answers maths b yr 11mathsbooks.net/maths quest 11b for queensland/answers.pdf · 2015-08-17 ·...

A n s w e r s 581

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CHAPTER 1 Modelling using linear functionsExercise 1A — Solving linear equations1 a 2 b −8 c −3 d 7

e −8 f 6 g 2 h −11i − j

2 a 7 b −6 c 5 d −11e −1 f 18 g − h −

3 a 9 b −4 c −7 d 4e −1 f −5 g 4 h 3

4 a 10 b 6 c −9 d −12e 14 f −18

5 a 6 b −2 c 5 d −5e −13 f 4 g 11 h −3

6 a 12 b −5 c 7 d 7e −9 f

7 a 3 b −4 c 5 d 8e −9 f −7

Exercise 1B — Rearrangement and substitution

1 a P = A − L b c

d e f

g

h or

i j k

l m n

o p

q or

r s t

2 a 56 b 30 c 80 d 16.97e 33.33 f 0.267 g 350 h 7i 13 100 j 2.498

3 a , 7.746 b , 6.204

c , 59.161 d , 4.167

e or , 17.108

f , 3.976 g , 10.247

h , 10.75 i , 2622

j , 4.706

4 a 42 cm b or c 40 mm

5 a 10 N b c 10.77 m/s2

6 a 240 m2

b or c 18 cm

7 a $1123.60

b c 41.4%

8 a 153° b 17.17 cm9 a 60.25 cm b 6 cm

10 a b c 150 cm

Exercise 1C — Gradient of a straight line1 a 2 b 5 c d −

e −4 f −1 g h −

2 a 2 b 5 c −4 d

e −6 f g h 2

i 0 j − k −2 l

3 a b −1 c d

e 1 f −12 g 0 h Undefined

4 a b 2 c d

e f − g −2 h 0

5 a 1 b c d −

e −6 f − g 12 h

6 a 1.192 b 3.078 c 0.176 d −0.577e −0.577 f 0 g 1 h 57.290

7 a 0.93 b 2.61 c −0.53 d −3.738 a D b C c A d B9 a B b E c D d D

10 e, b, a, c, d

Answers

73--- 13

6------

313

------ 72---

7659------

lAw----= t

dv---=

rC2π------= R

100APT

-------------= r3Vπh-------3=

I3

I4R4 I2R2 I1R2 E––+R3

-------------------------------------------------------=

α

R1

R2------ 1–

θ---------------= α

R1 R2–

R2θ------------------=

β E αθ–θ2

-----------------= rkQq

F----------= φ Et

n-----=

V 2

V 1N2

N1-------------= n

pVRT-------= a

2 s ut–( )t2

---------------------=

v2Fd mu2+

m---------------------------= r

µI2

2πF----------=

UV f 1

f 2---------- V–= U

V f 1 V f 2–

f 2--------------------------=

γ v2

rT------= w

S 2lh–2 l h+( )-------------------= H

S 2πr2–2πr

---------------------=

l A= r3V4π-------3=

vmg F–

k-----------------= a

v u–t

-----------=

hS

πr------ r–= h

S πr2–πr

-----------------=

l gT2π------

2= d l2 4 fl–=

VH U–

P---------------= c

1 α–( )Kα2

----------------------=

uH0v

Hi----------=

wP2--- l–= w

P 2l–2

--------------=

aFm----=

a2Ah

-------= b– a2A bh–

h-------------------=

r 100 AD---- 1–

100A D–

D----------------------

= =

fuv

u v+-----------= u

fvv f–------------=

14--- 1

3---

165

------ 209

------

12---

13--- 1

8---

12--- 5

2---

12--- 1

4--- 8

5---

34--- 1

2--- 7

6---

14--- 9

2---

43--- 1

5--- 11

5------

32--- 5

8---

1A➔

1C

Answers Maths B Yr 11 Page 581 Friday, October 26, 2001 10:54 AM

582 A n s w e r san

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s11 a b

c

12 13 14

15 a 4 b 31 c −5 d 316 a No b Yes c 224 cm

Exercise 1D — Equations of the form y = mx + c1

2 The higher the number, the steeper the graph. Positive values make the graph slope up when moving (or tracing) to the right; negative values make the graph slope down when moving to the right.

3 4

5 The number is where the graph cuts the y-axis (hence the name ‘y-intercept’).

6 a 5 b 6 c −9 d 2e −8 f 1 g −1 h 5i 3 j 0 k 0 l 0

7 a 7 b −4 c 1 d −13e −5 f 2 g −10 h 0i 0 j 17 k 2 l 0

8 a y = 2x + 7 b y = −3x + 1 c y = 5x − 2

d y = 3 e y = x f y = x − 5

g y = x + h y = − x − i y = −2x + 12

j y = x − 39 a 3, 9 b 7, −42 c −4, 12 d −5, −35

e 3, 10 f −6, 24 g −16, −15 h −9, −1i 1, −23 j 4, −18

10 a 4, 5 b 4, −8 c 4, −6 d −3, 1e −2, 4 f 3, 11 g −7, −9 h 2, 5

i − , −3 j − , −6 k − , l − ,

m , n − , −11 C 12 E13 y = −7x + • 14 y = • x − 615 3y + 5x = • 16 3y + • x = 17

17 a y = 4x + 2 b y = −3x − 5 c y = x − 2

d y = − x + 5 e y = 2x − 1 f y = −5x

18 a y = 10.7x b 84.7°

19 a a, b b − , c − , − d 2k, −3h

Exercise 1E — Sketching linear graphs using intercepts1

2

y

x

y

x

y

x

217------ –

17300--------- 2

25------

12---

23--- 1

3--- 3

4--- 1

2---

52---

12--- 2

11------ 8

3--- 2

3--- 3

4--- 13

4------

16--- 5

2--- 5

2--- 7

2---

a b

c d

e f

g h

a b

c d

e f

g h

65---

56---

ab--- c

b--- a

b--- c

b---

y

x–3

18

y

x

–21

7

y

x

12

12 — 5–

y

x

–3

3 – 2–

y

x

10

2

y

1

1

y

x

30

10 — 3

y

8

–16

y

x

2

3

y

x

4

5

y

x–2

5–4

y

x–3

6

y

x

–7

5

y

x–4

1–2

y

x

–2

2

y

x

11—6

11—2


A n s w e r s 583

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3

4

5

Exercise 1F — Simultaneous equations1

2

3

4 15 cents and 35 cents5 22 and 196 16 emus, 41 sheep7 Basketballs $9.45, cricket balls $3.058 Limousine $225 (sedan $75)

Exercise 1G — Formula for finding the equation of a straight line1 a i 3x − y − 1 = 0 ii y = 3x − 1

b i 2x − y − 4 = 0 ii y = 2x − 4c i 5x − y − 19 = 0 ii y = 5x − 19d i 4x − y + 11 = 0 ii y = 4x + 11e i x + y − 1 = 0 ii y = −x + 1f i 3x + y + 5 = 0 ii y = −3x − 5

g i x − 2y + 7 = 0 ii y = x +

h i 4x + 3y − 36 = 0 ii y = − x + 12

i i 4x − 5y + 19 = 0 ii y = x +

j i x + 6y − 60 = 0 ii y = − x + 10

k i 8x − 7y + 60 = 0 ii y = x +

l i 3x + 11y − 33 = 0 ii y = − x + 3

2 a i x − 2y − 1 = 0 ii y = x −

b i x − y = 0 ii y = x

c i x + 2y − 12 = 0 ii y = − x + 6

d i 3x + 2y − 2 = 0 ii y = − x + 1

e i 3x + y + 7 = 0 ii y = −3x − 7f i x + y − 4 = 0 ii y = −x + 4

g i 14x − 3y + 2 = 0 ii y = x +

h i 3x − 4y − 12 = 0 ii y = x − 3

i i 4x + 7y + 42 = 0 ii y = – x − 6

j i x + y − 1 = 0 ii y = −x + 1

8

9 y = − x −

a b

c d

e

a b

c

D 6 E 7 A

a b

c d

e f

g h

y

x

–6

–7

y

x

10

–4

y

x

4

16 — 3–

y

x2

–6

y

x

9

–2

y

x

(1, –1)

y

x

(1, 1)

y

x

(1, –2)

i j

a (1, 4) b (−2, 6) c (−4, −15) d (3, −15)

e (−7, −5) f (3, 3) g ( , ) h ( , )

i ( , − ) j (13, −3)

a (7, 9) b (−6, 5) c (6, 5) d (10, 1)

e (1, −2) f ( , ) g ( , ) h (− , − )

i ( , ) j ( , )

9 A 10 D

3 A 4 C 5 y = x − 6 6 y = 3x − 23 7 C

a y = x − b y = − x + 3

c y = 2x − 3 d y = − x −

59--- 17

9------ 12

5------ 32

5------

2314------ 20

7------

103

------ 43--- 1

2--- 19

2------ 9

10------ 3

5---

598

------ 218

------ 8467------ 99

67------

12--- 7

2---

43---

45--- 19

5------

16---

87--- 60

7------

311------

12--- 1

2---

12---

32---

143

------ 23---

34---

47---

25--- 43

5------ 1

2---

34--- 9

2---

52--- 3

2--- 1D➔

1G


584 A n s w e r san

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s10

Exercise 1H — Linear modelling1 a C = 2.5 + 5t

b c $40

2 a C = 60 + 8mb c $100

3 a P = 32 + 0.1nb c $197

6 a Opus $24, Belecom $20 b After 14 minutes7 a PinkCabs $28.50, NoTop $26

b After 6.7 km (6 km)8 After 4 rides9 6 visits

10 Savus would be cheaper for up to 9 days hire.

Chapter review1 D 2 A 3 −2 4 6 5 D

6 D 7 A 8 or

9 C 10 D 11 B 12 B

13 a b − c d −

14 − 15 4.331 16 Undefined 17 B

18 A 19 D

20 a 3, −7 b , 10 c , −2

21 y = x − 3 22 B 23 E

24

25 E 26 E27

28 ( , −5) 29 ( , )30 21 two dollar and 46 one dollar coins31 B 32 C33 y = −x + 4 34 y = x + 35 D36 a C = 75 + 65t

b c $302.50

37 No, the points are not co-linear. This may be shown by calculating gradients or equations for lines joining different pairs of points.

38 (−3, −4), (−1, 8), (3, 4)39 a C = 250 + 55j b 13 jumps

c This is open to question.

CHAPTER 2 Relations and functionsExercise 2A — Set notation1

2

3

4 E5

6

Exercise 2B — Relations and graphs

5

a y = x + b y = −x + 8

c y = − +

11 94 12 C = 22n + 280 13 H = 22 + 6t

4 $960 5 Yes ($410 compared to $450).

a b

67--- 82

7------

x8--- 39

4------

1 2

35

2530

Cost ($)

Time (h)

1 2

76

6068

Cost ($)

Time (min)

10 20

34

3233

Payment ($)

Numberof leaflets

23---

T4π2R3

GM---------------= 2πR

RGM---------

34--- 7

11------ 5

11------ 7

8---

73---

5–3

------ 12---

25---

y

x

24

8

y

x–40

5

c d

a b (−5, −5)

a ∅ b {4, 6} c {6}d {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}e {4, 5, 6, 7, 8} f {2, 8, 10, 12, 14} g {4, 5}

a {2, 3} b {−3, −2, −1} c {−1}d {−3, −2, −1, 0, 1}e {−3, −2, −1, 0, 1, 2, 3, 4}

a ∅ b {b, c, d, f, g, h} c {a, e, i}d {b, c, d, f, g, h} e {o, u}

a C b B

a T b F c T d Fe F f T g T h Fi T j F k F l T

1 B 2 A 3 E 4 C

a D b C c Cd D e D f C

y

x

–9

7

y

x

(1, –6)

53--- 63–

10--------- 33–

5---------

6–7

------ 347

------

1 2

205

75

140

C ($)

t (h)


A n s w e r s 585

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6

7

8 a

b

c The variables are discrete.

9 a

b i Approx. 110 km/h ii Approx. 320 km/h

10 a

b c The variables are discrete.

Exercise 2C — Domain and range1

2

3

4

7

8

9 a

Domain = (−∞, ∞), Range = (−∞, 2]b

Domain = [−2, 2], Range = [−7, 9]

a b

c d

e f

a b

c Because the variables are continuousd Approx. 11 minutes

n 0 1 2 3 4 5 6

P($) 300 340 380 420 460 500 540

n 15 16 17 18 19 20 21 22 23 24 25

C($) 140 146 152 158 164 170 176 182 188 194 200

70

60

Day

Cos

t (¢)

M T W T F S S

(D)

y

x0 32

9

1

4

1

(D)

y

x0 2–1

–2

1–1

–3

–4

–2

(D)

y

x0 2

–2

y = x – 2

(C)

y

x0 2

7

1

3456

2

1–1–2

–2

(D)

y

x0 2

6

1–1

4

2

–2

(C)

T (°C)

t (minutes)0

8070605040302010

2 4 6 8

T (°C)

t (minutes)0

8070605040302010

2 4 6 8 10

P ($)

n

550500450400350300250200

10 2 3 4 5 6

V (km/h)

t (s)0

35030025020015010050

1 2 3 4 5

a [−2, ∞) b (−∞, 5)c (−3, 4] d (−8, 9)e (−∞, −1] f (1, ∞)g (−5, −2] ∪ [3, ∞) h (−3, 1) ∪ (2, 4]a b

c d

e f

g h

i j

k l

a [−4, 2) b (−3, 1]

c d

e (3, ∞) f (−∞, −3]g (−∞, ∞) h [0, ∞)i (−∞, 1) ∪ (1, ∞) j (−∞, −2) ∪ (−2, ∞)k (−∞, 2) ∪ (3, ∞) l (−∞, −2] ∪ [0, ∞)a E b D

5 C 6 Ba i {3, 4, 5, 6, 7} ii {8, 10, 12, 14, 16}b i {1.1, 1.3, 1.5, 1.7} ii {1.4, 1.6, 1.8, 2}c i {3, 4, 5, 6} ii {110, 130, 150, 170}d i {M, T, W, Th, F} ii {25, 30, 35}e i {3, 4, 5} ii {13, 18, 23}f i R ii [−1, ∞)a R, R b R, (0, ∞) c [−2, 2], [0, 2]d [1, ∞), R e R, (0, 4] f R, (−∞, −3]g R\{0}, R\{0} h R, (−∞, 1] i R, R

C ($)

n

190200

180170160150140130

50 10 15 20 25

20–6 0–3–9

20 50

1010 720

1 30–2 20–8 6

410 0 5–1

0 2 0 1–2

1 3 ),–( –12--- 1

2-------,

y

x20

2

2–

y = x3 + 1

x ∈ [–2, 2]

y

x0

9

1

–7

–2 2

1H➔

2C

Answers Maths B Yr 11 85 Monday, October 29, 2001 12:27 PM

586 A n s w e r san

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sc Domain = (−∞, ∞),

Range = [− , ∞)

d Domain = [−2, 1], Range = [−4, 0]

e Domain = [−1, 4), Range = [−7, 3)

f Domain = (−∞, ∞), Range = [−6 , ∞)

10

Investigation — Interesting relations1 2

x2 + 2y2 = 9 x3 + y3 = 13 4

sin (x2 + y2) = 1 x2 – y2 = 15 6

7x2 – 6 xy + 13y2 = 16 x4 = x2 – y2

7 8

x2 + y2 < 25 x2 + y2 > 259 10

9 < x2 + y2 < 36 x sin x + y sin y < 1

Exercise 2D — Types of relations (including functions)1

2 b, c, d, e, f, h, i, j, k, l3 C45

Exercise 2E — Function notation and special types of function1

2

3

4 a, c, d, f, g, h5 i a, b, c, d, f, h, i, j, k, l ii c, h, i, k67 a b (−∞, 0) ∪ [1, ∞)

a R b [0, ∞) c [−4, 4]d R e R\{0} f R

y = x2 + 3x + 2

y

x0

2

–2 –1

14---

y = x2 – 4, x ∈ [–2, 1]

y

x0–2 –1

–3

–4

1

y = 2x – 5, x ∈ [–1, 4]

2

y

x0 3 4–1

–5

3

–7

1

y = 2x2 – x – 6

2

y

x0–1–2

–6

1

18---

–4

–4

4

y

x4

–2

–2

2

2

y

x

y

x10–10

–10

10

–10

–10

10

10

y

x

–2

–2

2

2

y

x

–1

–1

1

1

y

x

3

a One-to-many b Many-to-onec Many-to-one d One-to-onee One-to-one f Many-to-oneg Many-to-many h Many-to-onei One-to-one j Many-to-onek Many-to-many l Many-to-one

a E b D c Bb {−3, −1, 0, 1, 2}, {−2, −1, 1, 3}c {3, 4, 5, 6}, {−1} e R, {2}g R, R j [−1, ∞), [0, ∞)k R, R

a i 1 ii 7 iii –5 iv 16b i 2 ii 1 iii 3 iv 0c i 3 ii 2 iii 6 iv 9d i 9 ii 1 iii 16 iv a2 + 6a + 9e i 12 ii 6 iii −4 iv 2

a 3 b −3 or 3 c

d 2 or 3 e −4 or 1 f −1

a 3 b 3 c

d e f

a B b C

–10

–10

10

10

y

x

–10

–10

10

10

y

x

–10

–10

10

10

y

x

–10

–10

10

10

y

x

13---

5x--- 2x–

10x2------ x2– 10

x 3+------------ x– 3–

10x 1–----------- x– 1+

y

x0 2

2f (x)

1

1–1–2


A n s w e r s 587

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8 a

b [1, ∞)c i 3 ii 1 iii 2

9 a

b (−∞, 0] ∪ (4, ∞)c i −5 ii 0 iii −3 iv 0 v 7

10

11 f : [0, 1] → R, f (x) = with range [0, 1] or

f : [0, 1] → R, f (x) = – with range [−1, 0].

