JMAP REGENTS BY TYPE

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1 JMAP REGENTS BY TYPE The NY Algebra /Trigonometry Regents Exams Fall 009-June 0 Dear Sir I have to acknolege the reciept of your favor of May 4. in which you mention that you have finished the 6. first books of Euclid, plane trigonometry, surveying & algebra and ask whether I think a further pursuit of that branch of science would be useful to you. there are some propositions in the latter books of Euclid, & some of Archimedes, which are useful, & I have no doubt you have been made acquainted with them. trigonometry, so far as this, is most valuable to every man, there is scarcely a day in which he will not resort to it for some of the purposes of common life. the science of calculation also is indispensible as far as the extraction of the square & cube roots; Algebra as far as the quadratic equation & the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not to be indulged in by one who is to have a profession to follow for his subsistence. in this light I view the conic sections, curves of the higher orders, perhaps even spherical trigonometry, Algebraical operations beyond the d dimension, and fluxions. Letter from Thomas Jefferson to William G. Munford, Monticello, June 8, 799.

2 Algebra /Trigonometry Multiple Choice Regents Exam Questions Algebra /Trigonometry Multiple Choice Regents Exam Questions The relationship between t, a student s test scores, and d, the student s success in college, is modeled by the equation d = 0.48t Based on this linear regression model, the correlation coefficient could be ) between and 0 ) between 0 and ) equal to equal to 0 4 If a function is defined by the equation f(x) = 4 x, which graph represents the inverse of this function? ) 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! ) 4! 8! )!+! 8!!! ) 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 )

3 Algebra /Trigonometry Multiple Choice Regents Exam Questions 5 The expression i + i is equivalent to ) i ) i ) + i + i 6 What is the solution set of the equation x 5 48x = 0? ) {0, ±} ) {0, ±, } ) {0, ±, ±i} {±, ±i} 7 Expressed in simplest form, equivalent to 6y + 6y 54 ) (y 6)(6 y) y 9 ) y 6 ) y y y is 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 0 What is the fifteenth term of the sequence 5, 0, 0, 40, 80,...? ) 6, 840 ) 8, 90 ) 8,90 7,680 When x is divided by x, the quotient is ) ) x ) x (x ) 8 The sum of the first eight terms of the series is ), 07 ), 845 ) 9, 65, 55 The expression ( 7i) is equivalent to ) i ) 40 4i ) i 58 4i

4 Algebra /Trigonometry Multiple Choice Regents Exam Questions 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 6 What is the solution set for the equation 5x + 9 = x +? ) {4} ) { 5} ) {4, 5} { 5, 4} 4 Which equation represents the graph below? 7 Which is a graph of y = cotx? ) ) y = sinx ) y = sin x ) y = cosx y = cos x ) 5 When factored completely, the expression x 5x 48x + 80 is equivalent to ) (x 6)(x 5) ) (x + 6)(x 5)(x + 5) ) (x + (x (x 5) (x + (x (x 5)(x 5) )

5 Algebra /Trigonometry Multiple Choice Regents Exam Questions 8 Which function is not one-to-one? ) {(0, ), (, ), (,), (, } ) {(0, 0), (, ), (,), (, )} ) {(0, ), (, 0), (,), (, )} {(0, ), (, 0), (,0), (, )} 9 What is the middle term in the expansion of 6 x y? ) 0x y ) 5 4 x4 y ) 0x y 5 4 x4 y What is the value of x in the equation 9 x + = 7 x +? ) ) ) 4 In ABC, a = 5, b = 4, and c =, as shown in the diagram below. What is the m C, to the nearest degree? 0 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 ) 6,45 ),4,400 59,459,00 When factored completely, x + x 4x equals ) (x + )(x )(x ) ) (x + )(x )(x + ) ) (x (x + ) (x (x ) ) 5 ) 59 ) What is the common ratio of the geometric sequence shown below?, 4, 8,6,... ) ) ) 6 4

6 Algebra /Trigonometry Multiple Choice Regents Exam Questions 5 What is the common ratio of the geometric sequence whose first term is 7 and fourth term is 64? ) 4 64 ) 8 4 ) 7 6 Which graph represents the solution set of 6x 7 5? ) ) ) 9 Which expression always equals? ) cos x sin x ) cos x + sin x ) cosx sinx cosx + sinx 0 What is the sum of the first 9 terms of the sequence, 0, 7, 4,,...? ) 88 ) 97 ) 54 9 The expression log4m is equivalent to ) (log4 + logm) ) log4 + logm ) log4 + logn log6 + logm 7 What is the fifteenth term of the geometric sequence 5, 0, 5,...? ) 8 5 ) 8 0 ) Ms. Bell's mathematics class consists of 4 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 ) 5,040 9,04 8 What is the common difference of the arithmetic sequence 5, 8,, 4? 8 ) 5 ) ) 9 5

7 Algebra /Trigonometry Multiple Choice Regents Exam Questions Factored completely, the expression x 4 + 0x x is equivalent to ) x (4x + 6)(x ) ) (x + x)(x x) ) x (x )(x + ) x (x + )(x ) 6 Which equation is graphed in the diagram below? 4 The fraction ) ) ) a b b ab b ab a a b is equivalent to ) π y = cos 0 x + 8 ) π y = cos 5 x + 5 ) π y = cos 0 x + 8 π y = cos 5 x 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)? ) 65 ) 65 6 ) 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? ) 64 ) 75 )

8 Algebra /Trigonometry Multiple Choice Regents Exam Questions 8 What are the sum and product of the roots of the equation 6x 4x = 0? 4 Which graph represents one complete cycle of the equation y = sin πx? ) sum = ; product = ) sum = ; product = ) sum = ; product = sum = ; product = ) 9 In the diagram below, the length of which line segment is equal to the exact value of sin θ? ) ) ) TO ) TS ) OR OS 40 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 4 In ABC, m A = 74, a = 59., and c = 60.. What are the two possible values for m C, to the nearest tenth? ) 7.7 and 06. ) 7.7 and 6.7 ) 78. and and 68. 7

9 Algebra /Trigonometry Multiple Choice Regents Exam Questions 4 The expression ) ) ) 4 5 4(5 ) (5 + ) is equivalent to 46 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? ) 44 Which expression is equivalent to ) ) ) ? ) ) 45 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

10 Algebra /Trigonometry Multiple Choice Regents Exam Questions 47 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. 49 Which function is one-to-one? ) f(x) = x ) f(x) = x ) f(x) = x f(x) = sinx 50 Which equation is represented by the graph below? 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 ) ) ) y = 5 x ) y = 0.5 x ) y = 5 x y = 0.5 x 48 When x + x 4 is subtracted from x + x x, the difference is ) x + x 5x + 4 ) x + x + x 4 ) x + 4x + x 4 x x + 5x The value of x in the equation 4 x + 5 = 8 x is ) ) ) 5 0 9

11 Algebra /Trigonometry Multiple Choice Regents Exam Questions 5 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 = 40 ) (k ) k = 7 ) (k + ) k = 9 (k ) k = 55 Which equation is represented by the graph below? 5 In ABC, a =, b = 5, and c = 7. What is m C? ) ) 8 ) 60 0 ) y = cotx ) y = cscx ) y = secx y = tanx 54 Which summation represents ? 4 ) n n = 5 0 ) (n + ) n = 4 ) (n ) n = 4 (n n = 56 What is the number of degrees in an angle whose radian measure is 8π 5? ) 576 ) 88 ) 5 0

12 Algebra /Trigonometry Multiple Choice Regents Exam Questions 57 If n is a negative integer, then which statement is always true? ) 6n < 4n n ) 4 > 6n ) 6n < 4n 4n > (6n) 6 Which ordered pair is a solution of the system of equations shown below? x + y = 5 ) (, ) ) (5, 0) ) ( 5, 0) ( 4, 9) (x + ) + (y ) = 5 58 The expression 4ab b a 8b + 7ab 6b is equivalent to ) ab 6b ) 6ab b ) 5ab + 7ab 6b 5ab b + 7ab 6b 6 What is the domain of the function shown below? 59 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? 5 ) ) 5 75 ) ) x 6 ) y 6 ) x 5 y 5 60 What is the period of the function f(θ) = cosθ? ) π π ) π ) π 6 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, ).

