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P1: FXS/ABE P2: FXS

9780521740524ans-ch1-4.xml CUAU068-EVANS August 19, 2011 4:45

AnswersChapter 1

Exercise 1A

1 a 3 b 9 c 1 d 8 e 5 f 2 g 53

h72

i7

3j

20

3k

103

l14

5

2 a a + b b a b c ba

d ab ebc

a

3 a 7 b 5 c 3 d 14 e 72

f14

3

g 48 h3

2i 2 j 3 k 7 l 2

4 a4

3b 5 c 2

5 a 1 b 18 c 65

d 23 e 0 f 10

g 12 h 8 i 145

j12

5 k7

2

6 aba

be d

cc

c

a b d b

c ae

ab

b + a f a + b gb da c h

bd ca

7 a 18 b 78.2 c 16.75d 28 e 34 f

3

26

Exercise 1B

1 a x + 2 = 6, 4 b 3x = 10, 103

c 3x + 6 = 22, 163

d 3x 5 = 15, 203

e 6(x + 3) = 56, 193

fx + 5

4= 23, 87

2 A = $8, B = $24, C = $163 14 and 28 4 8 kg 5 1.3775 m2

6 49, 50, 51 7 17, 19, 21, 23 8 4200 L9 21 10 3 km 11 9 and 12 dozen

12 7.5 km/h 13 3.6 km 14 30, 6

Exercise 1C

1 a x = 1, y = 1 b x = 5, y = 21c x = 1, y = 5

2 a x = 8, y = 2 b x = 1, y = 4c x = 7, y = 1

23 a x = 2, y = 1 b x = 2.5, y = 1

c m = 2, n = 3 d x = 2, y = 1e s = 2, t = 5 f x = 10, y = 13g x = 4

3, y = 7

2h p = 1, q = 1

i x = 1, y = 52

Exercise 1D

1 25, 113 2 22.5, 13.53 a $70 b $12 c $34 a $168 b $45 c $155 17 and 28 6 44 and 127 5 pizzas, 25 hamburgers8 Started with 60 and 50; nished with 30 each9 $17 000 10 120 shirts and 300 ties

11 360 Outbacks and 300 Bush Walkers12 Mydney = 2800; Selbourne = 320013 20 kg at $10, 40 kg at $11 and 40 kg at $12.

Exercise 1E

1 a x < 1 b x > 13 c x 3 d x 12e x 6 f x > 3 g x > 2h x 8 i x 3

22

2

x < 2

1 0 1 2

a

2

x < 1

1 0 1 2

b

2 1 0 1 2

x < 1c

2 1 0 1 2 3 4

x 3d

719ISBN 978-1-107-67331-1 Photocopying is restricted under law and this material must not be transferred to another party.

Michael Evans et al. 2011 Cambridge University Press

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720 Essential Mathematical Methods 1 & 2 CAS

2 1 0 1 2 3 4

x < 4e

2 1 0 1 2 3 4

x > 1f

2 1 0 1 2 3 4

x < 3 12g

x 3

2 1 0 1 2 3 4

h

x >

0 1 2 3

16i

3 a x >12

b x < 2 c x > 54 3x < 20, x 3}

{x : x < 3

2

}

e

{x : 3

2< x < 2

3

}f {x : 3 x 2}

g

{x : x >

2

3

}{

x : x < 34

}

h

{x :

1

2 x 3

5

}i {x : 4 x 5}

j

{p :

1

2(5 41) p 1

2(5 + 41)

}k {y : y < 1} {y : y > 3}l {x : x 2} {x : x 1}

2 a i 5 < m < 5 ii m = 5iii m >

5 or m < 5

b i 0 < m 4

3or m < 0

c i 45

< m < 0 ii m = 0 or m = 45

iii m < 45

or m > 0

d i 2 < m < 1 ii m = 2 or 1iii m > 1 or m < 2

3 p >4

34 p = 1

25 2 < p < 8

Exercise 4K

1 a (2, 0), (5, 7) b (1, 3), (4, 9)c (1, 3), (3, 1) d (1, 1), (3, 3)

ISBN 978-1-107-67331-1 Photocopying is restricted under law and this material must not be transferred to another party.


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e

(1 + 33

2,3

33

),(

1 332

,3 +

33

)

f

(5 + 33

2, 23 + 3

33

),(

5 332

, 23 3

33

)

2 a Touch at (2, 0) b Touch at (3, 9)c Touch at (2,4) d Touch at (4,8)

3 a x = 8, y = 16 and x = 1, y = 7b x = 16

3, y = 37 1

3and x = 2, y = 30

c x = 45, y = 10 2

5and x = 3, y = 18

d x = 10 23, y = 0 and x = l, y = 29

e x = 0, y = 12 and x = 32, y = 7 1

2f x = 1.14, y = 14.19 and x = 1.68,

y = 31.094 a 13b i

x

y

20.3

03.3

ii m = 6 32 = 6 42

5 a c = 14 b c >

1

46 a = 3 or a = 1 7 b = 18 y = (2 + 23)x 4 23

and y = (2 23)x 4 + 23

Exercise 4L

1 2 2 a = 4, c = 83 a = 4

7, b = 24

74 a = 2, b = 1, c = 6

5 a y = 516

x2 + 5 b y = x2

c y = 111

x2 + 711

x d y = x2 4x + 3

e y = 54

x2 52

x + 3 34

f y = x2 4x + 66 y = 5

16(x + 1)2 + 3

7 y = 12

(x2 3x 18)

8 y = (x + 1)2 + 3 9 y = 1180

x2 x + 7510 a C b B c D d A11 y = 2x2 4x 12 y = x2 2x 113 y = 2x2 + 8x 614 a y = ax(x 10), a > 0

b y = a(x + 4)(x 10), a < 0c y = 1

18(x 6)2 + 6

d y = a(x 8)2, a < 015 a y = 1

4x2 + x + 2

b y = x2 + x 516 r = 1

8t2 + 2 1

2t 6 3

817 a B b D

Exercise 4M

1 a A = 60x 2x2b A

x

450

0 15 30c Maximum area = 450 m2

2 a E

x

100

0 0.5 1

b 0 and 1 c 0.5 d 0.23 and 0.773 a A = 34x x2b A

x

289

0 17 34c 289 cm2

4 a C($)300020001000

0 1 2 3 4 h

The domain depends on the height of thealpine area. For example in Victoria thehighest mountain is approx. 2 km highand the minimum alpine height wouldbe approx. 1 km, thus for Victoria,Domain = [1, 2].

b Theoretically no, but of course there is apractical maximum

c $ 1225



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5 a T(000)

0 8 16

t (0.18, 15.82)0.18 15.82

t

b 8874 units6 a

x 0 5 10 15 20 25 30

d 1 3.5 5 5.5 5 3.5 1

d

5

4

3

2

10

0 5 10 15 20 25 30 x

b i 5.5 mii 15 57 m or 15 + 57 m from the batiii 1 m above the ground.

7 a y = 2x2 x + 5 b y = 2x2 x 5c y = 2x2 + 5

2x 11

2

8 a = 1615

, b = 85, c = 0

9 a a = 721600

, b = 41400

, c = 5312

b Shundreds of

thousandsdollars 53

12

354.71 t (days)

c i S = $1 236 666 ii S = $59 259

Multiple-choice questions

1 A 2 C 3 C 4 E 5 B6 C 7 E 8 E 9 D 10 A

Short-answer questions (technology-free)

1 a

(x + 9

2

)2b (x + 9)2 c

(x 2

5

)2d (x + b)2 e (3x 1)2 f (5x + 2)2

2 a 3x + 6 b ax + a2 c 49a2 b2d x2 x 12 e 2x2 5x 12 f x2 y2g a3 b3 h 6x2 + 8xy + 2y2 i 3a2 5a 2j 4xy k 2u + 2v uv l 3x2 + 15x 12

3 a 4(x 2) b x(3x + 8) c 3x(8a 1)d (2 x)(2 + x) e a(u + 2v + 3w)f a2(2b 3a)(2b + 3a) g (1 6ax)(1 + 6ax)h (x + 4)(x 3) i (x + 2)(x 1)j (2x 1)(x + 2) k (3x + 2)(2x + 1)l (3x + 1)(x 3) m (3x 2)(x + 1)n (3a 2)(2a + 1) o (3x 2)(2x 1)

4 a

x

y

(0, 3)

0

b

x

y

(0, 3)

0

32

, 032

, 0

c

x

y

11

(2, 3)0

d

x

y

(2, 3)

0

11

e

x

y

29

0(4, 3)

32

+4 , 032

, 04

f

x

y

32

32

0

(0, 9)

g

x

y

0 (2, 0)

(0, 12)

h

x

y

(2, 3)

(0, 11)

0

5 a

x

y

1 0

5 5

(2, 9)

b

x

y

0 6

(3, 9)

c

x

y

4 234 + 23

(0, 4)

(4, 12)

0

d

x

y

2 6

2 + 6

(0, 4)

(2, 12)

0

e

x

y

2 + 72 7

(2, 21)

(0, 9)

0

f

x

y

15

0 5

(2, 9)



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6 a ii x = 72

x

y

(0, 6)

0 1 625

27

2,

b ii x = 12

x

y49

41

2,

(0, 12)

4 0 3

c ii x = 52

x

y814

5

2,

14

02 7

d ii x = 5

x

y

(0, 16)

0 2 8(5, 9)

e ii x = 14

x

y

18

1

4,

3 0 52

(0, 15)15

f ii x = 1312

x

y

1

35

2

50

124

13

12, 12

g ii x = 0

x

y

43

43

0

(0, 16)

h ii x = 0

x

y

52

52

0

(0, 25)

7 a 0.55,5.45 b 1.63,7.37c 3.414, 0.586 d 0.314,3.186e 0.719,2.781 f 0.107,3.107

8 y = 53

x(x 5)9 y = 3(x 5)2 + 2

10 y = 5(x 1)2 + 511 a (3, 9), (1, 1)

b (1.08, 2.34), (5.08, 51.66)c (0.26, 2), (2.6, 2)d

(1

2,

1

2

), (2, 8)

12 a m = 8 = 22b m 5 or m 5c b2 4ac = 16 > 0

Extended-response questions

1 a y = 0.0072x(x 50)

b

4

5

3

2

1

00 10 20 30 5040

x

y

c 10.57 m and 39.43 m(25 25

3

3m and 25 + 25

3

3m

)d 3.2832 me 3.736 m (correct to 3 decimal places)

2 a Width of rectangle = 12 4x6

m, length of

rectangle = 12 4x3

m

b A = 179

x2 163

x + 8c Length for square = 96

17m and length for

rectangle = 10817

m ( 5.65 6.35 m)3 a V = 0.72x2 1.2x b 22 hours4 a V = 10 800x + 120x2

b V = 46.6x2 + 5000x c l = 55.18 m5 a l = 50 5x

2

b A = 50x 52

x2

c

0 10 20 x

250

A(10, 250)

d Maximum area = 250 m2 when x = 10 m

6 x = 1 +

5

27 a

25 + x2

b i 16 x ii x2 32x + 265c 7.5 d 10.840 e 12.615

8 a i y = 64t2 + 100(t 0.5)2= 164t2 100t + 25

ii y(km)

