12
Pre-Calculus Grade Mathematics
Standards Test
Marking Guide
January 2009
Manitoba Education, Citizenship and Youth Cataloguing in Publication Data
371.26097127
Grade 12 Pre-Calculus Mathematics Standards Test :
Marking Guide (January 2009)
ISBN-13: 978-0-7711-4174-4
1. MathematicsExaminations, questions, etc.
2. MathematicsExaminations. 3. Educational tests and
measurementsManitoba. 4. MathematicsStudy and
teaching (Secondary)Manitoba. 5. CalculusStudy
and teaching (Secondary)Manitoba. 6. Mathematical
ability-testing. 7. MathematicsStudy and teaching
(Secondary). 8. CalculusStudy and teaching.
(Secondary). I. Manitoba. Manitoba Education,
Citizenship and Youth
Manitoba Education, Citizenship and Youth
School Programs Division
Winnipeg, Manitoba, Canada
Permission is hereby given to reproduce this document on a non-profit basis for
educational purposes, provided the source is cited.
After the administration of this test, print copies of this resource will be
available for purchase from the Manitoba Text Book Bureau. Order online at
<www.mtbb.mb.ca>.
This resource will also be available on the Manitoba Education, Citizenship
and Youth website at
<www.edu.gov.mb.ca/k12/assess/archives/index.html>.
Websites are subject to change without notice.
Ce document est disponible en franais.
Table of Contents
General Marking Instructions
Scoring Guidelines
1
3
Answer Key for Multiple-Choice Questions
5
Marking Key Including Student Exemplars
Part 1: Long-Answer Questions
8
Part 2: Multiple-Choice Questions
31
Part 2: Short-Answer Questions
36
Part 2: Long-Answer Questions
42
7
Appendices
69
Appendix A: Marking Guidelines
71
Appendix B: Irregularities in Standards Tests
Irregular Test Booklet Report 75
73
Appendix C: Summary of Marks for Short-Answer Questions
i
77
ii
General Marking Instructions
Please make no marks in the student test booklets. If the booklets have marks in them, the
marks need to be removed by departmental staff prior to sample marking should the booklet be
selected.
Please ensure that:
the student test booklet number and the number on the Answer/Scoring Sheet are identical
students and markers only use a pencil to complete the Answer/Scoring Sheets
the totals of each of the four parts are written at the bottom
each students final result is recorded, by test booklet number, on the corresponding
Answer/Scoring Sheet
the Answer/Scoring Sheet is complete
a photocopy has been made for school records
Once marking is completed, please forward the Answer/Scoring Sheets to Manitoba Education,
Citizenship and Youth in the envelope provided (for more information see the Grade 12 Pre-Calculus
Mathematics Standards Test: Administration Manual).
Scoring the Multiple-Choice Questions
Please follow the instructions in the Answer Key for Multiple-Choice Questions section on page 5.
Scoring the Short-Answer and Long-Answer Questions
The remainder of the Grade 12 Pre-Calculus Mathematics Standards Test is composed of short-answer
questions and long-answer questions. Short-answer questions are worth 1 mark each and long-answer
questions are worth 2 to 5 marks each.
Each question is designed to elicit a well-defined response according to the associated specific learning
outcome(s) and relevant mathematical processes. Their purpose is to determine whether a student
meets the standards for the course as they relate to the knowledge and skills associated with the
question.
To receive full marks, a students response must be complete and correct. Where alternative answering
methods are possible, the Marking Guide attempts to address the most common solutions. For general
guidelines regarding the scoring of students responses, see Appendix A.
Pre-Calculus Mathematics: Marking Guide (January 2009)
1
Irregularities in Standards Tests
During the administration of standards tests, supervising teachers may encounter irregularities. Markers
may also encounter irregularities during local marking sessions. Appendix B provides examples of such
irregularities as well as procedures to follow to report irregularities.
If an Answer/Scoring Sheet is marked with 0 and/or NR only (e.g., student was present but did not
attempt any questions) please document this on the Irregular Test Booklet Report.
Assistance
If, during marking, any marking issue arises that cannot be resolved locally, please call Manitoba
Education, Citizenship and Youth at the earliest opportunity to advise us of the situation and seek
assistance if necessary.
You must contact the Assessment Consultant responsible for this project before making any
modifications to the answer keys or scoring rubrics.
Allison Potter
Assessment Consultant
Grade 12 Pre-Calculus Mathematics Standards Test
Telephone: [phone]
Toll-Free: 1-800-282-8069, extension 7590
Email: [e-mail]
2
Pre-Calculus Mathematics: Marking Guide (January 2009)
Scoring Guidelines
Pre-Calculus Mathematics: Marking Guide (January 2009)
3
4
Pre-Calculus Mathematics: Marking Guide (January 2009)
Answer Key for Multiple-Choice Questions
If you are using this sheet by punching out holes corresponding to the correct answers and overlaying it
on the Answer/Scoring Sheet
1) check the students responses for multiple bubbles first (questions with multiple bubbles are to be
scored as 0)
2) overlay this page on the Answer/Scoring Sheet
3) count the number of correct responses, excluding questions with multiple responses, if any
You may also score the multiple-choice questions by making a transparency of this page and overlaying
it on the Answer/Scoring Sheet.
