Grade 11 MCR3U and MCF3M Exemplar Booklet for Mathematics

Exemplar Booklet 

for 

Grade 11 Mathematics Courses 

MCF3M and MCR3U 

2004 Revised Edition 

2002 Edition Prepared by: 

Leah Earl, Penny Elieff, Ron Gaudreau, Shawn Godin, 

Chris Noxon, Sheri Walker, Nancy Wyndham-Wheeler

Table of Contents 

Course of Study page 2 

Mathematical Communication page 10 

Mathematical Communication Rubric page 13 

Key Terms page 14 

Grade 11 General Formulae page 15 

Grade 11 Examination Formulae Sheet page 17 

Achievement Chart page 18 

Sample Examinations and Solutions 

Appendix A 

MCF 3M 2003 January Exam page 20 

Solutions to MCF 3M 2003 January Exam page 24 

MCF 3M 2003 Backup Exam page 31 

Solutions to MCF 3M 2003 Backup Exam page 34 

MCR 3U 2003 January Exam page 41 

Solutions to MCR 3U 2003 January Exam page 44 

MCR 3U 2003 Backup Exam page 52 

Solutions to MCR 3U 2003 Backup Exam page 55 

1

Ottawa-Carleton District School Board 

Mathematics Evaluation Project 

Common Course of Study 

Grade 11 Mathematics 

MCF3M and MCR3U 

Number of Periods 

(semestered length) 

Unit MCF3M MCR3U 

Strand: Tools for Operating and 

Communicating with Functions 

Introducing Functions and Transformations 12 10 

Quadratic Functions 16 14 

Strand: Trigonometric Functions 

Trigonometry 17 15 

Modelling Periodic Functions 11 9 

Strand: Investigations of Loci and Conics 

Investigation of Loci and Conics N/A 12 

Strand: Financial Applications of 

Sequences and Series 

Sequences and Series 14 12 

Financial Applications 12 10 

Review & End of Year summative tests and tasks 6 6 

Total 88 88 

Please note: The order of the units may be changed to follow the order used in a 

textbook or to suit a teacher’s preference. 

The MCF3M and MCR3U have differences in content and scope. The MCR3U 

course has the following additional expectations: Investigations of Loci and 

Conics strand, operations with complex numbers, and recursion formulas. 

N.B. The suggested number of periods allotted to each topic may. 

2

N.B. Communicating mathematical reasoning with precision and clarity is an overall expectation for the Tools for Operating and Communicating Functions 

Strand. In part, the specific expectations include: students will communicate solutions to problems clearly and concisely, present problems and their solutions to a 

group, demonstrate the correct use of mathematical language and conventions and use graphing technology effectively. Although not specifically addressed in 

the following course of study, the selection of questions and activities for this strand needs to address these expectations. 

Communicating solutions with clarity and justification using appropriate mathematics forms is an expectation throughout all the strands. 

INTRODUCING FUNCTIONS and TRANSFORMATIONS 

MCF3M - 12 periods 

MCR3U - 10 periods 

Periods Topic Specific Expectations/ Comments Nelson McGraw- 

3M 3U 

Hill 

3 2 Functions • understand concept & notation 3.1 3.1 

-relation, function, domain, 3.2 3.2 

range, vertical line test 

3.6 - part 1 

• investigate functions 

4.6 - part 2 

Addison- 

Wesley 

Skills 

p. 392 

7.1 

7.2 

Comments 

• investigation of absolute 

value function not 

required 

y = x 2 , y = 

x , y = 1 x 

1 1 Inequalities • solve first degree inequalities 

• graph solutions on a number line 

2.5 2 Inverse Functions • know properties 

e.g. ( x, y)→ ( y, x) 

• sketch/graph the inverse 

3.5 3 Transformations • apply transformations such as 

and Function 

translations, reflection, and 

Notation 

stretches 

• understand combinations of 

transformations e.g. 

y = af[ k( x − p) 

]+ q 

2 2 Review & 

Evaluation 

• state domain and range of 

transformed functions 

3.3 1.9 Skills 

p.394 

3.4 

(3.5) 

3.6 

p. 211, 

3.7 

3.5 7.5 

p.169, 

3.3, 

3.4, 

3.6 

3.7 

7.3 

7.4 

• single inequalities only 

• supplement A.W. material 

• leave trig transformations 

for later 

N.B. Textbook sections in parentheses are optional. 

3

QUADRATIC FUNCTIONS 

MCF3M - 16 periods MCR3U - 14 periods 

Periods Topic Specific Expectations/ Comments Nelson McGraw- Addison- Comments 

3M 3U 

Hill Wesley 

2 1 Extending Algebraic • simplify polynomials 

p. 303 1.4 

Skills 

Skills/Polynomials • factor 

4.12 p. 3 p.80 

2.1 

2 1.5 Simplifying Rational 

Expressions 

• state restrictions 4.8 1.5 2.2 

2.3 

2 1.5 Multiplying & • factor to simplify 4.9 1.6 2.4 

Dividing Rational 

Expressions 

2 2 Adding & 

Subtracting 

• determine L.C.D 4.11 1.7 

1.8 

2.5 

2.6 

Rational Expressions 

1 1 Completing The • extend to complex trinomials 4.1 p.99 Skills 

Square 

ax 2 + bx + c = 0, where a ≠ 1 

2.2 

p. 198, 

4.1 

2 1.5 Maxima/Minima • solve applications 4.2 2.2 4.1 • supplement A.W. 

applications 

2 1 Solving Quadratic 

Equations 

• solve applications 4.3 2.3 p.201 

4.2 

• supplement A.W. 

applications 

1 2.5 Complex Numbers • identify the complex number 

system 

• determine conjugate (omit for 

3M course) 

• add, subtract, multiply and 

divide (omit for 3M course) 

2 2 Review & 

Evaluation 

4.4 

4.5 

4.10 

2.1, 

2.5 


4.3 

4.4 

4


TRIGONOMETRY 


Periods Topic Specific Expectations/ Comments Nelson McGraw- Addison- 

3M 3U 

Hill Wesley 

1 1 Getting Ready • solve right triangles 

p. 397- 4.1 Skills 

• use trig ratios 

399 

p. 246-7 

2 2 Trig Functions of 

Angles in Standard 

Position 

4 3 Oblique Triangles & 

Applications 

• determine principal angle 

• determine related acute angle 

(reference angle) 

