nse mathematics 12 and advanced mathematics 12 item …hrsbstaff.ednet.ns.ca/matkins/dhs...

NSE Mathematics 12 and Advanced Mathematics 12

Item Release

2009/2010

Table of Contents Letter to educators ........................................................................................... 3

Item description tables ..................................................................................... 5

Quadratics unit items ....................................................................................... 9

Exponential Growth unit items ..................................................................... 26

Circle Geometry unit items ........................................................................... 44

Probability unit items .................................................................................... 53

1

Dear Mathematics Educator,

I am pleased to provide you with the 2009-2010 NSE Mathematics 12/Advanced Mathematics

12 Item Release booklet.

This booklet contains items that have appeared on previous NSE Mathematics and Advanced

Mathematics Examinations that will no longer be used. All Mathematics 12 and Advanced

Mathematics 12 teachers will receive this booklet so the items contained are not to be considered

“secure”. I would however ask you to refrain from posting the booklet in its entire form on the

web.

Accompanying the items is a list which indicates the outcomes that the question aimed to

evaluate as well as the cognitive level at which the question was classified. Please note that

outcomes that are listed with an asterisk beside them (*) are advanced outcomes.

Should you have any questions about the items please contact me by e-mail at

[email protected].

Sincerely,

Lennie Comeau

Mathematics Evaluation Coordinator,

Nova Scotia Department of Education

3

Question # Academic or Advanced? Outcomes Addressed Question Level1 advanced C4 1

2 academic/advanced C8/C9/C23/C31 2

3 academic C8 1

4 academic/advanced C31/C32 1

5 academic C31 2

6 academic/advanced C8 2

7 academic C9 1


9 advanced C32 1

10 advanced C32 2

11 academic A4/C15 2

12 academic C15/C32 1

13 academic C3/C8/C31 2

14 advanced C23 3

15 advanced C10*/C23/C29 3

16 academic C22 1

17 academic A7/C9/C22 1 and 2

18 academic/advanced C23 2

19 advanced C8 3

20 advanced C1/C23 3

21 advanced C8 2

22 academic C23 2

23 academic C22 2

24 advanced A4/C9/C15/C31 1

25 advanced C22 2


27 academic/advanced C23/F1 2


29 academic C8/C9/C23/C31 2

30 advanced A7/C9/C22/C32 1, 2 and 3


32 academic A5/B12 1

33 academic C34 1

5

Question # Academic or Advanced? Outcomes Addressed Question Level34 academic/advanced C24 1

35 academic B13/C24 1







42 academic B13 1

43 academic A5/B12/C24 2


45 advanced B13/C24 1


47 advanced C33 2

48 advanced B13 2

49 advanced A5/B12 1

50 advanced A7 2


52 advanced C33 2

53 advanced B13 2

54 advanced C4/C29 1


56 advanced C33 2

57 academic C19 1

58 academic C19 1


60 advanced C11/C35* 1


62 academic/advanced C2/C11/C25 2

63 academic/advanced A5/B13 3



66 advanced C2/C11/C25 2

6

Question # Academic or Advanced? Outcomes Addressed Question Level67 academic C2/C11/C25 2



70 academic C24 1

71 academic C2/C11/C25 2

72 advanced C24 1



75 academic A5/B12/B13 3


77 academic A5/B13/C4 3

78 academic C4 3


80 advanced A7/E13* 2

81 advanced E14*/E15* 1

82 advanced E8* 1

83 advanced E8* 1

84 academic E4 1

85 academic/advanced D1 2

86 advanced E3* 2

87 academic E12 1

88 academic E4 2

89 academic D1 1

90 academic/advanced E4 1

91 advanced E4/E8* 2


93 advanced E3* 2


95 academic D1 1

96 advanced E11 3

97 advanced E3*/E14*/E15* 2

98 academic D1/E5 2

99 advanced E11 3

7

Question # Academic or Advanced? Outcomes Addressed Question Level100 advanced A7/E14* 2