Investigation — A special relation

2

Exercise 2F — Circles1

2

3

4

5

6

1

x y x2 y2 x2 + y2

On the graph of

x2 + y2 = 25?

0430347

−4−5−4

139406

−3−2

5−3

5208437

−3035

−40

−3−5−6

4−5

0−4

016

909

1649162516

19

8116

036

94

259

2540

6416

949

909

2516

09

25361625

016

2520

96425259825252526258125257225292525

YesNoNoNoYesYesNoYesYesYesNoYesNoYesYesNoYesNoYesYes

y

x0 2

3g(x)

21

1–1–2

2 3

y

x0–1–2–3

–5–4

54321

1

f x( )x 2,+2x 1,+

=x 0≤x 0>

1 x2–

1 x2–

–8–6–4–2–10 –5

108642

105

y

x

a x2 + y2 = 9 b x2 + y2 = 1 c x2 + y2 = 25

d x2 + y2 = 100 e x2 + y2 = 6 f x2 + y2 = 8

g h

a Both [−3, 3] b Both [−1, 1]c Both [−5, 5] d Both [−10, 10]

e Both [− , ] f Both [−2 , 2 ]

g [−3, 3], [0, 3] h [−4, 4], [−4, 0]

a b

c d

e f

a b

c d

e f

g h

a D b B

a C b E

y 9 x2–= y 16 x2––=

6 6 2 2

y

x0 2

2

–2

–2

y

x0 4

4

–4

–4

y

x0 7

7

–7

–7

y

x0

7

– 7

7– 7

y

x0–2 3 2 3

–2 3

2 3

y

x0– 1–2

1–2

– 1–2

1–2

y

x0 3

3

–3

–3

y

x0 2–2

y

x0 1–1

y

x0

1–3

– 1–3

1–3

y

x0– 1–2

– 1–2

1–2

y

x0

5

5– 5

y

x0

10

– 10

10– 10

y

x0

3

– 3

– 3

2D➔

2F


588 A n s w e r san

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s7 a b

[−1, 1] and [−3, −1] [−2, 2] and [0, 4]c d

[1, 7] and [−3, 3] [−2, 6] and [–5, 3]e f

[−8, 2] and [−7, 3] [0, 6] and [−1, 5]g h

[−11, 1] and [−2, 10] [−1, 2] and [−3, 0]

8 ; Domain [−6, 6] and range [0, 6] or

; Domain [−6, 6] and range [−6, 0]

9 ; Domain [−3, 3] and range [2, 5] or

; Domain [−3, 3] and range [−1, 2]10 a 2 cm, 13.8 cm b 3.9 cm/s

Exercise 2G — Functions and modelling

1 a b

2 a

b

3 a

b Domain [0, 4]; range [0, 250]c i 60 km ii 170 km

4 a

b c $90

5 a T = 0.34x − 3978b

Domain [20 700, 38 000]; range [3060, 8942]c $6902

6 a P = 4x + 6b Domain (1, 6]; range (6, 30]

7 a A = x2 + 4xb Domain (0, 8]; range (0, 96]

8 a P = 100 000(1.02)t

b $121 8999 a 47

b 21c 9 weeksd No, as t increases approaches zero, so N

approaches 15.10 a T = 6000 + 100n − 50n2

b

c $11

Chapter review1 A 2 D 3 B 4 C5 a

b The number of cars is a discrete variable.c 120

y

x0 1

–2

–1

–3

–1

y

x0 2

4

2

–2

y

x0 4 71

3

–3

y

x0 62–2

3

–5

–1

y

x0 2–8 –3

3

–7

–2

y

x0 63

5

2

–1

y

0 1–5–11

10

4

–2

y

x0 2

–3

–1

– 3–2

1–2

y 36 x2–=y 36 x2––=y 2 9 x2–+=

y 2 9 x2––=

C t( )

40,

70,

110,

160,

=

0 t 1≤<1 t 2≤<2 t 4≤<4 t 6≤<

70

60

Day

Cos

t (¢)

M T W T F S S

(D)

C d( )

0.40,

0.60,

0.80,

1.70,

2.00,

=

0 d 50≤<50 d 100≤<100 d 200≤<200 d 700≤<d 700>

Cos

t ($)

Distance (km)0

2.00

0.600.80

1.70

100 200 700

0.40

d t( )60t ,

90,

80t 70,–

=0 t 1.5≤ ≤1.5 t 2≤ ≤2 t 4≤ ≤

Bn12------=

B (hours)

n0

10

5

60 120

8942

38 00020 700

3060

T ($)

x ($)

96t 3+-----------

T

n0

600050004000300020001000

1 2 3 4 5 6 7 8 9 101112

No.

of

cars

(n)

t (hours)0

500

400

300

200

100

21 3 4 5

Graph is not continuousas n ∈ N


A n s w e r s 589

answ

ers

6 a

b Domain = [−3, 3]; range = [−8, 1]7 E 8 C 9 B 10 E 11 A

12 D 13 D14 a x + 2, x ≥ 0

b Domain = [0, ∞); range = [2, ∞)15 E 16 B 17 C 18 B 19 C20 a, b, e21 A 22 D 23 E 24 A

25 a f : R \ {0} → R, f (x) =

b f : (–∞, 2] → R, f (x) = 26

27 a

Domain = [−1, 1]; range = [−1, 0]b

Domain = [−1, 5]; range = [−4, 2]28 D 29 E 30 C 31 B32 a

b f1: [−10, 10] → R, f (x) = with dom f = [−10, 10], ran f = [0, 10] and

f2: [−10, 10] R, f (x) = withdom f = [−10, 10], ran f = [−10, 0]

33 E34

35 a A = xy + 10y − x2 b P = 2x + 2y + 20 orP = 2(x + y + 10)

c A = 260 + 16x − 2x2 d (0, 13)e f 292 m2

CHAPTER 3 Other graphs and modellingExercise 3A — The parabola (turning point form)

y

x0 32

–3

1

1–1

–8

–3–2

y = 1 – x2

1x---

2 x–

2 3 4 5

y

x0–1–2

54321

1

y

x0 1–1

–1

y

x0 5

2

–4

–1(2, –1)

y

x0 10

10

–10

–10

x2 + y2 = 100

100( x2– )

100( x2– )–

0

100

25

50

75

1 2

Cos

t ($)

Number of truck loads

1 a Dilation by the factor of 2 in the y directionb Dilation by the factor of in the y directionc Dilation by the factor of 3 in the y direction,

reflection in the x-axisd Translation 6 units downe Dilation by the factor of 3 in the y direction,

translation of 4 units upf Dilation by the factor of in the y direction,

reflection in the x-axis, translation of 1 unit upg Translation of 2 units to the righth Reflection in the x-axis, translation of 3 units to

the lefti Dilation by the factor of 2 in the y direction,

translation of 3 units to the rightj Translation of 2 units to the left, translation of 1

unit downk Translation of 0.5 unit to the right, translation of

2 units upl Dilation by the factor of 2 in the y direction,

reflection in the x-axis, translation of 3 units to the left, translation of 1 unit up

m Dilation by the factor of 12 in the y direction, translation of 1.5 units to the right, translation of 0.25 units down

n Dilation by the factor of in the y direction, reflection in the x-axis, translation of units to the right, translation of 2 units up

2 D3 a (ii) b (v) c (i) d (iv) e (iii)4 a i (0, 0) (ii) ii Domain: R, range: y ≤ 0

b i (0, − ) ii Domain: R, range: y ≥ −c i (0, 2) ii Domain: R, range: y ≤ 2d i (6, 0) ii Domain: R, range: y ≥ 0e i (−2, 0) ii Domain: R, range: y ≤ 0f i (3, 0) ii Domain: R, range: y ≥ 0g i ( , 0) ii Domain: R, range: y ≥ 0h i (−3, −6) ii Domain: R, range: y ≥ −6i i (1, 1) ii Domain: R, range: y ≤ 1j i ( , 0) ii Domain: R, range: y ≥ 0k i (− , −5) ii Domain: R, range: y ≥ −5l i ( , 4) ii Domain: R, range: y ≤ 4

A (m2)

x (m)0

292

130

260

2 4 6 8 10 12 14

13---

12---

92---

43---

12--- 1

2---

12---

43---

12---

12---

2G➔

3A


590 A n s w e r san

swer

s5 a b

c d

e f

g h

i j

Exercise 3B — The cubic function of the form y = a(x - h)3 + k

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

7 E8

9 a y = x2

b y = −x2

c y = (x − 2)2 − 1d y = 3x2 − 2e y = −(x + 3)2

10 a y = (x − 3)2 − 4b y = −2(x + 1)2 + 1c y = (x + 3)2 − 4d y = − (x − 2)2 + 2e y = 3(x − 1)2 + 6f y = −4(x + 2)2 + 8

y

x

3

y

x1–2

1–2

–

–41

–11

y

x12

2

y

x

20

–3

2

y

x

2

1

–15

1–21 2 –

21

y

x

2.9

0.1

1

–8

1–21

y

x

1

9

– –34

y

x

4

–32 –

34

y

x

115

2

y

x

12---

–5

13

–3–4.6

–1.4

y

x

12---

13---

12---

11 a z = 3 or z = 15b y = 2(x − 3)2 − 8 or y = (x − 15)2 − 8

12 a 3b y = − (x + 4)2 + 3c x = −7, x = −1

13 1. f(x + 2) − 3, −4 ≤ x ≤ 02. f(x − 2) − 3, 0 ≤ x ≤ 43. f(x + 4), −6 ≤ x ≤ −24. f(x − 4), 2 ≤ x ≤ 65. −f(x + 4) + 6, −6 ≤ x ≤ −26. −f(x) + 6, −2 ≤ x ≤ 27. −f(x − 4) + 6, 2 ≤ x ≤ 68. −f(x + 2) + 9, −4 ≤ x ≤ 09. −f(x − 2) + 9, 0 ≤ x ≤ 4

1 a Dilation in the y direction by the factor of 7b Dilation in the y direction by the factor of ,

reflection in the x-axisc Translation by 4 units upd Reflection in the x-axis, translation by 6 units upe Translation by 1 unit to the rightf Reflection in the x-axis, translation by 3 units to

the leftg Dilation in the y direction by the factor of 4,

reflection in the y-axis, translation by 2 units to the right

h Dilation in the y direction by the factor of 6, reflection in the x-axis, reflection in the y-axis, translation by 7 units to the right

i Dilation in the y direction by the factor of 3, translation by 3 units to the left, translation by 2 units down

j Dilation in the y direction by the factor of , reflection in the x-axis, translation by 1 unit to the right, translation by 6 units up

k Dilation in the y direction by the factor of 2, translation by units to the left

l Dilation in the y direction by the factor of , reflection in the x-axis, translation by 8 units to the left, translation by 3 units up

2 a i, iv b iii, v c iid i, ii, iv e ii, v f iii, iv

3 a (0, 3) b (0, ) c (1, 0)

d (4, 0) e (−2, 4) f (1, −2)g (2, 1) h (−3, −4) i (−4, 1)

j ( , )

4 E

5 C

6 B

225------

13---

23---

12---

52---

14---

12---

16--- 2

5---


A n s w e r s 591

answ

ers

7 a b

c d

e f

g h

i j

k l

Exercise 3C — The hyperbola

8 a y = x3 b y = −(x + 5)3

c y = (x − 3)3 − 1 d y = 2x3 + 3e y = −(x + 1)3 − 1

9 a y = − x3 + 4 b y = 2(x − 1)3 + 2

c y = −3(x + 1)3 + 1 d y = − (x − 3)3

e y = 4(x + 1)3 −

10 E11 y = 2(x + 1)3 − 4

12 a y = − (2 − x)3 + 1

b Positive cubic

y

x0.8

1

y

x

2.08–6

y

x

4

–128

y

x

2

4

y

x1

4

y

x

1

y

x–0.3

21

–5

–2

–0.6

3

y

x

–4–6

–1 0.4

y

x

y

x

28

1

–2 – –23

y

x

2

245

3

3.6

y

x

3

35

4

5.8

12---

12---

13---

12---

12---

13 a (−3, 1) or (−1, 27)b

1 a Dilation in the y direction by the factor of 2b Dilation in the y direction by the factor of 3,

reflection in the x-axisc Translation by 6 units to the rightd Dilation in the y direction by the factor of 2,

translation by 4 units to the lefte Translation by 7 units upf Dilation in the y direction by the factor of 2,

translation by 5 units downg Translation by 4 units to the left, translation by 3

units downh Dilation in the y direction by the factor of 2,

translation by 3 units to the right, translation by 6 units up

i Dilation in the y direction by the factor of 4, reflection in the x-axis, translation by 1 unit to the right, translation by 4 units down

2 a v b iii c id v, iii e v, ii, iii f i, iiig v, i, iv h ii, iv

3 a i x = 0, y = 0 ii Domain: R\{0}iii Range: R\{0}

b i x = −6, y = 0 ii Domain: R\{−6}iii Range: R\{0}

c i x = 2, y = 0 ii Domain: R\{2}iii Range: R\{0}

d i x = 3, y = 0 ii Domain: R\{3}iii Range: R\{0}

e i x = 0, y = 4 ii Domain: R\{0}iii Range: R\{4}

f i x = 0, y = −5 ii Domain: R\{0}iii Range: R\{−5}

g i x = −6, y = −2 ii Domain: R\{−6}iii Range: R\{−2}

h i x = 2, y = 1 ii Domain: R\{2}iii Range: R\{1}

i i x = −n, y = −m ii Domain: R\{−n}iii Range: R\{−m}

4 a i x = 4, y = 0 ii Domain: R\{4} iii Range: R\{0}

b i x = 0, y = 2 ii Domain: R\{0} iii Range: R\{2}

c i x = 3, y = 2 ii Domain: R\{3} iii Range: R\{2}

d i x = −1, y = −1 ii Domain: R\{−1} iii Range: R\{−1}

y

x

2827

1–1–4

–3y = (x +1)3+27

y = (x +3)3+1

3B➔

3C


592 A n s w e r san

swer

s

6 a b

c d

e f

g h

i j

10 a b

c d

e

Exercise 3D — The square root function

e i x = m, y = n ii Domain: R\{m} iii Range: R\{n}

f i x = b, y = a ii Domain: R\{b} iii Range: R\{a}

5

7 E

8 C

9 a y = b y = − + 1

c y = − d y = − − 1

e y = + 2 f y = − 1

y

xx–32

x–32

– x–34

– x–34

x–3

x–3

x–1

x–1

–3

–31

y

x

y

x–2 –1–1

– –21

y

x1 5

– –43

–3–43

y

x–5 – –52

y

x–1

13

–3

y

x2

2

67–

21

–21

y

x2

1

1

–21

y

x–11 –1

4 –52

–52

y

x–1 –

4

4

–31

–23–

85

y

x

–1

– –31

–41 –

43

2x 2–----------- 3

x---

3x 4+------------ 4

x---

2x 4–----------- 6

x 1+------------

11 Domain: R\{0}, range: R\{3}

1 a Dilated in the y direction by the factor of 2b Dilated in the y direction by the factor of ,

reflected in the x-axisc Dilated in the y direction by the factor of 3,

translated 1 unit to the rightd Dilated in the y direction by the factor of 2,

reflected in the x-axis, translated 4 units to the left

e Translated 1 unit downf Dilated in the y direction by the factor of 3,

reflected in the x-axis, translated 2 units upg Translated 4 units to the right, translated 3 units

uph Dilated in the y direction by the factor of 2,

reflected in the x-axis, translated 3 units to the left, translated 6 units up

i Dilated in the y direction by the factor of , reflected in the x-axis, reflected in the y-axis, translated 2 units to the right and units up

2 a (0, 0) b (0, 0) c (1, 0)d (−4, 0) e (0, −1) f (0, 2)g (4, 3) h (−3, 6) i (2, )