5

0 t (h)



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iii t = 12

; 1.30 pm t = 982

; 1.07 pm

iv 0.305; 1.18 pm; distance 3.123 km

b i 0,25

41ii

25 226982

9 b 2x + 2y = bc 8x2 4bx + b2 16a2 = 0e i x = 6 14, y = 6 14ii x = y = 2af x = (5

7)a

4, y = (5

7)a

410 a b = 2, c = 4, h = 1

b i (x , 6 + 4x x2) ii (x , x 1)iii (0, 1) (1, 0) (2, 1) (3, 2) (4, 3)iv y = x 1c i d = 2x2 6x + 10ii

(0, 10)

(1.5, 5.5)

d

0 x

iii min value of d = 5.5 occurs when x = 1.511 a 45

5

b i y = 1600

(7x2 190x + 20 400)

ii

(190

14,

5351

168

)c

(20, 45) (40, 40)

(60, 30)

(30, 15)

y =1

2x

C

O x

D

Bd

A

d i The distance (measured parallel to they-axis) between path and pond.

ii minimum value = 47324

when x = 35

Chapter 5

Exercise 5A

1 a y

x0

(1, 1)

b y

x

(1, 2)

0

c y

x0

12

1,

d y

x0

(1, 3)

e y

x

2

0

f y

x0

3

g y

x0

4

h y

x

5

0

i y

x0

1 1

j y

x2

0

12

k y

x1

0

3

4

l y

x0 3

4

313

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 = 2k y = 3, x = 1 l y = 4, x = 3

Exercise 5B

1 a y

x

19

03

b y

x0

4



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c y

x14

0 2

d y

x

4

3

0 1

e y

x3

0

4

f y

x

112

0 2

g y

x

6

0

3

532

h y

x

2

11516

0 4

2 a y = 0, x = 3 b y = 4, x = 0c y = 0, x = 2 d y = 3, x = 1e y = 4, x = 3 f y = 1, x = 2g y = 6, x = 3 h y = 2, x = 4

Exercise 5C

a

0

3

x

y

x 0 and y 3

b y

0

(2, 3)x

x 2 and y 3c

011

(2, 3)

y

x

x 2 and y 3

d

0

1 + 2(2, 1)

y

x

x 2 and y 1e

0 7

3 2(2, 3)

y

x

x 2 and y 3

f

0

(2, 3)

22 3

14

y

x

x 2 and y 3

g

0 11

(2, 3)

y

x

x 2 and y 3

h y

x0

(4, 2)

x 4 and y 2i

0(4, 1)

y

x

x 4 and y 1

Exercise 5D

1 a x2 + y2 = 9 b x2 + y2 = 16c (x 1)2 + ( y 3)2 = 25d (x 2)2 + ( y + 4)2 = 9e (x + 3)2 + ( y 4)2 = 25

4f (x + 5)2 + ( y + 6)2 = (4.6)2

2 a C(1, 3), r = 2 b C(2,4), r = 5c C(3, 2), r = 3 d C(0, 3), r = 5e C(3,2), r = 6 f C(3,2), r = 2g C(2, 3), r = 5 h C(4,2), r = 19

3 a y

x

8

0

8

8 8

b y

x

4

7

10

c y

x7 2 0 3

d y

x

4

0

1

e y

x

52

32

0

f y

x3

0

g y

x

3

02

h y

x0

11

4



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i y

x3 0 3

j y

x0

5 5

k y

x4 0 2 8

l y

x

(2, 2)

0

4 (x 2)2 + ( y + 3)2 = 95 (x 2)2 + ( y 1)2 = 206 (x 4)2 + ( y 4)2 = 207 Centre (2, 3), radius = 68 2

21 (x-axis), 4

6 (y-axis)

9 a

2 2

2

y

x

2

0

b

1 1

1

y

x

10

c

5

y

x0 55

5

d

3 3

3

y

x

3

0

e

6

y

x0 66

6

f y

x022

22

22

22


1 E 2 B 3 E 4 A 5 A6 D 7 D 8 C 9 E 10 B

Short-answer questions (technology-free)

1 a y

x(1, 3)

0

b y

x(1, 2)

0

c y

x0

(2, 1)

(0, 1)x = 1

d y

x

(0, 3)y = 1

x = 1(3, 0) 0

e y

x0

f y

x(0, 1)

x = 1

0

g y

x

(0, 5)

x = 2

y = 3

0 103

hy

xy = 1

(3, 0) (3, 0)

i

02

y

x

j

0(3, 2)

y

x

k

01

(2, 2)

22 + 2

y

x

2 a (x 3)2 + ( y + 2)2 = 25b

(x 3

2

)2+(

y + 52

)2= 50

4

c

(x 1

4

)2+(

y + 14

)2= 17

8

d (x + 2)2 + ( y 3)2 = 13e (x 3)2 + ( y 3)2 = 18f (x 2)2 + ( y + 3)2 = 13

3 2y + 3x = 0 4 2x + 2y = 1 or y = x 52

5 a (x 3)2 + ( y 4)2= 25

y

x0

(3, 4)

b (x + 1)2 + y2 = 1y

x(1, 0)

0

c (x 4)2 + ( y 4)2= 4

y

x0

(4, 4)

d

(x 1

2

)2+(

y + 13

)2= 1

36y

x1

1

0

12

13

,



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6 a y

x3 30

b y

x5 1 3

0

c y

x1 1

(0, 2)

0

d y

x

(2, 3)

0


1 a (x 10)2 + y2 = 25 c m =

3

3

d P

(15

2,53

2

)e 5

3

2 a x2 + y2 = 16b ii m =

3

3; y =

3

3x 8

3

3,

y =

3

3x + 8

3

3

3 a4

3b

34

c 4y + 3x = 25 d 12512

4 a iy1x1

iix1y1

c

2x + 2y = 8 or 2x + 2y = 8

5 a y =

3

3x + 2

3

3a, y =

3

3x 2

3

3a

b x2 + y2 = 4a26 bii

y = 14

14

x

y

0

x

(14 ,

14

)c i

14

< k < 0 ii k = 0 or k < 14

iii k > 0

7 a 0 < k 1 c x = 67

10 a f : R R, f (x) = 3x + 2b f : R R, f (x) = 3

2x + 6

c f : [0,) R, f (x) = 2x + 3d f : [1, 2] R, f (x) = 5x + 6e f : [5, 5] R, f (x) = x2 + 25f f : [0, 1] R, f (x) = 5x 7

11 a y

x

(2, 4)

(1, 1)

0

Range = [0, 4]

b y

x

(2, 8)

(1, 1)0

2

Range = [1, 8]c y

x0

13

3,

Range =[

1

3,

)

d y

x(1, 2)

0

Range = [2, )

Exercise 6D

1 One-to-one functions are b, d, e and g2 Functions are a, c, d, f and g. One-to-one

functions are c and g.

3 a Domain = R, Range = Rb Domain = R+ {0}. Range = R+ {0}c Domain = R, Range = [1, )d Domain = [3, 3], Range = [3, 0]e Domain = R+, Range = R+f Domain = R, Range = (, 3]g Domain = [2, ), Range = R+ {0}h Domain =

[1

2,

), Range = [0, )

i Domain =(, 3

2

], Range = [0, )

j Domain = R \ { 12}, Range = R \ {0}k Domain = R \ { 12}, Range = (3, )l Domain = R \ { 12}, Range = R \ {2}

4 a Domain = R, Range = Rb Domain = R, Range = [2, )c Domain = [ 4, 4], Range = [ 4, 0]d Domain = R \ {2}, Range = R \ {0}

5 y = 2 x , Domain = (, 2],Range = R+ {0}y = 2 x , Domain = (, 2],Range = (, 0]

6 a y

x

222

b f1: [0,) R, f1(x) = x2 2,f2: (, 0] R, f2(x) = x2 2

Exercise 6E

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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2 a y

x

4

3

2

1

1 2 30

Range = (, 4]

3 y

x

1

2

1 2 3 54

3 2 1

1

2

3

4

5

0

4 ay

x(0, 1)

0

b Range = [1,)

5 a y

x3 3

0

9

b Range = R6 a

y

x

(1, 1)

0

b Range = (, 1]

7 f (x) =

x + 3, 3 x 1x + 1, 1 < x 21

2x, 2 x 4

Exercise 6F

1 a a = 3, b = 12

b 6

2 f (x) = 7 5x3 a i f (0) = 9

2ii f (1) = 3

b 34 a f (p) = 2p + 5 b f (p + h) = 2p + 2h + 5

c 2h d 25 26 b i 25.06 ii 25.032 iii 25.2 iv 267 f (x) = 7(x 2)(x 4)8 f (x) = (x 3)2 + 7, Range = [7, )9 a

(,15

8

]b

[3

7

8,

)c (, 20] d (, 3]

10 a y

x

5

0

(1, 8)

(6, 13)

b Range = [13, 8]11 a y

x

(8, 36)

(1, 9)

0 (2, 0)

b Range = [0, 36]

12 a Domain 3 x 3Range 3 y 3

b Domain 1 x 3Range 1 y 1

c Domain 0 x 1Range 0 y 1

d Domain 1 x 9Range 5 y 5

e Domain 4 x 4Range 2 y 6

13 a {2, 4, 6, 8} b {4, 3, 2, 1}c {3, 0, 5, 12} d {1, 2, 3, 2}

14 f (x) = 110

(x 4)(x 5); a = 110

, b = 910

,

c = 215 f (x) = 2(x 1)(x + 5)

g(x) = 50(x 1)(

x + 15

)

16 a k 100 km


1 B 2 E 3 B 4 C 5 E6 B 7 D 8 E 9 C 10 D

Short-answer questions(technology-free)

1 a 16 b 26 c 23

2 a y

x

(1, 7)(0, 6)

(6, 0)0

b Range = [0, 7]3 a Range = R b Range = [5, 4]

c Range = [0, 4] d Range = (, 9]e Range = (2, ) f {6, 2, 4}g Range = [0, ) h R \ {2}i Range = [5, 1] j Range = [1, 3]

4 a a = 15, b = 332

b Domain = R \ {0}5 a

(1, 1)

0 (2, 0)x

y b [0, 1]

6 a = 3, b = 5 7 a = 12, b = 2, c = 0

8 a R \ {2} b [2,) c [5, 5]d R \

{1

2

}e [10, 10] f (, 4]

9 b, c, d, e, f, g, and j are one-to-one10 a

(0, 1)0

(3, 9)

y

x

b(3, 9)

(0, 1)

0

y

x

11 a f 1(x) = x + 23

, Domain = [5, 13]b f 1(x) = (x 2)2 2, Domain = [2, )c f 1(x) =

x

3 1, Domain = [0, )

d f 1(x) = x + 1, Domain = [0, )12 a y = x 2 + 3 b y = 2x c y = x

d y = x e y =

x

3

Extended-response questions1 a

500400300200100

1 2 3 4 5 6 7

d (km)

t (hour)

Y

Z

0X

Coach starting from X :d = 80t for 0 t 4d = 320 for 4 < t 4 3

4d = 80t 60 for 4 3

4< t 7 1

4Range = [0, 520]Coach starting from Z :

d = 520 104011

t 0 t 5 12

Range = [0, 520]b The coaches pass 238

1

3km from X .

