Chapter answers
Chapter 1
1 a 42.6° b 67.35°
c 28.29° d 16.41°
e 7.1886 …°
2 a 19°12′ b 49°40′12″
c 56°58′12″ d 87°15′
e 22°9′36″
3 a 27.0 b 30.5° c 32.6
d 39.2 e 36.7° f 58.0°
4 a 103.3 m b 20 m c 6.9 m
d 15.6 m e 24.6 m f 1.82 km
g 9.6 m h 9.6 cm i 10.6 m
5 a 41.6° b 60° c 51.5°
d 26.4° e 32.9° f 54.1°
6 a 20 cm b 34.6 cm
7 a 8 mm b 13.9 mm c 21.2 mm
8 a 28 cm b 24.2 cm c 34.3 cm
d 10.2 cm
9 a 20 mm b 10 mm c 17.3 mm
d 34.6 mm
10 48.3 cm 11 1.6 m 12 72.8° m
13 24.1 m 14 69.5° 15 266.3 cm
16 7 m
17 a 96.4° b 44.7 cm
18 a 13.58 m b 26.95 m
19 No, first part of slope is about 30°.
20 26 cm (at least 25.1 cm)
1 a 230°T or S50°W b 100°T or N100°E
c 180°T or S d 000°T or N
2 a 090°T or E b 300°T or N60°W
c 101.5 m
d 287.2°T or N72.8°W
3 a 240°T or S60°W b 800 m
c 1.39 km
4 323°T or N37°W
5 65 km, 292.6°T or N67.4°W
6 a 60 km b hour
c 323.1°T or N36.9°W
7 226.7°T or S46.7°W, 26.1 km
8 5.59 km
9 217.35°T or S37.35°W
10 269.3 km, 248.2°T or S68.2°W
11 6.2 km from P, 7.9 km from Q Exercise 1.1
Exercise 1.2
1 2
---1 a a= 13.7, b= 13.4 b p= 13.4, q= 10.9
c h= 17.2, k= 13.9 d c= 34.3, d= 21.1
e m= 11.8, n= 6.6 f x= 38.5, y= 51.0
2 a 40.6° b 31.7° c 54.1°
d 38.1° e 27.1° f 17.5°
3 75.82 mm
4 a X= 77°, Y= 38°, x= 17.4 m
b R= 91.9°, r= 84.7 cm, p= 49.0 cm
c E= 27°, D= 42°, e= 6.08 km
d A= 37.2°, a= 178 cm, b= 244 cm
e Z= 57.36°, y = 11.13 m, z = 11.55 m
f L = 65.6°, m = 1.942 km, k = 2.727 km
5 D = 21.13° (21°8′) or 158.87° (158°52′),
e = 61.27 cm or 5.30 cm
6 A = 47.63° (47°38′) or 132.37° (132°22′)
7 31.4 cm 8 111.4° 9 6.69 km
10 a Her height is 2992 m above the hill, so her altitude is really about 3397 m.
b The altimeter is reading high.
1 a 5.4 cm b 7 m c 88.2 cm
d 55.1 m e 21.4 m f 43.8 cm
2 a 22.3° b 53.1° c 95.7°
d 78.0° e 109.5° f 134.4°
3 a m = 26 b z = 283 c f = 64.6
d g = 43.7 e s = 157 f q = 952.9
4 a A = 21.4°, G = 37.6°, d = 7 m
b U = 56.7°, V = 68.3°, w = 44.1 cm
c X = 33.4°, Y = 107.3°, Z = 39.4°
d A = 23.2°, B = 112°, C = 44.8°
e M = 81.7°, N = 56°, l = 10.5 km
f R = 5.2°, F = 6.6°, e = 341 mm
5 A = 89.6° 6 X = 24.15°
7 P = 90° 8 27 m
9 26.4° and 36.3° 10 170°T or S10°E
1 a 3.85 m b 17 mm c 50.25° d 41 cm
2 a 57.5 m b 10.8 m c 322 m d 9.2 cm
3 5.22 m 4 12.9 m
5 a 60.7 m b 76.12°
Exercise 1.3
132˚22′ 47˚38′
18 cm
C
A A′ 38˚ B
15 cm
Exercise 1.4
6 7.128 km/h 7 155 m
8 a 55 m b 29°
9 713 m
10 a 369.3 m b 143.6 m
11 4.31 km 12 68.6 m
13 a 361 km b 94.63°T
14 12.75 km 15 85 m 16 54 m
17 166 km 18 About 20 m 19 41.5 m
20 Tower ≈ 7.8 m, building ≈ 12.5 m
1 AYZ 51.34°, AZW 59.04°
2 a 12.04 m (south), 9 m (west)
b 44.9° (south), 53.13° (west)
c 15.03 m d 55.62°
3 a 401 m b 233.27°T or S233.25°W
4 76°T or N76°E at 12.78°
5 a 21.21 cm b 30.97°
c 30.97° d 40.32°
6 a 72.6° b 63.3°
7 3.16 m, 161.57°T or S18.43°E
8 a 73.3° b 7.5 m c 53.13°
d 6.3 m e 7.7 m
9 15.3°, 410.4 m 10 81.5 m
11 155 m 12 77.1 m
Chapter 2
1 a Number of DVDs, discrete
b Height, continuous
c Nationality, nominal
d Number of rooms, discrete
e Length, continuous
f Eye colour, nominal
g Blood pressure, continuous
h Collection size, discrete
i Chemical symbol, nominal
j Learner’s licence result, discrete (there are a fixed number of questions)
2 a People purchasing from a cinema snack bar, amount spent, mean amount
b Australian residents of Gympie, takeaway food eaten, preference of outlets
c Students, choice of calculator, preference of calculator
d Vehicles on Ipswich Motorway, traffic density, density for each type of vehicle
e Service stations near Gladstone, unleaded petrol prices, median price
Exercise 1.6
Exercise 2.1
3 Results from the sample should be a true reflection of the population.
4 Students 5130, 5204, 5278, 5352, 5426, 5500, 5574, 5648, 4978, 5052
5 a 50, 42, 76, 44, 43, 79, 47, 40
b 374 995, 380 868, 139 324, 347 249, 105 078, 110 178, 138 491, 336 824
c 7, 0, 2, 6, 4, 8, 3, 9
d 66, 39, 57, 63, 53, 32, 61, 67
6 a 3 men in board shorts, 2 men in briefs, 4 women in bikinis and 1 woman in a one-piece
b 7 soft-centred, 5 hard-centred, 3 liquid-centred and 4 nutty-liquid-centred chocolates (= 19)
c 2 fifteen-year-olds, 11 sixteen-year-olds, 3 seventeen-year-olds and 1 eighteen-year-olds (= 17)
7 a Area would probably have increasing numbers of children from families who need childcare facilities and are able to pay for it.
b Two incomes means that families probably need childcare and are more likely to be able to afford it.
c The proximity of other childcare centres and schools
d Conduct a survey themselves; pay someone else to conduct a survey.
8 a Less competition; high ‘passing’ trade
b Retail activity, population growth, income distribution, age distribution
c They could investigate retail activity and proximity of other fast-food outlets, etc. themselves.
d Proximity to factories, industrial areas, schools and other sources of potential customers
9 She should check the number and value of building approvals, and the number and value of buildings built in the last year, in different statistical divisions to ensure that the sales team targets the most likely areas.
