released items 2010

Pure Mathematics 30

2010 Released Diploma Examination Items

Released Items

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

Contents

Introduction ...............................................................................................................1

Released Written-Response Items

Question 1 .........................................................................................................2

Question 2 .........................................................................................................4

Question 3 .........................................................................................................7

Scoring Guides for Released Written-Response Items

Scoring Guide for Question 1 ............................................................................9

Scoring Guide for Question 2 .......................................................................... 13

Scoring Guide for Question 3 .......................................................................... 18

Alberta Education, Learner Assessment 1 Pure Mathematics 30

Introduction

This document contains written-response items and scoring guides from the January 2006 diploma examination. Teachers may wish to use these items in a variety of ways to improve the degree to which students develop and demonstrate an understanding of the concepts described in the Pure Mathematics 30 Program of Studies.


Use the following information to answer the first question.

In 2005, Alberta’s centennial year, the Alberta government began a program that gives each child born or adopted in Alberta in 2005 or later $500 to be deposited into a registered educational savings plan (RESP). Jordan was born in Alberta on January 3, 2005.

Written Response—10%

• What will be the value, to the nearest cent, of the province’s grant to Jordan on her 18th birthday if it has been invested on her birth date at 4%/a, compounded annually?

1.

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

Jordan’s family leaves Alberta in 2005. Her parents decide to contribute $500 to the same RESP on each of her birthdays, starting January 3, 2006.

Assume that the only contributions to the RESP are the initial grant from the provincial government and the amount that the parents contribute each year on Jordan’s birthday. The value of the RESP on Jordan’s nth birthday, V(n), including her parents’ contribution on that day, is given by the function below.

V(n) = 12 500(1.04n + 1 – 1), n W!

• Determine algebraically on which birthday the value of Jordan’s RESP will first exceed $8 500.



Tuition fees at a particular university have been increasing exponentially over a ten-year period.

Year Tuition Fees ($)

1995 2 600

2005 5 000

• Using the data in the table above, determine the average annual growth rate of tuition fees, to the nearest hundredth of a percent.


Assume that the earnings of student employees at their summer jobs are normally distributed with a mean of $2 240. It is known that 80% of students earn from $1 728 to $2 752, amounts that are symmetric about the mean.

• Determine the standard deviation of the distribution, to the nearest dollar, of student earnings.


Use the following information to answer the next question.

The roller-coaster at an indoor amusement park has a section of track that rises above the level of the roof. A glass dome covers this section, as shown below.

The side view of a particular section of the track has the shape of one period of a sinusoidal curve. At its highest point, this section of track is 14 m above the level of the roof. At its lowest point, it is 6 m below the level of the roof. A diagram of the side view of this section is represented on the coordinate plane shown below.

The y-axis represents the height of the track, in metres, with respect to the level of the roof. The x-axis represents the horizontal distance, in metres, to the right or left of the highest point on the track.



• State the period of the sinusoidal curve shown on the previous page.2.

•Determine the equation of the sinusoidal portion of the roller-coaster track in the form y = a cos[b(x – c)] + d for the domain x25 25# #- , where y is the height in metres and x is the horizontal distance in metres.

•Whentheequationofthesinusoidalportionoftheroller-coastertrackiswrittenin the form of y = [ ( )]sina b x c d- + , only one of the parameters a, b, c, and d is changed. Identify which parameter changes and state the new value of this parameter.



The outline of the cross section of the dome directly over the roller-coaster track forms a semi-ellipse. The cross section has a maximum width of 44 m and a maximum height of 20 m above the level of the roof. A diagram of the cross section of the dome and the roller-coaster track, with a coordinate plane superimposed on it, is shown below.

• Determine the horizontal distance, w, between the wall of the dome and the roller-coaster track at the level of the roof, to the nearest tenth of a metre.

• Statetheequationinstandardformoftheellipsethatmodelsthisdomefortherestriction y0 20# # .



A car dealership is promoting a particular model of car that has 8 different optional features available. Each optional feature can be purchased separately.


• How many different packages of 3 optional features are possible for this model of car?

3.

•Explain why the number of different packages of 5 optional features is the same as the number of different packages of 3 optional features for this model of car.