3 E4 D

y

x–2

–21

y

x–1

1

y

x–2

– –21

y

x

2

3

1 1–21

y

x–1

1

y

x

3

–31

13---

12---

23---

23---


A n s w e r s 593

answ

ers

8 a b

c d

e f

g h

i

10 a m = 1 b y = 2 − 4

Exercise 3E — The absolute value function1 a b

c d

e f

g h

i

4 a b

c d

e f

5 a Domain: x ≥ −1, range: y ≥ 0b Domain: x ≥ 3, range: y ≥ 0c Domain: x ≥ 0, range: y ≥ −3d Domain: x ≥ 0, range: y ≥ 4e Domain: x ≥ 0, range: y ≤ 5f Domain: x ≥ 1, range: y ≥ 3g Domain: x ≥ −2, range: y ≥ −1h Domain: x ≥ − , range: y ≤ 4i Domain: x ≥ , range: y ≤ 2j Domain: x ≤ 3, range: y ≥ −7k Domain: x ≤ 2, range: y ≥ 6l Domain: x ≤ 2, range: y ≤ 1

6 D7 D

9 E

11 y = 3 + 3

12 a p = 8 b y = −4 + 8c x = 3 d x ≥ −1e y ≤ 8 f

12---

43---

y

x–2

1.4

y

x

3

y

x2

4 (6, 1)

y

x

y

x

2

3.7

–3

y

x–4

–21

–1–21

–3–43

y

x–23

y

x

2

4.4

–2

y

x21

0.4

–1

x 1–

y

x

9

(4, 3)

4 x–

x 1+

y

x4

3

(–1, 8)

2 C

3 a Domain: R, range: y ≥ 0b Domain: R, range: y ≥ 1c Domain: R, range: y ≤ 4d Domain: R, range: y ≥ −2e Domain: R\{−1}, range: y > 1f Domain: R\{0}, range: y ≥ 0

y

x

y

x

1

1

y

x–21

3

y

x

6

66–

y

x

4

2–2

y

x

45

1 3 5

y

x

y

x

1

7

–2 –1

y

x1

2

y

x

y

x1–5–11 –1

–6

y

x1

7

3

y

x

2

(–1, 1) (1, 1)

y

x2–2 2– 2

2

y

x0.7

–2.7

–1

–1–2

–2

3D➔

3E


594 A n s w e r san

swer

sg h

i j

k l

Exercise 3F — Addition of ordinates

3 a b

c d

e f

4

5

6 a

b

c

d

7 E

Exercise 3G — Modelling

5 a y = , −2 ≤ x ≤ 2

b Yellow: y = 6 − , −2 ≤ x ≤ 2;

green: y = − 6, −2 ≤ x ≤ 2;

blue: y = − , −2 ≤ x ≤ 2

1 a R\{0} b [0, ∞) c [0, ∞)

d [−2, ∞) e R f R\{3}

g R\{−1} h (−∞, 1] i R\{0}

j [−1, 3]2 C

y

x

– –34

– –43

–34

y

x

–313

3

6

y

x– –

41

–4

–41

y

x

–1

–1 1

y

x

33.6

5

2–2

y

x2

35 63 99–5

(–1, –6)

32---x

32---x

32---x

32---x

y

x

h(x)

f(x)

g(x)

y

x

h(x)

f(x)g(x)

y

x

h(x)

f(x)

g(x)

y

x

h(x)

f(x)

g(x)

h(x)

f(x)

g(x)

y

x

y

x

h(x)

f(x)

g(x)

1 a y = ax3, a = 0.3 b y = ax2, a = −6

c y = a , a = 1.6 d y = , a = 5

e y = ax3, a = −1.5

2 a iii b ii c id iv

3 D

y

x

87654321

–4–5–6 –3 –2 –1 1 20

f(x)

g(x)

x2 + 5x + 6

y

x

y = x3 + x2 – 1

f(x)g(x)

1

2

–2

–2 23

3

y

x2

2

1

1

2 – xy = xy =

x +y = 2 – x

y

x

x y = –

y = 2x

x y = 2x –

–41

y

x

y = –x2

x – 3 y =

x – 3 – x2y =

y

x

52

–5 –5

5 – x y =x + 5 y =

5 – x x + 5 + y =

xax---


A n s w e r s 595

answ

ers

4 a b

c a = 2, b = −3.2

6 a b

c f =

Chapter review1 D 2 E3 a (3, −4) b Domain: R, range: y ≥ −4

c

4 C 5 D6 y = 1 − 3(x − 1)3

7 C 8 E

9 a x = −2, y = −1b Domain: R\{−2}, range: R\{−1}

c y = − 1

d

10 C 11 A 12 E 13 B 14 C

15 B

16 a

b

17 E 18 E 19 y =

5 y = x3 − 12

7 a

b I =

8 y = 3 + 4

9 a

b p = 2 + 4c 10.63, 10.93

y

x1 2 3 4 5–100

10

20304050

y

x25 10 15 20 25–100

10

20304050

14---

f

2 4 6 8 100

200

400600

800

1000

λ

f

0.5 1.5 2.51 2 3 3.50

200400600800

1000

λ1—

340λ

---------

I

d21 3 40

50

100

150

200

250

270

d2

---------

x

$

Month42 6 8 100

1

4

6

9

1110

5

32

87

Pric

e

m

y

x

–4

14

3

4.41.6

20 a i

A(−2, 0), B(0, −2), C (2, −3), D(4, −6)ii

A(2, 0), B(0, 2), C(−2, 3), D(−4, 6)iii

A(0, 0), B(2, 2), C(4, 3), D(6, 6)

4–x 2+------------

y

x–1

–3

–2–6

y

x

(f + g) (x)

f(x)

g(x)

y

x0

(f + g) (x)

f(x)

g(x)

100

x2

---------

y

x

A

A'

B

B'2

–2

–2C(2, 3)

C'(2, –3)

D(4, 6)

f(x)

f(x)

D'(4, –6)

y

xA'A

B

2–2

D(4, 6)D'(–4, 6)f(x)f(–x)

C(2, 3)C'(–2, 3)

y

x–2

f(x)f(x–2)

B'(2, 2)

C'(4, 3)C(2, 3)D'(6, 6)

D (4, 6)

A'A

2B

3F➔

3G


596 A n s w e r san

swer

s

CHAPTER 4 Triangle trigonometryInvestigation — Looking at the tangent ratio(Answers may vary slightly because of individual measurements.)1 a 8 mm b 16 mm c 0.502 a 13.5 mm b 26 mm c 0.523 a 18 mm b 35 mm c 0.514 a 23 mm b 44 mm c 0.52

Investigation — Looking at the sine ratio(Answers may vary slightly because of individual measurements.)1 a 8 mm b 17.5 mm c 0.462 a 13.5 mm b 29 mm c 0.47

3 a 18 mm b 38.5 mm c 0.474 a 23 mm b 49.5 mm c 0.46

Investigation — Looking at the cosine ratio(Answers may vary slightly because of individual measurements.)1 a 16 mm b 17.5 mm c 0.912 a 26 mm b 29 mm c 0.903 a 35 mm b 38.5 mm c 0.914 a 44 mm b 49.5 mm c 0.89

Exercise 4A — Calculating trigonometric ratios1 a 1.540 b 17.663 c 40.460

d 0.6572 a 0.602 b 2.092 c 15.246

d 51.8933 a 0.707 b 0.247 c 6.568

d 5.8964 a 0.5 b 0.9659 c 1

d 548.6 e 64 f 1.301g 5.306 h 1.374 i 15.77

5 a 0.42 b 1.56 c 0.09d 5.10 e 2.87 f 0.38g 7.77 h 73.30 i 0.87

6 10°7 a 44° b 80° c 57°8 86°40′9 a 42°57′ b 31°21′ c 16°5′

Exercise 4B — Finding an unknown side1 a

b

c

2 148.1 mm3 5.08 m4 30.0 cm5 a 12.1 cm b 55.2 m c 9.43 km6 a 12.5 m b 89.3 mm c 10.1 m7 a 5.42 m b 1.35 km c 2.06 km

d 18.4 mm e 3.20 cm f 66.5 mg 5.40 m h 5.39 km i 0.240 mj 41.6 km k 82.4 m l 13.2 cm

8 D 9 E10 6 m 11 4.2 m 12 20 km13 a b 30.3 cm

iv

A(−2, 3), B(0, 5), C(2, 6), D(4, 9)v

A(−2, 0), B(0, 4), C(2, 6), D(4, 12)vi

A(−3, 1), B(−1, −1), C(1, −2), D(3, −5)b Add multiples of 2, for example, f(x) + 2,

f(x) + 4, f(x) + 6, f(x) − 2 etc. and keep the domain fixed at [−3, 7].

21 a y = a(x − h)2 + kb h = 9c Straight line (negative gradient)d a = −0.55, k = 275e y = −0.55(x − 9)2 + 275f No, the prices started going down.g $266 000, $261 000h About 4 months

y

x

f(x)

f(x) + 3

A'(–2, 3)

C' (2, 6)

D'(4, 9)

C(2, 3)

D(4, 6)B'

–2

2

5

A

B

y

x

C'(2, 6)

D'(4, 12)

D(4, 6)

C(2, 3)B'4

A'2 BA

–2

2f(x)

f(x)

y

x

D'(3, –5)

C'(1, –2)B'(–1, –1)

A'(–3, 1)A

B C(2, 3)

D(4, 6)

f(x)

1 – f(x + 1)

2

hypopp

adjθ

hyp

opp

adjα

hyp

oppadjγ

24°13.5 cm


A n s w e r s 597

answ

ers

14 a b 1.6 m

15 9.65 m16 a b 58 m

c 15.5 m

Exercise 4C — Finding angles1 a 30° b 75° c 81°2 a 32°48′ b 45°3′ c 35°16′ 3 a 53°8′ b 55°35′ c 45°27′4 a 50° b 32° c 33°

d 21° e 81° f 34°5 a 39°48′ b 80°59′ c 13°30′

d 79°6′ e 63°1′ f 19°28′6 A 7 D 8 37°9 75°31′ 10 8°38′ 11 7° 12 4°35′

Exercise 4D — Applications of right-angled triangles

5 a 22.33 m b 13.27 m

7 a b 1319.36 m

8 22 m

9

10

11

14

15 201°48′ T16

17

18

19

20

Investigation — Fly like a birda i 572 m ii 715 mb i 143 m ii 4.29 km/h

Exercise 4E — Using the sine rule to find side lengths

1 a

b

c

2 a 14.8 cm b 1.98 km c 112 mm3 a 10.0 m b 22.1 cm c 39.6 km4 9.8 cm5 26.9 m6 37.8 m7 a b 43.2 m c 33 m

8 43.62 m9

1011 22.09 km from A and 27.46 km from B.

Investigation — Bearing east and west1

2 27.6 km 3 As for 1 4 27 km

Exercise 4F — Using the sine rule to find angle sizes1 a 43° b 34° c 27°

d 75° e 37° f 2°2 B3 B4 38°5 20°6 84°7 a 57° b 63°8 54°9 a 13.11 km b N20°47′W

Exercise 4G — The cosine rule

7 2218 m

89

12

16

1 571 m 2 30 m 3 91 m 4 43.18 m

6 2°44′

a 325° T b 227° T c 058° T d 163° T

a S66°W b S73°E c N39°W d N74°E12 a C b D 13 1691 m

a 5.39 km b N21°48′W

a 4.36 km b 156°35′ Ta 12.2 km b 348 T or N12°W

a 29.82 km b 38.08 km c 232° T

a 112.76 km b 5 hours 30 minutes

a 82.08 m b 136.03 m c 301°6′ T

60°

1.4 m

15°60 m

48°35°

2500 m

Helicopter

S1S2

5050 3

3------------- m–

a 6.97 m b 4 m

a 8.63 km b 6.48 km/h c 9.90 km

12 D 13 B 14 Yes, she needs 43 m altogether.

1 7.95 2 55.22 3 23.08, 41°53′, 23°7′ 4 28°57′5 88°15′ 6 A = 61°15′, B = 40°, C = 78°45′

a 12.57 km b S35°1′Ea 35°6′ b 6.73 m2

10 23° 11 89.12 m

a 130 km b S22°12′E13 28.5 km 14 74.3 km 15 70°49′

a 8.89 m b 76°59′ c x = 10.07 m17 1.14 km/h 18 E 19 C 20 B

asin A------------ b

sin B------------ c

sin C-------------= =

xsin X------------ y

sin Y------------ z

sin Z------------= =

psin P------------ q

sin Q------------- r

sin R------------= =

B

MN 20 m

49°34°

N

12°5°

W E8.1

4A➔

4G


598 A n s w e r san

swer

sChapter review1 a 0.7193 b 4.2303 c 2.7400

d 8.1955 e 21.9845 f 14.29982 a 54° b 51° c 53°3 a 78°31′ b 26°34′ c 14°54′4 a 37.9 cm b 3.8 m c 13.6 cm

d 11.7 cm e 14.7 cm f 14.6 mg 1.5 m h 4.7 cm i 15.6 mmj 7.5 m k 10.7 m l 5.3 km

5 8.5 m 6 2.5 km 7 63.9 m8 a 57° b 27° c 68°9 a 23°4′ b 61°37′ c 59°35′

10 39° 11 24°12 9.38 m13 a 12.59 km b S36°10′E14 2783 m15 a 1.67 cm b 81.7 mm c 9.81 km16 12.4 cm17 a 52° b 21° c 68°18 809 cm2

19 a 8.64 m b 8.80 m c 11.8 cm20 84.0 cm21 985 m

CHAPTER 5 Graphing periodic functionsExercise 5A — Period and amplitude of a periodic function12

3

45

6 A7 a Oestrogen

b 28 days

Investigation — Ferris wheeling1 2 2 0.5

Exercise 5B — Radian measure

1

2

34 E5 a 0.855 b 1.361 c −2.182

d 3.334 e 4.084 f 5.707g 2.967 h 3.787

6 a 20° b 84° c 180°d 55° e 894° f −155°g 233° h 458°

Exercise 5C — Exact values

1

2

3

4

5

67

8

Exercise 5D — Symmetry1234

a 4 b 1

a T = 4π A = 2b T = 2π A = 1c T = 3π A = 1.5

d A = 4

e T = 2π A = 2

f A = 3

g T = π A = 2.5

h A = 0.5

T = 2 A = 2

a Approximately periodicb 12 monthsc Februaryd Auguste 18°

T3π2

------=

T4π3

------=

T2π3

------=

14---

a b c

d e f

g h 2π i

j k l

m n

a 36° b 120° c 40° d 220°e 648° f −30° g −45° h 67.5°E

a b c d

e f g 1 h

a b c d

e 1 f g h

a P b P c P d Pe N f N g N h Ni P j N

a P b P c N d Ne N f P g P h Pi N j N

a P b P c N d Ne P f P g N h Ni N j P

a Quadrant 3 b Quadrant 1

a C b B c Ad B e C

a −1 b −1 c 0 d 0e 0 f 1 g Undefined h 0i −1 j 1

a 0.63 b −0.63 c −0.63 d −0.63

a −0.25 b −0.25 c 0.25 d 0.25

a −2.1 b −2.1 c 2.1 d −2.1

a –0.3 b −0.7 c −0.9 d −0.3e 0.3 f 0.7 g 0.7 h −0.9i 0.9

π6--- π

4--- π

3---

π9---

5π18------ π

2---

3π2

------ 5π6

------

5π–4

--------- 7π3

------ 5π3

------

4π15------ 2π

5------

32

-------1

2------- 1

3------- 1

2---

12--- 3

2------- 3

1

2------- 1

2--- 3

12---

32

-------1

2------- 1

3-------


A n s w e r s 599

answ

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5

6

7

8

9

Exercise 5E — Trigonometric graphs1

2

3 a i π ii 1

b i 2π ii 2

c i 4π ii 3

d i π ii 4

ei ii

f i 4π ii

g i 6π ii 5

h i 5π ii 4

i i ii 2

j i π ii 3

4

5 E

6

a b − c d −

e − f − g − h −

i j − k 0 l −1

a − b c −1 d

e f − g − h −

i − j k −1 l −1

a –0.383 b −0.924 c 0.414 d 0.924e 0.383 f −0.414

a 0.966 b −0.259 c −3.732 d −0.966e 0.259 f −3.732

a 0.644 b −0.765 c −0.842 d −0.644

a i 4π ii 2 b i π ii 1

c i 3π ii 1.5 d i ii 4

e i 2π ii 2 f i ii 3

g i π ii 2.5 h i ii 0.5

a i 2π ii 1 b i 2π ii 3

c i π ii 2 d i ii 4

e i 6π ii f i ii 2

g i 6π ii 0.4 h i ii 3

i i 8 ii 2.5 j i ii 1

k i 2 ii l i 4 ii

12---

1

2------- 3

1

3-------

32

------- 32

------- 3 32

-------

12---

1

2-------

1

2------- 1

2--- 3

2-------

12---

1

3------- 3

2------- 3

2-------

1

3------- 1

2-------

3π2

------

4π3

------

2π3

------

2π3

------

12---

π2---

2π5

------

π3---

15--- 1

4---

y

x

–1

1

0 π–2π

–2π —

2π3

y

x0

2

–2

2π

y

x

–3

3

0 π2 π4

a D b C c A

a y = 1.5 sin b y = 2 cos 2x

c y = 5 sin d y = 4 cos

e y = −sin f y = −3 cos 3x

π

y

x0

4

–4

–4π

–2π

—4π3

y

x02 —3 π–

3 π

1–2

1–2

–

2π3

------12---

y

x0 2π π 3π 4π

2–3

2–3–

23---

y

x0

5

–5

3 6π π

2π π 3π 4π

y

x0

4

–4

y

x0

2

–2–8π

–4π –

2π

—8π3

π2---

y

x0

3

–3

–2π π

2x3

------

x2--- 2x

3------

3x2

------ 5A➔

5E


600 A n s w e r san

swer

s7

8

9 a f: [0, ] → R, f(x) = 3 sin

b f: [0, 5π] → R, f(x) = cos

c f: [−1, 1] → R, f(x) = 2 sin πx

d f: [−1, 3] → R, f(x) = 1.8 cos

e f: [0, 3] → R, f(x) = −3 sin

f f: [− , 1] → R, f(x) = −2.4 cos

10

Investigation — How high?Check with your teacher.

