00 NQM11B SB TXT.prelim - mathsbooks.netmathsbooks.net/New Q Maths 11B/answers.pdfChapter answers 402 New QMaths 11B 9780170136570 Chapter 1 1a 42.6° b 67.35° c 28.29° d 16.41°

Chapter answers

402

New QMaths 11B

9780170136570

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

12---

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 mf 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 km10 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 cmd 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 = 64.6d g = 43.7 e s = 157 f q = 952.9

4 a A = 21.4°, G = 37.6°, d = 7 mb U = 56.7°, V = 68.3°, w = 44.1 cmc 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 kmf R = 5.2°, F = 6.6°, e = 341 mm

5 A = 89.6° 6 X = 24.15°7 P = 90° 8 27 m9 26.4° and 36.3° 10 170°T or S10°E

1 a 3.85 m b 17 mm c 50.25° d 41 cm2 a 57.5 m b 10.8 m c 322 m d 9.2 cm3 5.22 m 4 12.9 m5 a 60.7 m b 76.12°

Exercise 1.3

132˚22′47˚38′

18 cm

C

A BA′38˚

15 cm

Exercise 1.4

Exercise 1.5

13 NQM11B SB TXT.ch.ans.fm Page 402 Sunday, May 17, 2009 8:57 PM

Chapter answers 4039780170136570

6 7.128 km/h 7 155 m8 a 55 m b 29°9 713 m

10 a 369.3 m b 143.6 m11 4.31 km 12 68.6 m13 a 361 km b 94.63°T14 12.75 km 15 85 m 16 54 m17 166 km 18 About 20 m 19 41.5 m20 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°W4 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°E8 a 73.3° b 7.5 m c 53.13°

d 6.3 m e 7.7 m9 15.3°, 410.4 m 10 81.5 m

11 155 m 12 77.1 m

Chapter 2

1 a Number of DVDs, discreteb Height, continuousc Nationality, nominald Number of rooms, discretee Length, continuousf Eye colour, nominalg Blood pressure, continuoush Collection size, discretei Chemical symbol, nominalj Learner’s licence result, discrete (there are a

fixed number of questions)2 a People purchasing from a cinema snack bar,

amount spent, mean amountb Australian residents of Gympie, takeaway

food eaten, preference of outletsc Students, choice of calculator, preference of

calculatord Vehicles on Ipswich Motorway, traffic

density, density for each type of vehiclee 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, 40b 374 995, 380 868, 139 324, 347 249, 105 078,

110 178, 138 491, 336 824c 7, 0, 2, 6, 4, 8, 3, 9d 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-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’ tradeb Retail activity, population growth, income

distribution, age distributionc 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 groupii People using the resource centre will

probably share similar views.


404 New QMaths 11B9780170136570

iii Only one age groupiv Volunteers will probably have strong views

one way or another.v Probably the fairest methodvi Problems similar to ii and ivvii Could be biased depending on how students

are selectedviii Depends on how they are selectedix Similar to ivx Probably not representative because only

students entering school on a particular day during a particular period are asked

12 a Administration: males 1, females 3Factory: males 8, females 6

b Administration 4, factory 1413 a NSW 164, Vic. 124, Qld 99, SA 38, WA 50,

Tas.12, NT 5, ACT 8b Men: NSW 81, Vic. 61, Qld 50, SA 19, WA 25,

Tas. 6, NT 3, ACT 4Women: 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 schoolb 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 schoolb People in your suburb/townc People in your suburb/townd Students at your schoole Students at your schoolf People in your streetg People who watch current affairs programsh People who attend functions with disco

operatorsi People at the amusement parkj People who choose the washing detergent

for their householdii 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.

22 The people who subscribe to a health and fitness magazine are very unlikely to be typical of the population. The results will give biased tables.



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

45678

3 5 6 90 0 1 2 83 3 5 7 7 91 4 85 6

1 What is your sex? M F2 What is your age?

Under 10 11–20 21–40 Over 403 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 11 122 What is your sex? M F3 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, 44.5c 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 kg10 a 962 and 1530 hours

b 50 hours; 950–999, 1000–1049, …, 1500–1549 hours (others possible)

4 Year 11 handspans (cm)

Stem Leaf

17181920212223

2 91 52 6 8 81 80 7 83 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

TrucksOther

Motorcycles

vehicles vehicles

Total = 14.4 million

Key: 4 ⎪ 3 = 43

Key: 17 ⎪ 2 = 17.2


406 New QMaths 11B9780170136570

11 Table will vary depending on intervals chosen.

12 a

b

d Safety lightbulb lives (hours)

Stem Leaf

910111213

1415

62, 808, 10, 12, 40, 9220, 29, 39, 42, 43, 47, 59, 59, 765, 11, 27, 42, 44, 46, 77, 82, 85, 97, 9817, 21, 24, 26, 31, 32, 48, 52, 62, 74, 86, 87, 93, 992, 17, 40, 41, 53, 73, 9310, 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

Freq

uenc

y

20

15

10

5

0 10 20

% NESB students

30 40 50 60

% NESB students in schools

Freq

uenc

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

Freq

uenc

y

20

15

10

5

024 27

Number30 33 36 39

Occupational injuries per year

Occupational injuries per year

Freq

uenc

y

20

15

10

5

024 27

Number30 33 36 39

c

42

Freq

uenc

y

100

80

60

40

20

050 70

Girth (cm)90 170110 130 150

Girths of pine trees

b

Cum

ulat

ive

freq

uenc

y

50403020100

10 20 30Mark (%)

40 6050

6070

70

8090

80 90 100

Maths B students competition results



c 38%d 76 min (25%) or �75 min (20%) (other

answers possible)

1 a 22 b 33 c 37 d 18 e 25 f 31.32 a 112 b 82 c 50 d 149 e 127 f 1653 a Mean ≈ 10.9, median = 12, mode = 14

b Mean = 5.3, median = 3.5, mode = 2c Mean = 8.75, median = 9, mode = 9d 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 = 7–95 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

Cum

ulat

ive

perc

enta

ge

50403020100

60 65 70Time (minutes)

75 8580

6070

90

8090

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 = 7b Mean = 9.5, median = 8.5, mode = 15c 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 $3800c 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

Perc

enta

ge f

requ

ency

40

20

0

20003000

Income ($)4000

60

5000

80

100

60007000

8000


408 New QMaths 11B9780170136570

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 cm12 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 massesC

umul

ativ

e fr

eque

ncy

50403020100

0.8 1 1.2Mass (kg)

1.4 1.81.6

6070

2

8090

2.2 2.4 2.6

Life of batteries

Cum

ulat

ive

freq

uenc

y

50403020100

3 6 9Months

12 1815

60

21

70

24 27 30 3633 39 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 cmb Mean

17 32.5 years 18 $1106.67 19 $1.7020 a Mean ≈ $407, median ≈ $398,

modal class = $380–$419b 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 = 25.5c 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 ≈ 3.7c Mean ≈ 28.8, σ ≈ 7.8 d Mean ≈ 40.1, σ ≈ 5.7

4 a 11 motorists b 2 motoristsc 8 motorists d 6 motoristse 5.6 motorists f 3.50 motorists

5 a Continuousb Range = 3.5 kg, IQR = 0.95 kgc Mean = 2.7 kg, σ = 0.687 kg

6 Range = 9°C, IQR = 5°C, σ = 2.95°C7 Mean ≈ 11.91 s, σ = 0.650 s8 a 3.5 b 2.429 a Mean = 213, IQR = 42.5, σ ≈ 167.2

b Mean ≈ 144.4, IQR = 27.75, σ ≈ 29.5c 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 5 6 8 9



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, 44b Outliers: 198, 532, 503c Outliers: 56, 11, 3d 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 Median7 a Mean ≈ 87.25 kg, median = 86.64 kg,

modal class = 85–89 kgb 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 meand ($1800 × 20 + $2500 × 4) ÷ 24 ≈ $1917

10 a 72.4 kg b 73.2 kg11 10 people12 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 39 9 8 6 2 2 1 1

9 8 6 5 5 3 0 0 03

7 3 3 2 07 5 2 0 0

0123456

00 0 1 1 2 2 3 4 5 6 7 7 8 8 80 0 0 4 7 8 810 0 0 9 92 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 = 15619 Hill route: mean = 15.2 min, σ ≈ 2.04 min

Long way: mean = 15.9 min, σ ≈ 1.87 minThe 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 a Newspaper article Stem Computer article

