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)
MCF3M
MCR3U
Unit
Strand: Tools for Operating and
Communicating with Functions
Introducing Functions and Transformations
12
10
16
14
Trigonometry
17
15
Modelling Periodic Functions
11
9
N/A
12
14
12
12
10
Review & End of Year summative tests and tasks
6
6
Total
88
88
Quadratic Functions
Strand: Trigonometric Functions
Strand: Investigations of Loci and Conics
Investigation of Loci and Conics
Strand: Financial Applications of
Sequences and Series
Sequences and Series
Financial Applications
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
3M 3U
3
2
Topic
Specific Expectations/ Comments
Nelson
Functions
•
3.1
3.2
3.6 - part 1
4.6 - part 2
•
understand concept & notation
-relation, function, domain,
range, vertical line test
investigate functions
2
y = x ,y =
1
1
Inequalities
2.5
2
Inverse Functions
3.5
3
Transformations
and Function
Notation
x, y =
x
solve first degree inequalities
graph solutions on a number line
know properties
e.g. (x, y ) → (y , x)
•
•
sketch/graph the inverse
apply transformations such as
translations, reflection, and
stretches
understand combinations of
transformations e.g.
[
AddisonWesley
Skills
p. 392
7.1
7.2
Comments
•
investigation of absolute
value function not
required
3.3
1.9
•
•
single inequalities only
supplement A.W. material
3.4
(3.5)
3.6
3.5
Skills
p.394
7.5
p. 211,
3.7
p.169,
3.3,
3.4,
3.6
3.7
7.3
7.4
•
leave trig transformations
for later
1
•
•
•
•
McGrawHill
3.1
3.2
]
y = af k (x − p ) + q
•
2
2
state domain and range of
transformed functions
Review &
Evaluation
N.B. Textbook sections in parentheses are optional.
3
QUADRATIC FUNCTIONS
MCF3M - 16 periods
Periods
3M 3U
2
1
MCR3U - 14 periods
Topic
Specific Expectations/ Comments
Nelson
Extending Algebraic
Skills/Polynomials
•
•
simplify polynomials
factor
p. 303
4.12
McGrawHill
1.4
p. 3
Simplifying Rational
Expressions
Multiplying &
Dividing Rational
Expressions
Adding &
Subtracting
Rational Expressions
Completing The
Square
•
state restrictions
4.8
1.5
•
factor to simplify
4.9
1.6
AddisonWesley
Skills
p.80
2.1
2.2
2.3
2.4
•
determine L.C.D
4.11
1.7
1.8
2.5
2.6
•
extend to complex trinomials
2
ax + bx + c = 0, where a ≠ 1
4.1
p.99
2.2
Skills
p. 198,
4.1
4.1
2
1.5
2
1.5
2
2
1
1
2
1.5
Maxima/Minima
•
solve applications
4.2
2.2
2
1
•
solve applications
4.3
2.3
1
2.5
Solving Quadratic
Equations
Complex Numbers
•
identify the complex number
system
determine conjugate (omit for
3M course)
add, subtract, multiply and
divide (omit for 3M course)
4.4
4.5
4.10
2.1,
2.5
•
•
2
2
Review &
Evaluation
N.B. Textbook sections in parentheses are optional.
4
p.201
4.2
4.3
4.4
Comments
• supplement A.W.
applications
• supplement A.W.