101 advanced E14* 2

102 academic D1 2

103 academic D1 3

104 advanced G7/G8 1

105 academic G2 2

106 academic/advanced G3 2

107 academic/advanced A6 1

108 advanced A6 1



111 academic A6/G7/G8 1

112 advanced G5* 2

113 academic/advanced G2/G3 1

114 academic/advanced B8/G3 2


116 advanced G8 2

117 academic G7/G8 2





122 academic G7 3


124 academic B8/G2/G7/G8 2


8

1. If a sequence is constant in its second-level differences, , but not in its first, then it is2D

an arithmetic sequence a power sequencea geometric sequence a cubic sequence

A. B.

C. D.

2. The function describes the height of a volleyball h, in metres, as a function24.9 15.7 1h t t= − + +

of time t, in seconds, after it was hit. After how many seconds will the ball be at its maximumheight?

1 1.603.27 13.58

A. B.

C. D.

3. Given the mapping rule , which function is the image of ?( ) ( ), 4, 3 1x y x y→ + − + 2y x=

( )

23 4 1y x= + +

( ) ( )

211 4

3y x− = −

( )

23 4 1y x= − + +

( ) ( )

23 1 4y x− + = +

A. B.

C. D.

4. What is the equation of the axis of symmetry of the function ?23 12 9y x x= − − −

3x = 2x =

2x = − 3x = −

A. B.

C. D.

5. The function has a minimum value of 2 4 5= − +y x x

1 2

5 5−

A. B.

C. D.

6. If a quadratic function has a y-intercept at and a vertex at , which one of the(0, 11)− (4, 3)−

following is the transformational form of the function?

21

2( 3) ( 4)y x− + = −

22( 3) ( 4)y x− + = −

21

2( 3) ( 4)y x− − = +

22( 3) ( 4)y x− − = +

A. B.

C. D.

7. In order to find the y-intercept of

, Eleanor changed the equation to general( ) ( )21

5 32

y x+ = −

form. She should have obtained

22 12 4y x x= − +

22 13y x= +

22 12 13y x x= − +

22 23y x= −

A. B.

C. D.

9

8. A quadratic equation was solved by completing the square, as shown below. Which step containsthe first mathematical error?

Equation: 2 12 64 0x x− − + =

Solution: Step 1

→ ( )2 12 64x x− + = −

Step 2

→ ( )2 12 36 64 36x x− + + = − −

Step 3 → ( )

26 100x + =

Step 4 → 6 10x + =

Step 5 → 4x =

Step 1 Step 2 Step 3 Step 4

A. B.

C. D.

9. If p, q, and s are positive numbers, then which function could be represented by the graph below?

x

y

2y px qx s= − + +

2y px qx s= − + +

2y px qx s= − −

2y px qx s= + −

A. B.

C. D.

10

10. In the diagram below, Path 1 represents the path of a diver from the time she enters the wateruntil she resurfaces. Path 2 represents the path of the diver on her second trial.

y

x

Path 1

Path 2

Her coach graphs the quadratic function

to model the parabolic 21 ) ( )( − = −a y k x h

Path 1 of the dive. The coach then changes certain values in the given equation to model the

parabolic Path 2. Which value(s) did the coach NOT change ?

h h and aa k

A. B.

C. D.

11. Which graph could represent a quadratic function where has no real roots?( ) 0=f x

x

y

x

y

x

y

x

y

A. B.

C. D.

11

12. Which quadratic function has x-intercepts at and 8 and a maximum value?2−

( 2)( 8)= + −y x x ( 2)( 8)= − + −y x x

( 2)( 8)= − +y x x ( 2)( 8)= − − +y x x

A. B.

C. D.

13. The function undergoes the following transformations:2y x=

- a reflection in the x-axis;

- a vertical stretch of 2; - a horizontal translation of 1; - a vertical translation of 8.

Sketch the new parabola and label the vertex, x-intercept(s) and y-intercept. (4 points)

14. If one root of is three times the other, prove that

.

