Example of an appropriate solution - · PDF file · 2015-05-24x x x x x x x x x x or 3 2 or cos 1 2 1 cos 0 2cos 1 cos 1 ... P = 30x + 40y y 1500 1000 B (500, 1000) C (1000, 500)

Mathematics 568-536 Guide Page 9

15 /4

16 /4

Part C

Questions 15 to 25 4 marks each No marks are to be given if work is not shown. Examples of correct solutions are given. However, other acceptable solutions are possible.

Example of an appropriate solution Interval for which the height of the helicopter is at least 52 m

1023914114139

5190515190

689034

521209034

=−

≤≤

≤−≤−

≤−

−≥−−

≥+−−

tt

t

t

t

Answer: The helicopter was at least 52 m above the ground for 102 seconds. Note: Accept a graphical representation of the solution.

Students who graphed the function and indicated the line y = 52 have shown they have a partial understanding of the problem.

Example of an appropriate solution 4, 3 2, or 0 marks (h, k) is midpoint of 2 vertices (2, 13)

12a5a13

bac

5b13c

222

222

=

+=

+=

=

=

Answer: The equations of the asymptotes are ( ) 132125

+−+= xy and

( ) 132125

+−−= xy .

Note: Accept any other equivalent response.

Student who identified the values of b and c have shown a partial understanding of the problem.


17 /4 Example of an appropriate solution

( )

( )( )

π=π

=

−=−=

++=

++=

=−−

=−−

=−

xx

xx

xxxx

xx

xx

xx

or32

1cosor21cos

1cos1cos201cos3cos20

3cos3cos22

3cos3cos12

3cos3sin2

2

2

2

2

Answer: The exact values are π=π

= xx or32 .

Students who were able to arrive at 0 = (2 cos x + 1) (cos x + 1) have shown they have a partial understanding of the problem.


18 /4 Example of an appropriate solution Constraints:

00

000101051500

2

≥

≥

≥+

≤+

≤

yx

yxyxxy

Objective function: maximize profit P = 30x + 40y

1500

1000 B (500, 1000)

C (1000, 500)

2000 1500

A (400, 800)

x

y

Answer: The maximum profit the company can make is $55 000. Note: Deduct 1 mark if students use P = 35x + 50y

Students who have the inequalities and the correct graph have shown they have a partial understanding of the problem.

OR

Student who have the inequalities and the correct vertices have shown they have a partial understanding of the problem.

Vertices P = 30x + 40y

A (400, 800) $ 44 000

B (500, 1000) $ 55 000

C (1000, 500) $ 50 000


19 /4 Example of an appropriate solution Solution

26014

−=

+−=

+−=

a

a

khxay

Solving for x when y = 2

xx

x

x

=

=

−=−

+−=

42

24

622

giving (4, 2) Using the absolute value function

86

86

6800

144

6842

−=

−−=−

+−−=

=

=−

=−

+−=

x

x

x

yaa

a

x = 2 or x = 14 Answer: The plane spent 14 seconds in the air. Note: Students who found the point (4, 2) have shown they have a partial understanding of

the problem.


20 /4 Example of an appropriate solution Rule of distance of head to the ground

40Amplitudecleseconds/cy6Period

seconds60cycles10

Frequency

=

=

=

bP π=2

3

62

π=

π=b

( )

803

cos40

802401202minmax

cos

+π

−=

=

+=

+=

+−=

ty

k

khxbay

Distance to ground at t = 5

( )

cm60

8053

cos40

=

+π

−=

y

y

Answer: Five seconds after it is released, the head is at the height of 60 cm. Note: To account for rounding at different places, accept answers in the range of 59 to 61.

Students who use an equivalent rule to arrive at the same answer should not be penalized. Students who use an appropriate method to determine any 2 parameters have shown they have a partial understanding of the problem. .


21 /4 Example of an appropriate solution Since the radius is 4 cm, diameter AB is 8 cm. Segment AC is 4 cm, then triangle ABC is a 30-60-90 triangle. Therefore, m∠BAC = 60° and m∠ABC = 30°. Since arcs are double their inscribed angles,

m CDB = 120° and m AC = 60° Given m CD = 80°

∴ m DB = mCDB − MCD

m DB = 120° − 80°

m DB = 40° Since

m AC − mBD m∠DPB = 2

°=

°−°=∠∴

1024060DPBm

Accept any method to determine that measure of ∠BAC is 60°. Answer: The measure of ∠DPB is 10° .

