Introduction - Algebra II

Transcription

1 LIFORNI STNRS TEST lgebra II Introduction - lgebra II The following released test questions are taken from the lgebra II Standards Test. This test is one of the alifornia Standards Tests administered as part of the Standardized Testing and Reporting (STR) Program under policies set by the State oard of Education. ll questions on the alifornia Standards Tests are evaluated by committees of content eperts, including teachers and administrators, to ensure their appropriateness for measuring the alifornia academic content standards in lgebra II. In addition to content, all items are reviewed and approved to ensure their adherence to the principles of fairness and to ensure no bias eists with respect to characteristics such as gender, ethnicity, and language. This document contains released test questions from the alifornia Standards Test forms in 00, 00, 00, 006, and 007. First on the pages that follow are lists of the standards assessed on the lgebra II Test. Net are released test questions. Following the questions is a table that gives the correct answer for each question, the content standard that each question is measuring, and the year each question last appeared on the test. The following table lists each reporting cluster, the number of items that appear on the eam, and the number of released test questions that appear in this document. Some of the released test questions for lgebra II are the same test questions found in different combinations on the Integrated Mathematics and alifornia Standards Tests and the Summative High School Mathematics alifornia Standards Test. NUMER OF NUMER OF REPORTING QUESTIONS ON RELESE TEST LUSTER EXM QUESTIONS Polynomials and Rational Epressions 9 Quadratics, onics, and omple Numbers 7 0 Eponents and Logarithms 9 Series, ombinatorics, and Probability and Statistics 8 TOTL 6 80 In selecting test questions for release, three criteria are used: () the questions adequately cover a selection of the academic content standards assessed on the lgebra II Test; () the questions demonstrate a range of difficulty; and () the questions present a variety of ways standards can be assessed. These released test questions do not reflect all of the ways the standards may be assessed. Released test questions will not appear on future tests. For more information about the alifornia Standards Tests, visit the alifornia epartment of Education s Web site at This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

2 LIFORNI STNRS TEST lgebra II THE POLYNOMILS N RTIONL EXPRESSIONS REPORTING LUSTER The following five alifornia content standards are included in the Polynomials and Rational Epressions reporting cluster and are represented in this booklet by test questions. These questions represent only some ways in which these standards may be assessed on the lgebra II alifornia Mathematics Standards Test. LIFORNI ONTENT STNRS IN THIS REPORTING LUSTER lgebra II.0* Students solve equations and inequalities involving absolute value..0* Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices..0* Students are adept at operations on polynomials, including long division..0* Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes. 7.0* Students add, subtract, multiply, divide, reduce, and evaluate rational epressions with monomial and polynomial denominators and simplify complicated rational epressions, including those with negative eponents in the denominator. * enotes key standards This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

3 LIFORNI STNRS TEST lgebra II THE QURTIS, ONIS, N OMPLEX NUMERS REPORTING LUSTER The following seven alifornia content standards are included in the Quadratics, onics, and omple Numbers reporting cluster and are represented in this booklet by 0 test questions. These questions represent only some ways in which these standards may be assessed on the lgebra II alifornia Mathematics Standards Test. LIFORNI ONTENT STNRS IN THIS REPORTING LUSTER lgebra II.0* Students demonstrate knowledge of how real and comple numbers are related both arithmetically and graphically. In particular, they can plot comple numbers as points in the plane. 6.0* Students add, subtract, multiply, and divide comple numbers. 8.0* Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the comple number system. 9.0* Students demonstrate and eplain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a( b) + c. 0.0* Students graph quadratic functions and determine the maima, minima, and zeros of the function. 6.0 Students demonstrate and eplain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it. 7.0 Given a quadratic equation of the form a + by + c + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation. * enotes key standards This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

4 LIFORNI STNRS TEST lgebra II THE EXPONENTS N LOGRITHMS REPORTING LUSTER The following si alifornia content standards are included in the Eponents and Logarithms reporting cluster and are represented in this booklet by 9 test questions. These questions represent only some ways in which these standards may be assessed on the lgebra II alifornia Mathematics Standards Test. LIFORNI ONTENT STNRS IN THIS REPORTING LUSTER lgebra II Standard Set.0* Students prove simple laws of logarithms..* Students understand the inverse relationship between eponents and logarithms, and use this relationship to solve problems involving logarithms and eponents..* Students judge the validity of an argument according to whether the properties of real numbers, eponents, and logarithms have been applied correctly at each step..0* Students know the laws of fractional eponents, understand eponential functions, and use these functions in problems involving eponential growth and decay..0 Students use the definition of logarithms to translate between logarithms in any base..0 Students understand and use the properties of logarithms to simplify logarithmic numeric epressions and to identify their approimate values..0* Students determine whether a specific algebraic statement involving rational epressions, radical epressions, or logarithmic or eponential functions is sometimes true, always true, or never true. * enotes key standards This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

