5, 0. 5 C 5 =, 68, 760. PTS: REF: 067a STA: A.S. TOP: Combinations 9 ANS: PTS: REF: 06007a STA: A.S.9 TOP: Differentiating Permutations and Combinations 50 ANS: PTS: REF: 07a STA: A.S.9 TOP: Differentiating Permutations and Combinations 5 ANS: PTS: REF: 00a STA: A.S.9 TOP: Differentiating Permutations and Combinations 5 ANS: PTS: REF: 067a STA: A.S.9 TOP: Differentiating Permutations and Combinations 5 ANS: PTS: REF: 07a STA: A.S.9 TOP: Differentiating Permutations and Combinations 5

67 ID: A 5 ANS:,00. PTS: REF: fall095a STA: A.S. TOP: Sample Space 55 ANS: C 5 C = 0 = 0 PTS: REF: 06a STA: A.S. TOP: Combinations 56 ANS: π + π π = π π = PTS: REF: 008a STA: A.S. TOP: Geometric Probability 57 ANS: C C C PTS: REF: 0606a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 58 ANS: 6.%. 0 C C C PTS: REF: 0808a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 59 ANS: C C C PTS: REF: 08a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 6

68 ID: A 60 ANS: 5. 5 C 5 C 5 C 5 0 = 0 = 0 = PTS: REF: 068a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 6 ANS: C = 5 5 PTS: REF: 0a STA: A.S.5 TOP: Binomial Probability KEY: spinner 6 ANS: PTS: REF: 06a STA: A.S.5 TOP: Binomial Probability KEY: modeling 6 ANS: 7 C = = PTS: REF: 065a STA: A.S.5 TOP: Binomial Probability KEY: exactly 6 ANS: 5 C C PTS: REF: 07a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 65 ANS: 5 C C C PTS: REF: 068a STA: A.S.5 TOP: Binomial Probability KEY: at least or at most 66 ANS: a + 6 = a 0. a + 6 = a = a = a = 8 = 8 0 PTS: REF: 006a STA: A.A. TOP: Absolute Value Equations 7

69 ID: A 67 ANS: 6x 7 5 6x x 6x 7 5 6x x PTS: REF: fall0905a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 68 ANS: 6 x < 5 6 x >5 6 x > 5 or 6 x < 5 > x or < x. PTS: REF: 067a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 69 ANS: x 5 > x 5 > x > 8 x > or x 5 < x 5 < x < x < PTS: REF: 0609a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 70 ANS: x 7 or x 7 x x x 0 x 5 PTS: REF: 0a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 7 ANS: x > 5. x < 5 x > 6 x > x > x < PTS: REF: 0607a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 8

70 ID: A 7 ANS: x + 5 < x < 8 x > x + 5 > x > 8 x <.5 < x <.5 PTS: REF: 0a STA: A.A. TOP: Absolute Value Inequalities 7 ANS: x > 5 or x < 5 x > 8 x > x < x < PTS: REF: 060a STA: A.A. TOP: Absolute Value Inequalities 7 ANS: Sum b a = 5. Product c a = 5 PTS: REF: 0600a STA: A.A.0 TOP: Roots of Quadratics 75 ANS: sum: b a = 6 =. product: c a = 6 = PTS: REF: 009a STA: A.A.0 TOP: Roots of Quadratics 76 ANS: x x + 6 = 0. Sum b a =. Product c a = 6 = PTS: REF: 09a STA: A.A.0 TOP: Roots of Quadratics 77 ANS: Sum b a =. Product c a = PTS: REF: 068a STA: A.A.0 TOP: Roots of Quadratics 78 ANS: x 7x 5 = 0 c a = 5 PTS: REF: 06a STA: A.A.0 TOP: Roots of Quadratics 79 ANS: S = b a = ( ) =. P = c a = 8 = PTS: REF: fall09a STA: A.A. TOP: Roots of Quadratics KEY: basic 9

71 ID: A 80 ANS: b a = 6 =. c a = = PTS: REF: 0a STA: A.A. TOP: Roots of Quadratics KEY: basic 8 ANS: x 6x 7 = 0, b a = 6. c = 7. If a = then b = 6 and c = 7 a PTS: REF: 060a STA: A.A. TOP: Roots of Quadratics KEY: basic 8 ANS: sum of the roots, b a = ( 9) = 9. product of the roots, c a = PTS: REF: 0608a STA: A.A. TOP: Roots of Quadratics KEY: basic 8 ANS: b a = ( ) =. If the sum is, the roots must be 7 and. PTS: REF: 08a STA: A.A. TOP: Roots of Quadratics KEY: advanced 8 ANS: 6x x x = x(x + x 6) = x(x + )(x ) PTS: REF: fall097a STA: A.A.7 TOP: Factoring Polynomials KEY: single variable 85 ANS: x + 0x x = x (6x + 5x 6) = x (x + )(x ) PTS: REF: 06008a STA: A.A.7 TOP: Factoring Polynomials KEY: single variable 86 ANS: 0ax ax 5a = a(0x x 5) = a(5x + )(x 5) PTS: REF: 0808a STA: A.A.7 TOP: Factoring Polynomials KEY: multiple variables 87 ANS: t 8 75t = t (t 5) = t (t + 5)(t 5) PTS: REF: 06a STA: A.A.7 TOP: Factoring the Difference of Perfect Squares KEY: binomial 0

72 ID: A 88 ANS: x + x x x (x + ) (x + ) (x )(x + ) (x + )(x )(x + ) PTS: REF: 06a STA: A.A.7 TOP: Factoring by Grouping 89 ANS: x 5x 8x + 80 x (x 5) 6(x 5) (x 6)(x 5) (x + )(x )(x 5) PTS: REF: 07a STA: A.A.7 TOP: Factoring by Grouping 90 ANS: x (x + ) (x + ) (x )(x + ) (x + )(x )(x + ) PTS: REF: 06a STA: A.A.7 TOP: Factoring by Grouping 9 ANS: ± ( ) ()( 9) () = ± 5 = ± 5 PTS: REF: 06009a STA: A.A.5 TOP: Quadratics with Irrational Solutions 9 ANS: 7 ± 7 ()( ) () = 7 ± 7 PTS: REF: 08009a STA: A.A.5 TOP: Quadratics with Irrational Solutions 9 ANS: ± ( ) (6)( ) (6) = ± 76 = ± 9 = ± 9 = ± 9 6 PTS: REF: 0a STA: A.A.5 TOP: Quadratics with Irrational Solutions