10 For different locations, whether or not there are shops offering similar services already; the number of people that use the location for shopping, the age of the shoppers and the amount of disposable income they have (if possible)
11 a Students and teachers at your school.
b The number of people who favour and the number who are against the installation of a security system
c i Sample too small; only one age group
iii Only one age group
iv Volunteers will probably have strong views one way or another.
v Probably the fairest method
vi Problems similar to ii and iv
vii Could be biased depending on how students are selected
viii Depends on how they are selected
ix Similar to iv
x Probably not representative because only students entering school on a particular day during a particular period are asked
12 a Administration: males 1, females 3 Factory: males 8, females 6
b Administration 4, factory 14
13 a NSW 164, Vic. 124, Qld 99, SA 38, WA 50, Tas.12, NT 5, ACT 8
b Men: NSW 81, Vic. 61, Qld 50, SA 19, WA 25, Tas. 6, NT 3, ACT 4
Women: NSW 83, Vic. 63, Qld 50, SA 19, WA 25, Tas. 6, NT 2, ACT 4
14 Yes—the people who watched the show that week may have chosen to watch the show because it featured Phuket, so they may be particularly interested in similar resorts
15 a Students leaving school
b Post-school destination
16 Items 135, 90, 149, 188, 94, 105, 122, 97, 142, 207 (discarding the last digit of pairs)
17 Choose the people from one page only—quick, but could be biased to a particular group, such as Singhs, and they could be spread over a wide area.
Choose the first person from each page—not as quick as the first method; otherwise the same but bias unlikely.
Use a random number table to choose people from the 85 000—very slow, but not biased. Choose the first people to come up from a single street or suburb—quick, and it would be easy (cheap) to do the survey in a confined area, but likely to be biased.
18 Questions could vary, but should be as specific as possible.
a ‘What local TV station did you watch most last week?’
b ‘Do you know what you are going to do in your next holiday? If you have, what will you do?’
c ‘What local restaurant did you last visit?’
d ‘What are your mother’s and father’s occupations?’
e ‘If you do part-time work, for how many hours did you work last week?’
f ‘How many times did you have a fast-food meal last week?’
g ‘What current affairs programs did you watch in the last week?’
h ‘What disco operator played at the last three functions you went to?’
i ‘What age group are you in: under 10, 10–19, 20–29, 30–39, over 39?’
j ‘What washing detergent do you use? Why did you pick that one?’ (Probably restrict reasons after a pilot study.)
19 i a Students at your school
b People in your suburb/town
c People in your suburb/town
d Students at your school
e Students at your school
fPeople in your street
g People who watch current affairs programs
h People who attend functions with disco operators
i People at the amusement park
j People who choose the washing detergent for their household
ii and i i i Answers should show methods that give a fair sample, and the administration should be as quick as possible without introducing bias.
20 a People interviewed in their homes may be more relaxed, but choosing particular streets or a particular suburb could introduce bias. It is also expensive for the interviewer to go to people’s homes.
b This method is much easier and cheaper than
a, but obviously it is biased by being in the
CBD.
c This method is cheaper again, but it is biased towards people with a fixed phone line. Because people resent being badgered on the phone, the non-participation bias could be high.
d This method is very cheap indeed, but it is obviously very biased.
e This method is generally cheaper than a, but the small number in a focus group introduces some bias.
21 People may be in the middle of having a meal, or busy, or sick of being rung up at this time. The main advantage of phoning is that it is cheap, but it is biased towards those with a fixed line.
23 Questionnaires and reasons will vary.
24 Questions are imprecise and could be coded as follows:
b Medium 14, large 14, jumbo 8
1 a Topping f b Mexican
H 3
V 1
AM 3
S 8
M 5
2 a Mass (g) 50 51 52 53 54 55 56 57 58
f 5 3 6 1 4 3 0 2 3
Mass (g) 59 60 61 62 63 64 65 66
f 1 1 2 1 0 2 1 1
3 Hits on website
Stem Leaf
4 5 6 7 8
3 5 6 9 0 0 1 2 8 3 3 5 7 7 9 1 4 8 5 6
1What is your sex? M F 2 What is your age?
Under 10 11–20 21–40 Over 40 3 Leisure time is the time left affter you have
done the things you must do, like eating, sleeping, working, going to school and so on. What do you most prefer to do in your leisure time?
Circle the correct answer for each of the following.
1 What year are you in? 8 9 10 11 12 2 What is your sex? M F
3 How many hours did you spend on homework last week?
0 1 2 3 4 5 6 7 8 9 10 More than 10 4 How many As, Bs, Cs, Ds and Es did you get for last term’s level of achievement? (write the number)
A B C D E
5 How many more hours homework would you need to do in a week to improve your results?
0 1 2 3 4 5 6 7 8 9 10 More than 10
Exercise 2.2
5 a 4.5–9.5, 7 b 39.5–49.5, 44.5
c 23.5–27.5, 25.5 d 129.95–139.95, 134.95
6 75–79 (77), 80–84 (82), 85–89 (87), …, 125–129 (127) newtons (others possible)
7 3.25–3.49 (3.37), 3.50–3.74 (3.62), 3.75–3.99 (3.87), …, 4.50–4.74 (4.62) cm (others possible)
8
b 10 kg c 34.5, 44.5, 54.5, …, 84.5 kg
10 a 962 and 1530 hours
b 50 hours; 950–999, 1000–1049, …, 1500–1549 hours (others possible)
4 Year 11 handspans (cm)
Stem Leaf
17 18 19 20 21 22 23
2 9 1 5 2 6 8 8 1 8 0 7 8 3 2 7
9 a Weight (kg) f
30–39 3
40–49 7
50–59 9
60–69 11
70–79 7
80–89 3
c Hours f
950–999 2
1000–1049 4
1050–1099 1
1100–1149 6
1150–1199 3
1200–1249 6
1250–1299 5
1300–1349 7
1350–1399 7
1400–1449 4
1450–1499 3
1500–1549 2
Registered motor vehicles in Australia, March 2006
commercial Passenger
Light Trucks
Other Motorcycles
vehicles vehicles
Total = 14.4 million
Key: 4 ⎪ 3 = 43
11 Table will vary depending on intervals chosen.
12 a
b
d Safety lightbulb lives (hours)
Stem Leaf
9 10 11 12 13
14 15
62, 80
8, 10, 12, 40, 92
20, 29, 39, 42, 43, 47, 59, 59, 76 5, 11, 27, 42, 44, 46, 77, 82, 85, 97, 98 17, 21, 24, 26, 31, 32, 48, 52, 62, 74, 86, 87, 93, 99
2, 17, 40, 41, 53, 73, 93 10, 30
Energy (MJ) f
172 – 174 2
174 – 176 1
176 – 178 5
178 – 180 9
180 – 182 4
182 – 184 3
184– 186 1
13 a Injuries Midpoint f
23–25 24 13
26–28 27 11
29–31 30 10
32–34 33 6
35–37 36 6
38–40 39 2
Key: 10 ⎪ 12 = 1012
Frequenc
y
20
15
10
5
0 10 20
% NESB students
30 40 50 60
% NESB students in schools
Frequenc
y
20
15
10
5
0 10 20
% NESB students
30 40 50 60
% NESB students in schools
14
b 3750 pine trees (Include 140–160 as data is continuous.)
c 70 students
15 a Mark (%) f c.f.