Exercise 5F — Applications1 a i 1 kg ii 6 days

b W = cos + 3

2 a 110 beats/minb i 50 ii 60 min

c H = 50 sin + 110

3 a 1.6 m b i 1 m ii 0.7 m4 a 26°C at 2 pm

b i 18°C ii 22°C iii Approx. 11.1°C

5 a i 12 mm ii s

b 10c −11.41 mm; if the displacement is positive to the

right then the string is 11.41 mm to the left (or vice versa)

67

8 a i 6 m ii 3 m b Yes, by approx. 24 minutes.

Chapter review1 T = 3.5 A = 22 f =

3 A periodic function repeats itself over time.

4 a b c d

5 a 30° b 216° c 300° d 172°6 E 7 E 8 E 9 B 10 D

11 A 12 E13 a b c 1

14 a b c

15 a 0.69 b 0.69 c 0.6916 D 17 C 18 E 19 A20

21

a

b c

d

e

f

a b

c d

y

x0

1

–1

2π π–2π –π –2π

–4π –

4π

–2π

—4π7

—2π3

—4π5 —

4π3 —

4π3 —

4π5

—2π3

—4π7–

–

– –

–

–

y

x0

3

–3

4π–4π

y

x0

2

–2

π—3π2 —

3π4 2

y

x0

3

–3

ππ 2

y

x0

1.5

–1.56π–6π

y

x0

4

–4

ππ–2

y

x0 2π–π π

2–3

3–2

3 3——4–

–

–

y

x0

1.8

–1.8

6–6

y

x0 62 4

–1.4

1.4

y

x0

3

8

–3

8π6

------ 3x2

------

52---

2x5

------

πx2

------

2πx3

---------

13---

3πx2

---------

a 40 m b 3.9 s c 7.8 sa 60 s b 50 m c 100 md 314.16 m e i 50 m ii 75 m

a b

a i 2800 ii 1200 iii 2000b i 12 months ii 0.8 iii 1.25

y

x–1

1

0 π ππ

π–2

π3 —2

π–4

2π7 —4( , –1)

π5 —4

( , 1)

π –4

( , 1)

π3 —4( , –1)

(2 , 0)π ( , 0)

y = tan x

Vertical asymptotes

πt3-----

πt30------

110------

16---

π4--- 5π

6------ 5π

18------ 13π

9---------

12--- 1

2---

1

2------- 3

2------- 3

y

x0π– π π2 π3 π4π–2

4

–4

y

x0

1.5

–1.5

π–2π–π –

2π–


A n s w e r s 601

answ

ers

c

d 1600e 4 monthsf Approx. 5 months

CHAPTER 6 Trigonometric equationsExercise 6A — Simple trigonometric equations1 a 37° and 143° b 104° and 256°

c 80° and 260° d 238° and 302°e 79° and 281° f 140° and 320°g 199° and 341° h 41° and 319°

2 a 60° and 120° b 45° and 315°c 60° and 240° d 210° and 330°e 120° and 240° f 150° and 330°g 225° and 315° h 30° and 330°

3 C 4 Third quadrant, as sin is negative and tan is positive

in this quadrant.5 a The value of sin θ can never be greater than 1.

b There is no maximum value to tan θ.6 a 90° b 90° and 270°

c 0°, 180° and 360° d 0°, 180° and 360°e 180° f 270°

7 a 30° and 150° b 70° and 290°c 135° and 315° d 240° and 300°e 150° and 210° f 54° and 234°

Exercise 6B — Equations using radians1 a 0.93 and 2.21 b 2.09 and 4.19

c 0.98 and 4.12 d 5.95 and 3.47e 0.79 and 5.50 f 2.77 and 5.91

2 a and b and c and

d and e and f and

3 a 0, π and 2π b 0, π and 2π c and

d e 0 and 2π f π

g

4 a and b and c and

d and e and f and

5 D6 a 14° and 166° b 132° and 228°

c 74° and 254° d 270°e 78° and 282° f 108° and 288°

Exercise 6C — Further trigonometric equations1 a 2.29, 3.99, 8.57, 10.27

b 1.14, 2.00, 7.42, 8.28c 1.07, 5.21, 7.35, 11.49d 3.52, 5.90, 9.80, 12.19e 0.53, 3.67, 6.81, 9.95f 2.02, 5.16, 8.30, 11.44

2 a −6.01, −3.41, 0.27, 2.87b −3.88, −2.41, 2.41, 3.88c −3.59, −0.45, 2.69, 5.83d −2.57, −0.57, 3.71, 5.71e −4.90, −1.38, 1.38, 4.90f −5.03, −1.89, 1.25, 4.39

3 a , , , b , , ,

c , , , d , , ,

e , , , f , , ,

4 a −4.71, 1.57 b −6.28, 0, 6.28c −5.15, −1.13, 1.13, 5.15d −5.76, −3.67, 0.52, 2.62e −5.50, −0.79, 0.79, 5.50f −3.67, −2.62, 2.62, 3.67

5 a 3 b 2.29 seconds

Investigation — Fishing1.00 am, 5.00 am, 1.00 pm and 5.00 pm

Exercise 6D — Identities1

2 a 0.6 b 1.3333 a 0.954 b 3.184 a ±0.917 b ±0.714 c ±0.971 d ±0.436

5 a 2 b c

6 a 6 b c

7 a b c d

8 a B b D c C d A9 a 70 b 32 c 51 d 8

e 82 f 46 g 1 h 7310

0

2.82

1.2

3 6 9 12

P (thousands)

t (months)

12---

π3--- 2π

3------ 2π

3------ 4π

3------ π

4--- 5π

4------

π4--- 7π

4------ 2π

3------ 5π

3------ 5π

6------ 11π

6---------

π2--- 3π

2------

π2---

3π2

------

π6--- 5π

6------ π

6--- 11π

6--------- π

4--- 5π

4------

4π3

------ 5π3

------ 3π4

------ 5π4

------ 2π3

------ 5π3

------

q 30o 81o 129o 193o 260o 350o -47o

sin2 θ 0.5 0.99 −0.78 −0.22 −0.98 −0.17 −0.73

cos2 θ 0.87 0.16 −0.63 −0.97 −0.17 −0.98 −0.68

sin2 θ + cos2 θ 1 1 −1 −1 −1 −1 −1

sin q 0.8 0.28 0.954 0.77 0.954 0.573

cos q 0.6 0.96 0.3 0.64 0.477 0.82

tan q 1.33 0.29 3.18 1.2 2 0.7

11π6

---------– 7π6

------–π6--- 5π

6------ 3π

2------– π

2---–

π2--- 3π

2------

2π3

------– π3---–

4π3

------ 5π3

------ 4π3

------– π3---–

2π3

------ 5π3

------

7π4

------– π4---–

π4--- 7π

4------ 11π

6---------– 5π

6------–

π6--- 7π

6------

53

------- 23---

2 78

---------- 34---

513------– 4

5---–

2425------ 1

2---–

5F➔

6D


602 A n s w e r san

swer

s

11 a b c d 2

e f g h

i j k l

Investigation — Further trigonometric identitiesCheck with your teacher.

Exercise 6E — Using the Pythagorean identity

1 , , ,

2 a 0, , π, 2π b , π,

c 0, , , π, 2π d , π,

e f , ,

3 , ,

4 a , , b , ,

c , , , d , π,

e , π, f ,

Chapter review1 a 64°, 116° b 114°, 294° c 58°, 238°

2 a 240°, 300° b 60°, 300° c 45°, 225°

3 a 270° b 180° c 135° 315°

4 a −240°, −120°, 120°, 240°b −246°, −66°, 114°, 294°c −288°, −252°, 72°, 108°

5 a 0.76, 2.37 b 2.59, 3.69 c 0.20, 3.34

6 a , b , c ,

7 a , b , c ,

8 a 49°, 131° b 139°, 221° c 72°, 252°

9 a 2.05, 5.12, 8.33, 11.40b 3.67, 5.75, 9.95, 12.03c 2.44, 5.58, 8.72, 11.86

10 a −2.19, −0.95, 4.09, 5.33b −5.15, −1.13, 1.13, 5.15c −4.17, −1.03, 2.11, 5.25

11 a −4.10, −2.18, 2.18, 4.10

b −2π, −π, 0, π, 2π c , , ,

12 a −0.87 b 12

13 a 60 b 62

14 0, , , π, 2π

15 a 0.84, , , 5.44 b

c , , d 0.77, 2.26, 3.59, 5.83

CHAPTER 7 Exponential and logarithmic functions Exercise 7A — Index laws1

2

3 a b c 18u11v5 d 10e9f 2

e f x13 g −81x9y3 h 3m4p

i mp2 j

4

5 a A b E c B

67

8

9

10 E

Exercise 7B — Negative and rational powers

1

45--- 4

3--- 1

5-------

116

---------- 56--- 1

5------- 4

5---

56--- 2

5------- 3

5---–

115

----------

π3--- π

2--- 3π

2------ 5π

3------

π2--- π

2--- 3π

2------

4π3

------ 5π3

------ π3--- 5π

3------

π2--- π

2--- 7π

6------ 11π

6---------

π6--- 7π

6------ 11π

6---------

π2--- 7π

6------ 11π

6--------- π

6--- 5π

6------ 3π

2------

π4--- 3π

4------ 5π

4------ 7π

4------ π

3--- 5π

3------

π2--- 3π

2------ π

6--- 5π

6------

2π3

------ 4π3

------ 2π3

------ 5π3

------ 7π6

------ 11π6

---------

π3--- 2π

3------ 2π

3------ 4π

3------ 3π

4------ 7π

4------

5π3

------– 2π3

------–π3--- 4π

3------

a x10 b m5p5 c 518 d 8y11

e x7y8 f 46x18 g 6m9p17 h 57x10y13

a a5b3 b c d p11q2

e f a2b2 g h 5a3b3

i

a b c 16p17

d 27j3n2 e x3y4 f 12

a x3yn + 1z b x5y6m − 1

a 211 b 324 c 55 × 34

d 222 × 58 e f

a 8 b 59 049 c 16 d 1

e 5 f g 16 h

a 22 × 33n + 4 b 53n − 6

c 23x − 2 d 2−5n − 3 × 3−6n e 2n − 2 × 76n − 3

f 5−1 × 72 or g 25n − 6 × 39n − 3 h 3

i −

a b c d

e f g h

π6--- 5π

6------

π2--- 3π

2------ π

2---

π6--- 5π

6------ 3π

2------

a9b5

4----------- x3y9

2----------

8m2n5

7--------------- r10t

5---------

6x2

5--------

4 p10m8

3------------------ 9xy4

2-----------

2wt6

9-----------

u11v4

9-------------

154------

125k11d6

24----------------------

324-----

52---

15--- 1

3---

495

------

23---

163----- 1

54----- 52

32----- 45

75-----

921

646-------- 1

3---– 212

316-------


A n s w e r s 603

answ

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2

3

4 a D b C c E

5

Exercise 7C — Indicial equations1

2

3

4

5

6 A7 D8 B

9

10 E

Investigation — Simulating radioactivity

5 N =

Exercise 7D — Graphs of exponential functions1

2

a b c

d e f

a 3 b 3 c 5 d 2

e 4 f 27 g 625 h

i j k l

m n o p

a b c

d e f

g h i

j x k l x

m n o

pq r

a 5 b 4 c 5 d 2e −2 f −3 g 4 h 0

i 2 j k l

a b − c −1 d −

e f

a −3 b c − d

e f g − h

a b 7 c − d

e −3 f − g 1 h 5

i 3 j −9

a 0 or 1 b 1 or 2 c 2 or 3 d 0 or 1e 0 or 1 f 1 or 2 g 3 h 0 or 1

a 1.58 b 3.58 c 5.49 d 1.65e 1.28 f 1.66

22–1

xy2-------- p2

m-----

x10 213 38×1

4x6y2--------------

25---

23--- 125

64--------- 9

16------ 1

4---

127------ 9

4--- 11

4------ 6

5---

373---

x56---

x38---

x76--- y

12---

x23---

---- 274---

1

274---

-----1

323---

-----1

2 353---( )

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

64m10

x2 x 1+( )32---

x 1–

x12---

-----------

2 x 1+( )

x 2+( )12---

--------------------y 4–( )

32---

p 3+( )35---

12--- 3

4--- 4

3---

53--- 1

2--- 9

4---

116

------ 98---

107

------ 185

------ 34---

2111------ 4

11------ 5

2--- 25

9------

910------ 5

7--- 19

4------

111------

100 56---( )t

a b

c d

e f

g h

i j

k l

a

y = 0, (0, 2)

b

c

y = 0, (0, 0.5)

d

y = 3xy

x

3

1

10

y = 5xy

x

5

1

10

y = 6xy

x

6

1

10 x

y = 10xy

10

1

10

x

y = 2–xy

2

1

–1 0 x

y = 4–xy

4

1

–1 0

x

y = –3x

y

–3

–10 1

x

y = –2x

y

–2–1

0 1

x

y = –3–x

y

–3

–1 0–1

x

y = 0.5x

y

1

1

1–2

0

x

y = 2.7xy

1

2.7

10x

y

1

1

2–3

y = ( )x2–3

0

x

y = 2(3x)y

2

6

10 x

y = 3(2x)y

3

6

10

x

y = 0.5(4x)y

0.5

2

10 x

y = 4(5x)y

4

20

10 6E➔

7D


604 A n s w e r san

swer

s

3

4 a A b E

5

Investigation — A world population model1 The TI-83 gives the equation of the curve as

y = 0.121 × 0.00149x. Using the spreadsheet ‘Exponential model’, a close fit is obtained with y = 0.6 × 1.01x–1800.

2 Graphics calculator: 2.56 billionSpreadsheet: 4.6 billion.

3 The equation does not take into account other factors such as greater birth control or lack of food which may cause the population to level off.

4 Increasing A or a increases the steepness of the curve, b affects horizontal translation (+ = left, – = right), and B is the vertical translation (+ = up, – = down).

Investigation — Bode’s Law

Distance = 0.16 × 2Planet number

Exercise 7E — Logarithms1

2

3 D 4 C 5 A

6

7

8

9

14

e

y = 0, (0, )

f

a y = 0, (0, )

dom = R, ran = (0, ∞)

b y = 0, (0, 9)dom = R, ran = (0, ∞)

c y = 0, (0, 5)dom = R, ran = (0, ∞)

d y = 3, (0, 4)dom = R, ran = (3, ∞)

e y = −3, (0, −2)dom = R, ran = (–3, ∞)

f y = −1, (0, 7)dom = R, ran = (–1, ∞)

g y = 3, (0, 4)dom = R, ran = (3, ∞)

h y = 5, (0, 105)dom = R, ran = (5, ∞)

i y = −2, (0, −1 )

dom = R, ran = (–2, ∞)

j y = 1, (0, −3)dom = R, ran = (–∞, 1)

a 3.32 b 4.39 c −0.51d 1.30 e 0.90 f 1.65g 1.37 h 2.26 i 2.44, −2.86j 1.56, −3.99

x

y

1

1–4

y = (2x)1–4

1–2

0

14---

x

y

4

1

4–3

y = 4( )x1–3

0

12---

x

y y = 2x – 1

1–2

0x

y y = 3x + 2

0

9

x

y

5

y = 51 – x

0 x

y = 2x + 3

y

4

3

0

x

y = 3x – 3y

–2

–3

0

x

y = 2x + 3 – 1

y

–1

7

0

x

y = 6–x + 3

y

4

3

0 x

y = 102 – x + 5y

15

5

0 1

8081------

x

y = 3x – 4– 2

y

–2

0

(0, –1 )80—81

x

y = –2x + 2 + 1

y

0

1

–3

–1

–1

a log2 8 = 3 b log3 243 = 5c log5 1 = 0 d log10 0.01 = −2e logb a = n f log2 = −4

a 42 = 16 b 106 = 1 000 000

c 2−1 = d 33 = 27

e 54 = 625 f 27 = 128

g 3−2 = h bx = a

a 4 b 4 c 3d −2 e 3 f −5g −2 h −5 i 5j −6 k Undefined l 5a log2 80 b log3 105 c log10 100 = 2d log6 56 e log2 4 = 2 f log3 3 = 1g log5 12.5 h log2 3 i log4 5

j log10 k log3 4 l log2 3

m log3 20 n log4 2 =

a log10 250 b log2 1728 c log3 4 d log5 3 e log10 f log3 2

g log2 = −1 h log2 [x3(x − 4)]

i log3 64 j log10

a b 2 c 2 d

e f 2 g 3 h 6

i 3 j 610 E 11 B 12 D 13 C

a log3 81 = 4 b log4 1024 = 5 c log5

d log10 (100x3) e log2 f log3 4

g log6 9 h log10

116------

12---

19---

14---

12---

14---

12---

x 3+( )2

x 2–-------------------

23--- 3

2---

94---

825------

258

------

10x6( )


A n s w e r s 605

answ

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Exercise 7F — Solving logarithmic equations1

2

3

45

6 B7 C

Investigation — Logarithmic graphs1

2 a Reflectionb x = 0 (that is, the y-axis)

3 a Increasing A produces a steeper graph.b Increasing a produces a less steep graph.c b is the translation (of y = logax) in the x direction

(+ = left, – = right).d B is the y translation (+ = up, – = down).

