Nova Scotia Examinations Mathematics 12 Web Sample 2. Student Booklet

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1 Nova Scotia Eaminations Mathematics Web Sample Student Booklet

2 General Instructions - WEB SAMPLE* This eamination is composed of two sections with the following suggested time allotment: Selected-Response (Multiple-Choice) Questions Constructed-Response Questions Value 5 pts (appro. 40 min)* Value 75 pts (appro. 0 min) *note: there are 35 constructed response questions on the Math NSEs as of Januar 008 Total time: 3 hours (revision time included) Use these suggested times to guide ou in the completion of the eamination; however, ou might not find it necessar to spend the suggested time on each section. Plan our time to enable ou to complete the eamination. You are not permitted to use our own graphing calculator unless our teacher has cleared the memor immediatel prior to this eamination. The onl graphing calculators permitted are TI-8, TI-83, TI-83 Plus, TI-84, or TI-84 Plus. If the question indicates that ou are not to use a graphing calculator, ou are still permitted to use a calculator to perform arithmetic operations. Calculators are not to be shared. Graph paper, scrap paper, and formula sheets are provided at the end of this booklet. These pages can be removed from the booklet for our use during the eamination. Note: Diagrams are not necessaril drawn to scale. Mathematics Page

3 Selected-Response Questions - WEB SAMPLE (Total Value: 5 points) In this part of the eamination, there are 5 selected-response questions*, each with a value of point. Read each question carefull, and decide which of the responses best answers the question being asked. You are provided a separate student answer sheet. In the selected-response section of the student answer sheet, fill in the bubble that corresponds to our choice as shown in the eample below. Use an HB pencil onl. Eample. What are the roots of. A B D + 3 4= 0? A. 4 and B. 4 and 3 C. 4 and D. 4 and 3 (On student answer sheet) If ou wish to change an answer, please ensure that ou erase our first answer completel on the student answer sheet. Calculations or rough work on the selected-response pages of the eamination booklet will not be scored. *Note: As of Januar 008 there are 35 selected response questions on the NSE Math eams. Page Mathematics

4 . The function t = n + 3n is used to generate a quadratic sequence. What is the value of D? n A. B. C. 3 D What is the mapping rule that maps = onto ( 5) = ( + 7)? A. (, ) ( + 7, 5) B. C. (, ) ( + 7, 5) D. (, ) ( 7, + 5) (, ) ( 7, 5) 3. What are the coordinates of the verte of the quadratic function having a maimum value of 0 and -intercepts located at (5, 0) and (,0)? A. (3, 0) B. (, 0) C. (, 0) D. (3, 0) 4. The graph of a quadratic function = a + b+ c is given below. When solving for in the equation a + b + c = 0, the solution(s) represents A. B. C. D. the coordinate of the verte of the quadratic function the maimum or minimum value of the -intercepts of = a + b+ c the coordinate of the -intercept of the quadratic function = a + b+ c = a + b+ c Mathematics Page 3

5 5. In which of the following quadratics is the discriminant equal to zero? A. B C. D The verte of the parabola represented b = 4 4 is A. (, 6) B. (, 5) C. (, ) D. (, 6) 7. The ais of smmetr for the parabola defined b = 6+ is A. = 4 B. = 6 C. = 3 D. = 8. A cannon ball is shot into the air and its height in metres is represented b the equation h= t 4.9t where t is time in seconds. How high does the cannon ball go? A. 3.0 m B m C m D..36 m Page 4 Mathematics

6 9. A basketball plaer is tring to increase her shot accurac. She stas in the same position on the court and increases the arc of the flight path of the ball. Path II Path I Her coach graphs the quadratic function a ( k ) = ( h ) to model parabolic path I of the basketball. The coach then changes certain values in the given equation to graph path II. Which value(s) did the coach NOT change? A. a B. C. k D. h h and a 0. Which one of the following tables of values was generated b an eponential function? A. B C. D ( ) 3. The value of is: 000 A. B. C. 000 D Which of the following is equal to b if b 0? 3 b 3 b A. B. 3 b b3 C. D. Mathematics Page 5

7 3. Which graph represents = a b, where a >0 and 0< b <? A. B. C. D. 4. A particular bacteria population doubles ever 7 das. Which function determines the number of bacteria N after t das, given the initial amount of 500 bacteria? t A. N = 500(7) B. 7 N = 500() t N = t 500() N = 500(7) t 7 C. D What could be the net correct step in solving this equation ( )( ) =? = 3+ 5 = A. B. + 0 = C. 4 D. log (3 + 5) = log 6. log t r = s can also be epressed as A. s B. t = r t r C. r D. s = t t s = s = r 7. The value of in the equation log = 0 is A. 0 B. 5 C. D. 5 5 Page 6 Mathematics