13 Algebra /Trigonometry Multiple Choice Regents Exam Questions 64 What is the radian measure of an angle whose measure is 40? ) 7π ) 7π 6 7π ) 6 7π 68 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. 65 What is the fourth term in the binomial expansion (x ) 8? ) 448x 5 ) 448x 4 ) 448x 5 448x 4 66 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 4 and g(x) = x The roots of the equation x + 7x = 0 are ) and ) and ) 7 ± ± In which interval of f(x) = cos(x) is the inverse also a function? ) π < x < π ) π x π 67 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? ) 4 ) 7 ) 4 48 ) 0 x π π x π

14 Algebra /Trigonometry Multiple Choice Regents Exam Questions 7 What is the coefficient of the fourth term in the expansion of (a 4b) 9? ) 5, 76 ) 6 ) 6 5,76 7 The sum of 6a 4 b and 6a 4 b, expressed in simplest radical form, is 6 ) 68a 8 b 4 ) a b a b ) 4a 6ab 0a b 8 7 What is the period of the function y = sin x π? ) ) ) π 6π 74 When x + is divided by x +, the quotient equals ) ) x ) x 75 Which expression is equivalent to ) x 4 y 5 ) x 5 y 4 ) x 4 y 5 y 4 x 5 76 What is the product of ) ) ) x 8 9 x 6 9 x 8 x 6 9 x 6 x 6 9 x 4 x y 4 x 5 y? and x 4 + A 77 If r = B, then logr can be represented by C ) 6 loga + logb logc ) (loga + logb logc) ) log(a + B) C loga + logb logc? x

15 Algebra /Trigonometry Multiple Choice Regents Exam Questions 78 The table below shows the first-quarter averages for Mr. Harper s statistics class. 8 The value of csc8 rounded to four decimal places is ).76 ).408 ) Which ordered pair is in the solution set of the system of equations shown below? y x + = 0 ) (, 6) ) (, ) ) (, ) ( 6, ) y x = 0 What is the population variance for this set of data? ) 8. ) 8. ) The expression sin(θ + 90) is equivalent to ) sin θ ) cos θ ) sin θ cos θ 80 The roots of the equation x 0x + 5 = 0 are ) imaginary ) real and irrational ) real, rational, and equal real, rational, and unequal 8 Which values of x are in the solution set of the following system of equations? y = x 6 ) 0, 4 ) 0, 4 ) 6, 6, y = x x 6 84 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? ) 540 ) 567 )

16 Algebra /Trigonometry Multiple Choice Regents Exam Questions 85 Which graph represents the equation y = cos x? 87 In the diagram below of a unit circle, the ordered pair, represents the point where the terminal side of θ intersects the unit circle. ) ) ) What is m θ? ) 45 ) 5 ) What is the number of degrees in an angle whose measure is radians? 60 ) π π ) 60 ) 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 5

17 Algebra /Trigonometry Multiple Choice Regents Exam Questions 89 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 9 If m θ = 50, which diagram represents θ drawn in standard position? 90 Akeem invests $5,000 in an account that pays 4.75% 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? ) 0.0 ) 4.6 ). 4.0 ) ) 9 What is the graph of the solution set of x >5? ) ) ) ) 9 Which expression, when rounded to three decimal places, is equal to.55? ) sec 5π 6 ) tan(49 0 ) ) sin π 5 csc( 8 ) 94 What is the value of x in the equation log 5 x = 4? ).6 ) 0 ) 65,04 6

18 Algebra /Trigonometry Multiple Choice Regents Exam Questions 95 What is the range of f(x) = (x + + 7? ) y 4 ) y 4 ) y = 7 y 7 99 If sin A =, what is the value of cosa? ) ) ) 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 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) ) (4, ) ( 6, ) 97 What is the product of 5 x 4 y and 5 x + 4 y? 4 ) 5 x 9 6 y4 4 ) 5 x 9 6 y ) 5 x 4 y4 4 5 x 98 Expressed as a function of a positive acute angle, cos( 05 ) is equal to ) cos 55 ) cos 55 ) sin 55 sin 55 0 The equation x + y x + 6y + = 0 is equivalent to ) (x ) + (y + ) = ) (x ) + (y + ) = 7 ) (x + ) + (y + ) = 7 (x + ) + (y + ) = 0 0 A sequence has the following terms: a = 4, a = 0, a = 5, a 4 = 6.5. Which formula represents the nth term in the sequence? ) a n = 4 +.5n ) a n = 4 +.5(n ) ) a n = 4(.5) n a n = 4(.5) n 7

19 Algebra /Trigonometry Multiple Choice Regents Exam Questions 0 In the diagram below of right triangle KTW, KW = 6, KT = 5, and m KTW = In the diagram below of right triangle JTM, JT =, JM = 6, and m JMT = 90. What is the measure of K, to the nearest minute? ) ' ) 4' ) 55' 56' 04 Which equation is represented by the graph below? What is the value of cotj? ) ) ) 06 The conjugate of 7 5i is ) 7 5i ) 7 + 5i ) 7 5i 7 + 5i ) y = cosx ) y = sinx ) y = cos π x y = sin π x 07 In ABC, m A = 0, b = 0, and c = 8. What is the area of ABC to the nearest square inch? ) 5 ) 78 )

20 Algebra /Trigonometry Multiple Choice Regents Exam Questions 08 For which equation does the sum of the roots equal and the product of the roots equal? 4 ) 4x 8x + = 0 ) 4x + 8x + = 0 ) 4x x 8 = 0 4x + x = 0 Samantha constructs the scatter plot below from a set of data. 09 How many distinct triangles can be formed if m A = 5, a = 0, and b =? ) ) ) 0 0 The sides of a parallelogram measure 0 cm and 8 cm. One angle of the parallelogram measures 46 degrees. What is the area of the parallelogram, to the nearest square centimeter? ) 65 ) 5 ) 9 6 In PQR, p equals rsin P ) sin Q rsin P ) sin R rsin R ) sin P qsin R sin Q Based on her scatter plot, which regression model would be most appropriate? ) exponential ) linear ) logarithmic power If a = and b =, what is the value of the expression a b? ) 9 8 ) )

21 Algebra /Trigonometry Multiple Choice Regents Exam Questions 4 The expression log 5 5 ) ) is equivalent to 8 The number of minutes students took to complete a quiz is summarized in the table below. ) 5 In KLM, KL = 0, LM =, and m K = 40. The measure of M? ) must be between 0 and 90 ) must equal 90 ) must be between 90 and 80 is ambiguous 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 = x 6 + x x 7 = 6 + x 6 If sin 5 8 = A, then ) sin A = 5 8 ) sin A = 8 5 ) cosa = 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? ) 448 ) 504 ),40,50 cosa = The expression 4 6x y 7 is equivalent to 7 How many negative solutions to the equation x 4x + x = 0 exist? ) ) ) y ) x ) x 8 y y ) 4x 4x 8 y 8 0

22 Algebra /Trigonometry Multiple Choice Regents Exam Questions Which graph represents a one-to-one function? 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} ) ) 4 The quantities p and q vary inversely. If p = 0 when q =, and p = x when q = x +, then x equals ) 4 and 5 0 ) 9 ) 5 and 4 4 ) 5 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 Which function is one-to-one? ) k(x) = x + ) g(x) = x + ) f(x) = x + j(x) = x Which value of k satisfies the equation 8 k + 4 = 4 k? ) ) 9 4 ) 4 5

23 Algebra /Trigonometry Multiple Choice Regents Exam Questions 7 What is a formula for the nth term of sequence B shown below? B = 0,, 4, 6,... ) b n = 8 + n ) b n = 0 + n ) b n = 0() n b n = 0() n 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? ) 4.% ) 8.% ) 5.% 68.% 8 If tan Arccos ) ) ) k =, then k is Which graph does not represent a function? ) 9 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? 7 ) 49 ) 4 49 ) 49 ) ) 0 The roots of the equation 9x + x 4 = 0 are ) imaginary ) real, rational, and equal ) real, rational, and unequal real, irrational, and unequal

24 Algebra /Trigonometry Multiple Choice Regents Exam Questions 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? ).4 ) 6.0 ) In the interval 0 x < 60, tanx is undefined when x equals ) 0º and 90º ) 90º and 80º ) 80º and 70º 90º and 70º 7 If x + = 6x is solved by completing the square, an intermediate step would be ) (x + ) = 7 ) (x ) = 7 ) (x ) = (x 6) = 4 8 If log = a and log = b, the expression log 9 0 is equivalent to ) b a + ) b a ) b a + 0 b a + 5 If f(x) = x and g(x) = x + 5, what is the value of (g f)(? ) ).5 ) 6 6 If logx loga = loga, then logx expressed in terms of loga is equivalent to ) log5a ) log6 + loga ) log6 + loga log6 + loga 9 The equation y sin θ = may be rewritten as ) f(y) = sinx + ) f(y) = sinθ + ) f(x) = sinθ + f(θ) = sinθ + 40 The value of the expression (n + n ) is ) ) ) 4 6 n = 0

25 Algebra /Trigonometry Multiple Choice Regents Exam Questions 4 Which equation is sketched in the diagram below? 44 The expression x + 4 x + ) (x + x x ) (x + x x 4 ) x x + is equivalent to ) y = cscx ) y = secx ) y = cotx y = tanx 4 What is the solution set of the equation secx = when 0 x < 60? ) {45, 5, 5, 5 } ) {45, 5 } ) {5, 5 } {5, 5 } 4 The expression (x ) ) (x ) ) (x ) is equivalent to 45 When simplified, the expression equivalent to ) w 7 ) w ) w 7 w 4 w 5 w 9 46 If sin A = where 0 <A < 90, what is the value of sin A? 5 ) 5 ) ) is ) (x ) (x ) 4

26 Algebra /Trigonometry Multiple Choice Regents Exam Questions 47 The value of tan6 4 to the nearest ten-thousandth is ).407 ).408 ) Which graph does not represent a function? 50 Which problem involves evaluating 6 P 4? ) How many different four-digit ID numbers can be formed using,,, 4, 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 Which value of r represents data with a strong negative linear correlation between two variables? ).07 ) 0.89 ) ) 5 The area of triangle ABC is 4. If AB = 8 and m B = 6, the length of BC is approximately ) 5. ) 9. ) What is the number of degrees in an angle whose radian measure is π? ) 50 ) 65 ) Given the relation {(8, ), (, 6), (7, 5), (k, }, which value of k will result in the relation not being a function? ) ) ) 4 5

