PART I
Total Value: 50%
Answer all items. Shade the letter of the correct answer on the computer scorable answer
sheet.
1.
2.
Which will produce the same graph as −2( y − 3) = ( x + 5) 2 ?
(A)
(B)
y = −2( x + 5)2 − 3
(C)
y = − 12 ( x + 5) 2 − 3
(D)
y = − 12 ( x + 5) 2 + 3
y = −2( x + 5)2 + 3
Which function represents the transformation of y = x 2 under the mapping rule
( x, y ) → ( x − 3, − 13 y + 4) ?
3.
(A)
y = −3( x + 3) 2 + 4
(B)
y = − 13 ( x + 3)2 + 4
(C)
y = 13 ( x + 3) 2 − 4
(D)
y = 3( x + 3) 2 − 4
Which represents a sequence with a first level difference of − 12 ?
(A)
y
5
4
3
2
1
(B)
1
2
3
4
5
x
1
2
3
4
5
x
1
2
3
4
5
x
1
2
3
4
5
x
y
5
4
3
2
1
(C)
y
5
4
3
2
1
(D)
y
5
4
3
2
1
Page 1 of 16
Mathematics 3204 June 2009
4.
5.
Which has the greatest vertical stretch when compared to y = x 2 ?
(A)
(B)
− 4 y = ( x − 2)2
(C)
− 12 y = ( x − 2)2
(D)
− 14 y = ( x − 2)2
What is the range of the function y = 3( x − 4)2 + 5 ?
(A)
(B)
(C)
(D)
6.
{ y y ≤ −5, y ∈ R}
{ y y ≤ 5, y ∈ R}
{ y y ≥ −5, y ∈ R}
{ y y ≥ 5, y ∈ R}
What is the axis of symmetry for the quadratic function y − 4 = ( x + 2)2 ?
(A)
(B)
(C)
(D)
7.
− 2 y = ( x − 2)2
x = −4
x = −2
x=2
x=4
Which is the graph of a quadratic function with a negative leading coefficient and a
negative discriminant?
(A)
y
x
(B)
y
x
(C)
y
x
(D)
y
x
Mathematics 3204 June 2009
Page 2 of 16
8.
What is true of the roots of a quadratic function ax 2 + bx + c = 0 if b 2 − 4ac > 0 ?
(A)
(B)
(C)
(D)
9.
10.
What is the simplest form of
(A)
±i 6
(B)
±3i 6
(C)
1± i 6
(D)
1 ± 3i 6
12.
13.
3 ± −54
?
3
What is the transformational form of y = 4 x 2 + 8 x ?
(A)
− 14 y = ( x + 1) 2
(B)
− 14 ( y + 4) = ( x + 1) 2
(C)
1 y = ( x + 1) 2
4
1 ( y + 4) = ( x + 1) 2
4
(D)
11.
imaginary and equal
imaginary and unequal
real and equal
real and unequal
What is tn for the sequence
(A)
tn = 23 n + 11
3
(B)
tn = 23 n + 4
(C)
tn = 23 n + 14
3
(D)
tn = 23 n + 6
{143 , 163 , 6, 203 , 223 } ?
Which represents a quadratic relationship?
(A)
x
y
1
–3
2
–5
3
–8
4
–13
(B)
x
y
1
–2
2
0
3
2
4
4
(C)
x
y
1
0.25
2
0.5
3
1
4
2
(D)
x
y
1
10
2
4
3
2
4
4
y
Which equation describes the graph provided?
5
4
−2 ( y − 2 ) = ( x + 1)
2
(B)
−2 ( y + 2 ) = ( x + 1)
2
(C)
− 12 ( y − 2 ) = ( x + 1)
2
(D)
− 12 ( y + 2 ) = ( x + 1)
2
(A)
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
-2
-3
-4
-5
Page 3 of 16
Mathematics 3204 June 2009
14.
