Download 30s 2012 review Multiple Choice Identify the choice that best

30s 2012 review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Determine the first 4 terms of an arithmetic sequence, given the first term,
difference,
.
A. –4, –6, –8, –10
C. –2, 2, 6, 10
B. –4, –2, 0, 2
D. –2, –6, –10, –14
____
2. Determine t11 of this arithmetic sequence: –20, –35, –50, –65, ...
A.
C.
B.
D.
____
3. The sum of the first 15 terms of an arithmetic series is –210. The sum of the first 16 terms is –243.2. The
common difference is –2.4. Determine the first 4 terms of the series.
A.
C.
B.
D.
____
4. Which sequence could be geometric?
A.
2 2
6, 2,  ,  , ...
3 9
B.
9 18
6, , , 3, ...
2 5
C.
2 2
6, 2, , , ...
3 9
D.
3
6, 3, 2, , ...
2
____
5. Determine the 6th term of this geometric sequence: –9, 27, –81, 243, ...
A.
C.
B.
D.
____
6. In a finite geometric sequence,
A.
B.
____
7. Determine the 5th term of this geometric sequence: –16, 8, 4, 2, ...
A. 2
C. 1

2
B. 1
D. 1
2
____
8. The sum of the first 12 terms of which geometric series is 5460?
A.
C.
B.
D.
____
9. Which geometric series could this graph represent?
and
. Determine t10.
C.
D.
, and the common
Geom etric Series
320
280
240
Partial sums
200
160
120
80
40
0
1 2 3 4 5 6 7
Num ber of term s
A.
1
1
1
+
+
+ ...
12 24 48
B. 6 + 12 + 24 + 48 + ...
6+
C.
D. 6 + 18 + 54 + 162 + ...
____ 10. The common ratio of a geometric sequence is 1.5. Which graph could represent this geometric sequence?
A.
C.
Geom etric Sequence
Geom etric Sequence
120
Term value
Term value
80
40
0
–40
1 2 3 4 5 6 7
Term num ber
80
40
0
1 2 3 4 5 6 7
Term num ber
–80
B.
D.
Geom etric Sequence
Geom etric Sequence
120
Term value
Term value
80
40
0
–40
–80
1 2 3 4 5 6 7
Term num ber
80
40
0
1 2 3 4 5 6 7
Term num ber
____ 11. Determine the first 4 terms of an infinite geometric series with
and
.
A. t1 = –3; t2 = 15; t3 = 75; t4 = 375
B.
C.
3
3
3
t1 = –3; t2 = ; t3 =  ; t4 =
5
25
125
D.
3
3
3
t1 = –3; t2 =  ; t3 =  ; t4 = 
5
25
125
1
1
1
t1 = –3; t2 = ; t3 =  ; t4 =
5
75
1125
____ 12. An infinite geometric series has
and
. Determine
.
C. This series does not have a finite sum.
D.
A.
B.
____ 13. Use an infinite geometric series to express
A.
as a fraction.
C.
B.
D.
____ 14. Which point on the number line has an absolute value of 2.5?
W
–10
A. Y
____ 15. Evaluate
A. 7252
–8
–6
–4
–2
X
Y
0
2
4
B. X
when
B. 7259
Z
6
8
10
C. Z
D. W
C. 7189
D. 7217
C. 945
D. –17
C.
D.
C.
D.
C.
D.
C.
D.
.
____ 16. Evaluate:
A. 15
B. 47
____ 17. Write this mixed radical as an entire radical:
A.
B.
____ 18. Write this mixed radical as an entire radical:
A.
B.
____ 19. Write this entire radical as a mixed radical:
A.
B.
____ 20. Write this entire radical as a mixed radical:
A.
B.
____ 21. For which values of the variable, x, is this radical defined?
A.
C.
B.
D.
____ 22. Simplify by adding or subtracting like terms:
A.
B.
C.
D.
____ 23. Expand and simplify this expression:
A.
B.
C.
D.
____ 24. Rationalize the denominator:
A.
B.
C.
D.
____ 25. Expand and simplify this expression:
A.
B.
C.
D.
____ 26. Expand and simplify this expression:
A.
B.
____ 27. Solve this equation:
A.
9
x=
4
C.
D.
B.
x=
4
9
C.
x=
16
81
D.
x=
____ 28. Which statement is true for the equation
?
A. 7 and –1 are roots.
B. 7 is a root of the original equation and –1 is an extraneous root.
C. 1 is a root of the original equation and –7 is an extraneous root.
D. 7 and 1 are both extraneous roots.
____ 29. Solve by factoring:
A.
B.
C.
D.
____ 30. Without solving, determine the number of real roots of this equation:
A. 2
B. 0
C. 1
____ 31. Without solving, determine the number of real roots of this equation:
A. 0
B. 2
C. 1
____ 32. Calculate the value of the discriminant for this equation:
A. 20
B. 24
C. 5
D. –6
____ 33. Which statement below is NOT true for the graph of a quadratic function?
A. The vertex of a parabola is its highest or lowest point.
81
16
B. When the coefficient of
is positive, the vertex of the parabola is a minimum point.
C. The axis of symmetry intersects the parabola at the vertex.
D. The parabola is symmetrical about the y-axis.
____ 34. Identify the quadratic function that this table of values represents, then determine the value of y when
x = 7.
x
y
–1
–9
0
–3
3
–33
A.
B.
; –185
; –182
C.
D.
; 185
; 185
____ 35. A rectangular dog pen is to be enclosed with 20 m of fencing. The area of the dog pen, A square metres, is
modelled by the function
, where x is the width, in metres. What is the width that gives
maximum area? Write the answer to the nearest tenth, if necessary.
A. 5 m
C. 20 m
B. 25 m
D. 10 m
____ 36. Use the graph of
to determine the roots of
.
y
12
8
4
–6
–4
–2
0
2
4
6
x
–4
–8
–12
A.
B.
and
and
C.
D.
and
and
____ 37. Use graphing technology to approximate the solution of this equation:
Write the roots to 1 decimal place.
and
.
A. The roots are approximately
and
.
B. The roots are approximately
and
.
C. The roots are approximately
and
.
D. The roots are approximately
____ 38. Identify the coordinates of the vertex of the graph of this quadratic function:
B. (–4, –4)
A. (4, 4)
____ 39. Match the quadratic function
y
A.
to a graph below.
C.
6
–3
–2
–1
4
2
2
0
1
2
–1
–2
–1
0
–2
–4
–4
D.
y = h(x)
2
2
2
3 x
3 x
2
3 x
y
4
1
2
y = g(x)
6
4
0
1
–6
y = f(x)
y
–2
–3
3 x
–2
6
–3
y
6
4
–6
B.
D. (4, –4)
C. (–4, 4)
–3
–2
–1
0
–2
–2
–4
–4
–6
–6
y = k(x)
1
____ 40. A sports equipment company sells skates for $65 a pair. At this price, the company sells approximately
200 pairs a week. For every increase in price of x dollars, the company will sell 40x fewer pairs.
Determine the equation that should be used to maximize the revenue, R dollars.
A.
C.
B.
D.
____ 41. Two numbers have a difference of 10. The sum of their squares is a minimum. Determine the numbers.
A. 5 and 15
B. –2 and 8
C. 0 and 10
D. –5 and 5
____ 42. Does the quadratic function
value?
A. minimum value; 484
B. minimum value; 340
have a maximum value or a minimum value? What is that
C. maximum value; 484
D. maximum value; 340
____ 43. A rectangular dog pen is to be fenced with 24 m of fencing. Determine the maximum area and the width
of this rectangle.
A.
C.
;
;
B.
D.
;
;
____ 44. Three rectangular areas are to be fenced with 78 m of fencing. Which equation could be used to determine
the dimensions that enclose the maximum area?
l
w
w
w
w
l
A.
C.
B.
D.
____ 45. Represent the solution of this quadratic inequality on a number line:
A.
–5 –4 –3 –2 –1
0
1
2
3
4
5
–5 –4 –3 –2 –1
0
1
2
3
4
5
–5 –4 –3 –2 –1
0
1
2
3
4
5
–5 –4 –3 –2 –1
0
1
2
3
4
5
B.
C.
D.
____ 46. Which coordinates are a solution of the inequality
A. (2, 8)
B. (0, 1)
C. (3, 7)
?
D. (5, 8)
____ 47. Match the inequality –x + 2y  8 to its graph.
A.
C.
y
–6
–4
–2
y
6
6
4
4
2
2
0
2
4
6
x
–6
–4
–2
0
–2
–2
–4
–4
–6
–6
2
4
6 x
B.
D.
y
6
–6
–4
–2
y
6
4
4
2
2
0
2
4
–6
6 x
–4
–2
–2
–2
–4
–4
–6
–6
____ 48. Which ordered pair is a solution of the quadratic inequality
A. (0, 5)
B. (3, 12)
C. (2, 4)
____ 50. Which graph represents the inequality
–2
B.
?
D. (2, 19)
y
6
4
4
2
2
0
2
4
6 x
D. (1, 5)
C.
y
–4
4
?
6
–6
2
?
____ 49. Which ordered pair is a solution of the quadratic inequality
A. (3, 36)
B. (1, 1)
C. (–1, 2)
A.
0
–6
6 x
–4
–2
0
–2
–2
–4
–4
–6
–6
2
4
6 x
2
4
6 x
D.
y
6
y
6
4
4
2
2
–6
–4
–2
0
–2
2
4
6 x
–6
–4
–2
0
–2
–4
–4
–6
–6
____ 51. Write an inequality to describe this graph.
y
6
4
2
–3
–2
–1
0
1
2
3
x
–2
–4
–6
A.
B.
C.
D.
____ 52. Use a graphing calculator. Which graph represents this system of equations?
A.
C.
y
6
–3
–2
–1
y
6
4
4
2
2
0
1
2
3 x
–3
–2
–1
0
–2
–2
–4
–4
–6
–6
1
2
3 x
B.
