Nova Scotia Examinations Mathematics 12 Web Sample 1. Student Booklet

Transcription

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 Value 5 pts (appro. 40 min)* Constructed-Response Questions Value 8s (appro. 0 min) *note: there are 5 constructed response questions on the Math NSEs as of January 008 Total time: hours (revision time included) Use these suggested times to guide you in the completion of the eamination; however, you might not find it necessary to spend the suggested time on each section. Plan your time to enable you to complete the eamination. You are not permitted to use your own graphing calculator unless your teacher has cleared the memory immediately prior to this eamination. The only graphing calculators permitted are TI-8, TI-8, TI-8 Plus, TI-84, or TI-84 Plus. If the question indicates that you are not to use a graphing calculator, you 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 your use during the eamination. Note: Diagrams are not necessarily 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 carefully, 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 your choice as shown in the eample below. Use an HB pencil only. Eample. What are the roots of. A B D + 4= 0? A. 4 and B. 4 and C. 4 and D. 4 and (On student answer sheet) If you wish to change an answer, please ensure that you erase your first answer completely 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 January 008 there are 5 selected response questions on the NSE Math eams. Page Mathematics

4 . Which formula generates the sequence represented by the graph below? t n n A. = + B. tn n n C. = + D. tn n n t = n n+ n n t = n n+ 4. The equation of the ais of symmetry of the parabola represented by the function y = ( ) is A. = B. = C. = 0 D. y =. What is the nature of the roots of a quadratic equation having a discriminant equal to 0? A. two equal real roots B. C. two imaginary roots D. two unequal real roots no real roots 4. Given the equation + 5 = 0, then is equal to A. ± 5 B. 5 C. ± 5i D. 5 ± i Mathematics Page

5 5. Which graph has a vertical stretch factor of when compared to the graph of y =? A. B. y y C. D. y y For the function y = a + b+ c, the y-intercept is always A. b a B. C. c a D. c 4ac b 4a 7. The function representing a parabola with verte at (, ) and passing through the point (4, 4) is A. y = ( ) + B. ( ) C. y = + D. y = + ( ) ( ) y = + Page 4 Mathematics

6 8. The domain and range of the function represented by the graph below are: y 5 4 (, ) { R} and { y R} { R} and { y R y } A. B. { R} and { y R y } { R } and { y R y } C. D. 9. An olympic diver dives from the high diving board. The distance, d, in metres, from the surface of the water varies with the time, t, in seconds, that have passed since she left the board, according to the equation d = t + t+ 0. What is her maimum elevation above the water during the dive? (rounded to the nearest unit) A. metres B. 0 metres C. metres D. metres 0. Which of the following is NOT a geometric sequence? 9 { },,,... { 4, 8,.5,... } A. B { } 0.8, 0.08, 0.008,... {,,,...} C. D. 4 Mathematics Page 5

7 . The students at East High School are planning a surprise party for the principal. One student tells two other students who in turn each tell two other people and so on as indicated in the following diagram: What type of sequence is depicted by this situation? A. cubic B. C. quadratic D. geometric arithmetic. y 5 can also be epressed as 5 y 5 A. B. y 5 y C. D. y 5. Which of the following epressions is equivalent to log B log C+ log D? A. log BD C B. C. BD log C D. log log BD C B CD Page 6 Mathematics

8 4. The graph below shows the growth, in square millimetres, of a rare fungus, F, in a petrie dish after each day (t). F (4, 05) (, 4500) (, 6750) 000 (0, 000) The equation of the function is 4 5 t A. F = 4500(.5) t B. C. F = 4500(.5) t D. F = 000(.5) t F = 000(.5) t 5. If a person earns an annual interest of 7% compounded annually on an investment of $50.00, which function shows how much money the investor has at time, t, in years? A. Pt = 50(.07) t B. Pt = 50(.7) t t C. Pt () = (.07) 50 D. Pt = 50(0.07) t 6. The function y = 5() + has a horizontal asymptote at: A. y = 5 B. y = C. y = D. y = 5 7. An eponential equation that is equivalent to log 4 = is : 8 A. 8 = 4 B. 4 = 8 C. D. 4 = = Mathematics Page 7

