Math 20-2 Final Review

(1)

Math 20-2 Final Review

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 1. Guilia created the following table to show a pattern.

Multiples of 9 18 27 36 45 54

Sum of the Digits 9 9 9 9 9

Which conjecture could Guilia make, based solely on this evidence? Choose the best answer.

a. The sum of the digits of a multiple of 9 is equal to 9. b. The sum of the digits of a multiple of 9 is an odd integer. c. The sum of the digits of a multiple of 9 is divisible by 9.

d. Guilia could make any of the above conjectures, based on this evidence.. ____ 2. Siddartha made the following conjecture.

When you divide two whole numbers, the quotient will be greater than the divisor and less than the dividend.

Which choice, if either, is a counterexample to this conjecture?

1. 4

8 = 0.5

2. 12

4 = 3

a. Choice 1 only

b. Neither Choice 1 nor Choice 2

c. Choice 1 and Choice 2

d. Choice 2 only

____ 3. Which of the following choices, if any, uses deductive reasoning to show that the sum of three even integers is even?

a. x + y + z = 2(x + y + z)

b. 2 + 4 + 6 = 12 and 4 + 6 + 8 = 18 c. 2x + 2y + 2z = 2(x + y + z)

(2)

____ 4. Which type of reasoning does the following statement demonstrate? All birds have feathers.

Robins are birds.

Therefore, robins have feathers.

a. inductive reasoning

b. deductive reasoning

c. neither inductive nor deductive reasoning ____ 5. Determine the unknown term in this pattern.

3, 6, 12, 24, ____, 96, 192

a. 48 b. 102 c. 96 d. 36

____ 6. Which number should appear in the centre of Figure 4?

Figure 1 Figure 2 Figure 3 Figure 4

a. 36 b. 24 c. 41 d. 11

____ 7. In which diagram(s) is AB parallel to CD?

1. 2.

a. Choice 1 only

b. Choice 2 only

c. Choice 1 and Choice 2

(3)

____ 8. Which angle property proves ∠DAB = 120°?

a. alternate exterior angles b. corresponding angles c. vertically opposite angles d. alternate interior angles

____ 9. Which are the correct measures of the interior angles of ΔCDE?

a. ∠DCE = 56°, ∠CDE = 101°, and ∠CED = 23°

b. ∠DCE = 46°, ∠CDE = 101°, and ∠CED = 33°

c. ∠DCE = 32°, ∠CDE = 83°, and ∠CED = 65°

d. ∠DCE = 76°, ∠CDE = 91°, and ∠CED = 13°

____ 10. Which are the correct measures for ∠WXZ, ∠UZY, and ∠VYX?

a. ∠WXZ = 162°, ∠UZY = 106°, and ∠VYX = 88°

b. ∠WXZ = 166°, ∠UZY = 109°, and ∠VYX = 89°

c. ∠WXZ = 152°, ∠UZY = 116°, and ∠VYX = 88°

(4)

____ 11. Determine the sum of the measures of the angles in a 12-sided convex polygon.

a. 1080° b. 3600° c. 2160° d. 1800°

____ 12. Determine the sum of the measures of the angles in a 9-sided convex polygon.

a. 1720° b. 1440° c. 1260° d. 1080°

____ 13. Which expression describes the ratios of side-angle pairs in ΔQRS?

a. q(sin R) = r(sin S) = s(sin Q)

b. q

sin S = rsin Q = ssin R

c. s

sin S =

q

sin Q = rsin R

d. q(sin Q) = r(sin R) = s(sin S)

____ 14. Determine the measure of θ to the nearest degree.

(5)