Students who use an appropriate method to correctly determine the measure of AC have shown they have a partial understanding of the problem.


22 /4 Example of an appropriate solution Length of CE

( )

CEm51

CEm502

theoremproductConstantCEm55102

=

+=

+=

Length of CG

.bisectedisdiameteratolarperpendicuchordAcm5.7CGm = Length of DG

( )( ) ( )

xx

x

xxxx

x

=

=

=

=

=

=

3.475.18

325.56

theoremproductConstant35.75.74isdiameterthesince3FGm

DGmLet

2

2

Answer: To the nearest tenth of a centimetre, the length of DG is 4.3 cm. Note: Do not penalize students who did not round, or rounded incorrectly.

Students who have determined the length of CG have shown they have a partial understanding of the problem.


23 /4 Example of an appropriate solution Equation representing profit decrease for Company A

( )( )

( )

( ) ( )x

2

2

0

x

9.04

9.081.0

424.324.3,2

444,0ngSubstituti

=

=

=

=

=

=

=

xg

cc

c

aac

acxg

Equation representing growth for Company B

f(x) = acx + 15 Substituting (0, -10)

( )( )

cc

c

c

cxf

aac

≅

=

−=−

+−=

+−=

=−

+=−

92.042.0

255.10

5.4,10Point15255.4

1525

251510

10

10

10

x

0

∴ f(x) = -25(0.92)x + 15

Solving for f(11) and g(11)

( ) ( )

( ) ( )5125$orthousands255.1

9.0411

$5009orthousands009.51592.02511

11

11

=

=

=

+−=

g

f

Difference between the two companies

$5009 − $1255 = $3754 Answer: Company B would make $3754 more than Company A. Accept answers in the interval [ ]42003754, Accept also [ ]thousands4.0andthousands3.7 Note: Students who find the correct rule for either one of the two companies, have shown

they have a partial understanding of the problem.


24 /4 Example of an appropriate solution To convert from general form to standard form:

25)25()15(825625)25(225)15(

825503008255030

22

22

22

22

=−+−

−=−−+−−

−=−+−

=+−−+

yx

yx

yyxx

yxyx

! Formula for large circle C1: (x − 15)2 + (y − 25)2 = 25. Therefore the center is at (15, 25) and its radius is 5 m. ! Vertical distance from center of C1 to center of C2 is 25 m − 9 m = 16 m. Therefore the coordinates for the center of C2 are (15, 16) and its radius is 4 m. ! Vertex for the lower parabola is center of C2 – Radius of C2. Therefore coordinates for the vertex (h, k) of the lower parabola are (15, 12) ! Since center of C2 is also point on the directrix for lower parabola, then the value of

parameter “c” value is vertical distance from vertex of lower parabola to center of C2. Therefore “c” = -4

Substituting in (x − h)2 = 4c(y − k) we get:

(x − 15)2 = − 16(y − 12) ! The y-coordinate of P is 0; the x-coordinate must be found. Therefore, substitute y = 0 in the equation above:

( ) ( )( )

86.28or14.119215

19215

19215

12016152

2

≅≅

±=

±=−

=−

−−=−

xxx

x

x

x

Answer: The distance from point 0 to point P is 1.14 m. Note: Students who use an appropriate method to determine the vertex of the parabola,

(15, 12) have shown they have a partial understanding of the problem.

Do not penalize students who did not round or rounded incompletely.