73 ID: A 9 ANS: 60 = 6t + 5t = 6t + 5t + 5 t = 5 ± 5 ( 6)(5) ( 6) 5 ± PTS: REF: 06a STA: A.A.5 TOP: Quadratics with Irrational Solutions 95 ANS: b ac = 0 k ()() = 0 k 6 = 0 (k + )(k ) = 0 k = ± PTS: REF: 0608a STA: A.A. TOP: Using the Discriminant KEY: determine equation given nature of roots 96 ANS: b ac = (9)( ) = 9 + = 5 PTS: REF: 0806a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation 97 ANS: b ac = ( 0) ()(5) = = 0 PTS: REF: 00a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation 98 ANS: PTS: REF: 0a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation 99 ANS: b ac = ( 9) ()() = 8 = 9 PTS: REF: 0a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation 00 ANS: ( 5) ()(0) = 5 PTS: REF: 06a STA: A.A. TOP: Using the Discriminant KEY: determine equation given nature of roots

74 ID: A 0 ANS: ± 7. x x + = 0 x 6x + = 0 x 6x = x 6x + 9 = + 9 (x ) = 7 x = ± 7 x = ± 7 PTS: REF: fall096a STA: A.A. TOP: Completing the Square 0 ANS: x + = 6x x 6x = x 6x + 9 = + 9 (x ) = 7 PTS: REF: 06a STA: A.A. TOP: Completing the Square 0 ANS: PTS: REF: 06a STA: A.A. TOP: Completing the Square 0 ANS: (x + ) = 9 x + = ± 9 x = ± i PTS: REF: 008a STA: A.A. TOP: Completing the Square 05 ANS: PTS: REF: 0608a STA: A.A. TOP: Completing the Square 06 ANS: y x x 6 y (x )(x + ) PTS: REF: 0607a STA: A.A. TOP: Quadratic Inequalities KEY: two variables

75 ID: A 07 ANS: x x 0 > 0 (x 5)(x + ) > 0 x 5 > 0 and x + > 0 x > 5 and x > x > 5 or x 5 < 0 and x + < 0 x < 5 and x < x < PTS: REF: 05a STA: A.A. TOP: Quadratic Inequalities KEY: one variable 08 ANS: x < or x > 5. x x 5 > 0. x 5 > 0 and x + > 0 or x 5 < 0 and x + < 0 (x 5)(x + ) > 0 x > 5 and x > x > 5 x < 5 and x < x < PTS: REF: 08a STA: A.A. TOP: Quadratic Inequalities KEY: one variable 09 ANS: x x 6 = x 6 x x = 0 x(x ) = 0 x = 0, PTS: REF: 0805a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations 0 ANS: 9, and,. y = x + 5 y = x + 7x. x + 7x = x + 5 x + 6x 9 = 0 (x + 9)(x ) = 0 x = 9 and x = y = = and y = + 5 = PTS: 6 REF: 069a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations

76 ID: A ANS: x + y = y = 5 y = x + 5 y = 0 (x + ) + ( x + 5 ) = 5 x + 6x x x + = 5 x + x 0 = 0 x + x 0 = 0 (x + 5)(x ) = 0 x = 5, PTS: REF: 00a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations ANS: x = y. y (y) + = 0 y 9y = 8y = y = y = ±. x = ( ) = 6 PTS: REF: 06a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations ANS: x(x + ) = 0 x + x 0 = 0 (x + 5)(x ) = 0 x = 5, PTS: REF: 0a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations ANS: 9 x x +. x = x x = 9 x x x + = 9 x x + PTS: REF: 080a STA: A.N. TOP: Operations with Polynomials 5 ANS: PTS: REF: 0a STA: A.N. TOP: Operations with Polynomials 5

77 ID: A 6 ANS: 6y 7 0 y 5 y. y y y + 5 = 6y + 0 y y 5 y = 6y 7 0 y 5 y PTS: REF: 068a STA: A.N. TOP: Operations with Polynomials 7 ANS: The binomials are conjugates, so use FL. PTS: REF: 006a STA: A.N. TOP: Operations with Polynomials 8 ANS: The binomials are conjugates, so use FL. PTS: REF: 060a STA: A.N. TOP: Operations with Polynomials 9 ANS: PTS: REF: 0a STA: A.N. TOP: Operations with Polynomials 0 ANS: PTS: REF: 0607a STA: A.N. TOP: Operations with Polynomials ANS: ( ) = 9 8 = 8 9 PTS: REF: 0600a STA: A.N. TOP: Negative and Fractional Exponents ANS: 6n < n. Flip sign when multiplying each side of the inequality by n, since a negative number. 6 n < n 6 > PTS: REF: 06a STA: A.N. TOP: Negative and Fractional Exponents ANS: w 5 w 9 = (w ) = w PTS: REF: 080a STA: A.A.8 TOP: Negative and Fractional Exponents ANS: PTS: REF: 006a STA: A.A.8 TOP: Negative and Fractional Exponents 5 ANS: PTS: REF: 00a STA: A.A.8 TOP: Negative and Fractional Exponents 6 ANS: PTS: REF: 060a STA: A.A.8 TOP: Negative and Fractional Exponents 7 ANS: PTS: REF: fall09a STA: A.A.9 TOP: Negative and Fractional Exponents 6

78 ID: A 8 ANS: x x = x x = x x x = (x ) x x = x PTS: REF: 0808a STA: A.A.9 TOP: Negative Exponents 9 ANS: x y 9. x y 5 (x y 7 ) y 5 (x y 7 ) = x = y 5 (x 6 y ) x = x 6 y 9 x = x y 9 PTS: REF: 06a STA: A.A.9 TOP: Negative Exponents 0 ANS: x + x + = x + + x x + = x x + = x PTS: REF: 0a STA: A.A.9 TOP: Negative Exponents ANS: PTS: REF: 060a STA: A.A.9 TOP: Negative Exponents ANS: PTS: REF: 06a STA: A.A.9 TOP: Negative Exponents ANS:, PTS: REF: fall09a STA: A.A. TOP: Evaluating Exponential Expressions ANS: e ln = e ln = e ln 8 = 8 PTS: REF: 06a STA: A.A. TOP: Evaluating Exponential Expressions 5 ANS: A = 750e (0.0)(8) 95 PTS: REF: 069a STA: A.A. TOP: Evaluating Exponential Expressions 6 ANS: = 5000(.0075) PTS: REF: 00a STA: A.A. TOP: Evaluating Exponential Expressions 7 ANS: PTS: REF: 066a STA: A.A. TOP: Evaluating Exponential Expressions 7