27 Algebra /Trigonometry Multiple Choice Regents Exam Questions 54 The graph of y = f(x) is shown below. 56 What is the fourth term in the expansion of (x ) 5? ) 70x ) 40x ) 70x, 080x 57 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 > } 58 What is the range of the function shown below? Which set lists all the real solutions of f(x) = 0? ) {, } ) {, } ) {, 0, } {, 0, } 55 Which values of x are solutions of the equation x + x x = 0? ) 0,, ) 0,, ) 0,, 0,, ) x 0 ) x 0 ) y 0 y 0 59 The expression cos θ cosθ is equivalent to ) sin θ ) sin θ ) cos θ + cos θ 6

28 Algebra /Trigonometry Multiple Choice Regents Exam Questions 60 Which ratio represents csca in the diagram below? 6 Which graph represents the function log x = y? ) ) ) ) 6 If p varies inversely as q, and p = 0 when q =, ) what is the value of p when q = 5? ) 5 ) 5 ) 9 4 ) 6 What is the conjugate of + i? ) + i ) ) i + i i 7

29 Algebra /Trigonometry Multiple Choice Regents Exam Questions 64 What are the values of θ in the interval 0 θ < 60 that satisfy the equation tanθ = 0? ) 60º, 40º ) 7º, 5º ) 7º, 08º, 5º, 88º 60º, 0º, 40º, 00º 68 In parallelogram BFLO, OL =.8, LF = 7.4, and m O = 6. If diagonal BL is drawn, what is the area of BLF? ).4 ) 4. ) The solution set of 4 x + 4x = 6 is ) {, } ) {, } ) {, } {, } 69 What is the principal value of cos ) 0 ) 60 ) 50 40? 66 The expression 64a 6 ) 8a 4 ) 8a 8 ) 4a 5 a 4a a 5 is equivalent to 70 Which equation is represented by the graph below? 67 The expression 4 + (k x) is equal to ) 58 4x ) 46 4x ) 58 x 46 x 5 k = ) (x ) + (y + ) = 5 ) (x + ) + (y ) = 5 ) (x ) + (y + ) = (x + ) + (y ) = 8

30 Algebra /Trigonometry Multiple Choice Regents Exam Questions 7 Written in simplest form, the expression is equivalent to ) x ) x x ) x 4 x + x 4 x x Factored completely, the expression 6x x x is equivalent to ) x(x + )(x ) ) x(x )(x + ) ) x(x )(x + ) x(x + )(x ) 7 The product of i 7 and i 5 is equivalent to ) ) ) i i 74 In the right triangle shown below, what is the measure of angle S, to the nearest minute? ) 8 ' ) 8 4' ) 6 56' 6 9' 75 What is the domain of the function f(x) = x +? ) (, ) ) (, ) ) [, ) [, ) 76 What is the conjugate of + i? ) + i ) i ) i + i 77 Which expression is equivalent to x y? 4y 5 ) ) ) y x y x x y x y 78 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 9

31 Algebra /Trigonometry Multiple Choice Regents Exam Questions 79 The graph of y = x 4x + x + 6 is shown below. 8 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 What is the equation of the graph shown below? What is the product of the roots of the equation x 4x + x + 6 = 0? ) 6 ) 6 ) The solution set of the equation x + = x is ) {} ) {0} ) {, 6} {, } ) y = x ) y = x ) x = y x = y 8 Which value of r represents data with a strong positive linear correlation between two variables? ) 0.89 ) 0.4 ) What is the range of f(x) = x +? ) {x x } ) {y y } ) {x x real numbers} {y y real numbers} 0

32 Algebra /Trigonometry Multiple Choice Regents Exam Questions 85 Which graph shows y = cos x? 86 Which equation represents the circle shown in the graph below that passes through the point (0, )? ) ) ) (x ) + (y + = 6 ) (x ) + (y + = 8 ) (x + ) + (y = 6 (x + ) + (y = 8 ) 87 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

33 Algebra /Trigonometry Multiple Choice Regents Exam Questions 88 The expression x 5 is equivalent to ) x 5 5 ) x ) 5 x 5 x 9 If f(x) = 4x x and g(x) = x, then (f g) equal to 4 ) 7 ) 7 ) 4 is 89 What are the domain and the range of the function shown in the graph below? 9 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 ) {x x > 4}; {y y > } ) {x x 4}; {y y } ) {x x > }; {y y > 4} {x x }; {y y 4} 90 In simplest form, 00 is equivalent to ) i 0 ) 5i ) 0i i 5 9 The points (, ), 4, 4, 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 94 How many different six-letter arrangements can be made using the letters of the word TATTOO? ) 60 ) 90 ) 0 70

34 Algebra /Trigonometry Multiple Choice Regents Exam Questions 95 The solutions of the equation y y = 9 are ) ) ) ± i ± i 5 ± 5 ± 5 98 If A is acute and tana =, then ) cota = ) cota = ) cot(90 A) = cot(90 A) = 96 What is the formula for the nth term of the sequence 54, 8, 6,...? ) a n = 6 n ) a n = 6 n ) a n = 54 n a n = 54 n 97 What is the common ratio of the sequence 64 a5 b, a b 4 9, 6 ab5,...? ) b a ) 6b a ) a b 6a b 99 The conjugate of the complex expression 5x + 4i is ) 5x 4i ) 5x + 4i ) 5x 4i 5x + 4i 00 What is a positive value of tan x, when sin x = 0.8? ) 0.5 ) 0.4 ) What is the solution set of the equation 4a + 6 4a = 0? ) ) 0 ) 0,

35 Algebra /Trigonometry Multiple Choice Regents Exam Questions 0 Which calculator output shows the strongest linear relationship between x and y? 05 Which graph represents a relation that is not a function? ) ) ) ) ) 0 If fx = ) 5 x, what is the value of f( 0)? x 6 ) ) 5 4 ) The expression (x + i) (x i) is equivalent to ) 0 ) ) + 4xi 4xi 4

36 Algebra /Trigonometry Multiple Choice Regents Exam Questions 06 The expression sin θ + cos θ is equivalent to sin θ ) cos θ ) sin θ ) sec θ csc θ 07 The value of the expression ( r + r) is ) 8 ) ) r = 08 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 ) 4 ) π ) π π 09 If x = i, y = i, and z = m + i, the expression xy z equals ) mi ) 6 6mi ) mi 6 6mi 0 The expression logx ( logy + logz) is equivalent to ) log x y z ) log x z y ) log x yz log xz y Which graph represents the solution set of 4x 5 >? ) ) ) Which expression is equivalent to 9x y 6? ) xy ) xy ) xy xy 5

37 Algebra /Trigonometry Multiple Choice Regents Exam Questions When x x 4 is subtracted from 4 5 x x +, the difference is 4 ) x + x 5 7 Which graph best represents the inequality y + 6 x x? ) x x + 5 ) x x x x ) 4 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 ) 5 The product of ( + 5 ) and ( 5 ) is ) ) ) The lengths of 00 pipes have a normal distribution with a mean of 0.4 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 84 th percentiles ) between the 6 th and 50 th percentiles above the 84 th percentile ) 6

38 Algebra /Trigonometry Multiple Choice Regents Exam Questions 8 What is the solution set for cosθ = 0 in the interval 0 θ < 60? ) {0, 50 } ) {60, 0 } ) {0, 0 } {60, 00 } 9 The expression log 8 64 is equivalent to ) 8 ) ) 8 0 In MNP, m = 6 and n = 0. Two distinct triangles can be constructed if the measure of angle M is ) 5 ) 40 ) Which relation is not a function? ) (x ) + y = 4 ) x + 4x + y = 4 ) x + y = 4 xy = 4 A circle has a radius of 4 inches. In inches, what is the length of the arc intercepted by a central angle of radians? ) π ) ) 8π 8 Which equation has roots with the sum equal to 9 4 and the product equal to 4? ) 4x + 9x + = 0 ) 4x + 9x = 0 ) 4x 9x + = 0 4x 9x = 0 4 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 The simplest form of ) x ) x + x ) x x 4 x x 8 x is 7

39 Algebra /Trigonometry Multiple Choice Regents Exam Questions 6 In which graph is θ coterminal with an angle of 70? 8 The solution set of x + 6 = x + is ) {, 4} ) { 4, } ) {} { 4} ) 9 Which arithmetic sequence has a common difference of 4? ) {0, 4n, 8n, n,...} ) {n, 4n, 6n, 64n,...} ) {n +, n + 5, n + 9, n +,... } {n + 4, n + 6, n + 64, n + 56,... } ) ) 0 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 7 The expression cos4xcosx + sin 4xsin x is equivalent to ) sin x ) sin 7x ) cos x cos 7x The discriminant of a quadratic equation is 4. The roots are ) imaginary ) real, rational, and equal ) real, rational, and unequal real, irrational, and unequal 8

40 Algebra /Trigonometry Multiple Choice Regents Exam Questions 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 Which graph represents the function f(x + )? 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? ) ) ) ) 4 The graph below shows the function f(x). 9

41 Algebra /Trigonometry Multiple Choice Regents Exam Questions 5 What is the solution of the equation log 4 (5x) =? ) 6.4 ).56 9 ) 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 + 4 = 0 x 6x + 4 = 0 6 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? ) 40 ) 00 ) Which diagram represents a relation that is both one-to-one and onto? 9 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 4 ) ) ) 40 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 40