D.
y
6
–3
–2
–1
y
6
4
4
2
2
0
1
2
–3
3 x
–2
0
–1
–2
–2
–4
–4
–6
–6
1
2
3 x
____ 53. Two numbers are related:
The sum of twice the square of the first number plus the second number is 9.
The difference between twice the first number and the second number is 15.
Which system of equations represents this relationship?
A.
C.
B.
D.
____ 54. Solve this linear-quadratic system algebraically.
A. (5, 2)
B. (2, 5)
C. (–2, 1)
D. (2, 33)
____ 55. Solve this linear-quadratic system algebraically.
A. (1, –2) and (–1, 6)
B. (1, 6) and (–1, –2)
C. (6, 1) and (–2, –1)
D. (–2, 1) and (6, –1)
____ 56. A ladder leans against a wall. The base of the ladder is on level ground 1.4 m from the wall. The angle
between the ladder and the ground is 74. To the nearest tenth of a metre, how far up the wall does the
ladder reach?
A. 5.1 m
B. 4.9 m
C. 1.5 m
D. 0.4 m
____ 57. An angle  has its terminal arm in Quadrant 2. Which primary trigonometric ratio is greater than 0?
A. cos 
B. tan 
C. sin 
D. all 3 ratios
____ 58. Determine the reference angle for the angle 290° in standard position.
A. 290°
B. 20°
C. 110°
D. 70°
____ 59. Determine the exact value of tan 210.
A.
B.
D.
C.
____ 60. Point P(5, 0) is a terminal point of an angle  in standard position. Determine the value of cos .
A. undefined
B. 0
C. 5
D. 1
____ 61. In XYZ, XY = 6 cm and X = 33°. For which value of YZ is no triangle possible?
A. 6 cm
B. 3 cm
C. 4 cm
D. 12 cm
____ 62. In ABC, AB = 9.1 m and A = 41°. For which value of BC can one scalene triangle be drawn?
A. 9.1 m
B. 11.4 m
C. 6 m
D. 7.3 m
____ 63. In DEF, DE = 11 cm and EF = 9 cm. For which measure of D is it possible to draw two scalene
triangles?
A. 54°
B. 65°
C. 58°
D. 90°
____ 64. For DEF, determine the measure of E to the nearest degree.
E
7.4 cm
54°
D
F
4.2 cm
A. 155°
B. 27°
C. 25°
D. 58°
____ 65. In PQR, determine the measure of Q to the nearest degree.
P
R
44°
5.5 cm
3.5 cm
Q
A. 70°
B. 110°
C. 136°
D. 154°
B.
C.
D.
____ 66. Solve.
A.
____ 67. Solve.
A.
B.
C.
D. no solution
____ 68. This is the graph of the absolute value of a linear function.
y
4
2
–6
–4
–2
0
2
4
6
x
–2
–4
Which is the graph of the linear function?
A.
C.
y
y
4
4
2
2
–6
–6
–4
–2
0
2
4
6
–4
–2
0
2
4
6
x
2
4
6
x
–2
x
–2
–4
–4
B.
D.
y
y
4
4
2
2
–6
–6
–4
–2
0
2
4
6
–4
–2
0
–2
x
–2
–4
–4
____ 69. Sketch the graph of the absolute value function
.
A.
12
–8
–4
B.
–4
12
8
8
4
4
0
4
8
–8
x
–4
–4
–8
–8
D.
y
12
8
8
4
4
0
4
8
–8
x
–4
–4
–8
–8
C.
D.
4
8
x
4
8
x
y
0
–4
____ 70. Solve this equation:
and
A.
and
B.
y
0
–4
12
–8
C.
y
and
and
____ 71. The graph of the reciprocal of a linear function has this vertical asymptote. What is the linear function?
y
6
4
2
–6
–4
–2
0
2
4
6
x
–2
–4
–6
A.
B.
____ 72. Which graph represents the function
C.
D.
and its reciprocal function
?
A.
C.
y
–4
–2
4
4
2
2
0
2
4
–2
0
–2
–4
–4
D.
y
–2
–4
x
–2
B.
–4
y
4
2
2
2
4
4
x
2
4
x
y
4
0
2
–4
x
–2
0
–2
–2
–4
–4
____ 73. A linear function has a positive slope. Which graph below is a possible graph of its reciprocal function?
y
y
A.
C.
–4
–2
4
4
2
2
0
2
4
0
–2
–4
–4
D.
y
–2
–2
–2
B.
–4
–4
x
4
2
2
2
4
–4
x
–2
0
–2
–2
–4
–4
____ 74. The graphs of the reciprocal functions
the coordinates of any points of intersection?
4
x
2
4
x
y
4
0
2
and
are plotted on the same grid. What are
C. (–1, –6)
A.
1
(1,  )
6
B.
1
( , –1)
6
D.
1
( , 1 )
6
____ 75. This is the graph of the reciprocal of a linear function. Which graph below represents the linear function?
y
6
4
2
–6
–4
–2
0
2
4
6
x
–2
–4
–6
A.
C.
y
–8
–4
8
8
4
4
0
4
8
–4
0
–4
–8
–8
D.
y
–4
–8
x
–4
B.
–8
y
8
4
4
4
8
x
–8
–4
0
–4
–4
–8
–8
Short Answer
1. Could this sequence be arithmetic?
–1, –5, –10, –15, ...
8
x
4
8
x
y
8
0
4
2. The sum of the first n terms of an arithmetic series is:
Determine the first 4 terms of the series.
3. Determine r, t5, and t6 of this geometric sequence: 3125, 625, 125, 25, ...
4. Evaluate each expression, then order the values of the expressions from greatest to least.
i)
ii)
iii)
iv)
5. Evaluate each expression, then order the values of the expressions from least to greatest.
i)
ii)
iii)
iv)
6. Determine the root of each equation.
a)
b)
c)
d)
7. The total area of the large rectangle below is 24 m2. Determine the value of x.
x m
2m
2m
x m
x m
8. A baseball is hit upward. The approximate height of the baseball, h metres, after t seconds is modelled by
this formula:
When is the baseball 11 m high?
9. When 3 times a number is added to the square of the number, the result is 40.
Determine the number.
10. a) Calculate the value of the discriminant for the equation
b) Are the roots rational or irrational?
= 0.
11. Use a graphing calculator to graph the quadratic function
Determine:
a) the intercepts
b) the coordinates of the vertex
c) the equation of the axis of symmetry
d) the domain of the function
e) the range of the function
Round the answers to the nearest hundredth, if necessary.
.
12. The weekly profit, P hundred dollars, of a company is modelled by the equation
, where
x is the number of units produced per week, in thousands.
a) Use a graphing calculator to determine the number of units the company should produce per week to
earn the maximum weekly profit.
b) What is the maximum weekly profit?
13. The graph of a quadratic function is shown. What can you say about the discriminant of the corresponding
quadratic equation?
y
12
8
4
–6
–4
–2
0
2
4
6 x
–4
–8
–12
14. Determine an equation of a quadratic function with the given characteristics of its graph.
coordinates of the vertex: V(–4, 1); y-intercept 33
15. Use a graphing calculator or graphing software. Graph each quadratic function. How many x-intercepts
does the graph have?
a)
b)
c)
16. Use a graphing calculator or graphing software. Graph each quadratic function. How many x-intercepts
does the graph have?
a)
b)
c)
17. The equation of the axis of symmetry of the graph of a quadratic function is
. The graph passes
through the points R(3, 7) and T(–1, –25). Determine an equation of the function.
18. Does the quadratic function
have a maximum value or a minimum value? What is that
value?
19. A coffee shop sells coffee for $1.40 a cup. At this price, the store sells approximately 800 cups per day.
Research indicates that for every $0.05 increase in price, the store will sell 40 fewer cups. Determine the
price of a cup of coffee that will maximize the revenue.
20. Represent the solution of this quadratic inequality on a number line:
21. a) Determine the critical values of this quadratic inequality:
b) Complete the table.
Interval
Sign of
Value of x
c) What is the solution of the inequality?
22. a) Graph the inequality:
b) Write the coordinates of 2 points that satisfy the inequality.
23. Use technology to graph the inequality
. Sketch the graph.
24. Use a graphing calculator. Graph this system of equations.
a) Sketch the graphs on the same grid.
b) Write the coordinates of the point of intersection.
25. Use a graphing calculator. Graph this system of equations.
a) Sketch the graphs on the same grid.
b) Write the coordinates of the points of intersection.
26. Use a graphing calculator. Graph this system of equations.
a) Sketch the graphs on the same grid.
b) Write the coordinates of the point of intersection.
27. How is a linear-quadratic system different from a quadratic-quadratic system in terms of
the number of solutions the system may have?
28. Solve this quadratic-quadratic system algebraically.
29. Solve this quadratic-quadratic system algebraically.
30. A baseball is thrown upward at an initial speed of 9.5 m/s. The approximate height of the ball, h metres,
after t seconds is given by the equation
. At the same time, a golf ball is hit upward at
an initial speed of 12 m/s. The approximate height of the ball, h metres, after t seconds is given by the
equation
.
a) When are both balls at the same height?
b) What is this height? Give your answer to the nearest tenth of a metre.
31. Solve this quadratic-quadratic system algebraically. Give the solutions to the nearest tenth, when
necessary.
32. Solve this quadratic-quadratic system algebraically. Give the solutions to the nearest tenth, when
necessary.
33. A tree is supported by a guy wire. The guy wire is anchored to the ground 5.0 m from the base of the tree.
The angle between the wire and the level ground is 60. To the nearest tenth of a metre, how far up the
tree does the wire reach?
34. A flagpole is 11.0 m high. At a certain point, the angle between the ground and Jon’s line of sight to the
top of the flagpole is 61. To the nearest tenth of a metre, how far is Jon from the flagpole?
35. A ladder is 5 m long. It leans against a house. The base of the ladder is 1.4 m from the house. What is the
angle of inclination of the ladder to the nearest tenth of a degree?
36. A ski jump is 116 m long. It has a vertical rise of 54 m. What is the angle of inclination of the jump to the
nearest tenth of a degree?
37. A 2.8-m cable is attached to a sign. The cable is anchored to the ground 1.8 m from the base of the sign.
What is the angle of inclination of the cable to the nearest tenth of a degree?