2 a P = 12

n b

2 4 6 8

4

3

2

1

0

P(hours)

n

Domain = {n : n Z, 0 n 200}

2

nRange = : n Z, 0 n 200

3 a T = 0.4683x 5273.4266b

10180.473

8307.27Range = [8307.27, 10180.473]

29 30 31 32 33 x ($000)

T ($)

c $8775.57 (to nearest cent)4 a i C(n) = 1000 + 5n, n > 0

ii C (n)(1000, 6000)

1000

0 n

b i P(n) = 15n (1000 + 5n)= 10n 1000

ii P(n)

n

(1000, 9000)

10001000

5 V = 8000(1 0.05n) = 8000 400n6 a R = (50000 2500x)(15 + x)

= 2500(x + 15)(20 x)b

750 000

(2.5, 765 625)

0 20

R

x

c Price for max = $17.50



P1: FXS/ABE P2: FXS


Answ

ersAnswers 745

7 a A(x) = x4

(2a (6

3)x)

b 0 < x 3}e x 1 f x 1 g x > 4 h x 3

Exercise 7H

1 a y = 18

(x + 2)3 b y 2 = 14

(x 3)3

2 y = 2x(x 2)23 y = 2x(x + 4)24 a y = (x 3)3 + 2b y = 23

18x3 + 67

18x2 c y = 5x3

5 a y = 13

x3 + 43

x b y = 14

x(x2 + 2)6 a y = 4x3 50x2 + 96x + 270b y = 4x3 60x2 + 80x + 26c y = x3 2x2 + 6x 4d y = 2x3 3xe y = 2x3 3x2 2x + 1f y = x3 3x2 2x + 1g y = x3 3x2 2x + 1

Exercise 7I

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 = 5j 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

x4 0 5 6

(0.72, 503.46)480

c y

x

(1.89, 38.27)

03

d y

(3, 27)

4 x0



P1: FXS/ABE P2: FXS


Answ

ers


e y

(3.54, 156.25)(3.54, 156.25)

50x

f y

22

0

16

x

g y

9 9

(6.36, 1640.25) (6.36, 1640.25)

x0

h y

40 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(4.53, 20.25) (4.53, 20.25)

x

k y

0 2x

20

l y

(1.61, 163.71)

(5.61, 23.74)

5047 x

Exercise 7J

1 a f (n) = n2 + 3 b f (n) = n2 3n + 5c f (n) = 1

6n3 + 1

2n2 + 1

3n

d f (n) = 13

n3 + 12

n2 + 16

n

e f (n) = 2n3 52 a f (n) = n2 b f (n) = n(n + 1)

c f (n) = 13

n3 + 12

n2 + 16

n

d f (n) = 43

n3 13

n

e f (n) = 13

n3 + 32

n2 + 76

n

f f (n) = 43

n3 + 3n2 + 53

n

3 f (n) = 12

n2 12

n

4 f (n) = 13

n3 + 12

n2 + 16

n

5 f (n) = 14

n2(n + 1)2

Exercise 7K

1 a l = 12 2x, w = 10 2xb V = 4x(6 x)(5 x)

c

x (cm)10 2 3 4 5

100

(cm3)V d V = 80

e x = 3.56 or x = 0.51f V max = 96.8 cm3 when x = 1.81

2 a x = 64 h2 b V = h3

(64 h2)c

4.62

0

50

100

150

200

(m3)

1 2 3 4 5 6 7 8 h (m)

V d Domain = {h : 0 < h < 8}e 64

f h = 2.48 or h = 6.47g V max 206.37 m3, h = 4.62

3 a h = 160 2xb V = x2(160 2x), Domain = (0, 80)c

(cm3)

(cm)

50000

100000

150000

0 20 40 60 80 x53

V

d x = 20.498 or x = 75.63e V max 151 703.7 cm3 when x 53

Multiple-choice questions1 B 2 D 3 A 4 D 5 A6 C 7 B 8 B 9 D 10 B

Short-answer questions(technology-free)1 a y

x

2 + 13

(1, 2)(0, 3)

0

b

x

y

0

12 , 1

c y

x0 (1, 1)

13 + 13

(0, 4)

d y

x(1, 3)0

e y

x

(1, 4)

(0, 1)

0

313

f y

x

(2, 1)

+ 21330



P1: FXS/ABE P2: FXS


Answ

ersAnswers 749

g y

x(2, 3)

(0, 29) 23430

h y

x(2, 1)

(0, 23) 2133

0

2 a y

0

1

1x

b y

0

2

x , 121

c y

0

(1, 1)

2 x

d y

0 x

e y

0

1

x3

14 3

14

f y

0

15

1 3

(2, 1)

x

g y

0

1

(1, 3)

x 13

2 114

143

2

h y

03 1

(2, 1)

x2 +

12

14

2 12

14

3 a P

(3

2

)= 0 and P(2) = 0, (3x + 1)

b x = 2, 12, 3 c x = 1, 11,+11

d i P

(1

3

)= 0 ii (3x 1)(x + 3)(x 2)

4 a f (1) = 0 b (x 1)(x2 + (1 k)x + k + 1)5 a = 3, b = 246 a

(4, 0) (2, 0) 0 (3, 0)

(0, 24)

6 5 3 1 0 1 2 4

x

y4 2 3

b

(0, 24)

x

4 2 1 0

y

1 5 6

(3, 0) 0 (2, 0) (4, 0)

3 423

c1 1.53 2.5 1.5 0 1 2

x

y

(2, 0) 0

(0, 4)

0.5 0.52

, 0 23

, 0 12

d

x

y

5 4 3 2 1 0 1 4

36

0(6, 0) (2, 0) (3, 0)

6 2 3

7 a 41 b 12 c 439

8 y = 25

(x + 2)(x 1)(x 5)

9 y = 281

x(x + 4)210 a a = 3, b = 8 b (x + 3)(2x 1)(x 1)11 a y = (x 2)3 + 3 b y = 2x3

c y = x3 d y = (x)3 = x3

e y =( x

3

)3= x

3

27

12 a y = (x 2)4 + 3 b y = 2x4c y = (x + 2)4 + 3

13 a Dilation of factor 2 from the x-axis, translationof 1 unit in the positive direction of the x-axis,then translation of 3 units in the positivedirection of the y-axis

b Reection in the x-axis, translation of 1 unit inthe negative direction of the x-axis, thentranslation of 2 units in the positive directionof the y-axis

c Dilation of factor 12 from the y-axis, translationof 12 unit in the negative direction of the x-axisand translation of 2 units in the negativedirection of the y-axis


1 a v = 132 400

(t 900)2

b s = t32 400

(t 900)2

c

t (s)800600400200560105

0

1000

2000

3000(cm)

s Domain = {t : 0 < t < 900}

(300, 3333.3)

d No, it is not feasible since the maximum rangeof the taxi is less than 3.5 km (333 km).

e Maximum speed 2000105

= 19 m/s

Minimum speed 2000560

= 3.6 m/s2 a R 10 = a(x 5)3

b a = 225

c R 12 = 12343

(x 7)3



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Answ

ers


3 a 4730 cm2 b V = l2(2365 l)c

(cm3)

l (cm)

V

5000

10 000

15 000

20 000

10 20 30 40 500

d i l = 23.69 or l = 39.79ii l = 18.1 or l = 43.3

e V max 17 039 cm3, l 32.42 cm4 a a = 43

15 000, b = 0.095, c = 119

150,

d = 15.8b i Closest to the ground (5.59, 13.83),

ii furthest from the ground (0, 15.8)

5 a V = (96 4x)(48 2x)x= 8x(24 x)2

b

0 24

V

x

i 0 < x < 24ii Vmax = 16 384 cm3 when x = 8.00

c 15 680 cm3 d 14 440 cm3 e 9720 cm3

Chapter 8

Exercise 8A1 The lines are parallel.2 y = , x = 6 3 m = 44 a m = 5 b m = 35 a i m = 2 ii m = 4

b x = 4m+2 , y =

2(m + 4)m+2 m = 2

6 a x = 2, y = 3, z = 1b x = 3, y = 5, z = 2c x = 5, y = 0, z = 7

7 x = 6, y = 5, z = 18 x = 10 3w

2, y = 3(w + 2)

2, z = 2w + 2;

if w = 6 solution is (4,12, 14)9 a = 1, b = 2 and c = 3

10 b = 1, c = 2 and d = 511 x = 5, y = 2 and z = 112 b = 2, c = 0 and d = 3

Exercise 8B

1 a

[83

]b

[3a ba + 3b

]2 (1, 0) (2,4), (0, 1) (1, 3),

(3, 2) (4,6)3 a (2, 1), (4, 1) b (2, 0), (2, 2)

c (2, 3), (4,5)

4 a (6, 21) b (12, 7) c (6,7)d (6, 7) e (7, 6)

5 a

[2 33 1

]c

[1 21 2

]

6

[1 00 2

]

7 a

[1 00 1

]b

[0 11 0

]c

[0 1

1 0]

d

[1 00 2

]e

[3 00 3

]f

[3 00 1

]

8 a T =[

0 22 0

]b (4,6)

c a = 1 and b = 3.