2 0x px q− + = 23 16p q= ( )0, 0p q≠ ≠

(3 points)

12

15. The area of a circle is divided by a number of chords as shown below.

Number of parts: 2 4 7 11

(a) Algebraically, determine a function that models this situation. (4 points)

(b) According to this pattern, how many chords would be needed to divide a circle into 407parts? (2 points)

13

16. Given , use the quadratic formula to solve for the exact value(s) of x. Express26 7 3 5x x+ − = −

your answer(s) in simplest form. (2 points)

17. Given .2 6 16y x x= + −

(a) Express the above function in transformational form. (2.5 points)

(b) What is the domain of the function? (1 point)

(c) What is the range of the function? (1 point)

14

18. For the following problem, you are to set up and solve an equation that represents the situation.

The height of a triangle is 2 m more than the length of its base. If the area of the triangle is

17.5 m2, what is the length of its base?

(4 points)base heightArea of a triangle =

2

×

19. Write two different quadratic functions such that each has x-intercepts at -5 and 4. (2 points)

15

20. A cannonball attains a maximum height of 250 metres when it is accurately fired at a target that isa horizontal distance of 200 metres away at the same vertical height above the ground. When the

cannonball is directly above a landmark that is 20 metres from the target, at what height is thecannonball? Include a sketch of the graph that represents the given information. (5 points)

16

21. A quadratic function has x-intercepts at (–1, 0) and (5, 0). Find the general form of the quadraticfunction if the maximum y-value is 4. (4 points)

17

22. The function describes the height of a baseball h, in metres, as a function of25 20 1h t t= − + +

time t, in seconds, from the instant the ball is hit.

(a) Express this function in transformational form. (3 points)

(b) How long will it take the baseball to reach its maximum height? (2 points)

18

23. Given 2 2 0x x− − =

(a) Solve for x. (2 points)

(b) Solve for x using a different method. (2 points)

19

24. Given the function .22 4 5y x x= + −

(a) Write the equation in standard or transformational form. (2 points)

(b) What is the equation of the axis of symmetry? (1 point)

(c) Determine the nature of the roots for and explain what this tells you about22 4 5 0x x+ − =

the graph of the function

. (2 points)22 4 5y x x= + −

20

25. Given 26 2 0x x− − =

(a) Solve for x. (2 points)

(b) Solve for x using a different method. (2 points)

21

26. A lifeguard must join 3 shoreline ties and 3 anchored buoys (arranged as shown) with singlestrands of rope to form a rectangular swimming area in 2 equal sections (no rope runs along the

shoreline). The lifeguard uses 900 metres of rope in total.

Shoreline

Tie Tie Tie

Buoy Buoy Buoy

x xx

y

��

(a) Express the entire area of the swimming sections as a quadratic function in terms of either x or y. (2 points)

(b) Use the quadratic function obtained in (a) to determine the length and width that will produce

a maximum swimming area. (3 points)

22

27. The initial depth of a submarine is 40m below sea level. In the following graph, x represents time(in minutes) and y represents depth (in metres). The graph shows the distance the submarine rises

over a period of time.

1 2 3 4 5 6

-45

-40

-35

-30

-25

-20

-15

-10

-5

Time (minutes)

De

pth

(m

etr

es)

(1, -36.5)

(5, -12.5)

(4, -20)

(3, -26.5)

(2, -32)

(a) Is the relationship between x and y a quadratic relationship or is it an exponentialrelationship? Explain how you know. (2 points)

(b) Using regression, determine the best function that could be used to calculate the depth of the

submarine at any time during its rise to the surface. (1 point)

(c) When will the submarine reach the surface? (2.5 points)

23

28. Given the following quadratic function,

-2 -1 1 2 3 4 5 6 7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

(a) State the equation of the given parabola in transformational or standard form. (2 points)

(b) What is the equation of the axis of symmetry of the given parabola? (1 point)

29. An object is fired into the air and the function expresses the relationship25 30 1h t t= − + +

between height, h, in metres and time, t, in seconds.

