Grade 12 Pre-Calculus Mathematics (40S)
Standards Test
Student Booklet (Part 2)
January 2007
Manitoba Education, Citizenship and Youth Cataloguing in Publication Data
371.26097127 Grade 12 Pre-Calculus Mathematics (40S) Standards Tests. Student Booklet. (Part 2)
“January 2007”
ISBN-13: 978-0-7711-3617-4 ISBN-10: 0-7711-3617-6
1. Mathematics—Examinations, questions, etc. 2. Mathematics—Examinations. 3. Educational tests and measurements—Manitoba. 4. Mathematics—Study and teaching (Secondary)—Manitoba. 5. Calculus—Study and teaching (Secondary)—Manitoba. 6. Mathematical ability-testing. 7. Mathematics—Study and teaching (Secondary). 8. Calculus—Study and teaching. (Secondary). I. Manitoba. Manitoba Education, Citizenship and Youth
Copyright © 2007, the Crown in Right of Manitoba, as represented by the Minister of Education, Citizenship and Youth. Manitoba Education, Citizenship and Youth, Instruction, Curriculum and Assessment Branch, Winnipeg, Manitoba R3E 3J5
Permission is hereby given by the copyright owner to reproduce this document on a non-profit basis for educational purposes provided the source is mentioned.
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 1
Multiple-Choice Questions
Instructions
There are 15 multiple-choice questions.
Each question is worth 1 mark.
Read each question carefully.
Examine all the choices before you decide on your answer.
You may use the spaces beside each question for rough work.
Record your answers on the sheet provided.
Provide only one answer per question.
There is no penalty for guessing.
Answer all questions.
Calculators (scientific or graphing) are not allowed for this part of the test.
Page 2 Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007
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Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 3
Multiple-Choice Questions
10. The equation of an asymptote of the conic section whose equation is 2 2
14 9y x− = is:
a) 23
y x=
b) 32
y x=
c) 49
y x=
d) 94
y x=
11. The sum of the infinite geometric series 8060 403
− + is:
a) 20
b) 36
c) 100
d) 180
12. A value of θ that satisfies the equation ( )sin 2sin 1 0θ θ + = is:
a) 56π
b) 76π
c)2π
d) 32π
Page 4 Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007
13. The probability of Lisa winning a game is 13
. No game can end in a tie.
If she plays 2 games, what is the probability that she will lose them both?
a) 13
b) 23
c) 49
d) 89
14. The graph represented by the equation ( )3sin 4 5y θ= − + has a maximum value of:
a) 2
b) 3
c) 5
d) 8
15. If ( )3log 5 and log 4 , then log 4 5a a aT R= = is:
a) 3RT
b) 3R T+
c) 3R T+
d) 3RT
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 5
16. The expression sectan cot
θθ θ
is equivalent to:
a) sinθ
b) cosθ
c) 1sinθ
d) 1cosθ
17. Given ( ) 3 7f x x= + . An expression for ( )1f x− is:
a) 3 7x− −
b) 7 3x +
c) 13 7x +
d) 73
x −
18. The terminal point 41P8π− on the unit circle lies in quadrant:
a) I
b) II
c) III
d) IV
Page 6 Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007
19. An expression equivalent to 3
12 i
ix
= is:
a) 14x
b) 2 32 4 8x x x+ +
c) 2 32 2 2x x x+ +
d) 66x
20. The value of 5sec3π is:
a) 23
−
b) 23
c) 2−
d) 2
21. A party of 18 people is divided into 2 different groups consisting of 11 people and 7 people.The number of different ways this can be done is:
a) 18! 18!7! 11!
+
b) 18!11!7!
c) 18!18!7!11!
d) 11!7!2!
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 7
22. The coordinates of a point on the graph of ( )y f x= are ( )2, 3− − . Which of the following
points must be on the graph of ( )2y f x= − ?
a) ( )2, 6−
b) ( )2, 6− −
c) ( )4, 3−
d) ( )4, 3−
23. An ellipse has a major axis of 20 and a minor axis of 12. A possible equation for this ellipse is:
a)2 2
1400 144x y+ =
b)2 2
120 12x y+ =
c)2 2
1100 36x y+ =
d)2 2
110 6x y+ =
24. In the binomial expansion of ( )122 3a b− , the term containing 5b is the:
a) 4th term
b) 5th term
c) 6th term
d) 7th term
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Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 9
Short-Answer Questions
Instructions
There are 15 short-answer questions.
Each question is worth 1 mark.