9

[2 00 1

]X +

[34

]= X

X = 12

[1 00 2

](X

[34

]),

x = 12

(x 3), y = y 4

101

2

[ 1 121 5

]

Exercise 8C

1 a3

mb m 3 c f 1(x) = x + 3

m

d

(3

m 1 ,3

m 1)

e my + x = 3m

2 a c2

b c 2 c f 1(x) = x c2

d (c,c) e y = 12

x + c

3 a x = 0 and x = b b(b

2,b2

4

)c i (0, 0) and (1 b, 1 b)ii b = 1 iii b = 1

4 a

(2, 0)

b y = a and y = ac

(a2 + 8a 16 + a + 4

2,

a2 + 8a 16 + a 4

2

)and(

a2 + 8a 16 a 42

,

a2 + 8a 16 a 42

)



P1: FXS/ABE P2: FXS


Answ

ersAnswers 751

Exercise 8D

1 y = 2x2

9 2x

3 4 2 y = x

3

32 x

3 y = 3x + 184

4 y = x4

4

5 y = x4

+ 3 6 y = x + 214

7 y = 3x3

4 9x

2

2+ 14

8 y = 3x3

4 9x

2

2+ 8

9 a d = 1, a + b + c + d = 1,8a + 4b + 2c + d = 1,27a + 9b + 3c + d = 5

b

0 0 0 11 1 1 18 4 2 127 9 3 1

abcd

=

1115

c a = 1, b = 4, c = 5, d = 1d y = 2x3 + 8x2 10x + 2

10 a a = 2, b = 0, c = 4, d = 0b y = 2x3 + 4x


1 E 2 B 3 B 4 E 5 D6 B 7 C 8 C 9 D 10 C


1 a

[1 00 4

](1, 12) b

[3 00 1

](3, 3)

c

[1 00 1

](1,3) d

[1 00 1

](1, 3)

e

[0 11 0

](3,1)

2 x = 4, y = 1 and z = 73 a y = 2x + 2.

b i2a

ii 2 < a < 0

c

(1

a 1 ,1

a 1 + 3)

4 a a

(x + 1

a

)2+ a 1

ab

(1a

, a 1a

)c a = 1 d 1 < a < 1

5

[1 00 2

] [xy

]+[

23

]=[

x

y

], x = x 2 and

y = y 32

6

[3 00 1

] [xy

]+[2

3

]=[

x

y

], x = x

+ 23

and y = y 3

Extended-response questions1 a h = 1 22 b a = 22

c a = 8, b = 16

2 a 4b 5c d = 41, 2b 7c d = 53,4b + 3c d = 25

b x2 + y2 2x 4y 29 = 03 a c = b 8

b x = 0 or x = bc i y = 0 or y = b + 8 ii b = 8

4 a x a b(

4a + 1 12

,

4a + 1 1

2

)c a = 2 d a = 6 e a = c2 + c

5 a y + 5z = 15 and y + 5z = 15.b This indicates the solution is going to be a

straight line.c y = 5 15d x = 43 13

6 a y = 2 + 4zb x = 8 5, y = 2 + 4, z = R

7 u = ba

, v = ca

Chapter 99.1 Multiple-choice questions1 A 2 D 3 D 4 C 5 B6 C 7 A 8 E 9 B 10 A

11 E 12 B 13 D 14 D 15 E16 B 17 D 18 E 19 D 20 B21 D 22 D 23 A 24 B 25 D26 D 27 B 28 C 29 A 30 C31 A 32 B 33 C 34 D 35 E36 E 37 C 38 C 39 C 40 A41 A 42 B 43 E 44 A 45 B46 C

9.2 Extended-response questions1 a C = 3500 + 10.5x b I = 11.5x

c

x

CI = 11.5x

C = 3500 + 10.5x3500

35000

I and d 3500

e Prot

x

P

3500

3500

P = x 3500f 5500

2 a V = 45 000 + 40m b 4 hours 10 minutesc

V

m (minutes)

(litres)

250

55000

45000



P1: FXS/ABE P2: FXS


Answ

ers


3 a 200 L

b V ={

20t 0 t 1015t + 50 10 < t 190

3c

t (minutes)

(litres)

V

10

(63.3, 1000)

200

4 a Ar = 6x2 b As = (10.5 2.5x)2c 0 x 4.2d AT = 12.25x2 52.5x + 110.25e AT

x

(4.2, 105.84)

157

, 54

110.25

f 110.25 cm2 (area of rectangle = 0)g rectangle: 9 6, square: 3 3, (x = 3) or

rectangle:27

7 18

7; square:

51

7 51

75 a 20 m b 20 m c 22.5 m6 a A = 10x2 + 28x + 16b i 54 cm2 ii 112 cm2

c 3 cmd

x0

16

A

e V = 2x3 + 8x2 + 8xf x = 3 g x = 6.66

7 a i A = (10 + x)y x2ii P = 2( y + x + 10)

b i A = 400 + 30x 2x2ii 512

1

2m2 iii 0 x 20

iv

x (m)

(cm2)

A 12

12 7 , 512

(0, 400)

(20, 200)

0

8 a A = 6x2 + 7xy + 2y2c i x = 0.5 m ii y = 0.25 m

9 a 50.9 m b t = 6.12 secondsc h(t)

t6.2850

5

(3.06, 50.92)d 6.285 seconds

10 a x + 5 b V = 35x + 7x2c S = x2 + 33x + 70d

t

V V = 35x + 7x2

S = x2 + 33x + 70

0

70

and S

e 3.25 m f 10 cm11 a 2y + 3x = 22

b i B(0, 11) ii D(8, l)c 52 units2 d 6.45 units

12 a 25 km/h b tap A 60 min; tap B 75 minc 4 cm

13 a h = 100 3x b V = 2x2(100 3x)c 0 < x 2) c Pr(X 2)d Pr(X < 2) e Pr(X 2) f Pr(X > 2)g Pr(X 2) h Pr(X 2) i Pr(X 2)j Pr(X 2) k Pr(2 < X < 5)

3 a {2} b {3, 4, 5} c {2, 3, 4, 5}d {0, 1} e {0, 1, 2} f {2, 3, 4, 5}g {3, 4, 5} h {2, 3, 4} i {3, 4}

4 a1

15b

3

55 a 0.09 b 0.69

6 a 0.49 b 0.51 c 0.74

7 a 0.6 b 0.47 c2

38 a {HHH, HTH, HHT, HTT, THH, TTH,

THT, TTT}b

3

8c x 0 1 2 3

p(x)1

8

3

8

3

8

1

8

d7

8e

4

7

9 a {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} b1

6c

y 2 3 4 5 6 7 8 9 10 11 12

p(y)1

36

2

36

3

36

4

36

5

36

6

36

5

36

4

36

3

36

2

36

1

36



P1: FXS/ABE P2: FXS

9780521740524ans-ch8.xml CUAU068-EVANS August 19, 2011 4:48

Answ

ers


10 a {1, 2, 3, 4, 5, 6} b7

36c

1 2 3 4 5 6

1

36

3

36

5

36

7

36

9

36

11

36

11 a 0.09 b 0.4 c 0.5112 a

y 3 2 1 3

p(y)1

8

3

8

3

8

1

8

b7

8

Exercise 13B

1 0.378 228

57 0.491 3 12

13 0.923

460

253 0.237 5 0.930 6 0.109

Exercise 13C

1 a 0.185 b 0.060 2 a 0.194 b 0.9303 a 0.137 b 0.446 c 0.5544 a 0.008 b 0.268 c 0.4685 a 0.056 b 0.391 6 0.018

7 a Pr(X = x) =(

5x

)(0.1)x (0.9)5x

x = 0, 1, 2, 3, 4, 5 orx 0 1 2 3 4 5

p(x) 0.591 0.328 0.073 0.008 0.000 0.000

b Most probable number is 08 0.749 9 0.021 10 0.5398 11

175

25612 a 0.988 b 0.9999 c 8.1 101113 a 0.151 b 0.302 14 5.8%15 a i 0.474 ii 0.224 iii 0.078

b Answers will vary about 5 or more.16 0.014 17

18 19 a 5 b 820 a 13 b 2221 a 16 b 2922 a 45 b 59

23 a 0.3087 b0.3087

1 (0.3)5 0.309524 a 0.3020 b 0.6242 c 0.3225

Exercise 13D

1 Exact answer 0.1722 a About 50 : 50b One set of simulations gave the answer 1.9

Exercise 13E2 Exact answer 29.293 a One set of simulations gave the answer 8.3.b One set of simulations gave the answer 10.7.

4 Exact answer is 0.0009.5 a One set of simulations gave the answer 3.5.

Multiple-choice question

1 B 2 A 3 C 4 A 5 E6 C 7 A 8 D 9 B 10 E

Short-answer questions(technology-free)1 a 0.92 b 0.63 c 0.82

x 1 2 3 4

p(x) 0.25 0.28 0.30 0.17

3x 2 3 4

p(x)2

5

8

15

1

15

4 a 1st choice2nd choice 1 2 3 6 7 9

1

2

3

6

7

9

2

3

4

7

8

10

3

4

5

8

9

11

4

5

6

9

10

12

7

8

9

12

13

15

8

9

10

13

14

16

10

11

12

15

16

18

b { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 }c

x 2 3 4 5 6 7

Pr(X = x) 136

2

36

3

36

2

36

1

36

2

36

x 8 9 10 11 12 13

Pr(X = x) 436

4

36

4

36

2

36

3

36

2

36

x 14 15 16 18

Pr(X = x) 136

2

36

2

36

1

36

5 a 0.051 b 0.996 c243

256 0.949

6 a9

64b

37

647 a 0.282 b 0.377 c 0.3418 a 0.173 b 0.756 c 0.071

9 a( p

100

)15b 15

( p100

)14(1 p

100

)



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Answ

ersAnswers 759

c( p

100

)15+ 15

( p100

)14(1 p

100

)+ 105

(1 p

100

)2 ( p100

)1310 a

117

125b m = 5

Extended-response questions1 a

x 1 2 3 4

p(x) 0.54 0.16 0.06 0.24

b 0.46

2 a i 0.1 ii 0.6 iii2

3b i 0.0012 ii 0.2508

3 a3

5

b i7

40ii

3

10

c i11

40ii

11

174 a 0.003 b 5.320 1065 0.8 6 0.9697 a 0.401 b n 458 a 1 q2 b 1 4q3 + 3q4 c 1

3< q < 1

9 0.966 (exact answer)10 a 0.734 (exact answer)

b About 7 (by simulation)

11 a13

8b 3.7

12 b Pr(A) = 0.375, Pr(B) = 0.375,Pr(C) = 0.125, Pr(D) = 0.125(exact answer)

Chapter 1414.1 Multiple-choice questions

1 E 2 C 3 E 4 B 5 E6 E 7 C 8 C 9 B 10 D

11 D 12 D 13 E 14 A 15 E16 E 17 B 18 C 19 C 20 A21 E 22 E 23 C 24 D 25 D26 D 27 A 28 E 29 C

14.2 Extended-response questions

1 a i15

28ii

37

56iii

43

49

b i9

14ii

135

392

2 a1

2b

13

36

3 a3

8b

1

56c

3

28d

6

74 a 0.0027 b 0.12 c 0.17 d 0.72

5 a59

120b

45

59

6 a167

360b i

108

193ii

45

193

7 a i1

9ii

5

18

b i1

81ii

13

3248 a i m = 30, q = 35, s = 25

ii m + q = 65b

3

10c

7

129 a 0.084 b 0.52 c 0.68

10 a 60 b 8 c 0.1

11 a1

60b

1

5c

3

5d

6

1312 a i 10 000 cm2 ii 400 cm2 iii 6400 cm2

b i 0.04 ii 0.12 iii 0.64c i 0.0016 ii 0.000 64

13 a7

18b

13

36c

23

10814 a i 0.328 ii 0.205 iii 0.672

b i 11 ii 1815 a i 0.121 ii 0.851 iii 0.383

b i 9 ii 14

16 a20

81b

1

9c i

5

12ii

7

18d 0.6

Chapter 15

Exercise 15A

1

x

y

y = 2.4x

y = 1.8x1

0

y = 0.5x

y = 0.9x

All pass through (0, 1)base > 1, increasingbase < 1, decreasinghorizontal asymptote, y = 0