Remember to write the total score for the multiple-choice questions at the bottom of the Answer/Scoring
Sheet.
Multiple-Choice Questions / Questions Choix Multiple
11
14
17
20
23
12
15
18
21
24
13
16
19
22
25
Pre-Calculus Mathematics: Marking Guide (January 2009)
5
6
Pre-Calculus Mathematics: Marking Guide (January 2009)
M arking Key Including Student Exemplars
Pre-Calculus Mathematics: Marking Guide (January 2009)
7
Part 1: Long-Answer Questions
Question 1
E2
Consider the 8 letters ABDDDETX.
a)
How many arrangements are possible if you use all 8 letters?
b)
How many arrangements are possible if the A and the E must be together and if you use all 8 letters?
c)
How many arrangements are possible if the A and the E cannot be together and if you use all
8 letters?
Express your answer as a whole number.
Solution
a)
8!
= 6720 arrangements
3!
1 mark
b)
7!2!
= 1680 arrangements
3!
1 mark ( mark for 7!, mark for 2!)
mark for denominator
mark for evaluating factorials
2 marks
c)
8
6720 1680 = 5040 arrangements
1 mark
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplars
a)
Mark:
0 out of 1
b)
Mark: 2 out of 2
Note(s):
gave full marks [(b) was marked consistently with (a)]
c)
Mark: out of 1
Note(s):
gave full marks [(c) was marked consistently with (a) and (b)]
deducted mark for arithmetic error
Pre-Calculus Mathematics: Marking Guide (January 2009)
9
Q uestion 2
A5
Solve the following equation:
cot = 12
Give the general solution in radian measure correct to 3 decimal places.
Solution
Method 1
cot ( ) = 12
tan ( ) =
1
12
1 mark for reciprocal function
1
tan 1 = 0.083 141
12
= 0.083 + k , k I
1 mark for correct value of
1 mark for general solution
3 marks
Method 2
cot ( ) = 12
tan ( ) =
tan
1
12
1 mark for reciprocal function
1
1
= 0.083 141
12
= 0.083 141, 3.224 734
= 0.083 + 2k , k I
1 mark for general solution
= 3.225 + 2k , k I
10
1 mark ( mark for each correct value of )
3 marks
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 1 out of 3
Note(s):
gave 1 mark for reciprocal function
gave 1 mark for correct values of
deducted mark for notation errors
deducted mark for incorrect rounding
Pre-Calculus Mathematics: Marking Guide (January 2009)
11
Question 3
D8
A colony of insects is growing exponentially according to the formula:
P = 160 ( 2 )
t
3
where P is the population after t weeks.
a)
Find the population of insects at t = 4 weeks.
Express your answer correct to the nearest whole number.
b)
Find the length of time required for the population to reach 1000 insects.
Express your answer in number of weeks correct to 3 decimal places.
Solution
a) P = 160(2)
4
3
403
b)
1 mark
t
1000 = 160(2) 3
mark for substitution
t
1000
= 23
160
t
3
ln 6.25 = ln 2
ln6.25 =
t=
or
t
3
log 6.25 = log 2
t
ln2
3
or
log6.25 =
3ln6.25
ln2
or
t=
t
log 2
3
3log 6.25
log 2
mark for applying logarithms
1 mark for log theorem
mark for isolating t
t = 7.931 569
t = 7.932
mark for evaluating quotient of
logarithms
3 marks
12
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplars
a)
Mark:
0 out of 1
b)
Mark: 2 out of 3
Note(s):
gave full marks [concept error already deducted in part (a)]
deducted mark for notation error in line 3
Pre-Calculus Mathematics: Marking Guide (January 2009)
13
Question 4
H2
The 5th term of a geometric sequence is 324 and the 10th term is 78 732 .
Find the 11th term of this sequence.
Solution
Method 1
t r9
t
5
=1
=r
4
t
tr
5
1
10
r5 =
1 mark for ratio of terms
78 732
324
r = ( 243)
1
5
r = 3
t
11
1 mark for r
=t r
10
= ( 78 732 )( 3)
1 mark for t
= 236 196
11
3 marks
Method 2
78 732
t
324
t
1
6
1 mark for set-up
t = 324
1
t = 78 732
6
78 732 = 324r
5
5
r = 243
r = 3
t
11
1 mark for r
= ( 78 732 )( 3)
= 236 196
1 mark for t
11
3 marks
14
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 2 out of 3
Note(s):
gave 1 mark for set-up
gave 1 mark for r
gave 1 mark for t
11
deducted mark for notation error in set-up
(
deducted mark for arithmetic error 4 243
Pre-Calculus Mathematics: Marking Guide (January 2009)
)
15
Question 5
D6, D7
Solve for x:
ln6 + ln ( x 2 ) = 2
Express your answer correct to 3 decimal places.