79 ID: A 8 ANS: 8 = 6 PTS: REF: fall0909a STA: A.A.8 TOP: Evaluating Logarithmic Expressions 9 ANS: PTS: REF: 0a STA: A.A.8 TOP: Evaluating Logarithmic Expressions 0 ANS: y = 0 PTS: REF: 060a STA: A.A.5 TOP: Graphing Exponential Functions ANS: PTS: REF: 0a STA: A.A.5 TOP: Graphing Exponential Functions ANS: PTS: REF: 00a STA: A.A.5 TOP: Graphing Exponential Functions ANS: f (x) = log x PTS: REF: fall096a STA: A.A.5 TOP: Graphing Logarithmic Functions ANS: PTS: REF: 06a STA: A.A.5 TOP: Graphing Logarithmic Functions 5 ANS: PTS: REF: 0a STA: A.A.5 TOP: Graphing Logarithmic Functions 8

80 ID: A 6 ANS: logx ( logy + logz) = logx logy logz = log x PTS: REF: 0600a STA: A.A.9 TOP: Properties of Logarithms 7 ANS: PTS: REF: 060a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 8 ANS: logx = loga + loga logx = log6a logx = log6 logx = + loga log6 + loga logx = log6 + loga y z PTS: REF: 0a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 9 ANS: PTS: REF: 0607a STA: A.A.9 TOP: Properties of Logarithms KEY: antilogarithms 50 ANS: log9 log0 log log(0 ) log (log0 + log) b ( + a) b a PTS: REF: 06a STA: A.A.9 TOP: Properties of Logarithms KEY: expressing logs algebraically 5 ANS: logm = log + logm = log + logm PTS: REF: 06a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 5 ANS: logx = log + logx = log + logx PTS: REF: 066a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 9

81 ID: A 5 ANS: log (5x) = log (5x) = 5x = 5x = 8 x = 8 5 PTS: REF: fall09a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 5 ANS: x =, log x + x x + = x x + x x x + x x = (x + ) = x + 6x + 9 x + x = x + 6x + 9x 0 = 6x + 8x + 0 = x + x + 0 = (x + )(x + ) x =, PTS: 6 REF: 0809a STA: A.A.8 TOP: Logarithmic Equations KEY: basic 55 ANS: ln(t T 0 ) = kt ln(t 68) = 0.0(0) ln(50 68) = k() k +.78 k 0.0 ln(t 68) =.678 T T 08 PTS: 6 REF: 09a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 56 ANS: x = 5 = 65 PTS: REF: 0606a STA: A.A.8 TOP: Logarithmic Equations KEY: basic 0

82 ID: A 57 ANS: 800. x =.5 =. y = 5 y = 5 = 5. x y = 5 = 800 PTS: REF: 07a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 58 ANS: (x + ) = 7x x + 8x + 6 = 7x x 9x + 0 = 0 (x )(x 5) = 0 x =, 5 PTS: REF: 06a STA: A.A.8 TOP: Logarithmic Equations KEY: basic 59 ANS: x = 7 x = 8 x = 8 x = PTS: REF: 069a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 60 ANS: log (x + ) (x + )(x + 5) = 6 is extraneous (x + ) = (x + )(x + 5) x + 6x + 9 = x + x + 5 x + 7x + 6 = 0 (x + 6)(x + ) = 0 x = PTS: 6 REF: 09a STA: A.A.8 TOP: Logarithmic Equations KEY: applying properties of logarithms

83 ID: A 6 ANS: (5x ) = 5x = 6 5x = 65 x = PTS: REF: 06a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 6 ANS: = 5000e.075t ln.075 = e.075t ln = ln e.075t. t =.075t ln e.075 PTS: REF: 067a STA: A.A.6 TOP: Exponential Growth 6 ANS: 0 = 0() = () t 60 log = log() log = tlog log = t log 00 = t t 60 t 60 PTS: REF: 005a STA: A.A.6 TOP: Exponential Growth

84 ID: A 6 ANS: 0700 = 50e t 6 = e t ln 6 = ln e t ln 6 = tln e ln 6 = t. t PTS: REF: 0a STA: A.A.6 TOP: Exponential Growth 65 ANS: 000 = 500e.05t = e.05t ln = ln e.05t ln.05.9 t =.05t ln e.05 PTS: REF: 06a STA: A.A.6 TOP: Exponential Growth 66 ANS: x + x = 6. x + 8x = 6 ( ) x +x = 6 x + 8x = 6 x + 8x + 6 = 0 x + x + = 0 (x + )(x + ) = 0 x = x = PTS: REF: 0605a STA: A.A.7 TOP: Exponential Equations KEY: common base shown 67 ANS: 9 x + = 7 x +. ( ) x + = ( ) x + 6x + = x + 6 6x + = x + 6 x = x = PTS: REF: 08008a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown

85 ID: A 68 ANS: 6 x + = 6 x + ( ) x + = ( ) x + x + 6 = x + 6 x = 0 PTS: REF: 08a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 69 ANS: x + 5 = 8 x. x + 5 = x + 0 = 9x x + 0 = 9x 0 = 5x = x x PTS: REF: 0605a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 70 ANS: 8 x + x = 7 5x x + x = x + 8x = 5x x + 8x 5x = 0 x(x + 8x 5) = 0 x(x )(x + 5) = 0 5x x = 0,, 5 PTS: 6 REF: 069a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown

86 ID: A 7 ANS: 8 k + = k ( ) k + = ( ) k 9k + = k 9k + = k 5k = k = 5. PTS: REF: 009a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 7 ANS: 5 C (x) ( ) = 0 9x 8 = 70x PTS: REF: fall099a STA: A.A.6 TOP: Binomial Expansions 7 ANS: x 5 80x + 80x 0x + 0x. 5 C 0 (x) 5 ( ) 0 = x 5. 5 C (x) ( ) = 80x. 5 C (x) ( ) = 80x. 5 C (x) ( ) = 0x. 5 C (x) ( ) = 0x. 5 C 5 (x) 0 ( ) 5 = PTS: REF: 06a STA: A.A.6 TOP: Binomial Expansions 7 ANS: 9 C a6 ( b) = 576a 6 b PTS: REF: 066a STA: A.A.6 TOP: Binomial Expansions 75 ANS: C (x ) ( y) = 6x y PTS: REF: 05a STA: A.A.6 TOP: Binomial Expansions 76 ANS: 6 C x ( y) = 0 x 8 8y = 0x y PTS: REF: 065a STA: A.A.6 TOP: Binomial Expansions 77 ANS: 8 C x8 ( ) = 56x 5 ( 8) = 8x 5 PTS: REF: 008a STA: A.A.6 TOP: Binomial Expansions 5