5 LIFORNI STNRS TEST lgebra II THE SERIES, OMINTORIS, N PROILITY N STTISTIS REPORTING LUSTER The following 0 alifornia content standards are included in the Series, ombinatorics, and Probability and Statistics reporting cluster and are represented in this booklet by 8 test questions. These questions represent only some ways in which these standards may be assessed on the lgebra II alifornia Mathematics Standards Test. LIFORNI ONTENT STNRS IN THIS REPORTING LUSTER lgebra II 8.0* Students use fundamental counting principles to compute combinations and permutations. 9.0* Students use combinations and permutations to compute probabilities. 0.0* Students know the binomial theorem and use it to epand binomial epressions that are raised to positive integer powers..0 Students apply the method of mathematical induction to prove general statements about the positive integers..0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series..0 Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions..0 Students use properties from number systems to justify steps in combining and simplifying functions. Probability and Statistics PS.0 PS.0 PS7.0 Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces. Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces. Students compute the variance and the standard deviation of a distribution of data. * enotes key standards This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

6 LIFORNI STNRS TEST lgebra II What is the complete solution to the equation 6 =? What is the solution to the system of equations shown below? = ; = = ; = = ; = = ; = ST0007 ( 0,, ) y + z = 8 6 y z = y 9 z = What are the possible values of in =? = 0. or = < <. 0. > >. = 0. or =. 0 ST008 For a wedding, Shereda bought several dozen roses and several dozen carnations. The roses cost $ per dozen, and the carnations cost $8 per dozen. Shereda bought a total of 7 dozen flowers and paid a total of $9. How many roses did she buy? 6 dozen 7 dozen 8 dozen 9 dozen,, 0 no solution infinitely many solutions ST000 restaurant manager bought 0 packages of bagels. Some packages contained 6 bagels each, and the rest contained bagels each. There were 68 bagels in all. How many packages of bagels did the manager buy? ST009 ST This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

7 LIFORNI STNRS TEST lgebra II 6 What system of inequalities best represents the graph shown below? y 8 Which system of linear inequalities is represented by this graph? y y> and y> + y> and y< y< and y> + y< and y< + ST0000 y + y 7 Which point lies in the solution set for the y 6 system? y < 6 y + y y + y (, ) (, ) ( 0, ) (, ) + y y ST0079 ST009 7 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

8 LIFORNI STNRS TEST lgebra II 9 ) Which epression is equivalent to (6 y )( 6 y + )? 6y 6y 6y +y +y 6y +y y ST Which polynomial represents ( + )( )? ( + 6 +) ( + )= ST0009 ST00 ST00 What is the volume of the figure below? a + c = + ( a+ c )( a+ c )( a+ c) ( a c)( a + ac +c ) ( a c)( a + ac +c ) ( a+ c)( a ac + c ) + 6 ST08 ST008 8 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

9 LIFORNI STNRS TEST lgebra II The total area of a rectangle is 9 y. Which factors could represent the length times width? ( y)( + y) ( + y)( + y) ( y)( y) ( + y)( y) ST = Which product of factors is equivalent to ( + ) y? ST009 ( + +y) ( + y) ( + y)( y) ( + +y)( + y) 9 Which is a simplified form of abc? (a b c ) a bc 7 Which epression shows the complete factorization of 7? ST000 ab c bc ( 7)( + ) ( )( + 7) ( 7)( + 7) ab c ST0067 ( 7)( + 7) ST07 9 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

10 LIFORNI STNRS TEST lgebra II What is? 7 y y y 8 8y 7 y 8 8y 7 6 Which product is equivalent to? ( ) ( + ) ( ) ( + ) i = + ST0 ST00 What is the simplest form of y + 0 y + 0 y? y ( + ) ( + y) + y + y ST009 If i =, which point shows the location of i on the plane? Imaginary Real point point + point + point ST000 ST00 0 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

11 LIFORNI STNRS TEST lgebra II If i =, what is the value of i? i i 7 If i =, then i( 6i) = 8 8 ST008 ST00 6 Which of the following comple numbers is represented by the point on the graph below? Imaginary Real 8 What is an equivalent form of? + i i i i i ST000 9 What is the product of the comple numbers ( + i) and ( i)? + i i i + i ST i 0 6i ST008 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