4 a

b

c

Exercise 7G — Applications of exponential and logarithmic functions1 a P = 10 000(2t)

b i 80 000 ii 640 000 c 3.32 months2 a 15 000

b i 15 529 ii 16 077c 17 838 d 2022

3 a 3 kgb i 3.77 kg ii 5.1 kg iii 5.86 kgc d 26 weeks

4

56 a 90°C

b i 76.3°C ii 64.7°C c 12 min 37 s7 a 120

b i 145 ii 176 iii 213

c 11 years d

e Population will reach a limit at some stage.8 a 80 kg

b i 72.1 kg ii 52.8 kg c 67 weeksd i

ii No, the model suggests virtually no rubbish will be disposed of in 10 years or so which is unlikely.

9 a i 108 hectares ii 120 hectares iii 148 hectaresb 15 h

10 a i 51.18 m ii 51.65 m iii 51.95 m iv 52.2 mb c 67

11 a 400 b i 1005 ii 6340 iii 16 643 iv 17 146c 17.48 months

12 a h = 5(0.7n) b i 1.20 m ii 0.29 mc

13 a V = 5000(1.1t) b $8857.81 c 8 years

a 2 b 0 c 3 d 4e −1 f −2 g 6 h 2i 2 j 3a 8 b 9 c 625 d 10

e f g 64 h 10 000

i 30 j 5 k 5 l

m 24 n 10a 6 b 5 c 4 d 10e 2 f 25 g 2 h 1a B b A c E d Ca 3.459 b −0.737 c 2.727 d 0.483e 1.292 f −3.080 g 2.255 h 0.262i 0.661 j 0.431 k −0.423 l 2.138

18--- 1

27------

63

-------

y

x1 5432

1

y

x1084 62

1

432

–1

Asymptote

y

x1084 62

1

–2–2

–4

–4

–6

–8

–10Asymptote

y

x584 62

1

–1–1

–2

–3

–4

–5

76

Asymptote

a A = P(1.05)n b $16 288.95 c 20 years

a $25 000 b $14 427 c 10 years

W

t0

W = 3 log10

(8t + 10)3

–5–4

12--- N

t0

N = 120(1.1t)120

W

t0

W = 80(2–0.015t)80

d

n0

5152

d = 50 + log10(15n)

1 2 3 42–3

h

h = 5(0.7n), n ≥ 0, n ∈ J

n

54321

1 2 3 4 50

7E➔

7G


606 A n s w e r san

swer

sInvestigation — The decibel1 About 32 000 times as intense.2 Heavy traffic is about 32 times as intense.3 100 times as loud.

Investigation — The Richter scale1 a 1.1 b 1.3 c 1.42 d 1.77 e 2.43 f 3.12 No3 a 22 387 211 kJ

b 707 945 784 kJc 2.24 × 1010 kJ

4 Energy is multiplied by 32.5 The amount of energy in a magnitude 8 quake is

2.24 × 1013 kJ, which is 30 000 times more than for a magnitude 5 quake.

Chapter review1 B 2 C 3 4 A 5 B6 E78 x = 0.63 or 19 C 10 A 11 D 12 E

13

c

d dom = R ran = (1, ∞)14 C 15 B16

17

18 C 19 A 20 D 21 B 22 E23 C2425 0.86126 x = 0.36627 a 1500

b i 2800 ii 5200c 14.36 days

28 a i 20 ii 25b i L = 25, C = 28 ii L = 28, C = 30c Lions by 1 year 1 monthd 31 after 1 year 11 months

CHAPTER 8 Applications of exponential and logarithmic functions in financial mathematics

Exercise 8A — Geometric sequences1 a Not geometric

b Geometric, ratio = 3; t4 = 108; tn = 4 × 3n − 1

c Geometric, ratio = 2; t4 = 24; tn = 3 × 2n − 1

d Geometric, ratio = ; t4 = 13 ; tn = 3n − 123 − n

e Geometric, ratio = − ; t4 = ; tn = (−3)2 − n

f Geometric, ratio = −3; t4 = −54; tn = 2 × (−3)n − 1

g Geometric, ratio = ; t4 = ; tn = × n − 1

h Geometric, ratio = 2; t4 = 6; tn = 3 × 2n − 3

i Not geometric

j Geometric, ratio = −6; t4 = −54; tn = × (−6)n − 1

k Geometric, ratio = 2π; t4 = 16π4; tn = (2π)n

2 a tn = 5 × 2n − 1, t6 = 160, t10 = 2560b tn = 2 × 2.5n − 1, t6 = 195.31, t10 = 7629.39c tn = 1 × (−3)n − 1, t6 = −243, t10 = −19 683d tn = 2 × (−2)n − 1, t6 = −64, t10 = −1024e tn = 2.3 × (1.5)n − 1, t6 = 17.47, t10 = 88.42f tn = × 2n − 1, t6 = 16, t10 = 256

g tn = × n − 1, t6 = , t10 =

h tn = × n − 1, t6 = − , t10 = −

i tn = x × (3x3)n − 1, t6 = 243x16, t10 = 19 683x28

j tn = × n − 1

, t6 = , t10 =

3 a There are two possible answers because the ratio could be −3 or 3. The nth term is tn = 2 × 3n − 1 or tn = 2 × (−3)n − 1, t10 = ±39 366.

b There are two possible answers because the ratio could be −2 or 2. The nth term is tn = 2n − 1 or tn = (−2)n − 1, t10 = ±512.

c The nth term is tn = 5 × 2n − 1, t10 = 2560.d The nth term is tn = −1 × (−2)n − 1, t10 = 512.e There are two possible answers because the ratio

could be or . The nth term is tn = 35 − 3n or

tn = (−3)5 − 3n, t10 = ±3−25.

4 ±

5

6 m = 12, n = 48

7 m = 36, n =

8 a = 300, b = 0.759 t1 = 25, r = ±2, tn = 25 × 2n − 1 or tn = 25 × (−2)n − 1

10 t1 = , r = , tn = 3n − 221 − n

11 −6

12 2, , , or −2, , −

13 a b

14 k = 6

Exercise 8B — Geometric series1 a 31, 1023, 1 048 575

b 121, 29 524, 1.74 × 109

c 33, −1023, −1 048 575d −4, 103.8, 746.8

e , 46.5, 1534.5

f − , − , −

a x = 2.187 b x = 5

a (0, ) b y = 1

a −3 b

a log4 b 12

a x = 216 b x = 5 c x =

4y2

3x4--------

109

------

y = 3x – 2+ 1y

x

(2, 2)1

0

(0, )10—9

yx 1+( )2

x-------------------=

12536

---------

143

------

32--- 1

2---

13--- 1

9---

32--- 27

28------ 2

7--- 3

2---

14---

12---

13--- 1

4---

13072------------ 1

786 432-------------------

35--- 1

3---–

1405--------- 1

32 805----------------

1x--- 2

x---

32

x6

------ 512

x10

---------

127------–

127------

34---

3 2n 1–( )

2----------------

×

7294

---------

13--- 3

2---

12--- 1

8--- 1

2--- 1

8---

32---

24

2n------

32---

23--- 31

24------ 341

256---------


A n s w e r s 607

answ

ers

g 12 930, 70 972, 1 302 280h −309, 2239, −1865i 15 620, 48 828 120j −1.375, −1.332

2 a 3108 b , 63 , 66 535

3 458 4 9 5 121 875 0006 65 024 7 21 504

8 a 2 b c d 3 e

9 1.0; 50%, 25%, 12.5%

10 ; 75%, 18.75%, 4.6875%

11 4; 25%, 18.75%, 14.06%

12 a 6[1− 20] = 5.999 994 278

b 5.722 × 10−6

c 6

13 a 6 [1 − (− )9] = 6.750 343

b −3.425 × 10−4

c 6

14 6

15 16, 8, 4

16 ,

17 −

18

19 1, 0.9920 Various answers

Crossing the road1 No 2 No

Exercise 8C — Growth and decay functions1

2 Individual graphs to verify.3 a Not exponential growth

b, c, d Exponential growth

4

5

6

7

8

9

10 a A = 2000(1.08)T b $3173.7511 a V = 12 000(1.04)T b $17 762.93

15

16 Individual graphs to verify.17 a, b, d Exponential growth

c Not exponential growth18

19 B20

21

22 a N = 1400(0.98)12T b 1098 cockatoos23 a V = 899(0.7)T b $151.09

26 a Graph b 7 years c 1380

a V = 50 + 10T i $200 ii 5 yearsb P = 600 + 30T i 1320 ii 10 monthsc A = 1200 + 120T i 2160 ii 5 yearsd V = 1.560 + 78T i 2223 ii 5.64 years

a N = 1000(2)T b N = 2000(1.4)T

c N = 860(1.25)T d N = 1250(1.5)T

e N = 2300(2)T

a V = 700(1.1)T b V = 1100(1.05)T

c V = 5000(1.16)T d V = 2750(1.2)T

e V = 380(1.08)T

a C = 25 000(1.07)T b C = 1.5(1.03)T

c C = 250(1.05)T d C = 29.95(1.06)T

e C = 72(1.1)T

a A = 2000(1.16)T b A = 850(1.12)T

c A = 1900(1.06)T d A = 25 000(1.09)T

e A = 12 600(1.15)T

116------ 15

16------ 15

16------

23--- 3

2--- 3

5---

43---

12---

34--- 1

3---

34---

23---

4 15–3 5+( )15 3–( )

------------------------- 4 3 3 5+( )3

--------------------------------=

16---

13---

a b c d

T A ($) T V ($) T N T C ($)

0 1500 0 850 0 400 0 17 000

1 1620 1 901 1 408 1 18 700

2 1749.60 2 955.06 2 416.16 2 20 570

3 1889.57 3 1012.36 3 424.48 3 22 627

4 2040.73 4 1073.11 4 432.97 4 24 889.70

5 2203.99 5 1137.49 5 441.63 5 27 378.67

6 2380.31 6 1205.74 6 450.46

7 2570.74 7 1278.09 7 459.47

a 6605.40 b 738.75c $6580.94 d $1560.24e 4 017 915 bacteria

12 E 13 D 14 Ca N = 500 − 35T i 80 ii 7.14 yearsb V = 12 000 − 800T i $7200 ii 4 yearsc h = −5T + 300 i 225 cm ii 7 minutesd i 17 wombats/year ii 230 iii 12 yearse i $190/year ii $1170 iii 3 years

a A = 300( )T b V = 5000(0.75)T

c N = 2500(0.95)T d m = 900(0.8)T

e V = 850(0.92)T f N = 15 000(0.75)T

a 365.40 b 4343.88c 160.18 d 86.73e 3.67a b c d

T m T V T N T A

0 840 0 2600 0 290 0 1350

1 638.4 1 2080 1 263.9 1 945

2 485.18 2 1664 2 240.15 2 661.5

3 368.74 3 1331.2 3 218.54 3 463.05

4 280.24 4 1064.96 4 198.87 4 324.14

5 212.98 5 851.97 5 180.97 5 226.89

6 161.87 6 681.57 6 164.68 6 158.83

7 123.02 7 545.26 7 149.86 7 111.18

8 93.49 8 436.21 8 136.37 8 77.82

24 C 25 D

12---

14---

8A➔

8C


608 A n s w e r san

swer

sExercise 8D — Compound interest formula1

2

3

4

5

6

10 a I = $1760b I = $2071.83; Best optionc I = $2064.99

11 a A = $12 975.98b A = $13 743.08c A = $13 747.87; Best option

14

15

16

21

22

29 a i $2959.87 ii 12.36% p.a.b i $2997.20 ii 12.08% p.a.

30

31

39

Exercise 8E — Loan schedules1234567

8

9

10

11

12

13

1415

16

17

18

19 B 20 D 21 E 22 B

2324

Investigation — Loan schedules using spreadsheets1 $60 406 2 $71 1413 a 150 months b $134 566

c You can save $3186 by paying an extra $50 per month.

Investigation — Spreadsheets and investing for the future1 $74 066 2 $131 241 3 $725

a $583.20 b $1630.47 c $4472.27d $3764.86 e $939.15 f $1369.50a i $2519.42 ii $519.42 b i $8837.34 ii $1837.34c i $6615.42 ii $615 d i $3059.97 ii $1159.97a 5 b 20 c 8 d 72e 9a 1.5% b 2% c 5.5%d 1.5% e 1.75%a $3514.98 b $2687.83 c $8061.13d $3431.89 e $3073.14a $605.61 b $903.60 c $1314.83d $353.44 e $795.76

7 E 8 B 9 C

12 $764.08 13 $1880.88a $3542.13 b $2052.54 c $2969.18d $5000 e $3100a $2069.61 b $1531.34 c $2010.82d $3564.10 e $5307.05a $930.39 b $468.66 c $889.18d $2035.90 e $4692.95

17 B 18 D 19 $1351.56 20 $10 292a 8% b 6% c 12% d 7%e 7.8%a 13.98% b 5.93% c 5.62% d 6.84%e 10.04%

23 19.03% 24 9.06% 25 8.34%, i.e. D26 13.12%, i.e. B 27 Yes 28 Yes

a 19, 9 years b 7, 3 yearsc 20, 5 years d 24, 6 yearse 57, 4 yearsa 13, 3 years b 14, 7 yearsc 32, 8 years d 13, 6 yearse 35, 2 years 11 months

32 13, 13 years 33 5, 2 years 34 53, 4 years 5 months

35 26, 6 years 36 B 37 D 38 D

a 4 years b $9718.11

a $204 b $30.20a $306.61 b $49.85a $514.94 b $101a $189.01 b $68.60a $783.58 b $228.32a $666.85 b $287.44a i $18 750.82 ii $1249.18 iii $975.27

12--- 1

2---

34---

14---

12---

14--- 1

2---

12---

b i $16 048.36 ii $3951.64 iii $2769.91a i $29 714.91 ii $285.09 iii $747.16b i $27 994.62 ii $2005.38 iii $4748.82

a i $17 631.65 ii $2368.35 iii $953.10b i $19 362.07 ii $637.93 iii $987.37c i $19 556.52 ii $443.48 iii $991.22d i Decreases ii Increases iii Increases

a i $26 725.05 ii $3274.95 iii $3590.30b i $27 660.46 ii $2339.54 iii $3635.91c i $28 264.47 ii $1735.53 iii $3665.37d i Decreases ii Increases iii Increases

a i $49 493.44 ii $506.56 iii $1593.96b i $49 647.28 ii $352.72 iii $1595.80c i $49 836.73 ii $163.27 iii $1598.05d i $49 637.19 ii $362.81 iii $2194.07e i $49 767.60 ii $232.40 iii $2196.20f i $49 912.73 ii $87.27 iii $2198.57



a Increases b Increases c Increases

a i $262.27 ii $371.11b i $529.40 ii $103.98

a i $103.22 ii $398.64b i $427.87 ii $73.99

a i $264.10 ii $2384.84b i $2901.10 ii $557.84

a i $18.49 ii $449.82b i $66.97 ii $401.34c i $244.93 ii $223.38

a $50 356.80 b $50 179.46 c $522.14

a $81 433.37 b $81 640.94 c $833.73


A n s w e r s 609

answ

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Exercise 8F — The annuities formula1