____ 15. In ΔDEF, d = 23.9 cm, e = 16.8 cm, and f = 27.0 cm. Determine the measure of ∠D to the nearest degree.

a. 61°

b. 64°

c. 58°

d. 54°

____ 16. In ΔXYZ, x = 18 cm, y = 14 cm, and z = 17 cm. Determine the measure of ∠Y to the nearest degree.

a. 43°

b. 47°

c. 45°

d. 49°

____ 17. Which choice expresses these numbers as the product of two radicals?

1089 , 1764 , 225 , 1024

a. 3 • 11 , 6 • 7 , 3 • 5 , 4 • 8

b. 9 • 121 , 36 • 49 , 9 • 25 , 16 • 64

c. 33 • 33, 42 • 42, 15 • 15, 32 • 32 d. 3 • 11, 6 • 7, 3 • 5, 4 • 8 ____ 18. Which set contains like radicals?

a. 14 2 , 5 16 , – 8 , 21 6

b. 6 3 , 12 , –2 3 , 27

c. 25 , –9 15 , 5 2 , 3 5

d. 17( 643 ), 10 4 , 7 16 , – 4

____ 19. Which is the simplest form of 7 8 – 2 72 – 50 ?

a. 0

b. –3 2

c. 21 2

(6)

____ 20. How many solutions are there for 45+ 12x − 6 = x? a. one: 2

b. none

c. two: 3, –3 d. one: 3

____ 21. Environment Canada compiled data on the number of lightning strikes per square kilometre in Alberta and British Columbia towns from 1999 to 2008.

0.42 0.04 0.81 0.40 0.03 0.74

0.28 0.03 0.70 0.23 0.03 0.66

0.13 0.02 0.61 0.12 0.01 0.58

0.10 0.00 0.49 0.07 1.08 0.43

0.05 0.91 0.42 0.04 0.88

What value goes in the fourth row of this frequency table?

Lightning Strikes (per square kilometre) Frequency 0.00–0.19 13 0.20–0.39 2 0.40–0.59 6 0.60–0.79 0.80–0.99 3 1.00–1.19 1 a. 5 b. 6 c. 3 d. 4

____ 22. At the end of a bowling tournament, three friends analyzed their scores. Erinn’s mean bowling score is 92 with a standard deviation of 14. Declan’s mean bowling score is 130 with a standard deviation of 18. Matt’s mean bowling score is 116 with a standard deviation of 22. Who had the highest scoring game during the tournament?

a. Declan

b. Matt

c. Impossible to tell.

(7)

____ 23. A company measured the lifespan of a random sample of 30 light bulbs. Times are in hours. 985 1001 1024 1087 952 910 938 931 1074 1081 1078 1080 982 1108 1022 937 922 1017 1093 1115 880 1048 917 1086 935 936 986 1038 954 966

Determine the standard deviation, to one decimal place.

a. 58.5 h b. 38.5 h c. 68.5 h d. 48.5 h

____ 24. Which description does not describe the normal curve? a. starts off increasing

b. symmetrical

c. shaped like a bell d. always increasing

____ 25. A teacher is analyzing the class results for a physics test. The marks are normally distributed with a mean (µ) of 76 and a standard deviation (σ) of 4.

Determine Olivia’s mark if she scored µ – σ.

a. 84

b. 68

c. 80

d. 72

____ 26. Determine the percent of data between the following z-scores:

z = –0.45 and z = –0.15.

a. 76.68%

b. 44.04%

c. 32.64%

d. 11.40%

____ 27. In a recent survey of high school students, 42% of those surveyed said that the food in the cafeteria was overpriced. The survey is considered accurate to within 6 percent points, 19 times out of 20.

If a high school has 1000 students, state the range of the number of students who would agree with the survey.

a. 520–640

b. 360–420

c. 420–480

(8)

____ 28. Which relation is the factored form of f(x) = x2_{+ 2x – 8?}

a. f(x) = (x + 2)(x – 4)

b. f(x) = 2(x + 2)(x – 2)

c. f(x) = (x – 1)(x + 8)

d. f(x) = (x – 2)(x + 4)

____ 29. Which relation is the factored form of f(x) = x2_{– 4x + 4?}

a. f(x) = (x – 2)2

b. f(x) = (x + 4)(x – 1)

c. f(x) = 4(x – 1)2

d. f(x) = (x – 2)(x + 2)