9 8 8 3 28 7 7 7 4 3 2 29 8 5 3 3 2 2 2

6 1 1

1234

0 1 4 5 5 7 7 8 90 1 2 3 4 4 5 6 7 7 7 8 91 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

57Height (inches)

Rebound height of type 2 tennis balls


410 New QMaths 11B9780170136570

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 = −1d x = −3 e t = −1 f e = −3

4 a r = −17 b q = −2 c n = 2d 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 = 1.5c j = 12 d c = 4

7 a

b

Exercise 3.1

19--- 1

2---

23--- 4

11------

12---

2229------ 14

25------ 9

13------

23--- 11

12------

1 2 3

−6

2

−4

−8

4

810

6

4

y

x

y = 4x − 3

−2 −1

1 2 3

2

−4

4

810

6

4 x−2 −1

12y

−4 −3

y + 2x = 5

c

d

e

f

8 a

b

c

d

e

10 12 14 x2 4 6

2

−4

4

86

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

−12−10−8

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 + 613---

4x + 2y = 9

y − x + 11 = 012---



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°Cd −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 $900b 135 km

1

y + 4x = 123---

p5---

x m

1 m

0

500

400

Vol

ume

(mL

)

Temperature (°C)50 100 150 200

700

300

600

Volume of dry air

250

800

0

1500

1000

LG

W P

ork

Sun Minerals500 1000 1500 2000

500

Share mixtures for $5000

2500

Equalnumbers

150−1000

3000

1000

50 1000

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

810

6

8 x−4 −2

12

−6

14

1820

16

y

Hei

ght (

cm)

120

80

200

40

160

040 80 120 160 200

Mass (kg)

Thin Fat

Chi

ldre

n

2 4 6

12

45

3

8 x−4 −2

67

98

y

0

4 5 x

6

2

−4

4

−6

1 2 3−2 −1

−12−10

y

−8

−3−5 −4

2y = 4x − 7


412 New QMaths 11B9780170136570

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 Continuousc Continuous d Discretee Continuous f Discrete

1 a 99, 195, 387 b 100, 121, 142c 15, 21, 28 d 57, 92, 149e 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 term5 a, d, e and f are APs.

a d = 4 d d =

e d = −4 f d = 106 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, 2c −6, −14, −22, −30 d 24, 20, 16, 12e 2x, 2x + 9, 2x + 18, 2x + 27f −13m, −13m + 3n, −13m + 6n, −13m + 9ng 0, −4, −8, −12 h 11, 5, −1, −7

8 227 9 −16910 a 21 terms b 36 terms c 22 terms

d 20 terms e 14 terms f 24 termsg 19 terms h 18 terms

11 1023 creases; 10.24 cm12 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

13--- 1

5--- 1

7---

12--- 1

3--- 1

4--- 4

9--- 1

4---

38---

12--- 1

2--- 1

2--- 1

2---

34--- 3

4--- 1

2--- 1

4---

23--- 2

3---

34---

2

Exercise 3.4

43--- 2

5---

37---

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 −14 a 1 b 2 c d −5 e

f g h − i −2

5 e, b, c, a, f, d6 a 0.577 b 11.43 c −1.73 d 0.18

e 0.27 f 0 g −0.18 h −11.43i 1.54 j Undefined

7 a 1 b 0.532 c −1.428 d −2.475e 0 f 4.705

8 68° 9 108°10 a 4 b 2 c 10 d −211 a, c and d are collinear.12 No 13 Yes14 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, −6d 5, −7 e −7, 2 f −9, −4g 9, −2 h −7, no y-intercepti 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, 8e − , 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 mm18 Parallelogram 19 Right-angled triangle20 Rectangle

1 a y = 5 b y = −4 c y = −5d y = 7 e y = 6

2 a x = −4 b x = −7 c x = 7d x = −3 e x = −4

3 a 5x − y − 3 = 0 b 4x + y + 3 = 0c x + y − 5 = 0 d x + 2y + 10 = 0e 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,

12---

15--- 1

4---

32--- 4

3---

74--- 3

2--- 5

6--- 10

7------

45--- 2

3---

92--- 11

6------ 9

7---

34---

23--- 2

3--- 3

5---

Exercise 3.5

23---

12--- 1

2--- 1

5--- 1

2--- 1

2--- 1

2---

35--- 2

3--- 1

4--- 3

4---

13--- 1

3--- 1

2--- 4

7---



5 a Decreasing, above lineb Decreasing, on linec Decreasing, below lined Increasing, above linee Decreasing, on linef 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 = xe 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 = 0b i x + 4y + 11 = 0 ii 4x − y − 24 = 0c i 5x − y + 27 = 0 ii x + 5y − 31 = 0d i x + y + 17 = 0 ii x − y + 1 = 0e i 2x − 3y − 11 = 0 ii 3x + 2y − 10 = 0

9 a 4x + 5y − 20 = 0 b 2x − 3y + 6 = 0c x + y + 7 = 0 d x − 2y − 4 = 0e 3x + 5y − 15 = 0 f x − 10y − 4 = 0

10 a f (x) = 2x b f (3) = 6 c x = 1011 a

b x − 5 = 0, x − 3y + 7 = 0, 5x + 3y − 19 = 0c (4, 2) inside, (3, −1) outside

12 y = 0.0083x + 0.018

13 y = −0.401x + 21.0

1 a 13 b c 5d 17 e 3 f 13

23--- 1

3---

23---

58--- 7

8---

23--- 2

3---

14--- 3

4---

32---

y

x

(5, 4)(2, 3)

(5, −2)

0

20

10

Freq

uenc

y (k

Hz)

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 106 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 Pand (−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 = 018 (1, −8), (9, 10), (5, 0)

19 a Centre (4, 5), radius = 5b (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 = 1c k = −2, j = 3 d f = 3, g = −4e p = 5, q = −1 f f = −3, g = 6g h = 2, i = −3 h c = −1, d = −5

4 a m = 4, k = 3 b k = 5, j = 4c p = −2, q = −5 d a = −6, b = 2e x = −1, y = −3 f v = −2, w = 6g 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 = 6g b = −3, y = −6 h u = 10, p = 3

12--- 1

2--- 1

2---

12--- 1

2---

12--- 1

2--- 1

2---

12--- 1

2---

13---

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

12--- 1

2---

12---

14--- 2

3---


414 New QMaths 11B9780170136570

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 = 27 6 adults and 15 children8 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 pmc About 7:30 am d 24 hourse 45 ng/mL

2 a January b Julyc 12 months d 12 monthse About 4.3ºC f About 5.7ºCg Minimum temperatures

3 a Period = 20 ms, amplitude = 340 Vb 50 Hz

4 a About 2.3 m b About 0.3 mc About 1 m d About 12.3 h

5 a Yes b About 6.2 nmc About 0.6 ms

6 a Blue lineb Yes (approximately)c About 45 quolls and 24 foxesd Same for both—about 10 years

7

b Yes (approximately) c About 13ºCd About 11.5 months

1 a Period = 4 s, amplitude = 8 cmb −5.8 cmc 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

14--- 1

6---

Exercise 4.1

Surface water temperature

30

20

16 32Months

10

0

Tem

pera

ture

(°C

)

25

15

5

4 20 368 2412 28

Exercise 4.2

d

e About 52 cm f About 61 cm3 a Period = 20 ms, amplitude = 350 V

b −100 V c 200 Vd 0.002 s, 0.008 s, 0.022 s

4 a 4 times (if evenly spaced on cable)b 38 minutesc

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/minc It is the pulse rate. d 13 min 20 s

6 a 0.6 ms b 1.5 mNc 0.04 ms, 0.64 ms, 1.24 ms, 1.84 ms, 2.44 msd About 16 667 times

1 a

b

c

d

0

3020

Hei

ght (

m)