42 Algebra /Trigonometry Multiple Choice Regents Exam Questions 4 The value of sin(80 + x) is equivalent to ) sin x ) sin(90 x) ) sin x sin(90 x) 4 The expression a b ) ) ) a 6 b 5 b 5 a 6 a b a b a 4 b is equivalent to 4 Which expression represents the third term in the expansion of (x 4 y)? ) y ) 6x 4 y ) 6x 4 y x 4 y 4

43 Algebra /Trigonometry Point Regents Exam Questions Algebra /Trigonometry Point Regents Exam Questions x 4 y 5 44 Simplify the expression and write the (x y 7 ) answer using only positive exponents. 49 Evaluate: 0 + (n ) 5 n = 45 Find the sum and product of the roots of the equation 5x + x = Express the product of cos 0 and sin 45 in simplest radical form. 46 Write an equation for the graph of the trigonometric function shown below. x 5 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. 47 Express in simplest form: 4 d d + d 48 Evaluate e x ln y when x = and y =. 4

44 Algebra /Trigonometry Point Regents Exam Questions 5 The scores of one class on the Unit mathematics test are shown in the table below. 55 Determine the solution of the inequality x 7. [The use of the grid below is optional.] Find the population standard deviation of these scores, to the nearest tenth. 5 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. 56 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 %. Write an exponential regression equation for the data, rounding all values to the nearest thousandth. 57 Express the sum using sigma notation. 54 Determine the value of n in simplest form: i + i 8 + i + n = 0 58 Solve for x: 4x x = + x 4

45 Algebra /Trigonometry Point Regents Exam Questions 59 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, Two sides of a parallelogram are 4 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. 66 Factor the expression t 8 75t 4 completely. 60 Use the discriminant to determine all values of k that would result in the equation x kx + 4 = 0 having equal roots. 67 Express in simplest form: a 6 b 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. 68 Find the third term in the recursive sequence a k + = a k, where a =. 6 Determine the sum and the product of the roots of the equation x + x 6 = On the unit circle shown in the diagram below, sketch an angle, in standard position, whose degree measure is 40 and find the exact value of sin Express x as a trinomial. 64 If f(x) = x 6 and g(x) = x, determine the value of (g f)( ). 44

46 Algebra /Trigonometry Point Regents Exam Questions 70 Solve algebraically for x: log 7 (x ) = 4 74 Convert radians to degrees and express the answer to the nearest minute. 7 Express 08x 5 y 8 6xy 5 in simplest radical form. 75 Express cos θ(sec θ cos θ), in terms of sin θ. 7 Express 5 x 7x in simplest radical form. 76 Find the first four terms of the recursive sequence defined below. a = a n = a (n ) n 7 On the axes below, for x, graph y = x A circle shown in the diagram below has a center of ( 5, ) and passes through point (, 7). Write an equation that represents the circle. 45

47 Algebra /Trigonometry Point Regents Exam Questions 78 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. 84 Write an equation of the circle shown in the diagram below. 79 The heights, in inches, of 0 high school varsity basketball players are 78, 79, 79, 7, 75, 7, 74, 74, 8, and 7. Find the interquartile range of this data set. 85 Factor completely: 0ax ax 5a 80 Express cotxsin x as a single trigonometric secx function, in simplest form, for all values of x for which it is defined. 8 Solve algebraically for x: 6 x + = 64 x + 8 Express the product of y y and y + 5 as a trinomial. 86 Find, to the nearest minute, the angle whose measure is.45 radians. 87 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 Express with a rational denominator, in simplest radical form. 8 Starting with sin A + cos A =, derive the formula tan A + = sec A. 46

48 Algebra /Trigonometry Point Regents Exam Questions 89 The two sides and included angle of a parallelogram are 8,, and 60. Find its exact area in simplest form. 94 Evaluate: ( n 4 n) n = 90 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. 95 Write an equation of the circle shown in the graph below. 9 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 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. 96 Write a quadratic equation such that the sum of its roots is 6 and the product of its roots is 7. 9 Determine the sum of the first twenty terms of the sequence whose first five terms are 5, 4,,, Express the exact value of csc60, with a rational denominator. 98 Find the total number of different twelve-letter arrangements that can be formed using the letters in the word PENNSYLVANIA. 47

49 Algebra /Trigonometry Point Regents Exam Questions 99 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. 0 The graph below represents the function y = f(x). 00 If f(x) = x 6, find f (x). 0 The table below shows the number of new stores in a coffee shop chain that opened during the years 986 through 994. State the domain and range of this function. 0 Solve the equation 6x x = 0 and express the answer in simplest radical form. 04 Find, to the nearest tenth, the radian measure of 6º. 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. 05 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. 48

50 Algebra /Trigonometry Point Regents Exam Questions 06 If g(x) = ax x, express g(0) in simplest form. Solve algebraically for x: 4 x 5 = 07 Find, to the nearest tenth of a degree, the angle whose measure is.5 radians. 4 If sec(a + 5) =csc(a), find the smallest positive value of a, in degrees. 08 Find the number of possible different 0-letter arrangements using the letters of the word STATISTICS. 5 Determine the sum and the product of the roots of x = x If θ is an angle in standard position and its terminal side passes through the point (, ), find the exact value of cscθ. 0 Find, algebraically, the measure of the obtuse angle, to the nearest degree, that satisfies the equation 5cscθ = 8. Solve the equation tanc = tanc 4 algebraically for all values of C in the interval 0 C < 60. Find the solution of the inequality x 4x > 5, algebraically. 49

51 Algebra /Trigonometry 4 Point Regents Exam Questions Algebra /Trigonometry 4 Point Regents Exam Questions 6 Solve algebraically for x: x + x = 4 x 9 7 Write the binomial expansion of (x ) 5 as a polynomial in simplest form. 0 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 47 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.] 8 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. Graph the inequality 6 x < 5 for x. Graph the solution on the line below. 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. 9 A ranch in the Australian Outback is shaped like triangle ACE, with m A = 4, m E = 0, and AC = 5 miles. Find the area of the ranch, to the nearest square mile. 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. 50

52 Algebra /Trigonometry 4 Point Regents Exam Questions 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. 7 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 Determine the number of employees whose travel time is within one standard deviation of the mean. 4 Express in simplest form: 4 x x + 7x + x 4 x + 8 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. 5 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. 6 Solve the equation below algebraically, and express the result in simplest radical form: x = 0 x 9 Solve the equation 8x + 4x 8x 9 = 0 algebraically for all values of x. 0 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? 5

53 Algebra /Trigonometry 4 Point Regents Exam Questions 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. 5 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. 6 If log 4 x =.5 and log y 5 =, find the numerical value of x, in simplest form. y 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. 7 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. Solve algebraically for all values of x: log (x + (7x = 8 Express as a single fraction the exact value of sin 75. Solve x x + 4 = 0 by completing the square, expressing the result in simplest radical form. 9 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. 4 Find all values of θ in the interval 0 θ < 60 that satisfy the equation sin θ = sin θ. 5

54 Algebra /Trigonometry 4 Point Regents Exam Questions 40 If tana = and sin B = 5 and angles A and B 4 are in Quadrant I, find the value of tan(a + B). 4 Ten teams competed in a cheerleading competition at a local high school. Their scores were 9, 8, 9, 7, 45, 40, 4, 8, 7, and 48. How many scores are within one population standard deviation from the mean? For these data, what is the interquartile range? 4 The measures of the angles between the resultant and two applied forces are 60 and 45, and the magnitude of the resultant is 7 pounds. Find, to the nearest pound, the magnitude of each applied force. 5

55 Algebra /Trigonometry 6 Point Regents Exam Questions Algebra /Trigonometry 6 Point Regents Exam Questions 4 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. 44 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. 49 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 4.78 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.] 45 Solve algebraically for x: x x x 7x 50 Solve algebraically for all values of x: 8 x x 7 5x 46 Solve the following systems of equations algebraically: 5 y x 4x 7x y 4 5 Solve algebraically for all values of x: x 4 4x 4x 6x 47 Solve algebraically for x: log x x x x 48 Perform the indicated operations and simplify completely: x x 6x 8 x 4x x 4 x 4 x x x 8 6 x 54

56 ID: A Algebra /Trigonometry Multiple Choice Regents Exam Questions Answer Section ANS: Since the coefficient of t is greater than 0, r > 0. PTS: REF: 00a STA: A.S.8 TOP: Correlation Coefficient ANS: 4 PTS: REF: fall095a STA: A.S.0 TOP: Permutations ANS: 4 PTS: REF: 00a STA: A.S. TOP: Analysis of Data 4 ANS: f (x) = log 4 x PTS: REF: fall096a STA: A.A.54 TOP: Graphing Logarithmic Functions 5 ANS: i + i = ( ) + ( i) = i PTS: REF: 08004a STA: A.N.7 TOP: Imaginary Numbers 6 ANS: x 5 48x = 0 x(x 4 6) = 0 x(x + (x = 0 x(x + (x + )(x ) = 0 PTS: REF: 06a STA: A.A.6 TOP: Solving Polynomial Equations 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: S 8 = ( ( 8 ) ( = 96, = 9, PTS: REF: 0604a STA: A.A.5 TOP: Summations KEY: geometric 9 ANS: 5 π = 0π = 5π 6 PTS: REF: 065a STA: A.M. TOP: Radian Measure