applications
MCF3M - 17 periods
TRIGONOMETRY
Periods
3M 3U
1
1
Topic
Specific Expectations/ Comments
Nelson
Getting Ready
2
2
p. 397399
5.2
5.3
4
3
Trig Functions of
Angles in Standard
Position
Oblique Triangles &
Applications
•
•
•
•
2
1.5
Radian Measure
•
•
•
•
•
•
2
1.5
Trig Values of
Special Angles
•
2
2
•
2
2
Trigonometric
Identities
Trigonometric
Equations
2
2
•
solve right triangles
use trig ratios
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
understand the relation between
radian and degree measure
represent radian measure in exact
form and in approximate form
determine exact values for
o
o
0 ≤θ ≤ 2 π and 0 ≤ θ ≤ 360
prove simple trig identities
solve linear and quadratic
trigonometric equations
0 ≤θ ≤ 2 π
p. 400
6.1
6.2
MCR3U - 15 periods
McGrawHill
4.1
5.2,
p. 351
4.2
4.3,
4.4
AddisonWesley
Skills
p. 246-7
5.1,
5.3,
5.4
p. 248-252
5.2
Comments
•
supplement 3-D
applications
•
omit applications with
angular velocity
•
supplement A.W &
M. H.
solve equations involving
sin kx or cos kx
(i.e. 2cos(2x) =1) for 3U
5.4
5.1
5.6,
5.7
6.3
5.2
5.5
6.5
(6.4)
5.8,
6.6
5.7
5.9
5.8
5.8
•
Review & Evaluation
N.B. Textbook sections in parentheses are optional.
5
MCF3M - 11 periods
Periods
3M 3U
1
1
2.5
2
2.5
2
3
2
Topic
Specific Expectations/ Comments
Nelson
Modelling Periodic
Functions
Graphing
Trigonometric
Functions
•
5.1
Investigating the
Graphs of
Trigonometric
Functions
Applications of
Trigonometry
•
•
•
•
•
2
2
MODELLING PERIODIC FUNCTIONS
develop an understanding of a
periodic function
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
MCR3U - 9 periods
McGrawHill
5.3
AddisonWesley
6.1
5.3,
5.5
5.4
6.2,
6.6
5.6
5.5,
5.6
6.3,
6.4
5.7,
5.8
(5.9, 5.10)
5.5,
5.6
(p. 392)
6.5
Review &
Evaluation
N.B. Textbook sections in parentheses are optional.
6
Comments
•
supplement A.W.
•
use graphing calculators
or graphing software to
investigate the effect of
simple transformations
on y = sin x and y = cos x
•
supplement A.W. &
M.H.R applications
MCF3M – N/A
Periods
3M 3U
1
INVESTIGATION OF LOCI and CONICS
Topic
Specific Expectations/ Comments
Nelson
Investigating Loci
•
construct a geometric model to
represent a described locus of
points
The Circle
•
2
The Ellipse
•
•
2
The Hyperbola
•
•
2
The Parabola
•
•
identify in standard form with
centre (0,0) and (h,k)
solve applications
identify in standard form with
centre (0,0) and (h,k)
solve applications
identify in standard form with
centre (0,0) and (h,k)
solve applications
identify in standard form with
vertex (0,0) and(h,k)
solve applications
identify the nature of the conic in
general form and sketch
solve problems involving the
intersections of lines and conics
1
1
1
2
The General Form of
Conics
Intersection of Lines
& Conics
Review &
Evaluation
•
•
•
MCR3U - 12 periods
AddisonWesley
8.1 ,
8.5
Comments
7.1
McGrawHill
8.2,
(8.1),
(8.3)
7.2
8.4
p.510 – 512
9.5
•
7.4
(7.5)
8.5
7.9
8.6
7.6
(7.7)
8.7
7.10
8.8
8.2,
9.2,
9.5
8.3,
9.3,
9.5
8.4,
9.4,
9.5
9.6
7.11
8.9
N.B. Textbook sections in parentheses are optional.
7
9.1,
9.2 – 9.4
•
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]
MCF3M - 14 periods
SEQUENCES AND SERIES
Periods
3M 3U
1
1
Topic
Specific Expectations/ Comments
Nelson
Exploring Patterns
and Sequences
•
introduce sequences
0
1
Recursion Formulas
•
1.5
1
1.5
1
Arithmetic
Sequences
Geometric
Sequences
2
1.5
Rational Exponents
•
2
1.5
Solving Exponential
Equations
•
2
1.5
Arithmetic Series
•
2
1.5
Geometric Series
•
2
2
Review &
Evaluation
•
•
•
MCR3U - 12 periods
1.1
McGrawHill
6.1
AddisonWesley
p. 7
write terms of a sequence given a
recursion formula
omit for 3M course
identify the pattern and find the
general term
identify the pattern and find the
general term
1.3
6.4
1.4
1.6
6.2
1.7
6.3
1.1,
1.3
1.2, 1.3
use laws of exponents to simplify
and evaluate expressions
solve exponential equations
x
x +3
4 =8
,
e.g.,
2x
x
2
− 2 = 12
determine the sum of the terms
using appropriate formulas and
techniques
determine the sum of the terms
using appropriate formulas and
techniques
1.9,
1.10
1.11
1.1,
1.2
1.3
1.5,
1.6
1.6
2.1
6.5
1.7
(2.2),
2.3
6.6
1.8
N.B. Textbook sections in parentheses are optional.
