The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Wednesday, August 13, :30 to 11:30 a.m.

Transcription

1 The University of the State of New Yk REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Wednesday, August 13, 003 8:30 to 11:30 a.m., only Print Your Name: Print Your School s Name: Print your name and the name of your school in the boxes above. Then turn to the last page of this booklet, which is the answer sheet f Part I. Fold the last page along the perfations and, slowly and carefully, tear off the answer sheet. Then fill in the heading of your answer sheet. Scrap paper is not permitted f any part of this examination, but you may use the blank spaces in this booklet as scrap paper. A perfated sheet of scrap graph paper is provided at the end of this booklet f any question f which graphing may be helpful but is not required. Any wk done on this sheet of scrap graph paper will not be sced. Write all your wk in pen, except graphs and drawings, which should be done in pencil. This examination has four parts, with a total of 34 questions. You must answer all questions in this examination. Write your answers to the Part I multiple-choice questions on the separate answer sheet. Write your answers to the questions in Parts II, III, and IV directly in this booklet. Clearly indicate the necessary steps, including appropriate fmula substitutions, diagrams, graphs, charts, etc. The fmulas that you may need to answer some questions in this examination are found on page 19. When you have completed the examination, you must sign the statement printed at the end of the answer sheet, indicating that you had no unlawful knowledge of the questions answers pri to the examination and that you have neither given n received assistance in answering any of the questions during the examination. Your answer sheet cannot be accepted if you fail to sign this declaration. Notice... A graphing calculat, a straightedge (ruler), and a compass must be available f your use while taking this examination. DO NOT OPEN THIS EXAMINATION BOOKLET UNTIL THE SIGNAL IS GIVEN.

2 Part I Answer all questions in this part. Each crect answer will receive credits. No partial credit will be allowed. F each question, write on the separate answer sheet the numeral preceding the wd expression that best completes the statement answers the question. [40] 1 Which graph does not represent a function of x? Chance of Human Survival y Chance of Human Survival x Pollution Pollution ( 1 ) ( 3 ) y x Use this space f computations. Chance of Human Survival y Chance of Human Survival x Pollution Pollution ( ) ( 4 ) y x What is the value of x in the equation 5 x = 3i? (1) 1 (3) () 7 (4) 4 3 Which graph represents the solution set of x 1 <7? ( 1 ) ( ) ( 3 ) ( 4 ) Math. B Aug. 03 []

3 4 The strength of a medication over time is represented by the equation y = 00(1.5) x, where x represents the number of hours since the medication was taken and y represents the number of micrograms per millimeter left in the blood. Which graph best represents this relationship? y y Use this space f computations. x ( 1 ) ( 3 ) x y y x ( ) ( 4 ) x xy 5 Written in simplest fm, the expression 9 is equivalent to 3 xy (1) 1 (3) (3 + xy) 1 () (4) 3 + xy 3 + xy 6 Which graph represents data used in a linear regression that produces a crelation coefficient closest to 1? y y ( 1 ) x ( 3 ) x y y x x ( ) ( 4 ) Math. B Aug. 03 [3] [OVER]

4 + 3 7 Which expression is equal to? 3 Use this space f computations (1) (3) () (4) Which transfmation is not an isometry? (1) rotation (3) dilation () line reflection (4) translation 9 A dog has a 0-foot leash attached to the cner where a garage and a fence meet, as shown in the accompanying diagram. When the dog pulls the leash tight and walks from the fence to the garage, the arc the leash makes is 55.8 feet. Fence θ Garage (Not drawn to scale) What is the measure of angle between the garage and the fence, in radians? (1) 0.36 (3) 3.14 ().79 (4) 160 Math. B Aug. 03 [4]

5 10 In the accompanying diagram of parallelogram ABCD, DE BF. D E C Use this space f computations. G A F B Triangle EGC can be proved congruent to triangle FGA by (1) HL HL (3) AAS AAS () AAA AAA (4) SSA SSA 11 An architect commissions a contract to produce a triangular window. The architect describes the window as ABC, where m A = 50, BC = 10 inches, and AB = 1 inches. How many distinct triangles can the contract construct using these dimensions? (1) 1 (3) me than () (4) 0 1 The accompanying graph shows the relationship between a person s weight and the distance that the person must sit from the center of a seesaw to make it balanced. 1 y Distance (feet) Weight (pounds) x Which equation best represents this graph? (1) y = 1x (3) y = log x () y = 10x (4) y = 10 x Math. B Aug. 03 [5] [OVER]