8 8. The epression log 6 is equal to log ( ) A. 4 B. 3 C. 5 D Given the following figure: B (3, 5) A (-, ) C (7, 0) D (3, -3) Which of the following statements is false? AB = DC T A. B. The midpoint of AD is (, ). C. The diagonals AC and BD are of equal D. lengths. 0. The segment below is to be cut into 4 equal pieces. The diagonals AC and BD bisect each other. A(-4, 5) B C D E(8, -3) What are the coordinates of B? A. (,) B. (,) C. (, ) D. (,3). What is the converse of this statement? If two chords are equidistant from the centre of a circle, then the chords are congruent. A. B. C. D. If two chords are equidistant from the centre of a circle, then the are parallel. Two chords that are the same distance from the centre of a circle are congruent. If two chords of a circle are congruent, then the are equidistant from the centre of the circle. If two chords of a circle are congruent, then the pass through the centre of the circle. Mathematics Page 7

9 . Consider the following Venn diagram. Event K Event L Which of the following is correct? A. P(K or L) = P(K) + P(L) B. C. P(K or L) = P(K) P(L) D. P(K and L) = P(K) + P(L) P(K and L) = P(K) P(L) 3. Your math teacher gives our class a list of eight questions to stud. Five of the eight questions will be randoml selected for the net test. If ou stud onl the first five questions from the list, the probabilit that all of those five questions will be on the test is A. B. 8C5 8P C. D. 4. The value of 500! is 499! A. B..00 C. 500 D. undefined 5. A bag contains 9 marbles, 4 of which are red. What epression represents the probabilit of selecting three red marbles when three marbles are drawn at random? A. B. 9C3 9C3 4 C. D. 9P3 9P3 4 Page 8 Mathematics

10 Constructed-Response Questions (Total Value: 75 points) Read each question carefull, and be sure to write our response in the bo and space provided. If the answer bo indicates that ou are to show our work, then points will be awarded for our correct work and our correct final answer. The method used to solve a problem must clearl be shown even when using a graphing calculator. If the answer bo requires that just a final answer be provided, then points will be awarded for the correct answer onl. When working with decimal values, ou ma round off to the hundredths place in our final answer onl. If an decimal values are rounded prior to the final step of the solution, at least 4 decimal places must be kept. With the eception of the probabilit unit, all answers must be given in simplified form. Mathematics Page 9

11 6. (a) Write the function, in transformational form, that represents the following parabola. (b) State the domain and range of the function. ( points) Domain: Range: Page 0 Mathematics

12 7. Calculate the discriminant of = 0 and eplain what the result tells ou about the graph of = ( points) Mathematics Page

13 8. The severit of an automobile crash increases significantl as the speed increases. The table shows the relationship between the speed and a crash severit inde. Speed (km/h) Crash severit inde (a) Jimm claims that a quadratic function would best model this situation. Is Jimm's claim correct? Eplain. ( points) (b) What speed would have a crash severit inde of 50.40? Page Mathematics

14 9. Solve for in each of the following equations using a different algebraic method for each. (a) = + 6 (b) + 7 4= 0 Mathematics Page 3

15 30. A football is kicked into the air. The equation h= 4.9t + 9.8t+ epresses the relationship between height h in metres and time t in seconds. (a) Determine the maimum height reached b the football. (b) At what time(s) after the kick is the ball at a height of 5 m? Page 4 Mathematics

16 3. A golf ball is hit from ground level in a flat field and reaches a maimum height of 5 m. The ball first hits the ground 00 m awa while following a parabolic path. (a) Draw a diagram and include all important information needed to model this problem. Remember to label our aes. ( points) (b) How high is the golf ball above the ground when it is at a horizontal distance of 0 m from where it was hit? (4 points) Mathematics Page 5

17 3. Solve for in each of the three parts below. At least two of the three parts must be solved algebraicall. (a) 9 = (b) 3 = 7 (c) log 5( ) + log50 = log6 Page 6 Mathematics

18 ( ) 33. Given = (a) Determine the coordinates of the -intercept. ( point) (b) Write the equation of the horizontal asmptote. ( point) (c) Indicate whether the function above represents a growth curve or a deca curve. Eplain how ou know. ( points) 34. Epress the following epression as a single logarithm. Simplif our answer. log + 3(log log z) Mathematics Page 7