(a) What is the maximum height reached by this object? (3 points)

(b) When will the object be at a height of 30 m above the ground? (3 points)

24

30. Given .23 6 9y x x= − −

(a) Express the above function in transformational form. (2 points)

(b) Find the zeros of the function. (2 points)

(c) State the domain and range. (1 point)

(d) Write a specific and different quadratic function which has the same zeros as the givenfunction. (1 point)

25

31. If the zeros of a quadratic function are 3 and and its y-intercept is 42, determine the function7−

in general form. (3 points)

32. is equal to

1

38−

2−

3

1

8

1

2

1

2−

A. B.

C. D.

33. The equation of the horizontal asymptote of the graph of is5(3) 2xy = −

2y = − 2x = −

2y = 2x =

A. B.

C. D.

34. If

, then x is equal to( )3 2log 10 1x−=

1

3− 1

4

2

3

A. B.

C. D.

26

35. For all values of is equal to

101, log a

aa a>

10

a 10

a

a 10a

A. B.

C. D.

36. What type of function would best model the data in the table below?

x

y 64 32 16 8 4

15− 12− 9− 6− 3−

linear quadraticexponential logarithmic

A. B.

C. D.

37. What is a simplified form of

?3

( ), 0

( )≠

n

n

b bb

b

2

1nb

2

1

b

2 1− +n

b 2 1+n

b

A. B.

C. D.

38. Which of the following is equivalent to

? 1

32

( )

52−

52−

52−

( )

52

−−

A. B.

C. D.

39. Which of the following is equal to

?

3

21

16

1

24−

( )

1

24−

34−

( )

34−

A. B.

C. D.

40. When

, if , then x is equal to ( ) 13x

f x−

= ( ) 6f x =

1.63 32.63 243

A. B.

C. D.

27

41. Simplify

. 4

7

2a

a

32a

3

1

2a

3

2

a

31

2a

A. B.

C. D.

42. The expression is equivalent to5log 25k

2k 2k

25k 25k

A. B.

C. D.

43. Ryan used the following steps to solve the equation

. Which step contains the first3

log 272

x =

mathematical error?

Step 1 →3

2 27x =

Step 2 →3

227x =

Step 3

→ ( )2

3 27x =

Step 4 → 9x =

Step 1 Step 2Step 3 Step 4

A. B.

C. D.

44. John invested $10 000 at 8% interest compounded annually. If A is the amount of money indollars at time t, which function correctly models the growth of his investment?

( )10000 0.08

tA =

( )10000 1.8

tA =

( )10000 1.08

tA =

( )10000 0.92

tA =

A. B.

C. D.

45. If

, then x is equal to( )3 2log 1x

mm

−=

13

− 1

4 23

A. B.

C. D.

46. What is the solution to the equation ?( ) ( )2 2log 1 log 1 3x x− + + =

3x = 3x = −

3x = ± no solution

A. B.

C. D.

28

47. Which of the following statements is true about the variable b in the exponential function

, ?

xy ab= 0a >

If , then the function demonstrates decay; if , then the function demonstrates1b > 0 1b< <

growth.

If , then the function demonstrates growth; if , then the function demonstrates1b > 0 1b< <

decay.If b is positive, then the function demonstrates growth.If b is less than 3, then the function demonstrates decay.

A.

B.

C.

D.

48. Which expression is a simplified form of

?2 2

12 log log 3

2x y+ +

2log 3xy +

2log 2 3

2

yx

+ +

( )2

2log 8x y

( )

12 2

2 2log log 3x y

+

A. B.

C. D.

49. Which of the following has the largest value?

5502

3004

( )40016 2

501024

A. B.

C. D.

50. What is the domain of

? ( )2log 2 1 3y x= − +

{ }| 0x x∈ >�

{ }| 3x x∈ >�

1|

2x x

∈ ≥

�

1|

2x x

∈ >

�

A. B.

C. D.

51. Which is a simplified version of

?( )1

2 3x x−

−+

3

2

1x

x+ x

3

5 1

x

x + 2 3

1

x x−+

A. B.

C. D.

52. For , a vertical stretch of 2 is the same as:2xy =

a horizontal stretch of 2 a vertical translation of 1a horizontal stretch of -2 a horizontal translation of -1

A. B.

C. D.

29

53. If , thenlog log

b ba c d e f− =

log

log

a

b

d

b

cf

e=

( )logb ac de f− =

af

d

cb

e=

( ) log

b

ca d f

e

− =

A. B.

C. D.

54. Which sequence is geometric?

2, 5 2, 8 2 2, ...+ + 2, 4, 16, 128,....