Distance (cm)50 100 150 200

50

250 300

10

40

Reflector height60

0

150

100

Hei

ght (

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

y = 5 cos 3π2

------x



e

f

g

h

i

2 a

b

c

d

e

f

y = 9 sin 3π2

------x

y = 9 cos π2---x

y = 6 cos π2---x

y = 8 sin x

y = 3 cos x

y = 4 sin 2πx5

---------

y = 7 cos 3πx8

---------

y = 9 sin πx7

------

y = 2 sin 4πx5

---------

y = 6 cos 7πx8

---------

y = 5 sin πx5

------

3 a

b

c

d

e

f

4 a

b

c

d

e

y = 8 sin

y = 8 sin

πx3

------

2πx5

---------

y = 6 cos

y = 6 cos

2πx5

---------

4πx5

---------

y = 5 sin

y = 5 sin

2πx7

---------

2πx5

---------

y = 9 cos

y = 9 cos

2πx3

---------

2πx5

---------

y = 3 sin

y = 3 sin

2πx9

---------

5πx9

---------

y = 7 cos

y = 7 cos

2πx3

---------

7πx3

---------

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

------------------------

y = 7 sin 2π x 4+( )7

-------------------------


416 New QMaths 11B9780170136570

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/minc 30 m/s d 6.25 m/s2/Ne A$1.25/$US1 or $A1/$US0.80

2 a 0.4 mL/L, 250 L b 80 mLc 144 L d 1429 seedse 1908 bricks

3 a 1.9 t/m3 b m = 1.9v c 2.85 td 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

------------------------

2πx5

---------

2π x 4+( )5

------------------------- π x 2–( )3

---------------------

2πx3

--------- πx4

------

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/gd About 13 cm, the unstretched length

7 a

b The rate of change of temperature gradually decreases.

c 3.2°/hd 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/hc 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 legc 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/s13 a 0.72 b 0.46 c 0.40 d 0.9714 a 8 b 11.91 c 1 d 17 e 3.01

15 a 20 m b 1 m/m c l = 0.6h d 33 m e 1.14 m

16 a No

0.5 1.0 2.51.5 2.0

2

8

46

10

r

D

0

20

80 160Mass (g)

10

0

Len

gth

(cm

)

25

15

5

20 10040 12060 140

Spring stretch

24

12:20

Time

20

0Tem

pera

ture

(°C

) 26

22

18

11:20 12:40 1:0012 1:20

Temperature in car

11:00am

11:40am am noon pm pm pm pm

23---

100

Plant growth

Water per day (mL)200

10Gro

wth

(cm

)

50

50 150 250 300

60

403020

0



17 a Yes, within experimental error

18 5 g19 a 400 km/h b 169 km/h c 4 h

d Average speed = 300 km/h, average velocity = 0

20 Teacher to check21 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 ratec Increasing at an increasing rated Increasing at a decreasing ratee Decreasing at an increasing ratef Decreasing at a constant rateg Constant

Answers in this section based on tangents may vary a little from the stated answer.

1 −2, −2, −22

a −3 b 5 c 93

a 1 b −3 c −94

a 3 b 0 c 3

200

Sliding weight

Mass in pan (g)400

10Dis

tanc

e (c

m)

50

100 300

60

403020

0

Exercise 5.2

3−3 x

161284

−4

f(x)

f(x) = 2x2 + x − 4

20

84

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

−4f(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/day8 110°C/min, 0°C/min, −100°C/min9 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

128

f(x) = (x + 4)(x − 3)

−14

Exercise 5.3

4

s

12

5

3

t1 20

3

4

s

12

5

3

t20

4 6

6

s

t

2

10

−1

−2

−3

1 2 3 4

2

s

0.51

2.5

1.5

t20

3

1 3 4

2 4 6

2

0

Dis

plac

emen

t (m

)

4

Moving object

Time (s)−4

6

−2 8


418 New QMaths 11B9780170136570

4 a b

c

5 a C b D c B d A6 a

b 0.2 mc

7 Bottom at 12.4 m/s, water at 1 m/s8

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

Dis

plac

emen

t (m

)

1.2

Billiard ball

Time (s)−0.4

1.6

01.0

0.2 0.4 0.6 0.8

4

2

Vel

ocity

(m

/s)

6

Billiard ball

Time (s)−2

8

01.0

18

4000

1000

2000

5000

3000

20

Time (min)

6000

7000

4 6 8 10 12 14 16

Hei

ght (

m)

Airbus takeoff

18

400

100

200

500

300

20

Time (min)4 6 8 10 12 14 16

Rae

of

clim

b(m

/min

)

Airbus takeoff

1 a

b

c

2 The gradient is positive for x � −0.12, zero forx = −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 31−1−2 4 5

f(x) = x3 − 4x2 − x + 4

x

Gradient

x1

Gradient function2

Gradient

x−3

Gradient function 6

Gradient

x−3

Gradient function 6

Gradient

x1.5

Gradient function5



8 a 33.01 b 32.1001 c 32.010 0019 a 9 b −14 c 3

d 9 e −6210 a −10 b −16 c 230

d 5 e −5811 a 29.000 001 b 101.000 004

c 312

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/sc −19.6 m/s d −29.4 m/se −39.2 m/s

Chapter 6

1 a −3.5, 0.7 b −1.4, 4.8c −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 + 8and gradient function

f(x)

x1 2 3 4

f(x)

Exercise 6.1

−1 1 32−2−3 54 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 21 x−5−6

y = 5 − 4x − x2

3 421 65 x−1−2

−6

y

y = x2 − 4x − 6

1 32−2 −1 4 x

7

f(x)

f(x) = 3x + 7 − 2x2

−2 −1−3−4 21 x

f(x)

−8f(x) = 2x2 + 3x − 8

6 842 10 x−2

−7

y

y = − 4x − 7x2

2-----

f(x) = 2x2 − 5x −10

f(x) = 8 − − 4xx2

2-----


420 New QMaths 11B9780170136570

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 3.6c 4.63 and 4.64 d −8.64 and −8.63

5 a −0.33, 3 b −1.93, 2.59c −1.13, 1.63 d −3.10, 2.10e −6.14, 1.14 f −2.32, 4.32

6 a −1.15, 5.65 b −1.81, 3.14c −9.32, 0.32 d −3.80, 15.80e −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, 34 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) = + 7x − 15x2

5-----

Exercise 6.2

12---

13---

12--- 1

3---

25--- 1

2---

12--- 1

2--- 6

7--- 8

9---

12--- 5

7---

34--- 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 $2017 Larger pipe 10 h, smaller pipe 15 h 18 10 km/h19 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 + 10c y = x2 + 4x + 3 d y = −2x2 + 7x + 4

3 B4 a D b F c A

d B e C f E5 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 )

49---

18--- 1

3---

13--- 2

3---

10

11

3013--- 1

5---

16---

13--- 1

2---

12---

2l 2

3------- 32l

3--------

20m 1–------------- 20

m------ 20

j------ 20

j 1+------------

Exercise 6.3

14--- 1

8---

34--- 1

8---

12--- 1

4---

12--- 1

4---

12--- 1

4---

12--- 1

4---



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, 6c 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

12--- 1

4---

12--- 1

4---

12---

12---

12--- 1

4---

y

x

−12

(2, −16)

−2 6

y = x2 − 4x − 12

12--- 1

2--- 1

2---

y

x

24(− , 24 )

−4 3

12--- 1

2---

y = −2x2 − 2x + 24

x

(−3, −4)

y = x2 + 6x + 5

5

−5 −1

y

14---

y

x

(−1 , 2 )

y = −x2 − 3x(−4, −4)

(2, −10)

−3

12--- 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

y9

−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

( , −4)12--

12--11

2--−

y

8 y = −3x2 − 10x + 8

x

(−1 , 16 )23--

23--−4

13--

(−5, −17)

(2, −24)

13---


422 New QMaths 11B9780170136570

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 − 6b y = 1.5x2 − 1.5x − 3c f(x) = −0.8x2 + 0.8x + 4.8

13 Teacher to check proof14 2515 40°C

16 Teacher to check proof. The area is greatest when x = 20 m.

17 8.7518 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

Solu

bilit

y (g

/L)

Temperature, T (°C)20

10

3010

20

7040 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 solutionsd (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 21d 4 e 0 f 2.5

2 a 25 b Not defined c 4d 10 e Not defined f −3

3 a −7 b 12 c −4d 6 e 10 f 5.25

4 a 2 b 1 c 3 d 0 e 0f 10 g −5 h −3 i 7

5 a 11 b 11 c −8 d −4 e 12f −1228 g 25 h 110 i 7

6 a 9 b 1 c 6x d 0e 11 f 20x + 3 g −5 − 2x h 3 − 8xi 21x2 j 4x3

7 a y′ = −6x2 b f ′(x) = −21x2 c f ′(x) = 18xd g′(x) = −12x e f ′(x) = 3x2 f y′ = 6xg f ′(x) = 36x2 h y′ = 1 i g′(x) = −4j g′(x) = 11 k y′ = 15x2 l f ′(x) = −66x

8 a 9 b 2x − 7 c 6x + 8d 15x2 e 44x3 f −6x + 9

Exercise 6.4

12--- 1

2---

3 29+2

-------------------- 29,⎝ ⎠⎛ ⎞ 3 29–

2------------------- − 29,⎝ ⎠

⎛ ⎞

15 105+4

-------------------------- 5 19 105+( )4

----------------------------------,⎝ ⎠⎛ ⎞

15 105–4

------------------------- 5 19 105–( )4

----------------------------------,⎝ ⎠⎛ ⎞

12---

Exercise 7.1



9 a 4 b −8 c 2x d x + 2 e −2x + 5 f 4x − 3

10 a 7 b 25 c 6x − 5d The answers are the same because the general

derivative is the function that gives the values of the derivative at any point.