57 ID: A 0 ANS: a n = 5( ) n a 5 = 5( ) 5 = 8, 90 PTS: REF: 005a STA: A.A. TOP: Sequences ANS: x x = x x (x ) x = x x = x x = x PTS: REF: 0808a STA: A.A.9 TOP: Negative Exponents ANS: ( 7i)( 7i) = 9 i i + 49i = 9 4i 49 = 40 4i PTS: REF: fall090a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers ANS: PTS: REF: 060a STA: A.S. TOP: Analysis of Data 4 ANS: PTS: REF: 0606a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 5 ANS: x 5x 48x + 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 6 ANS: 5x + 9 = (x + ). ( 5) + shows an extraneous solution. 5x + 9 = x + 6x = x + x 0 0 = (x + 5)(x x = 5, 4 PTS: REF: 06a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions

58 ID: A 7 ANS: PTS: REF: 007a STA: A.A.7 TOP: Graphing Trigonometric Functions 8 ANS: 4 ( 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.4 TOP: Defining Functions 9 ANS: 6 C x ( y) = 0 x 8 8y = 0x y PTS: REF: 065a STA: A.A.6 TOP: Binomial Expansions 0 ANS: = 6, 45 5 C 8 PTS: REF: 080a STA: A.S. TOP: Combinations ANS: x + x 4x x (x + ) 4(x + ) (x (x + ) (x + )(x )(x + ) PTS: REF: 064a STA: A.A.7 TOP: Factoring by Grouping ANS: 4 9 x + = 7 x +. ( ) x + = ( ) x + 6x + = x + 6 6x + = x + 6 x = 4 x = 4 PTS: REF: 08008a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown

59 ID: A ANS: = (5)(cosC 69 = 4 40cosC 5 = 40cosC 5 40 = cosc 5 C PTS: REF: 060a STA: A.A.7 TOP: Law of Cosines KEY: find angle 4 ANS: 4 = PTS: REF: 004a STA: A.A. TOP: Sequences 5 ANS: 7r 4 = 64 r = 64 7 r = 4 PTS: REF: 0805a STA: A.A. TOP: Sequences 6 ANS: 6x 7 5 6x 7 5 6x x 6x x PTS: REF: fall0905a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 7 ANS: a n = 5 ( ) n a 5 = 5 ( ) 5 = 5 ( ) 4 = 5 7 = 8 5 PTS: REF: 0609a STA: A.A. TOP: Sequences 8 ANS: PTS: REF: 0600a STA: A.A.0 TOP: Sequences 9 ANS: PTS: REF: 008a STA: A.A.67 TOP: Proving Trigonometric Identities 4

60 ID: A 0 ANS: S n = n [a + (n )d] = 9 [() + (9 )7] = 54 PTS: REF: 00a STA: A.A.5 TOP: Summations KEY: arithmetic ANS: log4m = log4 + logm = log4 + logm PTS: REF: 06a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs ANS: 0 C 4 = 0 PTS: REF: 06a STA: A.S. TOP: Combinations ANS: 4 x 4 + 0x x = x (6x + 5x 6) = x (x + )(x ) PTS: REF: 06008a STA: A.A.7 TOP: Factoring Polynomials KEY: single variable 4 ANS: a b = a b b b = b ab = b ab PTS: REF: 0809a STA: A.A.5 TOP: Rationalizing Denominators KEY: index = 5 ANS: cos(a B) = = = 65 PTS: REF: 04a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: evaluating 6 ANS: 4 π b = 0 b = π 5 PTS: REF: 07a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 5

61 ID: A 7 ANS: x ± σ 5 ± 75 PTS: REF: 007a STA: A.S.5 TOP: Normal Distributions KEY: interval 8 ANS: sum: b a = 4 6 =. product: c a = 6 = PTS: REF: 009a STA: A.A.0 TOP: Roots of Quadratics 9 ANS: PTS: REF: 05a STA: A.A.55 TOP: Trigonometric Ratios 40 ANS: 9 sin A = is possible. +70 is not possible. sin 70 A = 58 PTS: REF: 00a STA: A.A.75 TOP: Law of Sines - The Ambiguous Case 4 ANS: period = π b = π π = PTS: REF: 0806a STA: A.A.70 TOP: Graphing Trigonometric Functions KEY: recognize 4 ANS: 59. sin 74 = 60. sin C C = 0.7 PTS: REF: 08006a STA: A.A.75 TOP: Law of Sines - The Ambiguous Case 4 ANS: = 4(5 + ) 5 = 5 + PTS: REF: 066a STA: A.N.5 TOP: Rationalizing Denominators 44 ANS: = = = PTS: REF: 060a STA: A.N.5 TOP: Rationalizing Denominators 45 ANS: PTS: REF: 00a STA: A.S.9 TOP: Differentiating Permutations and Combinations 6

62 ID: A 46 ANS: PTS: REF: 09a STA: A.A.5 TOP: Families of Functions 47 ANS: π + π π = π π = PTS: REF: 008a STA: A.S. TOP: Geometric Probability 48 ANS: PTS: REF: 04a STA: A.N. TOP: Operations with Polynomials 49 ANS: PTS: REF: 05a STA: A.A.4 TOP: Defining Functions 50 ANS: PTS: REF: 0608a STA: A.A.5 TOP: Identifying the Equation of a Graph 5 ANS: 4 x + 5 = 8 x. x + 5 = 4x + 0 = 9x 4x + 0 = 9x 0 = 5x = x x PTS: REF: 0605a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 5 ANS: PTS: REF: 0605a STA: A.A.4 TOP: Sigma Notation 5 ANS: 4 7 = + 5 ()(5)cosA 49 = 4 0cosA 5 = 0cosA = cosa 0 = cosa PTS: REF: 0807a STA: A.A.7 TOP: Law of Cosines KEY: angle, without calculator 7

63 ID: A 54 ANS: PTS: REF: 0605a STA: A.A.4 TOP: Sigma Notation 55 ANS: PTS: REF: 0600a STA: A.A.7 TOP: Graphing Trigonometric Functions 56 ANS: 8π 5 80 π = 88 PTS: REF: 060a STA: A.M. TOP: Radian Measure KEY: degrees 57 ANS: 6n < 4n. Flip sign when multiplying each side of the inequality by n, since a negative number. 6 n < 4 n 6 > 4 PTS: REF: 064a STA: A.N. TOP: Negative and Fractional Exponents 58 ANS: 4 4ab b a 9b b + 7ab 6b = 4ab b 9ab b + 7ab 6b = 5ab b + 7ab 6b PTS: REF: fall098a STA: A.A.4 TOP: Operations with Radicals KEY: with variables index = 59 ANS: 4 C = 5 5 PTS: REF: 0a STA: A.S.5 TOP: Binomial Probability KEY: spinner 60 ANS: π b = π PTS: REF: 06a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions KEY: period 8

64 ID: A 6 ANS: x + y = y = 5 y = x + 5 y = 0 (x + ) + ( x + 5 ) = 5 x + 6x x 4x + 4 = 5 x + x 40 = 0 x + x 0 = 0 (x + 5)(x = 0 x = 5, 4 PTS: REF: 00a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations 6 ANS: PTS: REF: 060a STA: A.A.5 TOP: Domain and Range 6 ANS: 4 PTS: REF: 09a STA: A.A.5 TOP: Properties of Graphs of Functions and Relations 64 ANS: 40 π 80 = 7π PTS: REF: 0800a STA: A.M. TOP: Radian Measure KEY: radians 65 ANS: 8 C x8 ( ) = 56x 5 ( 8) = 448x 5 PTS: REF: 008a STA: A.A.6 TOP: Binomial Expansions 66 ANS: PTS: REF: 0807a STA: A.A.44 TOP: Inverse of Functions KEY: equations 67 ANS: 68% 50 = 4 PTS: REF: 080a STA: A.S.5 TOP: Normal Distributions KEY: predict 68 ANS: 4 PTS: REF: 07a STA: A.S. TOP: Analysis of Data 69 ANS: 7 ± 7 4()( ) () = 7 ± 7 4 PTS: REF: 08009a STA: A.A.5 TOP: Quadratic Formula 70 ANS: PTS: REF: 064a STA: A.A.6 TOP: Domain and Range 9

65 ID: A 7 ANS: 9 C a6 ( 4b) = 576a 6 b PTS: REF: 066a STA: A.A.6 TOP: Binomial Expansions 7 ANS: 6a 4 b + (7 6)a 4 b a 6ab + a 6ab 4a 6ab PTS: REF: 09a STA: A.N. TOP: Operations with Radicals 7 ANS: 4 π b = π = 6π PTS: REF: 0607a STA: A.A.69 TOP: Properties of Graphs of Trigonometric Functions 74 ANS: x + x + = x + + x x + = x x + = x KEY: period PTS: REF: 0a STA: A.A.9 TOP: Negative Exponents 75 ANS: PTS: REF: 060a STA: A.A.9 TOP: Negative Exponents 76 ANS: The binomials are conjugates, so use FL. PTS: REF: 006a STA: A.N. TOP: Operations with Polynomials 77 ANS: 4 PTS: REF: 060a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 78 ANS: PTS: REF: fall094a STA: A.S.4 TOP: Dispersion KEY: range, quartiles, interquartile range, variance 0

