Camb Answers

8/19/2019 Camb Answers

1/75

Answers

Chapter 1

Exercise 1A

1 a 57.5 b −1.2 c 24.25 d −16e −4 f 2.75 g 6 h 1.5i

10

7 j 3 k 7 l −4

m −14.4 n −5 o 9 p −1q −1.2 r 1.4 s 51 t 82u −11.5 v 12 w 8 x −2.8y 3.5 z 2.4

2 a − ba

be − d

cc

c

a− b d b

a − c

eab

a+

b f a + b g

b − d a

−c

hbd − c

a

3 a −18 b −78.2 c 16.75 d 28e 34 f 0.1154

Exercise 1B

1 a x ( x + 3) b x ( x − 5) c x ( x + 1)d x (3 x − 4) e x (5 x − 1) f 5 x ( x − 3)g 3 x (2 x − 5) h − x ( x + 5) i −4 x ( x − 4)

2 a ( x + 4) ( x + 6) b ( x + 1) ( x + 8)c ( x

−3) ( x

−8) d ( x

−6) ( x

+5)

e ( x − 4) ( x − 5) f ( x + 3) ( x − 40)g ( x + 2) ( x − 9) h ( x − 3) ( x − 16)i ( x − 7) ( x + 12) j (5 x + 3) ( x + 4)

k (3 x − 2) (2 x − 1) l (5 x − 4) ( x − 3)m ( x + 4) (6 x − 5) n (3 x + 2) (5 x − 7)o ( x + 1) (15 x − 14)

3 a ( x − 7) ( x + 7) b ( x − 4) ( x + 4)c (1− x) (1+ x) d (2 x − 9) (2 x + 9)e 2 (5 x − 7) (5 x + 7) f 4 ( x − 3) ( x + 3)

4 a 5 ( x − 4) ( x + 4) b 6 ( x − 3) ( x + 3)c 2 (2 x − 5)(2 x + 5) d 4 ( x + 2) ( x + 3)e 3 ( x

+2) ( x

+3) f 2 ( x

−2) ( x

−7)

g 5 ( y + 2) ( y − 6) h 3 ( x + 3) (2 x + 5)i 2 (2 x + 7)(3 x − 4) j x ( x − 6) ( x + 1)

k x (5 x

−6) ( x

−2) l 3 x ( x

−4)2

Exercise 1C

1 a −3 or 3 b −6 or 6c −8 or 8 d −0.5 or 0.5e 4 or −4 f −0.25 or 0.25g −4.98 or 4.98 h 0

2 a 2 or 4 b −3 or 11c −7 or 2 d −16 or 4e −1.5 or −1 f 0.5 or 1.5g −3 or 8 h −1.5 or −2

3i−

1.5 or 2

3 a −7.606 or −0.394 b 0.7085 or 11.29c −3.245 or 9.245 d −6.071 or 1.071e −5.804 or 0.804 f −12.75 or 2.746

4 a −1.5 or −1 b 0.134 or 1.866c −5.266 or 6.266 d −5.727 or 1.881e −13.57 or 1.572 f −1.954 or 8.954

5 a −0.65 or 4.65 b −0.41 or 2.41c −1.23 or 1.90

6 a −8 or −1 b −11 or 3c −4 or 7 d −2.243 or 6.243e −2.281 or −0.219 f −1.5 or −0.5

g −2 or 8 h 0.5 or 5

3

i −1.886 or 2.386 j 56

or 3

k −1.5 or 3 l 0.5 or 0.6m −0.75 or 2

3 n 0.5

o10

3 or 0

p 0 or 3

q −5 or −3 r 0.2 or 27 4 or 9

8 3

9 2 or 2.375

10 1 or −

m

4

563

Cambridge University Press • Uncorrected sample pages • 9780521720618 • 2008 © Ousby, Cross, Bowman.

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564 Queensland Mathematics B Year 11

Exercise 1D

1 a

1 2

−2

−1

−1

1

2

3

4

x

y y = 3 x + 2

b

x

y

1 2

−1

1

2

3

4

5 y = 4 − 2 x

c

1 2 3 4 5 6

1

2

3

4

5

6

7

89

10

x

y

y = −1.5 x + 10

7

d

−2 −1 1 2

−4

−3

−2

−1

1

2

3

4

x

y y = x2 − 3

e

1 2 3 4 5

−2

−1

1

2

3

4

5

x

y

y = ( x − 1) ( x − 3)6

f

−1 1 2 3 4

−5

−4

−3

1

2

x

y

−2

−1

y = 3 x − x23

2 a

−3 −2 −1 1

−3

−2

−1

1

2

3

4

5

x

y

y = 0.5 x + 1

b

−3 −2 −1 1

−5

−4

−3

−2

−1

1

2

3

x

y

y = −2 − x


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Answers 565

c

−1 1 2 3−1

1

2

3

4

5

x

y

−2

y = −2.5 x + 56

d

−3 − −1 1

−4

−3

−2

−1

1

2

3

4

x

y

2

y = 3 x ( x + 2)

e

−4 −3 −2 −1

−4

−3

−2

−1

1

2

3

4

x

y y = ( x + 2)2 − 3

f

−2 −1 1 2

−3

−2

−1

1

2

3

4

5

x

y y = 5 − 2 x2

Exercise 1E

1 a

−3 −2 −1 1

−3

−2

−1

1

23

4

5

x

y y = 0 .75 x + 2

b

1 2 3 4

−5

−4

−3

−2

−1

1

2

x

y y = 0.5 x − 2

−6−6

c

−2 −1 1 2

−4

−3

−2

−1

1

2

3

4

x

y y = 2 x + 1

d

−2 −1 1 2

−5

−4

−3

−2

−1

1

2

x

y y = 3 x − 2

−6


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e

−1 1 2 3

−2

−1

1

2

3

4

5

6

x

y

y = −2 x + 4

f

−2 −1 1 2

−4

−3

−2

−1

1

2

3

4

x

y

y = −0.5 x

2 a

1 2 3 4

−3

−2

−1

1

2

3

4

5

x

y x + 2 y = 4

b y

−1 1 2 3

1

2

3

4

5

6

x

−1

73 x + y = 6

c

−3 −2 −1 1

1

2

3

4

5

6

7

x

y

−1

2 y − 3 x = 9

d

− 4 −3 −2 −1 1

−

−2

−1

1

2

3

4

5

x

y4 y − 2 x = 9

3

e

−4 −3 −2 −1

−3

−2

−1

1

2

3

4

x

y

4

x = 4 y − 4

−

f

−3 −2 −1 1−1

1

2

3

4

6

7

x

y

5

5 x − 2 y + 12 = 0


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Answers 567

3 a

−2 −1 1 2

1

2

3

4

5

7

x

y

−1

6

y = 6

b

−1 1 2 3

−6

−5

−4

−3

−2

−1

1

2

x

y

y = 4 x − 3

c

1 2 3 4

−5

−4

−3

−2

−1

1

2

x

y

−6

x − y = 4

d

−2 −1 1 2

−4

−3

−2

−1

1

2

3

4

x

y2 x − y = 0

e

−3 −2 −1

−3

−2

−1

1

2

3

4

x

y

−4

−4

x = −4

f

−2 −1 1 2

−4

−3

−2

−1

1

2

3

4

x

y2 y = 3 x

g

1 2−1−1

−2

−3

−2

1

2

3

4

x

y

−4

x = 0

h

1 2 3 4

−2

−1

1

2

3

4

5

x

y

−3

x + y = 3


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i

1 2−1−1

−2

−3

−2

1

2

3

4

x

y

−4

y = 0

4 a

x

y

–4

(2,2)

b

(−2,4)

x

y

–4

c

x

y

−

3

92

d

3

x

y

(1, –1)

e

–3.5

–7

x

y

f

–5

2

x

y

g

–2

8

x

y


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d

−4 x

y

(−1,−3)

e

x

y

−1

−3

1.5

f

x

y

−2

8

2

2 a y = ( x + 1)2 + 4

x

y

5

(−1,4)

b y = ( x − 2)2 + 2

x

y

6

(2,2)

c y = ( x − 1.5)2 + 2.75

x

y

4

5

(1.5,2.75)

d y = ( x + 3)2 − 9

x

y

(−3,−9)

e y = 2( x − 1)2 + 7

x

y

(1,7)

9


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Answers 571

f y = 3( x + 1)2 − 4

x

y

(−1,−4)

