mathsmethods1/2

8/13/2019 mathsmethods1/2

1/67

Answers

Chapter 1

Exercise 1A

1 a 3 b9 c1 d

8 e5 f2 g

5

3

h72

i7

3 j

20

3 k

103

l14

5

2 a a + b ba b c ba

dab ebc

a

3 a 7 b5 c3 d14 e 72

f14

3

g48 h3

2 i2 j3 k7 l2

4a4

3b5 c2

5 a1 b18 c 65

d23 e0 f10

g12 h8 i14

5 j

12

5 k7

2

6aba

be d

cc

c

a b d b

c ae

ab

b + a fa + b gb da c h

bd ca

7 a18 b78.2 c16.75d28 e34 f

3

26

Exercise 1B

1 a x + 2 = 6, 4 b3x= 10, 103

c3x + 6 = 22, 163

d3x 5 = 15, 203

e6(x + 3) = 56,193

fx + 5

4 = 23, 87

2A = $8,B= $24,C= $16314 and 28 48 kg 51.3775 m2

649, 50, 51 717, 19, 21, 23 84200 L

921 103 km 119 and 12 dozen127.5 km/h 133.6 km 1430, 6

Exercise 1C

1 a x= 1,y= 1 bx= 5,y= 21cx= 1,y= 5

2 a x= 8,y= 2 bx= 1,y= 4cx= 7,y= 1

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

cm= 2,n= 3 dx= 2,y= 1es= 2,t= 5 fx= 10,y= 13gx= 4

3,y= 7

2 h p = 1,q= 1

ix= 1,y= 52

Exercise 1D

125, 113 222.5, 13.53 a $70 b$12 c$3

4 a $168 b$45 c$15517 and 28 644 and 12

75 pizzas, 25 hamburgers8Started with 60 and 50; finished with 30 each9$17 000 10120 shirts and 300 ties

11360 Outbacks and 300 Bush Walkers12Mydney = 2800; Selbourne = 32001320 kg at $10, 40 kg at $11 and 40 kg at $12.

Exercise 1E

1 a x < 1 bx > 13 cx 3 dx 12ex 6 fx > 3 gx > 2hx

8 ix

3

22

2

x< 2

1 0 1 2

a

2

x< 1

1 0 1 2

b

2 1 0 1 2

x< 1c

2 1 0 1 2 3 4

x3d

708


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Ans

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Answers 709

2 1 0 1 2 3 4

x< 4e

2 1 0 1 2 3 4

x> 1f

2 1 0 1 2 3 4

x< 31

2g

x3

2 1 0 1 2 3 4

h

x>

0 1 2 3

16

i

3a x >12

bx < 2 cx > 5

43x < 20,x < 203

, 6 pages 587

Exercise 1F

1 a 18 b9 c3 d18e3 f81 g5 h20

2 a S= a + b + c b P=xy cC= 5pdT= dp + cq eT= 60a + b

3 a 15 b31.4 c1000

d12 e314 f720

4 a V= cp

ba= Fm

c P= Ir t

dr= wHC

et= S l PPr

fr= R(V 2)V

5 a T= 48 bb = 8 ch= 3.82 db = 106 a (4a + 3w) m b(h + 2b) m

c3whm2 d(4ah + 8ab + 6wb) m27 a iT= 2(p + q) + 4h ii88 + 112

b p = Ah

q

8 a D= 23

bb = 2

cn= 6029

dr= 4.8

9 a D= 12

bc(1 k2) bk=

1 2Dbc

ck=

2

3=

6

3

10 a P= 4b b A = 2bc c2 cb = A + c2

2c

11 a b = a2 a

2bx= ay

b

cr= 3q p2x2 dv= u2 1

x2

y2

Multiple-choice questions

1D 2D 3C 4A 5C6C 7B 8B 9A 10B

Short-answer questions

(technology-free)

1 a1 b32

c23

d27

e12 f44

13g

1

8 h31

2 at= a b b cd ba

cd

a+ c

dcb ac 1 e

2b

c a f1 cd

ad

3 ax 30 students

7adA=1

3t, dM= 57

3

10t

b

0

57

d(km)

90 t(min)

c10.30 am dAnne 30 km, Maureen 27 km

8b = 0.28 anda= 0.3, 257

m/s

Exercise 2G

1 a 135 b45 c26.57 d135

2 a 45 b135 c45 d135

e63.43(to 2 d.p.) f116.57(to 2 d.p.)3 a 45 b2634 c16134

d4924 e16134 f135

4 a 7134 b135 c45 d16134

5mBC= 35,mAB= 5

3

mBC mAB= 3

5 5

3= 1

ABCis a right-angled triangle6mRS= 12 ,mST= 2 R SST

mUT= 12 ,mST= 2 UT ST(Also needto showS R= UT.)RSTUis a rectangle.

7y= 2x + 28 a 2x 3y= 14 b2y + 3x= 89l

= 16

3 ,m

=80

3

Exercise 2H

1 a 7.07 b4.12 c5.83 d13229.27 3DN

Exercise 2I

1 a (5, 8) b

12, 1

2

c(1.6, 0.7) d(0.7, 0.85)

2MAB(3, 3).MBC8, 3 1

2 .MAC

6, 1 12

3Coordinates ofCare (6, 8.8)4 a PM= 12.04 bNo, it passes through

0, 3 1

3

5 a (4, 4) b(2, 0.2) c(2, 5) d(4, 3)

6

1 + a

2 ,

4 + b2

; a= 9, b = 6

Exercise 2J

1 a 3441 b45 c90 d4924 e2633


1A 2E 3C 4D 5B6E 7D 8C 9E 10E


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Ans

wers

Answers 713


(technology-free)

1a9

4 b10

11 cundefined d1 e b

a f

ba

2 a y= 4x by= 4x + 5 c y= 4x + 2d y= 4x 5

3 a a= 2 b20

344y + 3x= 7 53y + 2x= 56a= 3,b = 5,c = 147 a y= 11 b y= 6x 10 c3y + 2x= 38 a midpoint = (3, 2), length = 4

bmidpoint =

12,9

2

, length =

74

cmidpoint =

5,5

2

, length = 5

9

3y x= 3

3 2 10y +x= 1 113752

Extended-response questions

1 a

1

8

S

220 279 l

bSince the graph is a line of best fit answers

may vary according to the method used; e.g. if

the two end points are used then the rule isS= 7

59l 25.1

orl= 59

7S+ 1481

7

If a least squares method is used the rule is

l= 8.46S+ 211.73.c

33

41

C

220 273 l

dAgain this is a line of best fit. If the two end

points are used then

C= 853

l 1153

(orl= 53C 118

)

A least squares method givesl= 6.65C+ 0.6166.

2 a C= 110 + 38n b12 dayscLess than 5 days

3 a Cost of the plugbCost per metre of the cable

c1.8 d111

9m

4 a The maximum profit (whenx= 0)b43 seatscThe profit reduces by $24 for every seat empty.

5 a iC= 0.091n iiC= 1.65 + 0.058niiiC= 6.13 + 0.0356n

b

0

15

10

5

100

(50, 4.55)

(200, 13.25)

(300, 16.81)

200 300 n(kWh)

C($)

iFor 30 kWh,C= 2.73iiFor 90 kWh,C= 6.87

iiiFor 300 kWh,C= 16.81c389.61 kWh

6a y= 73x + 14 2

3 b20

1

3km south

7 as= 100 7xb

100

s(%)

0 14.3 x (%)1007

c5

7% d14

2

7%

eProbably not a realistic model at this value

ofs

f0 x 14 27

8 aAB,y=x + 2;CD,y= 2x 6bIntersection is at (8, 10), i.e. on the near bank.

9 a128

19 by= 199

190x + 128

19

cNo, since gradient ofA Bis20

19 (1.053),

whereas the gradient ofV Cis 1.047

10 a No b141

71km to the east ofH

11 ay=x 38 b B(56, 18)cy= 2x + 166 d(78, 10)

12 aL= 3n + 7b

10

61

L

0 1 18 n

13 aC= 40x + 30 000b$45 c5000 d R= 80xe

0

300

200

100

1000 2000 3000 x

R= 80x

C=40x + 30000

$000

f751 g P= 40x 30000


7/67

Answers

714 Essential Mathematical Methods 1 & 2 CAS

14 a Method 1: Cost = $226.75; Method 2:Cost = $227; Method 1 cheaper

b

0 1000 2000 3000

Method 1 100 181.25 262.50 343.75

Method 2 110 185 260 335

Cost the same for approx. 1600 unitsc

0

343.75

335

110

100

300

200

100

1000

1600

2000 3000 units

cost($)

dC1= 0.08125x + 100 (1)C2

=0.075x

+110 (2)

x= 160015 a (17, 12) b3y= 2x + 216a PD: y= 2

3x + 120;DC: y= 2

5x + 136;

CB: y= 52x + 600

AB: y= 25x + 20;AP: y= 3

5x + 120

bAt BandCsince product of gradients is 1e.g.mDC=

2

5,mCB=

5

2;

2

5 5

2= 1

17 a y= 3x + 2 b(0, 2) cy= 3x 8d(2, 2) eArea = 10 square unitsfArea = 40 square units

Chapter 3

Exercise 3A

1a2 2 b2 3 c1 4 d4 1

2a

1 0 0 0 10 1 0 1 0

0 0 1 0 00 1 0 1 0

1 0 0 0 1

b

1 1 1 1 11 1 1 1 1

1 1 1 1 11 1 1 1 1

1 1 1 1 1

3

1 0 0 0 00 1 0 0 0

0 0 1 0 00 0 0 1 0

0 0 0 0 1

Only the seats for top-leftto bottom-right diagonal

are occupied.

4

200 180 135 110 56 28110 117 98 89 53 33

5a[0 x] = [0 4] if x= 4

b

4 7

1 2

=x 7

1 2

ifx= 4

c

2 x 4

1 1 0 3

=

y 0 4

1 1 0 3

=

2 0 4

1 1 0 3

ifx= 0,y= 2

6a x= 2,y= 3 bx= 3,y= 2cx= 4,y= 3 dx= 3,y= 2

7

0 3 1 0

3 0 2 11 2 0 1

0 1 1 0

8

21 5 58 2 3

4 1 1

14 8 600 1 2

Exercise 3B

1X + Y =

4

2, 2X =

2

4,

4Y + X =

132

,X Y =

22

,

3A =3 36 9

,3A + B =

1 37 7

2a

6 1 2 4

8 4 2

b

5 1 018 7 13

c

5

3

1

3

0

6 7

3

13

3

32A =

2 20 4

,3A =

3 30 6

,

6A =6 6

0 12

4a Yes bYes

5a

6 44 4

b

0 912 3

c

6 58 1

d

6 1316 7

6a

0 1

2 3

b

2 36 3

c

3 31 7

7X =

2 4

0 3,Y =

92

232

12

11

8X + Y =

310 180 220 90

200 0 125 0

, representing

the total production at two factories in two

successive weeks.