• apply sine law 

• apply cosine law 

• understand ambiguous case 

• apply to 2-D & 3-D applications 

2 1.5 Radian Measure • understand the relation between 

radian and degree measure 

• represent radian measure in exact 

form and in approximate form 

2 1.5 Trig Values of 

Special Angles 

2 2 Trigonometric 

Identities 

2 2 Trigonometric 

Equations 

2 2 Review & Evaluation 

5.2 

5.3 

p. 400 

6.1 

6.2 

5.2, 

p. 351 

4.2 

4.3, 

4.4 

5.1, 

5.3, 

5.4 

5.4 5.1 5.6, 

5.7 

• determine exact values for 

0 ≤θ ≤ 2π and 0 o ≤ θ ≤ 360 o 6.3 5.2 5.5 

• prove simple trig identities 6.5 

(6.4) 

• solve linear and quadratic 5.8, 

trigonometric equations 

6.6 

0 ≤θ ≤ 2π 

5.7 5.9 

p. 248-252 

5.2 

Comments 

• supplement 3-D 

applications 

• omit applications with 

angular velocity 

5.8 5.8 • supplement A.W & 

M. H. 

• solve equations involving 

sin kx or cos kx 

(i.e. 2cos(2x) =1) for 3U 


5

MCF3M - 11 periods 

MODELLING PERIODIC FUNCTIONS 


Periods Topic Specific Expectations/ Comments Nelson McGraw- Addison- 

Comments 

3M 3U 

Hill Wesley 

1 1 Modelling Periodic 

Functions 

• develop an understanding of a 

periodic function 

5.1 5.3 6.1 • supplement A.W. 

2.5 2 Graphing 

Trigonometric 

Functions 

2.5 2 Investigating the 

Graphs of 

Trigonometric 

Functions 

3 2 Applications of 

Trigonometry 

2 2 Review & 

Evaluation 

• identify amplitude, period length 

and domain and range: 

y = sin x, 

y = cos x, 

y = tan x (asymptotes) 

• determine the effect of simple 

transformations ( translation, 

reflections and stretches) 

• sketch the graphs of simple 

sinusoidal functions 

[ e.g., y = a sin x, y = cos kx, 

y = sin( x + d) , 

y = a cos kx + c ] 

• determine the transformations 

from the sinusoidal equations in 

the form y = a sin(kx + d ) + c or 

y = a cos(kx + d ) + c 

• write the equation of a sinusoidal 

function, given its graph and 

given its properties 

5.3, 

5.5 

5.6 5.5, 

5.6 

5.7, 

5.8 

(5.9, 5.10) 

5.4 6.2, 

6.6 

5.5, 

5.6 

(p. 392) 

6.3, 

6.4 


• use graphing calculators 

or graphing software to 

investigate the effect of 

simple transformations 

on y = sin x and y = cos x 

6.5 • supplement A.W. & 

M.H.R applications 

6

MCF3M – N/A 

INVESTIGATION OF LOCI and CONICS 


1 The Circle • identify in standard form with 

centre (0,0) and (h,k) 

• solve applications 

2 The Ellipse • identify in standard form with 



2 The Hyperbola • identify in standard form with 



2 The Parabola • identify in standard form with 

vertex (0,0) and(h,k) 


1 The General Form of • identify the nature of the conic in 

Conics 

general form and sketch 

1 Intersection of Lines • solve problems involving the 

& Conics 

intersections of lines and conics 

2 Review & 

Evaluation 

Periods Topic Specific Expectations/ Comments Nelson McGraw- 

3M 3U 

Hill 

1 Investigating Loci • construct a geometric model to 7.1 8.2, 

represent a described locus of 

(8.1), 

points 

(8.3) 

Addison- 

Wesley 

8.1 , 

8.5 

7.2 8.4 p.510 – 512 

9.5 

7.4 

(7.5) 

8.5 8.2, 

9.2, 

9.5 

7.9 8.6 8.3, 

9.3, 

9.5 

7.6 

(7.7) 

8.7 8.4, 

9.4, 

9.5 

7.10 8.8 9.6 

7.11 8.9 9.1, 

9.2 – 9.4 

Comments 

• illustrate the conics as 

intersections of planes 

with cones, using 

concrete materials or 

technology 

• develop equations for 

conics from their locus 

definitions 

[e.g. determine the 

equation of the locus 

of points the sum of 

whose distances from 

(-3,0) and (3,0) is 10] 


7


SEQUENCES AND SERIES 


Periods 

3M 3U 

1 1 Exploring Patterns 

and Sequences 

Topic Specific Expectations/ Comments Nelson McGraw- 

Hill 

Addison- 

Wesley 

Comments 

• introduce sequences 1.1 6.1 p. 7 • supplement A.W. with 

patterning problems from 

A.W. 1.1 & 1.2 

0 1 Recursion Formulas • write terms of a sequence given a 

recursion formula 

• omit for 3M course 

1.5 1 Arithmetic 

• identify the pattern and find the 

Sequences 

general term 

1.5 1 Geometric 

Sequences 

• identify the pattern and find the 

general term 

2 1.5 Rational Exponents • use laws of exponents to simplify 

and evaluate expressions 

2 1.5 Solving Exponential • solve exponential equations 

Equations 

e.g., 4x = 8 x +3 , 

2 2x − 2 x = 12 

2 1.5 Arithmetic Series • determine the sum of the terms 

using appropriate formulas and 

techniques 

2 1.5 Geometric Series • determine the sum of the terms 

using appropriate formulas and 

techniques 

2 2 Review & 

Evaluation 

1.3 6.4 1.4 

1.6 6.2 1.1, 

1.3 

1.7 6.3 1.2, 1.3 

1.9, 1.1, 1.5, 

1.10 1.2 

1.6 

1.11 1.3 1.6 

2.1 6.5 1.7 

(2.2), 

2.3 

6.6 1.8 


• illustrate linear and 

exponential growth 

• solve application 

problems involving 

arithmetic and geometric 

sequences 

• solve application 

problems if time permits; 

otherwise, determine 

sums using appropriate 

formulas and techniques 

8

N.B. Financial Applications will not be evaluated as part of the District-Wide Formal Exam. This unit will be evaluated as part of the in-school 

summative test or task. 