12 LIFORNI STNRS TEST lgebra II 0 If i = and a and b are non-zero real numbers, what is? a+ bi a+ bi a + b a bi a + b a+ bi a b a bi a b ST07 There are two numbers with the following properties. ) The second number is more than the first number. ) The product of the two numbers is 9 more than their sum. Which of the following represents possible values of these two numbers? 6,,,, 6 ST009 What are the solutions to the equation + + = 0? = 0; = = 0; = i = + i; = i = + ; = What are the solutions to the equation + =? = + ; = ; = = + ST00 Jenny is solving the equation 8 = 9 by completing the square. What number should be added to both sides of the equation to complete the square? 8 6 Which of the following most accurately describes the translation of the graph y = ( + ) to the graph of y = ( ) +? ST0008 = + ; = ; = = + up and to the right down and to the right down and to the left up and to the left ST0097 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education. ST007

13 LIFORNI STNRS TEST 6 Which of the following sentences is true about the graphs of y = ( ) + and y = ( + ) +? Their vertices are maimums. The graphs have the same shape with different vertices. The graphs have different shapes with different vertices. One graph has a verte that is a maimum, while the other graph has a verte that is a minimum. lgebra II 7 What are the -intercepts of the graph of y =? and 6 and 6 and and ST0097 ST09 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

14 LIFORNI STNRS TEST lgebra II 8 Which is the graph of y = ( ) +? y y y y ST09 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

15 LIFORNI STNRS TEST 9 Which ordered pair is the verte of f( ) = + 6 +? (, ) (, ) (, 0) lgebra II Which of the following represents a parabola? + y = r y a + = b p = y (, 0 ) y a = b ST008 ST006 0 The graph of y = is a hyperbola. Which set of equations represents the y 6 0 y 9 = 0 What is the standard form of the equation of the conic given above? asymptotes of the hyperbola s graph? y =, y = y =, y = y =, y = y =, y = ( ) (y ) = (y + ) ( ) = (y ) ( + ) = 6 9 ( ) (y ) + = ST006 ST00 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

16 LIFORNI STNRS TEST lgebra II Which statement describes the graph of the equation + y + 6 y = 0? a hyperbola with center (, ) and vertices (, ) and (, ) a hyperbola with center (, ) and vertices (, ) and (, ) a circle with center (, ) and radius 8 a circle with center (, ) and radius ST07 What is the solution to the equation = 7? = = log 0 = log log 0 6 Which equation is equivalent to log =? 9 = 9 9 = = 9 9 = ST0 7 Which is the first incorrect step in simplifying log? 6 Step : log = log log 6 6 Step : = 6 Step : = log 0 7 = log0 Step ST00 Step Step If log 0 =, what is the value of? Each step is correct. ST007 = 0 = 0 = 00 =00 ST0 6 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

17 LIFORNI STNRS TEST 8 Jeremy, Michael, Shanan, and renda each worked the same math problem at the chalkboard. Each student s work is shown below. Their teacher said that while two of them had the correct answer, only one of them had arrived at the correct conclusion using correct steps. Jeremy s work Shanan s work 7 7 = = 7 7 = 0, 0 =, 0 Michael s work renda s work 7 7 = = 7 7 =, 0 =, 0 Which is a completely correct solution? Jeremy s work Michael s work Shanan s work renda s work ST00 lgebra II 9 student showed the following steps in his solution of the equation below, but his answer was not correct. log ( + ) log ( ) + log = 6 Step : log ( )( ) log ( ) + =6 Step : log ( )( ) log ( ) = Step : log ( ) = Step : = Step : = 6 In which step did he make his first error? Step Step Step Step ST06 0 certain radioactive element decays over t time according to the equation y = 00, where = the number of grams present initially and t = time in years. If 000 grams were present initially, how many grams will remain after 900 years? 00 grams 0 grams grams 6. grams 7 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education. ST0067

18 LIFORNI STNRS TEST lgebra II acteria in a culture are growing eponentially with time, as shown in the table below. ay acteria Growth acteria Which of the following equations epresses the number of bacteria, y, present at any time, t? Which table below correctly describes points of the eponential function f( ) =? 0 f() f() y =00 + t t y = (00) i () y = t t y = (00) i () ST0 If the equation y = is graphed, which of the following values of would produce a point closest to the -ais? 0 8 f() 9 0 f() 7 ST096 8 ST0 log 6 0 = log log 0 0 log 0 6 log 0 0 (log 0 6 )(log 0 0 ) log0 0 log 0 6 ST This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