66 ID: A 79 ANS: 4 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 80 ANS: b 4ac = ( 0) 4()(5) = = 0 PTS: REF: 00a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation 8 ANS: 4 PTS: REF: 067a STA: A.A.66 TOP: Determining Trigonometric Functions 8 ANS: 4 x = y. y (y) + = 0 y 9y = 8y = y = 4 y = ±. x = ( ) = 6 PTS: REF: 06a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations 8 ANS: x x 6 = x 6 x 4x = 0 x(x = 0 x = 0, 4 PTS: REF: 0805a STA: A.A. TOP: Quadratic-Linear Systems KEY: equations 84 ANS: 4 S n = n [a + (n )d] = [(8) + ( )] = 798 PTS: REF: 060a STA: A.A.5 TOP: Series KEY: arithmetic 85 ANS: PTS: REF: fall09a STA: A.A.65 TOP: Graphing Trigonometric Functions

67 ID: A 86 ANS: 80 π = 60 π PTS: REF: 00a STA: A.M. TOP: Radian Measure KEY: degrees 87 ANS: PTS: REF: 004a STA: A.A.64 TOP: Using Inverse Trigonometric Functions KEY: unit circle 88 ANS: 4 PTS: REF: 0607a STA: A.A.9 TOP: Properties of Logarithms KEY: antilogarithms 89 ANS: 4 5 C 5 =, C 5 = 5 C 0 = 5, 0. 5 C 5 =, 68, 760. PTS: REF: 067a STA: A.S. TOP: Combinations 90 ANS: = 5000e.0475t ln.0475 = e.0475t ln = ln e.0475t. t =.0475t ln e.0475 PTS: REF: 067a STA: A.A.6 TOP: Exponential Growth 9 ANS: x > 5. x < 5 x > 6 x > x > 4 x < PTS: REF: 0607a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 9 ANS: PTS: REF: 00a STA: A.A.66 TOP: Determining Trigonometric Functions 9 ANS: 4 PTS: REF: 0606a STA: A.A.60 TOP: Unit Circle 94 ANS: x = 5 4 = 65 PTS: REF: 0606a STA: A.A.8 TOP: Logarithmic Equations KEY: basic

68 ID: A 95 ANS: 4 PTS: REF: 06a STA: A.A.9 TOP: Domain and Range KEY: real domain 96 ANS: 4 PTS: REF: 060a STA: A.S. TOP: Analysis of Data 97 ANS: The binomials are conjugates, so use FL. PTS: REF: 060a STA: A.N. TOP: Operations with Polynomials 98 ANS: cos( ) = cos(55 ) PTS: REF: 0604a STA: A.A.57 TOP: Reference Angles 99 ANS: 4 cosa = sin A = = 9 = 7 9 PTS: REF: 0a STA: A.A.77 TOP: Double Angle Identities KEY: evaluating 00 ANS: PTS: REF: 080a STA: A.A.46 TOP: Transformations with Functions and Relations 0 ANS: x x + y + 6y = x x + + y + 6y + 9 = (x ) + (y + ) = 7 PTS: REF: 0606a STA: A.A.47 TOP: Equations of Circles 0 ANS: =.5 PTS: REF: 07a STA: A.A.9 TOP: Sequences 0 ANS: cosk = 5 6 K = cos 5 6 K ' PTS: REF: 060a STA: A.A.55 TOP: Trigonometric Ratios

69 ID: A 04 ANS: PTS: REF: 00a STA: A.A.7 TOP: Identifying the Equation of a Trigonometric Graph 05 ANS: 6 = 08 = 6 = 6. cotj = A O = 6 6 = PTS: REF: 00a STA: A.A.55 TOP: Trigonometric Ratios 06 ANS: 4 PTS: REF: 0a STA: A.N.8 TOP: Conjugates of Complex Numbers 07 ANS: K = (0)(8)sin0 = PTS: REF: fall0907a STA: A.A.74 TOP: Using Trigonometry to Find Area KEY: basic 08 ANS: S = b a = ( ) 4 = 4. P = c a = 8 4 = PTS: REF: fall09a STA: A.A. TOP: Roots of Quadratics KEY: basic 09 ANS: 0 sin 5 = sin B B 48, < < 80 PTS: REF: 0a STA: A.A.75 TOP: Law of Sines - The Ambiguous Case 0 ANS: K = (0)(8)sin46 9 PTS: REF: 080a STA: A.A.74 TOP: Using Trigonometry to Find Area KEY: parallelograms ANS: PTS: REF: 06a STA: A.A.7 TOP: Law of Sines KEY: side, without calculator ANS: PTS: REF: 067a STA: A.S.6 TOP: Regression ANS: ( ) = 9 8 = 8 9 PTS: REF: 0600a STA: A.N. TOP: Negative and Fractional Exponents 4 ANS: 4 PTS: REF: 04a STA: A.A.8 TOP: Evaluating Logarithmic Expressions 4

70 ID: A 5 ANS: 4 sin 40 = 0 sin M M < 80. (80 8) + 40 < 80 PTS: REF: 067a STA: A.A.75 TOP: Law of Sines - The Ambiguous Case 6 ANS: PTS: REF: 0a STA: A.A.64 TOP: Using Inverse Trigonometric Functions KEY: advanced 7 ANS: 4 PTS: REF: 06a STA: A.A.50 TOP: Solving Polynomial Equations 8 ANS: 4 PTS: REF: 064a STA: A.S. TOP: Average Known with Missing Data 9 ANS: = 448. 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 0 ANS: x y 7 = 6 x y = x 4 y PTS: REF: 0607a STA: A.A. TOP: Radicals as Fractional Exponents 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.4 TOP: Defining Functions ANS: PTS: REF: 068a STA: A.A.4 TOP: Defining Functions ANS: PTS: REF: 0a STA: A.A.9 TOP: Domain and Range KEY: real domain 5

71 ID: A 4 ANS: 0( ) = x( x + ) x x 40 = 0 40 = x + x x x 0 = 0 (x + (x 5) = 0 x = 4, 5 PTS: REF: 0a STA: A.A.5 TOP: Inverse Variation 5 ANS: PTS: REF: 06007a STA: A.S.9 TOP: Differentiating Permutations and Combinations 6 ANS: 4 8 k + 4 = 4 k. ( ) k + 4 = ( ) k 9k + = 4k 9k + = 4k 5k = 4 k = 4 5 PTS: REF: 009a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 7 ANS: common difference is. b n = x + n 0 = x + () 8 = x PTS: REF: 0804a STA: A.A.9 TOP: Sequences 8 ANS: tan0 =. Arc cos k = 0 k = cos0 k = PTS: REF: 06a STA: A.A.64 TOP: Using Inverse Trigonometric Functions KEY: advanced 9 ANS: PTS: REF: 06a STA: A.A.4 TOP: Completing the Square 6

72 ID: A 0 ANS: 4 b 4ac = 4(9)( = = 5 PTS: REF: 0806a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation ANS: 4.% + 9.% = 5.% PTS: REF: 0a STA: A.S.5 TOP: Normal Distributions KEY: probability ANS: 4 PTS: REF: 00a STA: A.A.8 TOP: Defining Functions KEY: graphs 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 4 ANS: 4 PTS: REF: 0a STA: A.A.56 TOP: Determining Trigonometric Functions KEY: degrees, common angles 5 ANS: f( = ( =. g( ) = ( ) + 5 = PTS: REF: fall090a STA: A.A.4 TOP: Compositions of Functions KEY: numbers 6 ANS: logx = loga + loga logx = log6a logx = log6 logx = + loga log6 + loga logx = log6 + loga PTS: REF: 04a STA: A.A.9 TOP: Properties of Logarithms KEY: splitting logs 7

73 ID: A 7 ANS: x + = 6x x 6x = x 6x + 9 = + 9 (x ) = 7 PTS: REF: 06a STA: A.A.4 TOP: Completing the Square 8 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 9 ANS: 4 y sin θ = y = sin θ + f(θ) = sin θ + PTS: REF: fall097a STA: A.A.40 TOP: Functional Notation 40 ANS: n 0 Σ n + n = + = + = 8 = 4 PTS: REF: fall09a STA: A.N.0 TOP: Sigma Notation KEY: basic 4 ANS: PTS: REF: 0a STA: A.A.7 TOP: Graphing Trigonometric Functions 8

74 ID: A 4 ANS: secx = secx = cosx = x = 5, 5 PTS: REF: 0a STA: A.A.68 TOP: Trigonometric Equations KEY: reciprocal functions 4 ANS: PTS: REF: 060a STA: A.A.0 TOP: Fractional Exponents as Radicals 44 ANS: 4 x + 4 x + x + x + = (x + ) x + x + = x + PTS: REF: 0a STA: A.A.5 TOP: Rationalizing Denominators KEY: index = 45 ANS: w 5 w 9 = (w 4 ) = w PTS: REF: 080a STA: A.A.8 TOP: Negative and Fractional Exponents 46 ANS: + cos A = sin A = sin AcosA cos A = 5 = 5 9 cosa = + 5, sin A is acute. = PTS: REF: 007a STA: A.A.77 TOP: Double Angle Identities KEY: evaluating 47 ANS: PTS: REF: 065a STA: A.A.66 TOP: Determining Trigonometric Functions 9

75 ID: A 48 ANS: 4 PTS: REF: fall0908a STA: A.A.8 TOP: Defining Functions KEY: graphs 49 ANS: π 80 π = 65 PTS: REF: 0600a STA: A.M. TOP: Radian Measure KEY: degrees 50 ANS: PTS: REF: 067a STA: A.S.9 TOP: Differentiating Permutations and Combinations 5 ANS: PTS: REF: 060a STA: A.S.8 TOP: Correlation Coefficient 5 ANS: 4 = (a)(8)sin6 4.5a a PTS: REF: 06a STA: A.A.74 TOP: Using Trigonometry to Find Area KEY: basic 5 ANS: PTS: REF: 005a STA: A.A.8 TOP: Defining Functions KEY: graphs 54 ANS: 4 PTS: REF: 06005a STA: A.A.50 TOP: Solving Polynomial Equations 55 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 56 ANS: 5 C (x) ( ) = 0 9x 8 = 70x PTS: REF: fall099a STA: A.A.6 TOP: Binomial Expansions 0