9 8. The table given shows how two dollars invested over a period of years grows in value when interest is compounded annually. What is the approimate annual interest rate? (years) y ($) A..0% B. % C. 6% D. 9.7% 9. Choose the converse of the following statement: "If two chords on a circle are equidistant from the centre of the circle, then they are congruent." A. B. C. D. Therefore chords on a circle are equidistant from the centre of the circle. If the centre of a circle is equidistant from two chords, then they are congruent. If two chords on a circle are congruent, then they are equidistant from the centre of the circle. Two chords on a circle are congruent if they are equidistant from the centre. 0. What are the coordinates of the centre of the circle that has the points (a, b) and (c, d) as endpoints of a diameter? A. a c, b d B. C. c a, d b D. a+ b c+ d, a+ c b+ d,. Given PQ with midpoint M(, -) and endpoint Q(-5, ), what are the coordinates of P? A. (.5, ) B. (, 4) C. (.5, ) D. (9, 7). Two eight-sided dice are to be rolled. On any roll what is the probability of getting two different numbers? A B. C D Page 8 Mathematics

10 . A bag contains green marbles, 0 red marbles and 8 white marbles. Bill removes marble from the bag. The probability that Bill removes a red marble from the bag is 4 0 A. B. 40 C. D. 4. If an event can succeed in s ways and fail in f ways, then the probability of success is s s s A. f f B. f + s C. s f D. f 5. In a school of 00 students, 80 have blood type O. If 5 students are chosen at random, what is the probability of selecting five students with type O blood? A. B C. D. P C 80 5 C C Mathematics Page 9

11 Constructed-Response Questions (Total Value: 80.5 points) Read each question carefully, and be sure to write your response in the bo and space provided. If the answer bo indicates that you are to show your work, then points will be awarded for your correct work and your correct final answer. The method used to solve a problem must clearly 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 only. When working with decimal values, you may round off to the hundredths place in your final answer only. If any 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 probability unit, all answers must be given in simplified form. Page 0 Mathematics

12 6. Given the quadratic function y = (a) Algebraically determine the coordinates of the verte of the parabola represented by the function above. ( points) (b) State whether the verte in (a) is a maimum point or a minimum point. Eplain how you know. ( point) Mathematics Page

13 7. Given the graph below, do the following tasks without using the regression feature on your graphing calculator. y (-5, 0) (, 0) (-, -4.5) -6 (a) Determine the transformational form of the function represented by the above graph. ( points) (b) On the same grid above, trace a parabola that has the same -intercepts as the given parabola and a maimum value of 9. Write the coordinates of the verte and other points on this parabola. (.5 points) Page Mathematics

14 8. Solve algebraically to find the eact roots of the following equations. Simplify where possible. (a) 5 + 7= 0 ( points) (b) (+ ) = 4 ( + ) (.5 points) Mathematics Page

15 9. A snowball is thrown into the air. The function h= 4.9t + 0t+.8 epresses the relationship between height, h, in metres and time, t, in seconds. (a) Algebraically determine the maimum height the snowball reaches. ( points) (b) How long is the snowball in the air? ( points) Page 4 Mathematics

16 0. A rectangular rink with dimensions of 5 m by 0 m is to be epanded by adding a rectangular strip of uniform width as shown below. If the new rink is to have an area of 644 m, what will be the width of the strip? (4 points) 0 5 Mathematics Page 5

17 . Bill kicks a football in Tom's direction. The football follows a parabolic path. Tom, who does not know it has been kicked, may be standing in the football's path. After having travelled a horizontal distance of 0 metres, the football reaches a maimum height of 8 metres. Will Tom, who is.8 metres tall, get hit by the football if he's standing 9.8 metres from the where the football was kicked? Solve this problem algebraically. (5 points) Page 6 Mathematics

18 . Algebraically solve for : (a) 5 = 5 + ( points) (b) + 8 = 0 ( points) (c) log ( ) + log0 = log8 ( points) Mathematics Page 7

19 . Michael was running a Biology eperiment where he was determining and recording the approimate number of bacteria over time. The following is a partial record of some of the readings. No. of hours No. of bacteria (a) What was the initial number of bacteria in the culture? ( point) (b) Determine an equation to calculate the number of bacteria at any time. (.5 points) (c) What is the bacteria count after 0 hours? ( point) Page 8 Mathematics

20 4. Suppose the cost of a parking permit increases by 4% annually. If the cost of parking is now $00 per year, how long will it take for the price to triple? Clearly define the variable(s) you use. (4 points) Mathematics Page 9

21 5. (a) Describe in words how the graphs of y= b and y= b for b> 0, and b are related. You must state a total of similarities and/or differences. ( points) (b) Given the function y = ab, for what values of 'a' and 'b' will the graph of the function be an eponential growth curve? ( points) 6. Susan tried to solve the equation = log. She got the error message 'NONREAL ANS' on her TI-8 calculator when trying to evaluate log. Eplain why. ( points) Page 0 Mathematics