38. To the nearest degree, which values of  satisfy this equation for
?
39. If AB = 7 cm, BC = 3 cm, and A = 70°, is it possible to construct ABC?
40. For JKL, can the Sine Law be used to determine the length of JL? If your answer is yes, determine the
length of JL to the nearest tenth of a centimetre. If your answer is no, explain why.
L
J
7 cm
3 cm
101°
K
41. Given the following information about ABC, determine how many triangles can be constructed.
a = 5.6 cm, c = 7.8 cm, A = 38°
42. In ABC, AB = 3.9 cm, C = 63°, and BC = 5.2 cm.
a) Determine how many triangles can be constructed. Justify your answer.
b) Sketch a diagram to show any possible triangles.
43. Complete this table of values.
x
–2
–1
0
1
2
3
–4
–3
–2
–1
0
1
44. Complete this table of values.
x
–2
–1
0
1
2
3
–16
–14
–8
2
16
34
45. Write this absolute value function in piecewise notation.
y
8
4
–4
–2
0
2
4
x
–4
–8
46. Write the absolute value function
in piecewise notation.
y
24
16
8
–8
–6
–4
0
–2
2
4
6
8 x
–8
–16
–24
47. Write this absolute value function in piecewise notation.
y
24
16
8
–12
–8
–4
0
4
8
12 x
–8
–16
48. Solve this equation:
y
20
16
12
8
4
–8
–6
–4
–2
0
–4
2
4
6
8
x
49. A student used this graph to solve an absolute value equation. What might the equation have been?
y
8
6
4
2
–8
–6
–4
–2
0
2
4
6
8 x
–2
–4
50. A student used this graph to solve an absolute value equation. What might the equation have been?
y
8
6
4
2
–8
–6
–4
–2
0
2
4
6
8
x
–2
–4
–6
–8
Problem
1. In this arithmetic sequence, k is a natural number:
a) Determine t6.
b) Write an expression for tn.
c) Suppose
; determine the value of k.
...
2. a) What is the sum of the first 124 multiples of 4?
b) What is the value of t124?
3. a) A geometric sequence has these terms:
1
t4 = 8, t5 = 2, t6 =
2
State the common ratio, then write the first 3 terms of the sequence.
b) Identify the sequence as convergent or divergent. Explain.
4. Show how you can use geometric series to determine this sum:
1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 + 32 + 48 + 64 + 96 + 128 + 192 + 256 + 384 + 512
1
the length of the preceding one.
4
A frog is programmed to jump to a position 12 m away in 12 jumps. Determine the horizontal distance of
each of the first 3 jumps. Give your answers to the nearest hundredth of a metre.
5. A robotic frog is programmed to make a sequence of jumps, each jump
6. Joel worked at Series Inc. for the summer. He earned $56 on the first day. His pay was increased at a rate
of 2.1% with each subsequent day. Determine the total amount Joel earned after 16 days.
7. Suppose the lengths of a frog’s jumps form a geometric series. After 12 jumps, the frog has travelled a
7
horizontal distance of approximately 6265.56 cm. The common ratio is .
5
Determine the length of the frog’s first jump.
8. Create the first 6 terms of a geometric sequence for this description of a graph: the term values and
increase in numerical value as more points are plotted.
9. a) Without graphing, describe the graph of this geometric sequence:
3, 9, 27, 81, 243, 729, ...
b) Without graphing, describe the graph of the partial sums of this geometric series:
...
c) Verify your descriptions by graphing. Sketch and label each graph on a grid below.
10. An infinite geometric series with
is represented by this equation:
a) Determine the first 4 terms of the series.
b) Determine whether the series diverges or converges.
c) If the series has a finite sum, determine the sum.
11. Three congruent rectangles are joined as shown. The length of each rectangle is 3 times its width. The
area of each rectangle is 39 square units. In simplest form, write a radical expression for the perimeter of
the shape formed. Show your work.
3x
A = 39 units 2
I
x
I
I
I
12. Expand and simplify this expression:
Show your work.
13. The side length of a square is
units.
Write, then simplify, a radical expression for the area of
the square.
Show your work.
14. Write this expression in simplest form:
Describe your strategy.
15. Determine whether the given value of x is a root of this equation. Justify your answer.
;
16. The volume of water in Lake Ontario is about 1710 km3.
a) To the nearest kilometre, determine the edge length of a cube with the same volume.
b) To the nearest kilometre, determine the radius of a sphere with the same volume.
17. Determine the lengths of the legs in this right triangle. Explain your strategy.
29 cm
x cm
x + 1 cm
18. Determine the values of k for which the equation
equation.
has two real roots, then write a possible
19. Determine the values of k for which the equation
equation.
has no real roots, then write a possible
20. Create two different quadratic equations whose discriminant is 64. Explain your strategy.
21. Use a graphing calculator to graph the quadratic function
Determine:
a) the intercepts
b) the coordinates of the vertex
c) the equation of the axis of symmetry
d) the domain of the function
e) the range of the function
Write the answers to the nearest hundredth, if necessary.
.
22. a) Identify the coordinates of the vertex, the domain, the range, the direction of opening, the equation of
the axis of symmetry, and the intercepts for this quadratic function:
b) Sketch a graph.
23. The arch of a bridge forms a parabola. The arch is 72 m wide and its maximum height is 27 m above its
base. Determine an equation to model this parabola. Explain your strategy.
24. A hospital sells raffle tickets to raise funds for new medical equipment. Last year, 2000 tickets were sold
for $24 each. The fund-raising coordinator estimates that for every $1 decrease in price, 125 more tickets
will be sold.
a) What decrease in price will maximize the revenue?
b) What is the price of a ticket that will maximize the revenue?
c) What is the maximum revenue?
Show and explain your work.
25. Consider this inequality:
a) Solve the inequality by factoring.
b) Illustrate the solution on a number line.
c) What do you notice about the solution of the inequality. Explain why.
26.
is a quadratic equation. For which values of b does the quadratic equation have:
a) 2 real roots?
b) exactly 1 root?
c) no real roots?
Show your work.
27. The length of a rectangular garden is 6 m greater than its width. The area of the garden is at least
20 m2. What are possible dimensions of the garden, to the nearest tenth of a metre? Show your work.
Verify the solution.
28. Graph each inequality for the given restrictions on the variables. Explain your strategy and describe the
solution.
a)
b)
; for
; for
29. For A(a, 3) to be a solution of
Show and explain your work.
, what must be true about a?
30. Use a graphing calculator. Graph this system of equations.
a) Sketch the graphs on the same grid.
b) Write the coordinates of the points of intersection.
31. Write the system of equations represented by this graph, then solve the system.
Explain your work.
y
12
10
8
6
4
2
–12
–10
–8
–6
–4
–2
0
2
4
6
8
10
x
–2
–4
–6
32. Two numbers are related in this way:
Twice the square of the first number minus the second number is 3.
The square of the sum of the first number and 5 is equal to the second number minus 2.
a) Create a system of equations to represent this relationship.
b) Solve the system to determine the numbers. Explain the strategy you used.
33. A football is kicked upward at an initial speed of 10 m/s. The approximate height of the ball, h metres,
after t seconds is modelled by the equation
. A soccer ball is kicked upward with an
initial speed of 12 m/s. The approximate height of the ball, h metres, after t seconds is modelled by the
equation
.
a) The soccer ball is kicked 1 s after the football is kicked. Determine when both balls reached the same
height.
b) What is this height?
Explain your strategy. Give your answers to the nearest tenth of a unit.
34. Determine the measures of A and B to the nearest tenth of a degree. Explain your strategy.
B
21
11
A
C
35. a) In BCD, identify the side opposite angle  and the side adjacent to angle  .
b) Determine sin  to the nearest tenth. Describe what the value of sin  indicates.
c) Determine the measure of  to the nearest tenth of a degree.
Show your work.
A
B
8 cm
D
14 cm

C
36. A coast guard patrol boat is due west of the Carmanah lighthouse. An overturned fishing boat is due north
of the lighthouse. The patrol boat travels 9.1 km directly to the fishing boat. The angle between due east
and the patrol boat’s path is 54°. To the nearest tenth of a kilometre, determine the distance between the
fishing boat and the lighthouse. Explain your work.
37. In
, AB = 35 cm, BC = 19 cm, and
.
a) Explain why it is possible to draw two different triangles with these measures.
b) Sketch a diagram to show both triangles.
38. In ABC, AB = 6 cm and
Length of BC (cm)
Value of
. Complete the chart below for your own values of BC.
How does
compare with sin A?
Description of possible
triangles
No triangles are possible.
1 isosceles triangle
1 scalene triangle
2 scalene triangles
39. A pair of campers paddle a canoe 3.4 km [W24°N], then 2.4 km [S28°E]. To the nearest tenth of a
kilometre, what is the straight-line distance from their start point to their end point? To the nearest degree,
what is the bearing of the end point from the start point? Show your work.
40. Solve ABC. Give angle measures to the nearest degree.
10 cm
C
A
9 cm
5 cm
B
41. Without solving the equation, explain how you know that this equation has no solution.
42. Explain how you would solve this rational equation:
43. Complete this table of values, then sketch the graphs of
x
–2
–1
–6
–4
0
2
1
0
and
2
3
2
4
on the same grid.
y
8
6
4
2
–8
–6
–4
0
–2
2
4
6
8 x
–2
–4
–6
–8
44. Write an equation for the absolute value function. Explain your strategy.
8
y
6
4
2
–8
–6
–4
–2
0
2
4
6
8 x
–2
–4
–6
–8
45. Write an equation for this absolute value function. Explain your strategy.
y
8
6
4
2
–8
–6
–4
–2
0
2
4
6
8 x
–2
–4
–6
–8
46. Solve the equation
. Show your work.
47. A motorcycle is travelling toward the Ontario-Quebec border. The motorcycle is 255 km from the border,
and is travelling at an average speed of 95 km/h.
a) Write an absolute value equation to represent the distance, d kilometres, of the motorcycle from the
border after t hours.
b) Determine how far the motorcycle is from the border after 1.5 h. Give distance to the nearest
kilometre.