Solution
Method 1
ln 6 ( x 2 ) = 2
ln ( 6 x 12 ) = 2
e2 = 6 x 12
1 mark for log theorem
1 mark for exponential form
6 x = e2 + 12
x=
e2 + 12
6
mark for isolating x
x = 2.022 556
x = 2.023
mark for evaluating exponential term
3 marks
Method 2
ln ( x 2 ) = 2 ln6
e2 ln6 = x 2
x = 2 + e2 ln6
1 mark for isolating ln ( x 2 )
1 mark for exponential form
mark for isolating x
x = 2.022 556
x = 2.023
mark for evaluating exponential term
3 marks
Note(s):
award a maximum of 2 marks if a base other than e is used
16
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 2 out of 3
Note(s):
gave 1 mark for exponential form
gave mark for isolating x
gave mark for evaluating exponential term
Pre-Calculus Mathematics: Marking Guide (January 2009)
17
Question 6
E4
14
4 3
In the expansion of 2 x + , a simplified term contains x31 .
x
Find and simplify this term completely.
Solution
Method 1
t:
C
1 14 0
0
4 14 3
(2x )
C 2x
t:
C
2 14 1
1
4 13 3
()
t:
3 14 2
x
x
x
x
2
4 12 3
()
2x
56
51
x
x
1 mark for set pattern or for identifying the correct
term
46
exponent drops by 5 each term
t contains x
31
6
t=
6
C
14 5
5
4 93
(2x )
(
x
= ( 2002 ) 2 x
9 36
)
= 249 080 832 x
14 5
35
x5
= 2002 ( 512 )( 243) x
2 marks for set-up
(1 mark for C , mark for each consistent factor)
31
1 mark for simplification ( mark for evaluating
combination, mark for power rule)
4 marks
31
Note(s):
deduct mark if answer is not completely simplified
18
Pre-Calculus Mathematics: Marking Guide (January 2009)
Question 6
E4
Method 2
(x )
4 14 k
x
(x )
1 k
56 4k
x
x
=x
=x
56 5k
31
=x
k
31
31
1 mark for set-up
56 5k = 31
5k = 25
k =5
t contains x
31
6
t=
6
C
14 5
5
4 93
()
2x
x
2 marks for set-up
(1 mark for C , mark for each consistent factor)
14 5
5
9 36 3
= ( 2002 ) 2 x
x5
1 mark for simplification ( mark for evaluating
combination, mark for power rule)
= 2002 ( 512 )( 243) x
4 marks
(
= 249 080 832 x
)
31
31
Pre-Calculus Mathematics: Marking Guide (January 2009)
19
E xemplar
Mark: 1 out of 4
Note(s):
gave 1 mark for consistent factors
gave mark for evaluating combination
20
Pre-Calculus Mathematics: Marking Guide (January 2009)
This page was intentionally left blank.
Pre-Calculus Mathematics: Marking Guide (January 2009)
21
Question 7
G5
The probability that Alex wears his boots is 0.35.
When he wears his boots, the probability that he jumps in a puddle is 0.6.
When he does not wear his boots, the probability that he jumps in a puddle is 0.3.
Given that Alex jumps in a puddle, what is the probability that he is wearing his boots?
Express your answer correct to 3 decimal places.
Solution
P(boots jump ) =
P(boots and jump)
P(jump)
( 0.35)( 0.6 )
=
( 0.35)( 0.6 ) + ( 0.65)( 0.3)
=
0.21
0.21 + 0.195
=
0.21
0.405
mark for simplification
3 marks
= 0.518 519
= 0.519 or
mark for P(boots and jump)
mark for complement of P(boots) [0.65]
mark for P(no boots and jump)
1 mark for conditional probability ratio (including
addition)
14
27
Note(s):
deduct mark if answer is not expressed as a single fraction or decimal
22
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 1 out of 3
Note(s):
gave mark for P(boots and jump)
gave mark for complement of P(boots) [0.65]
gave mark for P(no boots and jump)
Pre-Calculus Mathematics: Marking Guide (January 2009)
23
Question 8
G2
A mathematics class dropped a paper cup 500 times. The results of this experiment are shown in the
table.
Position of the cup
Frequency
open end up
50
open end down
115
on its side
335
A student plays a game where he drops a cup 3 times.
He wins the game if the cup lands in the same position all 3 times.
Using the data in the table above, determine the probability of winning this game.
Express your answer correct to 3 decimal places.
Solution
3
3
50
115
335
P ( 3 same ) =
+
+
500
500
500
up
down
3
2 marks for P(3 same)
(1 mark for probabilities,
1 mark for raising to the power of 3)
side
= 0.001 + 0.012 167 + 0.300 763
= 0.313 93
= 0.314
1 mark for adding cases
3 marks
24
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 2 out of 3
Note(s):
gave 1 mark for probabilities
gave 1 mark for adding cases
Pre-Calculus Mathematics: Marking Guide (January 2009)
25
Question 9
E3
Solve for n algebraically:
C = 36
n2
Solution
Method 1
n!