19 35. (a) Evaluate: ( points) (i) log 8 = (ii) - log 8 = (b) Evaluate: ( points) (i) log3 9 = (ii) - log 9 = 3 (c) Based on the answers obtained in parts (a) and (b), write an epression equivalent to log N. ( point) b 36. Marla tries to evaluate 0 3 (zero to the eponent negative 3) on her calculator and gets an error message. Eplain wh she got an error message. ( points) Page 8 Mathematics

20 5 37. Describe a situation that could be modelled b the function P = 000() t. 38. A circle has a diameter of 6 cm. A chord in the same circle measures 4 cm. What is the shortest distance from the centre of the circle to the chord? Mathematics Page 9

21 39. The coordinates of the corners of a stained glass window are A (, ), B(3, ), C (6, ) and D (, 5). (a) Show algebraicall that the diagonals have the same length. ( points) (b) Provide calculations to determine whether or not the diagonals of the window bisect each other. Page 0 Mathematics

22 40. Write an true statement for which the converse is NOT true. ( point) 4. There are 5 jellbeans randoml distributed in a jar; 5 are ellow and 0 are orange. You reach into the jar and, without looking, remove jellbeans. What is the probabilit that ou will remove ellow jellbeans? Mathematics Page

23 4. In a group of 5 people, 4 are left-handed and are right-handed. Seven people are selected at random from this group. (a) What is the probabilit that everone selected is right-handed? ( points) (b) If Sarah and Mike, two of the left handers, have alread been chosen, what is the probabilit that all the others members selected will be right-handed? ( points) 43. Create a real-life problem that demonstrates P(A or B) = P(A) + P(B) P(A and B) when the events A and B are NOT mutuall eclusive. (You don't have to solve the problem.) You have reached the end of the SAMPLE Mathematics Eamination. Please check our work to ensure ou have completed all questions. Page Mathematics

24 Formula Sheet Mathematics Quadratics Unit General form: Standard form: = + + a b c = a( h) + k Transformational form: If a + b+ c= 0, then ( k) ( h) a = ± = b b 4ac a Eponential Growth Unit = ab log ( ) = log + log a a a log a( ) = loga loga or log = log log b log = b(log ) a a a a a

25 Circle Geometr Unit d = ( ) + ( ) The coordinates of M are: +, + Δ m = Δ Probabilit Unit P(A and B) = P(A) P(B) P(A or B) = P(A) + P(B) P(A and B) n P r n! = ( n r)! n C r n! = r!( n r )!

26 Nova Scotia Eaminations Mathematics Web Sample Marking Guide Teacher Name:

27 Selected Response Answers. D. D 3. C 4. C 5. B 6. A 7. C 8. C 9. B 0. B. A. B 3. B 4. D 5. B 6. D 7. C 8. A 9. C 0. D. C. A 3. A 4. C 5. B

28 Question 6(a) a a ( 7) = ( 3) ( ) = ( ) To go from (7, 3) to (4, 5) there is a horizontal translation is and a vertical translation of -. If there was no vertical stretch the vertical translation should be, therefore the vertical stretch must be. = a a=.5 pt 7 3 = ( ) ( ) OR 7 3 ( ) = ( ).5 pt QuadReg on TI-83 = + + a b c a = b = c = + = ( 6) ( ) ( ) ( 7) ( 3) = + + = + + = = 3 = Question 6(b) { } ( ) Domain: or, { } Range: 7, or (,7]

29 Question 7 ( points) D= b 4ac ( 8) 4( 3)( 8) = = = 3 Since the discriminant is less than zero, the graph will have no -intercepts. Question 8 (a) ( points) Speed (km/h) Crash severit inde Yes Jimm is correct because the second-level differences, D, are constant OR Yes Jimm is correct because the QuadReg on the TI-83 shows an R value of. QuadReg TI-83 = a + b + c a = 0.0 b = 0.0 c = 0 R =

30 Question 8 (b) = a + b+ c c = 0 (obtained from table) = ( ) ( ) a Δ = a 00 = 00a = a 0.0 ( 0,. ) = = b ( ). = b. = + 0b 0. = 0b 0.0 = b = = 0 solve using an method to get: = 70 OR Using QuadReg on the TI-83: = Table of values on TI pt OR = = 0 solve using an method to get: = 70 OR Speed (km/h) Crash severit inde for st level differences = 70

31 Question 9 (a) ( ) ( )( )( ) () ± 4 6 = 6= 0 ( )( ) 3 + = 0 ± 5 = 3 = 0 = 3 + = 0 = + 5 = = 3 5 = = OR = 6 + = = 4 5 = 4 5 =± 5 = ± + 5 = + = 3 5 = + =

