secondary v june 2012 ccoommppeetteennccyy 22 · mathematics 506— competency 2 student booklet...

Mathematics — 565-506

Secondary V June 2012

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Science Option

Student Booklet (Parts A, B and C)

Name:

Group:

TEACHER USE ONLY

Part A _______ /24

Part B _______ /16

Part C _______ /60

Total ______ /100

Time: 3 hours

Mathematics 506— Competency 2 Student Booklet

BIM Société GRICS Page 1

Instructions

1. Write all the required information in this Student Booklet. Answers written in the

Question Booklet will not be evaluated; only the Student Booklet will be evaluated.

2. In Part C you must show all your work to justify your answers. The steps in your procedure must be organized and clearly presented. Always write down your procedure and reasoning along with aspects of the situation that you think are relevant. Do this even if you are unsure of how to complete your solution.

3. Permitted materials

Geometry set (ruler, compass, set square, protractor)

Calculator (with or without graphic display)

Handwritten memory aid prepared by the student prior to the evaluation situation. The memory aid consists of one letter-sized sheet of paper (8½” x 11”). Both sides of the sheet may be used. Any mechanical reproduction of this memory aid cannot be used.

Note: Figures have not necessarily been drawn to scale.

The following are the evaluation criteria required to complete the tasks in this booklet.

Evaluation Criteria Competency Two: Uses Mathematical Reasoning

Cr. 1 - Formulation of a conjecture suited to the situation

Cr. 2 – Correct use of appropriate mathematical concepts and processes

Cr. 3 – Proper implementation of mathematical reasoning suited to the situation

Cr. 4 – Proper organization of the steps in an appropriate procedure

Cr. 5 – Correct justification of the steps in an appropriate procedure



Part A

Questions 1 to 6 Darken the letter that corresponds to your answer.

Each question is worth 4 marks.

1. [A] [B] [C] [D]

2. [A] [B] [C] [D]

3. [A] [B] [C] [D]

4. [A] [B] [C] [D]

5. [A] [B] [C] [D]

6. [A] [B] [C] [D]

Part B

Questions 7 to 10 Write your answer in the space provided.

Each question is worth 4 marks.

7. The solution set of the equation is: _______________________. 4 0

8. The exact value of the expression is: _______________________. 4 0

9. xgf = __________________________ 4 0

10. a) The rule that defines the busy street is: _______________________. 4 2 0

b) The rule that defines the walking path is: _________________________.



Part C

This part of the examination consists of Questions 11 to 16.

For each question, you must show all your work to justify your answer. The work you show on all the pages of Part C will be considered.

Your work must be organized and clearly presented and cannot simply involve listing the calculator applications or programs used to obtain results or information.

Each question in Part C is worth 10 marks.

11. A Trigonometric Identity

Prove the trigonometric identity below

xxxxxxxx seccoseccossin2cot2costan2sin



Show your work or explain your reasoning.

Uses mathematical reasoning

Observable indicators correspond to level

Evalu

ati

on

Cri

teri

a

LEVEL A B C D E 0

Cr. 3 40 32 24 16 8 0

Cr. 2 Cr. 5

40 32 24 16 8 0

Cr. 4 20 16 12 8 4 0



12. The Displacement of a Squirrel

The following graph shows the displacement of a squirrel as it looks for food:

The path is also represented by the following piecewise function:

P31 4

139 93

94if 96

42 6

20 10

tbt

tkt

tta

tt

c

tt

f(t)

After how many seconds does the squirrel touch the ground?

f(t)

P

Height

(m)

t

Time (s)




ANSWER

The squirrel touches the ground after

____________________ seconds.



Evalu

ati

on

Cri

teri

a

LEVEL A B C D E

Cr. 3 40 32 24 16 8 0

Cr. 2 40 32 24 16 8 0

Cr. 4 Cr. 5

20 16 12 8 4 0



13. The Cell Phone Bill Josh is an avid cell phone user and his father pays his phone bill which is based on the

number of text messages and calling minutes.

Different constraints limit the number of text messages and calling minutes Josh can use.