6 13 If f and g are two functions defined by f(x) = 3x + 5 and g(x) = x + 1, then g(f(x)) is (1) x + 3x + 6 (3) 3x + 8 () 9x + 30x + 6 (4) 9x + 6 Use this space f computations. 14 What is the product of and 1 49, expressed in simplest a + bi fm? (1) i (3) i () 5 71i (4) 47 9i 15 The expression cos θ is equivalent to sin θ (1) csc (3) cot () sec (4) sin 16 If sin x =, cos y =, and x and y are acute angles, the value of 13 5 cos (x y) is 1 65 (1) (3) () (4) The amount of ketchup dispensed from a machine at Hamburger Palace is nmally distributed with a mean of 0.9 ounce and a standard deviation of 0.1 ounce. If the machine is used 500 times, approximately how many times will it be expected to dispense 1 me ounces of ketchup? (1) 5 (3) 80 () 16 (4) 100 Math. B Aug. 03 [6]

7 18 A commercial artist plans to include an ellipse in a design and wants the length of the hizontal axis to equal 10 and the length of the vertical axis to equal 6. Which equation could represent this ellipse? (1) 9x + 5y = 5 (3) x + y = 100 () 9x 5y = 5 (4) 3y = 0x Use this space f computations. 19 A function is defined by the equation y = x. Which equation defines the inverse of this function? (1) y = x + 3 (3) y = x + () y = x 3 (4) y = x In the equation ax + 6x 9 = 0, imaginary roots will be generated if (1) 1 < a < 1 (3) a > 1, only () a < 1, only (4) a < 1 Math. B Aug. 03 [7] [OVER]

8 Part II Answer all questions in this part. Each crect answer will receive credits. Clearly indicate the necessary steps, including appropriate fmula substitutions, diagrams, graphs, charts, etc. F all questions in this part, a crect numerical answer with no wk shown will receive only 1 credit. [1] 1 The height, h, in feet, a ball will reach when thrown in the air is a function of time, t, in seconds, given by the equation h(t) = 16t + 30t + 6. Find, to the nearest tenth, the maximum height, in feet, the ball will reach. Find the value of (x + ) 0 + (x + 1) 3 when x = 7. Math. B Aug. 03 [8]

9 3 Express in simplest fm: x 4 4 x 1 4 x 4 The triangular top of a table has two sides of 14 inches and 16 inches, and the angle between the sides is 30. Find the area of the tabletop, in square inches. Math. B Aug. 03 [9] [OVER]

10 5 Meteologists can determine how long a stm lasts by using the 3 function t(d) = 0.07d,where d is the diameter of the stm, in miles, and t is the time, in hours. If the stm lasts 4.75 hours, find its diameter, to the nearest tenth of a mile. 6 Tom sced 3 points in a basketball game. He attempted 15 field goals and 6 free throws. If each successful field goal is points and each successful free throw is 1 point, is it possible he successfully made all 6 of his free throws? Justify your answer. Math. B Aug. 03 [10]

11 Part III Answer all questions in this part. Each crect answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate fmula substitutions, diagrams, graphs, charts, etc. F all questions in this part, a crect numerical answer with no wk shown will receive only 1 credit. [4] 7 On the accompanying grid, graph and label AB, where A is (0,5) and B is (,0). Under the transfmation r x-axis r ( ) y-axis AB, A maps to A, and B maps to B. Graph and label AB. What single transfmation would map AB to AB? Math. B Aug. 03 [11] [OVER]

12 8 Express, in simplest a + bi fm, the roots of the equation x + 5 = 4x. 9 A ship at sea is 70 miles from one radio transmitter and 130 miles from another. The angle between the signals sent to the ship by the transmitters is Find the distance between the two transmitters, to the nearest mile. Math. B Aug. 03 [1]

13 30 A student attaches one end of a rope to a wall at a fixed point 3 feet above the ground, as shown in the accompanying diagram, and moves the other end of the rope up and down, producing a wave described by the equation y = a sin bx + c. The range of the rope s height above the ground is between 1 and 5 feet. The period of the wave is 4π. Write the equation that represents this wave. 3 ft 3 ft Math. B Aug. 03 [13] [OVER]

14 31 The table below shows the results of an experiment that relates the height at which a ball is dropped, x, to the height of its first bounce, y. Drop Height (x) (cm) Bounce Height (y) (cm) Find x, the mean of the drop heights. Find y, the mean of the bounce heights. Find the linear regression equation that best fits the data. Show that (x,y) is a point on the line of regression. [The use of the grid on the next page is optional.] Math. B Aug. 03 [14]