____ 30. Which quadratic function represents this parabola?

a. f(x) = 4(x + 1.5)2_{– 2}

b. f(x) = –4(x + 1.5)2_{+ 2}

c. f(x) = 4(x + 1.5)2_{+ 2}

(9)

____ 31. Which quadratic function defines this parabola in vertex form?

a. y = –(x + 5)2_{– 6}

b. y = –(x + 4)2_{– 5}

c. y = –(x + 6)2_{– 5}

d. y = –(x + 6)2_{– 4}

____ 32. Solve –2x2_{– 24x + 75 = 0 by graphing the corresponding function and determining the zeros.}

a. x = 5, x = –3

b. x = –3, x = –5

c. There are no zeros. d. x = 0, x = –8.333

____ 33. A bridge is supported by three arches. The function that describes the arches is

h(x) = –0.2x2_{+ 3.0x, where h(x) is the height, in metres, of the arch above the ground at any distance, x,}

in metres, from one end of the bridge. How tall is each arch?

a. 9.7 m

b. 11.3 m

c. 10.8 m

d. 13.5 m

____ 34. Which situations could be described using the rates $15.56/lb, 80 km/h, and $1.58/L? a. price of nails, average human running speed, price of sunflower oil

b. price of coffee, cruising speed of an airplane, price of milk c. price of lobster, highway speed limit, price of apple juice d. price of crude oil, average speed of a truck, price of cola

(10)

____ 35. A picture is 46 cm by 32 cm. A scale diagram of the picture must fit in a space that is 3 m by 2 m. Which scale would be the most reasonable one to use for the scale diagram?

a. 60%

b. 1 cm:60 cm

c. 6 cm:1 m

d. 1 cm:6 cm

____ 36. Data for triangle ABC is shown on the first line of the table. Triangle ABC is reduced so the height is 1.5 cm.

Which triangle is the reduction of triangle ABC?

Triangle Name Length of Base (cm) Height of Triangle (cm) Scale Factor Area (cm2₎

Area of Scaled Triangle Area of O riginal Triangle

ABC 5.0 3.0 1.0 7.5 1.00 DEF 10.0 1.5 2.0 2.5 4.00 GHI 1.5 1.5 0.5 3.0 0.25 JKL 2.5 1.5 0.5 1.875 0.25 MNO 2.5 1.5 0.25 1.75 0.50 a. DEF b. GHI c. JKL d. MNO

____ 37. Data for rectangle ABCD is shown on the first line of the table. Rectangle ABCD is reduced to an area of 13 cm2_.

Which rectangle is the reduction of rectangle ABCD?

Rectangle Name Length (cm) Width (cm) Scale Factor Area (cm2₎

Area of Scaled Rectangle Area of O riginal Rectangle

ABCD 9 13 1 117 1 EFGH 3 4 1 9 13 1 81 JKLM 4 _{3 1} 3 1 3 13 1 9 NOPQ 1 _{1 1} 3 1 9 13 1 3 RSTU 3 _{4 1} 3 1 3 13 1 9

(11)

____ 38. A large city map book will be changed so that it can be used as a street guide. To maintain the same number of pages, the page dimensions will be halved and the maps will be less detailed. The same type of paper will be used for the smaller map book. By what factor will the volume of the paper change?

a. 1 4 b. 1 2 c. 1 16 d. 1 8 Short Answer

39. What type of error occurs in the following deduction? Briefly justify your answer.

People wear hats to prevent sunstroke. Eldon is wearing a hat.

Therefore, Eldon is wearing the hat to prevent sunstroke.

.

40. Does the following statement demonstrate inductive reasoning or deductive reasoning? For the pattern 4, 13, 22, 31, 40, the next term is 49.

.

(12)

42. Solve for the unknown side length. Round your answer to one decimal place.

v

sin 84° = 15.0sin 40°

.