11 a 8x, 7 and 0 b 8x + 7 c 8x + 7d The derivative of the sum is the same as the

sum of the derivatives.12–15 Teacher to check proofs16 f ′(3) = 13 and g′(3) = 11, so f (x) is steeper at

x = 3.17 x � 618 f ′(x) = 6x − 5 and g′(x) = 6x + 1, so they are

parallel and f ′(x) is never higher.

1 a 24x3 − 15x2 + 4x + 8 b 27x2 + 40x3 − 15x4

c 3x2 + 2x + 1 d 36x8 − 80x9

e 180x11 + 180x14 − 180x4

f −7x6 + 63x8 + 10x9

g 40x7 + 108x8 − 60x9 + 12x11

h −3 + 2x + 24x2 − 4x3

i 20x3 − 17 + 48x2

j 175x6 − 48x5 + 70x4 + 2

2 a 12x3 − 12x b 12u3 − 12uc 30m5 + 40m4 − 24d 9 + 30y4 − 12y2 + 14y6

e 6z − 27z2 + 38 f 35t6 + 36t3 − 16g 27r2 − 13 − 12r h 4a − 24a3

i 8 − 12p3 − 42p6 − 45p8

j 8t − 6t2 + 44t3

3 a −5x−2 b c 12x−5 d

e −70x−11 f g h

i −144d−10 j

4 a b c 3.5

d 6 e −4 f

g −31 z or −31 z2

h i 8 d j 7

5 a 45(5a − 6)2 b −20(5x + 4)

c 49(y − 2)6 d −60(5q + 8)3

e −105(7 − 5a)2 f 360g2(5g3 + 3)5

Exercise 7.2

−8x2------ −6

t3------

21u8------ −20

y21--------- −60

v16---------

72z10-------

1

2 x---------- 1

3 m23--------------- x

12---–

23--- t

13--- 2

3--- p

53---– −1

2x x-------------

12--- z3 1

2--- z

−3

2 u34-------------- 2

3--- d6 1

2--- m

g 12(8x − 6)(4x2 − 6x + 4) or 48(4x − 3)(2x2 − 3x + 2)

h −64(3m2 − 4m)(m3 − 2m2 + 4)7 or −64m(3m − 4)(m3 − 2m2 + 4)7

i 18(9 − 3v)5

j 24(4g − 3g2)(2g2 − g3 + 3)5 or24g(4 − 3g)(2g2 − g3 + 3)5

6 a −5(x + 4)−6 b −12(3p + 5)−5

c 48(5 − 4x)−7 d −63(m + 8)−10

e −12p(3p2 − 4)−3 f

g h

i j

7 a 5(20x3 − 24x2)(5x4 − 8x3 + 7)4 or20x2(5x − 6)(5x4 − 8x3 + 7)4

b 7(21v2 + 4v + 1)(7v3 + 2v2 + v − 4)6

c d

e f

8 a (5z4 − 3z2 + 1)(9 + 3z2 − z4) + (z5 − z3 + z − 1)(6z − 4z3)

b 3(p2 − 2p − 4) + (3p + 4)(2p − 2) or3(p2 − 2p − 4) + 2(3p + 4)(p − 1)

c (15v2 − 6v + 1)(7v4 − 2v3 − v2 + 8v + 6)+ (5v3 − 3v2 + v − 9)(28v3 − 6v2 − 2v + 8)

d (6t − 2)(t3 + 4t2 + t − 6)(5t4 + t3 − 7t + 8) + (3t2 − 2t + 5)(3t2 + 8t + 1)(5t4 + t3 − 7t + 8) + (3t2 − 2t + 5)(t3 + 4t2 + t − 6)(20t3 + 3t2 − 7)

e (20b4 + 6b − 6)(b4 + 3b2 − 6b3 + 21) + (4b5 + 3b2 − 6b + 2)(4b3 + 6b − 18b2)or take common factor 2 out of last brackets

f (3 − 16u − 18u2)(7u3 − 4u2 + 9u − 6) + (3u − 8u2 − 6u3 + 8)(21u2 − 8u + 9)

9 a −6(v + 7)4(v − 6)3(9v − 2)b 3(6 − 5r)2(2r + 9)10(140r + 3)c 30(8 − 6h)6(−4 − 3h)5(−4 − 39h) or

1920(4 − 3h)6(4 + 3h)5(4 + 39h)d 20t + 1e −(3p + 10)6(8p + 1)7(360p + 661)f (4z + 15)5(28z − 57)

10 a b −

c − d −

e f

−3x 5+( )4

--------------------

−153 p 4–( )6

------------------------ 47 z–( )5

-------------------

−63m 4–( )3

------------------------- 163 7 4x–( )9---------------------------

6t 1+

2 3t2 t 5–+---------------------------------- 12h 3–

2 6h2 3h– 1+---------------------------------------

4

3 4v 5+( )23-------------------------------- 2y 2–

4 y( 2 2y– 3 )3+4---------------------------------------------

603 5q–( )2

----------------------- 262m 5–( )2

-------------------------

352x 5–( )2

----------------------- 10t 5–( )2

-------------------

1234 5t 8+( )2-------------------------- 21

5 2u 9–( )2---------------------------


424 New QMaths 11B9780170136570

11 a (15x4 − 6x)(2x2 + 5x + 2)+ (3x5 − 3x2 + 8)(4x + 5)− (8x3 + 15x2)(x5 − 6) − 5x4(2x4 + 5x3)

b

c 42(3m + 10)6 − 60(3m + 10)3 − 12(10 + 3m)−3

d − − +

e − + +

f 24(3w + 5)3(3w − 1)5 + 30(3w + 5)4(3w − 1)4 − 8(w − 6)7(1 − 3w)5 + 15(w − 6)8(1 − 3w)4

g

1 a −7 b y = −7x − 12c 81.9º d 1e y = x − 4 f 45º

2 a y = −23x − 37 b 1c y = x + 2 d 45º

3 a 45º b 73.06 c 25.02 d 87.5º4 a 4 b 76º c y = −48x − 72

d 88.8º e 2t + 4 − 12t2

f (− , −5 ) and ( , −2 )

g Not defined h y = −98x + 203

i −6 j y = x − 3

5 a y = x − 3 b y = x +

c y = − x − 23 d y = − x − 63

6 a 13 b 64 c 189 d −12 e 5337 a 22 b 102 c −10 d −18 e 1588 a −153

b i −15 ii −363 iii −135c The average rate is not the average of the

instantaneous rates.

9 a x � 4 b c y = x +

d e f (8, 2)

10 a −4 b −4 c Parallel d −25e −25 f Parallel g −1 h −1i Parallel j x = −2 ± k

11 a

b 9 − 10t

c6 63 20c–( )9 4c–( )3

--------------------------------

62 f 5–( )2

------------------------ 306 f 10+( )2

--------------------------- 20

5 f 11–( )2---------------------------

3t 6–( )4

------------------- 162t 7+( )3

---------------------- 2 67 18t–( )7 2t 9+( )5-----------------------------

2– 0 p5 30 p4 10 p3– 14 p2 102 p 56–+ + +5 p 1–( )7

-----------------------------------------------------------------------------------------------------------

Exercise 7.3

12--- 1

4--- 2

3--- 2

27------

16--- 1

6---

13--- 2

3--- 1

4--- 3

4---

129------ 2

29------ 1

52------ 3

52------

16--- 1

6--- 5

6---

1

2 a 4–-------------------- a 4–

a----------------

21 3 t (s)

h (m)

0

−10

10

−20h = 2 + 9t − 5t2

Pebble thrown into well

c

d −6 m/se The average speed is the average of the

instantaneous speeds.