15 Question 31 continued Math. B Aug. 03 [15] [OVER]

16 3 A company calculates its profit by finding the difference between revenue and cost. The cost function of producing x hammers is C(x) = 4x If each hammer is sold f $10, the revenue function f selling x hammers is R(x) = 10x. How many hammers must be sold to make a profit? How many hammers must be sold to make a profit of $100? Math. B Aug. 03 [16]

17 Part IV Answer all questions in this part. Each crect answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate fmula substitutions, diagrams, graphs, charts, etc. F all questions in this part, a crect numerical answer with no wk shown will receive only 1 credit. [1] 33 Given circle O with diameter GOAL; secants HUG and HTAM intersect at point H; m GM:m ML:m LT = 7:3:; and chd GU chd UT. Find the ratio of m UGL to m H. G U O M A L T H Math. B Aug. 03 [17] [OVER]

18 34 When Joe bowls, he can get a strike (knock down all the pins) 60% of the time. How many times me likely is it f Joe to bowl at least three strikes out of four tries as it is f him to bowl zero strikes out of four tries? Round your answer to the nearest whole number. Math. B Aug. 03 [18]

19 Fmulas Area of Triangle K = 1 ab sin C Law of Cosines a = b + c bc cos A Functions of the Sum of Two Angles sin (A + B) = sin A cos B + cos A sin B cos (A + B) = cos A cos B sin A sin B Functions of the Difference of Two Angles sin (A B) = sin A cos B cos A sin B cos (A B) = cos A cos B + sin A sin B Law of Sines a b c sin A = sin B = sin C Functions of the Double Angle sin A = sin A cos A cos A = cos A sin A cos A = cos A 1 cos A = 1 sin A Functions of the Half Angle sin cos 1 A =± 1 cos A 1 1 cos A =± + A Nmal Curve Standard Deviation 19.1% 19.1% 9.% 15.0% 15.0% 9.% 0.1% 0.5% 4.4% 4.4% 0.5% 0.1% 1.7% 1.7% Math. B Aug. 03 [19]

20 Tear Here Tear Here Scrap Graph Paper This sheet will not be sced.

21 Scrap Graph Paper This sheet will not be sced. Tear Here Tear Here

22 The University of the State of New Yk REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Tear Here Tear Here Wednesday, August 13, 003 8:30 to 11:30 a.m., only ANSWER SHEET Student Sex: Male Female Grade Teacher School Your answers to Part I should be recded on this answer sheet Part I Answer all 0 questions in this part Your answers f Parts II, III, and IV should be written in the test booklet. The declaration below should be signed when you have completed the examination. I do hereby affirm, at the close of this examination, that I had no unlawful knowledge of the questions answers pri to the examination and that I have neither given n received assistance in answering any of the questions during the examination. Signature Math. B Aug. 03 [3]

23 MATHEMATICS B Maximum Credits Rater s/scer s Question Credit Earned Initials Part I Part II Part III Part IV Rater s/scer s Name (minimum of three) Tear Here Maximum 88 Total Total Raw Checked by Scaled Sce Sce Notes to raters... Each paper should be sced by a minimum of three raters. The table f converting the total raw sce to the scaled sce is provided in the scing key f this examination. The scaled sce is the student s final examination sce. Tear Here Math. B Aug. 03 [4]

24 FOR TEACHERS ONLY The University of the State of New Yk REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Wednesday, August 13, 003 8:30 to 11:30 a.m., only SCORING KEY Mechanics of Rating The following procedures are to be followed f scing student answer papers f the Mathematics B examination. Me detailed infmation about scing is provided in the publication Infmation Booklet f Administering and Scing the Regents Examinations in Mathematics A and Mathematics B. Use only red ink red pencil in rating Regents papers. Do not attempt to crect the student s wk by making insertions changes of any kind. Use checkmarks to indicate student errs. Unless otherwise specified, mathematically crect variations in the answers will be allowed. Units need not be given when the wding of the questions allows such omissions. Each student s answer paper is to be sced by a minimum of three mathematics teachers. On the back of the student s detachable answer sheet, raters must enter their initials in the boxes next to the questions they have sced and also write their name in the box under the heading Rater s/scer s Name. Raters should recd the student s sces f all questions and the total raw sce on the student s detachable answer sheet. Then the student s total raw sce should be converted to a scaled sce by using the conversion chart printed at the end of this key. The student s scaled sce should be entered in the box provided on the student s detachable answer sheet. The scaled sce is the student s final examination sce. Part I Allow a total of 40 credits, credits f each of the following. Allow credit if the student has written the crect answer instead of the numeral 1,, 3, 4. (1) 1 (6) 4 (11) (16) () (7) 4 (1) 4 (17) 3 (3) 1 (8) 3 (13) (18) 1 (4) 1 (9) (14) 4 (19) 1 (5) 3 (10) 3 (15) 1 (0) 4 [1] [OVER]

