Download June 2007

PART I
Total Value: 50%
Answer all items. Shade the letter of the correct answer on the computer scorable answer
sheet.
1.
What is the range of the quadratic function y − 5 = ( x + 3) 2 ?
(A)
(B)
(C)
(D)
2
{ y y ≤ −5, y ∈ R}
{ y y ≥ −5, y ∈ R}
{ y y ≤ 5, y ∈ R}
{ y y ≥ 5, y ∈ R}
Which graph represents the transformation of y = x 2 under the mapping rule
( x, y ) → ( x + 1, − y ) ?
y
(A)
8
6
4
2
– 8 – 6 – 4 – 2
– 2
2
4
6
8 x
2
4
6
8 x
2
4
6
8 x
2
4
6
8 x
– 4
– 6
– 8
y
(B)
8
6
4
2
– 8 – 6 – 4 – 2
– 2
– 4
– 6
– 8
(C)
y
8
6
4
2
– 8 – 6 – 4 – 2
– 2
– 4
– 6
– 8
y
(D)
8
6
4
2
– 8 – 6 – 4 – 2
– 2
– 4
– 6
– 8
Page 1 of 16
Mathematics 3204 June 2007
3.
The graph of y = x 2 has been reflected in the x-axis, translated 3 units to the right, and
translated 2 units up, resulting in the graph of g ( x) . What is the nature of the roots of
g ( x) = 0 ?
(A)
(B)
(C)
(D)
4.
Which sequence below is arithmetic?
(A)
(B)
(C)
(D)
5.
imaginary and equal
imaginary and unequal
real and equal
real and unequal
{3−2(1), 3−2(2), 3−2(3), 3−2(4),…}
{3−1, 3+1, 3−2, 3+2,…}
{31, 32, 33, 34, …}
{3+2, 3+22, 3+23, 3+24, …}
A quadratic function f ( x ) has 1 ± 3i as its zeros. Which is the graph of f ( x ) ?
(A)
y
x
y
(B)
x
(C)
y
x
(D)
y
x
6.
What mapping rule was applied to y = x 2 to result in the quadratic function
3( y + 1) = ( x − 2 ) ?
2
(A)
(B)
(C)
(D)
( x, y ) → ( x − 2, 13 y + 1)
( x, y ) → ( x − 2,3 y + 1)
( x, y ) → ( x + 2, 13 y − 1)
( x, y ) → ( x + 2,3 y − 1)
Mathematics 3204 June 2007
Page 2 of 16
7.
What is the second level difference for the sequence {−1, −2, −7, −16, −29,...} ?
(A)
(B)
(C)
(D)
8.
What is the transformational form of y = 3 ( x − 2 ) + 5 ?
2
(B)
( y − 5) = ( x − 2 )
1 y+5 = x+2 2
) (
)
3(
(C)
3 ( y − 5) = ( x − 2 )
(D)
3 ( y + 5) = ( x + 2 )
(A)
9.
2
1
3
2
2
If f ( x) = 2 x 2 − 5 x − 12 , what are the roots of f ( x) = 0 ?
(A)
(B)
(C)
(D)
10.
−4
−1
1
4
{− 32 , − 4}
{− 32 , 4}
{ 32 , − 4}
{ 32 , 4}
Which table of values shows a quadratic relationship?
(A)
(B)
n
1
2
3
tn
7 11 15 19 23
n
1 2
tn
2 5 10 17 26
3
4
4
5
5
(C)
n 1 2 3 4 5
tn 1 3 9 27 81
(D)
n 1 2 3 4 5
tn 1 1 2 3 5
11.
What is the general form of y = 2( x + 3) 2 + 1 ?
(A)
(B)
(C)
(D)
12.
y = 2 x 2 + 19
y = 2 x 2 + 20
y = 2 x 2 + 12 x + 19
y = 2 x 2 + 12 x + 20
Which quadratic equation has imaginary roots?
(A)
(B)
(C)
(D)
x2 − 9 = 0
x2 + 4 x + 3 = 0
x2 − 3 = 0
x2 + 4 = 0
Page 3 of 16
Mathematics 3204 June 2007
13.
Which quadratic function has zeros of
(A)
(B)
(C)
(D)
14.
y = 2x2 − 7 x + 3
y = 2 x2 − 5x − 3
y = 2 x2 + 5x − 3
y = 2x2 + 7 x + 3
13
37
13
37
1
1
3
6
Given tn = n −
(A)
(B)
(C)
(D)
16.
and −3 ?