10–19 2 2
20–29 6 8
30–39 10 18
40–49 26 44
50–59 21 65
60–69 8 73
70–79 4 77
80–89 2 79
90–99 2 81
b
Frequenc
y
20
15
10
5
0
24 27
Number
30 33 36 39
Occupational injuries per year
Occupational injuries per year
Frequenc
y
20
15
10
5
0
24 27
Number
30 33 36 39
c
42
Frequenc
y
100 80 60 40 20 0
50 70
Girth (cm)
90 110 130 150 170
Girths of pine trees
b
Cumulati
v
e frequenc
y
50 40 30 20 10 0
10 20 30
Mark (%) 40 50 60 60
70
70 80
90
80 90 100
c 38%
d 76 min (25%) or 75 min (20%) (other answers possible)
1 a 22 b33 c37 d 18 e 25 f 31.3
2 a 112 b82 c50 d 149 e 127 f 165
3 a Mean ≈ 10.9, median = 12, mode = 14
b Mean = 5.3, median = 3.5, mode = 2
c Mean = 8.75, median = 9, mode = 9
d Mean = 11, median = 10. There are 4 scores with frequency 2 (7, 9, 11, 19), so there is really no mode
4 a Mean ≈ 15.6, median = 18, modal class = 15−19
b Mean ≈ 57.2, median = 57.5, modal class = 50–59
c Mean ≈ 9.4, median = 9, modal class = 7–9
5 Classes such as 5–9, 10–14, etc. have ends of 5, 9, 10, 14, etc. when considered as discrete, so have centres of 7, 12, etc. When the classes are considered as continuous, their ends are 4.5, 9.5, etc. so their centres are still 7, 12, etc. This means that it makes no difference to the mean. However, if continuous classes were to a greater accuracy, say, 5–9 to 1 decimal place or 5 to
10 etc., the centres for continuous data would become 4.95–9.95 or 5–10, so the class centres would be different and so the mean for continuous data would change.
16 a Time (min) f c.f. % c.f.
60–64 2 2 3
65–69 3 5 8
70–74 7 12 20
75–79 11 23 38
80–84 12 35 58
85–89 9 44 73
90–94 10 54 90
95–99 6 60 100
b
Cumulati
v
e percentage 50
40 30 20 10 0
60 65 70
Time (minutes)
75 80 85
60 70
90 80
90 100
95 100
Times for courier route
Exercise 2.3
The median could easily be different, because for discrete data it must be either a data item or
x.5 to represent the middle of two items. For
continuous data it is interpolated, so it can be any value.
The modal class must be the same.
6 a Mean ≈ 6.7, median = 7, mode = 7
b Mean = 9.5, median = 8.5, mode = 15
c Mean ≈ 5.6, median = 6, no mode (5 with f = 2)
d Mean ≈ 23.3, median = 24.5, no mode (4 with f = 2)
7 a Mean ≈ 5.0, median = 5, mode = 5
b Mean ≈ 8.3, median = 8, mode = 7
b $3800
c About $6000/month, as the top 10% earn between $5300 and $7500 per month
8 a Income ($) Cumulative frequency
2500–2999 12
3000–3499 41
3500–3999 85
4000–4499 124
4500–4999 162
5000–5499 187
5500–5999 204
6000–6499 213
6500–6999 218
7000–7499 220
7500–7999 221
9 a Mass (kg) Cumulative frequency
0.95–1.15 3
1.15–1.35 8
1.35–1.55 29
1.55–1.75 55
1.75–1.95 68
1.95–2.15 71
2.15–2.35 78
2.35-2.55 80
Employee monthly incomes
Percentage frequenc
y
40
20
0
2000 3000
Income ($)
4000
60
5000
80 100
b D3≈ 1.5 kg, P20≈ 1.45 kg, Q3≈ 1.8 kg, P84≈ 1.95 kg
c P33≈ 1.53 kg and P66≈ 1.73 kg, so make
small up to 1.5 kg, medium 1.6–1.7 kg and large over 1.7 kg.
10 a Note that it is possible that all the batteries shown as lasting 22–24 months lasted just 22 months, so the ogive class limits are somewhat unusual.
b P40= 25 months, P50= 26 months, P60= 27.5 months, P70= 29 months, P80= 30.5 months, P90= 33 months
c 21 out of 60 batteries fail before 24 months, so the manufacturer needs to recover that cost. Taking the pro-rata into account (blue line), it needs to add enough to replace about 15 out of 60 batteries at full cost. This is about 33%. (Almost any reasoned answer is okay.)
11 7.993 cm
12 a Discrete
b Mean ≈ 300.3, median = 301, mode = 301
b $1 480 000
13 a Shares traded ($ millions) f
0.8–0.9 5
1.0–1.1 3
1.2–1.3 6
1.4–1.5 4
1.6–1.7 8
1.8–1.9 7
2.0–2.1 2
2.2–2.3 1
Chicken masses
Cumulati
v
e frequenc
y
50 40 30 20 10 0
0.8 1 1.2
Mass (kg) 1.4 1.6 1.8 60
70
2 80
90
2.2 2.4 2.6
Life of batteries
Cumulati
v
e frequenc
y
50 40 30 20 10 0
3 6 9
Months 121518 60
21 70
24 27 30333639 42 45
14 Note that for age, 15–24 means 15–25. Mean = 44.8, median = 45.9,
modal class = 45–54 years
16 a Median = 168.4 cm, mean = 168.2 cm
b Mean
17 32.5 years 18 $1106.67 19 $1.70
20 a Mean ≈ $407, median ≈ $398, modal class = $380–$419
b The mean is higher than the median because a few high values affect the mean more than the median.
1 a Range = 20, IQR = 11, σ = 6.9
b Range = 87, IQR = 35, σ = 25.5
c Range = 23, IQR = 5, σ = 5.3
d Range = 18, IQR = 3, σ = 4.5
2 The range is greater than both the IQR and SD, but either of the IQR and SD can be the larger.
3 a Mean = 6.4, σ= 1.8 b Mean = 8.3, σ≈ 3.7
c Mean ≈ 28.8, σ≈ 7.8d Mean ≈ 40.1, σ≈ 5.7
4 a 11 motorists b 2 motorists
c 8 motorists d 6 motorists
e 5.6 motorists f 3.50 motorists
5 a Continuous
b Range = 3.5 kg, IQR = 0.95 kg
c Mean = 2.7 kg, σ= 0.687 kg
6 Range = 9°C, IQR = 5°C, σ= 2.95°C
7 Mean ≈ 11.91 s, σ= 0.650 s
8 a 3.5 b 2.42
9 a Mean = 213, IQR = 42.5, σ≈ 167.2
b Mean ≈ 144.4, IQR = 27.75, σ≈ 29.5
c Mean ≈ 178.7, IQR = 32.75, σ≈ 124 .9
10 Mean = 25, σ= 5
1
15 a Weight (kg) f c.f. b 78.95 kg
70–72 1 1
72–74 1 2
74–76 1 3
76–78 4 7
78–80 6 13
80–82 3 16
82–84 3 19
84–86 1 20
Exercise 2.4
Exercise 2.5
4 7 10
Spelling mistakes out of 20
2
3 Q1, Q2, Q3 found by interpolation. Extremes at
the start and ends of the first and last classes
4 a Outliers: 48, 6, 44
b Outliers: 198, 532, 503
c Outliers: 56, 11, 3
d Outliers: 26, 2, 30
5 a Mean ≈ 89.4%, median = 90%, mode = 90%
b 89.4 % (mean)
6 a Mean ≈ 3.27 accidents, median = 3 accidents (discrete), mode = 2 accidents
b Median
7 a Mean ≈ 87.25 kg, median = 86.64 kg, modal class = 85–89 kg
b Mean
8 a Mean ≈ 81.2 kg, median = 71 kg, mode = 65 kg
b Median (mean affected by some very high values)
9 a Mean = $1950, median = $1650, mode = $1750
b Mean = $1890, median = $1575, mode = $1680
c Only the mean
d ($1800 × 20 + $2500 × 4) ÷ 24 ≈ $1917
10 a 72.4 kg b 73.2 kg
11 10 people
12 a 44.3
b It is 44.3, because the total and the number will be in the same ratio.
b The Saabs are generally more expensive than the Volvos.