2y

y = 5 3x

y =

y = 2 3x

y =

5

20

2 3x

5 3x

For y = a bxy-axis intercept (0, a)c and d are reections of a and b in the x-axishorizontal asymptote, y = 0

3

(3.807, 14)

x

y

y = 2x

0

14

x = 3.807

4

(0.778, 6)

x

y

y = 10x

0

6

x = 0.778



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5 a y

0

5 y = 2

x

b y

0

y =

x

3

c y

0

(0, 3)y = 2

x

d y

y = 2

0x

e y

0

(0, 3)

x

fy

y = 2

0

(0, 4)

x

6 a y

0

2x

b y

0

1x

c y

0

1x

d y

y = 2

0

1x

Exercise 15B

1 a x5 b 8x7 c x2 d 2x3 e a6 f 26

g x2 y2 h x4 y6 ix3

y3j

x6

y4

2 a x9 b 216 c 317 d q8 p9

e a11b3 f 28x18 g m11n12 p2 h 2a5b2

3 a x2 y3 b 8a8b3 c x5 y2 d9

2x2 y3

4 a1

n4 p5b

2x8z

y4c

b5

a5d

a3b

c

e an + 2 bn + 1 cn 1

5 a 317n b 23 n c34n 11

22

d 2n + 133n 1 e 53n 2 f 23x 3 34g 36 n 25n h 33 = 27 i 6

6 a 212 = 4096 b 55 = 3125 c 33 = 27

Exercise 15C

1 a 25 b 27 c1

9d 16 e

1

2f

1

4g

1

25

h 16 i1

10 000j 1000 k 27 l

3

5

2 a a16 b

76 b a6b

92 c 3

73 5 76

d1

4e x6 y8 f a

1415

3 a (2x 1)3/2 b (x 1)5/2 c (x2 + 1)3/2d (x 1)4/3 e x(x 1) 12 f (5x2 + 1)4/3

Exercise 15D

1 a 3 b 3 c1

2d

3

4e

1

3f 4 g 2 h 3 i 3

2 a 1 b 2 c 32

d4

3e 1 f 8 g 3

h 4 i 8 j 4 k 3 12

l 6 m 71

2

3 a4

5b

3

2c 5

1

24 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

3c x 1

2d x < 3

e x 1 g x 3

Exercise 15E

1 a log2 (10a) b 1 c log2

(9

4

)d 1

e log5 6 f 2 g 3 log2 a h 92 a 3 b 4 c 7 d 3 e 4 f 3 g 4h 6 i 9 j 1 k 4 l 2

3 a 2 b 7 c 9 d 1 e5

2f logx a5 g 3 h 1

4 a 2 b 27 c1

125d 8 e 30

f2

3g 8 h 64 i 4 j 10

5 a 5 b 32.5 c 22 d 20

e3 + 17

2f 3 or 0

6 2 + 3a 5c2

8 10

9 a 4 b6

5c 3 d 10 e 9 f 2

Exercise 15F

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 < 1.46 c x < 1.15d x 2.77 e x 1.31

3 a y

02

3

x

y = 4

b y

0 14

x

y = 6



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ersAnswers 761

c y

02

0.2218

x

y = 5

log105

3

d y

0

log102

2

x

y = 4

0.3010

e y

3

01 x

y = 6

f y

0

1

x

0.2603log26

5

y = 6

4 d0 = 41.88, m = 0.094

Exercise 15G

1 a y

x

Domain = R+

Range = R

0.301

0 12

1

b y

x

Domain = R+

Range = R

0.602

0 1 2

c y

x

Domain = R+

Range = R

0.301

0 1 2 3 4

d y

x

Domain = R+

Range = R

0.954

0 113

e y

x

Domain = R+

Range = R

0 1

f y

x

Domain = R+

Range = R

01

2 a y = 2 log10 x b y = 1013 x

c y = 13

log10 x d y =1

310

12 x

3 a y = log3 (x 2) b y = 2x + 3

c y = log3(

x 24

)d y = log5 (x + 2)

e y = 13

2x f y = 3 2xg y = 2x 3 h y = log3

(x + 2

5

)

4 a y

0 5x

x = 4

Domain = (4, )

by

0

log23

2x

x = 3

Domain = (3, )

c y

0 12

x

Domain = (0, )

d y

0x =

1

1x

2

Domain = (2, )

e y

0 3x

Domain = (0, )

f y

012

x

Domain = ( , 0)5 a 0.64 b 0.40

6 yy = log10 (x2)

0 11x

y

y = 2log10 x

0 1x

7 y

x0

y = log10 x = log10 x for x (0, 10]12

8 yy = log10 (2x) + log10 (3x)

0 16

x

y

y = log10 (6x2)

016

x

16

9 a = 6(103

) 23

and k = 13

log10(

103

)

Exercise 15H

1 y = 1.5 0.575x 2 p = 2.5 1.35t3 a

Total thickness,Cuts, n Sheets T (mm)

0 1 0.21 2 0.42 4 0.83 8 1.64 16 3.25 32 6.46 64 12.87 128 25.68 256 51.29 512 102.410 1024 204.8



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b T = 0.2(2)nc T

n0 2 4 6 8 10

200

150

100

500.2

d 214 748.4 m4 a p, q

(millions)

t0

y = p(t)

y = q(t)

1.71.2

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


1 C 2 A 3 C 4 C 5 A6 B 7 A 8 A 9 A 10 A


1 a a4 b1

b2c

1

m2n2

d1

ab6 e3a6

2f

5

3a2

g a3 hn8

m4i

1

p2q4

j8

5a11k 2a l a2 + a6

2 a log2 7 b1

2log2 7 c log10 2

d log10

(7

2

)e 1 + log10 11 f 1 + log10 101

g1

5log2 100 h log2 10

3 a 6 b 7 c 2 d 0e 3 f 2 g 3 h 4

4 a log10 6 b log10 6 c log10

(a2

b

)

d log10

(a2

25 000

)e log10 y f log10

(a2b3

c

)5 a x = 3 b x = 3 or x = 0

c x = 1 d x = 2 or x = 36 a

x0

y = 2.2x

(1, 4)

(0, 2)

yb y

x0

(0, 3)

y = 3.2x

c

x

y = 5.2x

y

0

d y

x0

(0, 2)y = 2x + 1

y = 1

e

x0

y = 2x 1

y = 1

y f y

x0

(0, 3) y = 2

7 a x = 19 x = 3 10 a k = 1

7b q = 3

211 a a = 1

2b y = 4 or y = 20


1 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

2 n = 23 a

(1

2

)3nb

(1

2

)5n 2c n = 3

4 a 729

(1

4

)nb 128

(1

2

)nc 4 times

5 a Batch 1 = 15(0.95)n Batch 2 = 20(0.94)nb 32 years

6 a X $1.82 Y $1.51 Z $2.62b X $4.37 Y $4.27 Z $3.47c Intersect at t = 21.784 . . . and

t = 2.090 . . . therefore February 1997until September 1998



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ersAnswers 763

d February 1998 until September 1998,approximately 8 months.

7 a 13.81 years b 7.38 years8 a temperature = 87.065 0.94tb i 87.1 ii 18.56

c temperature = 85.724 0.94td i 85.72 ii 40.82

e 28.19 minutes9 a a = 0.2 and b = 5b i z = x log10 b ii a = 0.2 and k = log10 5

10 a y = 2 1.585x b y = 2 100.2xc x = 5 log10

( y2

)

Chapter 16

Exercise 16A

1 a

3b

4

5c

4

3d

11

6e

7

3f

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 2.81e 1.47 f 3.98 g 2.39 h 5.74

5 a 60 b 720 c 540 d 180e 300 f 330 g 690 h 690

6 a 2 b 3 c 43

d 4 e 116

f 76

Exercise 16B

1 a 0, 1 b 1, 0 c 1, 0 d 1, 0e 0, l f 1, 0 g 1, 0 h 0, 1

2 a 0.95 b 0.75 c 0.82 d 0.96e 0.5 f 0.03 g 0.86 h 0.61

3 a 0; 1 b 1; 0 c 1; 0 d 1; 0e 1; 0 f 0; 1 g 0; 1 h 0; 1

Exercise 16C

1 a 0 b 0 c undenedd 0 e undened f undened

2 a 34.23 b 2.57 c 0.97d 1.38 e 0.95 f 0.75 g 1.66

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

Exercise 16D

1 a 6759 b 4.5315 c 2.5357d 6.4279 e 5012 f 3.4202g 2.3315 h 6.5778 i 6.5270

2 a a = 0.7660, b = 0.6428b c = 0.7660, d = 0.6428c i cos 140 = 0.76604, sin 140 = 0.6428

ii cos 140 = cos 40

Exercise 16E

1 a 0.42 b 0.7 c 0.42 d 0.38e 0.42 f 0.38 g 0.7 h 0.7

2 a 120 b 240 c 60d 120 e 240 f 300

3 a5

6b

7

6c

11

6

4 a a = 12

b b =

3

2c c = 1

2

d d =

3

2e tan ( ) = 3

f tan () = 3

5 a

3

2b

1

2c 3 d

3

2e 1

2

6 a 0.7 b 0.6 c 0.4 d 0.6e 0.7 f 0.7 g 0.4 h 0.6

Exercise 16F

1 a sin =

3

2, cos = 1

2, tan =

3

b sin = 12, cos = 1

2, tan = 1

c sin = 12, cos =

3

2, tan = 1

3

d sin =

3

2, cos = 1

2, tan =

3

e sin = 12, cos = 1

2, tan = 1

f sin = 12, cos =

3

2, tan = 1

3

g sin =

3

2, cos = 1

2, tan =

3

h sin = 12, cos = 1

2, tan = 1

i sin =

3

2, cos = 1

2, tan =

3

j sin =

3

2, cos = 1

2, tan =

3

2 a

3

2b 1

2c 1

3d 1

2e 1

2

f

3 g

3

2h

12

i 13

3 a

3

2b 1

2c

13

d not dened

e 0 f 12

g12

h 1



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Exercise 16G

1 Period Amplitude

a 2 2

b 3

c2

3

1

2

d 4 3

e2

34

f

2

1

2

g 4 2

h 2 2

i 4 3

2 a y

x

3

0

3

2

Amplitude = 3, Period =

b y

x

2

0

2Amplitude = 2, Period =

2

32

3

43

2

c y

4

0

4

432

Amplitude = 4, Period = 4

d y

x02

,

32

32

212

1

21

3

4

Amplitude = Period =

e y

x02


4

3

2

3

2

3

4

f y

x0

5

5

2


43

23

45

47

4

2

g y

x432

0

3

3


h y

x0

2

2


43

87

83

85

8

2

4

2

i y

x0

2

2

6


3

2

3

2

9

3 a

x

y

0

1

22

1

2

323

2

2

b

x

y

0

2

2

6336

c

x

y

2

2

023

235

34

65

67

23

611

6

2

3

d

x

y

2

2

0 2

3

2

3

43



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ersAnswers 765

4

x

y

03

2,2

3

4

5

2

5

2

5

5 a dilation of factor 3 from the x-axisamplitude = 3, period = 2

b dilation of factor 15 from the y-axis

amplitude = 1, period = 25

c dilation of factor 3 from the y-axisamplitude = 1, period = 6

d dilation of factor 2 from the x-axisdilation of factor 15 from the y-axis

amplitude = 2, period = 25

e dilation of factor 15 from the y-axisreection in the x-axis

amplitude = 1, period =25

f reection in the y-axisamplitude = 1, period = 2

g dilation of factor 3 from the y-axisdilation of factor 2 from the x-axisamplitude = 2, period = 6

h dilation of factor 2 from the y-axisdilation of factor 4 from the x-axisreection in the x-axisamplitude = 4, period = 4

i dilation of factor 3 from the y-axisdilation of factor 2 from the x-axisreection in the y-axisamplitude = 2, period = 6