FINANCIAL APPLICATIONS 


Periods 

3M 3U 

2 1.5 Compound Interest • solve simple interest, compound 

interest, present value problems 

• solve problems involving linear 

and exponential growth 

4 2.5 Ordinary Annuities • solve simple and general 

annuities 

4 4 Mortgages & 

Financial Planning 

2 2 Review & 

Evaluation 

• solve problems in financial 

planning decision making 

• use spreadsheets or other 

appropriate technology 

• analyze effects of changing 

conditions of a mortgage 

2.5, 

2.7, 

2.8 

7.1, 

7.2, 

7.3, 

7.4 

7.5, 

7.6 

2.9 - 2.12 7.7, 

7.8 

Topic Specific Expectations/ Comments Nelson McGraw- 

Hill 

1.8, 

2.4 

Addison- 

Wesley 

p. 128 

3.1, 

3.2 

3.3, 

3.4 

3.5, 

3.6, 

3.7 

Comments 


9

MATHEMATICAL COMMUNICATION 

Content refers to the mathematical knowledge and skills taught in the course. 

A content mark is earned by demonstrating knowledge and skills related to a specific 

expectation. 

If there are several steps to a question, and you do not do the first step correctly, you may 

still earn marks by proving your ability in the rest of the solution. However, if your error 

made the problem easier or made it impossible, you would not earn all the remaining 

marks. 

Technical Correctness 

Most mathematical errors in a student's solution are accounted for in the content category of the 

marking scheme. Technical correctness considers other mathematical errors related to concepts 

learned in previous grades such as reducing fractions. 

Carelessness and incorrect notation lead to errors in mathematics. By knowing and using proper 

form, you can eliminate many mistakes. 

Presentation 

Your solutions should be presented in such a way that other grade 11 students can read 

them and can learn from them. Presentation considers your communication skills in 

mathematics. Lack of explanation in your solution is evaluated in this category. In 

mathematics, we are concerned not only with a correct mathematical solution, but also 

with the clarity of that solution. 

Be consistent. Be clear. 

A mathematical communication error will be indicated by 

C 

COMMON COMMUNICATION ERRORS 

CORRECT 

INCORRECT 

1. Given f(x) = x 2 + 4x, determine f( 3) . 

For f( 3), 3 has been substituted for x in the 

first step. 

f(x) = x 2 + 4x 

f(3) = (3) 2 + 4(3) 

= 21 

For f( 3), 3 has not been substituted in the 

first step. 

f(3) = x 2 + 4x 

f(3) = (3) 2 + 4(3) 

= 21 

2. The function has been written in terms of 

the proper independent variable t. 

f(t) = t 2 + 2t + 3 

The function has not been written in terms of 

the independent variable. 

f(x) = t 2 + 2t + 3 

10

CORRECT 

3. Given h(t) = −4.9(t − 3) 2 + 22.5 

Determine the maximum height and the 

time it is reached. 

a) the maximum height is 22.5 m 

b) the maximum height is reached 

at 3 s 

INCORRECT 

These are ordered pairs and do not indicate the 

proper value that is requested. 

a) the maximum height is ( 3, 22.5) 

b) the maximum height is reached 

at ( 3, 22.5) 

4. Solve: t 2 + 12t + 3 = 0 

Since the variable is t, x must not be used. 

t = 

− 12 ± ( 12 )2 − 4(1)(3) 

2 

etc. 

x = 

− 12± ( 12 )2 − 4(1)(3) 

2 

etc. 

5. This radical has been simplified, since it is 

rational. 

x = 16 

= 4 

6. This complex number has been simplified so 

that −1 is written as i . 

x = −16 

= 4i 

This radical has not been simplified. 

x = 16 

These complex numbers have not been 

simplified. 

x = −16 or 

x = i 16 or 

x = 4 −1 

7. Given t n = 3 + (n −1)(4) , determine t 5 . 

For t 5 , substitute 5 for n, since the indicated 

term is t 5 . 

t n = 3 + (n − 1)(4) 

t 5 = 3 + [(5) − 1](4) 

= 19 

For t 5 , 5 has not been substituted for n. 

t 5 = 3 + (n − 1)(4) 

= 3 + [(5) − 1](4) 

= 19 

11

CORRECT 

INCORRECT 

8. Solve sinθ =−0.5, 0≤θ ≤2π 

Since the domain is given in radian measure, 

the solution is in radian measure and within 

the given domain. 

θ = 7π 

6 , 11π 

6 

The first solution is not in radian measure. 

The second solution has a value not in the 

domain. 

θ = 210°, 330° or 

θ =− π 6 , 7π 6 , 11π 

6 

9. Given rectangle ABCD with 

∠ADB = 60°. State the measure 

of ∠DBC. 

A 

D 

α 

B 

C 

When more than one angle is at a vertex, then 

the angle must be named with three letters or 

labeled clearly with a variable on the diagram. 

∠DBC = 60° or α = 60° 

The angle is not clear from the diagram. 

(Whereas ∠A or ∠C would be clear) 

∠B = 60° 

N.B. Mixed and/or radicals will be accepted as exact values. 

e.g. 12 or 2 3 

e.g. 4 ± 24 

2 

or 2± 6 

N.B. Complex Numbers 

The symbol for the complex number system is § 

e.g. Solve the following: x 2 − 3x + 8 = 0, x ∈ §, indicates that the domain is the set of 

Complex numbers. 

If the domain is not indicated, then it is assumed that the domain is the set of Real numbers, 