2, 4, 6, 8, .... , 6, , 12, .... 3 2 6 2

A. B.

C. D.

55. Simplify ( )

222 x−

−

44 x+

4

14

x+

2 44 2x x

− −− +

2 4

4 14

x x− +

A. B.

C. D.

56. Which of the following statements is true given the exponential function ?3 xy b=

If , then the function demonstrates decay; if , then the function demonstrates1b > 0 1b< <

growth.

If , then the function demonstrates growth; if , then the function demonstrates1b > 0 1b< <

decay.If b is positive, then the function demonstrates growth.If b is less than 3, then the function demonstrates decay.

A.

B.

C.

D.

57. What is the exponential form of

?log =b

a c

=a

b c =b

c a

=c

b a =b

a c

A. B.

C. D.

58. What is the logarithmic form of ?a

b c=

log

cb a=

log

ba c=

log

ca b=

log

bc a=

A. B.

C. D.

30

59. If , what is the value of x?log10 5=

x

1 12

5 no solution

A. B.

C. D.

60. If the image of

is

then the mapping rule is: 3xy = 2( 5)( 2) 3 ,xy -- + =

12

( , ) ( 5, 2)x y x y→ + − + ( , ) (2 10, 2)x y x y→ − − +

51

2 2( , ) ( , 2)x y x y→ + −

12

( , ) ( 5, 2)x y x y→ + − −

A. B.

C. D.

61. Express the following as a single power with a base of 2. (3 points)

2x 1 x 1 2

x 2

(16 )(2 )

8

− +

−

62. The radioactive isotope rubidium-84 has a half-life of 32.9 days. How long would it take for 800mg of this isotope to decrease to 50 mg of rubidium-84? (4 points)

31

63. (a) Show that

. (2 points)3

5

1log 5

log 3=

(b) Given

where determine one value of a for which this1

loglog

a

b

ba

= 0, 0, and a b a b> > ≠

equation is false. Explain why. (2 points)

32

64. The population of a newly discovered organism can be described by the function ( )10500 3

t

P =

where P is the number of organisms and t is time (in minutes).

(a) In the equation

, what do the '3' and the '10' signify in the context of the( )10500 3t

P =

problem? (2 points)

(b) How long does it take the population to double? (3 points)

33

65. If and

, show that

. (3 points)2logb x n=

4logx b

n=

1

4n =

66. Bismuth 210 is radioactive chemical element. If at the end of 15 days, one eighth of a sampleremains, what is the half-life of Bismuth 210? (Provide units with your answer.)

(3 points)

34

67. Earth's population is increasing by approximately 2% each year. The population today isapproximately 6.51 billion.

(a) State the function that can be used to describe the population, P, of Earth after t years. (2 points)

(b) How long will it take Earth's population to reach 10 billion people? (3 points)

35

68. Every four hours after you take your medicine, the concentration of the medicine in yourbloodstream decreases by 60%. Your prescribed dose is 1 tablet. If the initial concentration from

one tablet is 64 micrograms/ml, how much medication would be left in your bloodstream after 12hours if no further medication is taken? (3.5 points)

36

69. A meteorite has hit Earth and scientists have begun to analyze its composition. To the amazementof the scientists, they have discovered a new element, which they call element-R. Scientists soon

discovered that element-R is radioactive. This graph shows the radioactive decay of element-Rover time.

x5 10 15 20

5

10

15

20

25

30

35

40

Time (days)

Amount (grams)

(0, 36)

(8, 18)

(16, 9)

y

(a) Determine the exponential function that best represents the curve. (2 points)

(b) When will there be only 4.5 grams of element-R left? (2 points)

37

70. Algebraically, solve for x in each of the following equations.

(a)

(2 points)2 1

13

9

x−

=

(b)

(2 points)( )8

2 3 34 2x+=

(c) (2 points)5 5 5log 2 log 7 log 4x = +

38

71. The depreciated value of a car was calculated over 4 years. This data is shown in the table below.

If the rate of depreciation is always the same, after how many years will the car first be worth less

than $1000? (5 points)

