Grade 12 Pre-Calculus Mathematics (40S) Standards Test

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.

Student Booklet (Part 2)—Pre-Calculus Mathematics (40S)—January 2007

NO MARKS WILL BE AWARDED FOR WORK DONE ONTHIS PAGE.


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


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


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


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!


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


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.


DO NOT USE

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)


DO NOT USE

116

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)


DO NOT USE

119

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)


DO NOT USE

122

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)


DO NOT USE

125

126

37. Evaluate:

3

4

log 27log 16

(1 mark)

38. Solve:15 5x− =

(1 mark)


DO NOT USE

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


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.


DO NOT USE

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)


DO NOT USE

130

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


DO NOT USE

132

133

43. a) Find the exact value of 5tan12π .

(3 marks)

b) Find the exact value of 5cot12π .

(1 mark)


DO NOT USE

134

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


DO NOT USE

136

45. The function ( )y f x= is shown below:

(2 marks)

Sketch the graph of 1( )

yf x

= .

y

1

1x

y

1

1x


DO NOT USE

137

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


DO NOT USE

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


DO NOT USE

14048. Prove the following identity: 3 2tan sec sin sin secθ θ θ θ θ− =

(3 marks)

Left-Hand Side Right-Hand Side


DO NOT USE

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 =


NO MARKS WILL BE AWARDED FOR WORK DONE ON THIS PAGE.

Grade 12 Pre-Calculus Mathematics (40S) Standards Test

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