12 a 6.786 m3 b 4.524 m3

13 a V(t) = 38 400 000 − 464 000t + 1320t2 − t3

b V ′(t) = −464 000 + 2640t − 3t2

c 431 469 m3 d 426 351 m3

14 a V = b πd2 c 1.61 cm3

15 a $46 640 b $777.33 c $432d i The average cost ($777.33) is higher than

the marginal cost ($432).ii The average cost ($620) is higher than the

marginal cost ($400).iii The average cost ($591.38) is lower than

the marginal cost ($628).

1 f1 ↔ g8, f2 ↔ g6, f3 ↔ g10, f4 ↔ g3, f5 ↔ g5,f6 ↔ g4, f7 ↔ g1, f8 ↔ g9, f9 ↔ g2, f10 ↔ g7

2–3 Teacher to check proofs4 a–f Decreasing g Stationary h–j Increasing5 a–c Decreasing d–e Increasing f–j Decreasing6 a Increasing b Stationary c Decreasing

d Decreasing e Decreasing f Stationaryg Increasing h Increasing

7 a Decreasing b Stationary c Stationaryd Decreasing e Decreasing f Decreasingg Stationary h Increasing

8 a (2, −5) b (2, 10)

c (− , 16 ), (2, −2) d (1, −7), (−2, 20)

e None f (1, −5), (−3, 27)9 a Maximum at (2, 1)

b Maximum at (−1, 5)c Maximum at (−2, 16), minimum at (2, −16)d Minimum at (−1, −2), maximum at (1, 2)e Maximum at (0, 7), minimum at (4, −57)

f Minimum at ( , − )

g Maximum at (− , 8 ), minimum at (2, −10)

h Maximum at (− , 1 + 4 ), minimum at

( , 1 − 4 )

i Minimum at (1, 2)

j Minimum at (0, 0), maximum at (5 , 75 )

0

−10

10

−20

21 3 t (s)

(m/s)

= 9 − 10tdhdt------

Pebble thrown into welldhdt------

πd3

3---------

Exercise 7.4

43--- 14

27------

18---

π16------

43--- 14

27------

2 2

2 2

13--- 23

27------



10 a y = 4x2 − 9x + 5

b g(x) = x2 − 7x + 12

c h(x) = 2x3 − 6x

d y = 2x3 − 12x2

e g(x) = x3 − 12x

f f (x) = x3 − 2x2 + x − 2

g h(x) = 3x3 − x2 − 7x − 3

h y = (x + 5)(x + 2)(x − 3)

(1 , − )116----1

8--

y

x

51 11

4--

(3 , − )14--1

2--

x

12

3 4

g(x)

(1, −4)

3

h(x)

x3–

(−1, 4)

(4, −64)

y

x6

(2, −16)

2 3

g(f )

x2 3–

(−2, 16)

(1, −2)

2

f (x)

x

( , −1 )13-- 23

27----

−2

79--(− , )104

243------

−3

h(x)

x1.87−1

(1, −8)

−0.54

−30

y

x

23--(−3 , 14 )

3−5

2227----

−2

(1, −36)

Chapter 8

1 a 120° b c 60°

d −60° e f

2 a b c

d e f

g h i

3 a 1.5708 b 0.9774 c 2.4435d 3.2289 e 4.8171 f 5.5851g 4.6740 h 9.8859 (or 3.6027)i 13.3678 (or 7.0846 or 0.8015)

4 a 120° b 150° c 315°d 98.2° e 396° (or 36°) f 292.5°g 337.5° h 1020° (or 660° or 300°)i 385.7° (or 25.7°)

5 a 57.3° b 126.1° c 45.8°d 189.1° e 118.0° f 312.3°g 4.0° h 318.0°i 452.1° (or 92.1°)

6 a, d, e and f are coterminal.7

1 a − b − c −1 d

e f

2 a b c − d −

e − f −

3 a 1 b −1 c 0 d 0e 0 f Not defined

4 a − b − c 1 d −1

e − f −

Exercise 8.1

π6---–

π4--- 7π

4------

π3--- π

4--- π

2---

3π4

------ 2π3

------ 5π9

------

π3--- or 7π

3------⎝ ⎠

⎛ ⎞ 2π5

------ 8π5

------

y

x

a

b

c

de

f

gh

i

π3 π

4π6

5π34π

3

5π4

7π6

2π3

3π4

Exercise 8.2

12---

32

------- 12---

1

3------- 3

2-------

1

2------- 1

2------- 3

2------- 1

3-------

32

------- 32

-------

32

------- 12---

3 1

2-------


426 New QMaths 11B9780170136570

5 a Quadrants 1 and 4 b Quadrants 1 and 3c Quadrants 3 and 4 d Quadrants 2 and 4e Quadrants 1 and 2 f Quadrants 2 and 3

6 a 1 b −1 c [−1, 1]7 a 1 b −1 c [−1, 1]8 a No b Real numbers

c Yes, , − , , , etc.

d Real numbers except odd multiples of

9 a sin θ = , cos θ = and tan θ =

b sin θ = , cos θ = − and tan θ = −

c sin θ = − , cos θ = and tan θ = −1

d sin θ = , cos θ = − and tan θ = −

e sin θ = − , cos θ = and tan θ = −

f sin θ = − , cos θ = − and tan θ = 1

10 a cos θ = and tan θ =

b sin θ = and tan θ =

c sin θ = and tan θ = −

d cos θ = − and tan θ =

e cos θ = and tan θ = −1

f sin θ = and tan θ = 1

g sin θ = − and tan θ =

h cos θ = − and tan θ = −

11 a sin β = , cos β =

b sin β = , cos β = −

c sin β = − , cos β =

d sin β = − , cos β = −

e sin β = − , cos β =

f sin β = , cos β = −

12 a − b − c − d − e −1

f g − h i −

13 a − b − c 1 d − e −

f − g 0 h i − j

k − l −

3π2

------– π2--- π

2--- 3π

2------

π2---

45--- 3

5--- 4

3---

513------ 12

13------ 5

12------

1

2------- 1

2-------

12---

32

------- 1

3-------

32

------- 12--- 3

1

2------- 1

2-------

35--- 4

3---

1213------ 12

5------

2425------ 24

7------

45--- 3

4---

1

2-------

1

2-------

725------ 7

24------

513------ 12

5------

45--- 3

5---

1213------ 5

13------

2425------ 7

25------

1

2------- 1

2-------

35--- 4

5---

513------ 12

13------

1

2------- 3

2------- 1

3------- 1

2-------

12--- 3 1

2---

32

-------

12---

32

------- 12---

1

3-------

1

2------- 1

2--- 3 1

2---

32

------- 32

-------

14 a 0 b c 1 d 0 e −

f 0 g − h − i − j −1

k − l

15 a b − c 0 d 0 e 0

f g 1 h −1 i − j −

k − l −

16 a 4 tan x b 0 c −6 sin 5yd cos x e −2 sin y f 4 cos gg 4 tan q h −sin m i sin c

1

2

3

4 a For both, period = 2π and amplitude = 1

b They appear the same, but one is shifted by along the x-axis from the other.

5 a

b Period = π, amplitude is infinite.