−1

g y = ( x + 3)2

x

y

−3

9

h y = 3( x − 1.5)2 + 1.25

x

y

(1.5,1.25)

8

i y = 9− 2 x2

x

y

(1,7)

9

3 a

x

y

−2

8

2

b

x

y

−3

−12

2

(−0.5,−12.5)

c

x

y

11

(2,3)

d

x

y

8

(3,−1)42


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e

x y

−10

(1.5,−12.25)

5−2

f

x

y

5

(2.5,−6.25)

g

x

y

−2

(−1,−2)

h

x

y

−12

(1.25,−15.125)

4−1.5

i

x

y

(0.5,0.75)

1

4 a

x

y

3 9,

8 16−

3

4

b

x

y

−8

8

3

8

3−

c

x

y

1

5 − 136

5 13

6

5 13,

6 12

–


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Answers 573

Exercise 1G

1 a x = −1 y = −1 b x = 5 y = 21c x = −1 y = 5 d x = 8 y = −2e x = 3 y = 4 f x = 7 y = 1/2

2 a x = 1 y = 5 b x = 2.5 y = −1c x = 2 y = −1 d s = −1 t = 4e p = 1 q = −1 f x = −1 y = 2.5g x = −1 y = 2

or

x = 3 y = 10

h x = −3 y = 8or

x = 1 y = 0i x = 1 y = 0 j x = −2 y = −4

or

x = 5 y = 17k x = 1 y = −1 l x = 0 y = 0

or

x = 1 y = 23 a x = 3 y = −2 b x = 7 y = 2

c x = 4 y = −3 d x = 7 y = 3e x = 4 y = −2 f x = 3 y = 1g x = 5 y = −2 h x = 3 y = −2i x = −2 y = −3

4 a x = 2 y = 4b x = −1 y = 7c x = 0 y = 8d x = −2.236 y = 0.528

or

x = 2.236 y = 9.472e x = −2.608 y = 3.804

or

x =

2.108 y =

1.446

f x = −0.781 y = 4.390or

x = 1.281 y = 3.360

Exercise 1H

1 a 20 b −12 c 16 d 9 e 412 a cross b neither c touch

d cross e neither f touch

3 a 2 b 0 c 1 d 2 e 1 f 0

4 a 1 rational root b 2 rational roots

c 2 irrational roots d 1 rational roote 2 irrational roots f no real roots

5 = m2 + 8m + 166 a m = 3 or −3 b −3 −1/4

Exercise 1J

1 a y = 3 x + 5 b y = −4 x + 6c y = 3 x − 4

2 a y = 3 x − 11 b y = −2 x + 93 a y = 2 x + 6 b y = 2.5 x − 24 a y = 4 x + 4 b y = −2

3 x

c y = − x − 2 d y = 12 x − 1

e y = 3.5 f x = −25 a 2 x + 3 y − 12 = 0 b 2 x + y + 6 = 0

c x + y − 8 = 0 d x + 2 y − 4 = 0e 2 x − 3 y − 2 = 0 f x = 3g 2 x − 2 y + 7 = 0 h y = 5i 2 x + 4 y − 1 = 0

6 yes

7 AB: 2 x − 3 y + 1 = 0 BC : 3 x + 2 y − 18 = 0 AC : x − 8 y − 6 = 0

Exercise 1K

1 a y = − 516

x2 + 5 b y = x 2

c y = 111

x ( x + 7) d y = ( x − 1) ( x − 3)

e y = −54

( x + 1)2 + 5 f y = ( x − 2)2 + 2

2 a y = 107

( x − 2) ( x + 4) b y = −4 x ( x − 4)

c y = 45

( x − 2) ( x − 5)

3 a y = 12

( x + 1)2 + 2

b y = −32

( x − 2)2 + 3

c y = 43 x2

4 a y = −2 x2 + x + 5b y = 1

4 x2 − 1

2 x + 3

c y = 4 x2 − 7 x5 y = 1

180( x − 90)2 + 30

Exercise 1L

1 a,b Check with your teacher

2 a C = 10, F = 50b −40◦ C = −40◦ F

3 a,b Check with your teacher

4 13

5 abc − ac − 1 b

2b

c − a

6

ab

a − b + c


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Answers 575

9 a

x

y

6

−3

b

x

y

3

9

c

x

y

7

2.8

10 a

x

y

2

−6

b

x

y

3

(1,1)

c

x

y

−10

6

11 a

x

y

(1,5)

3

b

x

y

29

(4,−3)


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c

x

y

6

−2 3

12 a y = ( x − 2)2 − 9

x

y

(2,−9)

−5

b y = −( x − 1.5)2 + 2.25

x

y

(1.5, 2.25)

c y = 2( x − 2)2 − 5

x

y

3

(2,−5)

13 a

x

y

1 6

6

(3.5, −6.25)

b

x

y

−25


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Answers 577

c

x

y

−5

13

12

−289

24,

1

3

5

2−

14 a x = −1 y = 2b x = −1 y = 4c x

=4 y

=1

15 a x = 2.5 y = −1b x = 4 y = −2c x = 1 y = −0.5

16 a x = 0.5 y = 6b x = −2.5 y = −3c x = 2.121 y = −0.257

or

x = −2.121 y = −8.74217 a x = 2 y = 4

b no solution

c x = 3.0769 y = −4.30718 a (−1, 1) and (3, 9)

b (−1.5, 4.5) and (4, 32)19 a 2 irrational roots b 1 rational root

c no real roots

20 Check with your teacher

21 a y = ( x + 4) ( x − 2)b y = 3 ( x − 1)2 − 3c y = −1

2( x − 1)2 + 2

22 a y = 43

( x + 1) ( x − 3)b y = −2 ( x − 3)2 + 2c y = 3 x2 + 4 x − 7

Chapter 2

Exercise 2A

1 a 4.10 b 0.87 c 2.94

d 4.08 e 33.69◦ f 11.92

240√

3cm

3 66.42◦, 66.42◦ and 47.16◦

4 23 m

5 a 9.59◦ b √ 35 m

6 a 60◦ b 17.32 m7 a 6.84 m b 6.15 m

8 12.51◦

9 182.7 m

10 1451 m

11 a 5√

2 cm b 45◦

12 3.07 cm

13 37.8 cm

14 31.24 m

15 4.38 m

16 57.74 m

Exercise 2B

1 a 8.15 b 3.98 c 11.75 d 9.46

2 a 56.32◦ b 36.22◦ c 49.54◦

3 a A = 48◦, b = 13.84 cm, c = 15.44 cmb a = 7.26, C = 56.45◦, c = 6.26c B = 19.8

◦

, b = 4.66, c = 8.27d C = 30◦, a = 5.41, c = 15.56

4 C = 26.69◦, A = 24.31◦, a = 4.185 554.26 m

6 35.64 m

7 1659.86 m

8 a 26.60 m b 75.12 m

Exercise 2C

1 5.93 cm

2 ∠ ABC

=97.90◦, ∠ AC B

=52.41◦

3 a 26 b 11.74 c 49.29◦ d 73e 68.70 f 47.22◦ g 7.59 h 38.05◦

4 2.626 km

5 3.23 km

6 a 8.23 cm b 3.77 cm

7 55.93 cm

8 a 7.326 cm b 5.53 cm

9 a 83.62◦ b 64.46◦

10 a 87.61 m b 67.7 m

Exercise 2D

1 400.10 m

2 34.77 m

3 575.18 m

4 109.90 m

5 16.51 m

6 027◦

7 056◦

8 a 034◦ b 214◦

9 a 3583.04 m b 353◦ or N7◦W

10 ∠ AS B = 113◦11 22.01◦

12 a ∠ B AC

=49◦ b 264.24 km

13 10.63 km


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Exercise 2E

1 a

3 b

4

5 c

4

3

d11

6 e

7

3 f

8

32 a 120◦ b 150◦ c 210◦ d 162◦

e 100◦ f 324◦ g 220◦ h 324◦

3 a 34.38◦ b 108.29◦ c 166.16◦ d 246.94◦

e 213.14◦ f 296.79◦ g 271.01◦ h 343.77◦

4 a 0.66 b 1.27 c 1.87 d 2.81

e 1.47 f 3.98 g 2.38 h 5.74

5 a 0.95 b 0.75 c −0.82 d 0.96e −0.50 f −0.03 g −0.86 h 0.61i −34.23 j 0.36

6 a 0, −1, 0 b −1, 0, undefined c 1, 0, undefined d −1, 0, undefined e −1, 0, undefined f 0, 1, 0g 0,

