JEFFERSON MATH PROJECT REGENTS BY PERFORMANCE INDICATOR: TOPIC

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1 JEFFERSON MATH PROJECT REGENTS BY PERFORMANCE INDICATOR: TOPIC NY Algebra /Trigonometry Regents Exam Questions from Fall 009 to August 00 Sorted by PI: Topic Dear Sir I have to acknolege the reciept of your favor of May. 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 TABLE OF CONTENTS TOPIC PI: SUBTOPIC QUESTION NUMBER GRAPHS AND STATISTICS A.S.-, -: Analysis of Data... - A.S.: Dispersion... - A.S.7: Regression A.S.8: Correlation Coefficient... 7 A.S.5: Normal Distributions PROBABILITY A.S.0: Permutations A.S.: Combinations... A.S.9: Differentiating Permutations and Combinations... A.S.: Sample Space... 5 A.S.6: Binomial Probability ABSOLUTE VALUE QUADRATICS A.A.: Absolute Value Inequalities... 8 A.A.0: Roots of Quadratics... 9 A.A.: Roots of Quadratics... 0 A.A.7: Factoring Polynomials... - A.A.5: Quadratic Formula A.A.: Using the Discriminant A.A.: Completing the Square... 8 A.A.: Quadratic Inequalities... 9 SYSTEMS A.A.: Quadratic-Linear Systems... 0 POWERS RADICALS RATIONALS A.N.: Operations with Polynomials... A.A.8-9: Negative and Fractional Exponents A.A.: Evaluating Exponential Expressions... 6 A.A.8: Evaluating Logarithmic Expressions... 7 A.A.5: Graphing Exponential Functions... 8 A.A.5: Graphing Logarithmic Functions... 9 A.A.9: Properties of Logarithms... 0 A.A.8: Logarithmic Equations... - A.A.7: Exponential Equations... - A.A.6: Binomial Expansions... 5 A.A.6, 50: Solving Polynomial Equations 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... 6 A.A.: Solving Rationals A.A.7: Complex Fractions i-

3 FUNCTIONS A.A.0: Functional Notation 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... 7 A.A.: Inverse of Functions A.A.6: Transformations with Functions and Relations SEQUENCES AND SERIES TRIGONOMETRY A.A.9-: Sequences A.A.: Recursive Sequences... 8 A.N.0, A.: Sigma Notation A.A.55: Trigonometric Ratios A.M.: Radian Measure A.A.60: Unit Circle A.A.6: Determining Trigonometric Functions... 9 A.A.6: Using Inverse Trigonometric Functions... 9 A.A.6: Arc Length A.A.58: Cofunction and Reciprocal Trigonometric Functions A.A.76: Angle Sum and Difference Identities A.A.77: Double and Half Angle Identities A.A.68: Trigonometric Equations A.A.69: Properties of Trigonometric Functions... 0 A.A.65, 70-7: Graphing Trigonometric Functions A.A.6: Domain and Range A.A.7: Using Trigonometry to Find Area A.A.75: Law of Sines - The Ambiguous Case... A.A.7: Law of Cosines... - A.A.7: Vectors... CIRCLES A.A.7, 9: Equations of Circles ii-

4 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. A.S.: DISPERSION The scores of one class on the Unit mathematics test are shown in the table below. A.S.: ANALYSIS OF DATA 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 Find the population standard deviation of these scores, to the nearest tenth.

5 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic The table below shows the first-quarter averages for Mr. Harper s statistics class. A.S.7: REGRESSION 5 The table below shows the number of new stores in a coffee shop chain that opened during the years 986 through 99. What is the population variance for this set of data? 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. 6 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. A.S.8: CORRELATION COEFFICIENT 7 Which value of r represents data with a strong negative linear correlation between two variables?