What is the discriminant of the quadratic equation 3x 2 − 5 x − 1 = 0 ?
(A)
(B)
(C)
(D)
15.
1
2
, what is t3 ?
5
6
17
6
19
6
17
2
The graph below shows the sales of a company over an eight month period. Between
what dates was there a negative average rate of change in sales?
Sales ($)
(A)
(B)
(C)
(D)
Jan. 1 → Mar. 1
Mar. 1 → May 1
May 1 → July 1
July 1 → Sept. 1
Jan. 1
Mar. 1
July 1
May 1
Sept. 1
Months
17.
The table below shows the height, in meters, of a dropped ball over time, in seconds.
Based on this data, what is the best approximation for the instantaneous rate of change, in
m/s, at 4 seconds?
(A)
(B)
(C)
(D)
18.
−39.2
−0.026
0
21.6
t
3.98
3.99 4.00 4.01
4.02
h(t ) 22.382 21.992 21.6 21.208 20.814
What type of function is illustrated by the graph shown?
y
(3,27)
(2,16)
(A)
(B)
(C)
(D)
(1,7)
cubic
exponential
linear
quadratic
Mathematics 3204 June 2007
(0,0)
Page 4 of 16
x
19.
Which graph represents the value of a car depreciating by 18% annually?
(A)
y
Dollar
Value
(B)
y
Years
x
Dollar
Value
(C)
Years
x
y
Dollar
Value
Years
(D)
x
y
Dollar
Value
Years
20.
Which represents a geometric sequence?
(A)
(B)
(C)
(D)
21.
(B)
1
12
1
3
(C)
(D)
3
4
What is the inverse of y = log 9 x ?
(A)
(B)
(C)
(D)
23.
⎧1 5 7 ⎫
⎨ ,1, , ,...⎬
⎩3 3 3 ⎭
⎧ 16 32 64 ⎫
⎨8, , , ,...⎬
⎩ 3 9 27 ⎭
⎧ 2 8 32 ⎫
⎨ , , 6, ,...⎬
3
⎩3 3
⎭
128 ⎫
⎧ 2 16
,...⎬
⎨ , ,18,
3
⎩3 3
⎭
Solve: log 5 6 − log 5 x = log 5 2 .
(A)
22.
x
x = y9
x = 9y
y = x9
y = 9x
Which describes the function y = 3 ( 2 ) + 4 ?
x
(A)
(B)
(C)
(D)
exponential decay, asymptote y = 4
exponential decay, asymptote y = − 4
exponential growth, asymptote y = 4
exponential growth, asymptote y = − 4
Page 5 of 16
Mathematics 3204 June 2007
24.
Which equation best represents the graph shown?
y
25.
(A)
y = ( 23 )
(B)
y = ( 32 )
(C)
y = 2 ( 23 )
(D)
y = 2 ( 32 )
x
– 8
x
– 6
– 4
– 2
– 2
– 4
– 6
– 8
– 10
2
−2
24
0
6
2
4
6
3
2
3
8
3
32
x
4
6
8
x
What is the y-intercept of y = 2(3) x + 1 ?
(A)
(B)
(C)
(D)
26.
10
8
6
4
2
x
(0, 2)
(0, 3)
(0, 4)
(0, 7)
Which function describes the data shown?
x
y
x
(A)
y = 6 ( 14 ) 2
(B)
y = 6 ( 14 )
(C)
y = 24 ( 14 )
(D)
y = 24 ( 14 ) 2
x
x
x
27.
28.
Which is equivalent to
(A)
(B)
(C)
(D)
3 − 4 x −1
3 4 x+1
3 −3 x −1
3 3 x+1
Solve:
3
(A)
(C)
1
6
2
3
3
2
(D)
6
(B)
29.
( 27 ) ( 3)
2x
?
7 = 49 x .
Given 2(5) x = 30 , what is the approximate value of x ?
(A)
(B)
(C)
(D)
30.
9x
0.5
1.5
1.7
3.0
The temperature, T, in degrees Celsius, inside a house at the time of a power failure cools
t
exponentially according to the function T = 15 ( 12 ) 30 + 5 , where t is time in minutes.
What is the temperature inside the house, in degrees Celsius, at the instant of the power
failure?
(A)
(B)
(C)
(D)
5
10
15
20
Mathematics 3204 June 2007
Page 6 of 16
−1
31.
32.
⎡⎛ 3 ⎞ −1
⎤
Evaluate: ⎢⎜ ⎟ − 60 ⎥ .