25 MATHEMATICS B continued Part II F each question, use the specific criteria to award a maximum of two credits. (1) [] 0.1, and appropriate wk is shown. [1] Appropriate wk is shown, but one computational rounding err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] The time when the ball reaches its maximum height is found crectly, but no further crect wk is shown. [1] 0.1, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 1 4 () [] 1 an equivalent answer, and appropriate wk is shown. [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] 1 1 an equivalent answer, but no wk is shown. 4 [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. []

26 MATHEMATICS B continued (3) [] x + 4, and appropriate wk is shown. 4 [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] x + 4, but no wk is shown. 4 [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. (4) [] 56, and appropriate wk is shown, such as sin 30. [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] 56, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 1 (5) [] 16.6, and appropriate wk is shown. [1] Appropriate wk is shown, but one computational rounding err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] A crect substitution of 4.75 f t is made, but no further crect wk is shown. [1] 16.6, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. [3] [OVER]

27 MATHEMATICS B continued (6) [] No, and a crect justification is given. [1] No, but an incomplete partially increct explanation is given. [0] No, but no explanation is given. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. [4]

28 MATHEMATICS B continued Part III F each question, use the specific criteria to award a maximum of four credits. (7) [4] AB and AB are graphed and labeled crectly, A (0, 5) and B (,0), and a crect transfmation is identified, such as R 180, R 180, r (0,0). [3] One err is made in graphing AB, but AB is graphed and labeled appropriately, and an appropriate transfmation is identified. [] AB is graphed and labeled crectly but one mistake is made in finding AB, but an appropriate transfmation is identified. [] Both AB and AB are graphed and labeled crectly, but the transfmation is missing is increct. [1] AB is graphed and labeled crectly, but one mistake is made in finding AB, and the transfmation is missing is increct. [1] One err is made in graphing AB, but AB is graphed and labeled appropriately, but the transfmation is missing is increct. [1] R 180, R 180, r (0,0), but no graph is drawn. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. [5] [OVER]

29 MATHEMATICS B continued (8) [4] ± i, and appropriate wk is shown. [3] Appropriate wk is shown, but one computational err is made, but the result is expressed as a complex number in simplest a + bi fm. [3] Appropriate wk is shown, but the roots are not expressed in simplest a + bi fm. [3] Appropriate wk is shown, but only one complex root, in simplest a + bi fm, is found. [] Appropriate wk is shown, but one computational err is made, resulting in a solution that is not a complex number. [] Appropriate wk is shown, but two me computational errs are made, but the result is expressed as a complex number in simplest a + bi fm. [] Appropriate wk is shown, but one conceptual err is made. [] An increct quadratic fmula is used, but the result is expressed as a complex number in simplest a + bi fm. [1] Increct substitution is made into the quadratic fmula, such as a = 1, b = 5, and c = 4, but the resulting equation is solved appropriately. [1] ± i, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. [6]

30 MATHEMATICS B continued (9) [4] 174, and appropriate wk is shown, such as the use of the Law of Cosines. [3] Appropriate wk is shown, but one computational rounding err is made. [] Appropriate wk is shown, but two me computational rounding errs are made. [] One conceptual err is made when applying the Law of Cosines, but an appropriate answer is found. [1] Crect substitution is made into the Law of Cosines, but no further crect wk is shown. [1] 174, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 1 (30) [4] y = sin x + 3 y = sin x + 3, and appropriate wk is shown. [3] The fact that c is equal to 3 is not recognized, resulting in an answer of 1 1 y = sin x y = sin x. [3] The values of a, b, and c are determined crectly, and appropriate wk is shown, but the equation is not written. [3] The value of a c is determined increctly, but the value of b is determined crectly, and appropriate wk is shown, and an appropriate equation is written. [] Only the value of b is determined crectly, but appropriate wk is shown, and an appropriate equation is written. [] Only the values of a and c are determined crectly, but appropriate wk is shown, and an appropriate equation is written. [1] The value of a c is determined increctly, and the value of b is not determined is determined increctly, but appropriate wk is shown, and an appropriate equation is written. [1] y = sin 1 x + 3 y = sin 1 x + 3, but no wk is shown. 1 [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. [7] [OVER]