76 ID: A 57 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.4 TOP: Quadratic Inequalities KEY: one variable 58 ANS: PTS: REF: 0608ge STA: A.A.5 TOP: Domain and Range 59 ANS: cos θ cosθ = cos θ (cos θ sin θ) = sin θ PTS: REF: 0604a STA: A.A.77 TOP: Double Angle Identities KEY: simplifying 60 ANS: PTS: REF: 0800a STA: A.A.55 TOP: Trigonometric Ratios 6 ANS: 0 = 5 p 5 = 5 p 5 = p PTS: REF: 06a STA: A.A.5 TOP: Inverse Variation 6 ANS: PTS: REF: 0a STA: A.N.8 TOP: Conjugates of Complex Numbers 6 ANS: PTS: REF: 06a STA: A.A.54 TOP: Graphing Logarithmic Functions 64 ANS: tanθ = 0.. tanθ = θ = tan θ = 60, 40 PTS: REF: fall090a STA: A.A.68 TOP: Trigonometric Equations KEY: basic

77 ID: A 65 ANS: 4 x + 4x = 6. x + 8x = 6 ( ) x +4x = 6 x + 8x = 6 x + 8x + 6 = 0 x + 4x + = 0 (x + )(x + ) = 0 x = x = PTS: REF: 0605a STA: A.A.7 TOP: Exponential Equations KEY: common base shown 66 ANS: 4 a 5 a = 4a 5 a PTS: REF: 0604a STA: A.A. TOP: Simplifying Radicals KEY: index > 67 ANS: ( x) + ( x) + (4 x) + (5 x) x + 9 x + x + 5 x 46 x PTS: REF: 065a STA: A.N.0 TOP: Sigma Notation KEY: basic 68 ANS: (7.(.8)sin6.4 PTS: REF: 08a STA: A.A.74 TOP: Using Trigonometry to Find Area KEY: basic 69 ANS: PTS: REF: 08007a STA: A.A.64 TOP: Using Inverse Trigonometric Functions KEY: basic 70 ANS: 4 PTS: REF: 068a STA: A.A.49 TOP: Equations of Circles 7 ANS: x 4 x x + 4 = x 4 4x x + 4 8x = (x + )(x ) 4x 8x (x + ) = x PTS: REF: fall090a STA: A.A.7 TOP: Complex Fractions 7 ANS: 4 6x x x = x(x + x 6) = x(x + )(x ) PTS: REF: fall097a STA: A.A.7 TOP: Factoring Polynomials KEY: single variable

78 ID: A 7 ANS: PTS: REF: 0609a STA: A.N.7 TOP: Imaginary Numbers 74 ANS: sin S = 8 7 S = sin 8 7 S 8 4' PTS: REF: 06a STA: A.A.55 TOP: Trigonometric Ratios 75 ANS: PTS: REF: fall09a STA: A.A.9 TOP: Domain and Range KEY: real domain 76 ANS: PTS: REF: 0804a STA: A.N.8 TOP: Conjugates of Complex Numbers 77 ANS: PTS: REF: 064a STA: A.A.9 TOP: Negative Exponents 78 ANS: PTS: REF: 06004a STA: A.A.5 TOP: Identifying the Equation of a Graph 79 ANS: The roots are,,. PTS: REF: 080a STA: A.A.50 TOP: Solving Polynomial Equations 80 ANS: PTS: REF: 0608a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 8 ANS: PTS: REF: 066a STA: A.S.8 TOP: Correlation Coefficient 8 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 8 ANS: PTS: REF: 00a STA: A.A.5 TOP: Graphing Exponential Functions 84 ANS: PTS: REF: 0a STA: A.A.9 TOP: Domain and Range KEY: real domain 85 ANS: PTS: REF: 069a STA: A.A.65 TOP: Graphing Trigonometric Functions 86 ANS: PTS: REF: 06a STA: A.A.49 TOP: Equations of Circles 87 ANS: PTS: REF: 060a STA: A.A.6 TOP: Domain and Range

79 ID: A 88 ANS: 4 5 x = x 5 = 5 x PTS: REF: 08a STA: A.A.0 TOP: Fractional Exponents as Radicals 89 ANS: PTS: REF: 0800a STA: A.A.5 TOP: Domain and Range 90 ANS: 00 = 00 PTS: REF: 06006a STA: A.N.6 TOP: Square Roots of Negative Numbers 9 ANS: 4 g = =. f() = 4() = 4 PTS: REF: 004a STA: A.A.4 TOP: Compositions of Functions KEY: numbers 9 ANS: PTS: REF: 066a STA: A.A.4 TOP: Compositions of Functions KEY: variables 9 ANS: =. 6 d = 4 4 = 6d = d = PTS: REF: 060a STA: A.A.5 TOP: Inverse Variation 94 ANS: 6 P 6!! = 70 = 60 PTS: REF: 04a STA: A.S.0 TOP: Permutations 95 ANS: 4 ± ( ) 4()( 9) () = ± 45 = ± 5 PTS: REF: 06009a STA: A.A.5 TOP: Quadratic Formula 96 ANS: 4 PTS: REF: 0606a STA: A.A.9 TOP: Sequences 4

80 ID: A 97 ANS: a b 4 6b = 64 a5 b a PTS: REF: 066a STA: A.A. TOP: Sequences 98 ANS: Cofunctions tangent and cotangent are complementary PTS: REF: 0604a STA: A.A.58 TOP: Cofunction Trigonometric Relationships 99 ANS: PTS: REF: 069a STA: A.N.8 TOP: Conjugates of Complex Numbers 00 ANS: If sin x = 0.8, then cosx = 0.6. tan x = = = 0.5. PTS: REF: 060a STA: A.A.77 TOP: Half Angle Identities 0 ANS: 4a + 6 = 4a 0. 4a + 6 = 4a = a = 4 a = 4 8 = 8 0 PTS: REF: 006a STA: A.A. TOP: Absolute Value Equations 0 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 0 ANS: 0 f0 = ( 0) 6 = 0 84 = 5 4 PTS: REF: 060a STA: A.A.4 TOP: Functional Notation 04 ANS: 4 (x + i) (x i) = x + xi + i (x xi + i ) = 4xi PTS: REF: 07a STA: A.N.9 TOP: Multiplication and Division of Complex Numbers 05 ANS: PTS: REF: 064a STA: A.A.8 TOP: Defining Functions KEY: graphs 5

81 ID: A 06 ANS: sin θ + cos θ = sin θ cos θ = sec θ PTS: REF: 06a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 07 ANS: n 4 5 Σ r + r + = = = 0 8 PTS: REF: 068a STA: A.N.0 TOP: Sigma Notation KEY: basic 08 ANS: s = θ r = π 8 6 = π PTS: REF: 06a STA: A.A.6 TOP: Arc Length KEY: arc length 09 ANS: (i)(i) (m + i) (i)(4i )(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 0 ANS: logx ( logy + logz) = logx logy logz = log x PTS: REF: 0600a STA: A.A.9 TOP: Properties of Logarithms ANS: 4x 5 > 4x 5 > 4x > 8 x > or 4x 5 < 4x 5 < 4x < x < y z PTS: REF: 0609a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 6

82 ID: A ANS: PTS: REF: 006a STA: A.A.8 TOP: Negative and Fractional Exponents ANS: PTS: REF: 04a STA: A.N. TOP: Operations with Polynomials 4 ANS: 6(x 5) = 6x 0 PTS: REF: 009a STA: A.A.4 TOP: Compositions of Functions KEY: variables 5 ANS: 4 ( + 5 )( 5 ) = 9 5 = 4 PTS: REF: 0800a STA: A.N.4 TOP: Operations with Irrational Expressions KEY: without variables index = 6 ANS: PTS: REF: fall095a STA: A.S.5 TOP: Normal Distributions KEY: interval 7 ANS: y x x 6 y (x )(x + ) PTS: REF: 0607a STA: A.A.4 TOP: Quadratic Inequalities KEY: two variables 8 ANS: 4 cosθ = cosθ = θ = cos = 60, 00 PTS: REF: 060a STA: A.A.68 TOP: Trigonometric Equations KEY: basic 7

83 ID: A 9 ANS: 8 = 64 PTS: REF: fall0909a STA: A.A.8 TOP: Evaluating Logarithmic Expressions 0 ANS: 6 sin 5 = 0 sin N N < 80 (80 7) + 5 < 80 PTS: REF: 066a STA: A.A.75 TOP: Law of Sines - The Ambiguous Case ANS: PTS: REF: 060a STA: A.A.8 TOP: Defining Functions ANS: 4 s = θ r = 4 = 8 PTS: REF: fall09a STA: A.A.6 TOP: Arc Length KEY: arc length ANS: sum of the roots, b a = ( 9) 4 = 9 4. product of the roots, c a = 4 PTS: REF: 0608a STA: A.A. TOP: Roots of Quadratics KEY: basic 4 ANS: PTS: REF: 06a STA: A.S.5 TOP: Binomial Probability KEY: modeling 5 ANS: 4 x x 8 x x = x 4x x x 8 = x(x (x (x + ) = x x + x PTS: REF: 0605a STA: A.A.7 TOP: Complex Fractions 6 ANS: 4 PTS: REF: 08005a STA: A.A.60 TOP: Unit Circle 7 ANS: PTS: REF: fall090a STA: A.A.76 TOP: Angle Sum and Difference Identities KEY: simplifying 8