⎢⎣⎝ 4 ⎠
⎥⎦
(A)
(B)
−4
− 14
(C)
1
3
(D)
3
Solve: log 2 8 + log 2 4 − log 2 2 = x .
(A)
(B)
(C)
(D)
33.
Solve: 3 x+1 = 5 .
(A)
(B)
(C)
(D)
34.
3
4
5
6
log 3
log 5
log 3
log 5
log 5
log 3
log 5
log 3
−1
+1
−1
+1
A baseball card valued at $100 appreciates 8% every 3 years. If this appreciation rate
continues, which function represents the value of the card, V, after t years?
t
3
(A)
V = 100(1.08)
(B)
V = 100(1.08)3t
(C)
V = 100(1.8) 3
(D)
V = 100(1.8)3t
t
35.
−5
Evaluate: ( 9 ) 2 .
(B)
1
243
1
15
(C)
(D)
15
243
(A)
x
36.
Which situation can be modeled by y = 11(0.5) 4 ?
(A)
(B)
(C)
(D)
4 g of a radioactive substance has a half-life of 11 years.
4 g of a radioactive substance increases by 50% every 11 years.
11 g of a radioactive substance has a half-life of 4 years.
11 g of a radioactive substance increases by 50% every 4 years.
Page 7 of 16
Mathematics 3204 June 2007
37.
What is the equation of the graph shown?
y
6
4
38.
(A)
⎡⎣ 14 ( x − 2 ) ⎤⎦ + ⎡⎣ 12 ( y + 3) ⎤⎦ = 1
(B)
⎡⎣ 14 ( x + 2 ) ⎤⎦ + ⎡⎣ 12 ( y − 3) ⎤⎦ = 1
(C)
⎡⎣ 4 ( x − 2 ) ⎤⎦ + ⎡⎣ 2 ( y + 3) ⎤⎦ = 1
– 2
(D)
⎡⎣ 4 ( x + 2 ) ⎤⎦ + ⎡⎣ 2 ( y − 3) ⎤⎦ = 1
– 4
(C)
(D)
2
2
2
2
– 6
– 4
– 2
2
2
2
4
x
6
– 6
If a quadrilateral is cyclic, then the vertices of that quadrilateral lie on a circle.
If a quadrilateral is not cyclic, then the vertices of that quadrilateral do not lie on a
circle.
If the vertices of a quadrilateral do not lie on a circle, then that quadrilateral is not
cyclic.
If the vertices of a quadrilateral lie on a circle, then that quadrilateral is not cyclic.
Which equation represents a circle?
(A)
(B)
(C)
(D)
40.
2
2
Which is the converse of, “If the vertices of a quadrilateral lie on a circle, then that
quadrilateral is cyclic.”?
(A)
(B)
39.
2
9 x 2 − 4 y 2 − 9 x − 4 y − 25 = 0
9 x 2 + 4 y 2 + 9 x + 4 y − 36 = 0
10 x 2 − 10 y 2 − x + 8 y − 121 = 0
10 x 2 + 10 y 2 − 3 x + y − 64 = 0
p is 68°. What is the
In the circle with centre O shown, the measure of minor arc BC
measure, in degrees, of ∠BOD ?
B
68°
(A)
(B)
(C)
(D)
102
112
136
146
O
D
C
A
41.
If a point P (1, 0) is rotated 120° from standard position on a unit circle, what are the new
coordinates of P ?
(A)
(B)
(C)
(D)
(−
(−
(−
(−
3
1
2 ,− 2
3 1
2 ,2
)
1 ,− 3
2
2
1, 3
2 2
)
Mathematics 3204 June 2007
)
)
Page 8 of 16
42.
In the circle with centre O shown, PA = 4 and PO = 2 . What is the length of the
diameter of the circle?
P
B
(A)
(B)
(C)
(D)
43.
44.
13
2 5
2 13
4 5
4
A
2
O
Given point P rotated through an angle θ from standard position as shown, which
equation is true?
y
(A)
5
sin θ = − 13
(B)
5
sin θ = − 12
(C)
5
cos θ = − 13
(D)
5
cos θ = − 12
θ
x
P(-5,-12)
In the circle with centre O shown, what is the length of YZ if YX is tangent to the circle
at X?
O
(A)
(B)
(C)
(D)
16
18
23
25
7
Y
X
45.
24
What is the length of the minor axis of the ellipse given by 9( x − 3) 2 + 16( y − 2) 2 = 144 ?
(A)
(B)
(C)
(D)
46.
Z
3
4
6
8
A circle with centre C (1, 6) has a diameter with endpoint (3, 9). What are the
coordinates of the other endpoint of this diameter?