13 a Costs of cars ($’000)
Volvos Stem Saabs
9 8 8 7 7 3 9 9 8 6 2 2 1 1 9 8 6 5 5 3 0 0 0 3 7 3 3 2 0 7 5 2 0 0
0 1 2 3 4 5 6
0
0 0 1 1 2 2 3 4 5 6 7 7 8 8 8 0 0 0 4 7 8 8
1 0 0 0 9 9 2 6
5 20
Number of strawberries in punnet
10 15
251 254 256
Mass of strawberries in punnet (g)
249 250 252 253 255
Key: 5 ⎪ 2 = 25 000 = 2 ⎪ 5
b The computer magazine article generally has shorter sentences than the newspaper article.
15 a
b Hybrid 315 is more suitable because its yield is more consistent and the median is higher.
16 a
b The ‘reading difficulty’ of the Australian is higher, so it may suit a more educated readership.
17 a
b A beginner may prefer Brand A balls as they will be slower since they don't bounce as much.
18 Total runs 12 × 50 = 600. Runs needed = 156
19 Hill route: mean = 15.2 min, σ≈ 2.04 min Long way: mean = 15.9 min, σ≈ 1.87 min The long way is more consistent but the hill route is generally quicker, so the ‘best’ way depends on whether Sonja values speed or consistency more.
20 Median = 15 push-ups, IQR = 12 push-ups, so 50 push-ups is an outlier and should be checked.
21 Mean ≈ 11.1 kg, σ≈ 13.5 kg, so 60 kg is an outlier and should be investigated.
Sentence length
14 aNewspaper article Stem Computer article 9 8 8 3 2
8 7 7 7 4 3 2 2 9 8 5 3 3 2 2 2 6 1 1
1 2 3 4
0 1 4 5 5 7 7 8 9 0 1 2 3 4 4 5 6 7 7 7 8 9 1 6
Key: 8 ⎪ 1 = 18 = 1 ⎪ 8
200 400 600 800 1000 1200
Hybrid 246
Hybrid 315
Mass of french beans per plant
Mass(g)
0 10 20 30 40 50
Australian
Courier-Mail
Sentence length in headline story
Words/sentence
53 54 55 56 58 59
Brand A
Brand B
57 Height (inches)
22 Mean ≈ 20.6 s, σ≈ 2.8 s, so no times are outliers. The evidence indicates all teachers were paying attention.
23 Dan was about 1.2 SDs above the mean in Modern History but only 0.4 SD above in Art, so he actually did much better in Modern History. His friend was wrong.
24 The height of 148 cm is slightly more unusual because it is about 1.8 SDs away from the mean, but the IQ of 125 is only about 1.7 SDs away.
25 3 years is about 1.4 SDs above the mean for Brand X, but 1.3 SDs above for Brand Y. From the table, there will be about 2% more of Brand Y batteries that last more than 3 years, compared to Brand X, so Brand Y batteries are more likely to last longer than 3 years.
Chapter 3
1 a and h are linear equations.
2 a g = 18 b b = 5 c q =−2
d u =−3 e r =−3 f w = −3
3 a k = 13 b a = 0 c u =−1
d x =−3 e t =−1 f e =−3
4 a r =−17 b q =−2 c n = 2
d x = 4 e w =−1 f d = 9
5 a k =− b y = 3 c x = 1
d r = 3 e q =−8 f w = 5
6 a s = 9.857 142 857 b y = 1.5
c j = 12 d c = 4
7 a
b
Exercise 3.1
1 9
--- 1
2
---2 3
--- 4
11
---1 2
---22 29
--- 14
25
--- 9
13
---2 3
--- 11
12
---1 2 3
−6 2
−4 −8 4 8 10
6
4 y
x y = 4x − 3
−2 −1
1 2 3
2
−4 4 8 10
6
4 x
−2 −1 12
y
−4 −3
y + 2x = 5
c
d
e
f
8 a
b
c
d
e
10 12 14 x
2 4 6
2
−4 4 8 6
8 −4 −2
y
−8 −6 −6
3y = 2x − 3
2
−4 4
−6
1 2 3 x
−2 −1
−12 −10 −14 y
−8 −3
y = 4x − 5
6
2
−4 4
−6
1 2 3
−2 −1
−12 −−108 y
4 5 6 7 x
2x = y + 8
6
2
−2 4 y
2 4 6 x
−4 −2 −6
3x + 4y = 8
y = 5x − 7
y = 8 − 3x
y = x 1 + 6 3
---4x + 2y = 9
---f
9 p = + 48; price $60, Matt $12
10 15s + 25(2s − 6) = 15 × 2s + 25(s + 6) where s is the weight of the pack originally lighter; 30 kg and 60 kg
11 4(x + 1) × 0.1 = 5; 11.5 m by 11.5 m
12
a 520 mL b 770 mL c 70°C
d −135°C e −270°C
13
About 750 shares of each (actually 757) for a total of 1500 shares
14
a i $1500 ii$2100 iii $300 iv $900
b 135 km
1
y + 4x = 1 2
3
---p
5
---xm
1m
0 500 400
V
olume (mL)
Temperature (°C)
50 100 150 200
700
300 600
Volume of dry air
250 800
0 1500
1000
LGW Pork
Sun Minerals 500 1000 1500 2000 500
Share mixtures for $5000
2500 Equal
numbers
150 −1000
3000
1000
50 100
0
Distance (km)
Prof
it ($)
2000
Profit on loads of furniture
Exercise 3.2
2
3
4
3
4
5
2
3 The relation could be ‘is 2 less than’.
4 a Domain is −4, −1, 0, 2, 5, 8. Range is −5, 1, 3, 7, 13, 19.
b Domain = [2, 240] kg (including babies and obese individuals), range = [25, 240] cm (including babies and giants)
c Domain = [−3, 7), range = (2, 8]
d Domain = (−4, 4), range = (−11.5, 4.5) 5.9
3.2 2.9
3.8 5.1
2 3 4
6 5
2 4 6
2
−4 4 8 10
6
8 x
−4 −2 12
−6 14 18 20
16 y
Height (cm)
120
80 200
40 160
0
40 80 120 160 200
Mass (kg)
Thin Fat
Children
2 4 6
1 2 4 5
3
8 x
−4 −2 6 7 9 8 y
0
4 5 x
6
2
−4 4
−6
1 2 3
−2 −1
−12 −10 y
−8 −3 −5 −4
5 a, c, e and h are functions.
6 a, b, d, e, f, g, i and j are functions.