25 /4 Example of an appropriate solution ( )( ) 123

343=

=

f

f therefore a point (x, y) on the ellipse is (3, 12)

Vertex on horizontal axis is at (4, 14) ∴ Center of the ellipse must be at (0, 14) = (h, k) a is 4 – 0 = 4 (x-value of vertex on horizontal axis – h) Substituting a, h, k, x and y we get:

( ) ( )

( ) ( )

02.37641674

14169

11412403

1

2

2

2

2

2

2

2

2

2

2

2

≅

=

=

=+

=−

+−

=−

+−

b

b

b

b

b

bky

ahx

Answer: The total height of the logo is 3.02 + 14 = 17.02 cm, to the nearest hundredth

centimetre. Note: Do not penalize students who did not round, or rounded incorrectly. Students who have used an appropriate method to determine the equation of the ellipse have shown they have a partial understanding of the problem

MATHEMATICS

568-536

Summative Examination

June 2008

Question Booklet

Secondary 5 Mathematics and Science & Technology Committee

3 hours

Mathematics 568-536 Question Booklet Page 1

INSTRUCTIONS 1. Write the required information on the cover page of your Answer Booklet. 2. Answer all 25 questions in the Answer Booklet. 3. Each question is worth 4 marks. 4. You may use a calculator (with or without graphing display), and a memory aid. 5. The following materials are allowed: graph paper, ruler, compass, set square,

and protractor. 6. The figures in this booklet have NOT been drawn to scale. 7. At the end of the exam period, hand in the Question Booklet and Answer Booklet.

Time allotted

3 hours


1

Part A

Questions 1 to 8 In the Answer Booklet, blacken the letter that corresponds to the answer chosen.

Which of the following scatter plots shows the weakest correlation? A)

B)

C)

D)


2 The manager of the Little Fry fast food restaurant noticed the following consistencies on any given day: • At least 250 orders of French fries were sold. • The restaurant sold no more than twice as many orders of large fries as small fries. Let l represent the number of orders of large fries

s represent the number of orders of small fries Which of the following systems of inequalities represents this situation? A)

00l2l250l

≥

≥

≥

≤+

s

ss

B)

002250

≥

≥

≤

≥+

s

ss

lll

C)

00l

l2250l

≥

≥

≤

≤+

s

ss

D)

00

2250

≥

≥

≥

≥+

s

ss

lll


3

60

y

- 60

30

- 30

The graph of a step function is shown below.

Which of the following rules defines this function? A)

!"

#$%

&= xy

20110

B) [ ]xy 2010 −−=

C)

!"

#$%

&−= xy

20110

D)

!"

#$%

&−−= xy

20110

x


5

6

4 Given f(x) =

xx−

+

132 2

and g(x) = -x + 2.

Which of the following represents (f ο g) (x)? A)

11182 2

−

+−

xxx

B)

3112 2

+

+

xx

C)

xxx

−

−−−

1122 2

D)

31182 2

+

+−

xxx

Which expression below is equivalent to the following?

10 log a − 3 log b + 21 log 9

A)

!"

#$%

&ba15

log B)

( )3103log ba

C)

( )ab135log −

D)

!!"

#$$%

&3

103log

ba

An ellipse has two vertices at (4, -3) and (4, -9), and one of its foci at (0, -6). Which of the following is the equation of the ellipse?

A)

( ) ( ) 1

96

254 22

=−

++ yx

B)

( ) ( ) 1

256

94 22

=−

++ yx

C)

( ) ( ) 1

96

254 22

=+

+− yx

D)

( ) ( ) 1

256

94 22

=+

+− yx


7

8

Consider vectors v and s below.

v

ur s

ur

What is the resultant vector of sv − ? A)

B)

C)

D)

In the diagram on the right,

°=∠

°=∠

60AEDm48ACDm

.

A B

C D

E 60°

48°

?

What is the measure of BC ?

A)

24°

B)

60°

C)

36°

D)

96°


11

10

9

12

Part B

Questions 9 to 14 Write your answer in the space provided in the answer booklet. Show your work, where required.

Given the function ( ) 173 +−= xxf . What is the rule for its inverse? Solve the following logarithmic equation:

( ) ( )4log33xlog 22 −−=+ x Prove the following trigonometric identity.

( ) xx

xx cos1cos1sinsin −=⎟⎟

⎠

⎞⎜⎜⎝

⎛

+

Consider 8=v and AB , whose vertices are A(-2, 4) and B(8, 28).

The scalar product of the two vectors is 104. What is the measure of the angle between the two vectors?


14

13 In the adjacent circle with centre O, CD is perpendicular to AB at point D BDm = 4 cm OCm = 13.5 cm

What is the area of triangle ABC? Round your answer to the nearest tenth.