1

2------- 3

2-------

1

2------- 1

2---

32

-------

1

3------- 3

2-------

12---

32

-------

1

3------- 1

2---

32

-------

1

3------- 1

2---

Exercise 8.3

y1

2π θ−2π

−1

π

y = sin θ

−π

y1

2π θ−2π

−1

π

y = cos θ

−π

y = tan θ

y2

−2

θ−π π

1

−1

π2---−π

2---

π2---

y = tan θ



1 a y = 2 sin x

b y = sin x

c y = 3 cos x

d y = 2 sin x + 1

2 a y = sin 2x

b y = sin x

c y = cos 3x

Exercise 8.4

x

y

π 3π2

------π2---

0

2

−2

2π

12---

x

y

3π2

------π2---

0

− 12

12

π 2π

x

y3

−3

π 2π0

3π2

------π2---

x

y

ππ2---

0

3

−1 2π

1

3π2

------

−π π–2

------ x

y

π2---

1

−1

π

12---

−π π–2

------ x

y

π2---

1

−1

π

−π π–2

------ x

y

π2---

1

−1

π

d y = sin 2x + 3

3 a y = 3 sin 4x

b y = cos 3x

c y = 5 sin 2x

d y = sin 8x + 4

4 a y = sin

b y = sin

−π π–2

------ x

y

π2---

4

π

32

0

x

y

3π8

------π8---

0

3

−3

π4--- π

2---

12---

x

y

π6---

0π3--- π

2---

12

− 12

2π3

------

x

y

3π4

------π4---

0

5

−5

π2--- π

13---

x

y

3π16------

4

4

3

π8---π

16------ π

4---

0

23

13

x π2---–⎝ ⎠

⎛ ⎞y1

−1

0x3π

2------π

2--- 2ππ

x π2---+⎝ ⎠

⎛ ⎞

x

y1

−1

π 2π3π2

------π2---

0


428 New QMaths 11B9780170136570

c y = cos

d y = cos

e y = 2 sin

f y = 2 sin

g y = 3 cos

h y = 3 cos

5 a y = 4 sin 2 + 2

x π6---–⎝ ⎠

⎛ ⎞

x3π4

------π2--- 2ππ

π6---

y1

−1

0

x π6---+⎝ ⎠

⎛ ⎞

y1

−1

0x3π

4------π

2--- 2ππ

11π6

---------

x π3---–⎝ ⎠

⎛ ⎞y2

1

−2

0

−1 x3π2

------π2--- 2ππ

π3---

x π3---+⎝ ⎠

⎛ ⎞

y2

−2

0x3π

2------π

2--- 2ππ

5π3

------

x π4---–⎝ ⎠

⎛ ⎞y3

−3

0x3π

2------π

2--- 2ππ

π4---

x π4---+⎝ ⎠

⎛ ⎞y3

−3

0x3π

2------π

2--- 2ππ

7π4

------

x π3---–⎝ ⎠

⎛ ⎞y6

2

−2

0x3π

4------π

4--- ππ

2---

b y = 2 cos 3 − 3

c y = 5 sin 4 − 2

d y = 3 cos 2 + 4

e y = 4 − 2 cos 4

f y = 3 − 3 sin 2

6 a y = 2 sin + 3

b y = 3 cos (4x + π) − 2

x π6---+⎝ ⎠

⎛ ⎞y0

−1

−5

−3

x2π3

------π6--- π

3--- π

2---

x π8---+⎝ ⎠

⎛ ⎞y3

0

−7

−2 x3π8

------π8--- π

4--- π

2---

x π4---–⎝ ⎠

⎛ ⎞y7

4

0x3π

4------π

4--- ππ

2---

1

x π3---+⎝ ⎠

⎛ ⎞y6

4

0x3π

8------π

8--- π

4---

2

π2---

x π4---–⎝ ⎠

⎛ ⎞y6

3

0x3π

4------π

4--- ππ

2---

2x π3---–⎝ ⎠

⎛ ⎞y5

3

0x3π

4------π

4--- ππ

2---

1

y10

−5

−2x3π

8------π

8--- π

4--- π

2---



c y = 4 − 3 sin

d y = 5 cos − 3

e y = 3 − 5 cos

f y = 4 − 3 sin

7 a

b 1:30 am d 6 hoursc 5:30 am, 9:30 am, 5:30 pm, 9:30 pm

8 a

b Jan, Feb, Oct, Nov and Dec

6x 3π2

------+⎝ ⎠⎛ ⎞

y7

4

01

xπ4---π

12------ π

6--- π

3---

2x 3π4

------–⎝ ⎠⎛ ⎞

y20

−8

−3x3π

4------π

4--- π

2--- π

3x π3---–⎝ ⎠

⎛ ⎞y8

0−2

3

xπ2---π

6--- π

3--- 2π

3------

4x π4---+⎝ ⎠

⎛ ⎞y7

4

01

x3π8

------π8--- π

4--- π

2---

d(t) = 5 + 2 cos ( )t 2

4

67

0

d(t) (m)

4 8 12 16 20 24 t (h)2 6 10 14 18 22

5

3

1

π6---

Tide on 18 March

S = 54.8 + 32.5 cos 20

60

100

0

S (h

undr

eds)

2 4 6 8 10 12 t (months)1 3 5 7 9 11

80

40

πt6-----

Sales

9 a

b 31°C, 19°C c 3 pm, 1 amd i 11:05 am, 6:55 pm, …

ii 11:08 pm, 2:52 am, …e Because the period was not 24 hours

10 T = 23.875 − 5.575 cos

11 T = −14 cos (t − 1) − 9

12 About 0.352 s, 0.648 s and 1.352 s

13 d = 4 + 0.9 sin . The boat with a draught

of 3.5 m can enter for 8.479 hours, then cannot for 3.854 hours. The boat with a draught of 4.5 m can enter for 3.854 hours, then cannot for 8.479 hours.

14 a s b y = 12 sin 400πt

Chapter 9

1 a b c d

e f g h

i j k l

21

25

2931

0

T (°C)

4 8 12 16 t (h)2 6 10 14 18 20

27

23

19

Temperature investigation

T = 25 + 6 sin 0.1πt

(10 am)

πt6-----

02 4 6 8 10 12

t (months)

20

30T (°C)

25

15

Mean monthly temperaturein Calcutta

π6---

−16

2 4 6 8 10 12t (months)

T (°C)8

0

−8

−24

Average temperature in Upernivik

6πt37--------

1200---------

Exercise 9.1

1a2----- 1

43----- 1

g7----- 1

54-----

1y6----- 1

c4d8----------- 4

h4----- 1

256h4---------------

2k--- 7

p4------ 8

q6----- v2

4-----


430 New QMaths 11B9780170136570

2 a y−4 b 4−3 c h−9

d k−6t4 e p−6y2 f 4i−6j8

g 3m−5y−2 h 2m−1z−4 i x8

j 2x4y4 k 3bk−3 l 4r−9w6

3 a b

c d or

e or f or

g or h or

i or

j or or or

k or or

l or or or or

4 a b c d

e f g h

i j k l

5 a b c 3p4 d

e f g h 5k18m14

i 16v11 j k l c3d19

6 a b c

d e f

g h

7 a 8 b 1 c 2 d 1e 6 f 9 g 1 h 9i 343 j 216 k 16 l 64

m n o 2 p 1

q r s 125 t

u v 4 w 100 000 x

8 a 24 b 5k + 1 c 3 × 23

d 2m32n + 3m e 32i + 2 × 73 f 23

9 a 192 b 196 608 c 24d 768 e 0.1875

10 a 51.7706 b 9.7865 c 7663.9046d 92.7454 e 0.8043

11 a 7.9222 b 4.1354 c 8.5928d 2.5397 e 21.0055

6 z4

2163 y35 y5( )3

493 49( )3 v73 v3( )7

873 83( )71654 164( )5

m5 m( )5

u−14 u4( )−1 1u---4

1

u4-------

19---⎝ ⎠

⎛ ⎞ −1 19---⎝ ⎠

⎛ ⎞ −19

p−35 p5( )−3 1p3------5

1

p35----------- 1

p5( )3----------------

k12---

314---

512---

415---

w13---

c32---

6443---

e34---

2849---

q65---

852---

1676---

1r2----- 12

y9------ 2

j6-----

16m13------------ 1

4k14----------- 15g

h2---------

4q17

p13----------- 4e12 f

7

3----------------- 1

2---

a41

2---

b31

2--- 6

n12---

-----

12g61

3---

12w32

3---p

212---

10m31

2---n

213---

21u3

s31

2---

------------ 24 p11

2---

q512------

---------------

14--- 16

9------

6427------ 1

36------ 2

5---

278

------ 110 000----------------

12 a The y-intercept is 1, the graph is increasing with increasing slope, and is always positive.

b The y-intercept is 5, the graph is increasing with increasing slope, and is always positive.

c The y-intercept is 4, the graph is decreasing with decreasing slope, and is always positive.

d The y-intercept is 10, the graph is increasing with increasing slope, and is always positive.

e The y-intercept is 40, the graph is decreasing with decreasing slope, and is always positive.

f The y-intercept is 1, the graph is increasing with increasing slope, and is always positive.

x

y

−3 −2 −1 0 1 2 3 4

20

15

10

5

y = 2x

x

y

−3 −2 −1 0 1 2 3

40

30

20

10

−4

y = 5 × 2x

x

y

−2 −1 0 1 2 3 4

20

15

10y = 4 × 0.6x

5

10

x

y

−3 −2 −1 0 1 2 3

60

403020

50

y = 10 × 1.8x

200

x

y

−2 −1 0 1 2 3 4

1000

600400

800y = 40 × 0.2x

x

y

−2 −1 0 1 2 3

1000

600400

800

200

y = 10x



13 a The y-intercept is 8.5, the graph is increasing with increasing slope, and is always positive.

b The y-intercept is 20, the graph is decreasing with decreasing slope, and is always positive.

c The y-intercept is 16, the graph is increasing with increasing slope, and is always positive.

d The y-intercept is 16, the graph is increasing with increasing slope, and is always positive.

e The y-intercept is 16, the graph is decreasing with decreasing slope, and is always positive.

f The y-intercept is 8, the graph is increasing with increasing slope, and is always positive.