What value of “c” makes x 2 + 14 x + c a perfect square?
(A)
(B)
(C)
(D)
15.
7
2
49
4
49
196
What is the value of D2 for the sequence represented by the graph provided?
y
(1,2)
x
(2,-1)
(A)
(B)
(C)
(D)
−4
−2
2
4
(3,-8)
(4,-19)
16.
What are the roots for the function 2 x 2 + 2 = 0 ?
(A)
(B)
(C)
(D)
17.
A lifeguard has 100 m of rope to enclose a rectangular swimming area. Which equation
represents the maximum area of the enclosure if the lifeguard uses the beach as one side
and the rope for the other three sides?
(A)
(B)
(C)
(D)
18.
− 1, −2
− 1,1
− i, i
i , −2
A = w(50 − w)
A = w(50 − 2 w)
A = w(100 − w)
A = w(100 − 2 w)
What rate of change is represented by the graph provided?
y
P
(A)
(B)
(C)
(D)
19.
negative average
negative instantaneous
positive average
positive instantaneous
x
A student invests $100 at 10% interest compounded annually. Using the data below,
what is the average rate of change in the value of the investment between year 4 and
year 8?
Year
0
2
4
6
8
Value 100 121 146.41 177.16 214.36
(A)
(B)
(C)
(D)
16.99
33.98
67.95
90.19
Mathematics 3204 June 2009
Page 4 of 16
20.
Which represents an exponential relationship?
(A)
x
y
1
3
2
–1
3
− 13
4
(B)
x
y
1
3
2
0
3
–3
4
–6
(C)
x
y
1
3
2
1
3
4
1
3
1
9
x
y
1
3
2
12
3
27
4
48
(D)
21.
How many blocks are required in figure 4 to make a geometric sequence?
(A)
(B)
(C)
(D)
22.
Figure 1
Figure 2
Figure 3
Figure 4
−2
(B)
(C)
(D)
1
16
16
25
25
16
16
1
Which is equivalent to
(A)
(B)
(C)
(D)
24.
2
4
8
16
Evaluate ( 30 + 14 ) .
(A)
23.
1
9
( 811 )
x+3
?
3−4 x −12
3−4 x −3
3−4 x +3
3−4 x +12
Which function describes the data provided?
x
4
(A)
y = 16 ( 12 )
(B)
y = 16 (2) 4
(C)
y = 13 ( 12 ) 4
(D)
y = 13 (2) 4
x −4 0
y 13 16
4
8
1
12
1
24
x
x
x
25.
Solve 4 x = 86 .
(A)
(B)
(C)
(D)
1
2
3
9
2
9
Page 5 of 16
Mathematics 3204 June 2009
26.
Solve: 25 x +1 = 54 x −3 .
(A)
(B)
(C)
(D)
27.
− 52
−2
2
5
2
x
Which represents y = ( 12 ) − 2 ?
(A)
y
x
(B)
y
x
y
(C)
x
y
(D)
x
28.
x
What is the range of ( y + 3) = ( 15 ) ?
(A)
(B)
(C)
(D)
29.
{ y y > −3, y ∈ R}
{ y y ≥ −3, y ∈ R}
{ y y > 3, y ∈ R}
{ y y ≥ 3, y ∈ R}
The temperature, T, of a cup of hot chocolate with respect to time, m,
in minutes is given by the equation below. What is the initial temperature, in degrees
Celsius, of the hot chocolate?
T = 73(0.90) m + 22
(A)
(B)
(C)
(D)
22
73
90
95
Mathematics 3204 June 2009
Page 6 of 16
30.
Which represents the inverse of the graph of f ( x) shown below?
y
x
(A)
y
x
(B)
y
x
(C)
y
x
y
(D)
x
31.
32.
Which is equivalent to log 5 ( x + 2) = y ?
(A)
5x = y − 2
(B)
5x = y + 2
(C)
(D)
5y = x − 2
5y = x + 2
Write as a single logarithm: 4 log A − log B3 + 2 log C .