6 a y

0 1 2

2

2

x12

34

b y

0 21

3

3

x12

34

7 y

x

y = sin x y = cos x

20

b

4,

5

4

Exercise 16H

1 a y

2

3

0

3

23

25

2

Period = 2, Amplitude = 3, y = 3b

y

2

1

0

1

Period = , Amplitude = 1, y = 1c

y

2

0

2

12

5

12

13

Period = 23

, Amplitude = 2, y = 2

d

y

0

3

3

2

3

2

Period = , Amplitude = 3, y = 3

e

x

y

0

3

3

2

Period = , Amplitude = 3, y = 3, 3

f

y

0

2

2

12

4

12

5

Period = 23

, Amplitude = 2, y = 2, 2



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g

y

0

2

2

6

53

4

3

Period = , Amplitude = 2, y = 2,2

h

x

y

3

3

0

2

Period = , Amplitude = 3, y = 3, 3

i

y

3

0

3

2

2

Period = , Amplitude = 3, y = 3,3

2 a f (0) = 12

f (2) = 12

b y

x0

1

1

, 1

2,0,

3

21

21

, 134

65

611

3 a f (0) =

3

2f (2) =

3

2b y

x0

1

1 2,

2 3

65

34

6113

23

4 a f () = 12

f () = 12

b y

x0

1

1

20, 1

2,

1

2,

1

5 a y = 3 sin x2

b y = 3 sin 2x

c y = 2 sin x3

d y = sin 2(

x 3

)e y = sin 1

2

(x +

3

)

Exercise 16I

1 a5

4and

7

4b

4and

7

4

2 a 0.93 and 2.21 b 4.30 and 1.98c 3.50 and 5.93 d 0.41 and 2.73e 2.35 and 3.94 f 1.77 and 4.51

3 a 150 and 210 b 30 and 150 c 120 and 240d 120 and 240 e 60 and 120 f 45 and 135

4 a 0.64, 2.498, 6.93, 8.781

b5

4,

7

4,

13

4,

15

4

c

3,

2

3,

7

3,

8

3

5 a3

4,3

4b

3,

2

3c

2

3,2

3

6

x

y

,

0

1

1

35

21

, 34

21

, 32

21

, 32

21

, 34

21

,35

21

,3

21

,3

21

7 a7

12,

11

12,

19

12,

23

12

b

12,

11

12,

13

12,

23

12

c

12,

5

12,

13

12,

17

12

d5

12,

7

12,

13

12,

15

12,

21

12,

23

12

e5

12,

7

12,

17

12,

19

12

f5

8,

7

8,

13

8,

15

88 a 2.034, 2.678, 5.176, 5.820b 1.892, 2.820, 5.034, 5.961c 0.580, 2.562, 3.721, 5.704d 0.309, 1.785, 2.403, 3.880, 4.498, 5.974

Exercise 16J

1 a

x

y

3

10

16

7

6

11



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b

x

y

2 3

2 33

0

6

7

3

6

3

4

c y

x

(0, 1 + 2)

1 20

24

3

4

5

d y

x0

2

4

4

4

5

e y

x

1 + 2

1 20 (2, 0)

2

3

2 a y22

x

2

4

0(2, 2)

(2, 2)

6

11

6

7

6

5

2

3

2

6

6

2

b

x2

(2, 1.414) (2, 1.414)2

2

0

2

y

12

23

4

12

12

512

912

1312

1712

2112

7

12

11

12

15

12

19

c

x

y

2 20

1

3

5

(2, 3)(2, 3)

d

x

(2, 3)

2

(2, 3)

2

y

3

01

1

3

5

3

4

3

2

3

2

3

5

3

4

3

3

e

(2, 2)2

(2, 2)2

y

x0

23

2

3

6

5

6

116

7

2

3

2

6

2

f

2

1 + 3

(2, 1 + 3)

10

3

x2

(2, 1 + 3)

y

12

19

12

7

4

5

12

5

4

7

12

17

4

3

4

3 a

10

3

x

(, 1 + 3) (, 1 + 3)1 + 3

y

127

4

125

43 b

10

3

x(, 3 + 1) (, 3 + 1)3 + 1

y

4

12

12

11

4

3



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c

0

3

x

(, 3) (, 3)

3 2

2 + 31 + 3

y

6

65

32

Exercise 16K

1 a 0.6 b 0.6 c 0.7 d 0.3 e 0.3f

10

7(1.49) g 0.3 h 0.6 i 0.6 j 0.3

2 a

3b

3c

5

12d

14

3 sin x = 45

and tan x = 43

4 cos x = 1213

and tan x = 512

5 sin x = 2

6

5and tan x = 26

Exercise 16L

1 a

4b

3

2c

2

2 ay

0x

2

34

2

x =34

x =4

x =4

x =

b

0x

y

56

x =56

23

23

x =

3

2

6

x =

3

x =6

x =2

x =

c y

0x

23

56

x =5

6x =

2

6

x = x =6

x =2x =

23

3

3

3 a7

8,3

8,

8,

5

8

b17

18,11

18,518

,

18,

7

18,

13

18

c5

6,3

,

6,

2

3

d13

18,718

,18

,5

18,

11

18,

17

18

4 a y

0x

6

5

6

2x =

2x =

(, 3)3

(, 3)

b y

0

2x

43

4

2x =

2x =

(, 2)(, 2)

c y

0(0, 3) (, 3)(, 3)

x

2

4

34

Exercise 16M

1 a 0.74 b 0.51c 0.82 or 0.82 d 0 or 0.88

2 y = a sin (b + c) + da a = 1.993 b = 2.998 c = 0.003

d = 0.993b a = 3.136 b = 3.051 c = 0.044

d = 0.140c a = 4.971 b = 3.010 c = 3.136

d = 4.971

Exercise 16N

1 a x = (12n + 1)6

or x = (12n + 5)6

b x = (12n 1)18

c x = (3n + 2)3

2 a x = 6

or x = 56

b x = 18

or x = 1118

c x = 23

or x = 53



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Answ

ersAnswers 769

3 x = n or x = (4n 1)4

;

x = 54

, , 4

, 0,3

4, or

7

4

4 x = n3

; x = , 23

, 3

, 0

5 x = 6n 112

or x = 3n + 26

;

x = 23, 7

12, 1

6, 1

12,

1

3,

5

12,

5

6,

11

12

Exercise 16O

1 a

0 3 6 12 18 24 t

D13107

b {t : D(t) 8.5} = {t : 0 t 7} {t : 11 t 19} {t : 23 t 24}

c 12.9 m2 a p = 5, q = 2b

0 6 12 t

D

753

c A ship can enter 2 hours after low tide.3 a 5 b 1

c t = 0.524 s, 2.618 s, 4.712 sd t = 0 s, 1.047 s, 2.094 se Particle oscillates about the point x = 3 from

x = 1 to x = 5.


1 C 2 D 3 E 4 C 5 E6 D 7 E 8 E 9 C 10 B


1 a11

6b

9

2c 6 d

23

4e

3

4

f9

4g

13

6h

7

3i

4

92 a 150 b 315 c 495 d 45 e 1350

f 135 g 45 h 495 i 1035

3 a12

b12

c 12

d

3

2

e

3

2f 1

2g

1

2h 1

2

4Amplitude Period

a 2 4

b 3

2

c1

2

2

3

d 3

e 4 6

f2

33

5 a2

0

2

y = 2sin 2x

y

x

b y

x0

3

3

point (6, 3)is the f inal point

3 6

y = 3cosx

3

c2

0

2

3

2

3

y

x

d

63

2

2

0x

y

e1

0

1

4

5

4

49

45

4

x

y

y = sin x

passes through

f

y

x

1y = sin x +

2

3

2

3

0

1 3

4

3

g y

x

y = 2cos x 56

0

2

2 34

6

5

6

14

6

17

611

h y

x6

43

116

56

3

3

3

0

6 a 23

,3

b 3

,6

,2

3,

5

6

c

6,

3

2d

7

6e

2,

7

6

Extended-response questions1 a i 1.83 103 hours

ii 11.79 hoursb 26 April (t = 3.86) 14 August (t = 7.48)

2 a 19.5 C b D = 1 + 2 cos(t

12

)c

0 24 t

D

6 12 18

3

2

1

1

d {t : 4 < t < 20}



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3 a

0 6 12 18 24 t (hours)

1.2

3

4.8

(m)d

b 3.00 am 3.00 pm 3.00 amc 9.00 am 9.00 pm d 10.03 ame i 6.12 pm ii 5 trips

4 b

0 8 16 t (hours)

D(m)

6

4

2

c t = 16 (8.00 pm)d t = 4 and t = 12 (8.00 am and 4.00 pm) depth

is 4 me i 1.5 m ii 2.086 mf 9 hours 17 minutes

Chapter 1717.1 Multiple-choice questions

1 B 2 B 3 B 4 E 5 D 6 A7 D 8 C 9 B 10 A 11 A 12 D

13 A 14 D 15 D 16 D 17 A 18 E19 D 20 D 21 E 22 A 23 E 24 B25 D 26 B

17.2 Extended-response questions

1 a

0 12 24 t (hours)

h (m)14

10

6

h = 10

b t = 3.2393 and t = 8.7606c The boat can leave the harbour for

t [0.9652, 11.0348]2 a 40 bacteriab i 320 bacteria ii 2560 bacteria

iii 10 485 760 bacteriac

N

0 t (hours)

(0, 40)(2, 320)