= 36
2!( n 2 ) !
1 mark for set-up
n!
= 72
( n 2 )!
n ( n 1)( n 2 ) !
= 72
( n 2 )!
n ( n 1) = 72
mark for correctly expanding numerator
mark for simplification
n 2 n = 72
n 2 n 72 = 0
( n + 8) ( n 9 ) = 0
n = 8 n = 9
mark for both values of n
mark for rejecting extraneous root
3 marks
Method 2
n ( n 1)
= 36
2!
1 mark for set-up
n ( n 1) = 72
mark for simplifying
two consecutive
positive numbers
= 72
9 ( 8) = 72
n = 9
1 mark for explanation
mark for the value of n
3 marks
Note(s):
give 1 mark for correct answer using trial and error
26
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 1 out of 3
Note(s):
gave 1 mark for trial and error
Pre-Calculus Mathematics: Marking Guide (January 2009)
27
Question 10
a) B6
The graph of y = f ( x ) is shown below.
a)
b)
b) B4
y
1
f ( x + 3) .
2
1
Sketch the graph of y =
.
f ( x)
Sketch the graph of y =
( 2, 3)
1
1
x
( 2, 1)
( 0, 3)
y
a)
( 5, 1.5)
1
1
( 3, 1.5)
x
( 1, 0.5)
1 mark for vertical compression
( mark for left branch, mark for right branch)
1 mark for horizontal shift
( mark for left branch, mark for right branch)
2 marks
28
Pre-Calculus Mathematics: Marking Guide (January 2009)
Question 10
a) B6
b) B4
y
b)
2, 1
3
1
x
1
( 2, 1)
mark for correctly placed asymptote
mark for left branch (arrowhead and endpoint)
1 mark for right branch ( mark for correct graph on ( 1, 0] , mark for correct graph on [ 0, 2] )
2 marks
Note(s):
any y-intercept between 1 and 0 is acceptable
deduct a maximum of mark for one or more endpoints or arrowheads incorrectly shown
deduct mark if graph crosses or curls away from the asymptote
Pre-Calculus Mathematics: Marking Guide (January 2009)
29
E xemplars
a)
Mark: 1 out of 2
Note(s):
gave 1 mark for horizontal shift
b)
Mark: 1 out of 2
Note(s):
gave mark for correctly placed asymptote
gave mark for left branch
gave mark for correct graph on [ 0, 2]
30
Pre-Calculus Mathematics: Marking Guide (January 2009)
Part 2: Multiple-Choice Questions
A2
11. If sec > 0 and cot < 0, then terminates in quadrant:
a)
b)
II
c)
III
d
C1
I
IV
12. An expression equivalent to sin cos tan is:
2
a)
b
sin 2
c)
cos 2
d)
A3
sec
tan 2
13. If sin =
a)
3
b)
3
and terminates in quadrant IV, the exact value of cot is:
2
1
3
1
3
c
d)
3
Pre-Calculus Mathematics: Marking Guide (January 2009)
31
A1
14. A central angle of 45 intercepts an arc 8 cm long. The radius of this circle is:
a
32
cm
b)
2 cm
c)
8
cm
45
d)
45
cm
8
D3 15. If f ( x ) = log x , then the inverse of this function is:
2
a)
f 1 ( x ) = x 2
b
f 1 ( x ) = 2 x
c)
f 1 ( x ) =
1
log x
2
d)
A6
f
1
( x ) = log 2 ( x 1)
16. All the x-intercepts of the graph f ( x ) = sin x can be described by:
a)
b)
k
, k
2
c
k , k
d)
32
+ k , k
2
0, , 2
Pre-Calculus Mathematics: Marking Guide (January 2009)
B3
17. If f ( x ) is an odd function, then:
a)
b)
f ( x ) = f 1 ( x )
c)
f (x) =
d
B1
f (x) = f ( x)
f (x) = f ( x)
1
f ( x)
18. If f ( x ) = x 2 and the graph is shifted 2 units to the right, an equation for the resulting
graph is:
a)
y = x2 + 2
b)
y = x2 2
c)
y = ( x + 2)
2
d
y = ( x 2)
2
D2 19. Solve the equation 3 i 9 x = 81 .
a)
x=3
b
x=
3
2
c)
x=
4
3
d)
x=
2
3
Pre-Calculus Mathematics: Marking Guide (January 2009)
33
D4 20. Which of the following curves could be the graph of y = log x ?
5
y
a)
b)
y
x
c
x
d)
y
y
x
E2
21. The expression
55 i 54 i 53
b)
55!
c)
55!
3!
d)
P
55 52
22. The number of ways to arrange 8 chairs of different colours in a circle is:
a)
8
b)
9!
c)
8!
d
34
P is equal to:
55 3
a
E2
x
7!