−1, 0 h 0,

−1, 0

Exercise 2F

1 a

√ 3

2 , −1

2, −

√ 3 b

1√ 2, − 1√

2, −1

c 0, −1, 0 d −√

3

2 , −1

2,√

3

e − 1√ 2,

1√ 2, −1 f 1

2,

√ 3

2 ,

1√ 3

g

√ 3

2 ,

1

2 , √ 3 h −1√ 2 , −

1√ 2 , 1

i

√ 3

2 ,

1

2,√

3 j −√

3

2 ,

1

2, −

√ 3

k −1, 0, undefined l 0, 1, 0

2 a

√ 3

2 b − 1√

2c 0 d −1

2

e 0 f √

3 g −√

3

2 h1√ 2

i − 1√ 3

j −1 k −1 l undefined

3 a −√

3

2 b − 1√

2c

1√ 3

d undefined

e 0 f − 1√ 2

g1√ 2

h −1

Exercise 2G

1 a 13 cm b 15.26 cm c 31.61◦ d 38.17◦

2 a 4 cm b 71.57◦ c 12.65 cm

d 13.27 cm e 72.45◦ f 266.39 cm2

3 17.58◦

4 1702.55 m

5 10.31◦ at B, 14.43◦ at A and C

6 45.04 m

7 a 24.78◦ b 65.22◦ c 20.44◦

8 42.40 m

9 1945.54 m

10 a 6.96 cm b 16.25 cm2

11 a 5 km b 215.65◦ c 6◦33

12 5 m

13 11.37 m

14 14 ≈ 401 km/h15 height = 8√

15= 2.07 m

Multiple-choice answers

1 D 2 C 3 C 4 D 5 B

6 C 7 A 8 C 9 B 10 E

Short-response answers

1 a7

6 b 0.51c c 2.06c

2 a 135◦ b 154◦42

3 a −12

b1√ 2

c −√

3

d 0 e −1

2 f undefined

g −12

h −12

i − 1√ 3

4 35.53 km

5√

91 ≈ 9.54 cm6 143◦

7 a 052.6◦

b ∠T QS = 33.14◦, bearing of T from Q is105.9◦

8 9.4 cm

9 a i 39◦ ii 9◦

b 1425.16 m c 1083.29 m

10 13.45 km11 a AC = 4.16km, BC = 2.4 km

b 57.6 km/h

12 804 m

13 a ∠ ACB= 12◦,∠CBO = 53◦,∠CBA= 127◦b 189.33 m c 113.94 m

14 a ∠TAB= 3◦,∠ ABT = 97◦,∠ ATB= 80◦b 2069.87 m

c 252.25 m

15 a 184.74 m b 199.71 m c 14.93 m

16 a 370.17 m b 287.94 m c 185.08 m

17 a 8√

2 cm b 10 cm c 10 cm d 68.90◦


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Answers 579

Chapter 3

Exercise 3A

1 a x5 b 8 x7 c x2 d 2 x3

e a6 f 26 g x2 y2 h x2 y6

i x3

y3 j x

6

y4 k 1 l 1

m3

2 n

23

52

2 a x9 b 216 c 317 d q8 p9

e a11b3 f 28 x18 g m11n12 p−2 h 2a5b−2

3 a x2 y3 b 8a8b3 c x5 y2 d9

2 x2 y3

4 a1

n4 p5 b

2 x8 z

y4 c

b5

a5 d

a3b

c

e an+2bn+1cn−1

5 a 1 b 317n c34n − 11

2

2

d 2n+133n−1 e 53n−2 f 23 x−3 × 3−4g 36−n × 2−5n h 33 = 27 i 6

j34n

2k

3n

5n l

1

2× 35n6 a 212 = 4096 b 55 = 3125 c 33 = 277 Check with your teacher

Exercise 3B

1 a 25 b 27 c1

9 d 16 e

1

2 f

1

4

g 125 h 16 i

110 000

j 1000

k 27 l3

5

2 a 253 b a

16 b

− 76 c a−6b

92 d 3

− 73 × 5−

76

e1

4f x6 y−8 g a

1415

3 a (2 x − 1)3/2 b ( x − 1)5/2 c ( x2 + 1)3/2d ( x − 1)4/3 e x√

x − 1f (5 x2 + 1)4/3

Exercise 3C

1 a 3 b 3 c 12

d3

4 e

1

3 f 4

g 2 h 3 i 3

2 a 1 b 2 c −32

d4

3 e −1 f 8

g 3 h −4 i 8 j 4 k 3

1

2 l 6

m 71

2

3 a4

5 b

3

2c 5

1

2

4 a 0 b 0,−2 c 1, 2 d 0, 15 a 2.32 b 1.29 c 1.26 d 1.75

6 a x > 2 b x >1

3 c x ≤ 1

2

d x 1

g x ≤ 37 a x ≤ 1.43 b x ≥ 0.77

c x > −1.89 d x > 2.718 0.5

9 x ≤ 210 a 2.5 b 2

Exercise 3D

1 y y = 2.4 x

y = 1.8 x

= 0.5 x

= 0.9 x

x

1

0

all pass through (0, 1)

base > 1, increasing

base >>


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A n s

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5 a–f Check with your teacher

6 a–d Check with your teacher

Exercise 3E

1 a 3 b 4 c −7 d −3 e 4f −3 g 4 h −6 i −9 j −1

k 4 l −22 a log2(10a) b 1 c log2

9

4

d 1 e − log5 6 f −2g 3 log2 a h 9

3 a 2 b 7 c 9 d 1

e5

2f 5 log x a g 3 h 1

4 a 2 b 27 c1

125 d 8

e 30 f 2

3 g 8 h 64

i 4 j 10

5 a 5 b 32.5 c 22 d 20

e3±

√ 17

2 f 3 or 0

6 2+ 3a − 5c2


8 10

9 a 4 b6

5 c 3

d 10 e 9 f 2

Exercise 3F

1 a 2.81 b −1.32 c 2.40 d 0.79 e −2.58f −0.58 g −4.30 h −1.38 i 3.10 j −0.68

2 a x > 3 b x >>


19/75

An s

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Answers 581

e y

x

3

2

1

−1−4 −3 −2 −1 1 2 3 4

−2

−3

−4

intercept: x = 1asymptote: x = 0

f y

x

3

21

−1−4 −3 −2 −1 1 2 3 4

−2

−3

−4

intercept: x = 1asymptote: x = 0

2 a i y = 2log10 x ii y = 13

log10 x

b i y = 1013 x

ii y = 13

1012 x

3 a y = log3( x − 2) b y = 2 x + 3

c y = log3

x − 24

d y = log5( x + 2)

e y = 13 × 2 x f y = 3× 2 x

g y = 2 x − 3 h y = log3

x + 25

4 a–f Check with your teacher

5 a 0.64 b 0.406 Check with your teacher

7 y = log10(√

x) = 12

log10 x for x ∈ (0, 10]

y

x0 1


9 a = 6

103

2/3 and k = 1

3 log10

10

3

10 a

f ( x)

y

x

g ( x) = f ( x) + 1

(1,0)

(1,1)

b y

x

h( x) = 2 f ( x)

(1,0)

f ( x)

11 Shift (translation) of cb

from y-axis parallel to

x-axis, followed by a 1

bstretch (dilation) also

from the y-axis parallel to the x-axis.

Exercise 3H

1 y = 1.5× 0.575 x2 p = 2.5× 1.35t 3 a

Total thickness,

Cuts, n Sheets T (mm)

0 1 0.2

1 2 0.4

2 4 0.8

3 8 1.6

4 16 3.2

5 32 6.4

6 64 12.8

7 128 25.6

8 256 51.2

9 512 102.4

10 1024 204.8

b T = 0.2(2)nc

200

T

150

100

500.2

0 2 4 6 8 10 n

d 214 748.4 m


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20/75

A n s

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4 an 0 1 2 3 4

M 0 1 3 7 15

b M = 2n − 1

n 5 6 7

M 31 63 127

c M

n0

30

20

10

1 2 3 4 5

dThree discs 1 2 3

Times moved 4 2 1

Four discs 1 2 3 4

Times moved 8 4 2 1

5 n = 2

6 a 12

3n

b 12

5n−2c n =

3

7 a 729

1

4

nb 128

1

2

nc 4 times

8 11.21% (2009)

9 H = 1.26× 10−11 , given H >>


21/75

An s

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Answers 583

e

0

y

x

f

y = 2(0,3)

0

y

x

7 x = 18 Check with your teacher 9 3

10 a k = 17

b q = 32

11 a a = 12

b y = −4 or y = 2012 a batch 1 = 15(0.95)n , batch 2 = 20(0.94)n

b 32 years

13 a

y = p(t )

p,q

y = q(t )

(millions)

1.7

1.2

0 t

b i t = 12.56 (i.e. mid 1962)ii t = 37.56 . . .(i.e. mid 1987)

14 a company X $1.82, company Y $1.51,

company Z $2.62

b company X $4.37, company Y $4.27,

company Z $3.47

c intersect at t = 21.784 and t = 2.090; hence,February 2006 until September 2007

d February 2007 until September 2007, approx.

