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ure Mathematics 30 Pure Mathematics 30 Pure Mathematics 30 Pure Mathematics 30 Pure Mathe

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Grade 12 Diploma ExaminationPure Mathematics 30

January 2002January 2002

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Copyright 2002, the Crown in Right of Alberta, as represented by the Minister of Learning, AlbertaLearning, Learner Assessment Branch, 11160 Jasper Avenue, Edmonton, Alberta T5K 0L2. All rightsreserved. Additional copies may be purchased from the Learning Resources Centre.

Special permission is granted to Alberta educators only to reproduce, for educational purposes andon a non-profit basis, parts of this examination that do not contain excerpted material only after theadministration of this examination.

Excerpted material in this examination shall not be reproduced without the written permission of theoriginal publisher (see credits page, where applicable).

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January 2002

Pure Mathematics 30Grade 12 Diploma Examination

Description

Time: This examination was developed tobe completed in 2.5 h; however, you maytake an additional 0.5 h to complete theexamination.

This is a closed-book examinationconsisting of

33 multiple-choice and 6 numerical-response questions, of equal value,worth 65% of the examination

3 written-response questions worth 35%of the examination

A tear-out formula sheet and twoz-score pages are included in thisbooklet.

All graphs on this examination arecomputer-generated.

Note: The perforated pages at the back ofthis booklet may be torn out and used for

your rough work. No marks will be givenor work done on the tear-out pages.

Instructions

You are expected to provide a graphingcalculator approved by Alberta Learning.

You are expected to have cleared yourcalculator of all information that is storedin the programmable or parametricmemory.

Use only an HB pencil for the machine-

scored answer sheet.

Fill in the information required on theanswer sheet and the examinationbooklet as directed by the presidingexaminer.

Read each question carefully.

Consider all numbers used in thequestions to be exact numbers and notthe result of a measurement.

If you wish to change an answer, eraseall traces of your first answer.

Do not fold the answer sheet.

The presiding examiner will collect youranswer sheet and examination bookletand send them to Alberta Learning.

Now turn this page and read the detailedinstructions for answering machine-scoredand written-response questions.

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ii

Multiple Choice

Decide which of the choices bestcompletes the statement or answersthe question.

Locate that question number on theseparate answer sheet provided and fillin the circle that corresponds to yourchoice.

Example

This examination is for the subject of

A. biologyB. physicsC. chemistryD. mathematics

Answer Sheet

A B C D

Numerical Response

Record your answer on the answer sheetprovided by writing it in the boxes andthen filling in the corresponding circles.

If an answer is a value between 0 and 1(e.g., 0.7), then be sure to record the 0before the decimal place.

Enter the first digit of your answerin the left-hand box. Any boxes onthe right that are not needed are toremain blank.

Examples

Calculation Question and Solution

Correct to the nearest tenth of a radian, 40

is equal to __________ rad.

40 = 0.6981317008 . . . rad

= 0.7

(Record your answer in the numerical-response

section on the answer sheet.)

0 0 0 0

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

6 6 6 6

7 7 7 7

8 8 8 8

9 9 9 9

. .

Record 0.7 on the

answer sheet 0 . 7

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iii

Correct-Order Question and Solution

When the following subjects are arranged in

alphabetical order, the order is ______ ,______ , ______ , and ______ .

1 biology

2 physics

3 chemistry

4 mathematics

(Record all four digits of your answer in thenumerical-response section on the answersheet.)

Answer: 1342

0 0 0 0

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

6 6 6 6

7 7 7 7

8 8 8 8

9 9 9 9

. .

Record 1342 on the

answer sheet 1 3 4 2

Written Response

Write your responses in theexamination booklet as neatly aspossible.

For full marks, your responses mustaddress all aspects of the question.

Descriptions and/or explanations ofconcepts must include pertinent ideas,diagrams, calculations, and formulas.

Your responses must be presentedin a well-organized manner usingcomplete sentences and correct units.

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iv

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1

Use the following information to answer the first question.