i.e., x ∈ ò 

12

Communication – Presentation and Technical Rubric 

TECHNICAL 

CORRECTNESS 

OF SOLUTIONS 

- using 

mathematical 

symbols & visuals 

-using 

mathematical 

conventions 

-using 

mathematical 

language 

Incomplete 

0 

All or most 

solutions are 

blank 

Unacceptable 

3.0 4.0 

No solutions are 

correct or many 

left blank 

• Numerous 

technical 

errors 

• Does not use 

any 

mathematical 

language or 

symbols 

• Many 

solutions left 

blank 

Poor 

5.2 5.5 5.8 

Few solutions are 

technically correct 

• Infrequently uses 

mathematical 

language, symbols, 

visuals and 

conventions 

correctly 

• Few solutions 

contain introductory 

statements 

• Few solutions 

contain all 

necessary steps 

and/or are illogical 

Acceptable 

6.2 6.5 6.8 

Some solutions are 


• Uses mathematical 



conventions 

correctly some of 

the time 

• Some solutions 

contain introductory 

statements 

• Some solutions 

contain all 

necessary steps and 

steps are in a logical 

sequence 

Good 

7.2 7.5 7.8 

Most solutions are 


• Uses mathematical 



conventions 

correctly most of the 

time 

• Most solutions 

contain clear 

introductory 

statements 

• Most solutions 

contain all 

necessary steps in a 

logical sequence 

Outstanding 

8.4 8.9 9.5 10 

All or almost all solutions 

are technically correct 

• Routinely uses 

mathematical 



conventions both 

correctly and 

efficiently 

• Solutions contain 

clear introductory 

statements 

• Solutions include all 

necessary steps in a 

logical sequence 

PRESENTATION 

OF SOLUTIONS 

communicating 

solutions 

All or most 

solutions are 

blank 

No evidence of 

presentation 

or many solutions 

left blank 

Solutions to few 

problems stand alone 

• Few solutions are 

clearly or neatly 

presented and little 

use of appropriate 

words 

• Layout is difficult to 

follow 

Solutions to some 

problems can stand 

alone 

• Some solutions are 

clearly or neatly 

presented 

using appropriate 

words 

• Layout of few 

solutions is easily 

followed and main 

ideas of solutions 

must be inferred 

Solutions to most 

problems can stand 

alone 

• Most solutions are 

clearly and neatly 

presented using 

appropriate words to 

clarify steps 

• Layout of most 

solutions is easily 

followed and legibly 

presented 

Solutions to all or almost 

all problems can stand 

alone 

• Solutions are clearly 

and neatly 

presented using 

appropriate words to 

clarify steps 

• Layout of solutions 

is easily followed 

and is legibly 

presented 

• Inclusion of any 

steps necessary for 

a peer to follow the 

solutions 

13

KEY QUESTION WORDS 

The following are indicator words for questions. See the glossary in your 

textbook for definitions of mathematical terms. 

1. CHECK/ VERIFY Use an appropriate method to demonstrate the 

correctness of the solution formally (using LS 

and RS) or using technology. 

2. COMPARE Tell what is the same and what is different. 

3. DESCRIBE Tell about something in a step-by-step manner. 

Use words, numbers, graphs, diagrams and/or 

symbols, to explain your thinking. 

4. EVALUATE Find a numerical answer. 

5. EXPLAIN Use words, numbers, graphs, diagrams and/or 

symbols, to make your solutions clear and 

understandable. 

6. GRAPH Draw the relationship between the variables 

on a well labeled, scaled set of axes. 

7. GIVE REASONS/ 

JUSTIFY YOUR ANSWERS 

Explain your reasoning in your own words. 

Give reasons and evidence to show your 

answer is correct and proper. 

8. PROVE Demonstrate the correctness of a statement 

using a formal method. 

9. REDUCE Divide out common factors in the 

numerator and denominator of a fraction 

leaving it in lowest terms. 

10. SHOW Indicate the plausibility of a solution 

without using a formal proof by way of 

examples or technology . 

11. SHOW YOUR WORK Record all calculations. Include all the steps 

you went through to get your answer. You 

may want to use words, numbers, graphs, 

diagrams and/or symbols, to explain your 

thinking. 

14

12. SIMPLIFY Perform all possible operations, remove any 

brackets, collect like terms, reduce fractions, . . . 

13. SKETCH Draw a reasonable likeness and identify key 

points. 

14. SOLVE/ ROOTS Determine the value(s) of the variable(s) 

that make the equation(s) true by showing all 

your work. 

15. STATE Write the answer only. 

16. EXACT SOLUTIONS Do not approximate or round answers. 

17. ZEROS of a FUNCTION The zeros of a function, f, correspond to the 

values of x such that f(x)= 0. 

Grade 11 General Formulae 

Quadratic formula: 

2 

If ax + bx + c = 0 and a ≠ 0 , then 

− b ± 

x = 

b 

2 − 4ac 

2a 

Trigonometric ratios: 

In a right triangle: 

For an angle in standard 

position: 

sin θ = 

cosθ = 

opposite 

hypotenuse 

adjacent 

hypotenuse 

tan θ = opposite 

adjacent 

y 

sin θ = 

r 

x 

cosθ 

= 

r 

y 

tanθ 

= 

x 

Distance between two points: 2 

( ) ( ) 2 

d = 

x 

2 

− x1 

+ y2 

− y1 

Sine law: 

Ambiguous case: 

sin A sin B sin C 

= = 

a b c 

b sin A < a < b 

Cosine law: a 2 = b 2 + c 2 − 2bc cosA 

15

Trigonometric Identities: 

2 

2 

sin θ + cos θ = 1 

sin θ 

tan θ = 

cos θ 

Arithmetic sequences: 

Arithmetic series: 

Geometric sequences: 

Geometric series: 

t n 

= a + ( n −1)d 

S n = n ( t 1 + t n) 

2 

or 

S n = n [ 

2 2a + ( n − 1 )d] 

t n 

= ar 

n−1 

( ) 

S n = a 1 − r n 

1− r 

( ) 

, r ≠ 1 or 

S n = a rn − 1 

r − 1 , r ≠ 1 

General Form of conic: 

2 2 

ax + by + 2gx 

+ 2 fy + c = 0 

Standard forms of conics: 

2 

2 

Circle: 

2 

( ) ( ) 

x − h + y − k = r 

circle if a = b 

ellipse if ab > 0 

parabola if ab = 0 

hyperbola if ab < 0 

radius r 

centre (h,k) 

Ellipse: 

2 

( x − h) ( y − k ) 

a 

2 

2 

( x − h) ( y − k ) 

b 

2 

+ 

+ 

b 

a 

2 

2 

2 

2 

= 1 

= 1 

centre (h,k) 

length of the major axis is 2a 

length of the minor axis is 2b 

Parabola: ( x − h) = 4 p( y − k ) 

2 

( y − k ) = 4p( x − h) 

2 vertex (h,k) 

distance from the vertex to 

the focus is |p| 

Hyperbola: 

2 

( x − h) ( y − k ) 

a 

2 

2 

( x − h) ( y − k ) 

b 

2 

− 

− 

b 

a 

2 

2 

2 

2 

= 1 

= −1 

centre (h,k) 

length of the transverse axis 

is 2a 

length of the conjugate axis 

is 2b 

16

Formulae Sheet to be provided for the MCR3U and MCF3M Examinations 

− b ± 

x = 

sinθ 

= 

cosθ 

= 

tanθ 

= 

y 

r 

x 

r 

y 

x 

b 

2 − 4ac 

2a 

sin A sin B sin C 

= = 

a b c 

a 2 = b 2 + c 2 − 2bccosA 

t n 

= a + ( n −1)d 

t 

n 

= ar 

n−1 

S n = nt ( 1 + t n) 

2 

( ) 

S n = a 1 − r n 

1− r 

or S n = n [ 

2 2a + ( n − 1)d 

] 

( ) 