8
Comments
•
supplement A.W. with
patterning problems from
A.W. 1.1 & 1.2
•
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
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.
MCF3M - 12 periods
Periods
3M 3U
2
1.5
Topic
Specific Expectations/ Comments
Nelson
Compound Interest
•
1.8,
2.4
•
4
2.5
Ordinary Annuities
•
4
4
Mortgages &
Financial Planning
•
•
•
2
2
FINANCIAL APPLICATIONS
solve simple interest, compound
interest, present value problems
solve problems involving linear
and exponential growth
solve simple and general
annuities
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
2.9 - 2.12
McGrawHill
7.1,
7.2,
7.3,
7.4
7.5,
7.6
AddisonWesley
p. 128
3.1,
3.2
7.7,
7.8
3.5,
3.6,
3.7
Review &
Evaluation
N.B. Textbook sections in parentheses are optional.
9
MCR3U - 10 periods
3.3,
3.4
Comments
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) = x2 + 4 x, determine f (3) .
For f (3), 3 has been substituted for x in the
first step.
f( x) = x2 + 4x
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 ( 3 ) = (3) 2 + 4(3)
= 21
The function has not been written in terms of
the independent variable.
f( x) = t 2 + 2t + 3
f(t) = t2 + 2t + 3
10
CORRECT
INCORRECT
3. Given h(t) = −4.9(t − 3) 2 + 22.5
Determine the maximum height and the
time it is reached.
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)
a) the maximum height is 22.5 m
b) the maximum height is reached
at 3 s
4. Solve: t 2 + 12t + 3 = 0
− 12 ± (12 )2 − 4(1)(3)
t=
2
Since the variable is t, x must not be used.
− 12 ± (12 )2 − 4(1)(3)
x=
etc.
2
etc.
5. This radical has been simplified, since it is
rational.
x = 16
=4
This radical has not been simplified.
6. This complex number has been simplified so
that −1 is written as i .
x = −16
These complex numbers have not been
simplified.
x = −16 or
x = 16
x = i 16 or
= 4i
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 .
For t 5 , 5 has not been substituted for n.
t 5 = 3 + (n − 1)( 4)
= 3 + [( 5) − 1]( 4)
t n = 3 + (n − 1)(4)
t 5 = 3 + [(5) − 1]( 4)
= 19
= 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 =
The first solution is not in radian measure.
The second solution has a value not in the
domain.
θ = 210°, 330°
or
7π 11π
,
6
6
θ =−
9. Given rectangle ABCD with
∠ADB = 60°. State the measure
of ∠DBC.
A
π 7π 11π
,
,
6 6
6
B
α
D
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.
e.g.