43. In ΔQRS, r = 4.1 cm, s = 2.7 cm, and ∠R = 88°. Determine the measure of ∠S to the nearest degree.

.

44. Convert 24 – 150 – 54 into mixed radical form. Then simplify.

.

45. Rationalize the denominator in 5 324

8 .

.

46. The ages of members in a hiking club are normally distributed, with a mean of 32 years and a standard deviation of 6 years. What percent of the members are between 32 and 44?

(13)

47. Is the data in this set normally distributed? Explain.

Interval 10–19 20–29 30–39 40–49 50–59 60–69

Frequency 1 8 11 13 9 3

.

48. Determine the quadratic function that contains the factors (x + 5) and (x – 9) and the point (5, –20). Express your answer in vertex form.

.

49. Determine the roots of the corresponding quadratic equation for the graph.

50. Determine the roots of the equation 4

5b2 – 4b + 5 = 0.

(14)

51. On Wednesday a crew paved 12 km of road in 7 h. On Thursday, the crew paved 8 km in 6 h.

On which day did they pave the road at the faster rate?

.

52. A standard music CD has an exterior diameter of 120 mm and an interior diameter of 15 mm. Draw a scale diagram of a CD using a scale factor of 60%.

.

53. Sooki enlarges this figure by a scale factor of 2.

(15)

54. Triangle A has an area of 19.00 cm2_{and similar triangle B has an area of 118.75 cm}2_{. Determine what}

scale factor makes triangle B an enlargement of triangle A.

.

55. The giant statue of a white fox in White Fox, SK, is 2.7 m long and 1.4 m tall.

Suppose the town council wants to create scale models of the statue that are 9 cm tall to sell to tourists.

What scale factor should they use?

.

Problem

56. Jamie created a math trick in which she always ended with 5. When Jamie tried to prove her trick, however, it did not work.

Jamie’s Proof

n I used n to represent any number.

3n Multiply by 3.

3n + 15 Add 15.

n + 15 Divide by 3.

15 Subtract your starting number.

Identify the error in Jamie’s proof and write the proof without error.

(16)

57. Prove: FG || HI

58. Given ∠z = 115°.

Determine the measures of y.

59. Rationalize each expression to compare the radical expressions. Show each step.

A. 7

3x B. 143 x

.

60. Gravity affects the speed at which objects travel when they fall. Suppose a rock is dropped off a 7.5 m cliff on Mars. The height of the rock, h(t), in metres, over time, t, in seconds could be modelled by the function h(t) = –1.9t2_{+ 3.0t + 7.5. (The acceleration due to gravity on Mars is 3.77 m/s.)}

a) How long would it take the rock to hit the bottom of the cliff?

b) The same rock dropped off a cliff of the same height on Earth could be modelled by the function h(t)

(17)

Math 20-2 Final Review

Answer Section

MULTIPLE CHOICE

1. ANS: D PTS: 1 DIF: Grade 11 REF: Lesson 1.1

OBJ: 1.1: Make conjectures by observing patterns and identifying properties, and justify the reasoning.

TOP: Conjectures and Inductive Reasoning KEY: conjecture| inductive reasoning

2. ANS: C PTS: 1 DIF: Grade 11 REF: Lesson 1.3

OBJ: 1.4 Provide and explain a counterexample to disprove a given conjecture. TOP: Disproving Conjectures: Counterexamples

KEY: conjecture| disproving conjectures| counterexamples

OBJ: 1.3 Compare, using examples, inductive and deductive reasoning.| 1.5 Prove algebraic and number relationships, such as divisibility rules, number properties, mental mathematics strategies or algebraic number tricks| 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs). TOP: Proving Conjectures; deductive reasoning

KEY: conjecture| proving conjectures| reasoning| deductive reasoning

4. ANS: B PTS: 1 DIF: Grade 11 REF: Lesson 1.6

OBJ: 1.9 Solve a contextual problem involving inductive or deductive reasoning.