(4, 2560)

d 40 minutes,

(= 2

3hours

)3 a 60 secondsb

y = 11

0 10 40 60 t (s)

h(m)20

11

2

c [2, 20]

d After 40 seconds and they are at this heightevery 60 seconds after they rst attain thisheight.

e At t = 0, t = 20 and t = 60 for t [0, 60]4 a V

120

120

1 t (s)60

130

b t = 1180

s c t = k30

s, k = 0, 1, 25 a i Period = 15 seconds

ii amplitude = 3 iii c = 215

b h = 1.74202c

1

2

5(metres)

hh(t) = 2 + 3sin

2 (t 1.7420)15

15 300 t (min)

6 a i 30 ii 49.5 iii 81.675b 1.65 c 6.792

d h(hectares)

t (hours)0

(0, 30)(1, 49.5)

h (t) = 30(1.65)t

7 at 0 1 2 3 4 5

100 60 40 30 25 22.5

b

0 t (min)

100(C)

c 1 minute d 27.0718 a PA = 70 000 000 + 3 000 000t

PB = 70 000 000 + 5 000 000tPC = 70 000 000 1.3

t10

b

70000000

PCPB

PA

i

ii

t

P

c i 35 years ii 67 years9 a i 4 billion ii 5.944 billion iii 7.25 billionb 2032

10 a V1(0) = V2(0) = 1000b

(25, 82.08)

V(litres)

1000

0 25 t

c 82.08 litres d t = 0 and t = 22.32



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11 a

y = h1(t)(6, 44)

h

(m)

18

28

8

0

(6, 18)

6 t (hours)

b t = 3.31 Approximately 3.19 am (correct tonearest minute)

c i 9.00 amd 8 + 6t metres

Chapter 18

Exercise 18A

Note: For questions 14 there may not be a singlecorrect answer.

1 C and D are the most likely. Scales shouldcome into your discussion.

2 height(cm)

Age (years)

3speed

(km/h)

0 1 1.25 distance from A (km)

4 C or B are the most likely.

5 a

time (seconds)10

100

0

distance(metres)

b

time (seconds)

speed(m/s)

10

10

0

6 a volume

0 height

b volume

height0

c volume

height0

d volume

height0

7 V

h0

8 D 9 C10 a [7, 4) (0, 3] b [7, 4) (0, 3]11 a [5, 3) (0, 2] c [5, 3) (0, 2]

Exercise 18B

14

3km/min = 80 km/h

150 m 0

200

d (km)

t (min)

2

100

100200300400500600US $

A $200 300 400 500 600 700 8000

3 a 60 km/h b 3 m/s

c 400 m/min = 24 km/h = 6 23

m/s

d 35.29 km/h e 20.44 m/s4 a 8 litres/minute b 50 litres/minute

c200

17litres/min d

135

13litres/min

5t 0 0.5 1 1.5 2 3 4 5

A 0 7.5 15 22.5 30 45 60 75

(1, 15)

1 5

A

t (min)

(L)75

0

6$200

13per hour = $15.38 per hour

7 2081

3m/s 8

(2, 16)

V

t (s)

(cm3)



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Exercise 18C

1 a 2 b 7 c12

d1 5

4

2 a25

7b

187

c 4 d4b

3a

3 a 4 m/s b 32 m/s4 a $2450.09 b $150.03 per year5 3.125 cm/min 6 C7

Car 2Car 1

10 20 30 40 50 60 70 80 time (s)

distance(km)

1

0

Exercise 18D

1 a1

3kg/month (answers will vary)

b1


c1


2 a 0.004 m3/s (answers will vary)b 0.01 m3/s (answers will vary)c 0.003 m3/s (answers will vary)

3 a1

80= 0.0125 litres/kg m

b1

60 0.0167 litres/kg m

4 a 8 years b 7 cm/year5 a 25C at 1600 hoursb 3C/h c 2.5C/h

6 0.59527 a

y

4 0 4 x

b 0 c 0.6 d 1.18 a 49 a 16 m3/min b 10 m3/min

10 a 18 million/min b 8.3 million/min11 a 620 m3/min owing out

b 4440 m3/min owing outc 284 000 m3/min owing out

12 7.1913 a 7 b 9 c 2 d 3514 28 b 12

15 a 10 b 4

16 a i2

0.637 ii 2

2

0.9003

iii 0.959 iv 0.998b 1

17 a i 9 ii 4.3246 iii 2.5893 iv 2.3293b 2.30

Exercise 18E

1 a 4 m/s b 1.12 m/s c 0 m/s

d (, 3) and (0, 3 ) e (1, 1)

2 a i 30 km/h ii20

3km/h iii 40 km/h

c

2 5 8 t (h)

V(km/h)

30

0

40

3 s

11

3

0

6

2 5 7 t

(2, 6)

(5, 3)

(7, 11)

4 a C b A c B5 a +ve slowing down b +ve speeding up

c ve slowing down d ve speeding up6 a gradually increasing speedb constant speed (holds speed attained at a)c nal speeding up to nishing line

7 a t = 6 b 15 m/s c 17.5 m/sd 20 m/s e 10 m/s f 20 m/s

8 a t = 2.5 b 0 t < 2.5c 6 m d 5 seconds e 3 m/s

9 a 11 m/s b 15 m c 1 s d 2.8 s e 15 m/s10 a t = 2, t = 3, t = 8 b 0 < t < 2.5 and t > 6

c t = 2.5 and t = 6Multiple-choice questions

1 C 2 B 3 D 4 E 5 D6 B 7 C 8 E 9 A 10 A


1 a

0 time

depth b

0 time

depth

c

0 time

depth d

0 time

depth



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ersAnswers 773

e

0 time

depth f

0 time

depth

2 a

0 180 t (min)

200

(km)d

constant speed = 2003

km/h

= 200180

km/min

= 109

km/min

b distance(m)30

5 10 15 20 time (s)

c distance5004804404003603202802402001601208040

1 2 3 4 5 6 time

(1, 40)

(4.5, 320)

(6.5, 500)

3 36 cm2/cm4 a 1 b 135 a 2 m/s b 12.26 m/s c 14 m/s

Extended-response questions1 a Yes, the relation is linear.b 0.05 ohm/C

2 a i 9.8 m/s ii 29.4 m/sb i 4.9(8h h2) ii 4.9(8 h) iii 39.2 m/s

3 a i1

4m/s2 ii 0.35 m/s2

b

60 160 180 time (s)

acceleration(m/s2)

4 a

1

50

40

30

20

10

2 3 4 5 6 7 8 n (days)

w (cm)

b gradient = 5 14 ; Average rate of growth of thewatermelon is 5 14 cm/day

c 4.5 cm/day

5 FullHalffull

Quarterfull

6 8 10 1214 16 1820 2224 time (h)

6 a b + a (a = b) b 3 c 4.017 a 2

2

3, 1

3

5; gradient = 1 1

15b 2.1053, 1.9048; gradient = 1.003c 1.000 025 d 1.000 000 3 e gradient is 1

8 69 a 3 13 kg/year b 4.4 kg/year

c {t : 0 < t < 5} {t : 10 < t < 12}d {t : 5 < t < 7} {t : 11 < t < 17 1

2}

10 a i 2.5 l08 ii 5 108b 0.007 billion/yearc i 0.004 billion/year ii 0.015 billion/yeard 25 years after 2020

11 a i 1049.1 ii 1164.3 iii 1297.7 iv 1372.4b 1452.8

12 a a2 + ab + b2 b 7 c 12.06 d 3b213 a B b A c 25 m d 45 s

e 0.98 m/s, 1.724 m/s, 1.136 m/s14 a i m ii cm iii m

b results are the same

Chapter 19

Exercise 19A

1 2000 m/s 2 7 per day3 a 1 b 3x2 + 1 c 20 d 30x2 + 1 e 54 a 2x + 2 b 13 c 3x2 + 4x5 a 5 + 3h b 5.3 c 56 a

12 + h b 0.48 c

12

7 a 6 + h b 6.1 c 6

Exercise 19B

1 a 6x b 4 c 0 d 6x + 4e 6x2 f 8x 5 g 2 + 2x

2 a 2x + 4 b 2 c 3x2 1d x 3 e 15x2 + 6x f 3x2 + 4x

3 a 12x11 b 21x6 c 5d 5 e 0 f 10x 3g 50x4 + 12x3 h 8x3 + x2 1

2x

4 a 1 b 0 c 12x2 3 d x2 1e 2x + 3 f 18x2 8 g 15x2 + 3x

5 a i 3 ii 3a2 b 3x2

6 ady

dx= 3(x 1)2 0 for all x R and

gradient of graph 0 for all xb

dy

dx= 1 for all x = 0 c 18x + 6



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7 a 1, gradient = 2 b 1, gradient = 1c 3, gradient = 4 d 5, gradient = 4e 28, gradient = 36 f 9, gradient = 24

8 a i 4x 1, 3,(

1

2, 0

)

ii1

2+ 2

3x,

7

6,

(3

4,

25

16

)iii 3x2 + 1, 4, (0, 0)iv 4x3 31, 27, (2, 46)

b coordinates of the point wheregradient = 1

9 a 6t 4 b 2x + 3x2 c 4z 4z3d 6y 3y2 e 6x2 8x f 19.6t 2

10 a (4, 16) b (2, 8) and (2, 8)c (0, 0) d

(3

2,5

4

)

e (2, 12) f(1

3,

4

27

), (1, 0)

Exercise 19C

1 b and d 2 a, b and e3 a x = 1 b x = 1 c x > ld x < l e x = 1

2

4 a (,3) (

1

2, 4

)

b

(3, 1

2

) (4,) c

{3, 1

2, 4

}5 a B b C c D d A e F f E6 a (1, 1.5) b (, 1) (1.5, )

c { l, l.5}7 a y

x0 3

y = f (x)

b y

x0

1y = f (x)

c y

x0

31

y = f (x)

d y

x0

y = f (x)

8 a (3, 0) b (4, 2) 9 a

(1

2,6 1

4

)b (0, 6)

10 a b

c

11 a (0.6)t2 b 0.6 m/s, 5.4 m/s, 15 m/s12 a height = 450 000 m; speed = 6000 m/s

b t = 25 s13 a a = 2, b = 5 b

(5

4,25

8

)

Exercise 19D

1 a 15 b 1 c 3 12

d 2 12

e 0 f 4 g 2 h 2

3

i 2 j 12 k 119

l1

42 a 3, 4 b 73 a 0 as f (0) = 0, lim

x0+f (x) = 0 but

limx0

f (x) = 2b 1 as f (1) = 3, lim

x1+f (x) = 3 but

limx1

f (x) = 1c 0 as f (0) = 1, lim

x0+f (x) = 1 but

limx0

f (x) = 04 x = 1

Exercise 19E

1 a y

x0

y = f (x)