12 or 2 3
4 ± 24
or 2 ± 6
2
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
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
technically correct
•
Uses mathematical
language, symbols,
visuals and
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
technically correct
•
Uses mathematical
language, symbols,
visuals and
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
language, symbols,
visuals and
conventions both
correctly and
efficiently
•
Solutions contain
clear introductory
statements
•
Solutions include all
necessary steps in a
logical sequence
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
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
- using
mathematical
symbols & visuals
-using
mathematical
conventions
-using
mathematical
language
PRESENTATION
OF SOLUTIONS
communicating
solutions
•
Layout is difficult to
follow
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
If ax 2 + bx + c = 0 and a ≠ 0 , then
− b ± b 2 − 4ac
x=
2a
Quadratic formula:
Trigonometric ratios:
In a right triangle:
sin θ =
opposite
hypotenuse
cos θ =
adjacent
hypotenuse
opposite
adjacent
y
sin θ =
r
x
cos θ =
r
y
tan θ =
x
tan θ =
For an angle in standard
position:
(x 2 − x1 ) 2 + ( y 2 − y1 )2
Distance between two points:
d=
Sine law:
sin A sin B sin C
=
=
a
b
c
b sin A < a < b
Ambiguous case:
a2 = b 2 + c 2 − 2bc cosA
Cosine law:
15
sin 2 θ + cos 2 θ = 1
sin θ
tan θ =
cos θ
Trigonometric Identities:
t n = a + (n − 1)d
Arithmetic sequences:
n(t1 + tn )
or
2
n
Sn = [2a + (n − 1)d]
2
t n = ar n −1
Arithmetic series:
Sn =
Geometric sequences:
Geometric series:
Sn =
Sn =
(
), r ≠ 1
(
), r ≠ 1
a 1 − rn
1− r
or
a rn − 1
r−1
General Form of conic:
ax 2 + by 2 + 2 gx + 2 fy + c = 0
circle if a = b
ellipse if ab > 0
parabola if ab = 0
hyperbola if ab < 0
Standard forms of conics:
Circle:
( x − h )2 + ( y − k ) 2
radius r
centre (h,k)
Ellipse:
(x − h )2 ( y − k )2
+
= r2
=1
a2
b2
(x − h )2 + ( y − k )2 = 1
b2
a2
centre (h,k)
length of the major axis is 2a
length of the minor axis is 2b
Parabola:
( x − h )2 = 4 p( y − k )
( y − k ) 2 = 4 p( x − h)
vertex (h,k)
distance from the vertex to
the focus is |p|
Hyperbola:
( x − h) 2 − ( y − k )2
centre (h,k)
length of the transverse axis
is 2a
length of the conjugate axis
is 2b
=1
a
b
2
( x − h) − ( y − k )2 = −1
b2
a2
2
2
16
Formulae Sheet to be provided for the MCR3U and MCF3M Examinations
− b ± b 2 − 4ac
2a
y
sin θ =
r
x
cos θ =
r
y
tan θ =
x
x=
tn = a + (n − 1)d
tn = ar n −1
sin A sin B sin C
=
=
a
b
c
a = b + c − 2bccosA
2
Sn =
Sn =
2
n (t1 + t n )
2
(
a 1 − rn
1− r
Conic Section Equations
d=
2
(x 2 − x1 )2 + ( y 2 − y1 )2
ax 2 + by 2 + 2 gx + 2 fy + c = 0
( x − h )2 + ( y − k )2 = r 2
(x − h )2 + ( y − k )2
=1
a2
b2
(x − h )2 + ( y − k )2 = 1
b2
a2
( x − h )2 = 4 p ( y − k )
( y − k )2 = 4 p ( x − h )
( x − h )2 − ( y − k ) 2
=1
a2
b2
(x − h )2 − ( y − k )2 = −1
b2
a2
17
or Sn =
), r ≠ 1
n
[2a + (n − 1)d]
2
or Sn =
(
), r ≠ 1
a rn − 1
r−1
Achievement Chart – Grades 11 and 12, Mathematics
Categories
50 –59%
(Level 1)
60 –69%
(Level 2)
70 –79%
(Level 3)
Knowledge/Understanding The student:
– understanding concepts
– demonstrates limited – demonstrates some
understanding of
understanding of
concepts
concepts
– performing algorithms
Thinking/Inquiry/
Problem Solving
– reasoning
– performs only
simple algorithms
accurately by hand
and by using
technology
The student:
– 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
– applies the steps of
an inquiry/problemsolving process with
limited effectiveness
– using mathematical
language, symbols, visuals,
and conventions
– infrequently uses
mathematical
language, symbols,
visuals, and
conventions correctly
The student:
– applies concepts and
procedures to solve
simple problems
relating to familiar
settings
Application
– applying concepts and
procedures relating to
familiar and unfamiliar
settings
The student:
– communicates with
limited clarity and
limited justification of
reasoning
80 –100%
(Level 4)
– demonstrates
considerable
understanding of
concepts
– performs algorithms – performs algorithms
with inconsistent
accurately by hand,
accuracy by hand,
mentally, and by using
mentally, and by using technology
technology