7 a R b −4 x 4 c R, x ≠−5
d R e R f R
g R, x ≠−3 h R, x ≠ 0
8 a Discrete b Continuous
c Continuous d Discrete
e Continuous f Discrete
1 a 99, 195, 387 b 100, 121, 142
c 15, 21, 28 d 57, 92, 149
e 3584, 14 336, 57 344 f 13, 21, 34
2 a −2, 1, 4, 7 b 3, 12, 27, 48
c 1, 8, 27, 64 d 1, , ,
e 2, 2 , 3 , 4 f 4, 1, ,
3 17th term 4 5th term
5 a, d, e and f are APs.
a d = 4 d d =
e d =−4 f d = 10
6 a d = 6, tn= 6n − 5 b d =−5, tn=−5n + 3
c d = 3 , tn= 3 n − 7 (or tn= (7n − 15))
d d = 4.2, tn= 4.2n + 1.1
e d = 2 , tn= 2 n + 2 (or tn= (11n + 10))
f d =− , tn= 2 − n g d = k, tn= m + k(n − 1) h d =−4, tn= 2r + (7 − 4n)
7 a 3, 7, 11, 15 b −7, −4, −1, 2
c −6, −14, −22, −30 d 24, 20, 16, 12
e 2x, 2x + 9, 2x + 18, 2x + 27
f −13m, −13m + 3n, −13m + 6n, −13m + 9n
g 0, −4, −8, −12 h 11, 5, −1, −7
8 227 9 −169
10 a 21 terms b 36 terms c 22 terms
d 20 terms e 14 terms f 24 terms
g 19 terms h 18 terms
11 1023 creases; 10.24 cm
12 a 21 − 5k b −3 − 8k
c 6k + 3 d 14 − 9k
e 3.5k + 6.7 f 3 k + 5
g h + 7k h p + k(q − p)
13 m = 4 − , p = 1
1 a − b c 0
d − e 6 f Not defined
Exercise 3.3
1 3 --- 1
5 --- 1
7
---1 2 --- 1
3 --- 1
4
--- 4
9 --- 1
4
---3 8
---1 2
--- 1
2 --- 1
2
--- 1
2
---3 4
--- 3
4 --- 1
2
--- 1
4
---2 3
--- 2
3
---3 4
---2
Exercise 3.4
4 3
--- 2
5
---3 7
---2 a −1 b c 0 d Not defined
e f − g 0 h −6
i j
3 a b − c − d e 0
f −1
4 a 1 b 2 c d −5 e
f g h − i −2
5 e, b, c, a, f, d
6 a 0.577 b 11.43 c −1.73 d 0.18
e 0.27 f 0 g −0.18 h −11.43
i 1.54 j Undefined
7 a 1 b 0.532 c −1.428 d −2.475
e 0 f 4.705
8 68° 9 108°
10 a 4 b 2 c 10 d −2
11 a, c and d are collinear.
12 No 13 Yes
14 a, c and d are parallel; b and e are perpendicular. 15 The x-intercept is stated first in each case.
a 4, 4 b −3, 8 c 2, −6
d 5, −7 e −7, 2 f −9, −4
g 9, −2 h −7, no y-intercept
i 7, 2 j no x-intercept, 6
16 The x-intercept is stated first in each case.
a 2, 10 b −6, 3 c −6, 8 d 4, 8
e − , 3 f 4, −4 g 4, −12 h −15, 5
i 2 , −8 j −2 , 1
17 a 6.24 m b No c 667 mm
18 Parallelogram 19 Right-angled triangle
20 Rectangle
1 a y = 5 b y =−4 c y =−5
d y = 7 e y = 6
2 a x =−4 b x =−7 c x = 7
d x =−3 e x =−4
3 a 5x − y − 3 = 0 b 4x + y + 3 = 0
c x + y − 5 = 0 d x + 2y + 10 = 0
e 5x − y = 0 f x − 5 = 0
4 Answers follow the form: x-intercept,
y-intercept, gradient.
a 2, −6, 3 b −8, −8, −1 c 6, 4, −
d 3, −4 , 1 e − , , 2 f −4, 2,
g , −3, 5 h −1 , −1 , −
i −4, 1 , j −3 , 2, 1
2
---1 5
--- 1
4
---3 2
--- 4
3
---7 4
--- 3
2
--- 5
6
--- 10
7
---4 5
--- 2
3
---9 2
--- 11
6
--- 9 7
---3 4
---2 3
--- 2
3 --- 3
5
---Exercise 3.5
2 3
---1 2 --- 1
2
--- 1
5 --- 1
2 --- 1
2
--- 1
2
---3 5
--- 2
3 --- 1
4 --- 3
4
---1 3 --- 1
3
--- 1
2 --- 4
---5 a Decreasing, above line
b Decreasing, on line
c Decreasing, below line
d Increasing, above line
e Decreasing, on line
f Decreasing, below line
6 a y = 6x − 1 b y = x +
c y =−x + 1 d y = 4
e x = 6 f y = x − 2
7 a y =− x + 3 b y =−x + 6
c y =−2x + 2 d y = x
e x = 3 f y = 2x
g y =−2x + 4 h y =− x + 3
i y = 6 j y = x + 7
8 a i 3x − 2y − 1 = 0 ii 2x + 3y − 18 = 0
b i x + 4y + 11 = 0 ii 4x − y − 24 = 0
c i 5x − y + 27 = 0 ii x + 5y − 31 = 0
d i x + y + 17 = 0 ii x − y + 1 = 0
e i 2x − 3y − 11 = 0 ii 3x + 2y − 10 = 0
9 a 4x + 5y − 20 = 0 b 2x − 3y + 6 = 0
c x + y + 7 = 0 d x − 2y − 4 = 0
e 3x + 5y − 15 = 0 f x − 10y − 4 = 0
10 a f (x) = 2x b f (3) = 6 c x = 10
11 a
b x − 5 = 0, x − 3y + 7 = 0, 5x + 3y − 19 = 0
c (4, 2) inside, (3,−1) outside
12 y = 0.0083x + 0.018
13 y =−0.401x + 21.0
1 a 13 b c 5
d 17 e 3 f 13
2 3 --- 1
3 ---2
3
---5 8 --- 7
8 ---2
3 --- 2
3
---1 4 --- 3
4
---3 2
---y
x (5, 4) (2, 3)
(5, −2)
0 20
10
Frequenc
y (kHz)
Age (years)
10 20 30
Hearing loss
Exercise 3.6
61
2 a (−1, 7) b (1, ) c (−4 , 2 )
d (2 ,−2) e (0,−3 ) f (8, 4)
g (3 , 1 ) h (−1 ,−1) i (6, 7)
j (−4, 6)
3 (−2, 2)
4 a AB = CD ≈ 10.2, AD = BC ≈ 6.7
b (1 , 0) c (1 , 0)
5 a OX = OY = OZ b 10
6 Show that adjacent sides are perpendicular and unequal.
7 Show that there is a right angle at (1, 1).
8 k =−3 9 m = 5
10 Show that two sides are parallel.
11 Show that the opposite sides are parallel.
12 a Show that PA = PB.
b Perpendicular bisector passes through P and (−2, 0).
13 No 14 k = 1 ± 3
15 Show that two adjacent pairs of sides are equal.
16 Sides: , , ,
Diagonals: =4 ,
17 4x − 3y + 16 = 0
18 (1,−8), (9, 10), (5, 0)
19 a Centre (4, 5), radius = 5
b (9, 10) and (3,−2) are on the circle but (−1,−1) is not.
20 a and
b Use gradient and length formulas.