8 months

15 a 13.81 years b 7.38 years

16 a Temperature = 87.065 × 0.94t b i 87.1◦ C ii 18.6◦ C

c Temperature = 85.724 × 0.94t d i 85.7◦ C ii 40.8◦ C

e 28.2 min

17 a a = 0.2 and b = 5b i z = x log10 b

ii a = 0.2 and k = log10 518 a y = 2× 1.585 x b y = 2× 100.2 x

c x = 5log10 y

2

Chapter 4

Exercise 4A

1 a

x

y

−1 1 2 3−1

−2

−3

1

2

3

b

x

y

−2 −1 1 2 3−1

−2

−3

1

2

3

c

x

y

−2 −1 1 2 3−1

−2

−3

1

2

3

d

x

y

−2−3 −1 1 2 3

1

2

3


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22/75

A n s

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2 a

x

y = −5

y

−2−3 −1 1 2 3−1

−2

−3

−4

1

2

3

4

b

x

x = −3

y

−2−3 −1 1 2 3 4 5−1

−2

−3

12

3

c

x

y = 1

y

−2−3 −1 1 2 3

1

2

3

4

5

6

d

x

y x = 2

1 2 3 4 5 6−1

−2

7 8 9 10

1

2

3 a

x

y

−2−4

−1 1 2 4 5 6−1

−2

−3

−4

−5

1

2

3

4

−3 3

b

x

y

−2−3 −1 1 2 3−1

−2

−3

1

2

3

c

x

y

−2−3−4 −1 1 2 3 4−1

−2

−3

−4

−5

1

2

3

4

5

−5

5

d

x

y

−2−4 −1 1 2 4 5−1

−2

−4

−5

−5

12

4

5

−3 3

3

−3


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23/75

An s

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Answers 585

4 a

x

x = 1

y

−2−3−4 −1 1 2 3 4 5 6−1

−2

−3

−4

−5

1

2

3

4

5

b

x

x = 2

y

−2−3−4 −1 1 2 3 4 5 6 7−1

−2

−3

−4

−5

1

2

3

4

5

c

x

x = 1

x = −

1

y

−2−3−4 −1 1 2 3 4 5−1

−2

−3

−4

−5

−5

1

2

3

4

d

x

y

−2−3−4 −1 1 2 3 4 5−1

−5

1

2

3

4

Exercise 4B

1

1 2 3 4 5 6

12

11

10

9

8

7

5

4

3

2

1

6

Cost

No. cartons7

2 y

1 2 3 4 5 6 x

7

5

4

3

2

1

6

3 a y

-781 2 3 4 5 6 7 x

8

7

5

4

3

2

1

6


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24/75

A n s

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b y

-781 2 3 4 5 6 7 x

8

7

5

4

3

2

1

6

4 y

-781 2 3 4 5 6 7 x

8

7

5

4

3

2

1

6

(4, 8)

5V

98 10

10.5

111 2 3 4 5 6 7

1200

1100

(4,1144)

(6,972)

(2,884)

(8,560)

(10,100)

Length of cut

1000

900

800

700

500

400

300

200

100

600

6 a y

5 x

5

b y

-7 9 1081 2 3 4 5 6 7

9

8

7

5

4

3

2

1

6

x

(5,4)

(7,6.3..)

(9,8.4..)

7

2

1

Cost ($)

50 100 150 200 250 500

Weight (g)

8Cost ($)

1 2 3 4 15 24

Time (h)2

456

8

10

12

14

36

Exercise 4C

1 a Independent: Number of cartons

Dependent: Cost

b Domain: {1, 2, 3, 4, 5}Range: {2, 4, 6, 8, 10}


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25/75

An s

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Answers 587

c

1

2

3

4

5

2

4

6

8

10

2 a Independent: Number on uppermost face

Dependent: Number on the table

b Domain: {1, 2, 3, 4, 5, 6}Range: {1, 2, 3, 4, 5, 6}

c

1

2

3

4

5

6

1

2

3

4

5

6

3 a i Independent: Height

Dependent: Base

ii Domain: 0


26/75

A n s

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2 a i 7 ii 15 iii −3 iv 10b i 2a + 5 ii a2 + 2a + 7

iii −6b − 1 iv b2 − 6b + 15c i 9.5 ii ±

√ 14

3 a i −5 ii 5 iii 4 iv 15b i 7− 3b ii 2a2 + 8a + 5

iii −

6a−

5 iv 8b2

+24b

+15

c i 21

3 ii ±54 a i −24 ii 4

b i 5a + 5h − a2 − 2ah − h2 ii 4c i no solution ii 1 or 4

5 a i

x

y

1−4

−4

ii

x

y

4

4−1

b i Domain: All real numbers

Range: y ≥ −6 14

ii Domain: All real numbers

Range: y ≤ 6 14

6 a i

x

y

−1 3

3

ii

x

y

3

(1,2)

b i Domain: All real numbers

Range: y ≤ 4ii Domain: All real numbers

Range: y ≥ 2Exercise 4F

1 a Domain: All real numbers

Range: All real numbers

b Domain: x ≥ 0Range: y ≥ 0

c Domain: All real numbers

Range: y ≥ 1d Domain: −3 ≤ x ≤ 3

Range: −3 ≤ y ≤ 0e Domain: x > 0

Range: y > 0

f Domain: All real numbers

Range: y ≤ 3g Domain: x ≥ 2

Range: y

≥0

h Domain: x ≥ −1Range: All real numbers

i Domain: x ≤ 1 12

Range: y ≥ 0 j Domain: x = −2

Range: y = 0k Domain: x > 5


l Domain: x = 12

Range: y = 0m Domain: x

= −2

Range: y = −4n Domain: x > 3

Range: y > 0

o Domain: x > −1Range: y > 0

2 a

x

y

(1,1)

−2


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27/75

An s

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Answers 589

b

x

y

(1,2)

−1

c

x

y

3

(−1,5)

d

x

y

(−2,4) (3,4)

e

x

y

(−2, 6)

4

2

f

x

y

−2

2

g

x

y

(1,3)

2

h

x

y

1

(10,1)

Exercise 4G

1 a y

x0

Range = [0, ∞)b y

x0

1

1

Range = [0, ∞)c y

x0

Range = (−∞, 0]d y

x0

Range = [1, ∞)e y

x

2

0

(1,1)

Range = [1, ∞)


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28/75

A n s

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2 a y

x

4

3

2

1

1 2 30

b Range = (−∞, 4]3

y

x

1

2

1 2 3 54

–3 –2 –1

–1

–2

–3

–4

–5

0

4 a y

x

(0,1)

0

b Range = [1,∞)5 a y

x –3 3

0

–9

b Range = R6 a y

x

(1,1)

0

b Range = (−∞, 1]

7

f ( x) =

x + 3, −3 ≤ x ≤ −1− x + 1, −1


29/75

An s

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Answers 591

12 f ( x) and h ( x) are even

g ( x) is odd

13 Offer A is cheaper if average number of calls is

>


30/75

A n s

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b i 2a + 11ii b2 − 4b + 6

c i 7.5

ii x = ±√

22

d

x

y

7

(−1,6)

6 a Domain: x ≥ −2Range: y

≥0

b Domain: x = 12

Range: y = 0c Domain: x > 8


d Domain : All real numbers


7 a

x

y

(3,8)

−1

b

x

y

−2 2

4

c

x

y

−1

(2,−3)

(2,−1)