– demonstrates
thorough
understanding of
concepts
– selects the most
efficient algorithm and
performs it accurately
by hand, mentally, and
by using technology
– follows arguments
of moderate
complexity and makes
simple arguments
– follows arguments
of considerable
complexity, judges the
validity of arguments,
and makes arguments
of some complexity
– applies the steps of – applies the steps of
an inquiry/probleman inquiry/problemsolving process with
solving process with
moderate effectiveness considerable
effectiveness
– follows complex
arguments, judges the
validity of arguments,
and makes complex
arguments
– communicates with
some clarity and some
justification of
reasoning
– communicates
concisely with a high
degree of clarity &
full justification of
reasoning
– routinely uses mathematical language,
symbols, visuals, and
conventions correctly
and efficiently
– uses mathematical
language, symbols,
visuals, and
conventions correctly
some of the time
– applies concepts and
procedures to solve
problems of some
complexity relating to
familiar settings
– communicates with
considerable clarity
and considerable
justification of
reasoning
– uses mathematical
language, symbols,
visuals, and
conventions correctly
most of the time
– applies the steps of
an inquiry/problemsolving process with a
high degree of
effectiveness
and poses extending
questions
– applies concepts and – applies concepts and
procedures to solve
procedures to solve
complex problems
complex problems
relating to familiar
relating to familiar and
settings; recognizes
unfamiliar settings
major mathematical
concepts and
procedures relating to
applications in
unfamiliar settings
Note: A student whose achievement is below 50% at the end of a course will not obtain a credit for the course.
18
Appendix A
Sample Examinations and Solutions
19
OTTAWA-CARLETON DISTRICT SCHOOL BOARD
MCF 3M Functions Final Examination
(January)
PART A (22 marks)
Each correct answer has a value of one (1) mark.
g ( x) = 3 − 2x , determine g (4x) .
1.
Given
2.
For the graph of the given relation, state:
y
(a)
the domain
(b)
the range
3
2
1
x
-3 -2 -1
-1
1
2
3
2 a+2
×
a
3
3.
State all restrictions :
4.
Evaluate: (express your answers as fractions)
3
−
(a)
16 4
3−1 + 30
(b)
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
2
-2
2
4
6
8
10
Express
State the period
(b)
State the amplitude
(c) Extend the graph of
the function for one more
cycle.
12
-2
7.
(a)
− 49 in terms of i .
20
8.
Convert 210° to a radian measure in terms of π.
9.
Solve for x:
10.
State the exact value of
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,
12.
− 3x < 15
cos π .
4
(a)
the related acute angle.
(b)
the value of θ.
The first term of a sequence is –5 and the common ratio is 2.
(a)
List the first three terms of this sequence.
(b)
State the general term.
3
x4
13.
Simplify:
14.
Determine the value sin θ .
1
x4
y
P(3,5)
θ
x
21
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 ∈ ÷:
2 x2 − 5x + 7 = 0
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
[1]
(a)
Explain in words why the student is incorrect.
[1]
(b)
Write the correct equation.
3.
Simplify. (It is not necessary to state restrictions)
[3]
(a)
x − 2
3x − 6 x 2 − 4
[3]
(b)
2x2
x 2 + 4x
÷
x + 4 x 2 + 8x + 16
[3]
4.
Sketch
5.
Prove the identity:
2
2
[3]
1
y = 3 cos x  + 1 for one cycle.
2 
(
)
tan θ = sin θ 1 + tan 2 θ
Solve for θ:
[3]
6.
(a)
[3]
(b)
tan 2 θ = 1, 0° ≤ θ ≤ 360°
7.
A soccer ball is kicked into the stands such that its height above the ground is given by h = −5t
where h is the height in metres and t is the time elapsed in seconds since the ball was kicked.
[3]
(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 sin θ + 1 = 0, 0 ≤ θ ≤ 2π
22
2
+ 15t
[2]
8.
(a)
[2]
(b)
[1]
[2]
[4]
Given the relation f as defined by
state the domain and the range of f .
−1
sketch the graphs of f and f
.
y = x−2,
(d)
does f represent a function? Explain your answer.
−1
determine the expression for f ( x) .
9.
Solve for x:
(c)
x
( )
1 10 x
16x + 3 = 2
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.
[1]
(a)
Identify the type of sequence. Explain.
[2]
[2]
(b)
If the last row has 80 seats, how many rows are in the theatre? (Use the appropriate formula).