Pre-Calculus Mathematics: Marking Guide (January 2009)
G5 23. The schools Math Club consists of 4 boys and 6 girls. If the names of 3 club members are
drawn at random, the probability they are all girls is:
a)
b
6!
10!
C
63
C
10 3
3
c)
d)
E2
6
10
6
3
10
24. The letters of the word EXCELLENCE can be arranged in:
a)
10! ways
b
10!
ways
4!2!2!
c)
10!
ways
2!
d)
10!
ways
8!
G3 25. Two events that cannot occur at the same time are:
a)
conditional
b)
dependent
c)
independent
d
mutually exclusive
Pre-Calculus Mathematics: Marking Guide (January 2009)
35
Part 2: Short-Answer Questions
Award half marks for incorrect answers that
have resulted from arithmetic or notation errors
as indicated in the marking guidelines.
Do not award any other half marks
unless indicated in this marking guide.
36
Pre-Calculus Mathematics: Marking Guide (January 2009)
Question 26
A1
1 mark
A2
1 mark
Find the measure of Y .
T
80
R
4
Solution
55 or
Y
11
36
Note(s):
4
= 45 or 80 =
4
9
deduct mark for missing degree symbol
give mark for
Question 27
1
If ,
2
y is a point on the unit circle, what is a possible value of y?
Solution
3
3
or
2
2
Question 28
A3
1 mark
11
Find the exact value of cos
.
4
Solution
2
1
or
2
2
Note(s):
award no marks for
2
1
or
2
2
Pre-Calculus Mathematics: Marking Guide (January 2009)
37
Question 29
B2
1 point mark
One on the graph of y = f ( x ) is ( 4, 8 ) . Give the coordinates of a point that must be on the
1
graph of y = f x .
4
Solution
(16, 8 )
Question 30
A3
1 mark
B3
1 mark
If sec = 2 , what is a possible value of ?
Solution
=
5
7
or
or
or ...
3
3
3
or
= 60 or 300 or 420 or ...
Note(s):
give mark for cos =
Question 31
1
2
Write an equation for the line formed by reflecting the graph of y = 4 x 1 over the x-axis.
Solution
y = ( 4 x 1)
or
y = 4x + 1
or
y = 4x 1
38
Pre-Calculus Mathematics: Marking Guide (January 2009)
Question 32
F3
1 mark
2
2
Find the centre of the circle with the equation ( x 2 ) + y = 9 .
Solution
( 2, 0 )
Question 33
D3
D5
Find the exact value of 5
1 mark
1 mark
log 7
5.
Solution
7
Question 34
Write log 6 log x as a single logarithm.
a
a
Solution
6
log
ax
Pre-Calculus Mathematics: Marking Guide (January 2009)
39
Question 35
H1
1 mark
The 5th term of a geometric sequence is 18 and the 6th term is 12. What is the common ratio?
Solution
12
2
or
18
3
Question 36
E1
1 mark
In Grade 12 there are 4 mathematics courses, 5 science courses, and 2 English courses.
In how many ways can a student choose one course in each of these three subjects?
Solution
CiCiC
41
51
21
or
4i5i2
or
40 ways
Question 37
E1
1 mark
Simplify:
11!
9!
Solution
11 i 10 or 110
40
Pre-Calculus Mathematics: Marking Guide (January 2009)
Question 38
F1
1 mark
Identify the conic section described by 3 y 2 + x 6 y 12 = 0 .
Solution
parabola
Note(s):
give mark for vertical parabola
Question 39
G2
1 mark
Mark is playing a game on his computer in which he wins one out of six times. If Mark plays this game
twice, what is the probability that he loses both times?
Solution
55
i
66
25
=
36
P ( lose, lose ) =
Question 40
C2
1 mark
5
5
Find the exact value of cos cos
+ sin sin
.
8
8
8
8
Solution
0
Note(s):
4
give mark for cos
or cos
8
2
Pre-Calculus Mathematics: Marking Guide (January 2009)
41
Part 2: Long-Answer Questions
Q uestion 41
A4, C1
Solve for x on the interval [ 0, 2] :
2sin 2 x + 3cos 2 x = cos x + 4
Solution
(
)
2 1 cos 2 x + 3cos 2 x cos x 4 = 0
1 mark for identity
2 2 cos 2 x + 3cos 2 x cos x 4 = 0
cos 2 x cos x 2 = 0
( cos x 2 ) ( cos x + 1) = 0
cos x = 2
no solution
cos x = 1
x=
mark for factoring
mark for solving for cos x
2 marks for consistent solutions of trigonometric
equations (1 mark for each solution)
4 marks
42
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 2 out of 4
Note(s):
gave 1 mark for identity
gave mark for solving for cos x
1
2
gave 1 mark for the consistent solution of cos x = 1
deducted mark for arithmetic error in line 2
gave mark for one consistent solution of cos x =
Pre-Calculus Mathematics: Marking Guide (January 2009)
43
Question 42
B4
Sketch a clearly labelled graph of at least one period of y = sec x .