84 ID: A 8 ANS: x + 6 = (x + ). 4 is an extraneous solution. x + 6 = x + 4x = x + x 0 = (x + (x ) x = 4 x = PTS: REF: 06a STA: A.A. TOP: Solving Radicals KEY: extraneous solutions 9 ANS: PTS: REF: 00a STA: A.A.0 TOP: Sequences 0 ANS: PTS: REF: 07a STA: A.S.9 TOP: Differentiating Permutations and Combinations ANS: 4 PTS: REF: 0a STA: A.A. TOP: Using the Discriminant KEY: determine nature of roots given equation ANS: 4 Students entering the library are more likely to spend more time studying, creating bias. PTS: REF: fall0904a STA: A.S. TOP: Analysis of Data ANS:. PTS: REF: 065a STA: A.S.8 TOP: Correlation Coefficient 4 ANS: PTS: REF: fall096a STA: A.A.46 TOP: Transformations with Functions and Relations 5 ANS: 4 log 4 (5x) = log 4 (5x) = 5x = 4 5x = 8 x = 8 5 PTS: REF: fall09a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 9

85 ID: A 6 ANS: 0 = 0() = () t 60 log = log() log = tlog log log = t 00 = t t 60 t 60 PTS: REF: 005a STA: A.A.6 TOP: Exponential Growth 7 ANS: 4 PTS: REF: 060a STA: A.A.4 TOP: Defining Functions 8 ANS: b a = 6 =. c a = 4 = PTS: REF: 0a STA: A.A. TOP: Roots of Quadratics KEY: basic 9 ANS: k + 5 = k + 4k + 6 k + = 4k + 6 = k + 44 k = 8 k = 4 PTS: REF: 06a STA: A.S. TOP: Average Known with Missing Data 40 ANS: 4 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 = 4 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 0

86 ID: A 4 ANS: PTS: REF: fall094a STA: A.A.9 TOP: Negative and Fractional Exponents 4 ANS: C (x4 ) ( y) = 6x 4 y PTS: REF: 05a STA: A.A.6 TOP: Binomial Expansions

87 ID: A Algebra /Trigonometry Point Regents Exam Questions Answer Section 44 ANS: x y 9. x 4 y 5 y = 5 (x y 7 ) (x y 7 ) x 4 = y 5 (4x 6 y 4 ) x 4 = x 6 y 9 x 4 = x y 9 PTS: REF: 064a STA: A.A.9 TOP: Negative Exponents 45 ANS: Sum b a = 5. Product c a = 5 PTS: REF: 0600a STA: A.A.0 TOP: Roots of Quadratics 46 ANS: y = sinx. 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 47 ANS: 4 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 48 ANS: e ln = e ln = e ln 8 = 8 PTS: REF: 06a STA: A.A. TOP: Evaluating Exponential Expressions 49 ANS: ( ) + ( ) + ( ) + (4 ) + (5 ) = = 0 PTS: REF: 0a STA: A.N.0 TOP: Sigma Notation KEY: basic 50 ANS: = 6 4 PTS: REF: 06a STA: A.A.56 TOP: Determining Trigonometric Functions KEY: degrees, common angles

88 ID: A 5 ANS: y = 0 PTS: REF: 060a STA: A.A.5 TOP: Graphing Exponential Functions 5 ANS: 7.4 PTS: REF: 0609a STA: A.S.4 TOP: Dispersion KEY: basic, group frequency distributions 5 ANS: y = 80.77(0.95 x PTS: REF: 06a STA: A.S.7 TOP: Exponential Regression 54 ANS: i + i 8 + i + n = 0 i + ( ) i + n = 0 + n = 0 n = PTS: REF: 068a STA: A.N.7 TOP: Imaginary Numbers 55 ANS: x 7 x 4 x or x 7 x 0 x 5 PTS: REF: 04a STA: A.A. TOP: Absolute Value Inequalities KEY: graph 56 ANS: A = 750e (0.0)(8) 95 PTS: REF: 069a STA: A.A. TOP: Evaluating Exponential Expressions

89 ID: A 57 ANS: 5 n = 7n PTS: REF: 0809a STA: A.A.4 TOP: Sigma Notation 58 ANS: 4x no solution. x = + x 4x x = 4(x ) x = 4 PTS: REF: fall090a STA: A.A. TOP: Solving Rationals KEY: rational solutions 59 ANS: 0700 = 50e t 64 = e t ln64 = ln e t ln64 = tln e ln64 = t.4 t PTS: REF: 0a STA: A.A.6 TOP: Exponential Growth 60 ANS: b 4ac = 0 k 4()( = 0 k 6 = 0 (k + (k = 0 k = ±4 PTS: REF: 0608a STA: A.A. TOP: Using the Discriminant KEY: determine equation given nature of roots 6 ANS: 7 C = = PTS: REF: 065a STA: A.S.5 TOP: Binomial Probability KEY: exactly

90 ID: A 6 ANS: Sum b a =. Product c a = PTS: REF: 068a STA: A.A.0 TOP: Roots of Quadratics 6 ANS: 4 9 x 4 x +. x = x x = 4 9 x x x + = 4 9 x 4 x + PTS: REF: 0804a STA: A.N. TOP: Operations with Polynomials 64 ANS: 7. f( ) = ( ) 6 =. g(x) = = 7. PTS: REF: 065a STA: A.A.4 TOP: Compositions of Functions KEY: numbers 65 ANS: K = absinc = 4 0 sin PTS: REF: 0604a STA: A.A.74 TOP: Using Trigonometry to Find Area KEY: parallelograms 66 ANS: t 8 75t 4 = t 4 (4t 4 5) = t 4 (t + 5)(t 5) PTS: REF: 06a STA: A.A.7 TOP: Factoring the Difference of Perfect Squares 67 ANS: a b 4 KEY: binomial PTS: REF: 0a STA: A.A. TOP: Simplifying Radicals KEY: index > 68 ANS: a =. a = () = 5. a = (5) = 9. PTS: REF: 06a STA: A.A. TOP: Recursive Sequences 4

91 ID: A 69 ANS: PTS: REF: 060a STA: A.A.60 TOP: Unit Circle 70 ANS: x = 7 x = 8 x = 8 x = 4 4 PTS: REF: 069a STA: A.A.8 TOP: Logarithmic Equations KEY: advanced 7 ANS: 08x 5 y 8 6xy 5 = 8x 4 y = x y y PTS: REF: 0a STA: A.A.4 TOP: Operations with Radicals KEY: with variables index = 7 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 5

92 ID: A 7 ANS: PTS: REF: 04a STA: A.A.5 TOP: Graphing Exponential Functions 74 ANS: 80 π PTS: REF: 05a STA: A.M. TOP: Radian Measure KEY: degrees 75 ANS: cosθ cosθ cos θ = cos θ = sin θ PTS: REF: 060a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 76 ANS:, 5, 8, PTS: REF: fall094a STA: A.A. TOP: Recursive Sequences 77 ANS: (x + 5) + (y ) = PTS: REF: 080a STA: A.A.49 TOP: Writing Equations of Circles 78 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 79 ANS: Ordered, the heights are 7, 7, 7, 74, 74, 75, 78, 79, 79, 8. Q = 7 and Q = = 7. PTS: REF: 0a STA: A.S.4 TOP: Dispersion KEY: range, quartiles, interquartile range, variance 6

93 ID: A 80 ANS: cotxsin x secx = cosx sin x sin x cosx = cos x PTS: REF: 064a STA: A.A.58 TOP: Reciprocal Trigonometric Relationships 8 ANS: 6 x + = 64 x + (4 ) x + = (4 ) x + 4x + 6 = x + 6 x = 0 PTS: REF: 08a STA: A.A.7 TOP: Exponential Equations KEY: common base not shown 8 ANS: 6y 7 0 y 5 y. y y y + 5 = 6y + 0 y 4y 5 y = 6y 7 0 y 5 y PTS: REF: 068a STA: A.N. TOP: Operations with Polynomials 8 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 84 ANS: r = + =. (x + 5) + (y ) = PTS: REF: 04a STA: A.A.49 TOP: Writing Equations of Circles 85 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 7

94 ID: A 86 ANS: 97º π PTS: REF: fall09a STA: A.M. TOP: Radian Measure KEY: degrees 87 ANS:, PTS: REF: fall09a STA: A.A. TOP: Evaluating Exponential Expressions 88 ANS: 5( + ) = 5( + ) 9 = 5( + ) 7 PTS: REF: fall098a STA: A.N.5 TOP: Rationalizing Denominators 89 ANS: K = absinc = 8 sin60 = 96 = 98 PTS: REF: 064a STA: A.A.74 TOP: Using Trigonometry to Find Area KEY: Parallelograms 90 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: 04a STA: A.S.5 TOP: Normal Distributions KEY: percent 9 ANS: 6 = 9w 8 = w PTS: REF: 00a STA: A.A.5 TOP: Inverse Variation 9 ANS: no. over 0 is more than standard deviation above the mean PTS: REF: 069a STA: A.S.5 TOP: Normal Distributions KEY: predict 8