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Ans

wers

Answers 715

Exercise 3C

1AX =

45

,BX =

4

1

,AY =

58

,

IX =

21

,AC =

0 11 2

,

CA =

1 10 1

, (AC)X =

10

,

C(BX) =

9

5

,AI =

1 21 3

,

IB =

3 2

1 1

,AB =

1 0

0 1

,

BA =

1 00 1

,A2 =

3 84 11

,

B2 = 11 8

4 3 ,A(CA) = 1 3

1 4 ,

A2C =2 5

3 7

2a AY,CIare defined,YA,XY,X2,XIare notdefined.

bAB =

0 0

0 0

3No

4LX = [7],XL =

4 26 3

5ABandBAare not defined unlessm= n.

6 1 0

0 1

7One possible answer is

A =

1 23 4

,B =

2 11.5 0.5

8One possible answer isA =

1 24 3

,

B =

0 1

2 3

,C =

1 22 1

,

A(B + C) =1 11

4 24,

AB + AC =1 11

4 24,

(B + C)A =

11 716 12

9

29

8.50

represents John spending 29 minutes

consuming food which cost him $8.50.29 22 128.50 8.00 3.00

Johns friends spent

$8.00 and $3.00 and took 22 and 12 minutes

respectively to consume their food.

10

6.008.00

2.0011.00

6.50

represents how much each studentspends in a week on magazines.

11a SC = s11c1 +s12c2 +s13c3s21c1

+s22c2

+s23c3

bSCrepresents the income from car sales foreach showroom.

cSC =s11c1+s12c2+s13c3 s11u1+s12u2+s13u3s21c1+s22c2+s23c3 s21u1+s22u2+s23u3

represents the income for each showroom fornew car sales and used car sales.

dCVgives the profit on each new car and each

used car for the three models.

Exercise 3D

1a 1 b

2 13 2

c2 d1

2

2 23 2

2a

1 14 3

b

2

7

114

1

7

3

14

c

1 0

0 1

k

d

cos sin sin cos

3A1 = 12 12

0 1

,B1 = 1 03 1,

AB =

5 13 1

, (AB)1 =

1

2

1

232

52

,

A1B1 =

1 12

3 1

,

B1A1

=

1

2

1

2

32

52

, (AB)1 = B

1A1

4a

12 32

1 2

b

0 7

1 8

c

5

2

72

11

2

212

5a

38

11

81

16

7

16

b

1116

17

16

1

4

3

4


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Answers


6

1

a110

0 1

a22

8

1 00 1

,

1 00 1

;

1 0k 1

,

1 0k 1

;

1 k0 1

,1 k

0 1

, kR

a b1 a2

ba

, b = 0

Exercise 3E

1a

310

b

517

2a

1

143

14

b 4

72

7

3a x= 17,y= 10

7 bx= 4,y= 1.5

cx= 307 ,y= 2

7dx= 2.35,y= 0.69

4(2,1) 5books $12, CDs $186a

2 34 6

x

y

=

36

b 2 34 6 is a singular matrix, not aregular matrix.

cThere is no unique solution for this system, buta solution can be found.

dThe solution set contains an infinite number ofpairs.

Multiple-choice questions1B 2E 3C 4E 5C

6A 7E 8A 9E 10D


(technology-free)

1a

1 5

3 9

b

1 13 1

c

0 103 29

d6 e

2

3

1

31

2 0

2a 0 012 8 b

0 0

8 8

3

a

2 34

a

, aR

4a ABdoes not exist,AC,CD,BEexist.

bDA =

14 0

,A1 = 1

7

1 23 1

5AB =

2 0

2 2,C1 =

2 13

2 1

2

6

1 23 5

7A2 =

4 0 0

0 4 0

0 0 4

,A1 =

1

2 0 0

0 0 1

2

0 1

2 0

88

9a i

3 55 8

ii

1 1818 19

iii1

7

3 1

1 2

bx= 2,y= 1


1a i

5 0

6 2

ii

1 24 6

iii

12 117 2

iv

72 21 7

b i

11 118 9

ii

1

13

4 11 3

iii1

13

13 213 7

iv

1

13

7 5

22 1

2a

8 2 115 3 1

14 18 7

b

2 6 63 3 3

15 12 3

c3 3 312 6 4

14 9 2

d

50

33

2

11

211

733

5

11

511

1

33

4

11

7

11

e

1

33

0 33 018 70 10

6 5 29

fA1CBC1 gC1B

3a i 2 34 1 xy = 3

5


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Ans

wers

Answers 717

ii14,1

14

1 3

4 2

iii1

7

91

iv

9

7,1

7

is the point of intersection

of the two lines

b i 2 14 2

xy =

3

8

ii0; Ais a singular matrix

clines represented by the equations are parallel

Chapter 4

Exercise 4A

1 a 2x 8 b2x + 8 c6x 12d12 + 6x ex 2 x f2x2 10x

2 a 6x + 1 b3x 6 cx + 1 d5x 33 a 14x 32 b2x2 11x

c32 16x d6x 114 a 2x2 11x b3x2 15x c20x 6x2

d6x 9x2 + 6x3 e2x2 x f6x 65 a 6x2 2x 28 bx 2 22x + 120

c36x2 4 d8x2 22x + 15ex 2 (

3 + 2)x + 2

3 f2x2 +

5x 5

6 a x 2 8x + 16 b4x2 12x + 9c36 24x + 4x2 dx 2 x + 1

4

ex 2 2

5x + 5 fx 2 4

3x + 12

7 a 6x

3

5x2

14x + 12 bx3

1c24 20x 8x2 + 6x3 dx 2 9e4x2 16 f81x2 121g3x2 + 4x + 3 h10x2 + 5x 2ix 2 +y2 z2 2xy

jax ay bx + by8 a ix 2 + 2x + 1 ii(x + 1)2

b i(x 1)2 + 2(x 1) + 1 iix 2

Exercise 4B

1 a 2(x + 2) b4(a 2) c3(2 x)d2(x 5) e6(3x + 2) f8(3 2x)2 a 2x(2x y) b8x(a + 4y)

c6b(a 2) d2xy(3 + 7x)ex (x + 2) f5x(x 3)g4x(x + 4) h7x(1 + 7x)ix (2 x) j3x(2x 3)

kx y(7x 6y) l2xy2(4x + 3)3 a (x2 + 1)(x + 5)

b(x 1)(x + 1)(y 1)(y + 1)c(a + b)(x +y) d(a2 + 1)(a 3)e(x a)(x + a)(x b)

4 a(x 6)(x + 6) b(2x 9)(2x + 9)c2(x 7)(x + 7) d3a(x 3)(x 3)e(x 6)(x + 2) f(7 +x)(3 x)g3(x 1)(x + 3) h5(2x + 1)

5 a(x 9)(x + 2) b(y 16)(y 3)c(3x 1)(x 2) d(2x + 1)(3x + 2)e(a

2)(a

12) f(a

+9)2

g(5x + 3)(x + 4) h(3y + 6)(y 6)i2(x 7)(x 2) j4(x 3)(x 6)

k3(x + 2)(x + 3) la(x + 3)(x + 4)mx (5x 6)(x 2) n3x(4 x)2ox (x + 2)

Exercise 4C

1 a2 or 3 b0 or 2 c4 or 3

d4 or 3 e3 or4 f0 or 1g

5

2

or 6 h

4 or 4

2 a0.65 or 4.65 b0.58 or 2.58c2.58 or 0.58

3 a4, 2 b11, 3 c4, 16

d2, 7 e32

, 1 f1

2,

3

2

g3, 8 h23,3

2 i3

2, 2

j5

6, 3 k3

2, 3 l

1

2,

3

5

m34,

2

3 n

1

2 o5, 1

p0, 3 q5, 3 r1

5, 2

44 and 9 53 62, 23

8713 850 96 cm, 2 cm

105 11$90, $60 1242

Exercise 4D

a i(0,4)iix= 0

iii(2, 0), (2, 0)

y

x202

(0, 4)

b i(0, 2)iix= 0

iiinone

y

x0

(0, 2)


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Answers


c i(0, 3)iix= 0

iii(

3, 0), (

3, 0) (0, 3)

y

x3 3

0

d i(0, 5)iix= 0

iii

5

2, 0

,

5

2, 0

(0, 5)

y

x0

25

25

e i(2, 0)iix= 2

iii(2, 0)

y

x

4

0 2

f i(3, 0)iix= 3

iii(3, 0)

y

x

9

3 0

g i(1, 0)iix= 1

iii(1, 0)

y

x0

1

1

h i(4, 0)

iix= 4iii(4, 0)

y

x0 4

8

i i(2,1)iix= 2

iii(1, 0)(3, 0)

y

x

3

1 3

0(2, 1)

j i(1, 2)

iix= 1iiinone

y

x

3

(1, 2)

0

k i(1,1)iix= 1

iii(2, 0)(0, 0)

y

x2 0

(1, 1)

l i(3, 1)iix= 3

iii(2, 0)(4, 0)

y

x0 2 4

(3, 1)

8

m i(2,4)iix= 2

iii(4, 0), (0, 0)

y

4(2, 4)

x

0

n i(2,18)iix= 2

iii(5, 0), (1, 0)

y

5

(2, 18)

x

100

1

o i(4, 3)iix= 4

iii(3, 0), (5, 0)

y

(4, 3)

x0 3 5

p i(5,2)iix= 5

iiinone

y

(5, 2) x

0

29

2

q i(

2,

12)

iix= 2iii(0, 0), (4, 0)

04

(2, 12)

x

y

r i(2, 8)iix= 2

iii(2

2, 0)(2 +

2, 0)

y

0

8

(2, 8)

x

2 2 2 + 2


12/67

Ans

wers

Answers 719

Exercise 4E

1 a x 2 2x + 1b x 2 + 4x + 4 cx 2 6x + 9dx 2 6x + 9 ex 2 + 4x + 4 fx 2 10x + 25gx 2 x + 1