(c)
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)
[4]
H
22°
200 m
T
18°
R
12.
F
Because of the tide, the depth of the water in a harbour is modelled by the equation
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
And so on.) The graph of the relation is shown below:
( )
d = −3 cos π t + 6 ,
6
= 1 means 1 A.M.,
d
A(3,
t
[2]
(a)
What is the missing coordinate of point A? What do the coordinates of point A represent?
[1]
(b)
State the maximum depth of the water.
[2]
(c)
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.
23
24
25
26
27
28
29
30
OTTAWA-CARLETON DISTRICT SCHOOL BOARD
MCF 3M Functions Final Examination
(Backup)
PART A (20 marks)
Write only your answer for each of the following questions in the space provided.
Each correct answer has a value of one (1) mark.
f ( x ) = 5x 2 − 2 , determine f (−3) .
1.
If
2.
For the given periodic relation, state:
(a) the period
y4
y = f (x
2
-2
2
4
6
8
(b) the amplitude
10
(c) the value of f (11) assuming the
relation continues in the same manner.
12 x
-2
−5
3.
Evaluate
4.
Given
8 3 . (Express answer as a fraction)
y = x − 5 , state:
(a) the domain
(b) the range
− 25 in terms of i .
5.
Express
6.
Given cos θ = -1, sin θ = 0, 0° ≤ θ ≤ 360°, state θ.
7.
State the restrictions for
8.
Given
θ=
x −3
x 2 ( x − 3)
.
π
, state:
6
(a) the measure of θ in degrees
(b) the exact value of
cos θ
Given the diagram below, state the exact measure of θ in radians.
9.
θ
10.
A point on the graph of
y = f (x ) is (8,−3) . The coordinates of the
corresponding image point
11.
(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
Given the sequence 2, 6, 10, …,
(a) state the next term
(b) state the general term
12.
Simplify:
5
3
(a)
x4 ⋅ x4
(b)
⎛ y 23 ⎞ 2
⎜ ⎟
⎝ ⎠
1
31
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.
[3]
1.
Solve and graph the solution set,
2( x − 4) ≥ 2 + 4( x − 2)
x∈ℜ.
-4
-3
-2
-1
0
1
2
3
2.
P (−1, −3) lies on the terminal arm of the angle in standard position with measure θ. Determine:
[1]
(a)
the value of r
[1]
(b)
the value of
[2]
(c)
the value of θ to the nearest degree, where
4
yè
x
-1
sin θ
r
0° ≤ θ ≤ 360° .
-1
-2
-3
P(-1,-3)
[3]
3.
[1]
[3]
4.
m+ 3 m + 2
÷
m − 3 9 − m2
(a)
Simplify
(b)
State the restrictions in (a).
Simplify completely: (It is not necessary to state restrictions.)
a +
10a
a + 3 a 2 + 4a + 3
5.
An arrow is shot from the roof of a building. Its height above the ground is modelled by
h (t ) = −5t 2 + 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]
(b)
When will the arrow reach the maximum height?
[1]
(c)
What is the maximum height?
[3]
(d)
When will the arrow hit the ground? (round answer to the nearest tenth of a second)
6.
Solve for θ:
[2]
[3]
(a)
tan θ = 3 , 0 ≤ θ ≤ 2π (exact values)
(b)
(3 cos θ − 1)(cos θ + 2) = 0 , 0° ≤ θ ≤ 360° (round answers to the nearest degree)
7.
The graph of a parabolic relation is shown.
[1]
(a)
State the domain.
[1]
(b)
Graph the inverse on the same grid
[1]
(c)
Is the inverse a function? Explain your answer.
4
3
2
1
-4
-3
-2
-1
1
2
3
-1
[1]
(d)
State the defining equation of the inverse.
-2
-3
-4
[2]
8.
If you were given a function in the form
equation of its inverse, namely
[3]
9.
y= f
y = f (x ) , explain how you would determine the defining
−1
( x) .
Prove the identity:
2
tan θ − 1 = 2 sin θ − 1
tan θ sin θ cos θ
32
4
[3]
10.
Solve for x:
x −2
−x
27
[4]
11.