48. The function
is linear. The line
line
intersects the graph of
the function
.
intersects the graph of
at x = 2 and x = 6. The
at x = 3 and x = 5. Use a graph to determine the equation for
y
20
16
12
8
4
–8
–6
–4
–2
0
–4
2
4
6
8
x
49. Use the graph of
to sketch a graph of
. Write the equation of the linear and reciprocal
functions. Show your work.
y
8
6
4
2
–6
–4
–2
0
2
4
6
x
–2
–4
–6
–8
50. a) Write a reciprocal function that describes the time, t hours, it takes an athlete to run 1 km, as a
function of her speed, s kilometres per hour.
b) What are the domain and range of the reciprocal function in part a?
c) Sketch a graph of the reciprocal function in part a.
t
10
8
6
4
2
0
2
4
6
8
10
s
30s 2012 review
Answer Section
MULTIPLE CHOICE
1. ANS:
LOC:
2. ANS:
LOC:
3. ANS:
LOC:
4. ANS:
LOC:
5. ANS:
LOC:
6. ANS:
LOC:
KEY:
7. ANS:
LOC:
8. ANS:
LOC:
9. ANS:
REF:
TOP:
10. ANS:
REF:
TOP:
11. ANS:
LOC:
12. ANS:
LOC:
13. ANS:
LOC:
14. ANS:
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15. ANS:
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TOP:
16. ANS:
REF:
TOP:
17. ANS:
REF:
TOP:
18. ANS:
A
PTS: 1
DIF: Easy
REF: 1.1 Arithmetic Sequences
11.RF9
TOP: Relations and Functions
KEY: Procedural Knowledge
B
PTS: 1
DIF: Easy
REF: 1.1 Arithmetic Sequences
11.RF9
TOP: Relations and Functions
KEY: Procedural Knowledge
A
PTS: 1
DIF: Difficult
REF: 1.2 Arithmetic Series
11.RF9
TOP: Relations and Functions
KEY: Procedural Knowledge
C
PTS: 1
DIF: Easy
REF: 1.3 Geometric Sequences
11.RF10
TOP: Relations and Functions
KEY: Procedural Knowledge
C
PTS: 1
DIF: Easy
REF: 1.3 Geometric Sequences
11.RF10
TOP: Relations and Functions
KEY: Procedural Knowledge
D
PTS: 1
DIF: Moderate
REF: 1.3 Geometric Sequences
11.RF10
TOP: Relations and Functions
Conceptual Understanding | Procedural Knowledge
B
PTS: 1
DIF: Easy
REF: 1.3 Geometric Sequences
11.RF10
TOP: Relations and Functions
KEY: Procedural Knowledge
D
PTS: 1
DIF: Moderate
REF: 1.4 Geometric Series
11.RF10
TOP: Relations and Functions
KEY: Procedural Knowledge
B
PTS: 1
DIF: Easy
1.5 Graphing Geometric Sequences and Series
LOC: 11.RF9 | 11.RF10
Relations and Functions
KEY: Procedural Knowledge
C
PTS: 1
DIF: Moderate
1.5 Graphing Geometric Sequences and Series
LOC: 11.RF9 | 11.RF10
Relations and Functions
KEY: Conceptual Understanding
C
PTS: 1
DIF: Easy
REF: 1.6 Infinite Geometric Series
11.RF10
TOP: Relations and Functions
KEY: Procedural Knowledge
B
PTS: 1
DIF: Easy
REF: 1.6 Infinite Geometric Series
11.RF10
TOP: Relations and Functions
KEY: Procedural Knowledge
B
PTS: 1
DIF: Moderate
REF: 1.6 Infinite Geometric Series
11.RF10
TOP: Relations and Functions
KEY: Procedural Knowledge
A
PTS: 0
DIF: Easy
2.1 Absolute Value of a Real Number
LOC: 11.AN1
Relations and Functions
KEY: Conceptual Understanding
B
PTS: 0
DIF: Moderate
2.1 Absolute Value of a Real Number
LOC: 11.AN1
Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
A
PTS: 0
DIF: Moderate
2.1 Absolute Value of a Real Number
LOC: 11.AN1
Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
C
PTS: 0
DIF: Easy
2.2 Simplifying Radical Expressions
LOC: 11.AN2
Relations and Functions
KEY: Procedural Knowledge
A
PTS: 0
DIF: Moderate
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
REF: 2.2 Simplifying Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Procedural Knowledge
ANS: A
PTS: 0
DIF: Moderate
REF: 2.2 Simplifying Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Procedural Knowledge
ANS: D
PTS: 0
DIF: Moderate
REF: 2.2 Simplifying Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Procedural Knowledge
ANS: A
PTS: 0
DIF: Moderate
REF: 2.2 Simplifying Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
ANS: C
PTS: 0
DIF: Moderate
REF: 2.3 Adding and Subtracting Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
ANS: D
PTS: 0
DIF: Easy
REF: 2.4 Multiplying and Dividing Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Procedural Knowledge
ANS: A
PTS: 0
DIF: Easy
REF: 2.4 Multiplying and Dividing Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Procedural Knowledge
ANS: C
PTS: 0
DIF: Moderate
REF: 2.4 Multiplying and Dividing Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
ANS: A
PTS: 0
DIF: Moderate
REF: 2.4 Multiplying and Dividing Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
ANS: C
PTS: 0
DIF: Easy
REF: 2.5 Solving Radical Equations
LOC: 11.AN3
TOP: Relations and Functions
KEY: Procedural Knowledge
ANS: B
PTS: 0
DIF: Moderate
REF: 3.2 Solving Quadratic Equations by Factoring
LOC: 11.AN3
TOP: Algebra and Number
KEY: Conceptual Understanding
ANS: B
PTS: 0
DIF: Moderate
REF: 3.2 Solving Quadratic Equations by Factoring
LOC: 11.RF5
TOP: Relations and Functions
KEY: Procedural Knowledge
ANS: B
PTS: 0
DIF: Easy
REF: 3.5 Interpreting the
Discriminant
LOC: 11.RF5
TOP: Relations and Functions
KEY: Conceptual Understanding
ANS: C
PTS: 0
DIF: Easy
REF: 3.5 Interpreting the
Discriminant
LOC: 11.RF5
TOP: Relations and Functions
KEY: Conceptual Understanding
ANS: A
PTS: 0
DIF: Easy
REF: 3.5 Interpreting the
Discriminant
LOC: 11.RF5
TOP: Relations and Functions
KEY: Procedural Knowledge
ANS: D
PTS: 0
DIF: Easy
REF: 4.1 Properties of a Quadratic Function
LOC: 11.RF4
TOP: Relations and Functions
KEY: Conceptual Understanding
ANS: C
PTS: 0
DIF: Moderate
REF: 4.1 Properties of a Quadratic Function
LOC: 11.RF4
TOP: Relations and Functions
KEY: Procedural Knowledge
35. ANS:
REF:
TOP:
36. ANS:
REF:
TOP:
37. ANS:
REF:
TOP:
38. ANS:
REF:
LOC:
39. ANS:
REF:
LOC:
40. ANS:
REF:
LOC:
41. ANS:
REF:
LOC:
42. ANS:
REF:
LOC:
43. ANS:
REF:
LOC:
44. ANS:
REF:
LOC:
45. ANS:
REF:
TOP:
46. ANS:
REF:
TOP:
47. ANS:
REF:
TOP:
48. ANS:
REF:
TOP:
49. ANS:
REF:
TOP:
50. ANS:
REF:
TOP:
51. ANS:
A
PTS: 0
DIF: Moderate
4.1 Properties of a Quadratic Function
LOC: 11.RF4
Relations and Functions
KEY: Problem-Solving Skills | Procedural Knowledge
B
PTS: 0
DIF: Easy
4.2 Solving a Quadratic Equation Graphically
LOC: 11.RF5
Relations and Functions
KEY: Conceptual Understanding
B
PTS: 0
DIF: Easy
4.2 Solving a Quadratic Equation Graphically
LOC: 11.RF5
Relations and Functions
KEY: Procedural Knowledge
D
PTS: 0
DIF: Easy
4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q
11.RF3
TOP: Relations and Functions
KEY: Conceptual Understanding
D
PTS: 0
DIF: Easy
4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q
11.RF3
TOP: Relations and Functions
KEY: Conceptual Understanding
B
PTS: 0
DIF: Easy
4.7 Modelling and Solving Problems with Quadratic Functions
11.RF4
TOP: Relations and Functions
KEY: Conceptual Understanding
D
PTS: 0
DIF: Moderate
4.7 Modelling and Solving Problems with Quadratic Functions
11.RF4
TOP: Relations and Functions
KEY: Procedural Knowledge
C
PTS: 0
DIF: Easy
4.7 Modelling and Solving Problems with Quadratic Functions
11.RF4
TOP: Relations and Functions
KEY: Procedural Knowledge
D
PTS: 0
DIF: Moderate
4.7 Modelling and Solving Problems with Quadratic Functions
11.RF4
TOP: Relations and Functions
KEY: Procedural Knowledge
B
PTS: 0
DIF: Moderate
4.7 Modelling and Solving Problems with Quadratic Functions
11.RF4
TOP: Relations and Functions
KEY: Procedural Knowledge
A
PTS: 0
DIF: Easy
5.1 Solving Quadratic Inequalities in One Variable
LOC: 11.RF8
Relations and Functions
KEY: Procedural Knowledge
A
PTS: 0
DIF: Easy
5.2 Graphing Linear Inequalities in Two Variables
LOC: 11.RF7
Relations and Functions
KEY: Procedural Knowledge
D
PTS: 0
DIF: Moderate