4 hx 2 3x + 9

4

2 a (x

2)2 b(x

6)2 c

(x

2)2

d2(x 2)2 e2(x 3)2 fx 1

2

2

g

x 3

2

2h

x + 5

2

23 a 1

2 b2

6 c3

7

d5

17

2 e

2

2

2 f1

3, 2

g1 1 k h 1

1 k2k

i3k

9k2 42

4 a y= (x 1)2 + 2t. pt (1, 2)

y

x(1, 2)

3

0

b y= (x + 2)2 3t. pt (2,3)

y

x

(2, 3)

1

0

c y=x 3

2

2 5

4

t. pt 3

2,5

4y

x

5

4

3

2 ,

1

0

d y= (x 4)2 4t. pt (4,4)

y

x

(4, 4)

12

0

e y=x 1

2

2 9

4

t. pt

1

2,9

4

y

x

9

4

1

2

2

0

,

fy= 2(x + 1)2 4t. pt (1,4)

y

x

(1, 4)

0

2

g y= (x 2)2 + 5t. pt (2, 5)

y

x

(2, 5)

1

0

h y= 2(x + 3)2 + 6t. pt (3, 6)

y

x

(3, 6)

0

12

i y= 3(x 1)2 + 9t. pt (1, 9)

y

x

12

0(1, 9)

Exercise 4F

1 a7 b7 c12 a2 b8 c43 a y

x01

1

b y

x6 3 0

(3, 9)

c y

x

25

5 0 5

d y

x2

4

0 2

e y

x2

,3

4

3

2

118

01

f y

x0 1 2

(1, 2)

g

x2 1 0

,3

4

32

1 18

h y

x

1

0

4 a y

x5 0 2

10

,3

2 121

4

b y

x

4

01 4

2 ,12

214

c y

x

(1, 4)

3 0 1

3

d y

x

(2, 1)3 1

3

0


13/67

Answers


e f(x)

x

1

0 1

,1

4

12

118

f

x

6

3 10 2

,1

2 6

1

4

f(x)

g f(x)

x3 2 0

6

2 ,12

1

4

h f(x)

x0

24

3 8

2 30,1

2

1

4

Exercise 4G

1 a i40 ii2

10 b i28 ii2

7

c i172 ii243 d i96 ii46e i189 ii3

21

2 a 1 +

5 b3

5

2

c1 +

5

2 d1 + 2

2

3 a3

13 b7

61

2 c

1

2, 2

d1 32

2 e2 3

2

2 f1

30

5

g1

2

2 h1,

32

i3

6

5

j13 145

12 k

2 4 2k22k

l2k

6k2 2k

2(1 k)4r= 2.16 m5 a

0

1

(2.5, 7.25)

0.195.19

y

x

b

0

10.28 1.78

(0.75, 2.125)

y

x

c

0

13.3

(1.5, 3.25)

0.3

y

x

d

0

4

3.24 1.24

(1, 5)

y

x

e

0

10.251

(0.625, 0.5625)

y

x

f

0 1

1

2

y

x2

3

2

3,

Exercise 4H

a1.5311 b1.1926 c1.8284 d1.4495

Exercise 4I

1 a 20 b

12 c25 d41 e41

2 a Crosses thex-axis bDoes not crosscJust touches thex-axis

dCrosses thex-axis eDoes not crossfDoes not cross

3 a 2 real roots bNo real roots c2 real roots

d2 real roots e2 real roots fNo real roots4 a = 0, one rational root

b = 1, two rational rootsc = 17, two irrational rootsd = 0, one rational roote = 57, two irrational rootsf = 1, two rational roots

5The discriminant=

(m+

4)2

0 for allm,

therefore rational solution(s).

Exercise 4J

1 a{x:x 2} {x:x 4}b{x: 3 3}

x:x < 3

2

e

x: 3

2 23 x:x <

3

4

h

x:

1

2 x 3

5

i{x: 4 x 5}

j

p :

1

2(5

41) p 1

2(5 +

41)

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

2 a i

5 < m

5 orm <

5

bi0 < m 43

orm < 0

ci45< m < 0 iim= 0 orm= 4

5

iiim < 45

orm > 0

d i2 < m < 1 iim=2 or 1iiim > 1 orm 4

3 4p = 1

2 5 2


14/67

Ans

wers

Answers 721

e

1 +

33

2 ,3

33

,

1

33

2 ,3 +

33

f5 +

33

2

, 23

+3

33,5

33

2 , 23 3

33

2 a Touch at (2, 0) bTouch at (3, 9)cTouch at (2,4) dTouch at (4,8)

3 a x= 8,y= 16 andx=1,y= 7bx= 16

3 ,y= 37 1

3andx= 2,y= 30

cx= 45,y= 10 2

5andx=3,y= 18

dx

=10

2

3

,y

=0 andx

=l,y

=29

ex= 0,y=12 andx= 32,y= 7 1

2fx= 1.14,y= 14.19 andx=1.68,y= 31.09

4 a13b i

x

y

20.3

03.3

iim= 6 32 = 6 42

5ac = 14 bc >

1

46a= 3 ora= 1 7b = 18y= (2 + 2

3)x 4 2

3

and y= (2 2

3)x 4 + 2

3

Exercise 4L

12 2a= 4,c = 83a

=

4

7,b

=

247

4a

= 2,b

=1,c

=6

5a y= 516

x2 + 5 b y=x2

c y= 111

x2 + 711

x d y=x2 4x+ 3

e y= 54x2 5

2x + 3 3

4

fy=x2 4x+ 66y= 5

16(x+ 1)2 + 3

7y= 12

(x2 3x 18)

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

x2 x + 7510a C b B c D d A

11y= 2x2 4x 12y=x2 2x 113y= 2x2 + 8x 614 ay= ax (x 10),a > 0

b y

=a(x

+4)(x

10),a < 0

cy= 118

(x 6)2 + 6d y= a(x 8)2,a < 0

15a y= 14x2 +x + 2

b y=x2 +x 516r= 1

8t2 + 2 1

2t 6 3

8

17 a B bD

Exercise4M

1 a A = 60x 2x2b A

x

450

0 15 30

cMaximum area = 450 m22 a E

x

100

0 0.5 1

b0 and 1 c0.5 d0.23 and 0.77

3 a A = 34x x2b A

x

289

0 17 34

c289 cm2

4 a C($)

3000

20001000

0 1 2 3 4 h

The domain depends on the height of thealpine area. For example in Victoria the

highest mountain is approx. 2 km high

and the minimum alpine height wouldbe approx. 1 km, thus for Victoria,

Domain = [1, 2].bTheoretically no, but of course there is a

practical maximumc$ 1225


15/67

Answers


5 a T(000)

0 8 16

t(0.18, 15.82)

0.18 15.82

t

b8874 units

6 a

x 0 5 10 15 20 25 30

d 1 3.5 5 5.5 5 3.5 1

d

5

4

3

2

1

0

0 5 10 15 20 25 30 xb i5.5 m

ii15 5

7 m or 15 + 5

7 m from the batiii1 m above the ground.

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

2x 11

2

8a= 1615

, b = 85, c = 0

9aa= 721600

, b = 41400

, c = 5312

b S

hundreds ofthousands

dollars 5312

354.71 t(days)

c iS=$1 236 666 iiS=$59 259


1A 2C 3C 4E 5B

6C 7E 8E 9D 10A

Short-answer questions (technology-free)

1a

x + 9

2

2b(x + 9)2 c

x 2

5

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

2 a3x + 6 bax + a2 c49a2 b2dx 2 x 12 e2x2 5x 12 fx 2 y2ga3 b3 h6x2 + 8xy + 2y2 i3a2 5a 2j4xy k2u + 2v uv l3x2 + 15x 12

3 a 4(x 2) bx (3x + 8) c3x(8a 1)d(2 x)(2 +x) ea(u + 2v + 3w)fa2(2b 3a)(2b + 3a) g (1 6ax )(1 + 6ax )

h(x + 4)(x 3) i(x + 2)(x 1)j(2x 1)(x + 2) k(3x + 2)(2x + 1)l(3x + 1)(x 3) m (3x 2)(x + 1)

n(3a 2)(2a + 1) o(3x 2)(2x 1)4 a

x

y

(0, 3)

0

b

x

y

(0, 3)

0

3

2, 0

3

2, 0

c

x

y

11

(2, 3)

0

d

x

y

(2, 3)

0

11

e

x

y

29

0(4, 3)

3

2

+4 , 0

3

2, 04

f

x

y

3

2

3

2 0

(0, 9)

g

x

y

0 (2, 0)

(0, 12)

h

x

y

(2, 3)

(0, 11)

0

5 a

x

y

1 0

5 5

(2, 9)

b

x

y

0 6

(3, 9)

c

x

y

4 23

4 + 23

(0, 4)

(4, 12)

0

d

x

y

2 6

2 + 6

(0, 4)

(2, 12)

0

e

x

y

2 + 72 7

(2, 21)

(0, 9)

0

f

x

y

1

5

0 5

(2, 9)


16/67

Ans

wers

Answers 723

6 a ii x= 72

x

y

(0, 6)

0 1 625

2

7

2

,

b ii x= 12

x

y

49

4

1

2,

(0, 12)

4 0 3

c ii x= 52

x

y

81

4

5

2,

14

02 7

d ii x= 5

x

y

(0, 16)

0 2 8(5, 9)

e ii x= 14

x

y

1

8

1

4,

3 0 52

(0, 15)15

f ii x= 1312

x

y

1

3

5

2

50

1

24

13

12,12

g ii x= 0

x

y

4

34

3 0

(0, 16)

h ii x= 0

x

y

5

2

5

2 0

(0, 25)

7 a0.55,5.45 b1.63,7.37c3.414, 0.586 d0.314,3.186e0.719,2.781 f0.107,3.107

8y= 53x(x 5)

9y= 3(x 5)2 + 210y= 5(x 1)2 + 511 a (3, 9), (1, 1)

b(1.08, 2.34), (5.08, 51.66)

c(0.26, 2), (2.6, 2)d

1

2,

1

2

, (2, 8)

12 a m=

8 = 2

2

bm

5 orm

5

cb2 4ac = 16 > 0


1 a y =0.0072x(x 50)

b

4

5

3

2

1

00 10 20 30 5040

x

y

c10.57 m and 39.43 m25 25

3

3 m and 25 + 25

3

3 m

d3.2832 m

e3.736 m (correct to 3 decimal places)