6 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.S.5: NORMAL DISTRIBUTIONS 8 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? 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 PROBABILITY A.S.0: PERMUTATIONS 0 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!!! Find the total number of different twelve-letter arrangements that can be formed using the letters in the word PENNSYLVANIA. 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. A.S.: COMBINATIONS 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 A.S.9: DIFFERENTIATING PERMUATIONS AND COMBINATIONS 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!! C 0 P 0 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. A.S.5: BINOMIAL PROBABILITY 6 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?

7 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 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. ABSOLUTE VALUE A.A.: ABSOLUTE VALUE INEQUALITIES 8 Which graph represents the solution set of 6x 7 5? QUADRATICS A.A.0: ROOTS OF QUADRATICS 9 Find the sum and product of the roots of the equation 5x + x = 0. A.A.: ROOTS OF QUADRATICS 0 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 A.A.7: FACTORING POLYNOMIALS Factored completely, the expression x + 0x x is equivalent to x (x + 6)(x ) (x + x)(x x) x (x )(x + ) x (x + )(x ) Factored completely, the expression 6x x x is equivalent to x(x + )(x ) x(x )(x + ) x(x )(x + ) x(x + )(x ) Factor completely: 0ax ax 5a A.A.5: QUADRATIC FORMULA The roots of the equation x + 7x = 0 are and and 7 ± 7 7 ± 7 5 The solutions of the equation y y = 9 are ± i ± i 5 ± 5 ± 5

8 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.: USING THE DISCRIMINANT 6 The roots of the equation 9x + x = 0 are imaginary real, rational, and equal real, rational, and unequal real, irrational, and unequal A.A.: QUADRATIC INEQUALITIES 9 Which graph best represents the inequality y + 6 x x? 7 Use the discriminant to determine all value of k that would result in the equation x kx + = 0 having equal roots. A.A.: COMPLETING THE SQUARE 8 Solve x x + = 0 by completing the square, expressing the result in simplest radical form. 5

9 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic SYSTEMS A.A.: QUADRATIC-LINEAR SYSTEMS 0 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 POWERS A.N.: OPERATIONS WITH POLYNOMIALS Express x as a trinomial. A.A.8: NEGATIVE AND FRACTIONAL EXPONENTS The expression a b a 6 b 5 b 5 a 6 a b a b is equivalent to a b If a = and b =, what is the value of the expression a b? When simplified, the expression equivalent to w 7 w w 7 w w 5 w 9 A.A.9: NEGATIVE AND FRACTIONAL EXPONENTS 5 When x is divided by x, the quotient is x x (x ) A.A.: EVALUATING EXPONENTIAL EXPRESSIONS 6 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. A.A.8: EVALUATING LOGARITHMIC EXPRESSIONS 7 The expression log 8 6 is equivalent to 8 8 is 6

10 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.5: GRAPHING EXPONENTIAL FUNCTIONS 8 The graph of the equation y = x has an asymptote. On the grid below, sketch the graph of y = x and write the equation of this asymptote. A.A.5: GRAPHING LOGARITHMIC FUNCTIONS 9 If a function is defined by the equation f(x) = x, which graph represents the inverse of this function? 7

11 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.9: PROPERTIES OF LOGARITHMS 0 The expression logx (logy + logz) is equivalent to log x y z log x z y log x yz log xz y A.A.8: LOGARITHMIC EQUATIONS What is the solution of the equation log (5x) =? A.A.6: BINOMIAL EXPANSIONS 5 What is the fourth term in the expansion of (x ) 5? 70x 0x 70x,080x A.A.6: SOLVING POLYNOMIAL EQUATIONS 6 Solve the equation 8x + x 8x 9 = 0 algebraically for all values of x. A.A.50: SOLVING POLYNOMIAL EQUATIONS 7 The graph of y = f(x) is shown below. Solve algebraically for x: log x + x + x x A.A.7: EXPONENTIAL EQUATIONS = The solution set of x + x = 6 is {,} {,} {, } {, } What is the value of x in the equation 9 x + = 7 x +? Which set lists all the real solutions of f(x) = 0? {,} {,} {,0,} {,0,} 8