Solution
y
1
2
3
2
2
1
2
3
2
2
x
1 mark for correctly placed asymptotes
mark for general shape
mark for correct range
2 marks
Note(s):
no additional deduction for missing horizontal scale (correctly placed asymptotes require a scale)
deduct mark if graph crosses or curls away from the asymptote
44
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 1 out of 2
Note(s):
gave full marks
deducted mark for not labelling the graph of y = sec x
Pre-Calculus Mathematics: Marking Guide (January 2009)
45
Question 43
A6, B3
Sketch a clearly labelled graph of at least one period of y = cos x .
Solution
y
1
2
2
x
mark for sinusoidal shape
mark for correct period
1 mark for the reflection without a shift
2 marks
46
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: out of 2
Note(s):
gave mark for sinusoidal shape
Pre-Calculus Mathematics: Marking Guide (January 2009)
47
Question 44
F3
The graph of a hyperbola is shown below.
a) Write the equation for this hyperbola.
y
6
4
2
4
2
2
x
4
2
4
6
8
b) Determine the range of the hyperbola drawn above.
Solution
a)
( y + 1) 2
16
x2
=1
9
mark for vertical hyperbola
mark for centre
mark for 4 2 underneath y component
mark for 3 2 underneath x component
2 marks
b)
Range: ( , 5] [ 3, )
1 mark for range consistent with given graph
1 mark
Note(s):
deduct mark for each bracket error in range to a maximum of 1 mark
48
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplars
a)
Mark: 1 out of 2
Note(s):
gave mark for centre
2
gave mark for 4 underneath y component
gave mark for 3 2 underneath x component
b)
Mark: 1 out of 1
Note(s):
another possible notation
Pre-Calculus Mathematics: Marking Guide (January 2009)
49
Question 45
B7
The equation y = A cos ( B ( x C ) ) + D is represented by the graph shown below. Give a possible set
of values for A, B, C, and D.
7
, 3
4
y
9
, 3
4
x
, 1
4
17
, 1
4
Solution
Method 1
A = 2
1
B=
2
C=
4
D =1
1 mark for each parameter
17
4
or
4 marks
Method 2
A=2
B=
1
2
C=
1 mark for each parameter
7
4
or
9
4
4 marks
D =1
Note(s):
deduct mark for sign error for the value of C
50
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 2 out of 4
Note(s):
gave 1 mark for A
gave 1 mark for C
gave 1 mark for D
deducted mark for transcription error in D
Pre-Calculus Mathematics: Marking Guide (January 2009)
51
Question 46
D4, B5
Sketch a clearly labelled graph of y = log x . Label all intercepts, if applicable.
Solution
y
1
x
mark for x-intercept at 1
1 mark for correct branch left of x-intercept
mark for correct branch right of x-intercept
2 marks
Note(s):
no deduction for missing y-scale
award a maximum of 1 mark if graph contains y-values less than zero
52
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 1 out of 2
Note(s):
gave 1 mark for correct branch left of x-intercept
gave mark for correct branch right of x-intercept
Pre-Calculus Mathematics: Marking Guide (January 2009)
53
Question 47
A3
Evaluate:
7
25
3
csc + tan
+ sin
6
4
2
Solution
= 2 +1 1
7
1 mark for csc
6
25
1 mark for tan
4
3
1 mark for sin
2
= 2
3 marks
Note(s):
deduct a maximum of 1 mark for sign error(s)
54
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 1 out of 3
Note(s):
7
gave 1 mark for csc
6
25
gave 1 mark for tan
4
deducted mark for not evaluating the final answer
Pre-Calculus Mathematics: Marking Guide (January 2009)
55
Question 48
C2
5
3
and sin = where and terminate in the same quadrant, calculate the
6
5
exact value of sin ( ) .