2 aWidth of rectangle = 12 4x6

m, length of

rectangle = 12 4x3

m

b A = 179

x2 163

x + 8

cLength for square = 9617

m and length for

rectangle = 10817

m ( 5.65 6.35 m)3 aV= 0.72x2 1.2x b22 hours4 aV= 10 800x+ 120x2

bV= 46.6x2 + 5000x cl= 55.18 m5 al= 50 5x

2

b A = 50x 52x2

c

0 10 20 x

250

A(10, 250)

dMaximum area = 250 m2 whenx= 10 m

6x= 1 +

5

2

7 a

25 +x2b i16

x ii

x2

32x

+265

c7.5 d10.840 e12.615

8 a iy=

64t2 + 100(t 0.5)2=

164t2 100t+ 25

ii y(km)

5

0 t(h)


17/67

Answers


iiit= 12

; 1.30 pmt= 982

; 1.07 pm

iv0.305; 1.18 pm; distance 3.123 km

b i0,25

41 ii

25 2

269

82

9 b2x + 2y= b

c8x

2

4bx + b2

16a2

= 0e ix= 6 14,y= 6 14

iix=y=

2a

fx= (5

7)a

4 ,y= (5

7)a

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

b i(x , 6 + 4x x2) ii(x ,x 1)iii(0, 1) (1, 0) (2, 1) (3, 2) (4, 3)ivy=x 1

c id= 2x2 6x + 10ii

(0, 10)

(1.5, 5.5)

d

0 x

iiimin value ofd= 5.5 occurs whenx= 1.511 a 45

5

bi y= 1600

(7x2 190x + 20 400)

ii

190

14 ,

5351

168

c

(20, 45) (40, 40)

(60, 30)

(30, 15)

y =1

2x

C

O x

D

Bd

A

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

iiminimum value = 47324

whenx= 35

Chapter 5

Exercise 5A

1 a y

x0

(1, 1)

b y

x

(1, 2)

0

c y

x0

1

21,

d y

x0

(1, 3)

e y

x

2

0

f y

x0

3

g y

x0

4

h y

x

5

0

i y

x0

1 1

j y

x2

0

12

k y

x

1 0

3

4

l y

x0 3

4

31

3

2 a y= 0,x= 0 b y= 0,x= 0cy= 0,x= 0 d y= 0,x= 0ey= 2,x= 0 fy= 3,x= 0g y= 4,x= 0 h y= 5,x= 0i y

=0,x

=1 jy

=0,x

= 2

ky= 3,x= 1 l y= 4,x= 3

Exercise 5B

1 a y

x

1

9

03

b y

x0

4


18/67

Ans

wers

Answers 725

c y

x

1

4

0 2

d y

x

4

3

0 1

e y

x3

0

4

f y

x

1

1

2

02

g y

x

6

0

3

53

2

h y

x

2

115

16

0 4

2 a y= 0,x=3 by=4,x= 0c y= 0,x= 2 dy= 3,x= 1e y=4,x=3 fy= 1,x= 2g y=6,x=3 hy= 2,x= 4

Exercise 5C

a

0

3

x

y

x 0 andy 3

b y

0

(2, 3)x

x 2 andy 3c

0

11

(2, 3)

y

x

x 2 andy 3

d

0

1 + 2

(2, 1)

y

x

x 2 andy 1e

0 7

3 2

(2, 3)

y

x

x 2 andy 3

f

0

(2, 3)

22 3

1

4

y

x

x 2 andy 3

g

0 11

(2, 3)

y

x

x 2 andy 3

h y

x0

(4, 2)

x 4 andy 2i

0(4, 1)

y

x

x 4 andy 1

Exercise 5D

1 ax 2 +y2 = 9 bx 2 +y2 = 16c(x 1)2 + (y 3)2 = 25d(x 2)2 + (y + 4)2 = 9e(x + 3)2 + (y 4)2 = 25

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

2 aC(1, 3), r= 2 bC(2,4), r=

5

cC(3, 2), r= 3 dC(0, 3), r= 5eC(3,2), r= 6 fC(3,2), r= 2gC(2, 3), r= 5 hC(4,2), r=

19

3 a y

x

8

0

8

8 8

b y

x

4

7

1

0

c y

x7 2 0 3

d y

x

4

0

1

e y

x

5

2

3

2

0

f y

x

3

0

g y

x

3

0

2

h y

x0

11

4


19/67


20/67

Ans

wers

Answers 727

6 a y

x3 30

b y

x5 1 3

0

c y

x1 1

(0, 2)

0

d

y

x

(2, 3)

0


1 a (x 10)2 +y2 = 25 cm=

3

3

d P152

,5

3

2 e53

2 a x 2 +y2 = 16bii m=

3

3 ;y=

3

3 x 8

3

3 ,

y=

3

3 x + 8

3

3

3a4

3 b

34

c4y + 3x= 25 d 12512

4 a iy1

x1ii

x1y1

c

2x +

2y= 8 or

2x +

2y= 8

5a y= 33

x + 233

a,y= 33

x 233

a

bx 2 +y2 = 4a26 bii

y = 14

14

x

y

0

x

14, 1

4

ci

14 < k< 0 iik= 0 ork< 1

4

iiik>

07a0 < k 1 cx= 6

710a f:RR, f(x) = 3x + 2

b f:RR, f(x) = 32x + 6

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

11a y

x

(2, 4)

(1, 1)

0

Range = [0, 4]

b y

x

(2, 8)

(1, 1)0

2

Range = [1, 8]c y

x0

1

33,

Range =

1

3,

d y

x(1, 2)

0

Range = [2, )

Exercise 6D

1One-to-one functions are b,d,eandg

2Functions area,c,d,fandg. One-to-onefunctions arecandg.

3a Domain =R , Range =RbDomain =R+ {0}. Range = R+ {0}cDomain =R , Range = [1, )dDomain = [3, 3], Range = [3, 0]eDomain =R+, Range =R+fDomain =R , Range = (, 3]gDomain = [2, ), Range =R+ {0}hDomain =

12,

, Range = [0, )

iDomain =

, 32

, Range = [0, )

jDomain =R \ 12

, Range =R \ {0}

kDomain =R \ 12

, Range = (3, )

lDomain =R \ 12

, Range =R \ {2}

4a Domain =R , Range =RbDomain =R , Range = [2, )cDomain = [ 4, 4], Range = [ 4, 0]dDomain =R \ {2}, Range =R \ {0}

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

6a y

x

222

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

, 0]

R, f2(x)

=x2

2

Exercise 6E

1a y

x0

Range = [0, )

b y

x0

1

1

Range = [0, )

c y

x0

Range = (, 0]

d

y

x0

Range = [1, )e y

x

2

0

(1, 1)

Range = [1, )


23/67

Answers


2a y

x

4

3

2

1

1 2 30

Range = (, 4]

3 y

x

1

2

1 2 3 54

3 2 1

1

2

3

4

5

0

4a y

x(0, 1)

0

bRange = [1,)

5a y

x3 3

0

9

bRange =R6a

y

x

(1, 1)

0

bRange = (, 1]

7 f(x) =

x+ 3, 3 x 1x+ 1, 1


24/67

Ans

wers

Answers 731

Exercise 6H

1 a y

xy = 3

03

b y

x

y =3

30

3

c

x =2

1

4

0

y

x

d

0 2

y

x

e

0

1

x=1

y

x

f

y =40

y

x

g

1

2

0

x=2

y

x

h y

x

1

3

x=3

0

i y

x

x=3

0

jy

x

x =4

1

16

0

k y

xy=1

x=1

0

l y

x

3

2

y=2

x=2

0

2 a y

x

1

x=1

0

by

xy=1

1 0

c y

x

1

3

x=3

0

d y

x

1

3

y=3

0

ey

x

x=1

0

fy

x

y=1

0

1

3 a y

x1

10

b y

x1

0

c y

x3 0

9

d y

x0

3

3 3

e y

x1 0

1

f y

x

1

1 10

4 a y

x

3

(1, 2)

0

b

y

x(3, 1)

0

10

c y

x3 5

(3, 5)

43 + 5

0

d y

x1 3 1 + 3

(1, 3)

0

e y

x1 2 1 + 2

01

(1, 2)

f y

x

24

4 6

0

(5, 1)

5 a y

xy=1

0

Range = (1, )

b y

x0

Range = (0, )c y

x

1

0

Range = (0, )

d y

x0

y=4

1

2

1

2

Range = ( 4, )

Exercise 6I

1 a i y= 4x2 iiy= x2

25 iii y= 2x

2

3

ivy= 4x2 vy= x2 viy =x2


25/67

Answers


b i y= 14x2

ii y= 25x2

iiiy= 23x2

ivy= 4x2

v y= 1x2

viy= 1x2

c i y= 12x

ii y= 5x

iiiy= 23x

ivy=4

x v y= 1

x viy= 1

x

d i y=

2x ii y=

x

5iiiy= 2

x

3

ivy= 4x vy= x vi y= x x 02 a y

x

(1, 3)

0

b y

x0

c y

x

(1, 3)

0

d y

x0

1, 12

e y

x0

3(1, )

f y

x0

1, 32

Exercise 6J

1 a y= 3x 2 by= x+ 3c y= 3x dy=

x

2

e y= 2x 2 3 fy=

x+ 22 3

2a y= 3x 2 by=

1x+ 3

c y

= 3

x

dy

=

2

xe y= 2

x 2 3 fy=2

x+ 2 3

3 a iA dilation of factor 2 from thex-axisfollowed by a translation of 1 unit in the

positive direction of thex-axis and

3 units in the positive direction of they-axis

iiA reflection in thex-axis followed by atranslation of 1 unit in the negative direction

of thex-axis and 2 units in the positivedirection of they-axis

iiiA dilation of factor 1

2

from they-axis

followed by a translation of 12unit in the

negative direction of thex-axis and2 units in the negative direction of the

y-axis

b iA dilation of factor 2 from thex-axisfollowed by a translation of 3 units in the

negative direction of thex-axisiiA translation of 3 units in the negative

direction of thex-axis and 2 units in thepositive direction of they-axis

iiiA translation of 3 units in the positivedirection of thex-axis and 2 units in the

negative direction of they-axis

c iA translation of 3 units in the negativedirection of thex-axis and 2 units in the

positive direction of they-axisiiA dilation of factor 1

3from they-axis

followed by a dilation of factor 2 from thex-axis

iiiA reflection in thex-axis followed by atranslation of 2 units in the positive direction

of they-axis

Exercise 6K

1 a i A = (8+x)y x2ii P= 2x+ 2y + 16

b i A = 192 + 16x 2x2ii0


26/67

Ans

wers

Answers 733

4 a iC1= 64+ 0.25x iiC2= 89b

10080604020

20 40 60 80 100 120

C($)

x (km)