The partial graph of a cubic function, f, is shown on the left, and the partialgraph of a transformation of f is shown on the right.

Partial Graph of Partial Graph of y = g(x),y = f(x) the Transformation of y = f(x)

3 2 1 1 2 3

6

4

2

2

4

6

x

y

f

3 2 1 1 2 3

6

4

2

2

4

6

x

y

g

1. Which of the following transformations of f results in graph g shown on theright?

A. )(1)( xfxg =

B. )()( xfxg =

C.)(

1)(xf

xg =

D.)(

1)(xf

xg =

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2

Use the following information to answer the next question.

The partial graphs of two functions, f and g, are shown below.

y

x

(1, )

f(0, 1)

y

x

(1, 2)g

(0, 1)

2. If graph g is a transformation of graph f, then the equation that would generategraph g is

A. )()( xfxg =

B. )()( 1 xfxg =

C. )()( xfxg =

D.)(

1)(xf

xg =

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3


The graph of function y = f(x) is shown below.

y

x

f

The design below was produced by using f and three differenttransformations of f.

y

x

f

The equations related to the three transformations are1 y = f(x)2 y = f(x)3 y = f(x)

Numerical Response

1. The equation that produced the portion of the design found in

quadrant II is equation number __________ (Record in the first column)quadrant III is equation number __________ (Record in the second column)quadrant IV is equation number __________ (Record in the third column)

(Record all three digits of your answer in the numerical-response section on the answer sheet.)

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3. The graph of y = f(x) = bx, where b > 1, is translated such that the equation ofthe new graph is expressed as y 2 = f(x 1). The range of the new function is

A. y > 2

B. y > 3

C. y > 1

D. y > 2

4. If y is replaced by y21 in the equation y = f(x), then the graph of y = f(x) will

be stretched

A. horizontally about the y-axis by a factor of21

B. horizontally about the y-axis by a factor of 2

C. vertically about the x-axis by a factor of21

D. vertically about the x-axis by a factor of 2

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5. The graph of y = x3 was transformed to the graph of y = (x 3)3 + 4. Which ofthe following statements describes the transformation?

A. The graph of y = x3 has been translated 4 units to the right and 3 unitsupward.

B. The graph of y = x3

has been translated 3 units to the left and 4 unitsdownward.

C. The point (x, y) on the graph y = x3 has been translated to point (x + 3, y + 4).

D. The point (x, y) on the graph y = x3 has been translated to point (x 3, y 4).

6. In the first stage of a letter-writing campaign, a person writes the same letter toeach of 3 pen pals. In the second stage, each of these 3 pen pals copies the letterand sends it to 3 other pen pals, and so on, following this pattern. If it is assumed

that no pen pal receives a letter twice, the least number of stages it would take untilthe minimum total sum of pen pals who received the letter is 9 800 is

A. 7

B. 8

C. 9

D. 10

7.A new coal mine produced 8 Mt (megatonnes) of coal in 1997, 7 Mt in 1998,and 6.125 Mt in 1999. If this geometric pattern were to continue indefinitely,then the total mass of coal produced by this mine would approach

A. 21.125 Mt

B. 64 Mt

C. 156.9 Mt

D. an infinite mass

Numerical Response

2. If32log7 =m , then the value of m, correct to the nearest hundredth,

is __________.

(Record your answer in the numerical-response section on the answer sheet.)

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8. In 1996, a particular car was valued at $27 500 and its value decreased exponentiallyeach year afterward. For each of the first 7 years, the value of the car decreasedby 24% of the previous years value. If t is the number of years and v is the valueof the car, then the equation for the cars value when t 7 is

A. v = 27 500(1.76)t

B. v = 27 500(1.24)t

C. v = 27 500(0.76)t

D. v = 27 500(0.24)t


The equation that defines the decibel level for any sound is

=

010log10 I

IL where L = loudness in decibels

I = intensity of sound being measured

I0 = intensity of sound at the threshold of hearing

9. Given that normal conversation is 1 000 000 times as intense as I0, then theloudness of normal conversation is

A. 5 decibelsB. 6 decibels

C. 16 decibels

D. 60 decibels

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7


The relationship between the length and mass of a particular speciesof snakes is

log m = log a + 3 log l

where a is a given constantm is the mass of the snake in gramsl is the length of the snake in metres