A

O

D C

B

13.5 cm

4 cm

Sally’s Z-score on a math test was -1.8. The marks of all the students in Sally’s math class are listed below. The standard deviation of the class is 6.98.

65 73 74 72 64 78 78 58 66 78 Billy’s mark in a different math class was 3 higher than Sally’s. His class average was 78 and the standard deviation for his class was 5. What was Billy’s Z-score?


17

18

15

16

Part C

Questions 15 to 25 ! Show all your work as well as your answer. The work shown is

taken into consideration when marks are awarded. ! Your written information must be legible, complete, and clearly

stated in correct language so the marker understands exactly what you have done.

Even if your answer is correct, no marks will be given unless acceptable work is shown.

In designing software for a helicopter simulation game, a computer programmer uses the following rule as a model to determine the height of the helicopter at any given

time:" ( ) 1209034

+−−= tth

where h is the height in metres and t is the elapsed time in seconds. For how many seconds was the helicopter at least 52 m above the ground? On a Cartesian plane, the vertices of a hyperbola are located at (2, 8) and (2, 18). The foci of the hyperbola are located at (2, 0) and (2, 26). What equations can be used to represent the asymptotes of the hyperbola? Solve the following trigonometric equation. Give the exact value(s).

2 sin2 x − 3 cos x = 3 x ∈ [ ]π,0 A company produces different games. The two most popular are the Memory game and the Construction game. The number of games that the company stocks is based on past sales, which indicate that it sells at most twice as many Construction games as Memory games. The company cannot have more than 1500 of these games in stock. It costs $5 to produce the Memory game and $10 for the Construction game. The company expects to spend a minimum of $10 000 to produce these games. The Memory game sells for $35 while the Construction game sells for $50. Let x represent the number of Memory games in stock

y represent the number of Construction games in stock What is the maximum profit the company can make selling these games?


19

20

Anthony received a remote-controlled airplane for his birthday. The plane’s altitude, as a function of time, is represented by a square root function followed by an absolute value function. The plane’s altitude follows a square root function until it first reaches 2 metres, at which point the altitude can be described by an absolute value function. Anthony begins by putting his plane into take-off position from an altitude of 6 metres. One second after take-off, the plane is 4 metres above the ground. The plane reaches its maximum altitude of 6 metres 8 seconds after take-off.

8

2

y

x

Altitude (m)

Time (seconds)

6

(1, 4)

How much time did the plane spend in the air? The diagram below depicts the head of a Jack-in-a-box used in the display window of a department store. The head is connected to a motor, and its up-and-down movement follows a sinusoidal curve. The head is compressed to 40 cm at t = 0 and it reaches a maximum height of 120 cm. It bounces with a frequency of 10 cycles per minute.

x

y

120 cm

40 cm

At what height is the head, 5 seconds after it is released?


21

The circle below with center O has a radius of 4 cm. The measure of segment AC is 4 cm, and the measure of arc CD is 80°. Segment CE is an altitude of triangle ABC whose side AB passes through the center of circle O.

P

B

D

O

C

E

A

80°

What is the measure of ∠DPB? Show and justify all of your work.


22

23

In the adjacent circle with centre O, • AB is a tangent to the circle at point A

cm5BCm

cm10ABm

=•

=•

• FD is a diameter • FDCE ⊥ • DGmOGm =

F

A B

C

D G

E

O

What is the length of DG? Round your answer to the nearest tenth of a centimetre. Company A has seen a decrease in profit since its competitor, Company B, opened its doors. The decrease can be estimated using an exponential function in the form of g(x) = acx. The profit of Company B can be estimated according to an exponential function in the form of f(x) = acx + 15.

x

y

4 Company A

(10, 4.5)

(Years since Company B opened its doors)

Company B

(2, 3.24)

Profit (thousands of $)

Comparison of Profit

-10

Based on these estimates, how much more profit would Company B make than Company A, 11 years after it opened its doors? Round your answer to the nearest dollar.


24 Courtney has been hired to paint lines on a field. The lines consist of a large circle C1, whose centre coincides with the centre of the rectangular field, two small (congruent) circles, and two (congruent) parabolas. The rectangular field, along with the lines Courtney must paint, are shown on the Cartesian plane below, which is scaled in metres. The equation of the large circle (C1) drawn on the field is:

x2 + y2 − 30x − 50y + 825 = 0 In addition, " Circles C1 and C2 are tangent to one another. Their centres are vertically aligned

9 metres apart. " The lower parabola is tangent to C2 at its vertex, which is directly below the center

of C2. " The center of C2 is a point on the directrix of the lower parabola. Courtney must begin the paint job at point P and she needs to know how far away from point 0 she should start.