87 ID: A 78 ANS: ±,. 8x + x 8x 9 = 0 x (x + ) 9(x + ) = 0 (x 9)(x + ) = 0 x 9 = 0 or x + = 0 (x + )(x ) = 0 x = x = ± PTS: REF: fall097a STA: A.A.6 TOP: Solving Polynomial Equations 79 ANS: x + x x = 0 x(x + x ) = 0 x(x + )(x ) = 0 x = 0,, PTS: REF: 00a STA: A.A.6 TOP: Solving Polynomial Equations 80 ANS: x 5 8x = 0 x(x 6) = 0 x(x + )(x ) = 0 x(x + )(x + )(x ) = 0 PTS: REF: 06a STA: A.A.6 TOP: Solving Polynomial Equations 8 ANS: x + x + x + 6x = 0 x(x + x + x + 6) = 0 x(x (x + ) + (x + )) = 0 x(x + )(x + ) = 0 x = 0, ±i, PTS: 6 REF: 069a STA: A.A.6 TOP: Solving Polynomial Equations 6

88 ID: A 8 ANS: x + 5x x 0 = 0 x (x + 5) (x + 5) = 0 (x )(x + 5) = 0 (x + )(x )(x + 5) = 0 x = ±, 5 PTS: REF: 067a STA: A.A.6 TOP: Solving Polynomial Equations 8 ANS: PTS: REF: 06005a STA: A.A.50 TOP: Solving Polynomial Equations 8 ANS: The roots are,,. PTS: REF: 080a STA: A.A.50 TOP: Solving Polynomial Equations 85 ANS: PTS: REF: 06a STA: A.A.50 TOP: Solving Polynomial Equations 86 ANS: ( + 5 )( 5 ) = 9 5 = PTS: REF: 0800a STA: A.N. TOP: Operations with Irrational Expressions KEY: without variables index = 87 ANS: a b PTS: REF: 0a STA: A.A. TOP: Simplifying Radicals KEY: index > 88 ANS: a 5 a = a 5 a PTS: REF: 060a STA: A.A. TOP: Simplifying Radicals KEY: index > 89 ANS: 5 x 7x = 5 x x 9x x = 5x x 6x x = x x PTS: REF: 060a STA: A.N. TOP: Operations with Radicals 7

89 ID: A 90 ANS: 6a b + (7 6)a b a 6ab + a 6ab a 6ab PTS: REF: 09a STA: A.N. TOP: Operations with Radicals 9 ANS: 7x 6x = x 6 = x = 6x PTS: REF: 0a STA: A.N. TOP: Operations with Radicals 9 ANS: ab b a 9b b + 7ab 6b = ab b 9ab b + 7ab 6b = 5ab b + 7ab 6b PTS: REF: fall098a STA: A.A. TOP: Operations with Radicals KEY: with variables index = 9 ANS: 08x 5 y 8 6xy 5 = 8x y = x y y PTS: REF: 0a STA: A.A. TOP: Operations with Radicals KEY: with variables index = 9 ANS: 5( + ) = 5( + ) 9 = 5( + ) 7 PTS: REF: fall098a STA: A.N.5 TOP: Rationalizing Denominators 95 ANS: = = = + 5 PTS: REF: 060a STA: A.N.5 TOP: Rationalizing Denominators 96 ANS: = (5 + ) 5 = 5 + PTS: REF: 066a STA: A.N.5 TOP: Rationalizing Denominators 8

90 ID: A 97 ANS: = = PTS: REF: 00a STA: A.N.5 TOP: Rationalizing Denominators 98 ANS: a b = a b b b = b ab = b ab PTS: REF: 0809a STA: A.A.5 TOP: Rationalizing Denominators KEY: index = 99 ANS: x + x + x + x + = (x + ) x + x + = x + PTS: REF: 0a STA: A.A.5 TOP: Rationalizing Denominators KEY: index = 00 ANS: x x x x + x x + x = x + x x x x = x(x + x ) x(x ) = x + x x PTS: REF: 065a STA: A.A.5 TOP: Rationalizing Denominators KEY: index = 0 ANS: PTS: REF: 0608a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 0 ANS: x + 6 = (x + ). is an extraneous solution. x + 6 = x + x + 0 = x + x 0 = (x + )(x ) x = x = PTS: REF: 06a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 9

91 ID: A 0 ANS: 7. x 5 = x 5 = x 5 = 9 x = x = 7 PTS: REF: 09a STA: A.A. TOP: Solving Radicals KEY: basic 0 ANS: 5x + 9 = (x + ). ( 5) + shows an extraneous solution. 5x + 9 = x + 6x = x + x 0 0 = (x + 5)(x ) x = 5, PTS: REF: 06a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 05 ANS: x + x = x x + x = 6x x = 5x 5x = x 5x + 0 = (x )(x ) x =, x 0 () + < 0 is extraneous PTS: 6 REF: 09a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 06 ANS: x = x x = x x + 0 = x 6x = (x )(x ) x =, PTS: REF: 0606a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 0

92 ID: A 07 ANS: PTS: REF: 060a STA: A.A.0 TOP: Fractional Exponents as Radicals 08 ANS: 5 x = x 5 = 5 x PTS: REF: 08a STA: A.A.0 TOP: Fractional Exponents as Radicals 09 ANS: 7 7 6x y 7 = 6 x y = x y PTS: REF: 0607a STA: A.A. TOP: Radicals as Fractional Exponents 0 ANS: 00 = 00 PTS: REF: 06006a STA: A.N.6 TOP: Square Roots of Negative Numbers ANS: 9 6 = i i = i PTS: REF: 060a STA: A.N.6 TOP: Square Roots of Negative Numbers ANS: PTS: REF: 0609a STA: A.N.7 TOP: Imaginary Numbers ANS: i + i = ( ) + ( i) = i PTS: REF: 0800a STA: A.N.7 TOP: Imaginary Numbers ANS: i + i 8 + i + n = 0 i + ( ) i + n = 0 + n = 0 n = PTS: REF: 068a STA: A.N.7 TOP: Imaginary Numbers 5 ANS: xi + 5yi 8 + 6xi + yi = xi + 5y 6xi + y = 7y xi PTS: REF: 0a STA: A.N.7 TOP: Imaginary Numbers 6 ANS: PTS: REF: 080a STA: A.N.8 TOP: Conjugates of Complex Numbers 7 ANS: PTS: REF: 0a STA: A.N.8 TOP: Conjugates of Complex Numbers 8 ANS: PTS: REF: 0a STA: A.N.8 TOP: Conjugates of Complex Numbers