After 1 year 2 years 3 years

Value 27360 20794 15803

4 years

12010

39

72. Algebraically, solve for x in each of the following.

(a) (2 points)( )( )2 53 9 27x x+

=

(b) (2 points)2 3 45 7x x− +

=

(c) (2 points)( )23 8 3 15 0

xx− + =

40

73. The number of cell phone users in Atlantic Canada has increased by 15% every year for the past

several years. If this rate continues and there are now 375 000 cell phone users, in how many

years will the number of cell phone users reach 1 000 000? Set up and solve an equation that

represents this situation. (4.5 points)

74. Algebraically, solve for x. (3 points)

5

1log 5 2

logx

x+ =

41

75. Explain why

given that and . (2 points)log 1 log 1a b

= 0a ≠ 0b ≠

76. Explain, in words, why is not a growth function for all positive values of b when . xy ab= 0a >

(2 points)

42

77. Using the laws of logarithms and the definition of an arithmetic sequence, show that is an

arithmetic sequence. (4 points)( ) ( ) ( ) ( )2 3log , log , log , loga ab ab ab

78. Show that if is a geometric sequence, then

. (3 points)3 ,3 ,3a b c

2

a cb

+=

43

79. The number of cell phone users in a city has increased by 10% every year for the past several

years. With this constant rate, if there are 500 000 cell phone users today, in how many years will

there be 800 000 cell phone users?

Set up and solve an equation(s) that represents this situation. (5 points)

80. The range of the ellipse

is

2 2

19 5

x y + =

{ }9 y 9y ∈ − ≤ ≤�

{ }9 y 5y ∈ − ≤ ≤�

{ }5 y 9y ∈ − ≤ ≤�

{ }5 y 5y ∈ − ≤ ≤�

A. B.

C. D.

81. If the equation of an ellipse is , then the centre of the ellipse is2 23 12 10 0x y x y+ − + =

(6, 5)− ( 6, 5)−

(2, 5)− ( 2, 5)−

A. B.

C. D.

44

82. C is the centre of the circle.

The value of x is

90o 45o

55o 35oA. B.

C. D.

83. Calculate x given that C is the centre of the circle below.

C

32o

x

o122

o64

o116

o128

A. B.

C. D.

84. If a chord in a circle measures 12 units and the radius of the circle measures 12 units, how far is

the chord from the centre?

12 units units6 3

6 units units6 5

A. B.

C. D.

85. has a midpoint at (m, 0) and the coordinates of P are (0, p). The coordinates of Q arePQ

(2m, 2p)

,

2 2

m p

(–m, –p) (2m, –p)

A. B.

C. D.

45

86. C is the centre of the ellipse drawn below.

x

y

1

1

C (3, 2)

Which equation best describes the graph?

( ) ( )

2 21 1

3 2 12 3

x y

− + − = ( ) ( )

2 21 1

3 2 13 2

x y

− + − =

( ) ( )

2 21 1

2 3 12 3

x y

− + − = ( ) ( )

2 21 1

2 3 13 2

x y

− + − =

A. B.

C. D.

87. Which of the following statements has a converse that is true?

If a quadrilateral is a square, then it has 4 right angles.

If a chord goes through the centre of a circle, then the chord is a diameter of the circle.

If you live in Halifax, then you live in Nova Scotia.

If it rains, then the grass is wet.

A.

B.

C.

D.

88. What is the length of the radius in the circle below, to the nearest hundredth, if the distance from

the centre of the circle to the chord is 2.20 cm ?AB

A

B

3.10 cm

2.69 cm 3.80 cm

5.38 cm 6.20 cmA. B.

C. D.

89. Given with A(2, –3) and B(–5, 1), what are the coordinates of the midpoint of segmentAB

?AB

(–1.5, –1) (–1, –4)

(3.5, –2) (9, –7)A. B.

C. D.

46

90. Given the diagram below.

A

D

C

B

Which of the following statements is TRUE?

If chord bisects chord , then chord goes through the centre of the circle.AB CD AB

If chord is perpendicular to chord , then chord goes through the centre of theAB CD AB

circle.