(A)
(B)
(C)
(D)
(−1, 3)
(2, 7.5)
(5, 12)
(9, −7)
Page 9 of 16
Mathematics 3204 June 2007
47.
What is the approximate area, in square units, of the shaded region in the circle with
centre O shown?
O
(A)
(B)
(C)
(D)
48.
48.87
69.81
244.35
314.16
80° 10
p is 140°. What is the
In the circle with centre O shown, the measure of minor arc HJ
measure, in degrees, of ∠GHJ ?
H
x°
(A)
(B)
(C)
(D)
49.
O
J
G
What mapping rule was applied to x 2 + y 2 = 1 to result in
(A)
(B)
(C)
(D)
50.
35
40
70
80
140°
1 ( x − 4) 2
9
1 ( y + 2) 2 = 1 ?
+ 25
( x, y ) → ( 3x − 4,5 y + 2 )
( x, y ) → ( 3x + 4,5 y − 2 )
( x, y ) → ( 9 x − 4, 25 y − 2 )
( x, y ) → ( 9 x + 4, 25 y + 2 )
In the circle with centre O shown, what is the value of x , in degrees?
B
(A)
(B)
(C)
(D)
O
80
90
100
120
C
160°
A
x°
D
Mathematics 3204 June 2007
Page 10 of 16
PART II
Total Value: 50%
Answer ALL items in the space provided. Show ALL workings.
Value
4
51.
Algebraically determine the exact roots in simplest form for
(3x − 2)( x + 4) = −13 .
4
52.
A baseball is hit and follows a parabolic path described by the function
h(t ) = −3t 2 + 12t + 1 , where t is time in seconds after the ball is hit and h(t) is
the height of the ball above the ground in metres. Algebraically determine the
maximum height reached by the ball and the time it takes the ball to reach its
maximum height.
Page 11 of 16
Mathematics 3204 June 2007
Value
4
53.
A rectangular swimming pool has length 30 m and width 20 m. There is a deck of
uniform width surrounding the pool. The area of the pool is the same as the area
of the deck. Write a quadratic equation to model this situation and use it to
determine the width of the deck.
x
30 m
20 m
x
x
x
4
54.
The height of a pebble fired by a sling shot is given by h(t ) = 25t − 4.9t 2 , where
h(t) is height in metres and t is time in seconds after the pebble leaves the sling
shot. Determine an approximation for the instantaneous rate of change in the
height of the pebble at 3 seconds and describe how the height of the pebble is
changing at that instant.
Mathematics 3204 June 2007
Page 12 of 16
Value
(
3
162 x
) = 64 .
1
2
55.
Solve for x:
4
56.
Solve for x: log 2 ( x − 6) + log 2 ( x) = 4 .
4
57.
Tom bought a car for $14 000. The value of the car depreciates at a rate of 21%
every two years. Write an equation that models the value of the car over time and
use it to determine the approximate value of the car after five years.
3
Page 13 of 16
Mathematics 3204 June 2007
Value
4
58.
A laboratory assistant decided to observe the reproductive properties of a new
strain of bacteria. The assistant started observing a population of 300 bacteria and
noted that the bacteria population doubled every 5 minutes. Write a function to
model this situation and use it to determine the time it will take for the population
to reach 18 000 bacteria.
4
59.
Find the centre of the circle defined by x 2 + y 2 + 10 x − 16 y − 11 = 0 and prove that
it lies on the line 3x − 4 y + 47 = 0 .
3
60.
Write 4 x 2 + y 2 − 8 x + 4 y − 8 = 0 in transformational form and state the coordinates
of the centre.
Mathematics 3204 June 2007
Page 14 of 16
Value
61.
4
Given the square ABCD with vertices A ( −1, − 1) , B ( 3, 2 ) , C ( 6, − 2 ) and
D ( 2, − 5 ) , prove that the diagonals are congruent.
y
B (3,2)
x
A (-1,-1)
C (6,-2)
D (2,-5)
4
62.
The circle shown has centre O and radius 5 cm. If AD is tangent to the circle,
q = 293° , and CD = 8 cm, determine the area of the shaded region.
ABC
A
5 cm
B
Page 15 of 16
O
C
8 cm
Mathematics 3204 June 2007
D
Value
4
63.
AB is a chord with midpoint M (6, 5) in the circle having centre O (5, 8) as
shown. Determine the equation of the line containing chord AB .
O (5,8)
M (6,5)
A
Mathematics 3204 June 2007
Page 16 of 16
B