10. This relationship can also be written as

A. log

3

al

m = 0

B. log

alm

3= 0

C. log

3

l

am = 0

D. log

l

am

3= 0

11. The x-intercept of the graph of y = logbx, where b > 0 and b 1, is

A. 0

B. 1

C. undefined

D. dependent on the value of b

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8


The two sets of numbers shown on the graph below represent the highest andlowest temperatures over 12 months at Waterton National Park in southernAlberta.

91 95 93

87

75 79

62

55 5651

3136

22 26

1118

3

1 11 7

2 7

36 32

J an F eb M ar A pr M a y J un Jul A ug Se pt O ct N ov D e c

Temperature(F)

C u r v e I I

100

90

80

70

60

50

40

30

20

10

0

10

20

30

40

C u r v e I

10 0

50

M o n t h

12. Each of these sets of numbers can be modelled by a sinusoidal curve of the formy = a sin [b (t c)] + d. The two parameters in the equation representing curve I thatwould change the most in value in the equation representing curve II are

A. a and b

B. b and c

C. c and d

D. a and d

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9

13. The value of

6

sin6

cos 22 is equal to the value of

A.12

cos

B.6

cos

C.3

cos

D.2

cos

14. In the equation 5 sin(2x) + 2 = cos(x) 1, the number of solutions for x, where

0 x < 2, is

A. 2

B. 3

C. 4

D. 5

15. When the following pairs of functions are graphed, the pair that could not be used

to solve the equation 4 sin x 1 = 0 is

A. y = sin x and y = 1

B. y = sin x and y =41

C. y = 4 sin x and y = 1

D. y = 4 sin x 1 and y = 0

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16. Three consecutive terms of a geometric sequence are tan, 2 sin, and4 sincos. The common ratio of this sequence is

A. 2

B. cos

C. 2 cos

D. sec21

17. The expression sin + (cot)(cos) is equivalent to

A. 1

B. csc

C. sec

D. 2cot

18. Which of the following statements does not describe the graph of

=

2sin3)( f ?

A. The amplitude is 3.

B. The period is 2.

C. The graph of

=

2sin3)( f is the same as the graph of

)sin(3)( =f with a phase shift of2 to the right.

D. The graph of

=

2sin3)( f is the same as the graph of

=

2sin)( f with a vertical translation of 3 units down.

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A computer-generated partial graph of the air pressure variation of a pianos

lowest note is illustrated below. The intercepts shown are at

50and,75,150,0

.

150

75

50

0.01

0.01

Time

Air pressure

Numerical Response

3. Given that pitch, which is measured in Hertz (Hz), is the reciprocal of the period,the pitch of this note, correct to the nearest tenth, is __________ Hz.


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12

19. The range of the ellipse 12581

22=+

yx is

A. 9 y 9

B. 5 y 9

C. 9 y 5

D. 5 y 5

20. When converted to general form, the equation3

)2(2

)3( 22 +

yx= 1 is

A. 3x2 2y2 18x 8y + 13 = 0

B. 3x2 2y2 18x 8y + 19 = 0

C. 3x2 2y2 18x + 8y + 29 = 0

D. 3x2 2y2 18x + 8y + 35 = 0

Numerical Response

4. The conic described by 1918

22=+

yx will have a positive x-intercept

at ( , 0),m n where m and n N and m > 1. The values of m and n are,

respectively, __________ and __________.