(A)
log ( A 4 B3C 2 )
(B)
log
(C)
log A4C2
log B3
(D)
8 log
( ABC )
4 2
3
( ACB )
3
Page 7 of 16
Mathematics 3204 June 2009
33.
What is the exact value of x for 3x +1 = 15 ?
(A)
(B)
(C)
(D)
34.
36.
log15
log 3
log15
log 3
−1
+1
−1
+1
4
±4
16
± 16
1 log 36 − log 2 .
3
3
2
(A)
(B)
(C)
1
2
log 3 12
(D)
log 3 36
Solve for x: 2 log 3 ( x + 1) = 4 .
1
5
8
40
In the circle with centre O shown, chord AB is 22 cm long and is 8 cm from the centre.
What is the length, in cm, of the diameter of the circle?
(A)
(B)
(C)
(D)
38.
log 3
log15
Evaluate:
(A)
(B)
(C)
(D)
37.
log15
Solve for x : 2 log 2 x + log 2 4 = log 2 64 .
(A)
(B)
(C)
(D)
35.
log 3
O
13.6
15.0
20.5
27.2
8 cm
B
22 cm
A
In the circle with centre O shown, ∠DAB = 98° . What is the value of x, in degrees?
B
A
98°
(A)
(B)
(C)
(D)
82
98
164
196
Mathematics 3204 June 2009
x°
O
D
C
Page 8 of 16
39.
In the circle with centre O shown, ∠ACB = 50° . What is the measure, in degrees, of
?
major arc ACB
C
50°°
O
(A)
(B)
(C)
(D)
40.
100
260
310
335
A
B
In the circle shown ∠ACD = 72° and DE is tangent to the circle at C. What is the
measure, in degrees, of ∠BAC ?
B
A
E
(A)
(B)
(C)
(D)
18
36
54
72
72°
C
D
41.
What are the coordinates of the centre of the circle represented by x 2 + y 2 − 4 y = 12 ?
(A)
(B)
(C)
(D)
42.
What is the length of the major axis of the ellipse represented by
4( x + 3) 2 + 25( y − 5) 2 = 100 ?
(A)
(B)
(C)
(D)
43.
2
4
5
10
What is the converse of the statement, “If a triangle is isosceles then the base angles are
congruent”?
(A)
(B)
(C)
(D)
44.
( −2, 0 )
( 0, −2 )
( 0, 2 )
( 2, 0 )
A triangle is isosceles iff the two base angles are congruent.
If a triangle is not isosceles then the base angles are not congruent.
If the base angles of a triangle are congruent then the triangle is isosceles.
The two base angles of a triangle are congruent iff the triangle is isosceles.
= 80° . What is the
In the circle with centre O shown, the radius is 6 units and minor AB
approximate area, in square units, of the shaded region?
C
O
(A)
(B)
(C)
(D)
4.19
8.38
12.57
25.13
6
A
B
80°°
Page 9 of 16
Mathematics 3204 June 2009
45.
if the circumference of
In the circle with centre O shown, what is the length of minor AB
the circle is 32π ?
O
(A)
(B)
(C)
(D)
46.
(B)
(C)
(D)
(B)
(C)
(D)
49.
(
(
(
(
−
3 1
2 ,2
−1 , 3
2 2
1,− 3
2 2
3 −1
2 , 2
)
)
)
)
( −2, 2 )
( − 4, 4 )
( 2,1)
( 8, −5 )
Which equation represents a circle?
(A)
4x2 − 4 y 2 = 0
(B)
4 x 2 + 10 y 2 − 8 x + 20 y − 70 = 0
(C)
12 x 2 − 20 y 2 − 2 x − 4 y − 6 = 0
(D)
16 x 2 + 16 y 2 − 12 x + 2 y − 18 = 0
The endpoints of the major vertical axis of an ellipse are ( 0, −5 ) and ( 0,5 ) . What is the
vertical stretch factor of the ellipse as compared to the unit circle?