1 1

b y

x0

3 2 4

c y

x0

y = f (x)

d y

x0

y = f (x)

e y

x01 1

f y

x01 1

2

332

0

y

x

y = f '(x)

f (x) ={2x + 3 if x 0

3 if x < 0



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3

0 1

y

x

(1, 2)

(1, 4)

f (x) ={

2x + 2 if x 12 if x < 1

4

3

y

x(1, 1)

(1, 2)

f (x) ={2x 3 if x 12 if x < 1


1 D 2 B 3 E 4 B 5 C6 C 7 A 8 E 9 A 10 D


1 a 6x 2 b 0 c 4 4xd 4(20x l) e 6x + l f 6x 1

2 a 1 b 0 c 4x + 74

d4x 1

3e x

3 a 1; 2 b 3; 4 c 5; 4 d 28; 364 a

(3

2,5

4

)b (2, 12)

c

(1

3,

4

27

), (1, 0) d (1, 8)(1, 6)

e (0, 1)

(3

2,11

16

)f (3, 0)(1, 4)

5 a x = 12

b x = 12

c x >1

2

d x 0

c5

2x

32 3

2x

12 ; x > 0 d x

12 5x 23

e 56

x116 f 1

2x

32 ; x > 0

2 a x(1 + x2)12 b

1

3(1 + 2x)(x + x2) 23

c x(1 + x2) 32 d 13

(1 + x) 23

3 a i4

3ii

4

3iii

1

3iv

1

3

4 a {x : 0 < x < 1} b{

x : x >

(2

3

)6}

5 a 5x 12 (2 5x) b 3x 12 (3x + 2)c 4x3 3

2x

52 d

3

2x

12 x 32

e15

2x

32 + 3x 12

Exercise 22D

1 a 6x b 0 c 108(3x + 1)2

d 14

x32 + 18x e 306x16 + 396x10 + 90x4

f 10 + 12x3 + 94

x12

2 a 18x b 0 c 12 d 432(6x + 1)2e 300(5x + 2)2 f 6x + 4 + 6x3

3 9.8 m/s2

4 a i 16 ii 4 m/s iii 74

m/s iv 32 m/sb t = 0 c 8 m/s

Exercise 22E

1 a

(1

2, 4

)(1

2,4

)b y = 15

4x + 1

2 12

31

24 a (4, 0) (1, 0) b y = x 5; x = 0

c (2, 1) min; (2, 9) max

(2, 9) (2, 1)

0x

y

5 36 4

(2, 4)0 x

y

y = x

7 a

(1, 2)

(1, 2)

x = 0

0x

y = x

y

3

b

(1.26, 1.89)

0

(1, 0)

x

y

y = x

y = x

x = 0

c

(4, 4)

(2, 0)

0

3

1

4

x

y = x + 1

x =

y

3

d

(3, 108)

(3, 108)

x

x = 0

y

0

e

x

y

y = x 5

0

(1, 3)

(1, 7)

5 + 2125 21

2

f

x

y

y = x 2

2

2

2

2 0

Multiple-choice questions1 B 2 D 3 A 4 A 5 A6 E 7 A 8 B 9 A 10 D

Short-answer questions(technology-free)1 a 4x5 b 6x 4 c 2

3x3d

4

x5

e15x6

f2x3

1x2

g2x2

h 10x + 2x2



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2 a1

2x12

b1

3x23

c2

3x43

d4

3x

13 e

1

3x43

f 13x

43

+ 65x

25

3 a 8x + 12 b 24(3x + 4)3 c 1(3 2x) 32

d2

(3 + 2x)2 e4

3(2x 1) 53f

3x(2 + x2) 32

g1

3

(4x + 6

x3

)(2x2 3

x2

) 234 a

1

6b 2 c 1

16d 2 e 1

6f 0

5

(1

2, 2

)and

(1

2,2

)6

(1

16,

1

4

)


1 a h = 400r 2

cd A

dr= 4r 800

r 2

d r =(

200

) 13 3.99 e A = 301

f600

105 r

AA = 2r2 +

A = 2r2

800r

(4, 300)

2 a y = 16x

c x = 4, P = 16d

x

P

0

50

10

10 50

P = 2x +32x

(4, 16)

3 a OA = 120x

b OX = 120x

+ 7

c OZ = x + 5 d A = 7x + 600x

+ 155

e x = 10

42

7 9.26 cm

4 a A(2, 0), B(0,2) b 12

x + 2c i

1

2ii 2y x = 3 iii 3

5

2

d x > 74

5 a h = 18x2

c x = 3, h = 2

d

x

A

0

100

20

2 10

A = 2x2 +108

x

(3, 54)

6 a y = 250x2

cd S

dx= 24x 3000

x2d S min = 900 cm2

Chapter 23

Exercise 23A

1 ax4

8+ c b x3 2x + c

c5x4

4 x2 + c d x

4

5 2x

3

3+ c

ex3

3 x2 + x + c f x

3

3+ x + c

gz4

2 2z

3

3+ c h 4t

3

3 6t2 + 9t + c

it4

4 t3 + 3 t

2

2 t + c

2 a y = x2 x b y = 3x x2

2+ 1

c y = x3

3+ x2 + 2 d y = 3x x

3

3+ 2

e y = 2x5

5+ x

2

2

3 a V = t3

3 t

2

2+ 9

2b

1727

6 287.83

4 f (x) = x3 x + 25 a B b w = 2000t 10t2 + 100 0006 f (x) = 5x x

2

2+ 4 7 f (x) = x

4

4 x3 2

8 a k = 8 b (0, 7) 9 8 23

10 a k = 4 b y = x2 4x + 911 a k = 32 b f (x) = 20112 y = 1

3(x3 5)

Exercise 23B

1 a 3x

+ c b 3x2 23x3

+ c

c4

3x

32 + 2

5x

52 + c d 9

4x

43 20

9x

94 + c

e3

2z2 2

z+ c f 12

7x

74 14

3x

32 + c

2 a y = 23

x32 + 1

2x2 22

3

b y = 2 1x

c y = 32

x2 1x

+ 92



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3 f (x) = x3 + 1x

172

4 s = 3t2

2+ 8

t 8 5 y = 5

6 a 2 b y = x2 + 17 y = x

3

3+ 7

3

Exercise 23C

1 a x = 2t2 6t b at the origin O c 9 cmd 0 cm/s e 3 cm/s

2 a x = t3 4t2 + 5t + 4, a = 6t 8b when t = 1, x = 6, when

t = 53, x = 5 23

27c when t = 1, a = 2 cm/s2,

when t = 53, a = 2 cm/s2

3 20 m to the left of O

4 x = 215 13, v = 73

5 a v = 10t + 25 b x = 5t2 + 25t c 2.5 sd 31

1

4m e 5 s

6 the 29th oor

Exercise 23D

1 a7

3b 20 c 1

4d 9 e

15

4

f297

6= 49.5 g 15 1

3h 30

2 a 1 b l c 14 d 31 e 21

4f 0

3 a 8 b 16 c 44 a 12 b 36 c 205

26

36 36 square units

7 3.08 square units8 a 24, 21, 45 b 4, 1, 39 4.5 square units

10 1662

3square units

1137

12square units

12 a4

3square units b

1

6square units

c 1211

2square units d

1

6square units

e 4

3 6.93 square unitsf 108 square units

Exercise 23E

1 a 13.2 b 10.2 c 11.72 Area 6 square units 3 3.13

4 a 36.8 b 36.755 a 4.371 b 1.1286 109.5 m2


1 C 2 D 3 A 4 D 5 B6 B 7 D 8 B 9 C 10 A


1 ax

2+ c b x

3

6+ c c x

3

3+ 3x

2

2+ c

d4x3

3+ 6x2 + 9x + c e at

2

2+ c

ft4

12+ c g t

3

3 t

2

2 2t + c

ht3

3+ t

2

2+ 2t + c

2 f (x) = x2 + 5x 253 a f (x) = x3 4x2 + 3x b 0, 1, 34 a

1x2

+ c b 2x52

5 4x

32

3+ c

c3x2

2+ 2x + c d 6x 1

2x2+ c

e5x2

2 4x

32

3+ c f 20x

74

7 3x

43

2+ c

g 2x 2x32

3+ c h

3x + 1x2

+ c

5 s = 12

t2 + 3t + 1t

+ 32

6 a 3 b 6 c 114 d196

3e 5

7 a14

3b 48

3

4c

1

2d

15

16e

16

158

x

y

0 21

Area = 154

square units

9 41

2square units 10 21

1

12square units

11 a (1, 3) (3, 3) b 6 c4

3


1 a y = 932

(x3

3 2x2

)+ 3



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b

x

y

(4, 0)

(0, 3)

0

c Yes, for the interval

[4

3,

8

3

]

2 a 27 square units b y = 325

(x 4)2

c189

25square units d

486

25square units

3 a i 120 L ii

60

R

0

(L/s)2

t (s)

b i 900 L ii

60

(60, 30)

t (s)

RR =

0

(L/s)t

2

iii 900a2 Lc i 7200 square unitsii volume of water which has owed iniii 66.94 s

4 a i & ii s(km/h)

0

60

2 t (h)

b i

0

1.5

s(km/min)

5 t (min)

(5, 1.5)

ii 3.75 km

c i 20 6t m/s2ii V

0 6 203

t

iii 144 metres

5 a i 4 m ii 16 m b i 0.7 ii 0.8c i

100

3ii

500

27d

3125

6m2

e i (15 + 533, 12)ii R = 6033 60, q = 20,

p = 15 + 5336 a i 9 ii y = 9x 3 iii y = 3x2 + 3xb i 12 + k ii k = 7iii f (x) = 3x2 7x + 12

7 a 6 m2

b i y = x 12

ii

(x2 1

4

)m2

c i P = (2, 2); S = (2, 2),equation y = 1

2x2

ii16

3m2

8 a y = 7 107x3 0.001 16x2 + 0.405x + 60b 100 mc i y

0x

ii (0, 60)

d 51 307 m2

9 a

x0

y

y =0 f (t)dtx

b x = 2.988

Chapter 24Multiple-choice questions1 D 2 E 3 C 4 D 5 E6 D 7 A 8 A 9 C 10 D

11 E 12 C 13 B 14 C 15 C16 E

Chapter 25Short-answer questions1 x = 4 2 t = 2d b

a 2c 3 x 3

24 a 12 b 3 c 1005 15 6 x 37

57 a = 7.9

8 a

(a + 8

2,

b + 142

)b a = 2, b = 6

9 a 4y 3x = 30 b 252

10 a

(2,

1

2

)b

445 c 11x + 18y = 31

d 22y 36x + 61 = 011

2

62

(2, 6)

0

y

x62 +



P1: FXS/ABE P2: FXS


Answ

ers


12 y = 98

(x 2)2 613 a = 214 a w = 1500 9x b V = 20x2(1500 9x)

c 0 x 5003

d 120 000 000 cm3

15 a16

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