21. A right circular cone is intersected by a plane that is parallel to a generator of thecone. The shape of the conic section formed in the plane is

A. a hyperbola

B. a parabola

C. an ellipse

D. a circle

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13

22. The equation of the conic 19

)2(25

)3( 22=

+

yxrepresents a hyperbola with

A. centre at (3, 2) and vertices at (3, 3) and (3, 7)

B. centre at (3, 2) and vertices at (3, 3) and (3, 7)

C. centre at (3, 2) and vertices at (8, 2) and (2, 2)

D. centre at (3, 2) and vertices at (8, 2) and (2, 2)


A paperboy who delivers papers on his bike can travel only on the trailsrepresented in the diagram below.

House A

House B

23. The number of different trails that the paperboy can take to get from house A tohouse B without backtracking is

A. 13

B. 32

C. 60

D. 72

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24. The number of different arrangements of 3 boys and 4 girls in a row, if the girlsmust stand together, is represented by

A. 4! 4!

B. 3! 4!

C. 4! 4! 2!D. 3! 4! 2!

25. The students in a music department have practised 6 contemporary and 5 traditionalchoruses. For their concert, they will choose a program in which they present 4 ofthe contemporary and 3 of the traditional choruses. How many different programscan be presented, if the order of the choruses does not matter?

A. 25

B. 35C. 150

D. 330

26. All telephone numbers are preceded by a 3-digit area code. In the original BellTelephone System of assigning area codes, the first digit could be any numberfrom 2 to 9, the second digit was either 0 or 1, and the third digit could be anynumber except 0. In this system, the number of different area codes possible was

A. 126

B. 144

C. 160

D. 576

Numerical Response

5. A term of the binomial expansion (ax + y)8, where a > 0, is 112x2y6 . The value

of a, correct to the nearest whole number, is __________.


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27. On an English test, the class mean was 40% with a standard deviation of 5%. Theteacher decided to add an additional 20% to each students test score. When theteacher recalculated the mean and standard deviation, he found that

A. both the mean and standard deviation had increased

B. both the mean and standard deviation remained unchangedC. the mean remained unchanged and the standard deviation had increased

D. the mean had increased and the standard deviation remained unchanged

28. Test scores for a particular examination are normally distributed with a meanof 67.4% and a standard deviation of 10.5%. What is the probability of a studentgetting less than 80% on the test?

A. 0.88

B. 0.51C. 0.38

D. 0.12


A standard normal distribution is shown below.

0.20z - scores 1.83

29. The area of the shaded region, correct to the nearest hundredth, is

A. 0.39

B. 0.55

C. 1.63

D. 2.03

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30. The mean examination mark of a random sample of 225 students is 64.0%, witha standard deviation of 7.5%. The symmetric 95% confidence interval forindividual student marks in this sample is from

A. 49.3% to 78.7%

B. 63.2% to 64.8%C. 63.5% to 65.5%

D. 64.0% to 78.7%

Numerical Response

6. A baseball pitcher has 65% of his pitches called strikes. For the next 200pitches he throws, the expected standard deviation of the number of those pitchesthat will be called strikes, correct to the nearest tenth, is __________.


31. From two vases that each contain purple and yellow tulips, a florist randomlyselected two tulips. If the selection constituted an independent event, then theflorist must have selected

A. one tulip from the first vase and one tulip from the second vase

B. one purple tulip and one yellow tulip from the second vase

C. two tulips of the same colour from the first vaseD. both tulips from the same vase

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1

4

2

3

1

4

2

3

Two identical spinners each have four equal areas. Each arrow is spun onceand stops in one of the numbered areas.

32. The probability that the sum of the two numbered areas in which the arrows stopsis greater than 4 is

A.41

B.21

C.85

D.1613


At a family reunion, door prizes are to be given out. At one table in thecommunity hall, 6 children, 3 teenagers, 4 adults, and 5 seniors areseated. The 3 winning tickets are held by 3 different people at this table.

33. The probability that the 3 winning tickets are held by 3 people in the same agegroup, correct to the nearest thousandth, is

A. 0.037

B. 0.043

C. 0.222

D. 0.980

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Written Response10%

1. Two students, Peter and Ken, performed a simulation that involved the tossingof 3 coins. They recorded the number of heads and tails from 10 trials, as shownbelow.