12 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 8 The graph of y = x x + x + 6 is shown below. A.A.: OPERATIONS WITH RADICALS 5 The expression ab b a 8b + 7ab 6b is equivalent to ab 6b 6ab b 5ab + 7ab 6b 5ab b + 7ab 6b A.N.5: RATIONALIZING DENOMINATORS What is the product of the roots of the equation x x + x + 6 = 0? RADICALS A.N.: OPERATIONS WITH RADICALS 9 The product of ( + 5) and ( 5) is Express 5 x 7x in simplest radical form. 5 Which expression is equivalent to ? 5 5 Express with a rational denominator, in simplest radical form. A.A.5: RATIONALIZING DENOMINATORS 5 The fraction a b b ab b ab a is equivalent to a b 9

13 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.: SOLVING RADICALS 55 The solution set of the equation x + = x is {} {0} {,6} {,} A.A.0: EXPONENTS AS RADICALS 56 The expression (x ) (x ) (x ) (x ) (x ) is equivalent to A.N.6: SQUARE ROOTS OF NEGATIVE NUMBERS 57 In simplest form, 00 is equivalent to i 0 5i 0i i 5 A.N.7: IMAGINARY NUMBERS 58 The product of i 7 and i 5 is equivalent to i i 59 The expression i + i is equivalent to i i + i + i A.N.8: CONJUGATES OF COMPLEX NUMBERS 60 What is the conjugate of + i? + i i i + i A.N.9: MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS 6 The expression ( 7i) is equivalent to 0 + 0i 0 i i 58 i RATIONALS A.A.: SOLVING RATIONALS 6 Solve for x: x x = + x 6 Solve algebraically for x: A.A.7: COMPLEX FRACTIONS x + x = x 9 6 Written in simplest form, the expression equivalent to x x x x x + 65 Express in simplest form: d d + d x x x + is 0

14 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic FUNCTIONS A.A.0: FUNCTIONAL NOTATION 66 The equation y sinθ = may be rewritten as f(y) = sinx + f(y) = sinθ + f(x) = sinθ + f(θ) = sinθ + A.A.8: DEFINING FUNCTIONS 68 Which graph does not represent a function? A.A.5: IDENTIFYING THE EQUATION OF A GRAPH 67 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 69 Which relation is not a function? (x ) + y = x + x + y = x + y = xy =

15 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.: DEFINING FUNCTIONS 70 Which graph represents a one-to-one function? A.A.5: DOMAIN AND RANGE 7 What are the domain and the range of the function shown in the graph below? {x x > }; {y y > } {x x }; {y y } {x x > }; {y y > } {x x }; {y y } A.A.: COMPOSITIONS OF FUNCTIONS 7 Which function is not one-to-one? {(0,),(,),(,),(,)} {(0,0),(,),(,),(,)} {(0,),(,0),(,),(,)} {(0,),(,0),(,0),(,)} A.A.9: DOMAIN AND RANGE 7 What is the domain of the function f(x) = x +? (, ) (, ) [, ) [, ) 7 If f(x) = x and g(x) = x + 5, what is the value of (g f)()?.5 6 A.A.: INVERSE OF FUNCTIONS 75 Which two functions are inverse functions of each other? f(x) = sinx and g(x) = cos(x) f(x) = + 8x and g(x) = 8x f(x) = e x and g(x) = lnx f(x) = x and g(x) = x +

16 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.6: TRANSFORMATIONS WITH FUNCTIONS AND RELATIONS 76 The graph below shows the function f(x). 77 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, ) SEQUENCES AND SERIES A.A.9: SEQUENCES Which graph represents the function f(x + )? 78 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 79 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 A.A.0: SEQUENCES 80 What is the common difference of the arithmetic sequence 5,8,,? 8 5 9

17 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.: SEQUENCES 8 What is the common ratio of the geometric sequence whose first term is 7 and fourth term is 6? A.A.: RECURSIVE SEQUENCES 8 Find the first four terms of the recursive sequence defined below. a = a n = a (n ) n A.N.0: SIGMA NOTATION 8 The value of the expression (n + n ) is 6 n = 0 A.A.: SIGMA NOTATION 8 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 = 85 Express the sum using sigma notation. TRIGONOMETRY A.A.55: TRIGONOMETRIC RATIOS 86 Which ratio represents csca in the diagram below?