1 a (−5,−1) b (−1,−1)
c (−1 ,− ) d (11,−15)
2 a (3, 2.5) b (−2, 3) c (2, 5)
d (2.5, 5) e (4, 1) f (5,−1)
3 a a = 2, b = 3 b x = 4, y = 1
c k =−2, j = 3 d f = 3, g =−4
e p = 5, q =−1 f f =−3, g = 6
g h = 2, i =−3 h c =−1, d =−5
4 a m = 4, k = 3 b k = 5, j = 4
c p =−2, q =−5 d a =−6, b = 2
e x =−1, y =−3 f v =−2, w = 6
g t =−4, u = 4 h f = 3, g = 7
5 a b = 1, u = 3 b a = , r = 3
c e = , m = 1 d y =−4, t =−6
e a = 5, n = 6 f i =−8, s = 6
g b =−3, y =−6 h u = 10, p = 3 1
2
--- 1
2 --- 1
2
---1 2
--- 1
2
---1 2 --- 1
2
--- 1
2
---1 2
--- 1
2
---1 3
---3
13 53 10 10
32 2 37
50 2
a+ f
2
--- b+g 2 ---,
⎝ ⎠
⎛ ⎞ j+ f
2 --- k+g
2 ---,
⎝ ⎠
⎛ ⎞
Exercise 3.7
1 2 --- 1
2
---1 2 ---1
4
--- 2
---6 a e = 7, f = 4 b x = 4, g = 3
c k =−6 , m =−17 d a = 11, b = 2
e a = 1, b = 2
7 6 adults and 15 children
8 In 16 years Petra will be 48 and Philippa 24.
9 15 kg of copper and 10 kg of zinc
10 Boat 12 km/h, river 6 km/h
Chapter 4
1 a Yes b About 7:30 pm
c About 7:30 am d 24 hours
e 45 ng/mL
2 a January b July
c 12 months d 12 months
e About 4.3ºC f About 5.7ºC
g Minimum temperatures
3 a Period = 20 ms, amplitude = 340 V
b 50 Hz
4 a About 2.3 m b About 0.3 m
c About 1 m d About 12.3 h
5 a Yes b About 6.2 nm
c About 0.6 ms
6 a Blue line
b Yes (approximately)
c About 45 quolls and 24 foxes
d Same for both—about 10 years
7
b Yes (approximately) c About 13ºC
d About 11.5 months
1 a Period = 4 s, amplitude = 8 cm
b −5.8 cm
c 1.1 s, 3.9 s, 5.1 s, 7.9 s, 9.1 s, 11.9 s, 13.1 s, 15.9 s, 17.1 s and 19.9 s
2 a 8 cm b 58 cm
c 66π≈ 207.3 cm 1 4
--- 1
6
---Exercise 4.1
Surface water temperature
30
20
16 32
Months 10
0
T
emperature (°C)
25
15
5
4 8 12 202428 36
Exercise 4.2
d
e About 52 cm f About 61 cm
3 a Period = 20 ms, amplitude = 350 V
b −100 V c 200 V
d 0.002 s, 0.008 s, 0.022 s
4 a 4 times (if evenly spaced on cable)
b 38 minutes
c
d One does 19 trips; 3 do 18 trips each.
e 730 people f 7 minutes
5 a 0.8 s b 1.25/s or 75/min
c It is the pulse rate. d 13 min 20 s
6 a 0.6 ms b 1.5 mN
c 0.04 ms, 0.64 ms, 1.24 ms, 1.84 ms, 2.44 ms
d About 16 667 times
1 a
b
c
d
0 30 20
Height (m)
Distance (cm)
50 100 150 200
50
250 300
10 40
Reflector height
60
0 150
100
Height (m)
Time (min)
10 20 30 40
250
50 50
200
60 70 80
Cable car progress
Exercise 4.3
y = 4 sin 2πx
y = 7 sin 2πx
y = 5 cos 2πx
e
f
g
h
i
2 a
b
c
d
e
f
y = 9 sin 3π 2 --- x
y = 9 cos π 2 --- x
y = 6 cos π 2 --- x
y = 8 sin x
y = 3 cos x
y = 4 sin 2πx 5
---y = 7 cos 3πx 8
---y = 9 sin πx 7
---y = 2 sin 4πx 5
---y = 6 cos 7πx 8
---y = 5 sin πx 5
---3 a
b
c
d
e
f
4 a
b
c
d
e
y = 8 sin
y = 8 sin πx
3
---2πx
5
---y = 6 cos
y = 6 cos 2πx
5
---4πx
5
---y = 5 sin
y = 5 sin 2πx
7
---2πx
5
---y = 9 cos
y = 9 cos 2πx
3
---2πx
5
---y = 3 sin
y = 3 sin 2πx
9
---5πx
9
---y = 7 cos
y = 7 cos 2πx
3
---7πx
3
---y = 4 sin 2π(x+1) 3
---y = 6 sin 2π(x–1) 5
---y = 8 sin 2π(x+2) 7
---y = 9 sin 2π(x–2) 5
---f
5 a
b
c
d
6 a y = 7 sin 2πx b y = 4 sin
c y = 3 sin 2πx d y = 6 sin
e y = 8 sin f y = 2 sin
g y = 9 sin h y = 5 sin
i y = 3 sin j y = 2 sin
7 a y = 4000 sin
b 2000 km south c About 3100 km
Chapter 5
1 a $2.45/kg b 45 rev/min
c 30 m/s d 6.25 m/s2/N e A$1.25/$US1 or $A1/$US0.80
2 a 0.4 mL/L, 250 L b 80 mL
c 144 L d 1429 seeds
e 1908 bricks
3 a 1.9 t/m3 b m = 1.9v c 2.85 t d 6.3 m3 e 1.52 t
4 Yes, rate = 0.15
y = 6 sin 2π(x–3) 11
---y = 4 sin
y = 4 sin
2π(x+1) 3
---2π(x–1) 3
---y = 8 sin
y = 8 sin
2π(x+1) 3
---2π(x+1) 9
---y = 7 sin
y = 7 sin
π(x+1) 2
---5π(x+1) 2
---y = 2 sin
y = 8 sin
2π(x+1) 3
---2π(x+1) 3
---2π(x–1) 3
---2πx
5
---2π(x+4) 5
--- π(x–2) 3
---2πx
3
--- πx
4
---2π(x+1) 3
--- π(x+3) 6
---2π(x+12.5) 90
---Exercise 5.1
5 Yes
6 a
b The graph is a straight line, within experimental error.
c About 0.07 cm/g
d About 13 cm, the unstretched length
7 a
b The rate of change of temperature gradually decreases.
c 3.2°/h
d It will probably be lowering the temperature about 1°C/h, so it won’t be much use.
e No
8 a About 88 000 km b About 3700 km/h
c Because of the motion of the Earth around the Sun, it has to orbit about 30° further than a complete orbit to appear at the same place in the sky.
9 a 13.3 km/h b The first leg
c The second leg d 8 km/h
10 About 91 km/h 11 4 m/s2
12 a 0.5 m/s b 1.5 m/s c 2.5 m/s d 1.5 m/s
13 a 0.72 b 0.46 c 0.40 d 0.97
14 a 8 b11.91 c1 d 17 e3.01
15 a 20 m b1 m/m cl = 0.6h d 33 me1.14 m
16 a No
0.5 1.0 1.5 2.0 2.5 2
8
4 6 10
r D
0
20
80 160
Mass (g) 10
0
Length (cm)
25
15
5
20 40 60 100120140
Spring stretch
24
12:20
Time 20
0
T
emperature (°C)
26
22
18
11:20 12 12:40 1:00 1:20 Temperature in car
11:00
am am 11:40am noon pm pm pm pm
2 3
---100
Plant growth
Water per day (mL) 200 10
Gro
wth (cm)
50
50 150 250 300
60
40 30 20
17 a Yes, within experimental error
18 5 g
19 a 400 km/h b 169 km/h c 4 h
d Average speed = 300 km/h, average velocity = 0
20 Teacher to check
21 0–10 s is about 9 kPa/s; 50–60 s is about 2 kPa/s.
22 a Increasing at a constant rate
b Decreasing at a decreasing rate
c Increasing at an increasing rate
d Increasing at a decreasing rate
e Decreasing at an increasing rate
f Decreasing at a constant rate
g Constant
Answers in this section based on tangents may vary a little from the stated answer.