22 7. When Drug enters the bloodstream, it gradually dilutes, decreasing eponentially, by 0% every 5 days. A second drug, after entering the bloodstream, also decreases eponentially, but by 0% every 7 days. If the initial amount of Drug is 00 mg and the initial amount of Drug is 50 mg, create and use functions to determine which drug has the greater amount remaining after days. (5 points) Mathematics Page

23 8. A chord AB is 9 cm from the centre of the circle whose radius is 5 cm. What is the length of AB? Points will be awarded for a relevant labelled diagram. (4 points) Page Mathematics

24 9. In the following diagram the points A(, ) and B(8, -7) are on the circumference of the circle. y A B (a) Determine the equation of the perpendicular bisector of chord AB. (5 points) (b) Determine algebraically if the perpendicular bisector of chord ( 4, ). AB passes through the point ( point) Mathematics Page

25 40. Prove that XYZ with vertices X (, 4), Y (4, ), and Z(, ) is an isosceles triangle and not an equilateral triangle. (4 points) Page 4 Mathematics

26 4. One card is selected from a standard deck of 5 playing cards. What is the probability that the card selected is a diamond or an ace? (.5 points) 4. From a group of 5 men and 6 women, what is the probability that a committee formed at random will consist of men and women? (.5 points) 4. Joe, Mary, and George are among the seven finalists for a random draw to win three different prizes. What is the probability that Joe will win st prize, Mary will win nd prize, and George will win rd prize? Epress your answer in fraction form. ( points) Mathematics Page 5

27 44. John, Amy, and Fred tried to solve the following problem: In a certain city, during a person's lifetime the probability of having diabetes is 0.0 and the probability of having cancer is What is the probability of a person having either diabetes or cancer in his/her lifetime? Suppose that event C is 'person having cancer' and event D is 'person having diabetes'. Their proposed solutions are as follows: John's solution: PC ( and D ) = = Amy's solution: PC ( or D ) = = 0.5 Fred's solution: PC ( or D ) = = 0.45 (a) Which student has the correct answer? ( point) (b) Eplain why the other two solutions are NOT correct. ( points) You have reached the end of the SAMPLE Mathematics Eamination. Please check your work to ensure you have completed all questions. Page 6 Mathematics

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

29 Circle Geometry Unit d = ( ) + ( y y ) The coordinates of M are: +, y + y Δy m = Δ Probability 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 )!

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

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

32 Question 6 (a) ( points) b = a = ( ) = 6 = y = f = + + = = = 8 OR y = y = 4 y = ( ) y = + 8 = ( y 8) ( ) Note: Last step may be omitted and then the final answer gets a value. The coordinates of the verte are (, 8). Question 6 (b) ( point) The verte is a maimum point because: - the a value is negative - graphing the function y = + + 6shows it is a maimum. (with accompanying sketch) - values for y decrease for values on either side of the verte. - any other correct eplanation

33 Question 7 (a) ( points) a 0 a = 9 a 4.5 a = 9 a = ( y+ 4.5) = ( + ) ( + ) = ( + ) OR To go from (, 4.4) there is a horizontal translation is and a vertical translation of 4.5. If there was no vertical stretch the vertical translation should be 9, therefore the vertical stretch must be..5 pt ( y+ ) = ( + ) 4.5 OR 4.5 ( y+ ) = ( + ) ( y+ ) = ( + ) 4.5 OR 4.5 ( y+ ) = ( + ).5 pt Question 7 (b) (.5 points) y (, 9) 0 8 ( 4, 5) 6 (0, 5) ( 6, 7) -6 (, 7) -8-0 Award for a correctly drawn parabola passing through the verte and the correct -intercepts. Award for each of the other points labeled on the parabola.

34 Question 8 (a) ( points) = = 5 ± ± = 0 = 49 0 Question 8 (b) (.5 points) 6 + = = 0 0 ± 4 0 = ± 89 = 0 OR 0 5 = 0 ( )( ) 5 + = = = 0 = 7 = 0 = = 0 = 5 = 0 =

35 Question 9(a) ( points) h= t + t+ ( t t) ( t t ) = = ( t ) = The maimum height is. metres. OR h = t 4.086t h+ + = t t+ 4.9 h+ 4.5 = ( t.0408) 4.9 ( h.078) = ( t.0408) The maimum height is. metres. b t = a 0 = 4.9 ( ).0408 h The maimum height is. metres. OR 4ac b h = 4a ( ) 4( 4.9) =. The maimum height is. metres. pt

36 Question 9(b) ( points) Note:.04 = 4.08 is not an acceptable answer. ( 4.9) 0 ± t = 0 ± 45.8 = 9.8 t 0.09 t 4.7 = 4.9 ( 0.078) ( t.0408) ( t ) 4.5 =.0408 ±.89 = t.0408 t =.0408 ±.89 t = 4.7 t = 0.09 The snowball is in the air for 4.7 seconds. The snowball is in the air for 4.7 seconds. Note: Deduct if 0.09 is not discarded. ( t ) = 0 ( t ) =.078 ( t ).0408 = 4.5 t.0408 =±.89 t =± t = 4.7 t = 0.09 OR y = Note: point must be shown for value to be given. The snowball is in the air for 4.7 seconds. Note: Deduct if 0.09 is not discarded. The snowball is in the air for 4.7 seconds.