5.2 Graphing Linear Inequalities in Two Variables
LOC: 11.RF7
Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
C
PTS: 0
DIF: Easy
5.3 Graphing Quadratic Inequalities in Two Variables
LOC: 11.RF7
Relations and Functions
KEY: Procedural Knowledge
D
PTS: 0
DIF: Easy
5.3 Graphing Quadratic Inequalities in Two Variables
LOC: 11.RF7
Relations and Functions
KEY: Procedural Knowledge
B
PTS: 0
DIF: Easy
5.3 Graphing Quadratic Inequalities in Two Variables
LOC: 11.RF7
Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
D
PTS: 0
DIF: Moderate
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
REF:
TOP:
ANS:
REF:
TOP:
ANS:
REF:
TOP:
ANS:
REF:
TOP:
ANS:
REF:
TOP:
ANS:
REF:
TOP:
ANS:
REF:
TOP:
ANS:
REF:
TOP:
ANS:
REF:
TOP:
ANS:
REF:
TOP:
ANS:
LOC:
ANS:
LOC:
ANS:
LOC:
ANS:
LOC:
ANS:
LOC:
KEY:
ANS:
LOC:
ANS:
LOC:
ANS:
LOC:
KEY:
ANS:
LOC:
5.3 Graphing Quadratic Inequalities in Two Variables
LOC: 11.RF7
Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
A
PTS: 0
DIF: Easy
5.4 Solving Systems of Equations Graphically
LOC: 11.RF6
Relations and Functions
KEY: Procedural Knowledge
B
PTS: 0
DIF: Easy
5.5 Solving Systems of Equations Algebraically
LOC: 11.RF6
Relations and Functions
KEY: Conceptual Understanding
C
PTS: 0
DIF: Easy
5.5 Solving Systems of Equations Algebraically
LOC: 11.RF6
Relations and Functions
KEY: Procedural Knowledge
A
PTS: 0
DIF: Easy
5.5 Solving Systems of Equations Algebraically
LOC: 11.RF6
Relations and Functions
KEY: Procedural Knowledge
B
PTS: 0
DIF: Moderate
6.1 Angles in Standard Position in Quadrant 1
LOC: 11.T2
Trigonometry
KEY: Procedural Knowledge | Problem-Solving Skills
C
PTS: 0
DIF: Easy
6.2 Angles in Standard Position in All Quadrants
LOC: 11.T1
Trigonometry
KEY: Conceptual Understanding
D
PTS: 0
DIF: Easy
6.2 Angles in Standard Position in All Quadrants
LOC: 11.T1
Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
B
PTS: 0
DIF: Moderate
6.2 Angles in Standard Position in All Quadrants
LOC: 11.T2
Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
D
PTS: 1
DIF: Moderate
6.2 Angles in Standard Position in All Quadrants
LOC: 11.T2
Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
B
PTS: 0
DIF: Easy
REF: 6.3 Constructing Triangles
11.T3
TOP: Trigonometry
KEY: Procedural Knowledge
B
PTS: 0
DIF: Easy
REF: 6.3 Constructing Triangles
11.T3
TOP: Trigonometry
KEY: Procedural Knowledge
A
PTS: 0
DIF: Moderate
REF: 6.3 Constructing Triangles
11.T3
TOP: Trigonometry
KEY: Procedural Knowledge
B
PTS: 0
DIF: Easy
REF: 6.4 The Sine Law
11.T3
TOP: Trigonometry
KEY: Procedural Knowledge
B
PTS: 0
DIF: Moderate
REF: 6.4 The Sine Law
11.T3
TOP: Trigonometry
Conceptual Understanding | Procedural Knowledge
D
PTS: 0
DIF: Easy
REF: 7.5 Solving Rational Equations
11.AN6
TOP: Algebra and Number
KEY: Procedural Knowledge
A
PTS: 0
DIF: Moderate
REF: 7.5 Solving Rational Equations
11.AN6
TOP: Algebra and Number
KEY: Procedural Knowledge
D
PTS: 0
DIF: Easy
REF: 8.1 Absolute Value Functions
11.RF2
TOP: Relations and Functions
Conceptual Understanding | Procedural Knowledge
A
PTS: 0
DIF: Moderate
REF: 8.1 Absolute Value Functions
11.RF2
TOP: Relations and Functions
KEY:
70. ANS:
REF:
TOP:
71. ANS:
REF:
TOP:
72. ANS:
REF:
TOP:
73. ANS:
REF:
TOP:
74. ANS:
REF:
TOP:
75. ANS:
REF:
TOP:
Procedural Knowledge | Communication
C
PTS: 0
DIF: Easy
8.2 Solving Absolute Value Equations
LOC: 11.RF2
Relations and Functions
KEY: Procedural Knowledge
C
PTS: 0
DIF: Moderate
8.3 Graphing Reciprocals of Linear Functions
LOC: 11.RF11
Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
A
PTS: 0
DIF: Moderate
8.3 Graphing Reciprocals of Linear Functions
LOC: 11.RF11
Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
A
PTS: 0
DIF: Moderate
8.3 Graphing Reciprocals of Linear Functions
LOC: 11.RF11
Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
A
PTS: 0
DIF: Difficult
8.3 Graphing Reciprocals of Linear Functions
LOC: 11.RF11
Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
C
PTS: 0
DIF: Moderate
8.3 Graphing Reciprocals of Linear Functions
LOC: 11.RF11
Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
SHORT ANSWER
1. ANS:
No, this could not be an arithmetic sequence because the difference between consecutive terms is not
constant.
PTS: 1
LOC: 11.RF9
2. ANS:
DIF: Easy
REF: 1.1 Arithmetic Sequences
TOP: Relations and Functions
KEY: Conceptual Understanding
PTS: 1
LOC: 11.RF9
3. ANS:
1
r = , t5 = 5, t6 = 1
5
DIF: Difficult
REF: 1.2 Arithmetic Series
TOP: Relations and Functions
KEY: Conceptual Understanding
PTS: 1
LOC: 11.RF10
4. ANS:
i)
ii)
1
iii) 4
5
iv)
DIF: Moderate
REF: 1.3 Geometric Sequences
TOP: Relations and Functions
KEY: Procedural Knowledge
1
The values of the expressions from greatest to least are: 49, 37, 4 , 4
5
PTS: 0
DIF: Moderate
REF: 2.1 Absolute Value of a Real Number
LOC: 11.AN1
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
5. ANS:
i)
ii)
iii)
iv)
The values of the expressions from least to greatest are: –8, 8, 10, 26
PTS: 0
DIF: Moderate
REF: 2.1 Absolute Value of a Real Number
LOC: 11.AN1
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
6. ANS:
a)
b)
c) The equation has no real root.
d)
PTS: 0
DIF: Moderate
REF: 2.5 Solving Radical Equations
LOC: 11.AN3
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
7. ANS:
PTS: 0
DIF: Moderate
REF: 3.2 Solving Quadratic Equations by Factoring
LOC: 11.RF5
TOP: Relations and Functions
KEY: Problem-Solving Skills | Procedural Knowledge
8. ANS:
The baseball is 11 m high after 1 s and after 2 s.
PTS: 0
DIF: Moderate
REF: 3.2 Solving Quadratic Equations by Factoring
LOC: 11.RF5
TOP: Relations and Functions
KEY: Problem-Solving Skills | Procedural Knowledge
9. ANS:
There are 2 numbers: 5 and –8
PTS: 0
DIF: Moderate
REF: 3.2 Solving Quadratic Equations by Factoring
LOC: 11.RF5
TOP: Relations and Functions
KEY: Problem-Solving Skills | Procedural Knowledge
10. ANS:
a)
b) The square root of the discriminant is rational, so the roots are rational.
PTS: 0
DIF: Moderate
REF: 3.5 Interpreting the Discriminant
LOC: 11.RF5
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
11. ANS:
a) x-intercepts: none
y-intercept: 9
b) vertex: (–2, 3)
c) axis of symmetry:
d) domain:
e) range:
,
PTS: 0
DIF: Moderate
REF: 4.1 Properties of a Quadratic Function
LOC: 11.RF4
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
12. ANS:
a) The company should produce 1000 units per week to earn the maximum weekly profit.
b) The maximum weekly profit is $600.
PTS: 0
DIF: Moderate
REF: 4.1 Properties of a Quadratic Function
LOC: 11.RF4
TOP: Relations and Functions
KEY: Problem-Solving Skills | Procedural Knowledge
13. ANS:
The graph intersects the x-axis at 2 points, so the related quadratic equation has 2 real roots. This means
that the discriminant is greater than 0.
PTS: 0
DIF: Moderate
REF: 4.2 Solving a Quadratic Equation Graphically
LOC: 11.RF5
TOP: Relations and Functions
KEY: Communication | Conceptual Understanding
14. ANS:
PTS: 0
DIF: Moderate
REF: 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q
LOC: 11.RF3
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
15. ANS:
a) The graph has 1 x-intercept.
b) The graph has 0 x-intercepts.
c) The graph has 2 x-intercepts.
PTS: 0
DIF: Moderate
REF: 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q
LOC: 11.RF3
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
16. ANS:
a) The graph has 2 x-intercepts.
b) The graph has 1 x-intercept.
c) The graph has 0 x-intercepts.