C2

C1

0cx >100 km


1B 2E 3B 4C 5E

6B 7D 8E 9C 10D


(technology-free)

1 a 16 b26 c23

2 a y

x

(1, 7)(0, 6)

(6, 0)0

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

cRange = [0, 4] dRange = (, 9]eRange = (2, ) f{6, 2, 4}gRange = [0, ) hR \ {2}iRange = [5, 1] jRange = [1, 3]

4 a a= 15,b = 332

bDomain =R \ {0}5 a

(1, 1)

0 (2, 0)x

y b[0, 1]

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

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

1

2

e[10, 10] f(, 4]

9 b, c, d, e, f, g,andjare one-to-one10 a

(0, 1)0

(3, 9)

y

x

b(3, 9)

(0, 1)

0

y

x

11a f1(x) = x+ 23

, Domain = [5, 13]b f1(x) = (x 2)2 2, Domain = [2, )

c f1(x) =

x

3 1, Domain = [0, )

d f1(x) = x+ 1, Domain = [0, )12 a y= x 2 + 3 by= 2x c y= x

dy= x ey= x3

Extended-response questions1 a

500

400

300

200

100

1 2 3 4 5 6 7

d(km)

t(hour)

Y

Z

0X

Coach starting fromX:d= 80t for 0 t 4d= 320 for 4


27/67

Answers


7 a A(x) = x4

(2a (6

3)x)

b0


28/67

Ans

wers

Answers 735

c

y

0 22 x

(0, 16)

d

y

0 22 x

16

e y

0x

81

3

f y

0 x

(1, 1)

(0, 1)

Exercise 7B

1a x 2 + 2x+ 3x 1 b2x

2 x 3+ 6x+ 1

c3x2 10x+ 22 43x+ 2

dx 2 x+ 4 8x+ 1

e2x2 + 3x+ 10+ 28x 3

f2x2 5x+ 37 133x+ 4 gx

2 +x+ 2x+ 3

2a1

2x2 + 7

4x 3

8+ 103

8(2x+ 5)bx 2 + 2x 3 2

2x+ 1c

1

3x2 8

9x 8

27+ 19

27(3x

1)

dx 2 x+ 4+ 13x 2

ex 2 + 2x 15f

1

2x2 + 3

4x 3

8 5

8(2x+ 1)

3a x 2 + 3x+ 8+ 9x 1

bx 2 x2+ 9

4+ 21

4(2x 1)

Exercise 7C

1 a 2 b29 c15 d4 e7f12 g0 h5 i8

2 a a=3 ba= 2ca= 4 da= 10

Exercise 7D

2 a 6 b28 c1

33 a (x 1)(x+ 1)(2x+ 1) b(x+ 1)3

c(x 1)(6x2 7x+ 6)d(x

1)(x

+5)(x

4)

e(x+ 1)2(2x 1) f(x+ 1)(x 1)2

g(x 2)(4x2 + 8x+ 19)h(x+ 2)(2x+ 1)(2x 3)

4 a(x 1)(x2 +x+ 1)b(x+ 4)(x2 4x+ 16)c(3x 1)(9x2 + 3x+ 1)d(4x 5)(16x2 + 20x+ 25)e(1 5x)(1 + 5x+ 25x2)f(3x+ 2)(9x2 6x+ 4)g(4m 3n)(16m2 + 12mn + 9n2)h(3b+ 2a)(9b2 6ab + 4a2)

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

6a= 3,b = 3,P(x) = (x 1)(x+ 3)(x+ 1)7 b inodd iineven

8 aa=

l,b=

l b i P(x)=

x3

2x2

+3

Exercise 7E

1 a1, 2, 4 b4, 6 c1

2, 3,2

3

d1, 1, 2 e2, 3, 5 f1,23

, 3

g1,

2,

2 h25, 4, 2

i12,

1

3, 1 j2,

3

2, 5

2 a6, 2, 3 b2, 2

3,

1

2 c3

d1 e1, 3 f3, 2

3

3 a2, 0, 4 b0,1 2

3

c5, 0, 8 d0, 1

17

4 a0,2

2 b1 + 2 3

2 c2d5 e 1

105 a1 b1 c5,

10 d4, a

6 a2(x 9)(x 13)(x+ 11)b(x+ 11)(x+ 3)(2x 1)

c(x+ 11)(2x 9)(x 11)d(2x 1)(x+ 11)(x+ 15)

Exercise 7F

1 a0 1 2 3

x+

x

y

1 2 30

b +

x2 1 0 1

22

1 1

y

x


29/67

Answers


c0 1 2

x+

3

1 2 30

y

x

d

2 1 0 1 2

+

x

3

1

2

3 2 1

612

y

x0 1 2

e +

x

3 2 1 0 1 2 3

3 2 1 1 2 30

y

x

f1 0

+

x

y

x1 0 1

g +

x

3210

y

x321

30

h + x321012

y

x321012

i+

x

21012

3 3

y

x21012

3

3 3

j 2

3 +

x

32101

y

x1

0 1 2 3

6

23

k 12

13 +

x101

1

1

0 1 12

13

y

x

2 a y

x

(1.02, 6.01)1.5

0.5

(3.02, 126.01)

0

5

by

x

(1.26, 0.94)1.5

1

(1.26, 30.94)

0

2.5

c y

x

(1.35, 0.35)

1.5

(0, 18)

(1.22, 33.70)

0

2.5 1.2

d y

x

(0.91, 6.05)

0

(2.76, 0.34)

e y

x0

(2, 8)

3

f y

x0

(2, 14)

3.28

6

3(x+ 1)(x+ 1)(x 3) = 0, Graph just touchesthex-axis atx= 1 and cuts itatx= 3.

Exercise 7G

a {x:x

2}

{x: 1

x

3}

b {x:x 4} {x:2 x 1}c {x:x


30/67

Ans

wers

Answers 737

e y

(3.54, 156.25)(3.54, 156.25)

50x

f y

2

2

0

16

x

g

y

9 9

(6.36, 1640.25) (6.36, 1640.25)

x

0

h

y

4

0 3

(3.57, 3.12)

(1.68, 8.64)

x

i y

5

0 4

(4.55, 5.12)

(2.20, 24.39)

x

j y

5 4 0 4 5(4.53, 20.25) (4.53, 20.25)

x

k y

0 2 x

20

l y

(1.61, 163.71)

(5.61, 23.74)

50

47x

Exercise 7J

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

c f(n) =1

6 n3

+1

2 n2

+1

3 n

d f(n) = 13

n3 + 12

n2 + 16

n

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

c f(n) = 13

n3 + 12

n2 + 16

n

d f(n) = 43

n3 13

n

e f(n) = 13

n3 + 32

n2 + 76

n

f f(n) = 43

n3 + 3n2 + 53

n

3 f(n) = 12

n2 12

n

4 f(n) = 13

n3 + 12

n2 + 16

n

5 f(n) = 14

n2(n + 1)2

Exercise 7K

1 a l

=12

2x,w

=10

2x

bV= 4x(6x)(5x)

c

x(cm)10 2 3 4 5

100

(cm3)V dV= 80

ex= 3.56 orx= 0.51fVmax = 96.8 cm3 whenx= 1.81

2 ax=

64 h2 bV= h3

(64 h2)c

4.62

0

50

100

150

200

(m3)

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

V dDomain = {h: 0


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Answers


g y

x(2, 3)

(0, 29) 2

3

43

0

h y

x(2, 1)

(0, 23) 21

33

0

2 a y

0

1

1 x

b y

0

2

x , 12

1

c y

0

(1, 1)

2 x

d y

0 x

e y

0

1

x3

14 3

14

f y

015

1 3

(2, 1)

x

g y

0

1

(1, 3)

x 1

3

2 114

143

2

h y

03 1

(2, 1)

x2 +

1

2

14

2 1

2

14

3a P32 = 0 andP(2) = 0, (3x+ 1)

bx= 2, 12, 3 cx= 1,

11,+

11

d i P

1

3

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

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

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

(0, 24)

6 5 3 1 0 1 2 4

x

y

4 2 3

b

(0, 24)

x

4 2 1 0

y

1 5 6

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

3 423

c

1 1.53 2.5 1.5 0 1 2

x

y

(2, 0) 0

(0, 4)

0.5 0.52

, 023

, 012

d

x

y

5 4 3 2 1 0 1 4

36

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

6 2 3

7 a41 b12 c43

9

8y= 25

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

9y= 281

x(x+ 4)2

10 a a= 3,b = 8 b(x+ 3)(2x 1)(x 1)11 a y= (x 2)3 + 3 by= 2x3

cy= x3 dy= (x)3 = x3

ey=x

3

3= x

3

27

12 a y= (x 2)4 + 3 by= 2x4

cy= (x+ 2)4

+ 313 a Dilation of factor 2 from thex-axis, translation

of 1 unit in the positive direction of thex-axis,then translation of 3 units in the positive

direction of they-axisbReflection in thex-axis, translation of 1 unit in

the negative direction of thex-axis, thentranslation of 2 units in the positive direction

of they-axis

cDilation of factor 12from they-axis, translation

of 12unit in the negative direction of thex-axis

and translation of 2 units in the negativedirection of they-axis


1a v= 132 400

(t 900)2

bs= t32 400

(t 900)2

c

t (s)800600400200560105

0

1000

2000

3000

(cm)s Domain = {t: 0 < t < 900}

(300, 3333.3)

dNo, it is not feasible since the maximum range

of the taxi is less than 3.5 km (333 km).eMaximum speed 2000

105 = 19 m/s

Minimum speed 2000560

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

ba=

2

25c R

12=

12

343(x

7)3


32/67

Ans

wers

Answers 739

3 a 4730 cm2 bV= l2(

2365 l)c

(cm3)

l (cm)

V

5000

10000

15000

20000

10 20 30 40 500

d il= 23.69 orl= 39.79iil= 18.1 orl= 43.3

eVmax 17 039 cm3,l 32.42 cm4aa= 43

15 000, b = 0.095, c = 119

150 ,

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

iifurthest from the ground (0, 15.8)