=9
Sketch one cycle of the following trigonometric function:
y = −2 sin 3⎛⎜ x + π ⎞⎟
6⎠
⎝
x
[2]
800 + 400 + 200 + 100 + K , using the appropriate formulas
12.
Given the series
(a)
determine t12 to 3 decimal places.
determine S12 to the nearest decimal place.
[2]
(b)
[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 ) = 3 cos( 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]
(a)
[2]
(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
34
35
36
37
38
39
40
OTTAWA-CARLETON DISTRICT SCHOOL BOARD
MCR 3U Functions and Relations Final Examination
(January)
PART A (21 marks)
Each correct answer has a value of one (1) mark.
g ( x) = 3 − 2x , determine g (4x) .
1.
Given
2.
For the relation defined by
2
2
x + y =1:
49 16
(a)
identify the type of conic
(b)
state the range
(c)
state the length of the major axis
2 ÷ x+ 2
x
3
3.
State all restrictions :
4.
Evaluate: (express your answers as fractions)
3
−
(a)
16 4
3
(b)
5.
−1
+3
0
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 = −3, t = 5, t = t
− t , determine
1
2
n
n −2
n −1
t3 .
− 2 + 3i .
7.
State the conjugate of
8.
State the equation of one asymptote for the graph of
9.
State the equation for the locus of points which are 5 units from
10.
State the exact value of
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.
x2 − y2 = 1.
cos 3π .
4
(a)
List the first three terms of this sequence.
(b)
State the general term.
3
13.
Simplify:
x4
1
x4
14.
(−1, 0) .
y
P(3,5)
Determine the value of sin θ.
θ
x
15.
For what value of c does the equation of the function defined by
y = x 2 − 6 x + c have only one x - intercept?
16.
Determine the number of zeroes of the function defined by
2
f ( x) = −3( x − 2) − 5 .
PART B (61 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.
[3]
2
3
(3 − 4i) − i(i ) + 2
i
1.
Simplify:
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
[1]
(a)
Explain in words why the student is incorrect.
[1]
(b)
Write the correct equation.
3.
Simplify. (It is not necessary to state restrictions)
[3]
(a)
x − 2
3x − 6 x 2 − 4
[3]
(b)
2 x + y ÷ 2 x + 3xy + y
x 2 + xy
2x 2
[3]
4.
Sketch
[3]
5.
Prove the identity:
2
2
1
y = 3 cos⎛⎜ x ⎞⎟ +1 for one cycle.
⎝2 ⎠
tan 2 θ = sin 2 θ
1 + tan 2 θ
6.
Solve for θ:
[3]
(a)
2 sin θ + 1 = 0, 0 ≤ θ ≤ 2π
[3]
(b)
4 tan 2 θ − 9 = 0, 0° ≤ θ ≤ 360° (answer to the nearest degree)
[2]
7.
(a)
[2]
(b)
[1]
(c)
[2]
(d)
does f represent a function? Explain your answer.
−1
determine the expression for f ( x) .
[4]
8.
Solve for x:
Given the relation f as defined by
state the domain and the range of f.
−1
.
sketch the graphs of f and f
(2 )
x 2
⎛ 1
= 64⎜⎜
⎝ 32 x
y = x−2 ,
x
⎞
⎟⎟
⎠
9.
A sporting goods store sells skates. During the first week, they sold 10 pairs of skates.
week they sold 14 pairs and in the third week they sold 18 pairs, and the pattern continues.
In the second
[1]
(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
8m
29°
R
11.
P
Q
Because of the tide, the depth of the water in a harbour is modelled by the equation
d = −3 cos⎛⎜ π t ⎞⎟ + 6 , where d represents the depth of the water in metres and
⎝6 ⎠
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:
A(3,
t
[2]
(a)
What is the missing coordinate of point A? What do the coordinates of point A represent?
[1]
(b)
State the maximum depth of the water.
[2]
(c)
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]
12.
(a)
Express
(b)
What are two advantages of writing the defining equation of a conic in standard form?
[2]
9x 2 − 4 y 2 − 36x − 8 y = 4 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.
y 10
8
6
4
2
-10
-8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
x
10
44
45
46
47
48
49
50
51
OTTAWA-CARLETON DISTRICT SCHOOL BOARD
MCR 3U Functions & Relations Final Examination
(Backup)
PART A (20 marks)
Write only your answer for each of the following questions in the space provided.