31 MATHEMATICS B continued (31) [4] x = 80, y = 0.8, and y = 0.5x + 0.8, and appropriate wk is shown to prove that ( x, y) is a point on the line of regression. [3] Appropriate wk is shown, but one computational err is made. [] Appropriate wk is shown, but two me computational errs are made. [] Appropriate wk is shown, but one conceptual err is made. [1] x = 80, y = 0.8, and y = 0.5x + 0.8, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. (3) [4] 9 hammers to make a profit and 45 hammers to make a profit of $100, and appropriate wk is shown. [3] Appropriate wk is shown, but one computational rounding err is made. [] Appropriate wk is shown, but two me computational rounding errs are made. [] Either the number of hammers to make a profit the number of hammers to make a profit of $100 is determined crectly, and appropriate wk is shown. [1] One conceptual and one computational err are made. [1] The crect equation and inequality the crect equations are written, but no further crect wk is shown. [1] 9 hammers to make a profit and 45 hammers to make a profit of $100, but no wk is shown. [0] 9 and 45, but no wk is shown and the answers are not labeled. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. [8]

32 MATHEMATICS B continued Part IV F each question, use the specific criteria to award a maximum of six credits. 1 (33) [6] :1 an equivalent ratio, and appropriate wk is shown. [5] Appropriate wk is shown, but one computational err is made, but an appropriate ratio is found. [5] Appropriate wk is shown, but the answer is not written as a ratio. [5] Appropriate wk is shown, but the ratio is reversed is simplified increctly. [4] Appropriate wk is shown, but two me computational errs are made, but an appropriate ratio is found. [4] Crect measures are found f all the arcs and the angles, and appropriate wk is shown, but no ratio is found. [4] Crect measures are found f all the arcs, but the measure of one angle is found increctly, but an appropriate ratio is found. [3] One conceptual err is made, but appropriate wk is shown, and an appropriate ratio is found. [3] Crect measures are found f all the arcs, but the measures of both angles are found increctly, but an appropriate ratio is found. [] Crect measures are found f all the arcs, but no further crect wk is shown. [1] Only the value of x is found crectly, and appropriate wk is shown. [1] :1 an equivalent ratio, but no wk is shown. 1 [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. [9] [OVER]

33 MATHEMATICS B concluded (34) [6] 19, and appropriate wk is shown, such as P(at least three) = 4 C 3 (0.6) 3 (0.4) + 4 C 4 (0.6) 4 and P(0) = (0.4) 4. [5] Appropriate wk is shown, but one computational err is made. [5] Crect probabilities are computed, but no answer an increct answer is found. [4] Appropriate wk is shown, but two me computational errs are made. [4] Only the probability f at least three strikes is found crectly, but an appropriate ratio is determined. [3] The probability f at least three strikes is found crectly, and no further crect wk is shown. [3] Only the probability f zero strikes is found crectly, but an appropriate ratio is determined. [] Only the probability f zero strikes is found crectly, and no further crect wk is shown. [] Only the equation f the probability f at least three strikes is written, and it is not solved. [1] Conceptual errs are made in finding the probabilities, but an appropriate ratio is determined, based on the increct probabilities. [1] 19, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. [10]

34 MATHEMATICS B Map to Learning Standards Key Ideas Item Numbers Mathematical Reasoning 10, 6 Number and Numeration 5, 7, 0, 3 Operations 14,, 8 Modeling/Multiple Representation 4, 1, 18, 1, 5, 9, 30, 3 Measurement 6, 9, 11, 16, 17, 4, 33 Uncertainty 31, 34 Patterns/Functions 1,, 3, 8, 13, 15, 19, 7

35 Regents Examination in Mathematics B August 003 Chart f Converting Total Test Raw Sces to Final Examination Sces (Scaled Sces) Raw Scaled Raw Scaled Raw Scaled Sce Sce Sce Sce Sce Sce To determine the student s final examination sce, find the student s total test raw sce in the column labeled Raw Sce and then locate the scaled sce that cresponds to that raw sce. The scaled sce is the student s final examination sce. Enter this sce in the space labeled Scaled Sce on the student s answer sheet. All student answer papers that receive a scaled sce of 60 through 64 must be sced a second time. F the second scing, a different committee of teachers may sce the student s paper the iginal committee may sce the paper, except that no teacher may sce the same open-ended questions that he/she sced in the first rating of the paper. The school principal is responsible f assuring that the student s final examination sce is based on a fair, accurate, and reliable scing of the student s answer paper. Because scaled sces cresponding to raw sces in the conversion chart may change from one examination to another, it is crucial that f each administration, the conversion chart provided in the scing key f that administration be used to determine the student s final sce. The chart above is usable only f this administration of the mathematics B examination. [1]