(A)
(B)
(C)
(D)
50.
A
A circle with centre C ( 6, −3) has a diameter PQ with one endpoint at P (10, −7 ) . What
are the coordinates of point Q?
(A)
48.
B
If point P (1, 0 ) is rotated 300° from standard position on a unit circle, what are the new
coordinates of point P?
(A)
47.
80°
7.1
11.2
22.3
44.6
1
5
1
5
5
5
What is the approximate measure of θ , in degrees, if the point P ( 8, −6 ) is on the
terminal arm of θ in standard position?
(A)
(B)
(C)
(D)
306.9
311.4
318.6
323.1
Mathematics 3204 June 2009
Page 10 of 16
PART II
Total Value: 50%
Answer ALL items in the space provided. Show ALL workings.
Value
4
4
4 x−6
=
.
x x −3
51.
Algebraically determine the exact roots in simplest form for
52.
A square and a rectangle have the same width x. The rectangle’s length is 1 cm
more than twice the width. When the area of the square is subtracted from the
area of the rectangle, the result is 56. Algebraically determine the width of the
square and the rectangle.
x
x
Page 11 of 16
Mathematics 3204 June 2009
Value
4
4
53.
54.
A soccer ball is kicked and follows a parabolic path described by the function
h(t ) = −5t 2 + 15t + 0.2 , where t is the time in seconds after the ball is kicked and
h(t) is the height of the ball above ground, in metres.
a)
What is the initial height of the ball? ____________
b)
Algebraically determine the maximum height reached by the ball.
A toy rocket is launched into the air and reaches a maximum height of 80 m after
a time of 4 seconds. If the rocket lands after 8 seconds, determine the quadratic
function that describes the flight path of the rocket. Use the function to determine
the height of the rocket at 6.5 seconds.
h
Path of rocket
80 m
(0,0)
Mathematics 3204 June 2009
Page 12 of 16
4s
8s
t
Value
4
55.
A ball is thrown vertically upward with an initial speed of 30 m/s. Its height, in
metres, t seconds after release is given by h(t ) = 1 + 30t − 5t 2 . Calculate the
instantaneous rate of change at 2.5 seconds and describe how the height of the
ball is changing at that instant.
4
56.
Algebraically solve for x:
4
57.
Algebraically solve for x:
1
log 2 125 + log 2 ( x + 2) = 4 .
3
( 361 )
2 x +1
− 6 = 210 .
Page 13 of 16
Mathematics 3204 June 2009
Value
4
58.
A radioactive substance has a half-life of 17 days. Write a function to model this
situation and use it to determine the time it will take for 300 g of this substance to
decay to 95 g.
4
59.
The table below shows the value of a house that is appreciating over time. Create
a function that describes the value of the house and use it to determine its value
after 18 years.
Time (years)
0
5
10
15
Value ($)
200 000 224 000 250880 280985.6
Mathematics 3204 June 2009
Page 14 of 16
Value
3
60.
Write x 2 + 4 y 2 − 10 x + 24 y − 3 = 0 in transformational form and state the
coordinates of the centre.
3
61.
Write, in transformational form, the equation for the ellipse shown.
y
(3,4)
(-1,2)
(7,2)
(3,0)
Page 15 of 16
x
Mathematics 3204 June 2009
Value
4
62.
In ∆ABC the coordinates of the vertices are A(0,0), B(4, 6), and C(8, -2) . Prove
that the segment joining the midpoints of AB and BC is one half the length of
AC .
y
B (4,6)
x
A (0,0)
C (8,-2)
4
63.
= 85° . Calculate the area of the
In the circle with centre O shown, minor AB
shaded region.
2 cm
B
O
85°
4 cm
A
Mathematics 3204 June 2009
Page 16 of 16