0

C1

C2

x

y

?

P

What is the distance from point 0 to point P? (Round your answer to the nearest hundredth metre.)


25

(3,y)

Lickety Splitz Cotton Candy Parlour is designing a logo for its letterhead. The sides of the paper cone are defined by the function:

( ) xxf 4= The cotton candy on top of the cone is in the shape of an ellipse whose centre is directly above the vertex of the cone. One vertex on the horizontal axis of the ellipse is at (4, 14). The ellipse intercepts the cone at x = 3. (Note that all units are in centimetres).

What is the total height of the logo? Round your answer to the nearest hundredth of a centimetre.

Name: Class: Teacher’s Name:

MATHEMATICS

568-536

Summative Examination

June 2008

Answer Booklet

Secondary 5 Mathematics and Science & Technology Committee

FOR TEACHER USE ONLY Part A ___________ /32 Part B __________ /24 Part C __________ /44 Total _________ /100

Mathematics 568-536 Answer Booklet Page 1

1

2

3

4

5

6

7

8

9

10

Part A

Questions 1 to 8 Blacken the letter that corresponds to the answer chosen.

Each question is worth 4 marks.

[A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D]

Part B Questions 9 to 15 Write your answer in the space provided.

The rule for the inverse of the function is 4 3 0 __________________________________________________ Answer : ____________________________________________ 4 2 0


11 4 2 0 Prove the following trigonometric identity. Show all your work.

( ) xx

xx cos1cos1sinsin −=⎟⎟

⎠

⎞⎜⎜⎝

⎛

+


12 4 3 1 0 Show all your work. Answer:

The measure of the angle between the two vectors is __________ °.


13 4 3 1 0 Show all your work.

A

O

D C

B

13.5 cm

4 cm

Answer:

The area of triangle ABC is __________ cm2.


14 4 2 0 Show all your work. Answer:

Billy’s Z-score was __________.


Part C

Questions 15 to 25 ! Show all your work as well as your answer. The work shown is

taken into consideration when marks are awarded. ! Your written information must be legible, complete, and clearly

stated in correct language so the marker understands exactly what you have done.

Even if your answer is correct, no marks will be given unless acceptable work is shown.

Note: Diagrams have not necessarily been drawn to scale.


15 4 3 2 1 0 Show all your work.

Answer:

The helicopter was at least 52 m above the ground

for _________________________ seconds.


16 4 3 2 1 0 Show all your work. Answer:

The equations of the asymptotes are: _____________________________ and _____________________________ .



2 sin2 x − 3 cos x = 3 x ∈ [ ]π,0 Answer:

The exact value(s) is (are) ___________________________ .



200

200

Answer:

The maximum profit the company can make is $_______________.



8

2

y

x

Altitude (m)

Time (seconds)

6

(1, 4)

Answer:

The plane spent _______________ seconds in the air.



x

y

120 cm

40 cm

Answer:

Five seconds after it is released, the head is at the height of __________ cm.



P

B

D

O

C

E

A

80°

Answer:

The measure of ∠DPB is __________° .



F

A B

C

D G

E

O

Answer:

To the nearest tenth of a centimetre, the length of DG is __________ cm.



x

y

4 Company A

(10, 4.5)

(Years since Company B opened its doors)

Company B

(2, 3.24)

Profit (thousands of $)

Comparison of Profit

-10

Answer:

Company B would make $ ____________ more than Company A.



0

C1

C2

x

y

?

P

Answer:

The distance from point 0 to point P is __________ m.



(3,y)

Answer:

The total height of the logo is ____________________ cm, to the nearest hundredth centimetre.

Documents

Example of an appropriate solution - · PDF file · 2015-05-24x x x x x x x x x x or 3 2 or cos 1 2 1 cos 0 2cos 1 cos 1 ... P = 30x + 40y y 1500 1000 B (500, 1000) C (1000, 500)