• Usinginformationfrompastsales,thecardealershipfoundthat62%ofitscustomers selected air conditioning as an option on one particular model of small car. Determine the probability, to the nearest hundredth, that at least 5 of the next 7 customers who purchase this model of car will select air conditioning as an optional feature.

• Anothermodelofcarhasn different optional features available. When 2 optional features are chosen for this model of car, there are 45 different packages available. Determine algebraically the number of optional features, n, that are available for this model of car.


Use the following information to answer the first question.

In 2005, Alberta’s centennial year, the Alberta government began a program that gives each child born or adopted in Alberta in 2005 or later $500 to be deposited into a registered educational savings plan (RESP). Jordan was born in Alberta on January 3, 2005.


• What will be the value, to the nearest cent, of the province’s grant to Jordan on her 18th birthday if it has been invested on her birth date at 4%/a, compounded annually?

A POSSIBLE SOLUTION for bullet 1

A = P(1 + i)n = 500(1.04)18

= $1 012.91

1.



Jordan’s family leaves Alberta in 2005. Her parents decide to contribute $500 to the same RESP on each of her birthdays, starting January 3, 2006.

Assume that the only contributions to the RESP are the initial grant from the provincial government and the amount that the parents contribute each year on Jordan’s birthday. The value of the RESP on Jordan’s nth birthday, V(n), including her parents’ contribution on that day, is given by the function below.

V(n) = 12 500(1.04n + 1 – 1), n W!

• Determine algebraically on which birthday the value of Jordan’s RESP will first exceed $8 500.


8 500 = 12 500(1.04n + 1 – 1)

1 + 12 5008 500 = 1.04n + 1

.log 1 6810 = (n + 1) .log 1 0410

..

loglog

1 041 68

10

10 – 1 = n

12.2275… = n

or log1.041.68 = n + 1

The value of Jordan’s RESP will not exceed $8 500 until after her 12th birthday. Therefore, Jordan’s RESP will first exceed $8 500 on her 13th birthday.



Tuition fees at a particular university have been increasing exponentially over a ten-year period.

Year Tuition Fees ($)

1995 2 600

2005 5 000

• Using the data in the table above, determine the average annual growth rate of tuition fees, to the nearest hundredth of a percent.


Compound Interest Formula

A = P(1 + i)n

5 000 = 2 600(1 + i)10

2 6005 000

10 = (1 + i)

1.0676 = 1 + i

0.0676 = i

or

Geometric Sequence

tn = arn – 1

5 000 = 2 600r11 – 1

2 6005 000

10 = r

1.0676 = r

System of Equations

y= a(bx)

2 600 = a(b)1995

5 000 = a(b)2005

b10 = 2 6005 000

b = 2 6005 000

10

b = 1.067 578…

b = 1.0676

or

Exponential Regression

•Inputyearvaluesintoalist.

•Inputtuitionfeevaluesinto another list.

•Performanexponential regression on the lists.

y = 2 600(1.067 578 123…)x y = 2 600(1.0676)x

The average annual growth rate of tuition fees is 6.76%.



Assume that the earnings of student employees at their summer jobs are normally distributed with a mean of $2 240. It is known that 80% of students earn from $1 728 to $2 752, amounts that are symmetric about the mean.

• Determine the standard deviation of the distribution, to the nearest dollar, of student earnings.


From Tables

z = –1.28

z = x -v

n

–1.28 = 1 728 2 240-v

v = .1 28

1 728 2 240-

-

v = $400

or

z = .1 29-

z = x -v

n

–1.29 = 1 728 2 240-v

v = $397

or Invnorm(0.10) = 1 728 2 240-v

= –1.281 551 567

v = $399.52

= $400



The roller-coaster at an indoor amusement park has a section of track that rises above the level of the roof. A glass dome covers this section, as shown below.

The side view of a particular section of the track has the shape of one period of a sinusoidal curve. At its highest point, this section of track is 14 m above the level of the roof. At its lowest point, it is 6 m below the level of the roof. A diagram of the side view of this section is represented on the coordinate plane shown below.

The y-axis represents the height of the track, in metres, with respect to the level of the roof. The x-axis represents the horizontal distance, in metres, to the right or left of the highest point on the track.



• State the period of the sinusoidal curve shown on the previous page.


The period is 50 m.

2.