PTS:
REF:
LOC:
KEY:
17. ANS:
0
DIF: Moderate
4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q
11.RF3
TOP: Relations and Functions
Conceptual Understanding | Procedural Knowledge
PTS: 0
DIF: Difficult
REF: 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q
LOC: 11.RF3
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
18. ANS:
maximum value; –2
PTS: 0
DIF: Easy
REF: 4.7 Modelling and Solving Problems with Quadratic Functions
LOC: 11.RF4
TOP: Relations and Functions
KEY: Conceptual Understanding
19. ANS:
$1.83
PTS: 0
DIF: Moderate
REF: 4.7 Modelling and Solving Problems with Quadratic Functions
LOC: 11.RF4
TOP: Relations and Functions
KEY: Problem-Solving Skills | Procedural Knowledge
20. ANS:
The solution is:
,
1.5
–9 –8 –7 –6 –5 –4 –3 –2 –1
PTS: 0
LOC: 11.RF8
21. ANS:
0
1
2
3
4
5
DIF: Moderate
REF: 5.1 Solving Quadratic Inequalities in One Variable
TOP: Relations and Functions
KEY: Procedural Knowledge
a) The critical values are
and
.
b)
Interval
Sign of
Value of x
positive
negative
positive
c) The solution is:
or
,
PTS: 0
DIF: Moderate
REF: 5.1 Solving Quadratic Inequalities in One Variable
LOC: 11.RF8
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
22. ANS:
a)
y
6
4
2
–6
–4
0
–2
2
4
6
x
–2
–4
–6
b) Sample response:
Two points that satisfy the inequality have coordinates: (0, 3), (–2, –1)
PTS: 0
DIF: Moderate
REF: 5.3 Graphing Quadratic Inequalities in Two Variables
LOC: 11.RF7
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
23. ANS:
y
6
4
2
–6
–4
–2
0
2
4
6
x
–2
–4
–6
PTS: 0
LOC: 11.RF7
24. ANS:
a) and b)
DIF: Moderate
REF: 5.3 Graphing Quadratic Inequalities in Two Variables
TOP: Relations and Functions
KEY: Procedural Knowledge
y
8
6
4
2
–3
(–1, 0)
–2
–1
0
1
2
3
x
–2
–4
PTS: 0
LOC: 11.RF6
25. ANS:
DIF: Moderate
REF: 5.4 Solving Systems of Equations Graphically
TOP: Relations and Functions
KEY: Procedural Knowledge
y
20
(–3, 16)
16
12
8
4
–10
–8
(–7, 0)
–6
–4
–2
0
2
4
6
8
x
–4
–8
PTS: 0
LOC: 11.RF6
26. ANS:
DIF: Moderate
REF: 5.4 Solving Systems of Equations Graphically
TOP: Relations and Functions
KEY: Procedural Knowledge
y
20
16
12
8
(2, 9)
4
–8
–6
–4
–2
0
2
4
6
8
x
–4
–8
PTS: 0
DIF: Moderate
REF: 5.4 Solving Systems of Equations Graphically
LOC: 11.RF6
TOP: Relations and Functions
KEY: Procedural Knowledge
27. ANS:
A linear-quadratic system may have 2 solutions, 1 solution, or no solution.
A quadratic-quadratic system may have 2 solutions, 1 solution, no solution, or infinite solutions.
PTS: 0
DIF: Easy
REF: 5.5 Solving Systems of Equations Algebraically
LOC: 11.RF6
TOP: Relations and Functions
KEY: Communication | Conceptual Understanding
28. ANS:
The solutions are: (2, 0) and (0, 4)
PTS: 0
DIF: Moderate
REF: 5.5 Solving Systems of Equations Algebraically
LOC: 11.RF6
TOP: Relations and Functions
KEY: Procedural Knowledge
29. ANS:
The solutions are: (0, 2) and (–1, –5)
PTS: 0
DIF: Moderate
REF: 5.5 Solving Systems of Equations Algebraically
LOC: 11.RF6
TOP: Relations and Functions
KEY: Procedural Knowledge
30. ANS:
a) After 0.4 s
b) Approximately 4.0 m
PTS: 0
DIF: Moderate
REF: 5.5 Solving Systems of Equations Algebraically
LOC: 11.RF6
TOP: Relations and Functions
KEY: Problem-Solving Skills | Procedural Knowledge
31. ANS:
The solutions are approximately: (1.7, –7.6) and (–0.4, 0.1)
PTS: 0
LOC: 11.RF6
32. ANS:
The solutions are:
DIF: Moderate
REF: 5.5 Solving Systems of Equations Algebraically
TOP: Relations and Functions
KEY: Procedural Knowledge
and
PTS: 0
DIF: Moderate
REF: 5.5 Solving Systems of Equations Algebraically
LOC: 11.RF6
TOP: Relations and Functions
KEY: Procedural Knowledge
33. ANS:
approximately 8.7 m
PTS: 0
DIF: Moderate
REF: 6.1 Angles in Standard Position in Quadrant 1
LOC: 11.T2
TOP: Trigonometry
KEY: Procedural Knowledge | Problem-Solving Skills
34. ANS:
approximately 6.1 m
PTS: 0
DIF: Moderate
REF: 6.1 Angles in Standard Position in Quadrant 1
LOC: 11.T2
TOP: Trigonometry
KEY: Procedural Knowledge | Problem-Solving Skills
35. ANS:
approximately 73.7
PTS: 0
DIF: Moderate
REF: 6.1 Angles in Standard Position in Quadrant 1
LOC: 11.T2
TOP: Trigonometry
KEY: Procedural Knowledge | Problem-Solving Skills
36. ANS:
approximately 27.7
PTS: 0
DIF: Moderate
REF: 6.1 Angles in Standard Position in Quadrant 1
LOC: 11.T2
TOP: Trigonometry
KEY: Procedural Knowledge | Problem-Solving Skills
37. ANS:
approximately 50.0
PTS: 0
DIF: Moderate
REF: 6.1 Angles in Standard Position in Quadrant 1
LOC: 11.T2
TOP: Trigonometry
KEY: Procedural Knowledge | Problem-Solving Skills
38. ANS:
and
PTS: 0
DIF: Moderate
REF: 6.2 Angles in Standard Position in All Quadrants
LOC: 11.T2
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
39. ANS:
It is not possible to construct ABC.
PTS: 0
DIF: Moderate
REF: 6.3 Constructing Triangles
LOC: 11.T3
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
40. ANS:
No, the Sine Law cannot be used because only one angle measure is given and the angle is contained
between the two given sides.
PTS: 0
DIF: Easy
REF: 6.4 The Sine Law
LOC: 11.T3
TOP: Trigonometry
KEY: Conceptual Understanding | Communication
41. ANS:
Two triangles can be constructed.
PTS: 0
DIF: Easy
REF: 6.4 The Sine Law
LOC: 11.T3
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
42. ANS:
a)
...
Since
, no triangle can be constructed.
b) No triangle can be constructed.
PTS: 0
DIF: Moderate
REF: 6.4 The Sine Law
LOC: 11.T3
TOP: Trigonometry
KEY: Conceptual Understanding | Communication
43. ANS:
x
–2
–1
0
1
2
3
–4
–3
–2
–1
0
1
4
3
2
1
0
1
PTS: 0
DIF: Easy
REF: 8.1 Absolute Value Functions
LOC: 11.RF2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
44. ANS:
x
–2
–1
0
1
2
3
–16
–14
–8
2
16
34
16
14
8
2
16
34
PTS: 0
DIF: Easy
REF: 8.1 Absolute Value Functions
LOC: 11.RF2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
45. ANS:
PTS: 0
DIF: Moderate
REF: 8.1 Absolute Value Functions
LOC: 11.RF2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
46. ANS:
PTS: 0
DIF: Moderate
REF: 8.1 Absolute Value Functions
LOC: 11.RF2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
47. ANS:
PTS: 0
DIF: Moderate
REF: 8.1 Absolute Value Functions
LOC: 11.RF2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
48. ANS:
The solutions are
and
.
PTS: 0
DIF: Easy
REF: 8.2 Solving Absolute Value Equations
LOC: 11.RF2
TOP: Relations and Functions
KEY: Conceptual Understanding
49. ANS:
The student might have used the graph to solve the equation:
PTS: 0
DIF: Moderate
REF: 8.2 Solving Absolute Value Equations
LOC: 11.RF2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
50. ANS:
The student might have used the graph to solve the equation:
PTS: 0
DIF: Difficult
REF: 8.2 Solving Absolute Value Equations
LOC: 11.RF2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge
PROBLEM
1. ANS:
a) The common difference, d, is:
b)
c)
PTS: 1
LOC: 11.RF9
2. ANS:
DIF: Difficult
REF: 1.1 Arithmetic Sequences
TOP: Relations and Functions
KEY: Problem-Solving Skills
a)
b)
PTS: 1
DIF: Moderate
REF: 1.2 Arithmetic Series
LOC: 11.RF9
TOP: Relations and Functions
KEY: Conceptual Understanding | Problem-Solving Skills
3. ANS:
a)
The first 3 terms are:
b) The sequence is convergent because the terms approach a constant value of 0.
PTS: 1
DIF: Moderate
REF: 1.3 Geometric Sequences
LOC: 11.RF10
TOP: Relations and Functions
KEY: Communication | Conceptual Understanding | Problem-Solving Skills
4. ANS:
Sample response:
This sum comprises 3 geometric series:
1 + 4 + 16 + 64 + 256
3 + 6 + 12 + 24 + 48 + 96 + 192 + 384
2 + 8 + 32 + 128 + 512
For the first series:
For the second series:
For the third series:
PTS: 1
DIF: Difficult
REF: 1.4 Geometric Series
LOC: 11.RF10
TOP: Relations and Functions
KEY: Conceptual Understanding | Problem-Solving Skills
5. ANS:
To determine t1, use:
The 1st jump is approximately 9.00 m.
To determine t2, use:
The 2nd jump is approximately 2.25 m.
To determine t3, use:
The 3rd jump is approximately 0.56 m.
PTS: 1
DIF: Difficult
REF: 1.4 Geometric Series
LOC: 11.RF10
TOP: Relations and Functions
KEY: Communication | Conceptual Understanding | Problem-Solving Skills
6. ANS:
The payments form a geometric series, with first term 56, common ratio 1.021, and number of terms 16.
Joel earned approximately $1051.94 after 16 days.
PTS: 1
DIF: Moderate
REF: 1.4 Geometric Series
LOC: 11.RF10
TOP: Relations and Functions
KEY: Communication | Conceptual Understanding | Problem-Solving Skills
7. ANS:
The length of the frog’s first jump was approximately 45 cm.
PTS: 1
DIF: Difficult
REF: 1.4 Geometric Series
LOC: 11.RF10
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
8. ANS:
Sample response:
For the term values to alternate between positive and negative,and increase in numerical value, the
25 125
common ratio must be
; for example, with
, a possible sequence is: –4, 5,  ,
,
4 16
625 3125

,
, ...