4

1

8 a {(3, 1), (4, 3), (6, 4), (8, 7)}

b f −1 ( x) = 12 x − 3

2

c g −1 ( x) = 34 − 1

4 x, x ≤ 3

9 a f −1 ( x) = √ x + 3 or f −1 ( x) = −√ x + 3

b f −1 ( x) =√

x2 + 9, x ≥ 0 or f −1 ( x) = −

√ x2 + 9, x ≥ 0

10 A ( x) = 25 − x2, 0 < x <√

50

11 V ( x) = x (30 − 2 x) (21 − 2 x) , 0 < x >>


31/75

An s

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Answers 593

4 a

P e r c e n t a g e o f s t u d e n t s

25

50

W a t c h

T V R e a d

L i s t e n

t o m u s i c

W a t c h

a v i d

e o

P h o n

e f r i e

n d s

O t h e

r

Leisure activity

b watching TV

Exercise 5C

1 Number 0 1 2 3 4 5 6

Frequency 4 4 4 4 3 2 1

2 a 4 b 2 c 5 d 28

3 a 0 b 48 c 60–69 d 334 a, b Temperatures Relative

(◦C) Frequency frequency

0 – 1 0.03

5 – 0 0

10 – 1 0.03

15 – 9 0.28

20 – 4 0.13

25 – 5 0.16

30 – 7 0.22

35 – 4 0.13

40 – 0 0

45 – 1 0.03

c

0

2

4

6

8

10

5 10 15 20 25 30 35 40 45

Temperature

N o . o f c i t i e s

d 47%

5 a

0

2

4

6

N o . o f b o o k s

Price

5 10 15 20 25 30 35 40 45

b $5.00–$5.99

cPrices ($) Cumulative frequency

less than 5 3

less than 10 9

less than 15 12

less than 20 15

less than 25 19

less than 30 19less than 35 20

less than 40 20

less than 45 21

Price

C u m u l a t i v e f r e q u e n c y

50

5

10

15

20

10 15 20 25 30 35 40 45

6 a

0 28 29 30 31 32 33 34

2

4

6

8

10

Measurement

F r e q u e n c y

b

0

28

28 29 30 31 32 33 34Measurement

C u m u l a

t i v e f r e q u e n c y

c The students’ estimates ranged from

28.9 cm to 33.3 cm, with most students (89%)

overestimating the 30 cm measure.

7 a

0 10 20 30 40 50 60 70 80 90 100

2

4

6

8

Marks

N o . o f s t u d e n t s

b

0 10 20 30 40 50 60 70 80 90 100

5

10

15

20

25

30

Marks


c The students’ marks ranged from 21 to 99,

with most students (over 70%) scoring more

than 50% on the test.


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32/75

A n s

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8 a

Length of hole

N o . o f h o l e s

0 240 260 280 300 320 340 360 380 400 420

2

4

6

8

10

12

14

b

Length of hole

0240 260 280 300 320 340 360 380 400 420

10

20

30

40

50


c i below 300 m = 425

ii no. of holes ≥ 360 m = 17, proportion = 17

50

iii approx. 280 m

Exercise 5D

1 a centre b neither c both

2 a positively skewed

b negatively skewed c symmetrical

3 symmetrical 4 symmetrical

5 approximately symmetrical

Exercise 5E

1 a 4 0 2 4 5 7 | 7 represents 7.7 days5 8 9

6 5 8

7 7

8 4 8

9 4

b four months

2 a 0 4

1

1 6 8 9 2 | 5 represents 25 hours2 1 1 3 (truncated)

2 5 5 5 6 7 7 9 9

3 1 1 2 3 3

3 6 9

4 1

4 6

b nine batteries

3 a 0 0

1 0 0 4 5 5 6 9

2 0 0 1 3 7 8 9

3 3 7 9

4 6 4 | 6 represents 46 minutes5

6 3 7

7 0

b three students

c positively skewed

4 a 2 5 8

3 5 6 9

4 5 6 9

5 2

6 8

7 5 5 6 8 9

8 2 4

9 5

10 16 | 4 represents $16411 (truncated)

12

13

14 9

15

16 4

17

18

19

20

21 0

b approximately symmetrical

5 a Father's age Mother's age

3 7 8 8 9

4 4 4 3 3 3 1 1 0 4 0 0 0 1 2 3 3 3 3 3 4 4

9 8 8 8 8 7 7 6 6 6 5 4 5 6 7 8 9 9 9

4 2 1 1 0 5 0 0 0

5 5

0 | 4 represents 40 years 4 | 0 represents 40 yearsb Both distributions are approximately

symmetrical. Fathers, with ages centred in the

late forties, tend to be older than mothers, with

ages centred in the early forties. The spread is

similar for both distributions.

6 a Class B Class A

3 2 1 9

2 2

3 9

4 5 7 8

5 5 8

9 6 5 8

6 4 3 3 2 2 1 0 0 7 1 6 7 9 9

8 8 4 4 3 2 1 1 0 0 8 0 1 2 2 5 5 9

8 1 9 1 9

9 | 6 represents 69 marks 7 | 1 represents 71 marksb six students in class A and two in class B

c Class B performed better as more students

scored in the higher values of 70s to 90s.

Exercise 5F

1 a 800 b 400 c 14 d 9

2 a 56 000 b 1988–89 c 1995–96 d 72%

e 1996–97


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33/75

An s

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Answers 595

3 a 51% b Jan 2005 c Aug 2007 d 128 000

e Wivenhoe

4 a 47 000 000 b 100 000 000

c 2004/05 d 21%

e Check with your teacher

5 a $8 billion b 4.7%

6 a 0.029 b 0.26 c 53

Exercise 5G

1 a mean = 18.36,median = 14b mean = 9.19,median = 10c mean = 7.41,median = 7.65d mean = 1.62,median = 1.15

2 a mean = 3.24,median = 3b mean

= −0.38,median

=0

3 mean = $193 386,median = $140000; themedian is a better measure of centre as it is

typical of more house prices.

4 mean = 4.06,median = 4; both are reasonablemeasures of centre in this example.

5 a range = 602, IQR = 455b range = 5.3, IQR = 3.2c range = 0.57, IQR = 0.21d range = 7, IQR = 3.5

6 a 145 b 42

7 a 2.4 kg b 1.0 kg

8 a

0 C u m u l a t i v e r e l a t i v e

f r e q u e n c y

Age

17 22 27 32 37 42 47

0.2

0.4

0.6

0.8

1.0

median = 18, IQR = 2b mean = 20.97, s = 7.37 c 92%

9 a 12.39 b 1.33 c 281.24 d 3.04

1 0 a i mean = 17.61, s = 15.96ii mean

=195.3, s

=52.9

b i 94% ii 100%

1 1 a i mean = 6.79,median = 6.75ii IQR = 1.8, s = 0.93

b i mean = 13.54,median = 7.35ii IQR = 1.81, s = 18.79

c The error does not affect the median or

interquartile range very much. It doubles the

mean and increases the standard deviation by

a factor of 20.

12 Approximately 95% of share prices lie between

$44 and $56.

13 About 95% of days lie in this interval.

Exercise 5H

1 a m = 154, Q1 = 141.5, Q3 = 161.5,min = 123,max = 180

b

120 130 140 150 160 170 180

c The distribution of heights is slightly

negatively skewed, centred at 154 cm, with the

middle 50% of heights ranging from 141.5 cm

to 161.5 cm.

2 a m = 3, Q1 = 0, Q3 = 13,min = 0,max = 52

b 38, 52

c * *

0 10 20 30 40 50

d The distribution of number of books borrowed

is positively skewed, centred at 3. While 75%of people borrowed 13 books or less, one

person borrowed 38 books and another

borrowed 52.

3 a

0 105 15 20 25 30 35 40 million

b The distribution of winnings is positively

skewed with a median value of

$5.3 million. The middle 50% of players won

between $2.9 million and $10.0 million. Roger

Federer is the outlier, winning

$39 million.4 a

0 5 10 15 20

b The distribution is symmetrical, centred at

$10.00. The middle 50% of students earn

between $8.15 and $11.85 per hour.

5 a

0 100 200 300 400 500 600 ,

000

*

d The distribution is approximately

symmetrical, centred at about 210 000, withan outlier at 570 000. The middle 50% of

papers have circulations from about 88 000 to

270 000.

Exercise 5I

1 a

0 10 20 30 40 50

*

*

*After

Before


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34/75

A n s

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b The distribution for the number of sit-ups is

negatively skewed before the course, centred

at 26. After the course, the distribution is more

symmetric, centred at 30, indicating that the

course has been effective. The distribution

after the course is more variable than before

the course, showing the course has not had the

same effect on all participants. There is one

outlier in the before group, who can achieve

46 sit-ups, and two in the after group,

recording 50 and 54 sit-ups, respectively.