, r ≠ 1 or S n = arn − 1 

r − 1 , r ≠ 1 

Conic Section Equations 

d 

= 

2 

( x − x ) + ( y − ) 2 

2 1 2 

y1 

ax 

2 

2 

+ by + 2gx 

+ 2 fy + c = 0 

2 

2 2 

( x − h) + ( y − k) = r 

2 

( x − h) ( y − k ) 

a 

2 

2 

( x − h) ( y − k ) 

b 

2 

+ 

+ 

b 

a 

2 

2 

2 

2 

= 1 

= 1 

2 

( x − h) = 4 p( y − k) 

2 

( y − k) = 4 p( x − h) 

2 

( x − h) ( y − k) 

a 

2 

2 

( x − h) ( y − k) 

b 

2 

− 

− 

b 

a 

2 

2 

2 

2 

= 1 

= −1 

17

Achievement Chart – Grades 11 and 12, Mathematics 

Categories 50 –59% 

(Level 1) 

Knowledge/Understanding The student: 

– understanding concepts – demonstrates limited 

understanding of 

concepts 

– performing algorithms – performs only 

simple algorithms 

accurately by hand 

and by using 

technology 

Thinking/Inquiry/ 

Problem Solving 

The student: 

– reasoning – follows simple 

mathematical 

arguments 

– applying the steps of an 

inquiry/problem-solving 

process (e.g., formulating 

questions; selecting strategies, 

resources, technology, 

and tools; representing in 

mathematical form; 

interpreting information and 

forming conclusions; 

reflecting on the 

reasonableness of results) 

Communication 

– communicating reasoning 

orally, in writing, and 

graphically 

– using mathematical 

language, symbols, visuals, 

and conventions 

Application 

– applying concepts and 

procedures relating to 

familiar and unfamiliar 

settings 

– applies the steps of 

an inquiry/problemsolving 

process with 

limited effectiveness 

The student: 

– communicates with 

limited clarity and 

limited justification of 

reasoning 

– infrequently uses 

mathematical 


visuals, and 

conventions correctly 

The student: 

– applies concepts and 

procedures to solve 

simple problems 

relating to familiar 

settings 

60 –69% 

(Level 2) 

– demonstrates some 


concepts 

– performs algorithms 

with inconsistent 

accuracy by hand, 

mentally, and by using 

technology 

– follows arguments 

of moderate 

complexity and makes 

simple arguments 



process with 

moderate effectiveness 


some clarity and some 

justification of 

reasoning 

– uses mathematical 




some of the time 



problems of some 

complexity relating to 

familiar settings 

70 –79% 

(Level 3) 

– demonstrates 

considerable 


concepts 

– performs algorithms 

accurately by hand, 

mentally, and by using 

technology 

– follows arguments 

of considerable 

complexity, judges the 

validity of arguments, 

and makes arguments 

of some complexity 



process with 

considerable 

effectiveness 


considerable clarity 

and considerable 

justification of 

reasoning 

– uses mathematical 




most of the time 



complex problems 

relating to familiar 

settings; recognizes 

major mathematical 

concepts and 

procedures relating to 

applications in 

unfamiliar settings 

80 –100% 

(Level 4) 

Note: A student whose achievement is below 50% at the end of a course will not obtain a credit for the course. 

– demonstrates 

thorough 


concepts 

– selects the most 

efficient algorithm and 

performs it accurately 

by hand, mentally, and 

by using technology 

– follows complex 

arguments, judges the 

validity of arguments, 

and makes complex 

arguments 



process with a 

high degree of 

effectiveness 

and poses extending 

questions 

– communicates 

concisely with a high 

degree of clarity & 

full justification of 

reasoning 

– routinely uses mathematical 

language, 

symbols, visuals, and 


and efficiently 



complex problems 

relating to familiar and 

unfamiliar settings 

18

Appendix A 

Sample Examinations and Solutions 

19

OTTAWA-CARLETON DISTRICT SCHOOL BOARD 

1. Given g( x) 

3− 

2x 

MCF 3M Functions Final Examination 

(January) 

PART A (22 marks) 

Each correct answer has a value of one (1) mark. 

= , determine ( 4x) 

g . 

2. For the graph of the given relation, state: 

y 

3 

2 

1 

-3 -2 -1 1 2 3 

-1 

x 

(a) 

(b) 

the domain 

the range 

3. State all restrictions : 

2 + 2 

× a 

a 3 

4. Evaluate: (express your answers as fractions) 

3 

16 − 

(a) 4 

(b) 

−1 

3 + 3 

0 

5. Describe the three transformations required to obtain the graph of 

y = −2 f ( x + 3) from the graph of a function defined by y = f (x) 

. 

(a) 

(b) 

(c) 

6. One cycle of the graph of a periodic function is shown below. 

4 

y 

(a) State the period 

2 

(b) 

State the amplitude 

-2 2 4 6 8 10 12 

-2 

(c) Extend the graph of 

the function for one more 

cycle. 

7. Express − 49 in terms of i .

8. Convert 210° to a radian measure in terms of π. 

9. Solve for x: − 3 x < 15 

10. State the exact value of 

cos π . 

4 

11. θ is the measure of an angle with its terminal arm in the fourth quadrant, 

where 0 ° ≤ θ ≤ 360° 

. If cos θ = 0. 423 , determine to the nearest 

degree, 

(a) 

the related acute angle. 

(b) the value of θ. 

12. The first term of a sequence is –5 and the common ratio is 2. 

(a) 

(b) 

List the first three terms of this sequence. 

State the general term. 

13. Simplify: 

x 

x 

3 

4 

1 

4 

14. Determine the value sin θ . 

y 

P(3,5) 

θ 

x

PART B (54 marks) 

Each of the following questions requires a short answer completion in the space provided. 

Show all work. Mark values for each question appear in the left margin. 

[2] 1. Solve for x, x ∈ : 

[1] 

2x 

2 − 5x + 7 = 0 

2. A graphing calculator shows the following for a sine function with a period of 2π. A student wrote the 

(a) 

π 

6 

equation as = 2 sin ( x − ) + 3 

y . 

Explain in words why the student is incorrect. 

[1] 

(b) 

Write the correct equation. 

[3] 

3. Simplify. (It is not necessary to state restrictions) 

(a) 

x 

− 

3x 

− 6 x 

2 

2 − 

4 

[3] 

(b) 

2 

2 

2x 

x + 4x 

÷ 

x + 4 2 

x + 8x 

+ 16 

1 

3 

⎛ ⎞ 

= ⎜ x⎟ 

+ 

⎝ 2 ⎠ 

[3] 4. Sketch y cos 1 for one cycle. 