2

3

4

5

6

7

89

10 D11

12

13

1415

16

17

1819 A, D20 D21

22

23

24 $38 231.1025 $6387.822627

Chapter review1 E2 a t1 = 1.4; t5 = 7.0875

b S10 = 158.66c 33.9%

3 a 134.2 milli-remb 3361.6 milli-rem

4 D 5 C6 89/997 a 4/3 b 3/58 B 9 D

10 $3.0911 A 12 D 13 C 14 D15 c16 $3842.4617 $41118 8 years 1 month19 B20 a i $149 900 ii $149 799 iii $149 696.99

b $4496.9921 A 22 C 23 B24 $60 205.4025 $46 741.4426 a i 0.6%, 1.1%, 1.0%, 0.7% ii 14%

b $2592c $406

27 a $186, $2470 b $1206, $246728 a $144.12 b $8541.6029 a

b During 1997c Annual increment of 1606 insectsd Annual growth rate of 1.095 96

a $41 475.93 b $18 419.91c $51 475.93 d $28 419.91e $8670.97 f $27 815.32g $52 435.82 h $71 419.61a $59 633.49 b $49 884.16c $32 172.59 d $15 200.98a $70 570.81 b $63 313.07c $51 420.42 d $31 932.92a $46 102.98 b $36 196.88c $19 556.12 d $7381.52a $27 564.36 b $29 291.80c $30 958.81 d $32 551.72a $5691.20 b $9087.53c $11 350.93 d $12 930.79e $14 115.16a $6310.40 b $10 774.07c $14 084.40 d $16 626.10e $18 630.33a C b Ba A b D

a $444.24 b $409.53c $403.54 d $323.81e $643.45 f $944.82g $2594.63 h $345.85a i $548.22 ii $1157.28b i $381.60 ii $1737.60c i $298.62 ii $2333.76d i $271.07 ii $2637.78e i $231.18 ii $3257.88a i $986.92 ii $33 430.40b i $874.12 ii $40 873.28c i $764 ii $52 520d i $693.18 ii $64 726.88e i $659 ii $73 160f i $600.76 ii $95 228a Decreases b Increasesa i $1069.67 ii $10 786.80b i $1169.78 ii $14 791.20c i $1274.76 ii $18 990.40d i $1301.74 ii $20 069.60e i $1329.01 ii $21 160.40f i $1412.51 ii $24 500.40a i $180.95 ii $24 070.50b i $204.91 ii $33 414.90c i $230.40 ii $43 356d i $237.01 ii $45 933.90e i $243.70 ii $48 543f i $264.27 ii $56 565.30a i $253.92 ii $170 076.80b i $507.97 ii $170 144.40c i $1101.28 ii $170 307.20d i $3311.33 ii $170 906.40e i $161.57 ii $74 032.80f i $323.23 ii $74 079.60

a Increases b Increases

a $365.84b i $259.59, $106.25 ii $339.75, $26.09a $464.49b i $134.03, $330.46 ii $365.64, $98.85a $323.73b i $46.23, $277.50 ii $299.28, $24.45

a $68 113.56 b $65 473.20 c $5774.12a $48 310.48 b $45 935.32 c $4676.88

Year n

Pop AGrowth

rate= 1.12

Pop BAnnual

increment = 1000 Difference

199019911992199319941995199619971998199920002001

123456789

101112

10 00011 2001254414 04915 73517 62319 73822 10724 76027 73131 05834 785

15 00016 00017 00018 00019 00020 00021 00022 00023 00024 00025 00026 000

5000480044563951326523771262-107

-1760-3731-6058-8785

8D➔

8F


610 A n s w e r san

swer

s30 a ii 96, 115.2, 138.24 ii 14 months

b ii Straight lineii Increase is a constant 1.25 hectares.

c ii $5378.24 ii 2 hectaresd i $14 880 — simple; $15 288.28 — compound

ii Simple. Less interest is charged compared to compound interest calculated on a monthly increasing loan amount

iii $406.44

CHAPTER 9 Presentation of dataInvestigation — Types of data1 Quantitative2 Discrete

Exercise 9A — Types of variables and data1 a Quantitative b Categorical c Quantitative

d Categorical e Quantitative f Quantitative2 a Continuous b Discrete c Continuous

d Continuous e Continuous3 a Quantitative and discrete

b Categoricalc Categoricald Quantitative and continuouse Quantitative and continuous f Quantitative and discrete

4 Categorical5 Categorical6 Quantitative and discrete7 B8 Categorical9 Quantitative and continuous

Exercise 9B — Collection of data1 Census — every member of the population

participates.2 Sample3 a Sample b Sample c Census

d Census e Sample 4 a Sample b Census c Census

d Sample5 Sample6 163, 176, 381, 495, 97 827, 211, 417, 554, 207, 26, 810, 781, 192, 3718 Check with your teacher.9 a Systematic b Stratified c Systematic

d Random e Stratified10 A11 C12 Year 8 — 11, Year 9 — 9, Year 10 — 8,

Year 11 — 7, Year 12 — 513 36 men and 24 women14

Investigation — Census or sample1 Sample — systematic sample 2 Census3 Sample — random sample 4 Census5 Sample — random sample

Investigations — Biased sampling, Women and work, Cost of a houseDiscuss with your teacher.

Exercise 9C — Stem plots1 a 1 2 5 8 12 13 13 16 16 17

21 23 24 25 25 26 27 30 32b 10 11 23 23 30 35 39 41 42 47

55 62c 101 102 115 118 122 123 123 136

136 137 141 143 144 155 155 156157

d 50 51 53 53 54 55 55 56 56 5759

e 1 4 5 8 10 12 16 19 19 21 2125 29

2

3

4

5 C6

7

8Age Male Female

20–29 10 7

30–39 7 8

40–49 12 3

50–59 1 2

Stem012345

Leaf51 8 93 7 91 2 5 6 7 91 2 3 52 Key: 0|5 = $5

Stem3456

Leaf7 92 9 91 1 2 3 7 8 91 3 3 8 Key: 3|7 = 37 yrs

Stem012345

Leaf8 96 8 91 3 6 6 7 8 90 0 2 51 3 54 0|8 = 8 dogs

Stem2233

Leaf35 8 8 9 9 90 0 2 2 2 3 3 45 5 7 8 9 Key: 2|3 = 23 yrs

Stem1111122222

Leaf134 56 78 8 8 9 9 9 90 0 1 1 12 2 3 3 35

9 Key: 1|1 = 11 people

Stem2233

Leaf0 1 2 45 6 8 9 9 91 1 2 3 4 4

Key: 2|0 = 20 hit outs


A n s w e r s 611

answ

ers

9

10

11 a

b

c

12 a

b

c

13 a

Key: 8|1 = 8.1 seconds

b

Key: 8|1 = 8.1 secondsc

Key: 8|1 = 8.1 seconds

Exercise 9D — Frequency histograms and bar charts

1 a

b

c

d

2 a

b

d

Stem1819202122

Leaf5 7 91 5 6 6 7 91 3 3 5 971 Key: 18|5 = 1.85 m

Stem2021222324252627282930

Leaf2 3 50 2 71 832 4 62 5

3 50 0 8

Key: 20|2 = $202 000Stem

45

Leaf3 7 7 8 8 9 9 90 0 0 0 1 2 2 3 Key: 4|3 = 43 cm

Stem4455

Leaf37 7 8 8 9 9 90 0 0 0 1 2 2 3

Key: 4|3 = 43 cm

Stem4444455555

Leaf

3

7 78 8 9 9 90 0 0 0 12 2 3

Key: 4|3 = 43 cm

Stem12

Leaf5 6 7 7 7 8 9 9 9 90 0 0 1 1 1 2 3 3 3

Key: 1|5 = 15 mm

Stem112

Leaf

5 6 7 7 7 8 9 9 9 90 0 0 1 1 1 2 3 3 3

Key: 1|5 = 15 mmStem

1111122222

Leaf

56 7 7 78 9 9 9 90 0 0 1 1 12 3 3 3

Key: 1|5 = 15 mm

Stem89

Leaf1 2 5 9 9 90 1 2 3 5 7

Stem8899

Leaf1 25 9 9 90 1 2 35 7

Stem8888899999

Leaf125

9 9 90 12 357

Score Frequency Score Frequency34567

12213

89101112

222

1

Class Frequency Class Frequency1–1.92–2.93–3.9

122

4–4.95–5.96–6.9

651

Class interval Frequency

Class interval Frequency

10–1415–1920–24

39

10

25–2930–3435–39

10101

Score Frequency Score Frequency0.30.40.50.60.70.8

121112

0.91.01.11.21.3

22111

3

123

4 5 6 7 8Score

Freq

uenc

y

9 10 11 12

123456

Freq

uenc

y

1 2 3 4Score

5 6 7

2468

10

Freq

uenc

y

10 15 20 25Score

30 35 40

c

0.2

12

0.4 0.6Score

Freq

uenc

y

0.8 1.0 1.2 1.4 9A➔

9D


612

A n s w e r s

answ

ers

3

Check your histograms against those shown in question

2

answers.

4

5

Investigations — Segmented bar charts, Looking at accidents

Discuss with your teacher.

Exercise 9E — Describing the shape of stem plots and histograms

1 a

Symmetric, no outliers

b

Negatively skewed, no outliers

c

Positively skewed, one outlier

d


e

Symmetric, one outlier

f

Positively skewed, no outlier

2 a


b


c


d


e

Negatively skewed, one outlier

f

Positively skewed, no outlier

3

E

4

D

5


6

Positively skewed. This tells us that most of the flight attendants in this group spend a similar number of nights interstate per month. A few stay away more than this and a very few stay away a lot more.

7 a


b

This tells us that there are few low-weight dogs and few heavy dogs but most dogs have a weight in the teens (in kg).

8 a


b

Most students receive about $8 (give or take $2). One student, however, is far removed from the rest and receives only $2.

9 a

Positively skewed, 2 outliers. This indicates that most workers do up to 3 hours of exercise per week. Very few do more.

b

The company has 2 fitness fanatics; one does 14 hours exercise a week and one does 19 hours exercise a week.

Exercise 9F — Cumulative data

1 a

Cumulative frequency

b

column: 9, 24, 44, 56, 64, 71, 75, 77

2 a

Cumulative frequency column: 1, 5, 20, 38, 50, 58, 60

b

3 a

Cumulative frequency column: 6, 18, 26, 33, 38, 39, 40

b

4 a

Cumulative frequency column: 13, 41, 87, 117, 134, 142, 149, 150

b

5

6 a

Cumulative frequency column: 2, 7, 13, 17, 18, 20

b

7 a

Cumulative frequency column: 2, 11, 18, 27, 29, 30

b

3

123

4 5 6 7 8Score

Freq

uenc

y

9 10 11 12

0

2468

1 2 3 4 5Number of hours

Num

ber

of s

tude

nts

6 7 8 9 10

Length (mm)

Ogive of lengthsof flathead

Cum

ulat

ive

freq

uenc

y

10

300 320 340 360 380

20304050607080

a

12

b

11

c

68%

d

8%

e

55

f

46

g

37

h

30 or less

Time (s)

Ogive of task times

Cum

ulat

ive

freq

uenc

y

6 10 14 18

60

40

20

Salary ($×1000)

Ogive of salaries

Cum

ulat

ive

freq

uenc

y

Cum

ulat

ive

freq

uenc

y (%

)

5

20 30 40 50

10152025303540

50%

100%c 11d 26e $33 000f $26 500g $21 500h $38 000

Size (cm)

Ogive of waist sizeC

umul

ativ

e fr

eque

ncy

Cum

ulat

ive

freq

uenc

y (%

)70 80 90 110100

25

50

75

100

125

150

50%

100%

c 60d 20e 96 cmf 83.5 cmg 25%

Time (s)

Ogive of timetrial results

Cum

ulat

ive

freq

uenc

y

Cum

ulat

ive

freq

uenc

y (%

)

65707580859095

5

0

10

15

20

50%

100%c 14d 90% of the

riders finished with a time 90 s or less.

e To qualify you need a time under 72 s.

Weight (kg)

Ogive of baby weights

Cum

ulat

ive

freq

uenc

y

Cum

ulat

ive

freq

uenc

y (%

)

2.4 3.2 4.0 4.8

51015

302520

50%

100%

c 75%d 3.45 kge 3%

Answers Maths B Yr 11 Page 612 Monday, June 24, 2002 1:10 PM

A n s w e r s 613

answ

ers

8 a Cumulative frequency column: 3, 12, 17, 19, 20

b

9 a Cumulative frequency column: 9, 19, 25, 29, 30b c

Chapter review1 a Categorical b Quantitative c Quantitative

d Quantitative e Categorical2 a Discrete b Continuous c Continuous

d Discrete e Continuous3 a Sample b Census c Census

d Sample4 Random sample — where the participants are

chosen by luckStratified sample — where the participants are chosen in proportion to the entire populationSystematic sample — where a system is used to select the participants

5 a Systematic b Random c Stratified6 Check with your teacher.7 Year 8 — 15, Year 9 — 14, Year 10 — 13,

Year 11 — 9, Year 12 — 98 a b

Key: 6|0 = $6c

9

10 Positively skewed, one outlier11 a 17 b 5 c 72%

d $88; 80% of people spent $88 or less on shoppinge $20

CHAPTER 10 Summary statisticsExercise 10A — Measures of central tendency1

2 a 1.0783 No, because of the outlier.b 17 Yesc 30.875 Yesd 15.57 No, because of the outlier.

3 12

4 D

5 A

6

7

8 24

9

10

11 a 17 b 148, 151 c No moded 72 e 2.6

12 a 4 b 8 c 42, 44

13 a 17–20 b 22–28

d 19 e 63% f 26% g 93° h 78°10 D 11 C

Stem678

Leaf0 1 5 71 1 3 8 8 92 3 4 4 7

Stem666667777788888

Leaf0 1

57

1 13

8 8 9

2 34 47

Key: 6|0 = $6

Number of passengers

Ogive of numberof passengers

Cum

ulat

ive

freq

uenc

y

Cum

ulat

ive

freq

uenc

y (%

)

70 90 110

5

10

15

20

50%

100%

c 13d 97 passengerse 89 passengersf 81 passengersg No

80 90 100

10

5

Freq

uenc

y

Temperature (°C)

Temperature at which paint blistered

Temperature (°C)

Ogive of temperatureat which paint blistered

Cum

ulat

ive

freq

uenc

y

Cum

ulat

ive

freq

uenc

y (%

)

80 90 100

10

20

30

50%

100%

Key: 6|0 = $6

Stem667788

Leaf0 15 71 1 38 8 92 3 4 47

a 6.75 b 7.125 c 4.9875

d e 0.8818

a Median b Meanc Median d Median

a 36.09 b 18.34c 168.25 d 18.55

Median

abcde

375

1142.5

628

Median

abcdefghi

6176

1018.54

194.5

23

50

2468

10

54 58

Freq

uenc

y

62 66 70 74 7652 56Speed (km/h)60 64 68 72

16.7̇.

9E➔

10A


614 A n s w e r san

swer

s14 a

b 42.2 c 16–30 d 16–30e Nof Check with your teacher.

15 a Player A: 34.3 Player B: 41.8b Player B c Player A: 32.5 Player B: 0d Player Ae Check with your teacher.

Investigation — Mean and median amount of soft drink3 Machine A: x– = 999.65, Median = 999.5

Machine B: x– = 1000.5, Median = 10014 No

Exercise 10B — Range and interquartile range

1

2

3 a 10 b 84 An example could be 2 3 6 8 95 An example could be 2 5 6 7 8 10 11 126 C

7

8

9 a 2 b 1 c 2d 1

10 a

b

c 1 d 2

Investigation — Range of soft drink amounts1 Machine A = 37

Machine B = 382 Machine A = 23

Machine B = 9.53 Yes. Machine B

Exercise 10C — The standard deviation1

2 As for question 1.

3 a Group A: mean = median = mode = 170 cmGroup B: mean = median = mode = 170 cm

b No c Group B d Group B e Group Bf Group A: range = 20, interquartile range = 0,

σ = 5.345Group B: range = 120, interquartile range = 20,σ = 32.51

4 a 60 b 110.48 c σ = 11.59

5 C

6 a 500 hours b Mean = 455.28, σ = 88.62c Mean = 455.28, s = 88.88

7 a Cumulative frequency column: 4, 26, 121, 245, 339, 358, 363, 365

b

ClassClass centre Frequency

Cumulative frequency

1–15 8 1 1

16–30 23 13 14

31–45 38 2 16

46–60 53 0 16

61–75 68 5 21

76–90 83 4 25

Range

abcde

5617181872

Range

abcdefghi

796

13147

179

21

MedianInterquartile

range Range

abc

2127.53.7

1883

45205.9

MedianInterquartile

range Range

ab

4232

217

9130

Score FrequencyCumulative frequency

0 26 26

1 31 57

2 22 79

3 8 87

4 3 90

a 2.288 b 2.195 c 20.17 d 3.066

00

10203040506070

1 2 3 4Score

Cum

ulat

ive

freq

uenc

y

8090

Temperature

Ogive of temperatures

Cum

ulat

ive

freq

uenc

y

Cum

ulat

ive

freq

uenc

y (%

)

0 20 3010 40

200

100

300

400

50%

75%

25%

100%

c QU = 21, QL = 13.5, interquartile range = 7.5

d 17e 17.55f 5.649g 40


A n s w e r s 615

answ

ers

8 a Frequency column: 4, 9, 9, 3, 4, 1b

9 a Electric Mateb Electric Mate = 197 hours, Hot Wire = 185 hoursc Electric Mate

10 Check with your teacher.

Investigation — Standard deviation of soft drink amountsDiscuss with your teacher.