Given that cos =
Solution
y
y
11
6
3
x
5
sin ( ) = sin cos cos sin
11 4 5 3
=
6 5 6 5
5
x
4
mark for 11
mark for 4
1 mark for sin
1 mark for cos
1 mark for substitution into correct formula
=
4 11 + 15
30
4 marks
Note(s):
deduct mark if answer is not expressed as a single fraction
56
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 2 out of 4
Note(s):
gave mark for 4
gave 1 mark for sin
gave 1 mark for substitution into correct formula
deducted mark for arithmetic errors
Pre-Calculus Mathematics: Marking Guide (January 2009)
57
Question 49
C1, C2
Prove the following identity: cos 2 =
cot 2 tan 2
sec 2 csc 2
Solution
Method 1
LHS = cos 2
2
2
= cos sin
1
1
=
2
sec csc 2
=
=
=
1 mark for identity
1 mark for identities
csc 2 sec 2
mark for common denominator
sec 2 csc 2
1 + cot 2 1 tan 2
sec 2 csc 2
cot 2 tan 2
2
1 mark for identities
mark for simplification
2
sec csc
= RHS
4 marks
Method 2
RHS =
cot 2 tan 2
sec 2 csc 2
( csc 1) (sec 1)
=
2
2
2
2
sec csc
=
=
=
csc 2 sec 2
2
mark for simplification
2
sec csc
csc 2
sec 2 csc 2
1
2
sec
1 mark for identities
sec 2
sec 2 csc 2
mark for separating fractions
1
csc 2
= cos 2 sin 2
= cos 2
1 mark for identity
= LHS
58
1 mark for identities
4 marks
Pre-Calculus Mathematics: Marking Guide (January 2009)
Question 49
C1, C2
Solution
Method 3
cot 2 tan 2
RHS =
sec 2 csc 2
2
cos
sin 2
1
=
cos 2
i
2
sin
cos 2
1
1 mark for identities
sin 2
cos 4 sin 4
sin 2 i cos 2
1
=
mark for common denominator
cos 2 sin 2
= cos 4 sin 4
mark for simplification
)( cos
= (1) ( cos sin )
(
= cos 2 + sin 2
2
2
sin 2
2
)
mark for factoring
mark for identity
= cos 2
1 mark for identity
= LHS
4 marks
Note(s):
other methods possible
Pre-Calculus Mathematics: Marking Guide (January 2009)
59
E xemplar
Mark: 1 out of 4
Note(s):
gave 1 mark for identities
gave mark for common denominator
60
Pre-Calculus Mathematics: Marking Guide (January 2009)
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Pre-Calculus Mathematics: Marking Guide (January 2009)
61
Question 50
F2
Given the relation 9 x 2 + 4 y 2 + 54 x 8 y + 49 = 0 , write the equation in standard form.
Solution
9 x 2 + 54 x + 4 y 2 8 y = 49
)(
)
9 ( x + 6 x + 9 ) + 4 ( y 2 y + 1) = 49 + 81 + 4
(
9 x 2 + 6 x + 4 y 2 2 y = 49
2
2
2
2
9 ( x + 3)
4 ( y 1)
36
+
=
36
36
36
( x + 3) 2
4
+
( y 1) 2
9
=1
1 mark for completing the square for x
( mark for left side, mark for right side)
1 mark for completing the square for y
( mark for left side, mark for right side)
1 mark for standard form
( mark for left side, mark for right side)
3 marks
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Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 2 out of 3
Note(s):
gave 1 mark for completing the square for x
gave 1 mark for completing the square for y
gave mark for standard form (right side)
deducted mark for notation errors
Pre-Calculus Mathematics: Marking Guide (January 2009)
63
Question 51
F3
Sketch a clearly labelled graph of the following:
( x + 3)
2
y
=1
9
+
16
y
2
1
x
1
Solution
y
7
6
5
4
i
i
-7
-6
-5
-4
i
-3
i
3
2
1
-2
-1
-1
i
1
2
3
4
5
6
7
x
-2
-3
-4
-5
-6
-7
mark for ellipse
mark for centre
mark for consistent endpoints of major axis
mark for consistent endpoints of minor axis
2 marks
Note(s):
give a maximum of 1 mark for correct graph of
64
( x + 3) 2
16
y2
= 1 or
9
2
y 2 ( x + 3)
=1
9
16
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: out of 2
Note(s):
gave mark for ellipse
Pre-Calculus Mathematics: Marking Guide (January 2009)
65
Question 52
H3
Evaluate:
3 k
k =1
Solution
3 k
=
k =1
1
1
1
+
+
+
2
33
33
t=
1
3
1 mark for t
r=
1
3
1 mark for r
1
S
=
1
1
3
1
1
3
1 mark for substitution into correct formula
3 marks
1
=3
2
3
=
1
2
Note(s):
deduct mark if answer is not expressed as a single fraction
66
Pre-Calculus Mathematics: Marking Guide (January 2009)
E xemplar
Mark: 1 out of 3
Note(s):
gave 1 mark for t
1
gave 1 mark for substitution into correct formula
deducted mark for arithmetic error
Pre-Calculus Mathematics: Marking Guide (January 2009)
67
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68
Pre-Calculus Mathematics: Marking Guide (January 2009)
Appendices
Pre-Calculus Mathematics: Marking Guide (January 2009)
69
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Pre-Calculus Mathematics: Marking Guide (January 2009)
Appendix A
MARKING GUIDELINES
Arithmetic error, deduct mark
Concept error, deduct 1 mark
Final probability answer greater than 1, deduct mark
Intercept stated as an ordered pair, deduct mark
Expression changed to an equation or vice versa, deduct mark
Units of measure missing or incorrect, deduct mark
Degree symbol in angle measure missing, deduct mark
Correct answer stated in degrees instead of radians or vice versa, deduct 1 mark
Variable introduced without being defined, deduct mark
Example: 2 sin x = 1
1
2
5
x= ,
over [ 0, 2 ]
66
5
+ 2 k or
+ 2 k , k not defined, deduct mark
x=
6
6
sin x =
(Note: Student can write k or k is an integer.)