64 256
PTS: 1
DIF: Moderate
REF: 1.5 Graphing Geometric Sequences and Series
LOC: 11.RF9 | 11.RF10
TOP: Relations and Functions
KEY: Communication | Conceptual Understanding | Procedural Knowledge
9. ANS:
a)
b)
c)
Geom etric Sequence
1000
800
600
Term value
400
200
0
–200
1 2 3 4 5 6 7
Term num ber
–400
–600
–800
–1000
To graph the geometric series, determine the partial sums.
S1
3
S2
S3
S4
S5
S6
1600
Geom etric Series
1200
Partial sums
800
400
0
–400
1 2 3 4 5 6 7
Num ber of term s
–800
–1200
–1600
PTS: 1
DIF: Moderate
REF: 1.5 Graphing Geometric Sequences and Series
LOC: 11.RF9 | 11.RF10
TOP: Relations and Functions
KEY: Communication | Conceptual Understanding | Procedural Knowledge
10. ANS:
a)
t2 is: –5
t3 is:
5
9
t4 is: 
b)
5
9

5
81
5
81
5
729
, so the series converges.
c)
PTS: 1
DIF: Moderate
REF: 1.6 Infinite Geometric Series
LOC: 11.RF10
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
11. ANS:
Use the formula for the area, A, of a rectangle:
Substitute:
and
Perimeter of shape formed
An expression for the perimeter of the shape formed is:
PTS: 0
DIF: Moderate
REF: 2.3 Adding and Subtracting Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Procedural Knowledge | Communication | Problem-Solving Skills
12. ANS:
PTS: 0
DIF: Easy
REF: 2.4 Multiplying and Dividing Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Procedural Knowledge | Communication
13. ANS:
Use the formula for the area, A, of a square:
PTS: 0
DIF: Moderate
REF: 2.4 Multiplying and Dividing Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Procedural Knowledge | Communication
14. ANS:
Simplify the denominators.
Rationalize the denominators.
PTS: 0
DIF: Difficult
REF: 2.4 Multiplying and Dividing Radical Expressions
LOC: 11.AN2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge | Communication
15. ANS:
Since the left side does not equal the right side,
is not a root of the equation.
PTS: 0
DIF: Easy
REF: 2.5 Solving Radical Equations
LOC: 11.AN3
TOP: Relations and Functions
KEY: Procedural Knowledge | Communication
16. ANS:
a) Use the formula for the volume, V, of a cube:
, where s represents the edge length of the cube.
A cube with the same volume as Lake Ontario would have an edge length of about 12.0 km.
b) Use the formula for the volume, V, of a sphere:
, where r represents the radius of the sphere.
A sphere with the same volume as Lake Ontario would have a radius of about 7.4 km.
PTS: 0
DIF: Difficult
REF: 2.5 Solving Radical Equations
LOC: 11.AN3
TOP: Relations and Functions
KEY: Procedural Knowledge | Problem-Solving Skills
17. ANS:
Use the Pythagorean Theorem.
Solve by factoring.
Since length cannot be negative,
.
The length of the shorter leg is 20 cm.
The length of the longer leg is:
PTS: 0
DIF: Difficult
REF: 3.2 Solving Quadratic Equations by Factoring
LOC: 11.RF5
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
18. ANS:
For an equation to have two real roots,
Substitute:
For
to have two real roots, k must be less than 3.
Sample response: A possible value of k is 2. So, an equation with two real roots is:
PTS: 0
DIF: Moderate
REF: 3.5 Interpreting the Discriminant
LOC: 11.RF5
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
19. ANS:
For an equation to have no real roots,
Substitute:
For
9
to have no real roots, k must be greater than .
8
Sample response: A possible value of k is 3. So, an equation with no real roots is:
PTS: 0
DIF: Moderate
REF: 3.5 Interpreting the Discriminant
LOC: 11.RF5
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
20. ANS:
Use guess and test to determine two values of a, b, and c so that:
Substitute:
This is satisfied by
So, one equation is:
and
.
and
.
Substitute:
This is satisfied by
So, another equation is:
PTS: 0
DIF: Difficult
REF: 3.5 Interpreting the Discriminant
LOC: 11.RF5
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
21. ANS:
a) x-intercepts: –1.70, 0.95
y-intercept: 3.25
b) vertex: (–0.38, 3.53)
c) axis of symmetry:
d) domain:
e) range:
,
PTS: 0
DIF: Moderate
REF: 4.1 Properties of a Quadratic Function
LOC: 11.RF4
TOP: Relations and Functions
KEY: Communication | Procedural Knowledge
22. ANS:
Compare
with the vertex form
.
a) a is positive, so the graph opens up.
and
, so the coordinates of the vertex are: (2, 4)
The equation of the axis of symmetry is
; that is
.
To determine the y-intercept, substitute
.
The y-intercept is 6.
To determine the x-intercepts, substitute
.
This equation has no solution, so there are no x-intercepts.
The domain is:
The graph opens up, so the vertex is a minimum point with y-coordinate 4. The range is:
b)
y
20
16
12
8
4
–12
–8
–4
0
4
8
12
x
–4
PTS: 0
DIF: Moderate
REF: 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q
LOC: 11.RF3
TOP: Relations and Functions
KEY: Communication | Conceptual Understanding | Procedural Knowledge
23. ANS:
Sketch the parabola on the coordinate plane so the line of symmetry is the y-axis.
The y-intercept represents the maximum height.
The vertex of the parabola is 27 m above the base, so the coordinates of the vertex are V(0, 27).
Since the bridge is 72 m wide, the x-intercepts are 36 and 36.
,
y
30
24
18
12
6
–36 –30 –24 –18 –12
–6
0
6
12
18
24
30
36 x
–6
The equation of the parabola has the form
, with vertex (p, q).
The coordinates of the vertex are V(0, 27), so
and
.
So, the equation of the parabola becomes
.
To determine the value of a, substitute the coordinates of an x-intercept: (36, 0)
An equation of the parabola is:
PTS: 0
DIF: Difficult
REF: 4.4 Analyzing Quadratic Functions of the Form y = a(x - p)^2 + q
LOC: 11.RF3
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
24. ANS:
Determine an equation to represent the situation.
For each $1 decrease in price, 125 more tickets will be sold. Let x represent the number of $1 decreases in
the price of a ticket.
When the price decreases by $1 x times:
• the price, in dollars, of a ticket is
• the number of tickets sold is
• the revenue, in dollars, is
.
.
.
Let the revenue be R dollars.
An equation is:
Use a graphing calculator.
Graph:
Use the CALC function to determine the coordinates of the vertex.
a) From the graph, the maximum revenue occurs when the number of $1 decreases is 4. So, the decrease
in price that will maximize the revenue is $4.
b) The price of a ticket that will maximize the revenue is:
c) Substitute
in
to determine the maximum revenue.
The maximum revenue is $50 000.
PTS: 0
DIF: Difficult
REF: 4.7 Modelling and Solving Problems with Quadratic Functions
LOC: 11.RF4
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
25. ANS:
a) Solve:
When
, such as
, L.S. = 1 and R.S. = 0; so
When
, such as
, L.S. = 1 and R.S. = 0; so
The solution is:
, x  2.5
b)
satisfies the inequality.
satisfies the inequality.
2.5
–7 –6 –5 –4 –3 –2 –1
0
1
2
3
4
5
6
7
c) There is only one number that does not satisfy the inequality.
Since the square of any non-zero number is positive, any real number but 2.5 satisfies the inequality.
PTS: 0
DIF: Moderate
REF: 5.1 Solving Quadratic Inequalities in One Variable
LOC: 11.RF8
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
26. ANS:
a) For
to have 2 real roots, its discriminant is greater than 0.
b) For
to have exactly 1 root, its discriminant is 0.
c) For
to have no real roots, its discriminant is negative.
, which is written
PTS: 0
DIF: Difficult
REF: 5.1 Solving Quadratic Inequalities in One Variable
LOC: 11.RF8
TOP: Relations and Functions
KEY: Communication | Conceptual Understanding
27. ANS:
Write an inequality to represent the problem.
Let x m represent the width of the garden.
Then its length is
m.
And its area, in square metres, is
.
The area is at least 20 m2.
So, an inequality is:
Use a graphing calculator to graph the corresponding quadratic function:
Determine the x-intercepts:
and
From the graph, the inequality is greater than or equal to 0 for
Since the width of the garden is positive, the solution of the problem is:
The width of the garden is greater than or equal to approximately 2.4 m and its length is greater than or
equal to approximately
, or 8.4 m.
Verify the solution.
The area of the garden with dimensions 2.4 m and 8.4 m is:
This is greater than 20 m2, so the possible dimensions are correct.
PTS: 0
DIF: Difficult
REF: 5.1 Solving Quadratic Inequalities in One Variable
LOC: 11.RF8
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
28. ANS:
a) Since
, the graph is in Quadrant 4.
The graph of the related function has slope –3 and y-intercept 4.
Draw a broken line to represent the related function in Quadrant 4.
Shade the region below the line.
The axes bounding the graph are broken lines.
y
6
4
2
–6
–4
–2
0
2
4
6 x
–2
–4
–6
b) Since
, the graph is in Quadrant 3.
Graph the related function.
When
,
.
When
,
.
Draw a solid line to represent the related function in Quadrant 3.
Use (0, 0) as a test point.
L.S. = 6; R.S. = 0
Since 6 > 0, the origin is not in the shaded region.
Shade the region below the line.
The axes bounding the graph are solid lines.
y
6
4
2
–6
–4
0
–2
2
4
6 x
–2
–4
–6
PTS: 0
DIF: Difficult
REF: 5.2 Graphing Linear Inequalities in Two Variables
LOC: 11.RF7
TOP: Relations and Functions
KEY: Communication | Procedural Knowledge
29. ANS:
In
, substitute:
,
Solve for a.
Divide both sides by –3.
Square both sides.