•Determine the equation of the sinusoidal portion of the roller-coaster track in the form y = a cos[b(x – c)] + d for the domain x25 25# #- , where y is the height in metres and x is the horizontal distance in metres.


a = max min2- = ( )

214 6- - = 10 m

d = max min2+ = ( )

214 6+ - = 4 m

b = period

2r = 502r =

25r (In degrees, b = 7.2)

c = 0 as there is no phase shift from y = cos x

Therefore, the equation is y = cos x1025

4+r

c m , where x25 25# #- .


•Whentheequationofthesinusoidalportionoftheroller-coastertrackiswrittenin the form of y = [ ( )]sina b x c d- + , only one of the parameters a, b, c, and d is changed. Identify which parameter changes and state the new value of this parameter.


Only parameter c would change as there would be a phase shift of y = sin x. This phase

shift would be to the left with a value of 41 of the period.

Therefore, c = 41 (–50) = –

225 = –12.5 m.

or

Only parameter c would change as there would be a phase shift of y = sin x. This phase shift can be determined by graphing both the cosine equation from the last bullet and the line y = 4 and determining the x-coordinate of the intersection point closest to the y-axis where the function is increasing.

Therefore, c = 41 (–50) = –

225 = –12.5 m.



The outline of the cross section of the dome directly over the roller-coaster track forms a semi-ellipse. The cross section has a maximum width of 44 m and a maximum height of 20 m above the level of the roof. A diagram of the cross section of the dome and the roller-coaster track, with a coordinate plane superimposed on it, is shown below.

• Determine the horizontal distance, w, between the wall of the dome and the roller-coaster track at the level of the roof, to the nearest tenth of a metre.


To find w, we need to find the x-intercept of the cosine function and subtract this from 22, the x-intercept of the cross section of the roof.

Graph the equation y1 = cos x1025r

c m + 4

Using the zero feature, find the x-intercept.x-intercept is approximately 15.774 7… m.

or

0 = cos x1025r

c m + 4

– 4 = cos x1025r

c m

1.982 313 = x25r

15.7747… = x

Therefore, w = 22 m – 15.77 m = 6.2 m.


• Statetheequationinstandardformoftheellipsethatmodelsthisdomefortherestriction y0 20# # .


The equation would have the form ax

b

y2

2

2

2

+ = 1.

The value of b = 20 m, as this would equal the height.

The value of a = 244 = 22 m.

Therefore, the equation is x y

22 202

2

2

2

+ = 1 or x y484 400

2 2

+ = 1.



A car dealership is promoting a particular model of car that has 8 different optional features available. Each optional feature can be purchased separately.


• How many different packages of 3 optional features are possible for this model of car?


Distinguishable choices of 3 options 8C3 = 56

3.

•Explain why the number of different packages of 5 optional features is the same as the number of different packages of 3 optional features for this model of car.


Each time 5 options are selected, 3 options remain that are not selected. Therefore, when forming a group of 5, a group of 3 is also formed. If all groups of 5 are counted, then there will be an equal number of groups of 3.

or8C5 =

( )! !!

8 5 58

-

= ! !!

3 58

and8C3 =

( )! !!

8 3 38

-

= ! !!

5 38


• Usinginformationfrompastsales,thecardealershipfoundthat62%ofitscustomers selected air conditioning as an option on one particular model of small car. Determine the probability, to the nearest hundredth, that at least 5 of the next 7 customers who purchase this model of car will select air conditioning as an optional feature.


P(5 or more) = 7C5(0.62)5(0.38)2 + 7C6(0.62)6(0.38)1 + 7C7(0.62)7 = 0.464 11…0 0.46

orP(5 or more) = binompdf(7, 0.62, 5) + binompdf(7, 0.62, 6) + binompdf(7, 0.62, 7)

= sum(binompdf(7, 0.62, {5, 6, 7}))0 0.46

orP(5 or more) = 1 – binomcdf(7, 0.62, 4) 0 0.46

• Anothermodelofcarhasn different optional features available. When 2 optional features are chosen for this model of car, there are 45 different packages available. Determine algebraically the number of optional features, n, that are available for this model of car.


nC2 = ( )! !

!n

n2 2-

= 45

( )!( ) ( )!

nn n n

21 2-

- - = 90

n n 902- - = 0

( ) ( )n n10 9- + = 0

n = 10 n 9=- , as n must be positive

There are 10 optional features available.

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