1 −2, −2, −2
2
a −3 b 5 c 9
3
a 1 b −3 c −9
4
a 3 b 0 c 3
200
Sliding weight
Mass in pan (g) 400 10
Distance (cm)
50
100 300
60
40 30 20
0
Exercise 5.2
3
−3 x
16 12 8 4
−4 f(x)
f(x) = 2x2+ x − 4 20
8 4
3
−3 x
f(x)
−8 −12 −16 −20 −4
f(x) = 5 − 3x − x2
−24
4
3
−3 x
f (x)
−8 −12 −16 −4
f(x) = x3 − 6
5
a −5 b −1 c 3
6 11 m/s2, −8 m/s2, −20 m/s2 7 10.5 mL/day, 4 mL/day
8 110°C/min, 0°C/min, −100°C/min
9 Velocity at 2 s is 25 m/s. It takes 5 s to go 100 m. Its velocity at 100 m is 10 m/s.
10 6
1 a
b
c
2
3
4
6
−6 x
f(x)
−8 −12 −4 12 8
f(x) = (x + 4)(x − 3)
−14
Exercise 5.3
4 s
1 2 5
3
t
1 2
0
3
4 s
1 2 5
3
t 2
0
4 6
6
s
t 2
1 0 −1 −2 −3
1 2 3 4
2 s
0.5 1 2.5
1.5
t 2
0 3
1 3 4
2 4 6
2 0
Displacement (m)
4
Moving object
Time (s) −4
6
4 a b
c
5 a C b D c B d A
6 a
b 0.2 m
c
7 Bottom at 12.4 m/s, water at 1 m/s
8
It would have taken 14 minutes.
9 About 19 cm/min
10 47.5 m 1
2
−2 t
v
−2 −1
−3
10 5
3
−3 t
v
−10 −5
4
t v
−8 −4 8
−3 3
0.2 0.4 0.6 0.8
0.8
0.4
Displacement (m)
1.2
Billiard ball
Time (s) −0.4
1.6
0
1.0
0.2 0.4 0.6 0.8
4
2
V
elocity (m/s)
6
Billiard ball
Time (s) −2
8
0
1.0
18 4000
1000 2000 5000
3000
2 0
Time (min) 6000
7000
4 6 8 10 12 14 16
Height (m)
Airbus takeoff
18 400
100 200 500
300
2 0
Time (min)
4 6 8 10 12 14 16
Rae of climb (m/min)
Airbus takeoff
1 a
b
c
2 The gradient is positive for x −0.12, zero for
x =−0.12, negative for −0.12 x 2.79, zero for x = 2.79 and positive for x 2.79.
3
4
5
6
7 24
Exercise 5.4
1
x
−2 −1 2
−3 3
Gradient of f(x)
3
−3 x
Gradient of f(x)
−4 −2
−6 −8
1
x Gradient of f(x)
2
−3 3
3 4
0
f(x) 4
2 3
1 −1
−2 4 5
f(x) = x3− 4x2− x + 4
x
Gradient
x 1
Gradient function 2
Gradient
x −3
Gradient function 6
Gradient
x −3
Gradient function 6
Gradient
x 1.5
8 a 33.01 b 32.1001 c 32.010 001
9 a 9 b −14 c 3
d 9 e −62
10 a −10 b −16 c 230
d 5 e −58
11 a 29.000 001 b 101.000 004
c 3 12
13
14
15 Yes, the functions in questions 4 and 5 have the same gradient function.
16 No. The values of a function are unique for each
x value, so the differences between the values at
two x values has only one possible answer, so the gradient of any chord has only one possible answer, so the limit of the gradients of chords used to get the gradient of a tangent is unique for each point.
17 d(1) = 2.5 m, v(1) = 4 m/s, d(2) = 8 m,
v(2) = 8 m/s
18 a The object is falling. b −9.8 m/s
c −19.6 m/s d −29.4 m/s
e −39.2 m/s
Chapter 6
1 a −3.5, 0.7 b −1.4, 4.8
c −1.5, 3.2 d −3.6, 2.7
2 a −4, 3
f(x) = 2x3− x2+ 2x − 5
and gradient function
f(x) =−x2− 3x + 8
and gradient function
f(x)
x
1 2 3 4
f(x)
Exercise 6.1
−1 1 2 3 −2
−3 4 5 x
−4 −5
f(x)
−12
f(x) = x2+ x − 12
b −5, 1
c −1.2, 5.2
d −1.3, 2.8
e −2.9, 1.4
f −1.5, 9.5
3 a −1.31, 3.81
b −9.66, 1.66
5 y
−2 −1 −3
−4 1 2 x
−5 −6
y = 5 − 4x − x2
3 4
2
1 5 6 x
−1 −2
−6 y
y = x2− 4x − 6
1 2 3
−2 −1 4 x
7 f(x)
f(x) = 3x + 7 − 2x2
−2 −1 −3
−4 1 2 x
f(x)
−8 f(x) = 2x
2+ 3x − 8
6 8
4
2 10 x
−2 −7
y
y =x 2− 4x − 7 2
---f(x) = 2x2− 5x −10
---c −0.64, 3.44
d −2.37, 4.37
e −1.74, 3.74
f −37.03, 2.03
4 a −2.3 and −2.2 b 3.5 and 3.6
c 4.63 and 4.64 d −8.64 and −8.63
5 a −0.33, 3 b −1.93, 2.59
c −1.13, 1.63 d −3.10, 2.10
e −6.14, 1.14 f −2.32, 4.32
6 a −1.15, 5.65 b −1.81, 3.14
c −9.32, 0.32 d −3.80, 15.80
e −1.12, 1.68 f −3.36, 1.69
1 a (3x − 5)(x − 2) b (x + 8)(x − 2)
c (7x − 6)(x + 1) d (2x − 3)(x − 4)
e (4x + 3)(x − 4) f (4x + 3)(2x + 3)
g (5x + 1)(x + 7) h (3x + 7)(x − 7)
i (7x + 4)(x − 8) j (5x − 4)(x − 8)
k (4x − 9)(5x + 2) l (5x − 8)(5x + 2)
2 a 0, 12 b −2, 5 c 0, 8
d −1 , 0 e −7,−2 f 3, 5
3 a −3, 1 b 1, 11 c −4,−1
d −1, 8 e −5,−2 f −1, 3
4 a −1, 5 b −2, 3 c −3, 4
d 3 e −2,− f −3,−
5 a −2, b 3 , 6 c −8, 8
d 3 e −1 ,− f −2,
6 a 2, 3 b −2,− c −7, 7
d 4 e −5,− f ,
y = 5x2− 14x −11
y = 6x + 31 − 3x2
y = 4x2− 8x −26
f(x) = x2+ 7x − 15 5
---Exercise 6.2
1 2
---1 3
---1 2
--- 1
3 ---2
5
--- 1
2
---1 2
--- 1
2 --- 6
7
--- 8
9
---1 2
--- 5
7
---3 4
--- 3
8 --- 1
2
---7 a 1, 7 b −3,−6 c −2,
d −3, 2 e 1 , 3 f −2 , 6
8 a −5, 1 b −7, 5 c 1, 8
d −11, 12 e −1 ,
f −2 ± ≈−5.16, 1.16
9 a −9, −5 b −17, −3 c −14, 21
d −2 ± ≈−5.32, 1.32
e −3 ± ≈−2.48, 8.48 f −7, 20
10 a −2 , 2 b −1 , 3
c − , 1 d −1.79, 2.79