If chord is a perpendicular bisector of chord , then chord goes through theAB CD AB

centre of the circle.

If chord goes through the centre of the circle, then it bisects chord .AB CD

A.

B.

C.

D.

91. C is the centre and is a diameter of the circle below.AD

C

A

BD

E

F

Which of the following statements is FALSE?

ABD DEA∠ ≅ ∠ AFC AEC∠ ≅ ∠

( )

1m m

2ECF EAF∠ = ∠

( )2 m mDAE DCE∠ = ∠

A. B.

C. D.

92. Which of the following statements has a converse that is true?

If a number is divisible by 2 and by 3, then it is divisible by 6.

If you live in Nova Scotia, then you live in Canada.

If a quadrilateral is a square, then it has congruent diagonals.

If an animal is a dog, then it is a 4-legged animal.

A.

B.

C.

D.

47

93. Sketch the graph of the ellipse represented by the equation

and indicate( )( )

2

2 12 1

3

yx

− − + =

the coordinates of the centre and the endpoints of both axes. (3

points)

1

1

-1-1

x

y

94. State the converse of each of the following and indicate whether or not the converse is true.

(a) If you are running too fast, then you become exhausted. (1 point)

(b) If two chords of a circle are congruent, then they are equidistant from the centre of the circle.

(1 point)

(c) If a quadrilateral is a square, then it has four right angles. (1 point)

48

95. Given: with A(–3, 11), B(5, 9) and C(1, –3). Show, using relevant calculations, that theABC∆

segment connecting the midpoints of and is half the length of . (5 points)AB BC AC

96. Given : with parallel to ABC� PQ AB

Prove : The sum of the three angles in the triangle ABC is (5 points)180°

A

QP

C

B

Statement(s) Reason(s)

49

97. Determine the coordinates of the centre and lengths of the major and minor axes of the ellipse

. (4 points)2 29 25 36 50 164 0x y x y+ − − − =

98. Show algebraically and state why the triangle with vertices X(1, 1), Y(2, 6), Z(7, 5) is isosceles.

(4 points)

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

X(1, 1)

Z(7, 5)

Y(2, 6)

x

y

50

99. In the figure below, E, D, and C are collinear and A, B, and C are collinear:

(4 points)

ABC

FD

E Statement(s) Reason(s)

Given:

CD CB

DE BA

≅

≅

Prove: E A∠ ≅∠

100. State the domain and range for

. (2 points)( ) ( )2 2

16 3 4 1 64x y+ + − =

51

101. Given the equation . Write the equation in transformational form.2 24 9 16 90 205 0x y x y+ − + + =

(4 points)

102. Line segment has endpoints atA(2, 11) and B(8, –7). Line segment has endpoints at AB CD

C(9, 8) and D(–1, –9). Determine whether or not the line segments and bisect eachAB CD

other. (3 points)

52

103. Given: with A(–3, 11), B(5, 9) and C(1, –3). Show, using relevant calculations, that theABC∆

segment connecting the midpoints of and is half as long as . AB BC AC

(5 points)

104. A bag contains 9 marbles, 4 of which are red. What expression represents the probability of

selecting exactly two red marbles when three marbles are drawn at random?

4 1 5 1

9 3

C C

C

×

4 2 5 1

9 3

C C

C

×

4 3 5 0

9 3

C C

C

×

4 2

9 3

C

C

A. B.

C. D.

105. If event X is "randomly choosing an ace from a standard deck of 52 playing cards", then is( )P X

equal to

1

52

4

52

48

52

51

52

A. B.

C. D.

106. If the probability of winning game A is 0.20 and the probability of winning game B is 0.35, what

is the probability of winning game A or game B?

0.07 0.15

0.48 0.55 A. B.

C. D.53

107. The exact value of

is100!

98!

50

492

9 900 undefined

A. B.

C. D.

108. The expression

can be simplified to( )

( )

( )

1 !!

2 ! 1 !

aa

a a

−×

− +

1

2a −

1

1

a

a

−

+

a 2

a

a −

A. B.

C. D.

109. There are 5 red, 4 blue, and 3 black marbles in a bag. Three marbles are chosen without

replacement. What is the probability of selecting a red one, then a blue one, then another red one?