Trial 1 HHT Trial 6 THHTrial 2 HHT Trial 7 HTHTrail 3 HTT Trial 8 TTTTrial 4 HHH Trial 9 HHHTrial 5 HTT Trial 10 TTT

According to the results from the simulation above, what is the experimentalprobability of getting 3 heads on a single trial?

Use the following information to answer the next part of the question.

Ken stated that the experimentalprobability of getting heads in thissimulation was higher than he hadexpected. He defined the samplespace as

HHH HHT HTH HTTTTT TTH THT THH

Peter suggested that the samplespace should be

3 heads 2 heads and 1 tail3 tails 2 tails and 1 head

Ken reasoned that because there were 8possible outcomes, the probability of

getting 3 heads on a single trial was81 .

Peter, reasoned that because therewere only 4 possible outcomes,the probability of getting 3

heads on a single trial was41 .

Determine who was correct and justify your choice.

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Use the formula nCk pk(1 p)n k and the sample space that you chose in the

previous bullet to determine, to the nearest thousandth, the theoretical probability

of flipping exactly 2 sets of HHH in 10 trials.

Ken suggests that the probability of flipping exactly 2 sets of HHH in 10 trialscan be modelled using a normal approximation of the binomial distribution.

Is Ken correct? Justify your answer.

Written-response question 2 begins on the next page.

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A square, S1, with sides of 10 cm is modelled below. A second

square, S2, with sides of cm25 is inscribed in S1 so that the

vertices of S2 lie at the midpoints of the sides of S1. A third

square, S3, is inscribed in S2 so that the vertices of S3 lie at the

midpoints of the sides of S2. This process is continued indefinitely.

10 cm

10cm

S1

S2

S3

S4

52

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Written Response15%

2. The lengths of the sides of S1, S2, S3, form a geometric sequence.

Determine the exact length of a side of S3.

The perimeter of each square forms a term in a geometric sequence.Determine the exact value of the perimeter of S4.

Written-response question 2 continues on the next page.

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If the process of drawing the squares were to continue indefinitely, then whatwould be the sum of the infinite geometric series of the squares perimetervalues, correct to the nearest hundredth of a centimetre?

Show how the areas of consecutive squares form a sequence that is geometric.

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Determine an expression that represents the sum of the first k terms of thegeometric sequence calculated in the previous bullet.

Written-response question 3 begins on the next page.

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The partial graph of f(x) = 8 cos x sin x is shown below.

5

6

3

2

1

4

1

3

4

6

2

5

y

x

232

Written Response10%

3. Identify the amplitude and the period of f.

Determine an equation in the form of g(x) = a sin[b(x c)] + d that defines f asa sine function.

Written-response question 3 continues on the next page.

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Determine an equation in the form of h(x) = a cos[b(x c)] + d that defines f asa cosine function.

Sketch the graph of )(xfy = as it would appear on a graphing calculator with

the viewing window [ ]1,6,6:2

,2,0:

yx .

56

3

2

1

4

1

3

4

6

2

5

y

x

2 3

22

Determine the roots of 0)( =xf , and express your answer as a general solution.

You have now completed the examination.If you have time, you may wish to check your answers.

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F

oldandtearalongperforation.

TearPa

Pure Mathematics 30 Formula SheetThe following information may be useful in writing this examination.

For 02 =++ cbxax ,

a

acbbx 2

42 =

For two points (x1, y1) and (x2, y2),

212

212 )()( yyxxd +=

Graphing Calculator Window Formatx: [x

min, x

max, x

scl]

y: [ymin, ymax, yscl]

Exponents, Logarithms,and Geometric Series

NMNM aaa loglog)(log +=

NMNM

aaa logloglog =

MnM an

a loglog =

b

cc

a

ab log

loglog =

1=

nn art

1,1

)1(

= r

r

raS

n

n

1,1

= r

r

artS nn

1,1

pm30_jan02_de

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