[3] 

[3] 

[3] 

[3] 

5. Prove the identity: 

2 2 

tan 

θ = sin 

6. Solve for θ: 

(a) 

θ 

2 

( 1+ 

tan θ) 

2sin θ + 1 = 0, 0 ≤ θ ≤ 2π 

(b) θ = 1, 0° 

≤ θ ≤ 360° 

tan 2 

7. A soccer ball is kicked into the stands such that its height above the ground is given by h = −5t 

2 + 15t 

where h is the height in metres and t is the time elapsed in seconds since the ball was kicked. 

(a) What is the maximum height of the ball 

[4] 

(b) 

As the ball is coming back down, a fan catches it 6 metres above ground level. 

How long was the ball in the air Express the answer to the nearest tenth of a second.

[2] 

[2] 

[1] 

[2] 

8. Given the relation f as defined by y = x − 2 , 

(a) state the domain and the range of f . 

(b) 

−1 

sketch the graphs of f and f . 

(c) 

does f represent a function Explain your answer. 

(d) determine the expression for f 

−1 

( x ) . 

x 

[4] 9. Solve for x: 

( ) 

10 x 

x + 3 

16 = 

1 

2 

[1] 

[2] 

[2] 

[4] 

10. In a theatre, seats are arranged so the first row has 29 seats, the second row has 32 seats, the third row has 

35 seats, and the pattern continues. 

(a) 

(b) 

(c) 

Identify the type of sequence. Explain. 

If the last row has 80 seats, how many rows are in the theatre (Use the appropriate formula). 

What is the total number of seats in the theatre (Use the appropriate formula). 

11. A helicopter, at H, is hovering 200 m directly above a forest observation tower, TR. From the helicopter, 

the angle of depression of a fire is 22°. From the top of the tower, the angle of depression of a fire at F is 

18°. How far is the fire from the base of the tower, R, to the nearest tenth of a kilometre (Hint: Find the 

length of TF first) 

H 

22° 

200 m 

T 

18° 

R 

F 

12. Because of the tide, the depth of the water in a harbour is modelled by the equation = −3 cos( t) + 6 

d 

where d represents the depth of the water in metres and t represents 

The number of hours after midnight. (i.e. t = 0 means midnight, t = 1 means 1 A.M., 

And so on.) The graph of the relation is shown below: 

A(3, 

π 

6 

d , 

[2] 

[1] 

[2] 

(a) 

(b) 

(c) 

What is the missing coordinate of point A What do the coordinates of point A represent 

State the maximum depth of the water. 

t 

Surfing is allowed between 8 A.M. (08:00 hrs) and 7 P.M. (19:00 hrs), but only when the depth of the 

water is 6 m or more. For how many hours is surfing allowed in one day Explain.


MCF 3M Functions Final Examination 

(Backup) 


Write only your answer for each of the following questions in the space provided. 


1. If f ( x) 

= 5x 2 − 2 , determine f (−3) 

. 

2. For the given periodic relation, state: 

(a) the period 

y 

4 

y = f (x 

(b) the amplitude 

2 

-2 2 4 6 8 10 12 x 

(c) the value of f (11) 

assuming the 

relation continues in the same manner. 

-2 

3. Evaluate 

5 

3 

8 − . (Express answer as a fraction) 

4. Given y = x − 5 , state: 

5. Express − 25 in terms of i . 

(a) the domain 

(b) the range 

6. Given cos θ = -1, sin θ = 0, 0° ≤θ≤ 360°, state θ. 

7. State the restrictions for 

x − 3 

. 

2 

x ( x − 3) 

8. Given 

θ = 

π 

, state: 

6 

(a) the measure of θ in degrees 

(b) the exact value of cosθ 

9. Given the diagram below, state the exact measure of θ in radians. 

θ 

10. A point on the graph of y = f (x) 

is ( 8, − 3) 

. The coordinates of the 

corresponding image point 

(a) on the graph of y = 2 f ( x) 

are 

(b) on the graph of y = f ( x + 2) 

are 

− 

(c) on the graph of y = f 

1 ( x) 

are 

11. Given the sequence 2, 6, 10, …, 

(a) state the next term 

(b) state the general term 

12. Simplify: 

5 

4 

3 

(a) x ⋅ x 

4 

(b) 

⎛ 

⎜ y 

⎝ 

2 

3 

⎞ 

⎟ 

⎠ 

1 

2 

31




[3] 1. Solve and graph the solution set, x ∈ R . 

2( 

x − 4) ≥ 2+ 

4( x − 2) 

-4 -3 -2 -1 0 1 2 3 4 

[1] 

[1] 

[2] 

[3] 

[1] 

2. P( 

− 1, −3) 

lies on the terminal arm of the angle in standard position with measure θ. Determine: 

(a) 

the value of r 

(b) the value of sin θ 

(c) the value of θ to the nearest degree, where 0 ° ≤ θ ≤ 360° 

. 

3. (a) Simplify 

(b) 

m + 3 m + 2 

÷ 

m − 3 2 

9 − m 

State the restrictions in (a). 

yè 

-1 

r 

-1 

-2 

-3 

P(-1,-3) 

x 

[3] 4. Simplify completely: (It is not necessary to state restrictions.) 

a 

+ 

10a 

a + 3 2 

a + 4a 

+ 3 

5. An arrow is shot from the roof of a building. Its height above the ground is modelled by 

2 

h ( t) 

= −5t 

+ 40t 

+ 20 , where h is the height in metres and t is the time elapsed in seconds, from the 

time the arrow was shot. 

[1] (a) From what height is the arrow shot 

[2] 

[1] 

[3] 

(b) 

(c) 

(d) 

When will the arrow reach the maximum height 

What is the maximum height 

When will the arrow hit the ground (round answer to the nearest tenth of a second) 

[2] 

[3] 


(a) tan θ = 3 , 0 ≤ θ ≤ 2π 

(exact values) 

(b) ( 3cosθ − 1)(cos θ + 2) = 0 , 0 ° ≤ θ ≤ 360° 

(round answers to the nearest degree) 

[1] 

7. The graph of a parabolic relation is shown. 

(a) State the domain. 

4 

3 

[1] 

(b) 

Graph the inverse on the same grid 

2 

1 

[1] 

[1] 

(c) 

(d) 

Is the inverse a function Explain your answer. 

State the defining equation of the inverse. 

-4 -3 -2 -1 1 2 3 4 

-1 

-2 

-3 

-4 

[2] 8. If you were given a function in the form y = f (x) 

, explain how you would determine the defining 

− 

equation of its inverse, namely y = f 

1 ( x) 

. 