That is,
PTS: 0
DIF: Moderate
REF: 5.3 Graphing Quadratic Inequalities in Two Variables
LOC: 11.RF7
TOP: Relations and Functions
KEY: Communication | Procedural Knowledge
30. ANS:
10000
y
8000
(40, 5400)
(50, 5000)
6000
4000
2000
–60
–40
–20
0
–2000
20
40
60
80
100
x
PTS: 0
DIF: Moderate
REF: 5.4 Solving Systems of Equations Graphically
LOC: 11.RF6
TOP: Relations and Functions
KEY: Problem-Solving Skills | Procedural Knowledge
31. ANS:
The line has slope 3 and y-intercept 4, so its equation is:
The parabola has vertex (5) and is congruent to
. So, its equation is:
The system of equations represented by the graph is:
The coordinates of the points of intersection are: (0, 4) and (5, 11)
PTS: 0
DIF: Difficult
REF: 5.4 Solving Systems of Equations Graphically
LOC: 11.RF6
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
32. ANS:
a) Let x represent the first number and let y represent the second number.
The statement that twice the square of the first number minus the second number is 3 can be
modelled with the equation:
The statement that the square of the sum of the first number and 5 is equal to the second number
minus 2 can be modelled with the equation:
So, the system of equations that represents this relationship is:


b) From equation , substitute
in equation .

So,
or
Substitute each value of x in equation .
When
:
When
:
The numbers are: 2 and 11; 12 and 291
Verify the solutions using the statement of the problem.
For
and
:
2
Two times (2) minus 11 is 3.
The square of
is equal to
These numbers satisfy the problem statement.
For
and
:
2
Two times (12) minus 291 is 3.
is equal to 291 minus 2.
These numbers satisfy the problem statement.
So, the numbers are: 2 and 11; 12 and 291
PTS: 0
DIF: Difficult
REF: 5.5 Solving Systems of Equations Algebraically
LOC: 11.RF6
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
33. ANS:
a) Write an equation that represents the height of the soccer ball t seconds after the football is kicked.
Solve the system formed by the 2 equations:


From equation , substitute
in equation .
The football and the soccer ball reached the same height after approximately 1.5 s.
b) Substitute
in equation .
After 1.5 s, both balls are at a height of approximately 4.9 m.
PTS: 0
DIF: Difficult
REF: 5.5 Solving Systems of Equations Algebraically
LOC: 11.RF6
TOP: Relations and Functions
KEY: Communication | Problem-Solving Skills
34. ANS:
First determine the measure of B.
In right ABC,
In a right triangle, when one acute angle is , the other acute angle is
.
So, B is approximately 58.4 and A is approximately 31.6.
PTS: 0
DIF: Moderate
REF: 6.1 Angles in Standard Position in Quadrant 1
LOC: 11.T2
TOP: Trigonometry
KEY: Conceptual Understanding | Communication
35. ANS:
a) In right BCD, BD is the hypotenuse.
BC is the side opposite angle  and CD is the side adjacent to angle .
b)
sin  is approximately 0.6. This means that in any right triangle similar to BCD, the length of the
side opposite angle  is approximately 0.6 times the length of the hypotenuse.
c)

 is approximately 34.8.
PTS: 0
DIF: Moderate
REF: 6.1 Angles in Standard Position in Quadrant 1
LOC: 11.T2
TOP: Trigonometry
KEY: Procedural Knowledge | Communication
36. ANS:
Draw a labelled diagram to represent the problem.
In right FLP, FP is the hypotenuse and FL is the side opposite P.
So, use the sine ratio to determine the length of FL.
F
9.1 km
Solve the equation for FL.
P
54°
L
The distance between the fishing boat and the lighthouse is approximately 7.4 km.
PTS: 0
DIF: Moderate
REF: 6.2 Angles in Standard Position in All Quadrants
LOC: 11.T2
TOP: Trigonometry
KEY: Communication | Problem-Solving Skills
37. ANS:
a)
, or 0.5428...
Since
, there are two possible triangles with the given measures.
b)
B
35 cm
19 cm
19 cm
30°
A
C1
C2
PTS: 0
DIF: Moderate
REF: 6.3 Constructing Triangles
LOC: 11.T3
TOP: Trigonometry
KEY: Communication | Problem-Solving Skills
38. ANS:
Possible solution:
Length of BC (cm)
Value of
How does
compare with sin A?
Description of possible
triangles
5
0.8333...
No triangles are possible.
6
1
1 isosceles triangle
7
1.1666...
1 scalene triangle
5.9
0.9833...
2 scalene triangles
PTS: 0
DIF: Moderate
REF: 6.3 Constructing Triangles
LOC: 11.T3
TOP: Trigonometry
KEY: Conceptual Understanding | Problem-Solving Skills
39. ANS:
Sketch a diagram to represent their trip from A, through B, to C.
Determine the measure of B in ABC.
B
24°
In ABC:
3.4 km
28°
2.4 km

24
C
Determine the measure of angle .
B
24°
3.4 km
28°
Determine the angle bearing, .
2.4 km

C
The straight-line distance is approximately 2.1 km.
The bearing of the end point from the start point is approximately 250°.
PTS: 1
DIF: Difficult
REF: 6.5 The Cosine Law
LOC: 11.T3
TOP: Trigonometry
KEY: Procedural Knowledge | Communication | Problem-Solving Skills
40. ANS:
Use:
24
Use:
PTS: 1
DIF: Moderate
REF: 6.5 The Cosine Law
LOC: 11.T3
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
41. ANS:
Factor the expressions and simplify.
Since the expressions have the same denominator but different numerators, the equation has no solution.
PTS: 0
DIF: Moderate
REF: 7.5 Solving Rational Equations
LOC: 11.AN6
TOP: Algebra and Number
KEY: Conceptual Understanding | Communication
42. ANS:
First, identify the non-permissible values and a common denominator:
The non-permissible values are:
and
A common denominator is:
Then, multiply each term in the equation by the common denominator and simplify.
Finally, solve the equation
using the quadratic formula.
The solutions are:
and
PTS: 0
DIF: Difficult
REF: 7.5 Solving Rational Equations
LOC: 11.AN6
TOP: Algebra and Number
KEY: Procedural Knowledge | Communication | Problem-Solving Skills
43. ANS:
x
–2
–1
0
1
2
3
–6
–4
–2
0
2
4
6
4
2
0
2
4
y
8
6
4
y=|f(x)|
2
–8
–6
–4
–2
y=f(x)
0
2
4
6
8 x
–2
–4
–6
–8
PTS: 0
DIF: Moderate
REF: 8.1 Absolute Value Functions
LOC: 11.RF2
TOP: Relations and Functions
KEY: Procedural Knowledge | Communication
44. ANS:
Choose two points on the line to determine the
slope of the linear function:
(–5, 5) and (–1, –3)
8
y
6
4
2
y-intercept: –5
An equation for the absolute value function is:
–8
–6
–4
0
–2
2
4
6
8 x
–2
–4
–6
–8
PTS: 0
DIF: Difficult
REF: 8.1 Absolute Value Functions
LOC: 11.RF2
TOP: Relations and Functions
KEY: Procedural Knowledge | Communication | Problem-Solving Skills
45. ANS:
Assume the middle piece of the graph was
y
reflected in the x-axis. So, the graph of the
8
quadratic function opens up and has vertex
(4, – 5). The equation has the form
6
Use the point (5, 0) to determine a.
4
2
An equation of the absolute value function is:
–8
–6
–4
–2
0
2
–2
–4
–6
–8
PTS: 0
DIF: Difficult
REF: 8.1 Absolute Value Functions
LOC: 11.RF2
TOP: Relations and Functions
KEY: Procedural Knowledge | Communication | Problem-Solving Skills
46. ANS:
Simplify the equation:
The absolute value function
creates two quadratic equations:
4
6
8 x
and
So, the solutions are
,
, and
.
PTS: 0
DIF: Moderate
REF: 8.2 Solving Absolute Value Equations
LOC: 11.RF2
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge | Communication
47. ANS:
a) After 1 h, the motorcycle has travelled 95 km. After t hours, the motorcycle has travelled 95t
kilometres. The distance from the border after t h can be represented by the equation:
b) Substitute: t = 1.5
So, the motorcycle is approximately 113 km from the border after 1.5 h.
PTS: 0
DIF: Moderate
REF: 8.2 Solving Absolute Value Equations
LOC: 11.RF2
TOP: Relations and Functions
KEY: Conceptual Understanding | Communication | Problem-Solving Skills
48. ANS:
The graph of
passes through the points (2, 6), (6, 6), (3, 3), and (5, 3).
Plot these points on the coordinate
y
grid.
The points are symmetrical about the
20
line
, so plot a point at (4, 0).
Join the points with straight lines.
16
An equation for the right branch of
the graph has the form
.
12
Use the points (4, 0) and (3, 3) to
8
determine the slope, m.
4
–16 –12
–8
–4
0
4
8
12
16 x
–4
An equation of the line is:
Use the point (2, 6) to determine the y-intercept, b.
So, an equation for the absolute value function is
PTS: 0
LOC: 11.RF2
.
DIF: Difficult
REF: 8.2 Solving Absolute Value Equations
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge | Communication
49. ANS:
An equation of the line has the form
.
Use the points
and
to determine m and b.
y
8
6
4
So, an equation of the linear function is
For the graph of the reciprocal function:
.
The equation is:
2
–6
Horizontal asymptote:
x-intercept is 0, so vertical asymptote is
–4
–2
0
2
4
6
x
–2
.
–4
–6
–8
Mark points at y = 1 and y =
on the graph of
.
Draw a smooth curve through each point so that the curve approaches the asymptotes but never touches
them.
PTS: 0
DIF: Difficult
REF: 8.3 Graphing Reciprocals of Linear Functions
LOC: 11.RF11
TOP: Relations and Functions
KEY: Conceptual Understanding | Procedural Knowledge | Communication
50. ANS:
a) Use the formula for speed:
Substitute: d = 1
b) Both time and speed are positive.
The reciprocal function has vertical asymptote
So, the domain is
and the range is
c) A sketch of the reciprocal function is:
and horizontal asymptote
.
.
t
10
8
6
4
2
0
2
4
6
8
10
s
PTS: 0
DIF: Moderate
REF: 8.3 Graphing Reciprocals of Linear Functions
LOC: 11.RF11
TOP: Relations and Functions
KEY: Procedural Knowledge | Communication | Problem-Solving Skills