14 a A = 3 × 1.05d

b i 3.4729 m2 ii 4.8867 m2

iii 7.9599 m2 c 104.9 days

15 a V = 3000 × 1.08y

b i $3779.14 ii $6476.77iii $20 545.43

c 9.01 years

16 a V = 24 900 × 0.85y

b i $15 291.71 ii $11 048.26iii $4902.17

c 19.8 years

y = 8.5 × 2.5x

y = 20 × 0.9x

y = 16 × 1.2x

y = 16 × 2x

y = 16 × 0.1x

y = 8 × 10x

17 a T = 500 − 480 × 0.9m

b i 216.6°C ii 332.6°Ciii 465.5°C

c 14.9 min

18 a p = 3 × 2m b 768

19 a q = 24 × 1.8x b 117.3402

20 y = 50 × 1.4x, 318.1810

1 a x = 1 b x = 4 c x = 2

d x = 1 e x = 3 f x ≈ 2 or 4g x ≈ 2 or 3 h x ≈ 2 or 3 i x ≈ −1 or 2j x ≈ 0, 1 or 2 k x = 1

2 a x ≈ 2.3 b x ≈ 0.8 c x ≈ 1.79

3

a x ≈ 2.73 b x ≈ 2.10 c x ≈ 4.46d x ≈ 0.93 e x ≈ −0.63 f x ≈ 0.37

4

a x ≈ 0.90 b x ≈ 1.90c x ≈ 2.08 d x ≈ 0.65

5

a x ≈ 1.32 b x ≈ 0.32c x ≈ −0.42 d x ≈ 0.71

6

a x ≈ 0.39 b x ≈ 2.39c x ≈ 0.69 d x ≈ 1.52

Exercise 9.2

78---

x

y

−1 0 1 2 3

25

15105

20y = 3x

x

y

−1 0 1 2 3

800600

400

200

y = 10x

1000

x

y

−2 −1 0 1 2

3

2 y = 0.5 x

4

1

200

150

100

50

01 2 x

y

3

y = 6 x


432 New QMaths 11B9780170136570

7 x ≈ 1.926 8 x ≈ 2.0879 x ≈ 1.473 10 x ≈ 0.733

11 x ≈ 0.905

1 a 1 b 3 c 0 d −1 e

f −3 g h 1 i 2

2 a 1.7634, 101.7634 ≈ 58b 2.2041, 102.2041 ≈ 160c 1.2041, 101.2041 ≈ 16d −1.3010, 10−1.3010 ≈ 0.05e 1.1761, 101.1761 ≈ 15f 2.9494, 102.9494 ≈ 890g 1.8261, 101.8261 ≈ 67h 1.7993, 101.7993 ≈ 63i 1.7959, 101.7959 ≈ 62.5

b

c

d At the same scale, the graphs are mirror images in the line y = x.

3 a x 5 10 15 20 25

log x 0.6990 1 1.1761 1.3010 1.3979

x 30 35 40 45 50

log x 1.4771 1.5441 1.6021 1.6532 1.6990

x 55 60 65 70 75

log x 1.7404 1.7782 1.8129 1.8451 1.8751

x 80 85 90 95 100

log x 1.9031 1.9294 1.9542 1.9777 2

Exercise 9.312---

13--- 1

2--- 1

2---

2

1.5

1

0.5

040 602010 30 50 70 80 90 100 x

y

y = log x

100

40

20

00.5 1 1.5 2 x

y

60

80

y = 10 x

b

c


b

4 a x 0.1 0.2 0.3 0.4

log x −1 −0.6990 −0.5229 −0.3979

x 0.5 0.6 0.7 0.8

log x −0.3010 −0.2218 −0.1549 −0.0969

x 0.9 1.0 1.1 1.2

log x −0.0458 0 0.0414 0.0792

x 1.3 1.4 1.5 1.6

log x 0.1139 0.1461 0.1761 0.2041

x 1.7 1.8 1.9 2

log x 0.2304 0.2553 0.2788 0.3010

5 a x 0.001 0.005 0.01 0.05 0.1

log x −3 −2.301 −2 −1.301 −1

x 0.5 1 5 10 50

log x −0.301 0 0.699 1 1.699

x 100 500 1000

log x 2 2.699 3

0.60.40.2

x−0.2−0.4−0.6−0.8

1 1.4 1.60.2

y

0.4 0.8 1.2 1.8 2

−1

0

y = log x

x

y

−0.4 0 0.4

2

1

−0.8

y = 10x

−1

x800200 400 600 1000

2

1

−1

−2

−3

y

0

3y = log x



c


6 a log 3.5 ≈ 0.5441, log 350 ≈ 2.5441b log 350 = log 3.5 + log 100

7 a log 18 ≈ 1.2553, log 3 ≈ 0.4771, log 6 ≈ 0.7782

b log 18 = log 3 + log 6

8

The graphs are mirror images in the line y = x.9 a Domain = real numbers,

range = positive numbersb Domain = positive numbers,

range = real numbers10–14 Teacher to check proofs15 a 1.7559 b 3.7559 c −1.244116 a 1.4533 b 3.4533 c −0.546717 a 4.9883 b −0.0117 c −3.011718 a 2.0899 b 4.0899 c −0.910119 a x ≈ 1.5943 b x ≈ 0.5943

c x ≈ 17.6730 d x ≈ 9.9031e x ≈ 3.7310 f x ≈ 2.2295g x ≈ 1.7055 h x ≈ 2.7055i x ≈ −1.0766 j x ≈ 1.5349

1 a P = 1005 × 0.89h b 97.7 hPac 358.4 hPa d 1271 hPae 19.76 km f 13.85 km

2 a A = 100 × 0.8n b 32.77%c 3.106 days d 13.43 days

3 a P = 30 000 − 25 000 × 0.7y

b 21 425 crabs c 25 798 crabsd 4.51 years e 9.02 years

4 a i $20 110 ii $20 220.61 ii i $20 331.82 iv V = 20 000 × 1.0055t

b $38 625.93 c $41 345.95d 4 years 4 months e About $26 600f $18 217.16

x

y

1 2 3

1000

800

600

400

200

−3 −2 −1 0

y = 10x

y = 10 x

y = log x

Exercise 9.4

5 3.1 cm; L = 100 × ( )n

6 a 250 people b 200 people c 27 peopled 29 years e 52 people

7 a About 65°C b 57 × 0.96 t

c T = 15 + 57 × 0.96t d About 53°Ce 20.2 min

8 9% p.a. 9 2 days10 8 days 11 104 h 52 min

Chapter 10

1 a 48 trials b 22 c 0.458d 0.104 e 9 students

2 a 0.075 b 0.025 c 0.2753 a 0.2037 b 0.1697 c 0.23104 a 100 trials

c 0.325 a 0.06, 0.337, 0.26, 0.287, 0.02, 0.037

b 0.287 c 7 people d 8 people6 a 0.0135 b 0.0016 c 0.0025

d 0.000 02 e 2 children7 a 52 elements b Yes

c {Q♥, Q♦, Q♣, Q♠}

d ≈ 0.077 e = 0.25 f ≈ 0.019

8

9 a 0.078 b Text page c 0.279d 0 e 0.273

10 a {red, green, pink, orange, blue}

b = 0.2 c = 0.4

11 P(A) = , P(B) = , P(C) =

b Number in basket f

0 7

1 14

2 32

3 23

4 11

5 5

6 6

7 1

8 0

9 1

Total 100

12---

Exercise 10.1

113------ 1

4--- 1

52------

14---

15--- 2

5---

47--- 2

7--- 1

7---


434 New QMaths 11B9780170136570

12 Straight up 0.027, split 0.054, street 0.081, corner 0.108, six line 0.162, dozen 0.324, column 0.324, even chances 0.486