The polygon of constraints ABC below represents the number of text messages and calling

minutes that Josh must respect each month when he uses his cell phone. Each side of the

polygon and its corresponding relation are identified by the same number.

where x : number of text messages

y : number of calling minutes

Josh’s cell phone plan charges $0.25 per text message and $0.40 per calling minute (to

the nearest second) for calls. Josh always manages to spend the maximum amount

permitted each month.

During Spring Break, Josh missed his curfew so his father reduced his cell phone

privileges for the next month—the sum of the number of text messages and the number

of calling minutes combined could now be no more than 420.

What amount did Josh’s cell phone bill decrease by, in the month following Spring

Break?

1. y ≤ 3x -300 2. x + y ≤ 600 3. x ≤ 2y

Vertices of the Polygon of Constraints

A (120,60)

B (225,375)

C (400,200)

Josh’s Text Messages and Calling Minutes y

x

1 2

3



Show your work or explain your reasoning. ANSWER

Josh’s cell phone bill decreased by

____________________ in the month following

Spring Break.



Evalu

ati

on

Cri

teri

a

LEVEL A B C D E

Cr. 3 40 32 24 16 8 0

Cr. 2 40 32 24 16 8 0

Cr. 4 Cr. 5

20 16 12 8 4 0



14. Modern Art The city wishes to construct a modern art sculpture using one semi-elliptical and two

parabolic metal bars. The artist’s sketch of the sculpture is pictured below:

The semi-elliptical bar has a major axis of 14 m. The vertex of the lower parabolic bar intersects the vertex of the minor axis of the

semi-ellipse, and it is also the focus of the upper parabolic bar.

The lower parabolic bar has a point located at (2,3) and the distance from its vertex to

its focus is 0.5 m.

The ends of the upper parabolic bar are situated at the foci of the semi-ellipse. Due to city regulations, the sculpture cannot be more than 7 m in height. Will the modern art sculpture respect the city height regulations?

x

y




ANSWER

The modern art sculpture will respect

the city’s height regulations.

The modern art sculpture will NOT

respect the city’s height regulations.



Evalu

ati

on

Cri

teri

a

LEVEL A B C D E

Cr. 3 40 32 24 16 8 0

Cr. 2 40 32 24 16 8 0

Cr. 4 Cr. 5

20 16 12 8 4 0



15. Where the Wild Things Are Over the years rats have unfortunately made their way onto the Hawaiian Islands through

shipping and trading of goods. In order to curb this problem, the government of Hawaii

decided to introduce snakes to the island of Maui in order to control the rat population.

However with no natural predators, the snake population increases exponentially. After

introducing 4 snakes to an area of Maui, the snake population is expected to double every

3 years. Meanwhile, the graph below demonstrates how the rat population (y) is declining

exponentially over time (x) in years:

How long will it take for the snake population to equalize the rat population? Round your answer to the nearest hundredth of a year

x

y

(1,270)

(2,243)

A

B

Time (years)

Population (animals)

(1, 270)

(2, 243)




ANSWER

It will take ________________ years for the

snake population to equalize the rat

population.



Evalu

ati

on

Cri

teri

a

LEVEL A B C D E

Cr. 3 40 32 24 16 8 0

Cr. 2 40 32 24 16 8 0

Cr. 4 Cr. 5

20 16 12 8 4 0



16. The Angle Between Two Vectors A student has studied only right-angled trigonometry and needs to find the angle

between two vectors.

Formulate a conjecture to determine the angle between two vectors that have

components (a,b) and (c,d).

*N.B. All the components have the same sign.

a

b

(a,b)

e.g.

c

d

(c,d)



Show your work or explain your reasoning. CONJECTURE:



Evalu

ati

on

Cri

teri

a

LEVEL A B C D E

Cr. 3 40 32 24 16 8 0

Cr. 2 20 16 12 8 4 0

Cr. 4 Cr. 5

20 16 12 8 4 0

Cr. 1 20 16 12 8 4 0

secondary v june 2012 ccoommppeetteennccyy 22 · mathematics 506— competency 2 student booklet...

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