TOP: reasoning to solve problems KEY: reasoning| inductive reasoning| deductive reasoning

5. ANS: A PTS: 1 DIF: Grade 11 REF: Lesson 1.6

OBJ: 1.1 Generalize, using inductive reasoning, the relationships between pairs of angles formed by transversals and parallel lines, with or without technology. | 1.5 Verify, with examples, that if lines are not parallel the angle properties do not apply. TOP: Parallel lines

KEY: parallel lines| transversals

OBJ: 1.2 Prove, using deductive reasoning, properties of angles formed by transversals and parallel lines, including the sum of the angles in a triangle. | 1.4 Identify and correct errors in a given proof of a property involving angles. | 2.1 Determine the measures of angles in a diagram that involves parallel lines, angles and triangles, and justify the reasoning. | 2.2 Identify and correct errors in a given solution to a problem that involves the measures of angles. | 2.3 Solve a contextual problem that involves angles or triangles. | 2.4 Construct parallel lines, using only a compass or a protractor, and explain the strategy used. | 2.5 Determine if lines are parallel, given the measure of an angle at each intersection formed by

the lines and a transversal. TOP: Angles formed by parallel lines

KEY: parallel lines| transversals| angles

(18)

OBJ: 1.2 Prove, using deductive reasoning, properties of angles formed by transversals and parallel lines, including the sum of the angles in a triangle.| 2.1 Determine the measures of angles in a diagram that involves parallel lines, angles and triangles, and justify the reasoning.

TOP: Angles in triangles KEY: angles| triangles

OBJ: 1.3 Generalize, using inductive reasoning, a rule for the relationship between the sum of the interior angles and the number of sides (n) in a polygon, with or without technology 1.4 Identify and correct errors in a given proof of a property involving angles.| 2.2 Identify and correct errors in a given solution to a problem that involves the measures of angles.| 2.3 Solve a contextual problem that involves angles or triangles. TOP: Angle properties in polygons KEY: polygons| angle properties

OBJ: 3.3 Solve a contextual problem that requires the use of the sine law or cosine law, and explain the

reasoning. TOP: Side-angle relationships in acute triangles

KEY: primary trigonometric ratios

reasoning. TOP: Proving and applying the sine law

KEY: sine law

reasoning. TOP: Proving and applying the cosine law

KEY: cosine law

reasoning. TOP: Proving and applying the cosine law

KEY: cosine law

OBJ: 3.1 Compare and order radical expressions with numerical radicands. | 3.2 Express an entire radical with a numerical radicand as a mixed radical. | 3.3 Express a mixed radical with a numerical radicand as an

entire radical. TOP: Mixed and entire radicals KEY: radical

OBJ: 3.4 Perform one or more operations to simplify radical expressions with numerical or variable

radicands. TOP: Adding and subtracting radicals KEY: radical

(19)

OBJ: 4.1 Determine any restrictions on values for the variable in a radical equation. | 4.2 Determine, algebraically, the roots of a radical equation, and explain the process used to solve the equation. | 4.3 Verify, by substitution, that the values determined in solving a radical equation are roots of the equation. | 4.5 Solve problems by modelling a situation with a radical equation and solving the equation.

TOP: Solving simple radical equations KEY: radical

TOP: Frequency tables, histograms, and frequency polygons KEY: frequency distribution | histogram | frequency polygon

OBJ: 1.3 Explain, using examples, the properties of a normal curve, including the mean, median, mode,

standard deviation, symmetry and area under the curve. TOP: Standard deviation

KEY: mean | standard deviation

OBJ: 1.2 Calculate, using technology, the population standard deviation of a data set. | 1.3 Explain, using examples, the properties of a normal curve, including the mean, median, mode, standard deviation,

symmetry and area under the curve. TOP: Standard deviation

KEY: standard deviation

OBJ: 1.3 Explain, using examples, the properties of a normal curve, including the mean, median, mode, standard deviation, symmetry and area under the curve. TOP: The normal distribution