93 ID: A 9 ANS: PTS: REF: 069a STA: A.N.8 TOP: Conjugates of Complex Numbers 0 ANS: (x + yi)(x yi) = x y i = x + y PTS: REF: 06a STA: A.N.8 TOP: Conjugates of Complex Numbers ANS: ( 7i)( 7i) = 9 i i + 9i = 9 i 9 = 0 i PTS: REF: fall090a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers ANS: (x + i) (x i) = x + xi + i (x xi + i ) = xi PTS: REF: 07a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers ANS: (i)(i) (m + i) (i)(i )(m + i) (i)( )(m + i) ( i)(m + i) mi i mi + PTS: REF: 069a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers ANS: (x + 6) x. x (x ) + 6(x ) x x x x x x + x 8 6 x (x + 6)(x ) x(x ) (x ) x (x ) ( + x)( x) (x + )(x ) (x + 6) x PTS: 6 REF: 09a STA: A.A.6 TOP: Multiplication and Division of Rationals KEY: division

94 ID: A 5 ANS: (x ) (x + )(x + ) x + (x ) = (x + )(x ) x + (x ) = (x + ) (x + ) PTS: REF: 066a STA: A.A.6 TOP: Multiplication and Division of Rationals KEY: division 6 ANS: x + 9x x ( x) = (x + )(x ) (x + )(x ) x = x PTS: REF: 0a STA: A.A.6 TOP: Multiplication and Division of Rationals KEY: Division 7 ANS: y y y = y y 6 9 y 6 = y 9 y 6 = (y ) (y ) = PTS: REF: 05a STA: A.A.6 TOP: Addition and Subtraction of Rationals 8 ANS: x no solution. x = + x x x = (x ) x = PTS: REF: fall090a STA: A.A. TOP: Solving Rationals KEY: rational solutions 9 ANS: x + x = x 9 x + + x = x 9 x + (x + ) (x + )(x ) = x + x + 6 = x = x = (x + )(x ) PTS: REF: 0806a STA: A.A. TOP: Solving Rationals KEY: rational solutions

95 ID: A 0 ANS: x = 0 x. x = 0 ± 00 ()() () = 0 ± 8 = 0 ± = 5 ± = 0x x x 0x + = 0 PTS: REF: 066a STA: A.A. TOP: Solving Rationals KEY: irrational and complex solutions ANS: 0 (x + )(x ) + (x + )(x ) (x + )(x ) = 5(x + ) (x )(x + ) 0 + x 9 = 5x + 5 x 5x + 6 = 0 (x )(x ) = 0 x = is an extraneous root. PTS: REF: 067a STA: A.A. TOP: Solving Rationals KEY: rational solutions ANS: x + 6 x 7(x ) x 0 6x + 0 = 0 6x = 0 x = 6x x x = 5 6() + 0 = 6() + 0 =, which is less than 0. If x =, 6(6) + 0 = 6 6 =, which is less than 0. x = 0. Check points such that x <, < x < 5, and x > 5. If x =, = =, which is greater than 0. If x = 6, PTS: REF: 0a STA: A.A. TOP: Rational Inequalities ANS: x x x + = x x x + 8x = (x + )(x ) x 8x (x + ) = x PTS: REF: fall090a STA: A.A.7 TOP: Complex Fractions ANS: d d + d = d 8 d d + d d = d 8 d d 5d = d 8 5 PTS: REF: 0605a STA: A.A.7 TOP: Complex Fractions

96 ID: A 5 ANS: x x 8 x x = x x x x 8 = x(x ) (x )(x + ) = x x + x PTS: REF: 0605a STA: A.A.7 TOP: Complex Fractions 6 ANS: a + b ac + b c c d b = = ac + b c cd b c cd b = ac + b cd b c c PTS: REF: 005a STA: A.A.7 TOP: Complex Fractions 7 ANS: + x 5 x x x x = x + x x 5x = x(x + ) (x 8)(x + ) = x x 8 PTS: REF: 066a STA: A.A.7 TOP: Complex Fractions 8 ANS: 6 = 9w 8 = w PTS: REF: 00a STA: A.A.5 TOP: Inverse Variation 9 ANS: 0 = 5 p 5 = 5 p 5 = p PTS: REF: 06a STA: A.A.5 TOP: Inverse Variation 0 ANS: 0( ) = x( x + ) x x 0 = 0 0 = x + x x x 0 = 0 (x + )(x 5) = 0 x =, 5 PTS: REF: 0a STA: A.A.5 TOP: Inverse Variation 5

97 ID: A ANS: =. 6 d = = 6d = d = PTS: REF: 060a STA: A.A.5 TOP: Inverse Variation 6

98 ID: A Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic Answer Section ANS: 0 = 5t 8 = t PTS: REF: 0a STA: A.A.5 TOP: Inverse Variation ANS: y sin θ = y = sin θ + f(θ) = sin θ + PTS: REF: fall097a STA: A.A.0 TOP: Functional Notation ANS: f0 = 0 ( 0) 6 = 0 8 = 5 PTS: REF: 060a STA: A.A. TOP: Functional Notation 5 ANS: g(0) = a(0) x = 00a ( 9) = 900a PTS: REF: 06a STA: A.A. TOP: Functional Notation 6 ANS: PTS: REF: 09a STA: A.A.5 TOP: Families of Functions 7 ANS: PTS: REF: 09a STA: A.A.5 TOP: Properties of Graphs of Functions and Relations 8 ANS: As originally written, alternatives () and () had no domain restriction, so that both were correct. PTS: REF: 0605a STA: A.A.5 TOP: Properties of Graphs of Functions and Relations 9 ANS: PTS: REF: 0600a STA: A.A.5 TOP: Identifying the Equation of a Graph 50 ANS: PTS: REF: 0608a STA: A.A.5 TOP: Identifying the Equation of a Graph 5 ANS: PTS: REF: fall0908a STA: A.A.8 TOP: Defining Functions KEY: graphs 5 ANS: PTS: REF: 060a STA: A.A.8 TOP: Defining Functions 5 ANS: PTS: REF: 00a STA: A.A.8 TOP: Defining Functions KEY: graphs 5 ANS: PTS: REF: 06a STA: A.A.8 TOP: Defining Functions KEY: graphs