37 Question 0 (4 points) ( )( ) = = = 0 ( )( ) + 48 = = 0 = 0 = 48 = OR The width of the strip is m. ( ) () 45± = 45± 60 = 45± 5 = = = 48 Note: Deduct if 48 is not discarded. Question (5 points) y = a a 0,.5.5 = = 00a 0.65 = a ( ) y = = =.5.5 m >.8 m therefore the football will not hit Tom.

38 Question (a) ( points) + ( 5 ) = ( 5 ) 5 = = 6 4= 4 = OR ( ) log 5 = log5 + + log 5 = log5 log5 + = 5 + =.5 + = = = Question (b) ( points) log ( ) + = 5 + = log5 + log = log5 log5 + = log = 5 log 5 = = 0 ( ) + 6 = log = log0 log0 + 6 = log OR + = = 0 log 0 =

39 Question (c) ( points) log 0 = log 8 log 0 = 4 4 = 0 6 = 0 8 = 5 OR log 0 = log 8 log 0 = 4 log 0 = log 6 0 = 6 8 = 5 Question (a) ( point) P = P t 00 = P 00 = 4P 5 = P 0 4 Note: answer only is worth full value. The initial number of bacteria in the culture was 5. Question (b) (.5 points) P = 5 t y = 5(.4) Ep Reg on TI-8 a = 5 y = a b OR ( 4, 00) 00 = 5( b) 4 4 = b 4 4 = b.4 b 4 y = 5(.4)

40 Question (c) ( point) 0 y = 5.4 = There are approimately 47 bacteria after 0 hours. OR y y y y = = = 5 = 5600 There are approimately 5600 bacteria after 0 hours. Question 4 (4 points) 900 = t =.04 log.04 = t 8.0 t t It will take 8.0 yrs for the price to triple. 900 = t =.04 log = t log.04 log = t log t t It will take 8.0 yrs for the price to triple. C = C.04 =.04 log.04 = t 8.0 t It will take 8.0 yrs for the price to triple. t t OR C = C.04 t =.04 log = t log.04 log = t log t t It will take 8.0 yrs for the price to triple.

41 Question 5 (a) ( points) Note: The answers must refer to the graphs and NOT the equations. Here are some eamples: - same y-intercept - when y = b is increasing, y = b is decreasing - each is a reflection of the other in = 0 - same horizontal asymptotes - neither have -intercepts Question 5 (b) ( points) a > 0 and b > Question 6 ( points) log = = is always positive therefore it cannot equal. OR The domain of y= log a is > 0,. OR Susan has her calculator set to give only real number answers.

42 Question 7 (5 points) Drug M = = = 7.07 d 5 5 Drug M = = = 5. d 7 7 Drug has a greater amount remaining after days. Question 8 (4 points) 5 = cm 5 cm B 5 9 = 44 = = A The length of the chord is 4 cm.

43 Question 9(a) (5 points) ( 5, ) Midpoint of AB =, = m AB 7 = 8 8 = 6 = slope of bisector is 5, y= + b = ( 5) + b 6= 5+ b = b = b The equation of the perpendicular bisector is y= +. Question 9(b) ( point) ( 4, )? = ( 4) +? 4 = + The point is not on the line since it does satisfy the equation of the perpendicular bisector.

44 Question 40 (4 points) XY ( 4 ) ( 4) d = + XZ ( ) ( 4 ) d = = + 5 = = 6 = 5 XY ( 4 ) ( ) = 5+ = 6 d = XYZ has only equal sides and not equilateral. it is isoceles Question 4 (.5 points) ( D or A) = ( D) + ( A) ( D and A) P P P P 4 = = or 0. 5 Question 4 (.5 points) = = C C C = or

45 Question 4 ( points) P = 0 7 OR = Question 44(a) ( point) Fred has the correct answer. Question 44(b) ( points) Student s answers should address the following: - John s solution is not correct because he calculated the probability of C and D, not C or D. - Amy s solution does not discard the probability of C and D occuring.