KEY: normal curve

OBJ: 1.7 Solve a contextual problem that involves the interpretation of standard deviation. | 1.9 Solve a contextual problem that involves normal distribution. TOP: The normal distribution

KEY: normal distribution | mean | standard deviation

OBJ: 1.8 Determine, with or without technology, and explain the z-score for a given value in a

normally distributed data set. TOP: Applying the normal distribution: z-scores

KEY: z-score | standard normal distribution

OBJ: 2.1 Explain, using examples, how confidence levels, margin of error and confidence intervals may vary depending on the size of the random sample. | 2.2 Explain, using examples, the significance of a confidence interval, margin of error or confidence level. TOP: Confidence intervals

KEY: margin of error | confidence interval | confidence level

OBJ: 2.6 Express a quadratic equation in factored form, given the zeros of the corresponding quadratic function or the x-intercepts of the graph of the function.

TOP: Factored form of a quadratic function KEY: quadratic relation | zero

(20)

OBJ: 1.3 Determine the coordinates of the vertex of the graph of a quadratic function, given the equation of the function and the axis of symmetry, and determine if the y-coordinate of the vertex is a maximum or a minimum. | 2.6 Express a quadratic equation in factored form, given the zeros of the corresponding quadratic function or the x-intercepts of the graph of the function.

TOP: Vertex form of a quadratic function KEY: quadratic relation | parabola | zero

TOP: Solving problems using quadratic function models KEY: quadratic relation | parabola

OBJ: 1.5 Sketch the graph of a quadratic function. | 2.1 Determine, with or without technology, the intercepts of the graph of a quadratic function.

TOP: Solving quadratic equations by graphing KEY: quadratic equation | roots

OBJ: 1.1 Determine, with or without technology, the coordinates of the vertex of the graph of a quadratic function. | 1.6 Solve a contextual problem that involves the characteristics of a quadratic

function. TOP: Solving problems using quadratic equations

KEY: quadratic equation | roots

OBJ: 1.1 Interpret rates in a given context, such as the arts, commerce, the environment, medicine or recreation. | 1.8 Describe a context for a given rate or unit rate.

TOP: Solving problems that involve rates KEY: rate

OBJ: 2.2 Determine, using proportional reasoning, the scale factor, given one dimension of a 2-D shape or a 3-D object and its representation. TOP: Scale diagrams

KEY: scale | scale diagram

OBJ: 2.2 Determine, using proportional reasoning, the scale factor, given one dimension of a 2-D shape or a 3-D object and its representation. | 3.1 Determine the area of a 2-D shape, given the scale diagram, and justify the reasonableness of the result.

TOP: Scale factors and areas of 2-D shapes KEY: scale | scale factor | area

OBJ: 2.2 Determine, using proportional reasoning, the scale factor, given one dimension of a 2-D shape or a 3-D object and its representation. | 3.1 Determine the area of a 2-D shape, given the scale diagram, and justify the reasonableness of the result.

TOP: Scale factors and areas of 2-D shapes KEY: scale | scale factor | area

OBJ: 3.6 Explain, using examples, the relationships among scale factor, area of a 2-D shape, surface area of a 3-D object and volume of a 3-D object. | 3.8 Solve a contextual problem that involves the

relationships among scale factors, areas and volumes. TOP: Scale factors and 3-D objects

(21)

SHORT ANSWER

39. ANS:

There is an error in reasoning: there are many other reasons why Eldon might wear a hat, such as style, or being on a baseball team

PTS: 1 DIF: Grade 11 REF: Lesson 1.5

OBJ: 1.7 Determine if a given argument is valid, and justify the reasoning. | 1.8 Identify errors in a

given proof; e.g., a proof that ends with 2 = 1. TOP: invalid proofs; deductive reasoning

KEY: valid proofs| invalid proofs| deductive reasoning 40. ANS:

inductive reasoning

TOP: reasoning to solve problems KEY: reasoning| inductive reasoning| deductive reasoning 41. ANS:

3

OBJ: 2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. | 2.2 Identify and correct errors in a solution to a puzzle or in a strategy for winning a game.| 2.3 Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game.