1

4

4

11

1

10

1

1 0

2

33

A. B.

C. D.

110. An electronic lock on a door has 6 buttons numbered 1–6. The door will open only when a

certain 4-digit sequence is entered. The digits in the sequence must all be different. What is the

probability that a random sequence of 4 different digits will open the door?

6 5 4 3× × × 6 4

1

C

6 4

1

P

4

6!

A. B.

C. D.

111. In how many different ways can the seven letters in the word "clothes" be rearranged without

repetition of letters?

7 7P 7 7

C

77 7! 1−

A. B.

C. D.

54

112. A card is drawn from a standard deck of 52 playing cards. If you know it is a face card, what is

the probability that it is a King?

1

3

1

13

1

4

3

13

A. B.

C. D.

113. A fair eight-sided die is rolled. The faces of the die are numbered 1 through 8. On any given roll,

what is the probability of rolling a 1 or a number larger than 6?

3

8

2

64

2

8

3

16

A. B.

C. D.

114. There are 10 red marbles and 20 blue marbles in a bag. Two marbles are randomly removed from

the bag at the same time. What is the probability that both are blue?

2 2

20 2

C

C

2

20

20 2

30 2

C

C

20

30

A. B.

C. D.

115. If a letter is picked at random, from the English alphabet, what is the probability that the letter is

in the word MATH and in the word EXAM?

1

2

1

13

2

13

3

13

A. B.

C. D.

55

116. At High School XY, the Math Club has 7 student members.

(a) Three of the seven students from the Math Club will be randomly selected to attend a

provincial mathematics competition. In how many ways can these three students be chosen?

(2 points)

(b) An executive of president, secretary, and treasurer will be selected from this Math Club. In

how many ways can this executive be created? (2 points)

(c) Determine the value of k for which the equation is true. Support your answer.( ) n r n rP k C=

(2 points)

56

117. A group of students consists of 10 boys and 8 girls. Four students will be randomly chosen to

help organize an upcoming event at the school.

(a) What is the probability that Billy, Sue, Karla, and Derek will be chosen in that

specific order? (2 points)

(b) What is the probability that four boys will be chosen if the order does not matter?

(3 points)

118. From 5 men and 3 women, a committee of 4 is to be chosen. What is the probability that the

committee will contain

(a) exactly 2 women? (2 points)

(b) at least 2 women? (2 points)

57

119. If a bag holds 3 green, 5 red, and 7 blue marbles, what is the probability of randomly selecting 4

blue marbles? (2.5 points)

120. A box contains 50 white balls, 110 green balls, and 40 red balls. Suppose that 2 balls are selected

from the box without replacement.

(a) In how many ways can 2 red balls be chosen? (1.5 points)

(b) In how many ways can 2 balls be chosen? (1.5 points)

(c) What is the probability that 2 red balls will be chosen? (1 point)

58

121. Three students are selected from a Math Club to be the president, the secretary, and the

treasurer.

(a) In how many ways can this group be selected if there are 20 people in the club?

(1.5 points)

(b) What is the probability that Beth will be president, Kyle will be secretary, and Dawn will be

treasurer? (1.5 points)

122. (a) Explain the meaning of . (2 points)7 3P

(b) Why does not make sense? (1 point)3 7P

59

123. An internet company generates passwords composed of 6 letters with no letter repeated. Ray has

to determine the total number of possible passwords. Explain why he should use permutations

and not combinations. (3 points)

124. (a) Four students are selected from the student council to form a committee. If there are 20

students on the student council, how many different committees of four students are possible?

(1.5 points)

(b) A meeting will be held to randomly select a four-member executive from 20 student

council members. What is the exact probability that council members LeBlanc, Smith,

Brown, and Legere will be assigned the positions of president, vice-president, secretary,

and treasurer respectively? (2 points)

60

125. Graeme rolls a fair 8-sided die.

(a) What is the probability that he rolls an '8' on the first roll and an odd number on the second

roll? (2 points)

(b) What is the probability that he rolls an '8' on the first roll or the second roll? (3 points)

60

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