Exercise 10D — Boxplots1234

5

9 a (22, 28, 35, 43, 48) b

10 a (10, 13.5, 22, 33.5, 45)b

11 a (18, 20, 26, 43.5, 74) b

12 a (124 000, 135 000, 148 000, 157 000, 175 000)b

13 a Key: 12 1 = 121

b

14 a Key: 1* 7 = 17 years

b

15 C

Exercise 10E — Back-to-back stem plots1 Key: 2 3 = 23

German French2 1 1 0 2 3 47 6 5 5 2 5 5 8

3 2 1 0 0 3 0 1 4 49 8 7 7 3 5 6 8 8 9

2 1 4 2 3 4 45 4 6 8

2 Key: 2 7 = 2.7 (kg)Boys Girls

2 6 74 4 3 0 1 1 2 3

8 7 6 3 6 73 2 4 09 8 4

0 5

3 a A B2 1 1 0

7 7 6 6 5 1 5 64 3 2 1 0 2 0 1 3

7 5 2 5 6 8 93 0 1 23 5

b For supermarket A the mean is 19, the median is 18.5, the standard deviation is 4.9 and the interquartile range is 7. The distribution is symmetric.For supermarket B the mean is 24.4, the median is 25.5, the standard deviation is 7.2 and the inter-quartile range is 10. The distribution is symmetric.The centre and spread of the distribution of super-market B is higher than that of supermarket A.There is greater variation in the number of trucks arriving at supermarket B.

4 a Females Males1 0

3 2 1 2 35 5 4 4 1 4 4 5

7 6 1 71 9

b For the marks of the females, the mean is 14.5, the median is 14.5, the standard deviation is 1.6 and the interquartile range is 2. The distribution is symmetric.For the marks of the males, the mean is 14.25, the median is 14, the standard deviation is 2.8 and the interquartile range is 3.5. The distribution is symmetric.The centre of each distribution is about the same. The spread of marks for the boys is greater, however. This means that there is a wider variation in the abilities of the boys compared to the abilities of the girls.

5 a 1998 19992 22 6 7 8

1 0 3 0 1 1 3 49 7 5 3 6

3 2 1 1 46 4

a 13 b 5 c 26a 122 b 6 c 27a 49.0 b 5.8 c 18.6a 140 b 56 c 90d 84 e 26a 58 b 31 c 43d 27 e 7

6 C 7 C 8 D

Stem12131415161718

Leaf1 5 6 91 2 43 4 8 80 2 2 2 5 73 52 91 1 1 2 3 7 8

Stem1*22*33*44*

Leaf7 7 8 8 8 9 90 0 0 1 2 2 2 2 3 3 3 3 4 4 45 5 8 91 2 3

8

Number of fruit

Ogive of numberof fruit on each tree

Cum

ulat

ive

freq

uenc

y

Cum

ulat

ive

freq

uenc

y (%

)7030 50 90

5

152025

10

30

50%

100%

c Median = 52, QU = 62, QL = 44

d 18e 54f 13.25g 13.48h 45

20 30 40 50 Sales

0 10 20 30 40 50 Rainfall (mm)

10 30 50 70 Age

120 140 160 180 ($×1000)

120 140 160 180 Number sold

15 25 35 45

×

Age

10B➔

10E


616 A n s w e r san

swer

sb The distribution of marks for 1998 and for 1999 are

each symmetric.For the 1998 marks, the mean is 38.5, the median is 40, the standard deviation is 5.2 and the interquartile range is 7. The distribution is symmetric.For the 1999 marks, the mean is 29.8, the median is 30.5, the standard deviation is 4.2 and the inter-quartile range is 6.The spread of each of the distributions is much the same but the centre of each distribution is quite different with the centre of the 1999 distribution quite a lot lower. The work may have become a lot harder!

6 a Female Male4 3 2 2

8 7 6 5 2 51 0 3 0 1

3 6 74 24 6

b For the distribution of the females, the mean is 26.75, the median is 26.5, the standard deviation is 2.8 and the interquartile range is 4.5.For the distribution of the males, the mean is 33.6, the median is 33.5, the standard deviation is 8.2 and the interquartile range is 12.The centre of the distributions is very different: it is much higher for the males. The spread of the ages of the females who attend the fitness class is very small but very large for males.

7 a Kindergarten Prep.3 0 5

4 3 1 2 78 5 2 5 76 2 3 2 57 1 4 4 6

0 5 2b For the distribution of scores of the kindergarten

children, the mean is 28.9, the median is 30, the standard deviation is 15.4 and the interquartile range is 27.For the distribution of scores for the prep. children, the mean is 29.5, the median is 29.5, the standard deviation is 15.3 and the interquartile range is 27.The distributions are very similar. There is not a lot of difference between the way the kindergarten children and the prep. children scored.

Exercise 10F — Parallel boxplots1 a

b Clearly, the median height increases from Year 9 to Year 11. There is greater variation in 9A’s distribution than in 10A’s. There is a wide range of heights in the lower 25% of the distribution of 9A’s distribution. There is a greater variation in 11A’s distribution than in 10A’s, with a wide range of heights in the top 25% of the 11A distribution.

2 a

b Clearly, there is a great jump in contributions to superannuation for people in their 40s. The spread of contributions for that age group is smaller than for people in their 20s or 30s, suggesting that a high proportion of people in their 40s are conscious of superannuation. For people in their 20s and 30s, the range is greater, indicating a range of interest in contributing to super.

3 a

b Overall, the biggest sales were of multi-vitamins, followed by vitamin B, then C and finally vitamin A.

4 a C b B c E d C

Chapter review1 a 5.2 b 64.875 c 7.7 d 35.8

2

Mean = 38.23 a 31.1 b 23.2 c 0.4454 a 29.9 b 26.4 c 18.65 a 27 b 6 c 3.2

d 5.5 e 1286 a 2 b 56 c 68.57 46–49 8 a 27.8 b 24.5 c 28

d Median9 Check with your teacher.

10 a 7 b 159 c 1.411 a i 25 ii 24 iii 27.5 iv 3.5

b i 62.5 ii 43 iii 84 iv 41c i 1.1 ii 0.7 iii 1.5 iv 0.8

12 a 2 b Lower = 1, upper = 3 c 2

8 B 9 C

120 130 140 150 160 170 180 190 200

11A

9A

10A

Height (cm)

ClassClass centre Frequency

21–24 22.5 3

25–28 26.5 9

29–32 30.5 17

33–36 34.5 31

37–40 38.5 29

41–44 42.5 25

45–48 46.5 19

49–52 50.5 10

Σ f = 143

10 15

40-49 age group

20-29 age group

30-39 age group

0 5

Annual superannuation contribution (× $1000)

15 205 10

Number of jars sold

Multi-vitamin

B

C

A


A n s w e r s 617

answ

ers

13 a 61 b Lower = 42, upper = 70 c 28

14

15

16 a 43 b 43 c 1417

18 a Full-time Volunteer1 0

2 2 04 4 3 3 0

6 5 00 81 0 1 11 2 3 31 4 511

b Both distributions are symmetric with the same spread. The centre of the volunteers’ distribution is much higher than that of the full-time firefighters’ distribution. Clearly, the volunteers needed more counselling.

19

CHAPTER 11 Introduction to probabilityExercise 11A — Informal description of chance1 a Probable b Unlikely c Impossible

d Fifty-fifty2 a Impossible b Certain c Even chance

d Even chance e Probable f Unlikelyg Impossible h Even chance

3 Check with your teacher.4 More likely during school term5 a More likely b Equally likely c Less likely

d More likely e Less likely6 Rolling a 6

Rolling a number less than 3Rolling an even numberRolling a number greater than 2

7 Winning a raffle with 5 tickets out of 30Selecting a court card from a standard deckDrawing a green marble from a bag containing 4 red, 5 green and 7 blue marblesRolling a die and getting a number less than 3Tossing a coin and having it land Heads

8 Australia9 Carl Bailey because he has better past performances.

10 B 11 B 12 D13 Probable 14 Unlikely 15 Fifty-fifty

Investigation — What will the weather be?Check with your teaacher.

Exercise 11B — Single event probability1 S = {Heads, Tails}, 12 a S = {1, 2, 3, 4, 5, 6}, 1

b S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, 3c S = {a, b, c, d, e, . . ., y, z}, 5d S = {Sun, Mon, Tue, Wed, Thu, Fri, Sat}, 2e S = {Jan, Feb, Mar, . . ., Nov, Dec}, 3

3 a 26, 52 b 1, 15 c 1, 44d 5, 1500 e 3, 11

4

5 a b c

d e f

6 a b c

d e f

g g i

7 a b c

d e f

8 a b c

9 a b c

d e f

10

11 a b c

d e

12 C 13 D 14 C 15 D

16 a b

17 a b c

18 a b c

d e f

19 Check with your teacher.20 a Because there are two numbers which could go in

the last place.b A number less than 400

Exercise 11C — Relative frequency1 0.74 2 0.79 3 0.3754 a 0.45 b 0.555 4%6 a 0.03 b 0.977 a 0.96 b 0.048 A 9 E

10 a 0.525 b 0.4375 c 0.037511 a 6.67% b 8012 a 0.02 b $40013 Yes, the relative requency is 27%.

a 4.9 b 5.5 c 2 and 6d 8 e 5 f 2.625a 65.781 b 60– c 70–d 60 e 13.727

2 6 10 14 180 4 8 12 16 20

90 100 110 120 130 140 150

Team C

Team A

Team B

IQ

12---

16--- 1

6--- 1

2---

12--- 2

3--- 1

3---

145------ 1

45------ 22

45------

2345------ 1

5--- 1

3---

1945------ 2

9--- 2

15------

152------ 1

13------ 1

4---

12--- 12

52------ 3

13------

112------ 4

12------ 7

12------

14--- 1

4--- 1

4---

34--- 1

2--- 3

4---

1012------

14--- 3

4--- 1

4---

14--- 2

4---

51000------------ 4

999---------

3384160 000------------------- 6768

160 000------------------- 10 152

160 000-------------------

15--- 1

5--- 4

5---

15--- 4

5--- 1

5---

10F➔

11C


618 A n s w e r san

swer

s14 a 2.5% b 51.5% c 17.5%15 40 000 km

16 a

b Win = 0.375, Loss = 0.35, Draw = 0.275

Exercise 11D — Modelling probability1 Answers will vary.2 Answers will vary.3 Answers will vary.4 a One way is to use randInt(0,1,10) to generate

10 values that are either equal to 0 or 1, and let 0s represent Heads, and 1s represent Tails.

b Answers will vary.c Answers will vary.

5 Generally, the histogram for 100 tosses will be more even than that for 10 tosses.

6 a, b The player can expect to win about once every 15 games, spending $15 to win $10 (a loss of $5).

7 Answers will vary.8 Answers will vary.

Exercise 11E — Long-run proportion1 0.12 0.4993 a, b Note the steady improvement from about 0.25

(25% success) to about 0.36 (36%) success.

4 D 5 D6 0.708 This result supports the suspicion that the coin

is biased because the expected result would be 0.5.7 0.78 8 Yes9

There is a huge bias for 2s and 4s, against 3s and 5s, 1s and 6s OK.

10 Bread shop 3, proportions were 0.75, 0.84 and 0.86 respectively. However, Bread shop 1 wasted only 22 loaves, versus 32 for Bread shop 2 and 42 for Bread shop 3. This could reduce Bread shop 3’s true effectiveness.

11 a, b Answers will vary.12 a Left-handers caught 6 out of 14 (0.43),

right-handers caught 24 out of 56 (0.43)— no difference.

b Caught 30 out of 70, bowled 30 out of 70 — no difference.

c Left-handers — not out 4 out of 14 (0.29), right-handers — not out 6 out of 56 (0.11) — more likely to be not out against left-handers.

d The relatively small number of left-hander observations means comparisons are not very accurate. However, there seems little difference between left- and right-hander effectiveness.

13 1329

Chapter review1 Marcia will probably get a higher card.2 a Probable b Impossible c Even chance

d Unlikely e Unlikely3 Check with your teacher.4 Hot weather5 Rolling a die and getting a number greater than 1

Selecting a picture card from a standard deckSelecting a blue marble from a bag containing 14 blue, 15 red and 21 green marblesWinning the lottery with 1 ticket out of 100 000 tickets sold

6 Mark is most likely to win based on past performances.

7 S = {1, 2, 3, 4, 5}, E = {3, 4, 5}

8 a b c

d e f

9 a b c

d e f

10 a b c

11 a b c

d e

12 a b

13 0.0214 a 0.15 b $75015 randInt(−20, 20, 10)16 They can expect to win an average 50c/game.

CHAPTER 12 Rates of changeExercise 12A — Constant rates1 c, d, g, h, j2 b, d, e, g, h, j3

4 D 5 C6

78

Result Number

WinLossDraw

151411

Sales 7 11 17 25 41 53 60 72 84 97

Houses 28 47 68 93 135 164 186 217 244 270

Proportion 0.25 0.234 0.25 0.269 0.304 0.323 0.323 0.332 0.344 0.359

1 2 3 4 5 6

0.180 0.233 0.074 0.246 0.092 0.175i b, e, h ii g, j

iii d

a i 50 ii 50 m/hiii d = 50t

b i 7 ii 7 L/miniii v = 7t + 50

c i −1 ii −1 kg/weekiii w = 100 − t

d i 0 ii 0 m/yriii h = 75

e i 0.75 ii 0.75 g/miniii w = 0.75t + 10

f i −0.25 ii −0.25 g/Liii w = −0.25v + 35

a D b Ba $24/h b Restc $18/h d The picker is tiring a

little or fruit is scarcer.e $156

125------ 1

25------ 13

25------

15--- 9

25------ 16

25------

152------ 1

13------ 1

4---

12--- 4

13------ 10

13------

720------ 1

10------ 3

4---

124------ 3

4--- 1

4---

34--- 1

2---

1400--------- 4

1999------------


A n s w e r s 619

answ

ers

9

10

11

12

13 Check with your teacher.

Exercise 12B — Variable rates1 a, c, d, e, g, h 2 b, d, e, f3 a

b B c D d A e C f D4

5

6

Exercise 12C — Average rates of change1

2 a, c3

4

5

6 D 7 C8

9

10

11

a 1000 m b 70 sc 80 s d 30 se

a b 0.2 kg/cm

c W = 0.2x

a 6 L/h b 6 h

a iv b ic ii d iii

Interval

Constant or variable

rate?

Positive, negative or zero rate?

OP Constant Positive

PQ Variable Positive

QR Constant Zero

RS Variable Negative

ST Variable Positive

TU Variable Negative

UV Constant Negative

VW Variable Negative

a b Form a straight line

c A constant d 3 kg/mina b No

c Variable d i 2 kg/minii 3 kg/min iii 7 kg/min

d (metres)

t (s)

1000

820

420

0 70 150 180

w (kg)

x (cm)

54321

5 10 15 20 250

23---

W (kg)

t (min)

12963

1 2 3 40

W (kg)

t (min)

201612

84

1 2 3 40

a Constant b Variablec Constant d Variablee Variable f Variable

a b −4

a ii 30 iii 30 km/hb ii 10 iii 10 km/hc ii 15 iii 15 km/hd ii −5 iii −5 km/he ii 0 iii 0 km/hf ii −10 iii −10 km/ha 2 b 2°C/minc 2°C/min because the rate of change is constant.a 1.5 b 0.5c i 1.5 kg/m ii 0.5 kg/md No, because it is a variable rate of change.

a, b

c i 170 ii 20 iii 0d i 170 people/h ii 20 people/h

iii 0 people/he Most people arrive in the morning, few in the

middle of the day and nobody later in the afternoon.

a b i 5 km/h

ii 3 km/h iii 0 km/hc A rest or meal break d The first half houre 1 to 1.5 hoursa iii 110 m/min iii 70 m/min

iii 45 m/min iv 15 m/minv 5 m/min b Decreasing

a 20°Cb, c d 10

e 10°C/s

25---

00 10 am 11 am 12 pm 1 pm 2 pm 3 pm 4 pm 5 pm T

N

300

500400

200100

600

00 1 2 3 4 t

d

123456789

T (°C)

t (s)2 4 6 8 100

120

20

(8, 84)

(2, 24) 11D➔

12C


620 A n s w e r san

swer

s12

13 72 km 14 min

Exercise 12D — Instantaneous rates1 a, c, d, h2

3

4

5 a, b

c i 4 ii 0 iii −46

7 6 cm/min8

9

Exercise 12E — Motion graphs1

2

3

4

5

6

a 200 g b i 1 g/weekii −2 g/week iii 10 g/weekiv 28 g/week

a 1 b 2.5c −1 d 0.25a 3 g/min b 8 g/minc −25 g/min d 12 g/mina, b c C

d C e A

a A b Ic A d Ie A f Ag A h Ii I j I

a 85 kg after 35 weeks b approx. −1 kg/weekc Approx. 4.4 kg/weeka Approx. 6 hours b Approx. 2.5°C/hour

a C b Bc D d Aa False b Truec True d Falsea 40 mb −2 m (or 2 m below the platform)c 0.5 m/sd −0.025 m/s (or 0.025 m/s downwards)a x = 1 b x = −3c Right d t = 2e C f Dg A h Di C j B