TOP: analyzing puzzles and games KEY: reasoning| solving puzzles

42. ANS: 23.2

reasoning. TOP: Side-angle relationships in acute triangles

KEY: primary trigonometric ratios 43. ANS:

∠S = 41°

reasoning. TOP: Proving and applying the sine law

KEY: sine law 44. ANS:

2 6 − 5 6 − 3 6 = −6 6

(22)

45. ANS:

45 2

2

OBJ: 3.4 Perform one or more operations to simplify radical expressions with numerical or variable radicands. | 3.5 Rationalize the monomial denominator of a radical expression.

TOP: Multiplying and dividing radicals KEY: radical | rationalize the denominator 46. ANS:

47.5%

OBJ: 1.7 Solve a contextual problem that involves the interpretation of standard deviation. | 1.9 Solve a contextual problem that involves normal distribution. TOP: The normal distribution

KEY: normal distribution | mean | standard deviation 47. ANS:

Yes. The graph of the data has a rough bell shape.

OBJ: 1.4 Determine if a data set approximates a normal distribution, and explain the reasoning.

TOP: The normal distribution KEY: normal distribution | mean | standard deviation

48. ANS:

y = 0.5(x + 2)2_{– 24.5}

OBJ: 1.6 Solve a contextual problem that involves the characteristics of a quadratic function. | 2.6 Express a quadratic equation in factored form, given the zeros of the corresponding quadratic function or the x-intercepts of the graph of the function.

TOP: Solving problems using quadratic function models KEY: quadratic relation | parabola 49. ANS:

x = 0, x = 4

OBJ: 2.1 Determine, with or without technology, the intercepts of the graph of a quadratic function. | 2.4 Explain the relationships among the roots of an equation, the zeros of the corresponding function, and the x-intercepts of the graph of the function. | 2.5 Explain, using examples, why the graph of a quadratic function may have zero, one or two x-intercepts.

TOP: Solving quadratic equations by graphing KEY: quadratic equation | roots

50. ANS:

b = 5

2

OBJ: 2.2 Determine, by factoring, the roots of a quadratic equation, and verify by substitution.

(23)

51. ANS:

Wednesday: 1.7 km/h Thursday: 1.3 km/h

They paved the road faster on Wednesday.

OBJ: 1.1 Interpret rates in a given context, such as the arts, commerce, the environment, medicine or recreation. | 1.3 Determine and compare rates and unit rates. TOP: Comparing and interpreting rates KEY: rate | unit rate

52. ANS:

The CD should have an exterior diameter of 72 mm and an interior diameter of 9 mm.

OBJ: 2.3 Determine, using proportional reasoning, an unknown dimension of a 2-D shape or a 3-D object, given a scale diagram or a model. | 2.4 Draw, with or without technology, a scale diagram of a given 2-D shape, according to a specified scale factor (enlargement or reduction).

TOP: Scale diagrams KEY: scale | scale diagram | scale factor

53. ANS: 46 units2

OBJ: 3.1 Determine the area of a 2-D shape, given the scale diagram, and justify the reasonableness of

the result. TOP: Scale factors and areas of 2-D shapes

KEY: scale | scale factor | area 54. ANS:

2.5

OBJ: 2.2 Determine, using proportional reasoning, the scale factor, given one dimension of a 2-D shape or a 3-D object and its representation. | 3.6 Explain, using examples, the relationships among scale

(24)

55. ANS: 9

140 or about 0.064

OBJ: 2.2 Determine, using proportional reasoning, the scale factor, given one dimension of a 2-D shape or a 3-D object and its representation. TOP: Similar objects: scale models and scale diagrams

KEY: scale | scale factor | similar objects

PROBLEM

56. ANS:

For example, in the 4th step, Jamie did not divide the term 15 by 3, which she should have done. This throws out the rest of the calculations. Here is the correct proof.

n I used n to represent any number.