[3] 9. Prove the identity: 

2 

tan θ − 

1 

= 

2sin θ − 1 

tan θ sin θ cosθ 

32


x 2 −x 

27 = 9 

[4] 11. Sketch one cycle of the following trigonometric function: 

y = −2 sin 3⎜ 

⎛ x + 

π 

⎟ 

⎞ 

⎝ 6 ⎠ 

x 

[2] 

[2] 

12. Given the series 800 + 400 + 200 + 100 + K , using the appropriate formulas 

(a) determine t 12 to 3 decimal places. 

(b) determine S 12 to the nearest decimal place. 

[3] 13. You have the opportunity to work between 1 and 50 hours during the March Break. You can choose the 

method of payment from the following: 

Choice 1: You can be paid $15 per hour 

Choice 2: You can be paid $1 for the first hour, $2 for the second hour, $3 for the third hour, and 

the pattern continues. 

What are the advantages of each choice Justify your answers. 

14. The inside temperature of a building is modelled by T ( t) 

= 3cos(0.262t) 

+ 22 , where T is the 

temperature in °C and t is the number of hours elapsed since 5 A.M. The graph is shown below. 

[2] 

[2] 

(a) 

(b) 

Using an appropriate calculation, explain why the coefficient of t in the 

equation is 0.262. 

In another building, the temperature fluctuates in a similar manner except that the maximum temperature 

is 27°C and the minimum temperature is 23°C. Determine the defining equation that models the 

temperature in this other building. 

T 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

5 a.m. 5 p.m. 5 a.m. 

33


MCR 3U Functions and Relations Final Examination 

(January) 



1. Given g( x) 

= 3− 

2x 

, determine g ( 4x) 

. 

2. For the relation defined by 2 2 

+ y = 1 

49 16 

(a) identify the type of conic 

(b) state the range 

(c) state the length of the major axis 

3. State all restrictions : 

2 

÷ x + 2 

x 3 

4. Evaluate: (express your answers as fractions) 

3 

(a) 16 − 4 

(b) 

−1 

0 

3 + 3 

5. Describe the transformations required to obtain the graph of 

y = − f ( x + 3) from a graph of a function defined by y = f (x) 

. 

(a) 

(b) 

6. Given the recursion formula defined by 

t 

1 

= − 3, t 

2 

= 5, t 

n 

= t 

n−2 

− t 

n , determine 

−1 

t . 

3 

7. State the conjugate of − 2+ 3i 

. 

2 2 

8. State the equation of one asymptote for the graph of x − y = 1. 

9. State the equation for the locus of points which are 5 units from (− 1, 0) 

. 

10. State the exact value of cos 

3π . 

4 

11. θ is the measure of an angle with its terminal arm in the fourth quadrant 

such that cos θ = 0. 423 . Determine the value of θ to the nearest degree, 

0 ° ≤ θ ≤ 360° 

. 

12. The first term of a sequence is –5 and the common ratio is 2. 

(a) 

(b) 

List the first three terms of this sequence. 

State the general term. 

13. Simplify: 

3 

x4 

1 

x4 

14. Determine the value of sin θ. 

y 

P(3,5) 

θ 

x 

15. For what value of c does the equation of the function defined by 

2 

y = x − 6x 

+ c have only one x - intercept 

16. Determine the number of zeroes of the function defined by 

2 f ( x ) = − 3( x − 2) − 5 .


Each of the following questions requires a short answer completion in the space provided. Show 

all work. Mark values for each question appear in the left margin. 

[3] 1. Simplify: 

2 3 

(3 − 4 i) 

− i( 

i ) + 

2 

i 

[1] 

2. A graphing calculator shows the following for a sine function with a period of 2π. 

A student wrote the equation as y = 2 sin ( x − 

π) + 3 . 

6 

(a) Explain in words why the student is incorrect. 

[1] 

(b) 

Write the correct equation. 

[3] 

[3] 

3. Simplify. (It is not necessary to state restrictions) 

(a) 

x 

− 

2 

2 

3x 

− 6 x − 4 

(b) 

2 

2 

2x 

+ y 

÷ 

2x 

+ 3xy+ 

y 

2 

2 

2x 

x + xy 

1 

[3] 4. Sketch y 3 cos 

⎛ ⎞ 

= ⎜ x⎟ 

+ 1 for one cycle. 

⎝ 2 ⎠ 


2 

tan θ 2 

= sin θ 

2 

1+ 

tan θ 


[3] 

[3] 

(a) 2sin θ + 1 = 0, 0 ≤ θ ≤ 2π 

2 

(b) 4 tan θ − 9 = 0, 0° 

≤ θ ≤ 360° 

(answer to the nearest degree) 

[2] 

[2] 

[1] 

[2] 

7. Given the relation f as defined by y = x − 2 , 

(a) state the domain and the range of f. 

(b) 

−1 

sketch the graphs of f and f . 

(c) does f represent a function Explain your answer. 

(d) 

−1 

determine the expression for f ( x ) . 

x 


x 2 

( ) 

⎟ ⎞ 

⎜ ⎛ 1 

2 = 64 

x 

⎝ 32 ⎠ 

[1] 

9. A sporting goods store sells skates. During the first week, they sold 10 pairs of skates. In the second 

week they sold 14 pairs and in the third week they sold 18 pairs, and the pattern continues. 

(a) Identify the type of sequence. Explain. 

[4] 

(b) 

How many weeks did it take to sell a total of 1450 pairs of skates (Use the appropriate formula.)

[4] 10. Determine the length of PQ, to the nearest metre. 

C 

12 m 

8 m 

29° 

R 

P 

11. Because of the tide, the depth of the water in a harbour is modelled by the equation 

d = −3 cos 

⎛π ⎞ 

⎜ t + 6 

6 

⎟ , where d represents the depth of the water in metres and 

⎝ ⎠ 

t represents the number of hours after midnight. (i.e. t = 0 means midnight, 

t = 1 means 1 A.M., and so on.) 

d 

The graph of the relation is shown below: 

Q 

A(3, 

t 

[2] 

[1] 

[2] 

(a) 

(b) 

(c) 

What is the missing coordinate of point A What do the coordinates of point A represent 

State the maximum depth of the water. 

Surfing is allowed between 8 A.M. (08:00 hrs) and 7 P.M. (19:00 hrs), but only when the depth of the 

water is 6 m or more. For how many hours is surfing allowed in one day Explain. 

[3] 

2 2 

12. (a) Express 9x − 4y 

− 36x 

− 8y 

= 4 in standard form. 