Calculators are not allowed for this part of the test.
Write your final answer in the space provided.
Work need not be shown in order to obtain full marks for a correct answer. However, in the case of an incorrect final answer, part marks may be given only if work is shown.
Solutions need not have rationalized denominators.
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Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 11
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113
114
115
Short-Answer Questions
25. What is the x-coordinate of 4P π3
on the unit circle?
(1 mark)
26. A circle has a radius of 10 cm. An arc on this circle is intercepted by a central angle of 90° . Calculate the length of this arc, in centimetres.
(1 mark)
27. In how many ways can 10 people be seated at a round table? Leave your answer in factorial notation.
(1 mark)
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117
118
28. State the measure of an angle, in radians, formed by the two hands of a clock at 7:00 p.m. (1 mark)
29. Express coscotsin
θθθ
in terms of a single trigonometric function.
(1 mark)
30. State the x-intercept of ( ) 1f x x= − .(1 mark)
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 13
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120
121
31. The probability that Stephan passes Spanish is 0.5 and the probability that he passes Math is 0.4. What is the probability that he passes Spanish or Math?
(1 mark)
32. The parabola whose equation is ( ) ( )25 2x a y− = + opens to the left.
State a possible numeric value for a.(1 mark)
33. State the domain of the function: 1cosy x−=
(1 mark)
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123
124
34. Find the exact value of sin 47 cos17 cos 47 sin17° ° − ° ° .(1 mark)
35. If the period of ( )5cosy bx= is π4
, find the value of b .
(1 mark)
36. Simplify:
( )( )
1 !1 !
nn
+−
(1 mark)
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 15
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125
126
37. Evaluate:
3
4
log 27log 16
(1 mark)
38. Solve:15 5x− =
(1 mark)
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127 39. Complete the graph of ( )y f x= if it is an odd function. (1 mark)
-4
-5
( )y f x=
y
x
4
5
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 17
Long-Answer Questions
Instructions
There are 10 long-answer questions worth a total of 35 marks.
Calculators are not allowed for this part of the test.
Write each solution in the space provided.
For full marks, your answers must show all pertinent diagrams, calculations, and explanations.
Your solutions should be neat, clear, and well organized.
If any curve contains asymptotes, the asymptotes should be included in the graph.
Solutions need not have rationalized denominators.
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Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 19
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128
129
Long-Answer Questions
40. Solve:
( ) ( )log 5 log 1 2x x+ − =(4 marks)
41. The equation of a circle is given by 2 2 2 6 0x y x y k+ − + + = .
Find the value of k if the radius of the circle is 4. (3 marks)
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131
42. a) State the y-intercept of xy e= .(1 mark)
The y-intercept is
b) Sketch a clearly labelled graph of 4xy e= − .(2 marks) y
x
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 21
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133
43. a) Find the exact value of 5tan12π .
(3 marks)
b) Find the exact value of 5cot12π .
(1 mark)
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135
44. The equation of a conic section is given by 24 16 19x y y= − + .
a) Write the equation in standard form. (2 marks)
b) Sketch a clearly labelled graph of the conic section.
(2 marks) y
x
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 23
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45. The function ( )y f x= is shown below:
(2 marks)
Sketch the graph of 1( )
yf x
= .
y
1
1x
y
1
1x
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46. The graph of a function ( )y f x= is shown below.
Sketch the graph of:
a) (2 marks) 2
xy f=
y
x
y
x
( )2, 2( )2, 2−
( )3, 2− −
( )f x
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 25
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138
139
Sketch the graph of:
b) (2 marks)
47. Find the exact value of: 2 25 11sin sec tan csc3 6 4 6π π π π− −
(4 marks)
( )2 1y f x= +
y
x
Page 26 Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007
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14048. Prove the following identity: 3 2tan sec sin sin secθ θ θ θ θ− =
(3 marks)
Left-Hand Side Right-Hand Side
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 27
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14149. A sinusoidal function has a maximum at ( )4, 6 .
The following maximum is at ( )16, 6 .
The range of this function is [ ]12, 6− .
Write an equation to represent this function. (4 marks)
y =
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Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 29
NO MARKS WILL BE AWARDED FOR WORK DONE ON THIS PAGE.
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NO MARKS WILL BE AWARDED FOR WORK DONE ON THIS PAGE.
Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007 Page 31
NO MARKS WILL BE AWARDED FOR WORK DONE ONTHIS PAGE.
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NO MARKS WILL BE AWARDED FOR WORK DONE ONTHIS PAGE.