99 ID: A 55 ANS: PTS: REF: 005a STA: A.A.8 TOP: Defining Functions KEY: graphs 56 ANS: PTS: REF: 0609a STA: A.A.8 TOP: Defining Functions KEY: graphs 57 ANS: () fails the horizontal line test. Not every element of the range corresponds to only one element of the domain. PTS: REF: fall0906a STA: A.A. TOP: Defining Functions 58 ANS: () and () fail the horizontal line test and are not one-to-one. Not every element of the range corresponds to only one element of the domain. () fails the vertical line test and is not a function. Not every element of the domain corresponds to only one element of the range. PTS: REF: 0800a STA: A.A. TOP: Defining Functions 59 ANS: PTS: REF: 05a STA: A.A. TOP: Defining Functions 60 ANS: PTS: REF: 068a STA: A.A. TOP: Defining Functions 6 ANS: PTS: REF: 060a STA: A.A. TOP: Defining Functions 6 ANS: PTS: REF: 007a STA: A.A. TOP: Defining Functions 6 ANS: PTS: REF: fall09a STA: A.A.9 TOP: Domain and Range KEY: real domain 6 ANS: PTS: REF: 06a STA: A.A.9 TOP: Domain and Range KEY: real domain 65 ANS: PTS: REF: 0a STA: A.A.9 TOP: Domain and Range KEY: real domain 66 ANS: PTS: REF: 0a STA: A.A.9 TOP: Domain and Range KEY: real domain 67 ANS: PTS: REF: 06a STA: A.A.9 TOP: Domain and Range KEY: real domain 68 ANS: PTS: REF: 0800a STA: A.A.5 TOP: Domain and Range 69 ANS: D: 5 x 8. R: y PTS: REF: 0a STA: A.A.5 TOP: Domain and Range 70 ANS: PTS: REF: 060a STA: A.A.5 TOP: Domain and Range 7 ANS: PTS: REF: 0608ge STA: A.A.5 TOP: Domain and Range 7 ANS: PTS: REF: 068a STA: A.A.5 TOP: Domain and Range

100 ID: A 7 ANS: f() = () =. g( ) = ( ) + 5 = PTS: REF: fall090a STA: A.A. TOP: Compositions of Functions KEY: numbers 7 ANS: 6(x 5) = 6x 0 PTS: REF: 009a STA: A.A. TOP: Compositions of Functions KEY: variables 75 ANS: 7. f( ) = ( ) 6 =. g(x) = = 7. PTS: REF: 065a STA: A.A. TOP: Compositions of Functions KEY: numbers 76 ANS: g = =. f() = () = PTS: REF: 00a STA: A.A. TOP: Compositions of Functions KEY: numbers 77 ANS: PTS: REF: 066a STA: A.A. TOP: Compositions of Functions KEY: variables 78 ANS: h( 8) = ( 8) = = 6. g( 6) = ( 6) + 8 = + 8 = 5 PTS: REF: 00a STA: A.A. TOP: Compositions of Functions KEY: numbers 79 ANS: f(g(x)) = (x + 5) (x + 5) + = (x + 0x + 5) x 5 + = x + 7x + 6 PTS: REF: 069a STA: A.A. TOP: Compositions of Functions KEY: variables 80 ANS: PTS: REF: 0807a STA: A.A. TOP: Inverse of Functions KEY: equations

101 ID: A 8 ANS: y = x 6. f (x) is not a function. x = y 6 x + 6 = y ± x + 6 = y PTS: REF: 06a STA: A.A. TOP: Inverse of Functions KEY: equations 8 ANS: PTS: REF: fall096a STA: A.A.6 TOP: Transformations with Functions and Relations 8 ANS: PTS: REF: 080a STA: A.A.6 TOP: Transformations with Functions and Relations 8 ANS: PTS: REF: 065a STA: A.A.6 TOP: Transformations with Functions and Relations 85 ANS: PTS: REF: 0606a STA: A.A.9 TOP: Sequences 86 ANS: common difference is. b n = x + n 0 = x + () 8 = x PTS: REF: 080a STA: A.A.9 TOP: Sequences 87 ANS: 0 =.5 PTS: REF: 07a STA: A.A.9 TOP: Sequences

102 ID: A 88 ANS: 9 7 = = x + ( ) = 9 x + = 9 x = 7 a n = 7 + (n ) PTS: REF: 0a STA: A.A.9 TOP: Sequences 89 ANS: PTS: REF: 0600a STA: A.A.0 TOP: Sequences 90 ANS: PTS: REF: 00a STA: A.A.0 TOP: Sequences 9 ANS: (a + ) (a + ) = a + PTS: REF: 00a STA: A.A.0 TOP: Sequences 9 ANS: PTS: REF: 06a STA: A.A.0 TOP: Sequences 9 ANS: 7r = 6 r = 6 7 r = PTS: REF: 0805a STA: A.A. TOP: Sequences 9 ANS: = PTS: REF: 00a STA: A.A. TOP: Sequences 95 ANS: a b 6b = 6 a5 b a PTS: REF: 066a STA: A.A. TOP: Sequences 96 ANS: a n = 5( ) n a 5 = 5( ) 5 = 8, 90 PTS: REF: 005a STA: A.A. TOP: Sequences 5

103 ID: A 97 ANS: a n = 5 ( ) n a 5 = 5 ( ) 5 = 5 ( ) = 5 7 = 8 5 PTS: REF: 0609a STA: A.A. TOP: Sequences 98 ANS:, 5, 8, PTS: REF: fall09a STA: A.A. TOP: Recursive Sequences 99 ANS: a =. a = () = 5. a = (5) = 9. PTS: REF: 06a STA: A.A. TOP: Recursive Sequences 00 ANS: n 0 Σ n + n = + = + = 8 = PTS: REF: fall09a STA: A.N.0 TOP: Sigma Notation KEY: basic 0 ANS: ( ) + ( ) + ( ) + ( ) + (5 ) = = 0 PTS: REF: 0a STA: A.N.0 TOP: Sigma Notation KEY: basic 0 ANS: n 5 Σ r + r + = 6 + = = 0 8 PTS: REF: 068a STA: A.N.0 TOP: Sigma Notation KEY: basic 0 ANS: 0. PTS: REF: 00a STA: A.N.0 TOP: Sigma Notation KEY: basic 6