2

0 1 3 5 72 4 6 8 9

Year 12

Year 8

a Year 12 b Year 12

3 a

15 20 30 40 5025 35 45

1990

1970

b The distributions of ages in both groups are

slightly positively skewed, with the mothers in

1970 (median = 24.5) generally younger thanthe mothers in 1990 (median = 28). Thevariability in both groups is the same

(IQR = 10 for both groups).

Exercise 5J

1 a

1.0 2.0 3.0 4.0 5.0 6.00

10203040506070

Drug dose (mg)

R e s p o n s e t i m e ( m i n )

b negative association c no outliers

2 a

100 200 300 400 500 600

5

10

15

0 B u s i n e s s ( $ ’ 0 0 0 )

Advertising ($)

b positive association c no outliers

3 a

100 200 300 400

400

500

600

700

800

No. of seats

A i r s p e e d ( k m / h )

b positive association

c (122, 378) is an outlier

4 a

2 4 6

5

10

15

0 8 10 12

P r i c e

( $ ’ 0 0 0 )

Age (years)

b negative association

c (10, 8700) is an outlier

Exercise 5K

1 a no correlation

b weak negative correlation

c strong negative correlation

d weak positive correlation

e strong positive correlation

f strong negative correlation

g strong positive correlation

h no correlation

i strong negative correlation

j weak positive correlation

k strong positive correlation

l moderate negative correlation

2 a 0.71 b 0.78 c 0.82 d 0.92

3 a −0.6 b moderate negative correlation4 a 0.67 b moderate positive correlation

5 a 1 b strong positive correlation

6 a −0.43 b weak negative correlation

Exercise 5L

1 a no linear relationship

b weak negative linear relationship

c strong negative linear relationship

d weak positive linear relationship

e strong positive linear relationship

f strong negative linear relationship

g strong positive linear relationship

h no linear relationship

i moderate negative linear relationship

j weak positive linear relationship


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35/75

An s

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Answers 597

k perfect positive linear relationship

l perfect negative linear relationship

2 a 0.8 b 0.8 c 0.7

d 0.8 e −0.7 f −0.23 a −0.86

b strong negative linear relationship

4 a 0.95b strong positive linear relationship

5 a 0.77

b strong positive linear relationship

6 a −0.77b strong negative linear relationship

7

50

50

55

55

60

60

65

65

70

70

40 45Attempt 1

A t t e m p t 2

a a strong positive relationship

b Yes, the data are numerical and the

relationship is linear. There are no outliers.

8

210230250270290310330

200 240220 260280 300 320 3400

T e s t 2

Test 1

a There is a strong positive linear relationship

between the scores on Test 1 and Test 2.

b Yes, the data are numerical and the

relationship is linear.

c q = 0.71, r = 0.87d q: moderate positive linear relationship

r : strong positive linear relationship

e i q = 0.43, r = −0.004ii The error in the data has a much greater

effect on Pearson’s correlation coefficient.

Exercise 5M

Note: Answers will vary for lines drawn by eye.

1

x

y

1 2 3 4 5 6 7 8

5

10

15

0

y = 1 + 2 x

2

x

y

0 1 2 3 4

5

–5 –1 –2 –3

–20

–15

–10

y = – 4.5 – 3.75 x

3 a

2000

2000

4000

4000

6000

6000

K B A

J D

E CG H

I

F

Year 1

Y e a r 2

0

b y = 424 + 0.794 xc The positive slope indicates that districts with

high rates in Year 1 also had high rates in

Year 2.

4 a

85

90

95

36 40 44 48 52 56 60

H e i g

h t ( c

m )

Age (months)

b y = 72+ 0.4 xc The intercept (72 cm) is the predicted height

at age 0. The slope predicts an increase of 0.4 cm in height each month.

d i 89 cm ii 158 cm

e Part i is reasonable as it is a value close to the

data. Part ii is not reliable as the relationship

may no longer be linear here.

5 a

150

150

160

160

170

170

180

180

Mother

D a u g h t e r

b y = 18.3+ 0.91 x c 173 cm6 a

160 180100

2.5

3.0

3.5

4.0

200140120

C o s t ( $ ’ 0 0 0 )

Number of MP3 players


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b y = 1300 + 13 x c ≈ $1300d ≈ $13

7 a y = 70− 14 xb The intercept is the predicted time taken to

experience pain relief if no drug is given.

From the slope we predict a reduction of

14 minutes in time taken to experience pain

relief for each mg of drug administered.

c −14 min, which is not a realistic answer 8 a y = 18.2 x + 1000

b Intercept predicts $1000 of sales if nothing is

spent on advertising. The slope means that, on

average, each $1 spent on advertising is

associated with an increase of $18.20 in sales.

c i $19 200 ii $1000

Exercise 5N

1 a y = 68.2+ 0.46 xb The y-intercept is the predicted height at birth.

From the slope, we predict an increase in

height of 0.46 cm each month.

c i 88 cm ii 168 cm

d The height at 42 months is reliable since this

is within the range of data given. The height at

18 years is less reliable since this is outside the

range of data given.

2 y = 487.6+ 0.77 x3 a y = 50.2+ 0.72 x

b An increase of 1 cm in the mother’s height is

associated with an increase of 0.72 cm in thedaughter’s height, on average.

c 172 cm (to the nearest cm)

4 a y = 1330 + 12 xb $1330

c $12

5 a response time = 57.0− 10.2× drug doseb The intercept of 57.0 minutes is the predicted

time for pain relief when no drug is given.

From the slope, we predict a 10.2 minute

decrease in response time for each 1 mg of

drug given.

c −4.2 min, which is not a realistic answer.6 a business = 1123.8+ 18.9× advertising

b Intercept is the volume of business with no

advertising. From the slope we predict an

increase in business of $18.90 for every dollar

spent on advertising.

c i $20 044 (to the nearest dollar)

ii $1124 (to the nearest dollar)

Exercise 5O

1 72.667

2 8.5 years

3 $34 000

4 $12 000

5 $21 000

6 9% p.a.

7 a–d Check with your teacher


9 102



12 a Method 1

Method 2

Method 3

50 60 70 80 90 100

b The distributions of scores are negatively

skewed for methods 1 and 3 and symmetrical

for method 2. The scores for method 1 are

higher than for methods 2 and 3 (90, 79 &70, respectively), and are also less variable

than method 2. They show similar variation

to the scores for method 3.

c Thus training method 1 would be

recommended, as it consistently produces

higher scores.

13 a First-born

Second-born

Third-born

10 20 30 40 50

b The distribution for the first-born is

symmetrical, while for the second and

third-born the distributions are positively

skewed. The centre for the first born is higher

than for the second, which is higher than the

third (35, 21 & 12, respectively), whilst the

variability is most for the first-born, followed

by the second-born and then the third-born.

14 a yes

b 60; this is reliable because it is an

interpolation within the data.

15 4750; this is not reliable as it is an extrapolationfar beyond the data.

Multiple-choice answers1 D 2 B 3 D 4 C 5 D

6 C 7 A 8 D 9 C 10 A

11 D 12 E 13 A 14 B 15 B

16 E 17 C 18 C 19 A 20 D


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37/75

An s

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Answers 599

Short-response answers1 a continuous b ordinal

c discreate d nominal

2 a numerical b categorical

3 a Class composition by gender

Girls

Boys

b Responses to survey

0

2

4

6

8

10

12

14

16

18

Agree Neutral Disagree

F r e q u e n c y

4

0

F r e q u e n c y

0

2

4

68

10

12

10 20

No. of cigarettes smoked

30 40

5 a 2 2

3 9 4 | 7 represents 47 minutes 4 3 4 5 7 9

5 0 1 1 2 2 4 5 6 6 7 9

6 5 8 9 7 2

b m = 52, Q1 = 47, Q3 = 576 x = $283.57,m = $267.507 a 92.9%

b yes, it is close to 95%

8

0 5 10 15 20 25

9

0 10 20 30 40 50

*

10 a numerical

b 0 0 5 6 6 8 9

1 4 4 5 8 9 3 | 2 represents 32 %2 5 6 7 8

3 2 2

4 4

5 3

c positively skewed

d 21.1%

e x = 20.05,m = 18f

0 10 20 30 40 50 60

0

1

2

3

4

5

6

Divorce rate

F r e q u e n c y

i positively skewed

ii 5

g

10 20 30 40 50 600

0.0

20.0

40.0

60.0

80.0

100.0

Divorce rate

C u m

u l a t i v e f r e q u e n c y %

i 58%

ii ≈ 17%


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38/75

A n s

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11 a

600

2

4

6

8

10

12

70 80 90

Travel time

F r e q u e n c y

i 21.4% ii positively skewed

iii 38.1%

b x = 69.60, s = 9.26,min = 57,Q1 = 62,m = 68, Q3 = 76,max = 90

c i 69.60 ii 68

iii 33, 14 iv 76

v 9.26 vi 51.08, 88.12

d

50

Hillside

Met

60 70 80 90 100

e The distributions of travel times are both

positively skewed. The travel times for the

Met (median = 70) tend to be longer thanthe travel times for Hillside trains

(median = 68). The spread of times is alsolonger for the Met (IQR

=24) than the travel

times for Hillside trains (IQR = 14).12 a

30 40

40

50 60

60

70

80

100

120

140

Inside 50

S c o r e ( p o i n t s )

b positive c 0

13 0.927

14 weight ≈ −200 + 2× height15 errors = 14.9− 0.533 × time16 a Intercept: no sensible interpretation. Slope:

For each additional second taken to complete

the task, on average, the number of errors is

reduced by about 1

2.

b 10

17 a IV = Exam scoreDV

= Number of new clients

b

7.00

70.00

5.00

8.00

80.00

6.00

60.00

9.00

10.00

11.00

65.00 75.00 85.00

Exam score

N u m b e r o f n e w c l i e n t s

c positive d 1, strong positive

e 0.748, moderate positive

f number of new clients=−4.00+ 0.173 × examscore

g Intercept: no sensible interpretation. Slope:

On average, each extra 1 mark in the final

exam is associated with an increase of 0.173

clients.

h 13

i Not very reliable as it is outside the range of

the data.

Chapter 6

Exercise 6A

1 a k = 2.6 b k = 92 a k = 1.3 b k = 4.83 $55.44 4 $6.40/kg 5 k

=38.88

6 a w (ox ) = 45

ox b 60 moles c 10.8 moles

7 3.75 barrels

8 1+ 2 ≈ 2.57


10 k = 1411 Check with your teacher

12 a A = klb, k = 1 b V = kr 2h, k = 3

c A = kd 2, k = 8

13 a F = km1m2d 2

b new force = old force ×4

9

Exercise 6B

1 a y

x0

(1,1)


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39/75

An s

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Answers 601

b y

x

(1,2)

0

c y

x0

1

21,

d y

x0

(1,–3)

e y

x

2

−12

0

f y

x0

–3

13

g y

x0

–4

12

h y

x

5

0

110

i y

x0

–1 1

j y

x –2

0

– 12

k y

x

–1 0

3

4

43

–

l y

x0 3

–4

– 31

352

2 a y = 0, x = 0 b y = 0, x = 0c y = 0, x = 0 d y = 0, x = 0e y = 2, x = 0 f y = −3, x = 0g y = −4, x = 0 h y = 5, x = 0i y = 0, x = 1 j y = 0, x = −2

k y = 3, x = −1 l y = −4, x = 3


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40/75

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Exercise 6C

1 a

x

f ( x) = 2 x − 3

y

1.5

−3

b

x

y

5

5

f ( x) = 5 − x

c

x

y

3

f ( x) = 3

d

x

y

(3,2)

f ( x) =2 x3

e

−2

x

y

f ( x) = −2

f

x

y4

(4,1)

f ( x) = 4 − 3 x4

163

2 a

x

y

−1 2

−2

b

x

y

−1 3

−3

c

x

y

−2−3

6

d

x

y

2 3

−6

e

x

y

−1 1 22

−6

f

x

y

(1,3)

212


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41/75

An s

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Answers 603

3 a

x

y

4

(1, −3)

b

x

y

−2 2

−4

c

x

y

1 3

3

d

x

y

−2 4

−8

e

x

y

42

8

f

x

y

−2 −1

4

g

x

y

−1.5 1.5

9

h

x

y

2

−10

56

−

Exercise 6D

1 a x2 + 2 x + 3 x − 1 b 2 x

2 − x − 3+ 6 x + 1

c 3 x2 − 10 x + 22− 43 x + 2

d x2 − x + 4− 8 x + 1

e 2 x2 + 3 x + 10 + 28 x − 3

f 2 x2 − 5 x + 37 −133

x + 4 g x2 + x +2

x + 3

2 a1

2 x2 + 7

4 x − 3

8 + 103

8(2 x + 5)b x2 + 2 x − 3− 2

2 x + 1c

1

3 x2 − 8

9 x − 8

27 + 19

27(3 x − 1)d x2 − x + 4+ 13

x − 2e x2 + 2 x − 15

f

1

2 x

2

+3

4 x −3

8 −5

8(2 x + 1)

Exercise 6E

1 a ( x − 1)( x + 1)(2 x + 1) b ( x + 1)3c ( x − 1)(6 x2 − 7 x + 6)d ( x − 1)( x + 5)( x − 4)e ( x + 1)2(2 x − 1) f ( x + 1)( x − 1)2g ( x − 2)(4 x2 + 8 x + 19)h ( x + 2)(2 x + 1)(2 x − 3)


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42/75

A n s

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2 a ( x − 1)( x2 + x + 1)b ( x + 4)( x2 − 4 x + 16)c (3 x − 1)(9 x2 + 3 x + 1)d (4 x − 5)(16 x2 + 20 x + 25)e (1− 5 x)(1+ 5 x + 25 x2)f (3 x

+2)(9 x2

−6 x +

4)

g (4m − 3n)(16m2 + 12mn + 9n2)h (3b + 2a)(9b2 − 6ab + 4a2)

3 a ( x + 2)( x2 − x + 1)b (3 x + 2)( x − 1)( x − 2)c ( x − 3)( x + 1)( x − 2)d (3 x + 1)( x + 3)(2 x − 1)

4 a = 3, b = −3, P ( x) = ( x − 1)( x + 3)( x + 1)

Exercise 6F

1 a

x

y

1 2 30

b

–2 –2

–1 1

y

x

c

1 2 3

−6

0

y

x

d

–3 –2 –1

612

y

x1 2

e

–3 –2 –1 1 2 3

y

x

f y

x –1 0 1

g y

x321

–30

h y

x32

18

1 –1 –2

i y

x210 –2 –1

–3

– √3 √3

j y

x –1

01 2 3

–6

– – 23

k

–1

1

0 1 – – 12

1

3 –

y

x

2 a y

x

(1.02, 6.01)1.5

0.5

(–3.02, –126.01)

0

–5

–15


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43/75

An s

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Answers 605

b y

x

(1.26, 0.94)1.5

1

(–1.26, –30.94)

0

–2.5

–15

c y

x

(1.35, 0.35)

1.5

(0, –18)

(–1.22, –33.70)

0

–2.5 1.2

d y

x

(–0.91, –6.05)

–2.5 –3 0

(–2.76, 0.34)

e y

x0

(–2, 8)

–3

f y

x0

(–2, 14)

–3.28

6

3 ( x + 1)( x + 1)( x − 3) = 0,∴ graph just touches the x-axis at x = −1 and cuts it at x = 3.