e −0.28, 1.95 f −2.64, 1.14
11 a −7.36, 1.36 b −1 ,
c −2.12, 6.12 d 3.413, 2.733
e 2.54, 9.46 f −2,−1
g −28.79,−0.21 h 0.11, 1.76
12 l2− 0.9l = 16.2 or w2+ 0.9w = 16.2,
3.6 m by 4.5 m
13 l2+ 12l = 160, 8 m
14 6h2+ 32h = 88 or + = 88,
2 cm × 4 cm × 6 cm
15 = + 1 or = + 1,
Jon 4 km/h, Marnie 5 km/h
16 $20
17 Larger pipe 10 h, smaller pipe 15 h
18 10 km/h
19 40 ohms and 60 ohms
1 a Minimum, (3, −12) b Maximum, (−1 , 13 )
c Minimum, (−2, −1) d Maximum, (1 , 10 )
2 a y = x2− 6x − 3 b y = −2x2− 5x + 10
c y = x2+ 4x + 3 d y = −2x2+ 7x + 4
3 B
4 a D b F c A
d B e C f E
5 a Minimum, (−1,−2) b Minimum, (2, 3)
c Minimum, (1, 9) d Maximum, (−3, 1)
e Maximum, (−2, 60) f Maximum, (4,−2)
6 a Minimum, (1 ,−1 )
b Maximum, (3 , 14 )
c Minimum, (2 ,−4 )
d Minimum, (1 ,−20 )
4 9
---1 8
--- 1
3
---1 3 --- 2
3
---10
11
30 1 3
--- 1
5
---1 6
---1 3 --- 1
2
---1 2
---2l2 3 --- 32l
3
---20
m–1
--- 20
m
--- 20
j
--- 20
j+1
---Exercise 6.3
1 4 --- 1
8
---3 4 --- 1
8
---1 2 --- 1
4
---1 2 --- 1
4
---1 2 --- 1
4
---1 2 --- 1
---e Minimum, (3 ,−7 )
f Maximum, (1 , 3 )
7 a Minimum, (−3, 1) b Minimum, (−4,−4)
c Maximum, (5, 0) d Minimum, (6 ,−48)
e Minimum, (2 , 16) f Maximum, (−4,−9)
g Minimum, (2,−16) h Maximum, (−4 , 6 )
8 a −12 b −2, 6
c Minimum, (2,−16)
d x = 2
Domain: x ∈ real numbers. Range: y −16
9 a 24 b −4, 3
c Maximum, (− , 24 ) d x =−
Domain: x ∈ real numbers. Range: y 24.5
10 a Range: y –4
b Range: −10 y 2
c Range: y −3 1 2 --- 1
4
---1 2 --- 1
4
---1 2
---1 2
---1 2 --- 1
4
---y
x
−12
(2,−16)
−2 6
y = x2− 4x − 12
1 2 --- 1
2
--- 1
2
---y
x 24
(− , 24 )
−4 3
1 2 --- 1
2
---y =−2x2− 2x + 24
x (−3,−4)
y = x2+ 6x + 5
5
−5 −1
y
1 4
---y
x (−1 ,2 )
y =−x2− 3x (−4,−4)
(2,−10) −3
1 2 --- 1
4
---y
x
(1,−3) −0.73
y = x2− 2x − 2
2.73 −2
d Range: −9 y 7
e Range: y 9
f Range: −63 y 12
g Range: y 12
h Range: y −4
i Range: −24 y 16 y
−9
−3 3
y = x2− 9
(−4, 7) (4, 7)
x
y 9
−3 3
y = 9 − x2
x
1
y =−3x2− 6x + 9 (−1, 12) 9
x
y
(4,−63) (−2, 9)
y
4
y = −3x2+ 12x (2, 12)
x
y
−3
y = 4x2− 4x − 3
x
(1,−4)
2
--1 2
--1
1 2
--−
y
8 y = −3x2− 10x + 8
x (−1 , 16 )2
3
--2 3
--−4
1 3
--(−5,−17)
---j Range: y 12
k Range: y 16
l Range: y −27
11 a y-intercept: 2; zeros: −5.65, −0.35;
minimum, (−3, −7)
b f(x)-intercept: 9; zeros: −3.71, 1.21;
maximum, (−1.25, 12.125)
c y-intercept: −1; zeros: 0.04, 5.71;
maximum, (2.875, 32.0625)
d f(x)-intercept: 7; zeros: none;
minimum, (1.25, 3.875)
12 a y =−x2+ 4x − 6 b y = 1.5x2− 1.5x − 3
c f(x) =−0.8x2+ 0.8x + 4.8
13 Teacher to check proof
14 25
15 40°C
16 Teacher to check proof. The area is greatest when x = 20 m.
17 8.75
18 Maximum, (2.5, 33.25); h-intercept: 2;
t-intercepts: −0.08, 5.08. It takes 2.5 s to reach
maximum height and 5.08 s to reach the ground. y
4 y = −3x2+ 12x (2, 12)
x
y
y = −4x2+ 16x
x (2, 16)
4
y
−15
y = 3x2− 12x − 15
x
(2, −27)
−1 5
Solubility (g/L)
Temperature, T (°C) 20
10
30 10 20
70 40 50 60 30
0
Solubility of compound
1 a (−5.8, 4.6) and (2.8, −12.5)
b (−3.1, 17.3) and (0.6, 9.7)
c (−2.6, −12.7) and (2.6, 2.7)
d (−0.6, −5.2) and (6.6, 9.2)
2 a (2.5, 9.5) and (−3,−7)
b (2, 5) and (−3,−15)
c About (3.1, 11.2) and (−0.1, 4.8)
d About (6.1, 8.1) and (−1.1, 0.9)
3 a (1.5, 12.5) and (−4, 18) b (1.67, 6.67)
c No solutions
d (1.19,−0.19) and (−4.19, 5.19)
4 a (−2.5, 30) b No solutions
c (4, 6) and (−1 , ) d (1, 3) and (4, 6)
e and
≈ (4.19, 5.39) and (−1.19,−5.39)
f and
≈ (6.31, 36.56) and (1.19, 10.94)
g No solutions h (−1,−1) and (−5,−9)
5 (−1 , 4) and (5, 17), PQ ≈ 14.5
6 (−1, 1) and (2, 4)
Chapter 7
1 a 7 b −2 c 21
d 4 e 0 f 2.5
2 a 25 b Not defined c 4
d 10 e Not defined f −3
3 a −7 b 12 c −4
d 6 e 10 f 5.25
4 a 2 b 1 c 3 d 0 e 0
f 10 g −5 h −3 i 7
5 a 11 b 11 c −8 d −4 e 12
f −1228 g 25 h 110 i 7
6 a 9 b 1 c 6x d 0
e 11 f 20x + 3 g −5 − 2x h 3 − 8x
i 21x2 j 4x3
7 a y′=−6x2 b f′(x) =−21x2c f′(x) = 18x d g′(x) =−12x e f′(x) = 3x2 f y′= 6x g f′(x) = 36x2 h y′= 1 i g′(x) =−4 j g′(x) = 11 k y′= 15x2 l f′(x) =−66x 8 a 9 b 2x − 7 c 6x + 8
d 15x2 e 44x3 f −6x + 9 Exercise 6.4
1 2 --- 1
2
---3+ 29 2
---, 29
⎝ ⎠
⎛ ⎞ 3– 29
2
---, − 29
⎝ ⎠
⎛ ⎞
15+ 105 4
--- 5 19( + 105) 4 ---,
⎝ ⎠
⎛ ⎞
15– 105 4
--- 5 19( – 105) 4 ---,
⎝ ⎠
⎛ ⎞
1 2