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

2

b

0 24

V

x

i0


33/67

Answers


Exercise 8D

1y= 2x2

9 2x

3 4 2y= x

3

32 x

3y= 3x+ 184

4y= x4 4

5y=x

4+ 3 6y=x

+21

4

7y= 3x3

4 9x

2

2 + 14

8y= 3x3

4 9x

2

2 + 8

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

b

0 0 0 1

1 1 1 18 4 2 1

27 9 3 1

a

b

c

d

=

111

5

ca= 1,

b = 4,

c = 5,

d= 1dy= 2x3 + 8x2 10x+ 210 a a= 2, b = 0, c = 4, d= 0

by= 2x3 + 4x


1E 2B 3B 4E 5D

6B 7C 8C 9D 10C


(technology-free)

1 a1 0

0 4

(1, 12) b 3 00 1 (3, 3)c

1 00 1

(1,3)d

1 00 1

(1, 3)

e

0 11 0

(3,1)

2x= 4,y= 1 andz= 73 a y= 2x+ 2.

b i2a

ii2


34/67

Ans

wers

Answers 741

3 a 200 L

bV=

20t 0 t 1015t+ 50 10


35/67

Answers


e13 face, king, queen, jack

g4 h16

4 a {BB,BR,RB,RR}b {H1,H2,H3,H4,H5,H6,T1,T2,T3,

T4,T5,T6}c {MMM,MMF,MFM,FMM,MFF,FMF,

FFM,FFF}

5 a {0, 1, 2, 3, 4, 5}b {0, 1, 2, 3, 4, 5, 6} c {0, 1, 2, 3}

6 a {0, 1, 2, 3, . . .} b {0, 1, 2, 3, . . . , 41}c {1, 2, 3, . . .}

7 a {2, 4, 6} b {FFF} c

8

H

H

T

T

1

2

3

4

5

6

{HH,HT,T1, T2,

T3, T4, T5, T6}

9 a {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2,3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4),

(4, 1), (4, 2), (4, 3), (4, 4)}b {(2, 4), (3, 3), (4, 2)}

10 a {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2),(2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)}

b {(l, 1), (2, 2), (3, 3)}

11 a {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}b {(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6),

(5, 2), (5, 4), (5, 6)}

12 a

R

R

B

B

W

W

RB

WR

BW

RBW

RB

W

RB

W

RBW

RBW

RBW

RB

W

RB

W

RB

W

RBW

RBW

BR

W

{(RR), (RBR), (RBB), (RBWR), (RBWB),(RBWW), (RWR), (RWBR), (RWBB),(RWBW), (RWW), (BRR), (BRB), (BRWR),

(BRWB), (BRWW), (BB), (BWRR), (BWRB),

(BWRW), (BWB), (BWW), (WRR), (WRBR),(WRBB), (WRBW), (WRW), (WBRR),

(WBRB), (WBRW), (WBB), (WBW), (WW)}

b4

13 aS

S

HH

D

D

C

SHC

D

SHC

D

SHCD

SHC

{(SHS), (SHH), (SHCS), (SHCH), (SHCC),

(SHCDS), (SHCDH), (SHCDC), (SHCDD ),(SHDS), (SHDH), (SCDCS), (SHDCH),

(SHDCC), (SHDCD ), (SHDD )}b5

Exercise10B

1a17

50 b

1

10 c

4

15 d

1

2002 a No banswers will vary

canswers will vary dYes

eAs the number of trials approaches infinity therelative frequency approaches the value of the

probability.3 a No banswers will vary

canswers will vary dYes

eAs the number of trials approaches infinity therelative frequency approaches the value of the

probability.

4Pr(a 6 from first die) 78500

= 0.156

Pr(a 6 from second die) 102700

0.146 choose the first die.

5 a 0.702 b0.722

cThe above estimates for the probability should

be recalculated.d0.706

6Pr(4) = 13

7Pr(2) = Pr(3) = Pr(4) = Pr(5) = 213

,

Pr(6) = 413

, Pr(1) = 113

8Pr(A) = 0.225 9Pr(A) = 0.775

Exercise10C

1a 13

b 18

c 14

d 518

2 a 0.141 b0.628 c0.769

3a1

365 b

30

365 c

30

365 d

90

365

e364

365 f

334

365

4 a1

4 b

1

2 c

4

13 d

3

4

5 a9

13 b

10

13 c

5

13 d

1

13

6a1

2

b1

18

c5

18


36/67

Ans

wers

Answers 743

7 a1

12 b

1

2 c

7

12

81

49 a R

BG

R

BG

RBG

RBG

RBG

RBG

RBG

RBG

RBG

R

B

G

R

B

G

R

B

G

R

B

G

b i1

27 ii

2

9 iii

1

3 iv

2

9

10 a

Bla

Bla

Bla

Bla

Bla

Bla

Bla

Bla

Bla

B

R

R

R

R

R

R

R

R

R

R

B

B

B

B

B

B

B

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

B

B

Bla

b i0.25 ii12

24= 0.5 iii 1

12

11a5

13 b

11

13

12 a

1st ball

2nd ball

1 2 3 4 5

1

2

3

4

5

(1, 1) (1, 2) (1, 3) (1, 4) (1, 5)

(2, 1) (2, 2) (2, 3) (2, 4) (2, 5)

(3, 1) (3, 2) (3, 3) (3, 4) (3, 5)

(4, 1) (4, 2) (4, 3) (4, 4) (4, 5)

(5, 1) (5, 2) (5, 3) (5, 4) (5, 5)

b i4

25 ii

4

5 iii

3

25

13 a

3 cm

b9

25

14

415a

4

25 b1 4

25

Exercise10D

1 a{1, 2, 3, 4, 6} b {2, 4}c{5, 6, 7, 8, 9, 10} d {l, 3}e{1, 3, 5, 6, 7, 8, 9, 10} f{5, 7, 8, 9, 10}

2 a{1, 2, 3, 5, 6, 7, 9, 10, 11}

b {1, 3, 5, 7, 9, 11} c{2, 4, 6, 8, 10, 12}d {1, 3, 5, 7, 9, 11} e{1, 3, 5, 7, 9, 11}

3 a{E, H, M, S} b {C, H, I, M}c{A, C, E, I, S, T} d {H, M}e{C, E, H, I, M, S} f{H, M}

4 a20 b45

5 a6 bl c18 d2

6 a2

3 b0 c

1

2 d

5

6

7a1

2 b

1

3 c

1

6;

2

3

8 a1 b4

11 c

9

11 d

6

11 e

7

11 f

4

11

Exercise 10E

1 a0.2 b0.5 c0.3 d0.72 a0.75 b0.4 c0.87 d0.48

3 a0.63 b0.23 c0.22 d0.774 a0.45 b0.40 c0.25 d0.70

5 a0.9 b0.6 c0.1 d0.96 a95% b5%

7 a A = {J , Q , K , A , J, Q, K, A,J , Q , K , A , J, Q, K, A}

C=

{2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ,10 , J , Q , K , A }

b iPr(a picture card) = 413iiPr(a heart) = 1

4

iiiPr(a heart picture card) = 113

ivPr(a picture card or a heart) = 2552

vPr(a picture card or a club, diamond or

spade) = 4352

8a8

15 b

7

10 c

2

15 d

1

3

9 a0.8 b0.57 c0.28 d0.08

10 a0.81 b0.69 c0.74 d0.86


37/67

Answers


11 a 0 b1 c1

5 d

1

3

12 a 0.88 b0.58 c0.30 d0.12


1B 2C 3A 4C 5D

6E 7E 8D 9A 10B


(technology-free)

1a1

6 b

5

6 20.007

3a1

3 b

1

4 c

1

2

4 a 0.36 b87

2455

4

15

6 a {156, 165, 516, 561, 615, 651}

b2

3

c1

37a

5

12 b

1

4

8No

9 a 0.036 b0.027 c0.189 d0.729

10a1

27 b

4

27 c

4

9 d

20

27 e

2

5

Extended-response questions1 a b c1 +

d1 e1 + f1 2 a 0.15 b0.148

c T 18 19 20 21 22 23 24

Pr 0.072 0.180 0.288 0.258 0.148 0.046 0.008

3 a 0.204 b0.071 p 0.214 c0.0224 a 0.6 b

1

3 c

2

7 d0.108

e3

20

Chapter 11

Exercise 11A

1 14

2a65

284 b

137

568 c

21

65 d

61

2463 a 0.06 b0.2

4 a4

7 b0.3 c

15

22

5 a 0.2 b0.5 c0.4

6 a 0.2 b10

27 c

1

3

7 a 0.3 b0.75

8a1

2 b

3

4 c

1

2dl e

2

3 f

1

2

916% 101

5

11 a1

16 b

1

169 c

1

4 d

16

169

12a1

17 b

1

221 c

25

102 d

20

221

130.230 808

0.231

14a15

28 b

1

2 c

1

2 d

2

5

e3

7 f

8

13 g

5

28 h

3

14

15 a 0.85 b0.6 c0.51 d0.51

160.4; 68%

17 a i0.444 ii0.4 iii0.35 iv0.178 v0.194b0.372 c i0.478 ii0.425

18 a i0.564 ii0.05 iii0.12 iv0.0282 v0.052

b0.081 c0.35

19 a1

6 b

53

90 c

15

53

20 a BA b A B= c A B

Exercise 11B

1 a Yes bYes cNo

20.6 4No5 a 0.6 b0.42 c0.88

6 a 0.35 b0.035 c0.1225 d0.025

7a4

15 b

1

15 c

133

165 d

6

11 e

4

15; No

9 a 0.35 b0.875

10 a18

65 b

12

65 c

23

65 d

21

65

e4

65 f

8

65 g

2

15 h

8

21; No

11 a i0.75 ii0.32 iii0.59

bNo cNo

12 b i1

8 ii

3

8 iii

7

813 b i0.09 ii0.38 iii0.29 iv0.31

14a1

216 b

1

8 c

1

2 d

1

36

15 a1

32 b

1

32 c

1

2 d

1

16

16a 16

b 130

c 16

d 56

e 16

17a1

2 b

1

8 c

1

2

Exercise 11C

1a

0.6 0.45

0.4 0.55

b0.525

2a

3

5

1

32

5

2

3

b

7

15


38/67

Ans

wers

Answers 745

3a

0.43 0.33

0.57 0.67

b0.385

4a

Pr(Wi+1)Pr(L i+1)

=

0.6 0.50.4 0.5

Pr(Wi )Pr(L i )

b0.552

5a Pr(L i+1)Pr(T

i+1) =

0.25 0.100.75 0.90

Pr(L i )Pr(T

i)

b0.84

6

Pr(Ai+1)Pr(Ei+1)