3n Multiply by 3.

3n + 15 Add 15.

n + 5 Divide by 3.

5 Subtract your starting number.

OBJ: 1.7 Determine if a given argument is valid, and justify the reasoning. | 1.8 Identify errors in a

given proof; e.g., a proof that ends with 2 = 1. TOP: invalid proofs; deductive reasoning

KEY: valid proofs| invalid proofs| deductive reasoning 57. ANS:

∠FHG +∠GHI + ∠IHJ = 180° Sum of angles in triangle is 180°

94° + ∠GHI + 73° = 180° Substitute known values.

∠GHI = 180° – 94° – 73° Determine ∠GHI. ∠GHI = 13°

∠GHI = ∠FGH

Therefore, FG || HI equal alternate interior angles

(25)

58. ANS:

∠z + ∠a = 180° Supplementary angles

115° + ∠a = 180° Substitute

∠a = 65°

∠n + ∠a + ∠a = 180° Angle sum of a triangle

∠n + 65° + 65° = 180° Substitute

∠n = 50°

∠n + ∠m + 90° = 180° Angle sum of a triangle

50° + ∠m + 90° = 180° Substitute

∠m = 40°

∠m + ∠b + ∠b = 180° Angle sum of a triangle

40° + 2∠b = 180° Substitute

2∠b = 140° ∠b = 70°

∠b + ∠y = 180° Supplementary angles

70° + ∠y = 180° Substitute

∠y = 110°

(26)

59. ANS: A. 7 3x ⋅ 3x 3x = 21x 9x2 7 3x ⋅ 3x 3x = 21x 3x 7 3x ⋅ 3x 3x = 21 x 3x B. 14 3 x = 143 x ⋅ x x 14 3 x = 14 x 3 x2 14 3 x = 14 x 3x

The restriction of both expressions is x > 0 since x cannot be 0 or a negative number. Since the denominators are the same, compare the numerators.

Since both numerators contain x and 21 < 14, then 7

3x < 143 x .

OBJ: 3.4 Perform one or more operations to simplify radical expressions with numerical or variable radicands. | 3.5 Rationalize the monomial denominator of a radical expression. | 3.6 Identify values of the variable for which the radical expression is defined.

TOP: Simplifying algebraic expressions involving radicals KEY: radical | restrictions | rationalize the denominator

(27)

60. ANS: a) 0 = –1.9t2_{+ 3t + 7.5} a = –1.9, b = 3.0, c = 7.5 t= −b± b2 − 4ac 2a t= −3.0 ± (3.0) 2 _{− 4 −1.9}₍ _{) 7.5}₍ ₎ 2(−1.9) t= −3± 66 −3.8 t= −3+ 66 −3.8 t= −1.348... or t= −3− 66 −3.8 t= 2.927...

Time cannot be a negative value, so the time it takes for the rock to hit the bottom of the cliff on Mars is 2.9 s.

b) 0 = –4.9t2_{+ 3.0t + 7.5} a = –4.9, b = 3.0, c = 7.5 t= −b± b2 − 4ac 2a t= −3.0 ± (3.0) 2 _{− 4 −4.9}₍ _{) 7.5}₍ ₎ 2(−4.9) t= −3± 156 −9.8 t= −3+ 156 −9.8 t= −0.968... or t= −3− 156 −9.8 t= 1.580...

Time cannot be a negative value, so the time it takes for the rock to hit the bottom of the cliff on Earth is 1.6 s.

Difference in times = 2.927... s – 1.580... s Difference in times = 1.3 s

The difference in times between the two planets is 1.3 s.

OBJ: 2.3 Determine, using the quadratic formula, the roots of a quadratic equation. | 2.7 Solve a contextual problem by modelling a situation with a quadratic equation and solving the equation. TOP: Solving quadratic equations using the quadratic formula