[2] 

(b) 

What are two advantages of writing the defining equation of a conic in standard form 

[3] 13. The receiver of a parabolic satellite dish is at the focus. The focus is 72 cm from the vertex. If the dish is 

240 cm in diameter, find the depth of the dish. 

[5] 14. A hyperbola has centre (2, -1) and one of its foci at (2, 4). Its transverse axis has a length of 8 units. 

Sketch the graph of the hyperbola. 

10 

y 

8 

6 

4 

2 

-10 -8 -6 -4 -2 2 4 6 8 10 

-2 

x 

-4 

-6 

-8 

-10


MCR 3U Functions & Relations Final Examination 

(Backup) 


Write only your answer for each of the following questions in the space provided. 


1. If f ( x) 

= 5x 2 − 2 , determine f (−3) 

. 

2. For the given periodic relation, state: 

(a) the period 

4 

2 

y = f 

-2 2 4 6 8 10 12 

-2 

(b) the amplitude 

(c) the value of f (11) 

assuming the relation 

continues in the same manner. 

3. Evaluate 

5 

3 

8 − . (Express answer as a fraction) 

4. Given y = 2 x − 5 , state: 

(a) the domain 

(b) the range 

5. Express − 25 in terms of i . 

6. Evaluate 

6 

i . 

7. State the restrictions for 

8. Given 

θ = 

5π 

, state: 

6 

x − 3 

. 

2 

x ( x − 3) 

(a) 

the measure of θ in degrees 

(b) the exact value of cos θ . 

9. y Given the diagram below, state the exact measure of α in radians. 

α 

x 

10. A point on the graph of y = f (x) 

is ( 8, − 3) 

. The coordinates of the 

corresponding image point 

(a) on the graph of y = 2 f ( x) 

are 

(b) on the graph of y = f ( x + 2) 

are 

− 

(c) on the graph of y = f 

1 ( x) 

are 

11. Given the recursion formula defined by t 1 = 5 , t n = 2t n −1 − 3 , determine 

t 2 . 

2 

12. Given the conic defined by y = −8x 

, determine: 

(a) 

(b) 

the coordinates of the focus. 

the equation of the directrix. 

5 3 

13. Simplify a 4 ⋅ a 4 

52




[3] 1. Find the defining equation of the conic whose graph is shown below. Express your answer in standard 

form. 

7 

6 

5 

4 

3 

2 

1 

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 

-1 

1 2 

-2 

[3] 2. 

5 + 3i 

Simplify 

4 − i 

3. P( 

− 2, −3) 

lies on the terminal arm of the angle in standard position with measure θ. Determine: 

[2] 

[2] 

(a) the exact value of sin θ . 

(b) the value of θ to the nearest degree, where 0 ° ≤ θ ≤ 360° 

. 

[3] 4. Simplify completely: (It is not necessary to state restrictions.) 

a 

+ 

9a 

a + 3 2 

3a 

+ 8a 

− 3 

[4] 5. Simplify and state the restrictions 

2m 

+ 3 m + 3 

÷ 

2 

2m 

− 3 9 − 4m 

[4] 6. An arrow is shot from the roof of a building. Its height above the ground is modelled by 

2 

h ( t) 

= −5t 

+ 40t 

+ 20 , where h is the height in metres and t is the time elapsed in seconds, from the 

time the arrow was shot. For what length of time is the arrow more than 35 m above the ground 

Express your answer to the nearest tenth of a second. 


2 

tan θ − 

1 

= 

2sin θ − 1 

tan θ sin θ cosθ 


[2] (a) tan θ − 3 = 0 , 0 ≤ θ ≤ 2π 

(exact values) 

[3] (b) 

2 

3cos θ − 7cos θ + 2 = 0 , 0 ≤ θ ≤ 2π 

(round answers correct to 2 decimal places) 

[2] 

9. If you were given a function in the form y = f (x) 

, explain how you would determine the defining 

− 

equation of its inverse, namely y = f 

1 ( x) 

. 

10. The graph of a parabolic relation is shown. 

4 

3 

[1] 

(a) 

State the domain. 

2 

1 

[1] 

[1] 

(b) 

(c) 

[3] 11. Solve for x: 

Graph the inverse on the same grid. 

Consider the statement: “Since the given relation is not a function, then its 

inverse is not a function.” Is this statement true Explain your answer. 

x 

27 

−2 1 

= 

x 

9 

-4 -3 -2 -1 1 2 3 4 

-1 

-2 

-3 

-4 

53

[4] 12. Sketch one cycle of the following trigonometric function: 

y = −2sin 

⎜ 

⎛3x 

+ 

π 

⎟ 

⎞ 

⎝ 2 ⎠ 

x 

13. Given the series 800 + 400 + 200 + 100 + K , using the appropriate formulas, 

[2] 

[2] 

(a) determine t 12 to 3 decimal places. 

(b) determine S 12 to the nearest decimal place. 

[4] 14. Two guy wires as shown in the diagram support a microwave tower. What is the height, h metres, of the 

tower, to the nearest metre 

T 

75 

h 

50 

P R Q 

100 

[3] 15. You have the opportunity to work between 1 and 50 hours during the March Break. You can choose the 

method of payment from the following: 

Choice 1: You can be paid $15 per hour 

Choice 2: 

You can be paid $1 for the first hour, $2 for the second hour, $3 for the third hour, and 

the pattern continues. 

What are the advantages of each choice Justify your answers. 

[2] 

[2] 

16. The inside temperature of a building is modelled by T ( t) 

= 3cos(0.262t) 

+ 22 , where T is the 

temperature in °C and t is the number of hours elapsed since 5 A.M. The graph is shown below. 

(a) Using an appropriate calculation, explain why the coefficient of t in the 

equation is 0.262. 

(b) 

In another building, the temperature fluctuates in a similar manner except 

that the maximum temperature is 27°C and the minimum temperature is 

23°C. Determine the defining equation that models the temperature in this 

other building. 

17. A radar screen shows the activity within a circular region of radius 60 km. 

T 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

5 5 5 

[1] 

(a) 

Assuming the centre of the screen is (0, 0), write the equation that represents this circle. 

[4] 

[2] 

(b) A small aircraft flies on a path given by the equation x + 2 y = 140 . Is this small aircraft detected on the 

radar screen Explain your answer algebraically. 

18. 

2 2 

Given the conic defined by 25x 

+ 9 y −100x 

+ 18y 

−116 

= 0 , determine: 

(a) the coordinates of the centre 

[3] 

(b) 

the coordinates of the foci. 

54

Grade 11 MCR3U and MCF3M Exemplar Booklet for Mathematics

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