32 Question 9 (b) ( ) ( )( )( ) ( ) 7± = = 7± = 4 = 4 = = = 0 ( ) ( ) ( )( ) = = 0 = 0 = = + 4= 0 = 4 OR = = = + 7= = = = = =± = ± = = 4 4 = 4

33 Question 30 (a) h= t + t ( t t) ( t t ) = = ( t ) = The maimum height is 5.9 metres. ( h 5.9) ( t ) ( ) ( ) h = 4.9 t t h 4.9= 4.9 t t+ ( t ) h 5.9 = 4.9 = 4.9 h= t + t+ The maimum height is 5.9 metres. b t = a 9.8 = 4.9 = ( ) () () h = = 5.9 The maimum height is 5.9 metres. OR h = = 4ac b 4a ( )( ) ( ) 4( 4.9) = 5.9 The maimum height is 5.9 metres. pts OR (, 5.9) pts The maimum height is 5.9 metres.

34 Question 30(b) + + = 4.9t 9.8t 5 + = 4.9t 9.8t 4 0 t = = ( ) ( )( )( ) ( 4.9) 9.8 ± ± OR t + t= t t t t ( t ) t = t = t ( t ) t ± t t t 0.57 t.43 The ball will be at 5 m at 0.57 s and at.43 s. OR = 4.9t + 9.8t 4 = 4.9t + 9.8t + = 5 (0, 0.57) (0,.43) (5, 0.57) (5,.43) The ball will be at 5 m at 0.57 s and at.43 s. The ball will be at 5 m at 0.57 s and at.43 s.

35 Question 3 (4 points) (0, 0) (50, 5) (00, 0) for each of the following: the graph has a parabolic form, opens downward and aes are labeled. Passes through point (0, 0) and point is labeled. Passes through (00, 0) and point is labeled. Verte identified and labeled at (50, 5) Question 3(b) (4 points) ( ) = a ( ) a( ) 0, 0 0 = = 500a 0.0 = a ( ) = ( ) = = 6 The ball will be at a height of 6 m at a horizontal distance of 0 m.

36 Question 3 (a) 3 ( 3 ) = ( 3 ) = 3 + 4= = = 7 ( + ) = ( + ) ( + ) log9 3 log7 + = 3 log7 log9 ( )( ) + = = = 0.5 = 7 OR = 9 + = = 9 7 (0.4, 0) = 0.4 = 0.4

37 Question 3 (b) log 3 = log7 log7 3 = log = 3 7 OR = 3 = 7 (.70, 0) (.70, 7) =.70 =.70

38 Question 3 (c) 5 ( ) log 0 = 4 log 0 = = = 65 = 3.5 = log + log = log + log = 4 OR (3.5, 0) (3.5, 4) = 3.5 = 3.5

39 Question 33 (a) ( point) ( 0, 4.5) Question 33 (b) ( point) = 4 Question 33 (c) ( points) It is a growth curve. The function is of the form = ab + c, which is a growth curve if a > 0 and b >.

40 Question 34 log + 3 log z 3 log + log 3 z log 3 z 3 OR log + 3log 3log z log + log log z log log 3 z log z Question 35(a) ( points) (i) 3 (ii) 3 Question 35(b) ( points) (i) (ii) Question 35(c) ( point) log N = log b N b

41 Question 36 ( points) = = 3 Division b 0 is undefined. Question 37 Initial amount of 000 Doubles Doubling period is 5 units of time Eamples: A $000 investment doubles ever 5 ears. The number of bacteria starts at 000 and doubles ever 5 hours.

42 Question 38 h = + 3 = = = 5 = The chord is 5 cm from the centre. Question 39(a) ( points) B (3, ) AC ( 6 ( ) ) ( ( ) ) D = + A (-, -) C (6, -) = 49 + = 50 D (, -5) = 5 BD ( 3 ) ( ( 5) ) D = + = + 49 = 50 = 5 Question 39(b) ( ) Midpoint of AC =, 5 3 =, ( ) Midpoint of BD =, 5 3 =, Since the diagonals have the same midpoint, the bisect each other.

43 Question 40 ( point) Some eamples: If a quadrilateral is a rhombus, then the diagonals are perpendicular. If a quadrilateral is a square, the diagonals are congruent. If ou have a million dollars, then ou are rich. Question 4 C 0 C = 05 = 5 5 OR = = 5 4 0

44 Question 4(a) ( points) for numerators C 330 C = 6435 = OR = for denominators Question 4(b) ( points) for numerators C OR 3C = 5 87 = = for denominators Question 43 Eamples: What is the probabilit of being 6 3 or left handed? What is the probabilit of becoming a teacher or a mother? What is the probabilit of being over 30 and having O positive blood?

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