LHS and RHS equated throughout the proof of an identity, deduct 1 mark
Variable omitted more than once in a trig identity or equation, deduct mark
Variable changed without being redefined, in either an equation or an identity, deduct mark for
poor notation
cos 2 = 1
Example: 2sin = 1
1
2
5
x= ,
66
sin x =
or
x2 = 1
x = 1
sin x 2 written instead of sin 2 x , deduct mark
Incorrect precision or rounding, deduct mark
Pre-Calculus Mathematics: Marking Guide (January 2009)
71
In a binomial theorem expansion, answer given as an entire term when only the numerical
coefficient is requested, deduct mark
Parentheses omitted when using the log of a power theorem such as:
a)
log3x +1 = x + 1 log 3 = x log 3 + log3, deduct mark
b)
log3x +1 = x + 1 log 3 = x + log 3, deduct 1 mark
11
+
Unsimplified fractions such as 2 3 , deduct mark
2
(Note: Unreduced fractions such as
6
or unrationalized fractions such as
8
3
are
2 1
acceptable.)
Error made while simplifying fractions, deduct mark
Cases in permutations, combinations, or probability problems not briefly described, deduct mark
Brackets in a fraction omitted but assumed, deduct mark
Example:
22 4
=
3
9
Endpoints or arrowheads incorrectly shown, deduct mark
Asymptotes shown as solid lines or not shown, deduct mark
Graphs of functions drawn to cross or curl away from an asymptote, deduct mark
Axes not labelled or scale values not indicated (tick marks are not assumed to be one unit), deduct
mark
Domain or range written in incorrect order, deduct 1 mark
Example: ( , 0] written as [ 0, )
Bracket error made when stating the domain or range, deduct mark to a maximum of 1 mark per
question
72
Pre-Calculus Mathematics: Marking Guide (January 2009)
Appendix B
IRREGULARITIES IN STANDARDS TESTS
A GUIDE FOR LOCAL MARKING
The Administration Manual and the Policies and Procedures for Standards Tests are distributed
to teachers and administrators to ensure consistency in the administration of standards tests.
These documents deal with the issues of test material security and the Departments policy
regarding adaptations and exemptions.
During the marking of standards tests, markers have occasionally encountered test booklets in
which there have been irregularities. The following list provides some examples of such
irregularities:
Completely different penmanship in the same test booklet
Incoherent work with correct answers
Notes from a teacher indicating how he or she has assisted students during the test
administration
Student offering that he or she received teacher assistance for a question
Student submitting work on unauthorized paper
Evidence of plagiarism or cheating
If a students mark on the test is 0% because he or she answered with all non-responses
(NR), all inappropriate responses (0), or a combination of both, please complete an
Irregular Test Booklet Report to confirm that the result is accurate (i.e., the student was not, in
fact, absent or exempted).
Please note that student comments or responses indicating that the student may be at personal
risk of being harmed or of harming others are personal safety issues. This type of student
response requires an immediate and appropriate follow-up action at the school level. In this
case, please ensure the Department is made aware that a follow-up has taken place by
completing an Irregular Test Booklet Report.
Except in the case of cheating or plagiarism where the result is a standards test mark of 0%, it
is the responsibility of the division or the school to determine how they will proceed with
irregularities. Once an irregularity has been confirmed, the marker prepares an Irregular Test
Booklet Report documenting the situation, the people contacted, and the follow-up. The original
copy of this report is to be retained by the local jurisdiction and a copy is to be sent to the
Instruction, Curriculum and Assessment Branch along with all other test materials.
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Pre-Calculus Mathematics: Marking Guide (January 2009)
Irregular Test Booklet Report
Test: ________________________________________________________________________
Date marked: ________________________________________________________________
Booklet ID No.: _______________________________________________________________
Problem(s) noted: ____________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Question(s) affected: _________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Action taken or rationale for assigning marks: ___________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Pre-Calculus Mathematics: Marking Guide (January 2009)
75
Follow-up: ___________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Decision: _____________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Markers Signature: ___________________________________________________________
Principals Signature: _________________________________________________________
For Department Use OnlyAfter Marking Complete
Consultant: ______________________________________________________
Date: ___________________________________________________________
76
Pre-Calculus Mathematics: Marking Guide (January 2009)
Appendix C
SUMMARY OF MARKS FOR SHORT-ANSWER QUESTIONS
26 *
55 or
11
36
33
7
34
27
3
3
or
2
2
28 *
2
1
or
2
2
29
35
12
2
or
18
3
36
CiCiC
41
(16, 8 )
30 *
=
6
log
ax
5
7
or
or
or ...
3
3
3
or
51
or
4i5i2
or
40 ways
21
37
11 i 10 or 110
= 60 or 300 or 420 or ...
31
y = ( 4 x 1)
or
y = 4x + 1
or
38 *
parabola
39
55
i
66
25
=
36
P ( lose, lose ) =
y = 4x 1
32
( 2, 0 )
40 *
0
* refer to marking notes in the short answer section of this guide
Pre-Calculus Mathematics: Marking Guide (January 2009)
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Pre-Calculus Mathematics: Marking Guide (January 2009)