19 LIFORNI STNRS TEST Jonathan wrote the equation log 6 ( ) = 0 on the board. He needs one clue for problem solving. Which fact provides the correct information that he needs to solve the equation? 6 0 = 6 = 6 = 0 6 = 6 What is the value of log 7? 6 9 ST08 ST009 7 If log 0. 0 and log 0. 77, what is the approimate value of log 7? lgebra II 9 On a recent test, Jeremy wrote the equation 6 = +. Which of the following statements is correct about the equation he wrote? The equation is always true. The equation is always true, ecept when =. The equation is never true. The equation is sometimes true when =. 60 Given the equation y = n where >0 and n <0, which statement is valid for real values of y? y > 0 y = 0 y < 0 y 0 ST ST06 ST00 6 If is a real number, which best describes the values of for which the inequality >0 is true? 8 If is a real number, for what values of is the 9 equation = true? all values of some values of no values of impossible to determine all > 0 all 0 all values of no values of ST0096 ST000 9 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

20 LIFORNI STNRS TEST lgebra II 6 Which of the following conclusions is true about the statement below? = The statement is always true. The statement is true when is negative. The statement is true when = 0. The statement is never true. 6 Teresa and Julia are among 0 students who have applied for a trip to Washington,.. Two students from the group will be selected at random for the trip. What is the probability that Teresa and Julia will be the students selected? ST06 6 belardo wants to create several different 7-character screen names. He wants to use arrangements of the first letters of his first name (abe), followed by arrangements of digits in 98, the year of his birth. How many different screen names can he create in this way? ST ST00 6 train is made up of a locomotive, 7 different cars, and a caboose. If the locomotive must be first, and the caboose must be last, how many different ways can the train be ordered? 00 8,0 6,880 8, ST09 66 ( y ) = 8y 08y + y y + 8y +08y y y + 8y y 08y y + 8y + y 08y y + ST How many terms does the binomial epansion of ( + y ) 0 contain? ST06 0 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

21 LIFORNI STNRS TEST lgebra II 68 What are the first terms in the epansion of (+ ) 6? ST00 7 Which epression represents f( g( )) if f = and g = +? ( ) ( ) ST00 69 What is the sum of the infinite geometric series ? 8 6 ( ) ( ) 7 Given that f = and g = 6, what is g f )? ( ( ) ST What is the nth term in the arithmetic series below? n + n n + n ST067 7 ( ) and g ( )= If f = + + ( +), which is an equivalent form of f( )+ g( )? ST00 ST078 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

22 LIFORNI STNRS TEST lgebra II 7 math teacher is randomly distributing rulers with centimeter labels and 0 rulers without centimeter labels. What is the probability that the first ruler she hands out will have centimeter labels and the second ruler will not have labels? 76 One bag contains green marbles and white marbles, and a second bag contains green marbles and white marble. If Trent randomly draws one marble from each bag, what is the probability that they are both green? 6 ST08 ST0 7 On a certain day the chance of rain is 80% in San Francisco and 0%in Sydney. ssume that the chance of rain in the two cities is independent. What is the probability that it will not rain in either city? 7% % % 0% ST bo contains 7 large red marbles, large yellow marbles, small red marbles, and small yellow marbles. If a marble is drawn at random, what is the probability that it is yellow, given that it is one of the large marbles? SN00 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

23 LIFORNI STNRS TEST lgebra II 78 The probabilities that Jamie will try out for various sports and team positions are shown in the chart below. Jamie Try out for basketball team Try out for baseball team Try out for guard 80% Try out for forward 0% Try out for catcher 70% Try out for pitcher 0% Jamie will definitely try out for either basketball or baseball, but not both. The probability that Jamie will try out for baseball and try out for catcher is %. What is the probability that Jamie will try out for basketball? 0% 60% 80% 90% ST00 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

24 LIFORNI STNRS TEST lgebra II 79 small-business owner must hire seasonal workers as the need arises. The following list shows the number of employees hired monthly for a -month period.,,, 6, 9 If the mean of these data is approimately 7, what is the population standard deviation for these data? (Round the answer to the nearest tenth.) , 6,,, 7, James found the mean and standard deviation of the set of numbers given above. If he adds to each number, which of the following will result? The mean will be multiplied by. The standard deviation will increase by. The mean will not change. The standard deviation will not change. SN007 ST00 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

25 LIFORNI STNRS TEST lgebra II Question Number orrect nswer Standard Year of Release This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

26 LIFORNI STNRS TEST lgebra II Question Number orrect nswer Standard Year of Release This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.

27 LIFORNI STNRS TEST lgebra II Question Number orrect nswer Standard Year of Release PS PS PS PS PS PS PS This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 008 alifornia epartment of Education.