18 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 87 In the diagram below of right triangle KTW, KW = 6, KT = 5, and m KTW = 90. A.A.60: UNIT CIRCLE 9 In which graph is θ coterminal with an angle of 70? What is the measure of K, to the nearest minute? ' ' 55' 56' A.M.: RADIAN MEASURE 88 What is the number of degrees in an angle whose radian measure is π? Find, to the nearest minute, the angle whose measure is.5 radians. 90 What is the radian measure of an angle whose measure is 0? 7π 7π 6 7π 6 7π 5

19 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 9 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 sin0. A.A.58: COFUNCTION & RECIPROCAL TRIGONOMETRIC RELATIONSHIPS 96 If A is acute and tana =, then cota = cota = cot(90 A) = cot(90 A) = A.A.76: ANGLE SUM AND DIFFERENCE IDENTITIES A.A.6: DETERMINING TRIGONOMETRIC FUNCTIONS 9 If θ is an angle in standard position and its terminal side passes through the point (,), find the exact value of cscθ. A.A.6: USING INVERSE TRIGONOMETRIC FUNCTIONS 9 What is the principal value of cos? A.A.6: ARC LENGTH 95 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 97 The expression cosxcosx + sinxsinx is equivalent to sinx sin7x cosx cos 7x 98 If tana = and sinb = 5 and angles A and B are in Quadrant I, find the value of tan(a + B). A.A.77: DOUBLE AND HALF ANGLE IDENTITIES 99 The expression cos θ cosθ is equivalent to sin θ sin θ cos θ + cos θ A.A.68: TRIGONOMETRIC EQUATIONS 00 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 6

20 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 0 Solve the equation tanc = tanc algebraically for all values of C in the interval 0 C < Find all values of θ in the interval 0 θ < 60 that satisfy the equation sinθ = sinθ. A.A.70: GRAPHING TRIGONOMETRIC FUNCTIONS 0 Which graph represents one complete cycle of the equation y = sinπx? A.A.69: PROPERTIES OF TRIGONOMETRIC FUNCTIONS 0 What is the period of the function y = sin x π? π 6π 7

21 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.65: GRAPHING TRIGONOMETRIC FUNCTIONS 05 Which graph represents the equation y = cos x? A.A.7: GRAPHING TRIGONOMETRIC FUNCTIONS 06 Which equation is represented by the graph below? y = cotx y = cscx y = secx y = tanx A.A.6: DOMAIN AND RANGE 07 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 8

22 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic A.A.7: USING TRIGONOMETRY TO FIND AREA 08 In ABC, m A = 0, b = 0, and c = 8. What is the area of ABC to the nearest square inch? 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? 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. A.A.75: LAW OF SINES - THE AMBIGUOUS CASE A.A.7: VECTORS 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. A.A.7: EQUATIONS OF CIRCLES 5 The equation x + y x + 6y + = 0 is equivalent to (x ) + (y + ) = (x ) + (y + ) = 7 (x + ) + (y + ) = 7 (x + ) + (y + ) = 0 A.A.9: EQUATIONS OF CIRCLES 6 Write an equation of the circle shown in the graph below. 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 68. A.A.7: LAW OF COSINES In ABC, a =, b = 5, and c = 7. What is m C? 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. 9

23 Algebra /Trigonometry Regents Exam Questions by Performance Indicator: Topic 7 A circle shown in the diagram below has a center of ( 5,) and passes through point (,7). Write an equation that represents the circle. 0