1 a ≈ 0.064 b

2 3

4 a b

5 P(D) =

6 a b

7 a

Exercise 10.2

8125--------- 44

125---------

13--- 8

663---------

155------ 41

55------

13--- 2

5--- 2

15------

118------1

6---1

3---

13--- 3

8---

18---

× =

× =

=×

13---

13---

13---

25---

16---

56---

38---

58---

1

2

3

35---

D

N

D

N

D

N

113360---------

311

−1311

−11

−1311

−11

−1

2

0

2

0

0

−22

0

0

−2

3

1

1

−1

1

−1

2

0

0

−2

1

−1

12---

BF

E

B

D

F

C

D

E

F

C

F

C

B

C

E

BDE

C

E

D

DBA

b 0

8

9 a b

10

11 a i ii

b

12 13

14 (includes 00)

1 a n = 3, p = , q = b 0.016

c 0.422 d 0.1562 a 0.296 b 0.988

c 0.593

3 a = 0.3125 b = 0.343 75

4 a 0.555 b 4 times

5 = 0.6875

6 a 1.708 × 10−6 ≈ 0 b 0.6797 a P(0) ≈ 0.735, P(1) ≈ 0.232, P(2) ≈ 0.031,

P(3) ≈ 0.002, P(4) ≈ 0.0001, P(5) ≈ P(6) ≈ 0b P(0) ≈ 0.430, P(1) ≈ 0.383, P(2) ≈ 0.149,

P(3) ≈ 0.033, P(4) ≈ 0.005, P(5) ≈ 0.0004, P(6) ≈ P(7) ≈ P(8) ≈ 0

c P(0) = P(7) ≈ 0.008, P(l) = P(6) ≈ 0.055, P(2) = P(5) ≈ 0.164, P(3) = P(4) ≈ 0.273

8 a 0.663 b 0.051 c 0.3379 a 0.012 b 0.16 c 0.841

10 a 0.000 061 b 0.133c 0.173 d 0.244

11 0.89512 a n = 3, p = 0.2, q = 0.8 b 0.008

c 0.512 d 0.48813 0.262 14 0.63315 0.000516 a 0.147 b 0.832

17

18 a 0.276 b 0.003 c 0.180

19 0.678

14---

1140------ 3

8---

38---

38--- 5

8---

A B

x

z

y

512------ 5

6---

140------

Exercise 10.3

14--- 3

4---

516------ 11

32------

1116------

577117 649--------------------



Chapter 11

1 Answers may differ by a constant.

a 2x3 + 2x2 + x, 2x3 + 2x2 + x + 5b 4t2 − 7t + 3, 4t2 − 7t − 3

c z4 + 1 z2 − z5 − 1, z4 + 1 z2 − z5 + 8

d 4x + 2, 4x + 5

e − − y + 4, − − y − 4

f v4 − v5 + v3 − 2v + 1,

v4 − v5 + v3 − 2v + 8

g 4.9t2 − 0.1t3 − 6t + 3, 4.9t2 − 0.1t3 − 6t − 3h 1.5m3 + 0.75m2 + 0.89m,

1.5m3 + 0.75m2 + 0.89m + 1

i 0.025d5 + 3.5d2 − 0.38d + 2, 0.025d5 + 3.5d2 − 0.38d + 25

j k3 − k4 + k2 − 5k − 6,

k3 − k4 + k2 − 5k − 5000

2 a i4 − 2i3 − i2 − 3i + c

b x4 − 2x5 − 2x3 + 12x + c

c u2 − u4 − u3 + 8u + c

d m3 − 1.575m4 − 0.35m2 + 4m + c

e 0.035t2 + t3 + c

f x3 − 24.5x2 + 122.5x + c

3 a x3 − 5x + c b 1.5t2 + 2t + c

c 7u − u4 + u3 + c d h5 + c

e p3 + 4p2 + 16p + c

f z5 − z4 + z3 − z2 + z + c

4 a y = x4 + 2x3 − 3x2 + 7x + c

b y = 3x4 − 4x3 + 6x2 − 12x + c

c y = 1.5t2 + 5t + c

d y = z2 − z4 + 8z + c

e y = 0.25x2 − 0.01x4 + x3 − 0.52x + c

f y = 0.35t2 − t3 + 0.6t + c

5 y = 2x2 − 5x − 1, y(5) = 24

6 7.6 7 −3 8 −41.25

9 m(r) = r + 6 − or

m(r) = or

m(r) ≈ 2.67r − 7.54

10 y = 2(x + 4) − 54

Exercise 11.1

12--- 1

2---

y3

3----- y2

2----- y3

3----- y2

2-----

34--- 1

5--- 4

3---

34--- 1

5--- 4

3---

53--- 1

2---

53--- 1

2---

14---

54--- 4

3---

0.26

0.3

1.6313---

14--- 1

3--- 1

5---

13---

15--- 1

4--- 1

3--- 1

2---

12--- 1

4---

0.23

0.13

83--- r 16 2

3-------------

8r r 18 16 2–+3

----------------------------------------------

r

x 4+

1 For graphs of indefinite integrals, the x-axis may be higher or lower than shown, but the shape should be similar.a

b

c

d

e

Exercise 11.2

105

−5

−10

−15

−20

2 4 6 x

F(x)

c = −5c = 0

c = 5

F(x) = x2 − 8x + c

2 4 6 z−5

252015

G(z)

105

c = 5 c =

0 c = −5

G(z) = 4z + c

−2 2 4 m−2

−4

1086

D(m)

42

12

−6

c = 6

c = 3

c = 0

D(m) = 6m − 2m2 + c

−2 2 4 x

−4

−6

4

Q(x)

2

−2

−8

−10

6c = 3

c = 0

c = −3

Q(x) = 5x − 2x2 + c

−2 2 4 x

−4

−6

8

6

4

2

−8

c = 0

c = −2

−2

P(x) = x3 − 3x2 + cP(x)

c = 2


436 New QMaths 11B9780170136570

f

g

h

i

j

−2 2 y−1

−2

43

W(y)

2

1

−3

c = 2

c = 0

c = −2

W(y) = 4y − 3y2 + c

105

−5

−10

−15

−20

4 8 12 t

V(t)

c = 15

c = 0

c = 5

−25

15V(t) = t2 − 7t + c1

2---

−3 −2 −1 m

−4

−6

864

G(m)

2

−2

10

−8

c = −4

c = 0

c = 5

G(m) = 3m2 + 6m + c

−6 −4 −2 x−4

2016

12

8

4

c = −4

c = 0

c = 84036

32

28

24

P(x) P(x) = x3 + 6x2 + c

−4 −2 2 h−4

1612

K(h)

84 c = −4

c = 0

c = 8 K(h) = 5h2 + h3 + c53---

2 a

b

c

d

e

f

3 a

b

c

F(x) = x4 + x2 + 5x + c32--- 3

2---

M(t) = 2t2 − 3t + c

G(z) = z4 − 3z + z3 + c152------ 10

3------

H(x) = 7x − x3 + c43---

P(y) = + + + y + cy4

4----- y3

3----- y2

2-----

S(x) = 0.05x4 − 0.06x5 + 0.16x3 − 2x + c.

y

x

y

x

y

x



4 a

b

c

y

x

y

x

y

x

1 a v = −10t + 5 b 0.5 s

c h = −5t2 + 5t + 6 d 7.25 m

e −5t2 + 5t + 6 = 6, t = 1, v = −5 m/s2 a About 12.6 m/s, 45.5 km/h

3 2.31 m/s2

4 154 L, 223 L

5 a C = 10n2 + 20 000, P = 1800n − 10n2 − 20 000b 90 TVs

6 Assuming that an average person has an energy of 10 500 J at 60 km/h:

a 3.15 × 1010, 3.15 × 107, 1.17 × 106

b B = 3.15 × 1010x2, B = 3.15 × 107x2, B = 1.17 × 106x2

c 3 150 000 N, 315 000 N, 105 000 N. The further the body’s stopping distance, the lower the force. For 100 km/h, the force is much greater.

Exercise 11.3


Documents

00 NQM11B SB TXT.prelim - mathsbooks.netmathsbooks.net/New Q Maths 11B/answers.pdfChapter answers 402 New QMaths 11B 9780170136570 Chapter 1 1a 42.6° b 67.35° c 28.29° d 16.41°