104 ID: A 0 ANS: + ( x) + ( x) + ( x) + (5 x) + 6 x + 9 x + x + 5 x 6 x PTS: REF: 065a STA: A.N.0 TOP: Sigma Notation KEY: advanced 05 ANS: (a ) + (a ) + (a ) + (a ) (a a + ) + (a a + ) + (a 6a + 9) + (a 8a + 6) a 0a + 0 PTS: REF: 0a STA: A.N.0 TOP: Sigma Notation KEY: advanced 06 ANS: PTS: REF: 0605a STA: A.A. TOP: Sigma Notation 07 ANS: 5 n = 7n PTS: REF: 0809a STA: A.A. TOP: Sigma Notation 08 ANS: PTS: REF: 0605a STA: A.A. TOP: Sigma Notation 09 ANS: PTS: REF: 060a STA: A.A. TOP: Sigma Notation 0 ANS: S n = n [a + (n )d] = [(8) + ( )] = 798 PTS: REF: 060a STA: A.A.5 TOP: Series KEY: arithmetic ANS: S n = n 9 [a + (n )d] = [() + (9 )7] = 5 PTS: REF: 00a STA: A.A.5 TOP: Summations KEY: arithmetic 7

105 ID: A ANS: a n = 9n. S n = a = 9() = 5 a 0 = 9(0) = 76 0(5 + 76) = 80 PTS: REF: 08a STA: A.A.5 TOP: Summations KEY: arithmetic ANS: S 8 = ( ( )8 ) ( ) = 96, = 9, PTS: REF: 060a STA: A.A.5 TOP: Summations KEY: geometric ANS: cosk = 5 6 K = cos 5 6 K ' PTS: REF: 060a STA: A.A.55 TOP: Trigonometric Ratios 5 ANS: PTS: REF: 0800a STA: A.A.55 TOP: Trigonometric Ratios 6 ANS: 6 = 08 = 6 = 6. cotj = A O = 6 6 = PTS: REF: 00a STA: A.A.55 TOP: Trigonometric Ratios 7 ANS: PTS: REF: 05a STA: A.A.55 TOP: Trigonometric Ratios 8

106 ID: A 8 ANS: sin S = 8 7 S = sin 8 7 S 8 ' PTS: REF: 06a STA: A.A.55 TOP: Trigonometric Ratios 9 ANS: 5 π = 0π = 5π 6 PTS: REF: 065a STA: A.M. TOP: Radian Measure 0 ANS: 97º π PTS: REF: fall09a STA: A.M. TOP: Radian Measure KEY: degrees ANS: π 80 π = 65 PTS: REF: 0600a STA: A.M. TOP: Radian Measure KEY: degrees ANS: 0 π 80 = 7π PTS: REF: 0800a STA: A.M. TOP: Radian Measure KEY: radians 9

107 ID: A ANS:.5 80 π. PTS: REF: 09a STA: A.M. TOP: Radian Measure KEY: degrees ANS: 80 π = 60 π PTS: REF: 00a STA: A.M. TOP: Radian Measure KEY: degrees 5 ANS: 6 π 80.8 PTS: REF: 06a STA: A.M. TOP: Radian Measure KEY: radians 6 ANS: 80 π PTS: REF: 05a STA: A.M. TOP: Radian Measure KEY: degrees 7 ANS: 8π 5 80 π = 88 PTS: REF: 060a STA: A.M. TOP: Radian Measure KEY: degrees 8 ANS: 5 80 π 86 PTS: REF: 07a STA: A.M. TOP: Radian Measure KEY: degrees 9 ANS:.5 80 π ' PTS: REF: 06a STA: A.M. TOP: Radian Measure KEY: degrees 0

108 ID: A 0 ANS: PTS: REF: 060a STA: A.A.60 TOP: Unit Circle ANS: PTS: REF: 08005a STA: A.A.60 TOP: Unit Circle ANS: PTS: REF: 0606a STA: A.A.60 TOP: Unit Circle ANS: If cscp > 0, sin P > 0. If cotp < 0 and sin P > 0, cosp < 0 PTS: REF: 060a STA: A.A.60 TOP: Finding the Terminal Side of an Angle ANS: PTS: REF: 06a STA: A.A.60 TOP: Finding the Terminal Side of an Angle 5 ANS: PTS: REF: 0a STA: A.A.56 TOP: Determining Trigonometric Functions KEY: degrees, common angles 6 ANS: = 6 PTS: REF: 06a STA: A.A.56 TOP: Determining Trigonometric Functions KEY: degrees, common angles 7 ANS:. sin θ = y x + y = ( ) + =. cscθ =. PTS: REF: fall09a STA: A.A.6 TOP: Determining Trigonometric Functions

109 ID: A 8 ANS: PTS: REF: 065a STA: A.A.66 TOP: Determining Trigonometric Functions 9 ANS: PTS: REF: 00a STA: A.A.66 TOP: Determining Trigonometric Functions 0 ANS: PTS: REF: 067a STA: A.A.66 TOP: Determining Trigonometric Functions ANS: PTS: REF: 08007a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: basic ANS: PTS: REF: 00a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: unit circle ANS: PTS: REF: 0a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: advanced ANS: tan0 =. Arc cos k k = 0 = cos0 k = PTS: REF: 06a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: advanced

110 ID: A 5 ANS: If sin A = sin A, cosa =, and tana = 5 cosa = 5 5 = 7 PTS: REF: 0a STA: A.A.6 TOP: Using Inverse Trigonometric Functions KEY: advanced 6 ANS: cos( ) = cos(55 ) PTS: REF: 060a STA: A.A.57 TOP: Reference Angles 7 ANS: s = θ r = = 8 PTS: REF: fall09a STA: A.A.6 TOP: Arc Length KEY: arc length 8 ANS: s = θ r = π 8 6 = π PTS: REF: 06a STA: A.A.6 TOP: Arc Length KEY: arc length 9 ANS: π 8 50'.6 radians s = θ r = PTS: REF: 05a STA: A.A.6 TOP: Arc Length KEY: arc length 50 ANS: Cofunctions tangent and cotangent are complementary PTS: REF: 060a STA: A.A.58 TOP: Cofunction Trigonometric Relationships 5 ANS: sin θ + cos θ = sin θ cos θ = sec θ PTS: REF: 06a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 5 ANS: cosθ cosθ cos θ = cos θ = sin θ PTS: REF: 060a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships

111 ID: A 5 ANS: a a = 90 a + 5 = 90 cotxsin x secx a = 75 a = 5 PTS: REF: 00a STA: A.A.58 TOP: Cofunction Trigonometric Relationships 5 ANS: cosx sin x sin x = cosx = cos x PTS: REF: 06a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 55 ANS: cosθ secθ sin θ cotθ = sin θ cosθ sin θ = PTS: REF: 08a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 56 ANS: cosx cotx cscx = sin x = cosx sin x PTS: REF: 060a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 57 ANS:. If sin 60 =, then csc60 = = PTS: REF: 05a STA: A.A.59 TOP: Reciprocal Trigonometric Relationships 58 ANS: sin A cos A + cos A cos A = cos A tan A + = sec A PTS: REF: 05a STA: A.A.67 TOP: Proving Trigonometric Identities 59 ANS: PTS: REF: 008a STA: A.A.67 TOP: Proving Trigonometric Identities 60 ANS: PTS: REF: fall090a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: simplifying

112 ID: A 6 ANS: cos B + tan(a + B) = cos B + sin B = 5 = cos B + 5 = cos B = 6 cosb = = tanb = sin B cosb = = = 5 = 5 PTS: REF: 0807a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: evaluating 6 ANS: sin(5 + 0) = sin5cos0 + cos5sin0 = + = 6 + = 6 + PTS: REF: 066a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: evaluating 6 ANS: cos(a B) = = = 65 PTS: REF: 0a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: evaluating 6 ANS: sin(80 + x) = (sin80)(cosx) + (cos80)(sinx) = 0 + ( sin x) = sin x PTS: REF: 08a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: identities 65 ANS: sin(θ + 90) = sin θ cos90 + cosθ sin 90 = sin θ (0) + cosθ () = cosθ PTS: REF: 0609a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: identities 5

113 ID: A 66 ANS: cos(x y) = cosxcosy + sin xsin y = b b + a a = b + a PTS: REF: 06a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: simplifying 67 ANS: cos θ cosθ = cos θ (cos θ sin θ) = sin θ PTS: REF: 060a STA: A.A.77 TOP: Double Angle Identities KEY: simplifying 68 ANS: cos A = 5 9 cosa = + + cos A = 5, sin A is acute. sin A = sin AcosA = 5 = 5 9 PTS: REF: 007a STA: A.A.77 TOP: Double Angle Identities KEY: evaluating 69 ANS: If sin x = 0.8, then cosx = 0.6. tan x = = 0..6 = 0.5. PTS: REF: 060a STA: A.A.77 TOP: Half Angle Identities 70 ANS: cosa = sin A = = 9 = 7 9 PTS: REF: 0a STA: A.A.77 TOP: Double Angle Identities KEY: evaluating 6

114 ID: A 7 ANS: tanθ = 0 tanθ = θ = tan θ = 60, 0.. PTS: REF: fall090a STA: A.A.68 TOP: Trigonometric Equations KEY: basic 7 ANS: 0, 60, 80, 00. sin θ = sin θ sin θ sin θ = 0 sin θcosθ sin θ = 0 sin θ(cosθ ) = 0 sin θ = 0 cosθ = 0 θ = 0, 80 cosθ = θ = 60, 00 PTS: REF: 0607a STA: A.A.68 TOP: Trigonometric Equations KEY: double angle identities 7 ANS: 5, 5 tanc = tanc = tanc tan = C C = 5, 5 PTS: REF: 080a STA: A.A.68 TOP: Trigonometric Equations KEY: basic 7

115 ID: A 7 ANS: cosθ = cosθ = θ = cos = 60, 00 PTS: REF: 060a STA: A.A.68 TOP: Trigonometric Equations KEY: basic 75 ANS: secx = secx = cosx = x = 5, 5 PTS: REF: 0a STA: A.A.68 TOP: Trigonometric Equations KEY: reciprocal functions 76 ANS: 5cscθ = 8 cscθ = 8 5 sin θ = 5 8 θ PTS: REF: 06a STA: A.A.68 TOP: Trigonometric Equations KEY: reciprocal functions 77 ANS: sin x + 5 sinx = 0 ( sin x )(sin x + ) = 0 sin x = x = π 6, 5π 6 PTS: REF: 06a STA: A.A.68 TOP: Trigonometric Equations KEY: quadratics 8

116 ID: A 78 ANS: secx = cosx = cosx = x = 5, 5 PTS: REF: 06a STA: A.A.68 TOP: Trigonometric Equations KEY: reciprocal functions 79 ANS: π b = π = 6π PTS: REF: 0607a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions 80 ANS: π b = π PTS: REF: 06a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions 8 ANS: π b = π KEY: period KEY: period b = PTS: REF: 05a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions 8 ANS: π 6 = π PTS: REF: 06a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions 8 ANS: π b = 0 KEY: period KEY: period b = π 5 PTS: REF: 07a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 9

117 ID: A 8 ANS: y = sin x. The period of the function is π, the amplitude is and it is reflected over the x-axis. PTS: REF: 065a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 85 ANS: PTS: REF: 00a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 86 ANS: PTS: REF: 0606a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 87 ANS: PTS: REF: fall09a STA: A.A.65 TOP: Graphing Trigonometric Functions 88 ANS: PTS: REF: 069a STA: A.A.65 TOP: Graphing Trigonometric Functions 89 ANS: period = π b = π π = PTS: REF: 0806a STA: A.A.70 TOP: Graphing Trigonometric Functions KEY: recognize 90 ANS: PTS: REF: 0600a STA: A.A.7 TOP: Graphing Trigonometric Functions 9 ANS: PTS: REF: 0a STA: A.A.7 TOP: Graphing Trigonometric Functions 9 ANS: PTS: REF: 007a STA: A.A.7 TOP: Graphing Trigonometric Functions 9 ANS: PTS: REF: 060a STA: A.A.6 TOP: Domain and Range 0