=

0.7 0.50.3 0.5

i0.60.4

a0.620

b0.624 c0.625

7a0.762 b0.7963 c0.2033 83

7

Exercise 11D

1 a i

6238

ii

68.6

31.4

iii

70.6

29.4

b 0.71498 0.712550.28502 0.28745

ci

68.631.4

ii

70.6

29.4

iii

71.421728.5783

2a i

187.5

202.5

ii

210180

iii

223.5

166.5

b

0.6425 0.5958

0.3575 0.4042

ci

210

180

ii

223.5

166.5

iii

236.5153.5

3a i

0.390.61

ii

0.4563

0.5437

iii

0.4698

0.5302

bi

0.560.44

ii

0.48520.5148

iii

0.47000.5300

4a i

0.125

0.875

ii

0.16290.8371

iii

0.1794

0.8206

bi

0.4286

0.5714

ii

0.25510.7449

iii

0.1797

0.8203

5a

0.87 0.23

0.13 0.77

b63.1% at the indoor pool, 36.9% at the

outdoor pool

6a 0.96 0.98

0.04 0.02

b96.1%

7

71.4286

28.5714

8

237.5

142.5

9a

0.8 0.140.2 0.86

b675 people

10a

0.93 0.110.07 0.89

b44.0% school A, 56.0% school Bc61.0% school A, 39.0% school B

11a

0.5 0.37

0.5 0.63

bi 0.4266 ii0.4244

c0.4253

12a

0.25 0.650.75 0.35

bi 0.45 ii0.61

c0.53613 a 51.8% to Dr Black, 48.2% to Dr White

14a

0.74 0.14

0.26 0.86

b530 garage A, 959 garage Bc521 garage A, 968 garage B


1E 2C 3A 4B 5C6D 7E 8D 9E 10C


(technology-free)

1a2

7 b

32

63 c

9

16

2 a0.65 bNo

30.999894 a0.2 b0.4

5 a0.7 b0.3 c1

3 d

2

3

6a

0.9

0.1

b

0.83

0.17

70.4888


1a A : 3

28 B :

3

4 b A :

9

64 B :

49

64c0.125 d0.155

2 aBis a subset ofAbAandBare mutually exclusive

cAandBare independent

3a1

4b

1

3

ci1

16ii

1

4n

d1

4

4a

0.9 0.20.1 0.8

b

122 Melbourne

78 Tullamarine

c 133 Melbourne

67 Tullamarine

5a

0.25 0.2

0.75 0.8

bi

0.2105

0.7895

ii

0.21050.7895

c

0.2105

0.7895

Chapter 12

Exercise 12A

1 a11 b12 c37 d29

2 a60 b500 c350 d512

3 a128 b160420 563 626 72408260 000 917 576 000 1030


39/67

Answers


Exercise 12B

1 a 6 b120 c5040 d2 e1 f12 a 20 b72 c6 d56 e120 f720

3120 45040 524 67207720 8336

9 a 5040 b210 10 a120 b120

11 a 840 b2401 12 a480 b151213 a 60 b24 c252

14 a 150 b360 c156015 a 720 b48

Exercise12C

1 a 3 b3 c6 d42 a 10 b10 c35 d35

3 a 190 b100 c4950 d31125

4 a 20 b7 c28 d1225

51716 62300 7133 784 560

88 145 060 91810 a 5 852 925 b1 744 200

11100 386 12 a792 b33613 a 150 b75 c6 d462 e81

14 a 8 436 285 b3003 c66 d2 378 37615186 1632 17256 1831 1957

20 a20 b21

Exercise12D

1 a 0.5 b0.5 20.3753 a 0.2 b0.6 c0.3

40.2 5 329858

6 a27

28 1 0.502 b56

255 c

73

85

7a5

204 b

35

136

8 a25

49 b

24

49 c

3

7 d0.2

9 a1

6 b

5

6 c

17

21 d

34

35

10 a 0.659 b0.341 c0.096 d0.282

11 a5

42 b

20

21 c

15

37


1E 2D 3A 4D 5C

6B 7C 8A 9E 10E


(technology-free)

1 a 499 500 b1 000 000 c1 000 000

2648 3120 48n 55416636 750 711 025

8 a 10 b32 91200

10a1

8 b3

8 c3

28


1 a 2880 b80 6402 a 720 b48 c336

3 a 60 b364 a 210 b100 c80

5 a 1365 b210 c11556 a 3060 b330 c1155

7Division 1: 1.228 107Division 2: 1.473 106Division 3: 2.726 105

Division 4: 1.365 103

Division 5: 3.362 1038 a 1.290 10 4

b6.449 10 4

Chapter 13

Exercise 13A

1 a no bno cyes dno eno2 a Pr(X= 2) bPr(X>2) cPr(X 2)

dPr(X2)gPr(X 2) hPr(X 2) iPr(X 2)jPr(X

2) kPr(2 < X


40/67

Ans

wers

Answers 747

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

36c

1 2 3 4 5 6

1

36

3

36

5

36

7

36

9

36

11

36

11 a 0.09 b0.4 c0.5112 a

y 3 2 1 3

p(y) 1

8

3

8

3

8

1

8

b7

8

Exercise 13B

10.378 228

57 0.491 3 12

13 0.923

460

253 0.237 50.930 60.109

Exercise 13C

1 a 0.185 b0.060 2 a0.194 b0.930

3 a 0.137 b0.446 c0.5544 a 0.008 b0.268 c0.468

5 a 0.056 b0.391 60.018

7aPr(X= x) =

5

x

(0.1)x (0.9)5x

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

p(x) 0.591 0.328 0.073 0.008 0.000 0.000

bMost probable number is 080.749 90.021 100.5398 11

175

25612 a 0.988 b0.9999 c8.1 101113 a 0.151 b0.302 145.8%

15 a i0.474 ii0.224 iii0.078bAnswers will vary about 5 or more.

160.014 17

18 19 a5 b820 a13 b22

21 a16 b2922 a45 b59

23 a 0.3087 b0.3087

1 (0.3)5 0.309524 a 0.3020 b0.6242 c0.3225

Exercise 13D

1Exact answer 0.172

2 a About 50 : 50bOne set of simulations gave the answer 1.9

Exercise 13E

2Exact answer 29.29

3 aOne set of simulations gave the answer 8.3.bOne set of simulations gave the answer 10.7.

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

Multiple-choice question

1B 2A 3C 4A 5E6C 7A 8D 9B 10E


(technology-free)

1 a0.92 b0.63 c0.82

x 1 2 3 4

p(x) 0.25 0.28 0.30 0.17

3x 2 3 4

p(x) 2

5

8

15

1

15

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

1

2

3

67

9

2

3

4

7

8

10

3

4

5

8

9

11

4

5

6

9

10

12

7

8

9

12

13

15

8

9

10

13

14

16

10

11

12

15

16

18

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

cx 2 3 4 5 6 7

Pr(X=x) 136

2

36

3

36

2

36

1

36

2

36

x 8 9 10 11 12 13

Pr(X=x) 436

4

36

4

36

2

36

3

36

2

36

x 14 15 16 18

Pr(X=x) 136

2

36

2

36

1

36

5 a0.051 b0.996 c243

256 0.949

6 a9

64 b

37

64

7 a0.282 b0.377 c0.341

8 a0.173 b0.756 c0.071

9a p

10015

b15 p

10014

1 p

100


41/67

Answers


c p

100

15+ 15

p100

141 p

100

+ 105

1 p100

2 p100

13

10a117

125bm= 5

Extended-response questions1 ax 1 2 3 4

p(x) 0.54 0.16 0.06 0.24

b0.46

2 a i0.1 ii0.6 iii2

3b i0.0012 ii0.2508

3 a3

5

b i7

40 ii

3

10

c i 1140

ii 1117

4 a 0.003 b5.320 10650.8 60.9697 a 0.401 bn 458 a 1 q2 b1 4q3 + 3q4 c 1

3


42/67

Ans

wers

Answers 749

5 a y

0

5 y = 2

x

b y

0

y=

x

3

c y

0

(0, 3)y=2

x

d y

y= 2

0x

e y

0

(0, 3)

x

f y

y=2

0

(0, 4)

x

6 a y

0

2

x

b y

0

1

x

c y

0

1

x

d y

y= 2

0

1

x

Exercise 15B

1 a x 5 b8x7 cx 2 d2x3 ea6 f26

gx 2y2 hx 4y6 ix3

y3j

x6

y4

2 a x 9 b216 c317 dq 8p9

ea11b3 f28x18 gm11n12p2 h2a5b2

3 a x 2y3 b8a8b3 cx 5y2 d9

2x2y3

4a1

n4p5 b

2x8z

y4 c

b5

a5d

a3b

cean + 2 bn + 1 cn 1

5 a 317n b23 n c34n 11

22

d2n + 133n 1 e53n 2 f23x 3 34g36 n 25n h33 = 27 i6

6 a 212 = 4096 b55 = 3125 c33 = 27

Exercise 15C

1 a 25 b27 c1

9d16 e

1

2 f

1

4g

1

25

h16 i1

10 000 j1000 k27 l3

5

2a a16 b

76 ba6b

92 c3

73 5

76

d1

4 ex 6y8 fa

1415

3 a(2x 1)3/2 b(x1)5/2 c(x2 + 1)3/2d(x 1)4/3 ex (x 1) 12 f(5x2 + 1)4/3

Exercise15D

1 a3 b3 c1

2d

3

4e

1

3 f4 g2 h3 i3

2 a1 b2 c32

d4

3e1 f8 g3

h4 i 8 j4 k3 12

l6 m71

2

3a4

5 b

3

2 c5

1

2

4 a0 b0, 2 c1, 2 d0, 15 a2.32 b1.29 c1.26 d1.75

6 ax >2 bx >1

3

cx

1

2

dx 1 gx 3

Exercise 15E

1 alog2(10a) b1 clog2

9

4

d1

elog56 f2 g3 log2a h92 a3 b4 c7 d3 e4 f3 g4

h6 i9 j1 k4 l23 a2 b7 c9 d1 e

5

2

flogxa5

g3 h1

4 a2 b27 c1

125d8 e30

f2

3g8 h64 i4 j10

5 a5 b32.5 c22 d20

e3 +

17

2f3or0

62 + 3a 5c2

810

9 a4 b6

5c3 d10 e9 f2

Exercise 15F

1 a2.81 b1.32 c2.40 d0.79 e2.58f0.58 g4.30 h1.38 i3.10 j0.68

2 ax >3 bx


43/67

Answers


c y

02

0.2218

x

y=5

log10

5

3

d y

0

log10

2

2

x

y= 4

0.3010

e y

3

01 x

y= 6

f y

0

1

x

0.2603log2 65

y=6

4d0= 41.88,m= 0.094

Exercise15G

1 a y

x

Domain =R+

Range =R

0.301

0 12

1

b y

x

Domain =R+

Range =R

0.602

0 1 2

c y

x

Domain =R+

Range =R

0.301

0 1 2 3 4

d y

x

Domain =R+

Range =R

0.954

0 113

e y

x

Domain =R+

Range =R

0 1

f y

x

Domain =R+

Range =R

01

2 a y= 2 log10x by= 1013x

c y= 13

log10x dy=1

310

12x

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

c y= log3

x 24

dy= log5(x + 2)

e y=

1

32x fy

=3

2x

g y= 2x 3 hy= log3

x + 25

4 a y

0 5x

x= 4

Domain = (4, )

b y

0

log23

2 x

x=3

Domain = (3, )

c y

0 12

x

Domain

=(0,

)

d y

0x=

1

1x

2

Domain

=(

2,

)

e y

0 3x

Domain = (0, )

f y

012

x

Domain = ( , 0)5 a 0.64 b0.40

6y

y =log10 (x2)