1207

---------

y

x

16

12

8

4

43210

y

x

9

5

3210 4–2–3–4 –1–2

–7

a i x = 0 ii Rightiii t = 2, x = 8 iv t = 5, x = −3

b i x = 4 ii Rightiii t = 4, x = 12 iv t = 6, x = 10

c i x = 0 ii Rightiii t = 3, x = 12 and t = 6, x = 3iv t = 8, x = 10

d i x = 0 ii Leftiii t = 1, x = −5 iv t = 3, x = 18

e i x = −3 ii Left

iii t = 1 , x = −6 iv t = 5, x = 5

f i x = 2 ii Leftiii t = 3, x = −5 and t = 5, x = 5iv t = 6, x = 4

a i x = 0

ii Right iii Noiv x = 10

b i x = −2


c i x = 0

ii Left iii Yes, t = 1, x = −1iv x = 15

d i x = 0

ii Right iii Yes, t = 1, x = 1iv x = −8

12---

x

t

108642

1 2 3 4 50

x

t

16

60–2

x

t

15

1 2 3 4 50–1

x

t

1

1 2 3 4 50

–8


A n s w e r s 621

answ

ers

7

8

9

10

11

Exercise 12F — Relating the gradient function to the original function1 a b

c d

e f

2 It is a straight line.3 Quadratic functions.4 The gradient function of sin x is cos x, the gradient

function of 2x is 0.7 × 2x.5 There are stationary (turning) points there.6

Exercise 12G — Relating velocity–time graphs to position–time graphs1

2 C3

e i x = 4

ii Left iii Yes, t = 2, x = 0iv x = 9

f i x = −12


a b i 4

ii 2 iii 0iv −2 v −4

c i 4 m/s ii 2 m/siii 0 m/s iv −2 m/sv −4 m/s d

a i Positive ii Speeding upb i Negative ii Slowing downc i Negative ii Speeding upd i Positive ii Slowing downe i Positive ii Neithera C b Ec B d Fe A f D

a 45 m

b t = 5 c t = 2d 20 m/sa Alan b Billc 15 metres d 2 m/se 40 s f Twice

x

t

9

4

1 2 3 4 50

x

t

18

1 2 3 4 50

–12

x

t1 2 3 4 50

–5

4

x

t1 2 3 4 50

–6–4–2

42

x (m)

t (s)1 2 3 4 50

25

45

a b

a b

y

x0–4 –2 2 4

4

–4

8

12

Gradient function

y

x0–4 –2 2 4

4

–4

8

12

Gradient function

y

x0–4 –2 2 4

4

–4

8

12

Gradientfunction

y

x0–2 –1 1 2

2

4

6

8

Gradientfunction

y

x02 4 6 8

1

–1

Gradientfunction

1–1–2–3

2345y

x

1

2 3–1–2–3–4–5

t (s)

2

2 4 6 8

v

t0 3 6

3

–3

v

t0 1 2

2

–2

v

t0 321

5

v

t0 4321

8

4

–4

–812D➔

12G


622 A n s w e r san

swer

s

4

5 a

6

7 A

8 a It travels at a constant velocity of 20 m/s.b It starts at rest, increasing its velocity at a

constant rate for 15 seconds when it reaches 18 m/s. It then maintains this velocity.

c It starts from rest, increasing its velocity at a constant rate for 10 seconds when it reaches

25 m/s. It maintains this velocity for a further 15 seconds then decreases its velocity at a constant rate for 5 seconds bringing it back to rest.

9

10

Exercise 12H — Rates of change of polynomials1 a

b i 6 ii 5 iii 4.5 iv 4.1

c 4

2 a

b i 7 ii 4.75 iii 3.31 iv 3.0301

c 3

3 a

4 C 5 E 6 B

c d

a b

c d

e f

t 0 1 2 3 4

V −4 −2 0 2 4

Gradient of x–t graph −4 −2 0 2 4

b

a b

c d

v

t0 21

2

–2

v

t

12

1 2 3 40

3

x

t40

12

x

t40

–8

x

t1 2 3 4 50

x

t1 2 3 40

x

t1 2 3 40

x

t1 2 3 40

x

t1 2 3 40

x

t40

x

t40 2

x

t40

x

t40

a b

a i 15 m/s ii 45 m/sb 60 m/s c 34 seconds

x 3 2 1.5 1.1 1.01 1

y 7 3 1.75 1.11 1.0101 1

b i 3 ii 2iii 1.5 iv 1.1v 1.01 c 1

v (m/s)

t (s)100 20

35

v (m/s)

t (s)150 20

30

y

x

16

12

8

4

43210

i

ii

iii

iv

(4, 16)

(3, 9)

(2, 4)

(1, 1)

(2.5, 6.25)

(2.1, 4.41)

y

x

8

7

3

2

6

5

4

1

210

i

ii

iiiiv

(2, 8)

(1.5, 3.375)

(1.1, 1.331)(1, 1) (1.01, 1.030301)

Answers Maths B Yr 11 Page 622 Monday, October 29, 2001 12:27 PM

A n s w e r s 623

answ

ers

7

8

9

10

Chapter review1 C 2 D

3

4 B 5 B

6

7 A 8 E

9

10 C 11 B 12 a D b C c B

13

14

15 D 16 C 17 C

18

19 E 20 D

21

22 a

b 240c Approx. 3 pmd Approx. 75 cans/hour

CHAPTER 13 Differentiation and applicationsExercise 13A — Introduction to limits1 8

2 Circle

3 B

4

5 a

b C

6

7 C

8

9

10 D

11 A

12

Investigation — Sneaking up on a limit1 a

b 3

a 2 b 12c 6 d 4e −7 f 4.75g 31 h −2.5

a 19.6 m/s b 29.4 m/s

a 16°C/min b 700°C/min

a 3.75 b 2.81c 0 d 0

a 30 m3/h b 28 hours

a b Variable, as the graph is not a straight line.

c Approx. 12

a 1 kg/h b 5 kg/h

a b x = 5 m

c t = 1 s and x = 4 m d 14 m

a x = 30 m b 3 sc 34 m/s d −50 m/se 370 m f 30.83 m/s

a i 4 m/s ii 0 m/siii –8 m/sb The height of the golf ball increases during first

6 seconds then decreases after that.

M

t

23

1 2 3 4 50

–9

9 Tangentat t = 2

x

t1 2 3 4 50

54

10

15

20

x (t)

x

t3 60

a 0 b

n 1 2 3 4 5 6 10

S 1 1 1 1 1 1 1

x 2.95 2.99 2.995 3 3.005 3.01 3.05

f(x) 7.95 7.99 7.995 8 8.005 8.01 8.05

a 14 b −3 c 0d 5 e 4 f 15

a 9 b 2 c −11d 50 e 2 f −3g 40 h −12 i −2j 19 k 0 l −27

a 3 b 0 c 7 d 16

x 1.5 1.2 1.1 1.01 1.001 1.0001 1

3.5 3.2 3.1 3.01 3.001 3.0001 undefined

Time

Num

ber o

f can

s

0

60

120

180

240

300

360

420

480

540

600

8.00 am 10.00 am 12.00 pm 2.00 pm 4.00 pm 6.00 pm

1n---

n ∞→lim 0=

12--- 3

4--- 7

8--- 15

16------ 31

32------ 511

512---------

12---

x2

x 2–+x 1–

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

12H➔

13A


624 A n s w e r san

swer

sc

d 3e When x = 1 is substituted, zero is produced on the

denominator, so the expression is undefined.2 8 3 7

Exercise 13B — Limits of discontinuous, rational and hybrid functions1 b, c, d, f23

4 a Undefined because you cannot divide by 0.

5

6

7

8

9

10

Investigation — Secants and tangents1 a 3 b 2.5 c 2.1 d 2.01 e 2.0012 23 a 2 b 3 c 3.5 d 3.9 e 3.9994 4

Exercise 13C — Differentiation using first principles12

345

Exercise 13D — Finding derivatives by rule

1

2

3

4 a b

c

d

e f

5

6

7

8

x 0.5 0.6 0.9 0.95 0.99 0.999 1

2.5 2.6 2.9 2.95 2.99 2.999 undefined

b 4 c 0 d −1 f 3a 4 b

b x = 0 c x(x + 1) d f (x) = x + 1e f 1

a f (x) = x + 3, x ≠ 0 b f (x) = 6, x ≠ 3c f (x) = x − 5, x ≠ 0 d f (x) = x + 1, x ≠ −4e f (x) = x − 1, x ≠ 6 f f (x) = x2 − 2x + 4, x ≠ −2g f (x) = x + 4, x ≠ 1 h f (x) = x2 + 3x + 9, x ≠ 3a 3 b 6 c −5 d −3e 5 f 12 g 5 h 27a b

c d

a i 5 ii 4 iii Does not existb i 5 ii −2 iii Does not existc i 0 ii 0 iii 0d i 2 ii 3 iii Does not exista Does not exist b −2c Does not exist d Does not existe 1 f 0a 12 b 7 c 4d −11 e −2 f Does not existg 0 h −3 i 6j −7 k 5 l 2

x2

x 2–+x 1–

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

y

x0 4

f (x) = x discontinuous at 4

y

x0

f (x)

1

–1

y

x0

f (x)345

–3 2

y

x0

g(x)5

–2–1 1

y

x0

h(x)

1

–2

–1

y

x0

p(x)

21

3

1

a 5 b 2x + 10 c 2x − 8 d 3x2 + 2a 2x b 2x − 3 c 8xd −2x e 6 − 4x f 3x2 + 5a 2x − 6 b x = 3

a 3x2 b x = −2 or 2a 7 b 10 c 3 d 27

6 C, E 7 C 8 A

a b

c 5 d

a 12x3 b 56x6 c 25x4

d −24x5 e −6x2 f −7a D b B c E d Fe G f A g C

a x3 b − x

c 6x6 d x2 + x − 3

e 2x4 + 3x3 + x2 f 12x2 − x −

a 2x + 3 b 12x − 15c 2x + 8 d –432 + 162x

e 3x2 + 12x + 12 f 24x2 − 120x + 150

a 2x, x ≠ 0 b 4, x ≠ 0c 6x + 2, x ≠ 0 d 10x + 1, x ≠ 0a −4x−5 b −7x−8 c −12x−5

d −40x−9 e 24x−7 f 15x−6

g h i

j k l

m n o

p q r

s −

dydx------ 6x5=

dydx------ 14x=

dydx------ 16x3 4

3---x 5–+=

dydx------ 6x5 6x+= dy

dx------ 20x3 21x2 6+–=

dydx------ 11x10 18x5 20x4 6x+ +–=

dydx------ 50x4 12x3 6x2 8–+–=

dydx------ 0= dy

dx------ 32x3=

83--- 5

4---

94---

12--- 8

7--- 5

6---

4x5-----–

9x10-------– 15

x4------–

60x7------– x

12---– 2

3--- x

13---–

x34---– 6

5--- x

35---– 1

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

1

2 x3------------– 2

x------- 1

3--- x

23---–

23--- x

43---–


A n s w e r s 625

answ

ers

9

10 a x = 2, x = 3b At x = 2 gradient = −1, at x = 3 gradient = 1

c i x = 2 ii x = 6 iii x = 1

11 a x = −3, x =

b At x = −3 gradient = −7, at x = gradient = 7

c (−1 , −6 )

1213

14 y = x + 5 or 2y = x + 10

15 a x = 2 b y = – x +

16 y = –x + 5

Investigation — Graphs of derivatives1

2 Turning points, intercepts

Exercise 13E — Rates of change1 a 13 b f ′(x) = 2x + 5 c f ′(5) = 152 a i V = 0 cm3 ii V = 800 cm3

b 80 cm3/sc i 0 cm3/s ii 120 cm3/s iii 0 cm3/s

3 E 4 C 5 C6 a h′(t) = 18 − 6t

b i 6 m/s ii 0 m/s iii –6 m/sc The ball stops rising, that is, it reaches its highest

point.d 12 m/s

7 a b i 20 m/s ii 4 m/s iii –4 m/s

c The lift changed direction.d t = 10 s and x = 200 m

8 a i 4000 ii 15 000b 5500 people per hourc i 3500 people/hour ii 4500 people/hour

iii 6500 people/hour iv 7500 people/hourd More people arrive closer to starting time.

9 a 80 kg b

c i 9 kg/week ii 6 kg/week iii 3 kg/weekd Decreasing e 20 weeks

10 a

b i $37.50 ii −$9.38 iii −$12.00 c n = 911 a i −20.2 cm3/s ii −21 cm3/s iii −22 cm3/s

b No, because the volume is always decreasing.

12 a A = πr2 b

c i 20π m2/m ii 100π m2/m iii 200π m2/m

d Yes, because is increasing.

13 a V = πr3 b

c iii 0.04π m3/m or 0.13 m3/miii 0.16π m3/m or 0.50 m3/miii 0.36π m3/m or 1.13 m3/m

14 a Length = 2h, width = 2h

b V = 4h3

c i 12 m3/m ii 48 m3/m iii 108 m3/m

15 a x = 2h b

c i ii

16 a = −0.000 06x2 + 0.012x

b i 0.384 ii 0.6 iii 0.384 iv 0.216c x = 50 and x = 150 d 12.5 < y < 67.5

17 a hectares/hour

b i 0 ii 566 iii 864 iv 900 v 864 vi 576vii 0 (all hectares/hour)

c The fire spreads at an increasing rate in the first 10 hours, then at a decreasing rate in the next 10 hours.

d The fire is spreading, the area burnt out by a fire does not decrease.

e The fire stops spreading; that is, the fire is put out or contained to the area already burnt.

f t = 6 and t = 14 hours.

Exercise 13F — Solving maximum and minimum problems1 a x = 20 m, y′(19) > 0 and y′(21) < 0 (a maximum)

b y = 11.2 m2 a t = 10 min

b V ′(5) = −6 and V ′(15) = 18 (a minimum)c V = 160 litresd t = 15 min

3 a h = 12.25 m (when t = 1.5 s)b h′(1) = 5 and h′(2) = −5 (a maximum)

4 32 m2

5 a 16 − x b A = x(16 − x)c Both numbers are 8.d P ′(7) = 2, P ′(9) = −2, (a maximum)

a i 13 ii −17 iii 3b i 5 ii −4 iii 0c i 4 ii −14 iii −2d i 10 ii 19 iii 7

a (0, 0) and (2, −4) b (1, −2)a y = 6x − 2 b x + 6y = 25

12---

12---

12---

14--- 1

8---

12---

12--- 7

2---

x0

10

5

–15

2 4 6

–10

–5

y

f (x)

f'(x)

a b

x0

10

5

2–2 4

–10

–5

y

f (x)

f'(x)

c d

x0

10

5

–15

2–2 4

–10

–5

y

f (x)

f'(x)

x0

10

5

2–2 4

–10

–5

yf (x)

f'(x)

dxdt------ –4t 40+=

dWdt

-------- 12 0.6t–=

dPdn------- 4.5 1.5n

12---

–=

dAdr------- 2πr=

dAdr-------

43---

dVdr------- 4πr2=

V 6 3h2=dVdh------- 6 3= dV

dh------- 12 3=

dydx------

dAdt------- 180t 9t2–=

13B➔

13F


626 A n s w e r san

swer

s6 a 10 − x b A = x(10 − x) c x = 5

d Length and width = 5 cm e 25 cm2

7 a Length and width = 15 m b 225 m

8 a P = 60n − 250 − 1.2n2 b 25 c 500

9 a P = 800 + 240n − 20n2 b n = 6, p = 1520

10 Both numbers are 5.

11 a x ∈(0, 6) or 0 < x < 6b i x ii 12 − 2x iii 12 − 2xc V = x(12 − 2x)(12 − 2x) d 128 cm3

12 a 7.36 cm by 25.28 cm by 35.28 cmb 6564.23 cm3

13

14 10 cm × 10 cm × 10 cm

15 a 7500 b

c V = 400 km/h and C = $7200

Investigation — When is a maximum not a maximum?

3 Maximum temperature at t = 50 is 31.8°C.

Chapter review1 E 2 A 3 −6 4 3 5 B

6 A 7 B 8 C

9 a f (x) = x + 4, x ≠ −3 b 1

10 a

11 A 12 B 13 D 14 2 15 4 − 6x16 E 17 D 18 B 19 A 20 C21 A 22 E 23 C24 a g ′(x) = x2 − 7x − 8

b i −20 ii 10

c (−1, 5 ) and (8, −116 )

25 a ( , − ) b ( , 0) c (1, −1) and (5, 27)

26 E 27 D

28 a 3 cm b v = = −4t + 8 cm/s

c v = = −8 cm/s d t = 2 s and x = 11 cm

e Left f 10 cm29 A3031 a 400 cm2

b Yes, a circle of circumference 80 cm has an area of approx. 509.3 cm2

32 2000π cubic units33 a Gradient is 0.16 at the beginning and −0.38 at the

end.b (2, 2.04)d Maximum height is 2.04 km.e (0, 1.8), minimum height is 1.8 km.

34 350 cm2

35

a b

c 8 cm × 8 cm × 4 cm d 192 cm2

1 36 2 −16

h256L2

---------= A L2 1024L

------------+=

C1 440 000

V-----------------------= 9V+

y

x0

40

20

20 40

a 2 litres/hour b 3 hours

f (x)

x0

1

2

–1

b No limit exists.

16--- 1

3---

54--- 9

8--- 1

2---

dxdt------

dxdt------

y

x0

g(x)

6

3


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