Each correct answer has a value of one (1) mark.
f ( x ) = 5x 2 − 2 , determine f (−3) .
1.
If
2.
For the given periodic relation, state:
(a) the period
4
y=f
2
-2
2
4
6
8
(b) the amplitude
10
(c) the value of f (11) assuming the relation
continues in the same manner.
12
-2
5
3.
Evaluate
4.
Given
−
8 3 . (Express answer as a fraction)
y = 2 x − 5 , state:
(a) the domain
(b) the range
− 25 in terms of i .
5.
Express
6.
Evaluate
7.
State the restrictions for
8.
Given
9.
i6 .
x −3
2
x ( x − 3)
.
θ = 5π , state:
6
(a)
the measure of θ in degrees
(b)
the exact value of
cos θ .
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
11.
(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
Given the recursion formula defined by t1
= 5 , t n = 2t n −1 − 3 , determine
t2 .
12.
Given the conic defined by
(a)
the coordinates of the focus.
(b)
the equation of the directrix.
13.
Simplify
5
y 2 = −8 x , determine:
3
a 4 ⋅a 4
52
PART B (67 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.
[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.
5 + 3i
4 −i
P ( −2,−3) lies on the terminal arm of the angle in standard position with measure θ. Determine:
[2]
(a)
the exact value of
[2]
(b)
the value of θ to the nearest degree, where
[3]
4.
Simplify completely: (It is not necessary to state restrictions.)
[3]
2.
Simplify
sin θ .
0° ≤ θ ≤ 360° .
a +
9a
2
a + 3 3a + 8a − 3
[4]
5.
Simplify and state the restrictions
2m + 3
2m − 3
÷
m+3
9 − 4m 2
[4]
6.
An arrow is shot from the roof of a building. Its height above the ground is modelled by
h (t ) = −5t 2 + 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.
[3]
7.
Prove the identity:
2
tan θ − 1 = 2 sin θ − 1
tan θ sin θ cos θ
8.
Solve for θ:
[2]
(a)
tan θ − 3 = 0 , 0 ≤ θ ≤ 2π (exact values)
[3]
(b)
3 cos 2 θ − 7 cos θ + 2 = 0 , 0 ≤ θ ≤ 2π (round answers correct to 2 decimal places)
9.
If you were given a function in the form
equation of its inverse, namely
y= f
y = f (x ) , explain how you would determine the defining
−1
( x) .
[2]
4
10.
The graph of a parabolic relation is shown.
[1]
(a)
State the domain.
[1]
(b)
Graph the inverse on the same grid.
3
2
1
-4
-3
-2
-1
1
-1
[1]
(c)
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.
-2
-3
-4
[3]
11.
Solve for x:
27 x − 2 = 1
9x
53
2
3
4
[4]
12.
Sketch one cycle of the following trigonometric function:
y = −2 sin ⎛⎜ 3x + π ⎞⎟
2⎠
⎝
x
800 + 400 + 200 + 100 + K , using the appropriate formulas,
13.
Given the series
[2]
(a)
determine t12 to 3 decimal places.
[2]
(b)
determine
[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
S12 to the nearest decimal place.
75
P
[3]
15.
50
h
R
100
Q
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]
T ( t ) = 3 cos( 0.262t ) + 22 , where T is the
16.
The inside temperature of a building is modelled by
(a)
temperature in °C and t is the number of hours elapsed since 5 A.M. The graph is shown below.
T
Using an appropriate calculation, explain why the coefficient of t in the
equation is 0.262.
[2]
(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.
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
5
5
5
17.
A radar screen shows the activity within a circular region of radius 60 km.
[1]
(a)
Assuming the centre of the screen is (0, 0), write the equation that represents this circle.
[4]
(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.
Given the conic defined by
[2]
(a)
the coordinates of the centre
[3]
(b)
the coordinates of the foci.
25 x 2 + 9 y 2 − 100 x + 18 y − 116 = 0 , determine:
54
55
56
57
58
C anglemeasured in degrees
59
60
61
62