4 a y = −18

( x + 2)3 b y − 2 = −14

( x − 3)3

5 y = 2 x( x − 2)26 y = −2 x( x + 4)27 a y = ( x − 3)3 + 2

b y = 2318

x3 + 6718

x2 c y = 5 x3

Exercise 6G

1 a x = 0 or x = 3b x = 2 or x = −1 or x = 5 or x = −3c x = 0 or x = −2 d x = 0 or x = 6e x = 0 or x = 3 or x = −3f x = 3 or x = −3g x = 0 or x = 4 or x = −4h x = 0 or x = 4 or x = 3i x = 0 or x = 4 or x = 5

j x = 2 or x = −2 or x = 3 or x = −3k x = 4 l x = −4 or x = 2

2 a

5

y

x0

(3.15, –295.24)

b y

x –4 0 5 6

(5.53, –22.62)

480 (0.72, 503.46)

c y

x

(–1.89, –38.27)

0 –3

d y

(3, –27)

4 x

0

e y

(3.54, –156.25)(–3.54, –156.25)

5 –5 x

0


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44/75

A n s

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f y

2

–2

0

16

x

g y

–9 9

(–6.36, –1640.25) (6.36, –1640.25)

x

0

h y

4

0 3

(3.57, –3.12)

(1.68, 8.64)

x

i y

5

0 4

(4.55, –5.12)

(2.20, 24.39)

x

j y

–5 –4 0 4 5

400

(–4.53, –20.25) (4.53, –20.25)

x

k y

0 2 x

–20

l y

(1.61, –163.71)

(–5.61, 23.74)

50

–4

–14

–7 x

Exercise 6H

1 a

x

y

−2−3−4 −1 1 2−1

1

2

3

4(2,4)

shift left 2

b

x

y

−2 2

−2

−4

2

4

4−4

shift down 2

c

x

y

−2−4 2

(3,3)

−1

11

2

3

4

shift left 3


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45/75

An s

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Answers 607

d

x

y

1 2

(1,1)

(1,−3)

3 4 5 6−1

−2

−3

−4

1

2

shift down 4

e

x

y

2

(1,0)(1,0)

(10,−1)(10,−1)

(10,1)(10,1)

4 6 8−1

−2

−3

−4

10

1

(1,−2)(1,−2)

shift down 2

f

x

y

−2−3−4 −1 1

(1,2)(1,2)(−1,1)(−1,1)

2−1

1

22

3

4

5

shift left 12 a

x

y

−2−3 −1 1

(1,1)

(−1,−3)

−1

−2

−3

1

2

shift left 1, shift down 3

b

x

y

1 2 3

(3,4)

(3,1)

(2,3)

(1,1)

4−1

5 6

1

2

3

4

shift right 2, shift up 3

c

(2,1)(0,1)

(0,0)

2 4−4− 6 −2

2

−2

−4

x

y


d

x

y

−2 1 2 4−1

1

2

3

4(1,4)

(−1,−1)

3−1

−3

−2


e

x

y

1 2 4−1

−2

−3

−4

6 8 10

1

(1,0)

(2,−2)

shift right 1, shift down 2

f

x

y

−2−3 −1 1 2 3−1

1

2

3

(3,4)

(1,3) (3,3)

4

5

(2,2)

(2,1)



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46/75

A n s

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3

x

y

−1 1 2 3

(2,−3)

(−1,1)

(2,1)

(3,−4)

4 5

−1

−2

−3

−4

1

2

3

shift right 3, shift down 4

4

x

y

−2−4−1

−2

−6

1

2

(−3,1)

(−4,−1) (−1,−1)

3

4

(−3,0)

shift left 3, shift up 1

5

x

y

−2−3−4 −1 1 2

(1,−1)(−1,−1)

(−2,−3)

(−1,−4)

3

−2

−4

−6

2


6

x

y

−2 2 4

−4

6

(3,−1)(0,−1)

(1,1) (3,1) (4,1)

(4,3)

2

4

−2


Exercise 6I

1 a

x

y

−2 −1 1

(1,−1)

(1,1)

2−1

−2

−3

−4

12

3

4

reflect about the x-axis

b

x

y

2

(2,2)

(1,1)

4

2

4

−2−4

−2

−4

dilate by 4 from the x-axisc

x

y

−2−3 112

2

1

2

3

4

−1−13

4,( ) 1

214

,( )

dilate by 3 from the x-axis

d

x

y

1 2

34 , 2

(4 ,2)

3 4−1

5 6

1

2

3

4

( )

dilate by 13 from the y-axis


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47/75

An s

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Answers 609

e

x

y

2 4

(4,2)

(4,−2)

−1

−2

−3

−4

6 8

1

2

3

4

reflect about the x-axis

f

x

y

−2−3−4 −1 1 2−1

12

2(−1, 3)

3

4

5

2(−1, 1)


2 a

x

y

−2 −1 1

(1,−3)

(1,1)

(1,3)

2−1

−2

−3

−4

1

2

3

4

dilate by 3 from the x-axis, reflect about the

x-axis

b

x

y

1 2 3

(4, 2)

4

−2

−4

5 6

2

4

3( 4 ), −2

3( 4 ), −2

dilate by 13 from the y-axis, reflect about the

x-axis

c

x

y

2 4

−2

2

4

(2,2)

(2,1)

(2,−1)

−2−4−4

−4


x-axis

d

x

y

−2−3 2 3

(2,−4)

(2,−2)

−1

−2

−3

−4

−5

−6

1

−1 1


e

x

y

21

12

4−1

−2

−3

−4

6 8 10

1

2

3

4

dilate by 3 from the x-axis, dilate by 12 from the

y-axis


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48/75

A n s

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f

x

y

−2−3 −1 1 2 3

(1,2)

−1

−2

−3

−4

−5

−6

2

4

6

3( 1 ), 2

3( 1 ),−2


x-axis3

x

y

−2−3 1 2 3−1

−3−4

1

2

3

4

−1

−2


x-axis

4

x

y

−2−3−4−5 −1 1 2−1

−2

−3

−4

1

2

3

4

2( ),−1 1 81 1

2( ),−1 1 4−212( ),−1 1 8−11


x-axis

5

−5

x

y

−2 −1 1 2

(1, 2)

(2, −2)

3 4−1

−2

−3

−4

1

2

3

4

(1, −2)


x-axis

6

x

y

−1 1 2 3

(4,3)

(4,2)

4−1

−1

−2

−3

−4

5 6

1

2

3

4

12

dilate by 11

2 from the x-axis

Exercise 6J

1 a

x

y

−2−3 −1 1

(1,1)

(−1,3)

2−1

1

2

3

4

5

6

shift left 1, dilate by 2 from the x-axis, shift

up 3


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49/75

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Answers 611

b

x

y

1 2 3 4

−2

−4

−6

−8

5

2

4

(4,−4)

(4,2)

(4,4)

(4,−8)

dilate by 2 from the x-axis, reflect about the x-axis, shift down 4

c

3 5

2

x

y

21 4

−2

4

(2, −1)

−2

−4

shift right 2, reflect about the x-axis,

shift up 3

d

x

y

−2 −1 1

(−1,1)

2 3−1

−2

−3

−4

1

2

3

4

shift right 1, dilate by 3 from the x-axis, reflect

about the x-axis

e

x

y

2 4−1

−2

−3

−4

−5

6 8 10

1(5,1)

(10,1)

(5,−2)

dilate by 12 from the y-axis, shift down 3

f

x

y

−2−3 −1 1−1

−2

−3

−4

2

4

2( 1 ), −1

(1,3)2( 1 ),3

dilate by 12 from the y-axis, shift down 4

2

x

y

−2−3 −1 1

(1,1)

2−1

−2

(−2,−4)

(−1,−1)

−3

−4

2

4

shift left 2, dilate by 3 from the x-axis, shift

down 4

3

x

y

2 4

(2,3)

−2

−4

6 8 10

2

4

shift right 1, dilate by 2 from the x-axis, reflect

about the x-axis, shift up 3


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A n s

w e r s


4

x

y

−2 −1 1 2 3

12

(1,1)

4−1

−2

−3

−1

−4

−5

1

2

( ),

dilate by 12 from the y-axis, dilate by 3 from the

x-axis, shift down 4

5 a

x

y

−2−3 −1 1

(−2,4)(−1,4)

2

1

2

3

4

5

(1,3)

6

dilate by 12 from the y-axis, shift right 1, shift

up 3

b

x

y

2 4

(2,1)

6 8 10−1

−2

−3

−4

−2

1

2

(5,1)(10,1)

dilate by 12 from the y-axis, shift left 3, shift

down 1

c

x

y

−2−3−4 −1 1 2

6

8

2

(1,3)3

4 12

,3( )

dilate by 12 from the y-axis, shift left 1

2, dilate

by 2 from the x-axis

6 a Dilate from y-axis by a factor of 13; translate

right 2 units; dilate from x-axis by a factor of

4; translate upwards 1 unit

b Dilate from y-axis by a factor of 12; translate

left 5 units; dilate from x-axis by a factor of 8;

reflect about the x-axis; translate downwards

7 units

c Dilate from y-axis by a factor of 16; translate

right 1 unit; dilate from x-axis by a factor of 2;

translate downwards 5 units

Exercise 6K

1 a 7 b 6 c 5

d 6 e 11 f −42 a

x −2 −1 0 1 2 y 6 3 0 3 6

6

5

4

3

2

1

−4 −3 −2 −1

−1

1 2 3 4

y = 3| x|

y

x

b x −2 −1 0 1 2 y 1 2 3 2 1

6

5

4

3

2

1

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