0 11 x

y

y =2log10x

0 1 x

7y

x0

y = log10x = log10x forx (0, 10]12

8 y

y =log10 (2x) + log10 (3x)

0 1

6

x

y

y =log10 (6x2)

01

6

x

1

6

9a= 6103

23

andk= 13

log10

103

Exercise15H

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

Total thickness,

Cuts,n Sheets T(mm)

0 1 0.21 2 0.4

2 4 0.83 8 1.6

4 16 3.25 32 6.4

6 64 12.87 128 25.6

8 256 51.29 512 102.4

10 1024 204.8


44/67

Ans

wers

Answers 751

bT= 0.2(2)nc T

n0 2 4 6 8 10

200

150

100

50

0.2

d214 748.4 m

4 a p,q(millions)

t0

y=p(t)

y= q(t)

1.71.2

b it= 12.56 . . . (mid 1962)iit

=37.56 . . . (mid 1987)


1C 2A 3C 4C 5A6B 7A 8A 9A 10A


(technology-free)

1 a a4 b1

b2 c

1

m2n2

d1

ab6 e

3a6

2 f

5

3a2

ga3 h n8

m4 i 1

p2q4

j8

5a11 k2a la2 + a6

2 a log27 b1

2log27 clog102

dlog10

7

2

e1 + log1011 f1 + log10101

g1

5log2100 hlog210

3 a 6 b7 c2 d0

e3 f2 g3 h4

4 a log106 blog106 clog10a2

b

dlog10

a2

25 000

elog10 y flog10

a2b3

c

5 a x= 3 bx= 3 orx= 0cx= 1 dx= 2 orx= 3

6 a

x0

y = 2.2x

(1, 4)

(0, 2)

yb

y

x0

(0, 3)

y =3.2x

c

x

y= 5.2x

y

0

d y

x0

(0, 2)

y = 2x + 1

y= 1

e

x0

y = 2x 1

y =1

y fy

x0

(0, 3)y = 2

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

7 bq= 3

2

11a a= 12

by= 4 ory= 20


1 an 0 1 2 3 4

M 0 1 3 7 15

bM= 2n 1

n 5 6 7

M 31 63 127

c M

n0

30

20

10

1 2 3 4 5

dThree discs 1 2 3

Times moved 4 2 1

Four discs 1 2 3 4

Times moved 8 4 2 1

2n= 23a

1

2

3nb

1

2

5n 2cn= 3

4a 729

1

4

nb128

1

2

nc4 times

5 aBatch 1 = 15(0.95)n Batch 2 = 20(0.94)nb32 years

6 aX $1.82 Y $1.51 Z $2.62

bX $4.37 Y $4.27 Z $3.47cIntersect att= 21.784 . . . and

t=2.090 . . . therefore February 1997

until September 1998


45/67

Answers


dFebruary 1998 until September 1998,approximately 8 months.

7 a 13.81 years b7.38 years

8 a temperature = 87.065 0.94tb i87.1 ii18.56

ctemperature = 85.724 0.94td i85.72 ii40.82

e28.19 minutes9 a a= 0.2 andb = 5b iz=xlog10b iia= 0.2 andk= log105

10 a y= 2 1.585x by= 2 100.2xcx= 5 log10

y2

Chapter 16

Exercise 16A

1 a

3 b

4

5 c

4

3 d

11

6e

7

3 f

8

3

2 a120 b150 c210 d162

e100 f324 g220 h324

3 a34.38 b108.29 c166.16 d246.94

e213.14 f296.79 g271.01 h343.77

4 a0.66 b1.27 c1.87 d2.81

e1.47 f3.98 g2.39 h5.74

5 a60 b720 c540 d180e300 f330 g690 h690

6 a2 b3 c43

d 4 e116

f76

Exercise16B

1 a 0, 1 b1, 0 c1, 0 d1, 0e0, l f1, 0 g1, 0 h0, 1

2 a 0.95 b0.75 c0.82 d0.96e0.5 f0.03 g0.86 h0.61

3 a 0; 1 b1; 0 c1; 0 d1; 0e1; 0 f0; 1 g0; 1 h0; 1

Exercise16C

1 a 0 b0 cundefined

d0 eundefined fundefined2 a34.23 b2.57 c0.97d1.38 e0.95 f0.75 g1.66

3 a 0 b0 c0 d0 e0 f0

Exercise16D

1 a 6759 b4.5315 c2.5357d6.4279 e5012 f3.4202g2.3315 h6.5778 i6.5270

2 a a= 0.7660,b = 0.6428bc = 0.7660,d= 0.6428c icos 140

= 0.76604, sin 140

=0.6428

iicos 140 = cos 40

Exercise 16E

1 a0.42 b0.7 c0.42 d0.38e0.42 f0.38 g0.7 h0.7

2 a 120 b240 c60d120 e240 f300

3 a

5

6 b

7

6 c11

6

4aa= 12

bb =

3

2 cc = 1

2

dd=

3

2 etan ( ) =

3

ftan () =

3

5a

3

2b

1

2 c

3 d

3

2 e1

2

6 a0.7 b0.6 c0.4 d0.6e0.7 f0.7 g0.4 h0.6

Exercise 16F

1a sin =

3

2 , cos = 1

2, tan =

3

bsin = 12, cos = 1

2, tan = 1

csin = 12, cos =

3

2 , tan = 1

3

dsin =

3

2 , cos = 1

2, tan =

3

esin = 12, cos = 1

2, tan = 1

fsin = 12, cos =

3

2 , tan = 1

3

gsin =

3

2 , cos = 1

2, tan =

3

hsin = 12, cos = 1

2, tan = 1

isin =

3

2 , cos = 1

2, tan =

3

jsin = 32 , cos = 1

2, tan = 3

2a

3

2 b 1

2c 1

3d1

2 e 1

2

f

3 g

3

2 h

12

i 13

3a

3

2 b 1

2c

13

dnot defined

e0 f 12

g12

h1


46/67

Ans

wers

Answers 753

Exercise 16G

1 Period Amplitude

a 2 2

b 3

c2

3

1

2

d 4 3

e2

3 4

f

2

1

2

g 4 2

h 2 2

i 4 3

2 a y

x

3

0

3

2

Amplitude = 3, Period =

b y

x

2

0

2Amplitude = 2, Period =

2

32

3

43

2

c y

4

0

4

432

Amplitude = 4, Period = 4

d y

x0

2

,

3

2

32

212

1

2

1

3

4

Amplitude = Period =

e y

x02

Amplitude = 4, Period =4

4

3

2

3

2

3

4

f y

x0

5

5

2


4

3

2

3

4

54

74

2

g y

x432

0

3

3


h y

x0

2

2


4

3

87

83

85

8

2

4

2

i y

x0

2

2

6


3

2

3

2

9

3 a

x

y

0

1

22

1

2

32

32

2

b

x

y

0

2

2

6336

c

x

y

2

2

023

23

53

4

6

56

72

36

116

2

3

d

x

y

2

2

02

3

23

43


47/67

Answers


4

x

y

03

2,23

4

5

2

5

2

5

5 a dilation of factor 3 from thex -axis

amplitude = 3, period= 2bdilation of factor 1

5from they-axis

amplitude = 1, period= 25

cdilation of factor 3 from the y-axis

amplitude = 1, period= 6ddilation of factor 2 from thex -axis

dilation of factor 15from they-axis

amplitude = 2, period= 25

edilation of factor 15from they-axis

reflection in thex-axis

amplitude = 1, period=2

5freflection in they-axis

amplitude = 1, period= 2gdilation of factor 3 from the y-axis

dilation of factor 2 from thex -axis

amplitude = 2, period= 6hdilation of factor 2 from the y-axis

dilation of factor 4 from thex -axis

reflection in thex-axisamplitude = 4, period= 4

idilation of factor 3 from the y-axisdilation of factor 2 from thex -axis

reflection in they-axisamplitude = 2, period= 66 a y

0 1 2

2

2

x1

2

3

4

b y

0 21

3

3

x

12

34

7 y

x

y= sinx y= cosx

20

b

4,

5

4

Exercise16H

1 ay

2

3

0

3

2

32

52

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

y

2

1

0

1

Period= , Amplitude = 1,y= 1c

y

2

0

2

12

5

12

13

Period= 23

, Amplitude = 2,y= 2

d

y

0

3

3

2

3

2

Period= , Amplitude =

3,y=

3

e

x

y

0

3

3

2

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

f

y

0

2

2

12

4

12

5

Period

=

2

3

, Amplitude

=2,y

= 2, 2


48/67

Ans

wers

Answers 755

g

y

0

2

2

6

53

4

3

Period= , Amplitude =

2,y=

2,

2

h

x

y

3

3

0

2

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

i

y

3

0

3

2

2

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

2a f(0) = 12

f(2) = 12

b y

x0

1

1

, 1

2,0,

3

2

1

2

1

, 13

4

6

56

11

3 a f(0) =

3

2 f(2) =

3

2

b y

x0

1

1 2,

2 3 6

5 34 6

113

2

3

4a f() = 12

f() = 12

by

x0

1

1

20, 1

2,

1

2,

1

5 a y=

3 sinx

2

by

=3 sin 2x

cy= 2 sin x3

dy= sin 2x

3

ey= sin12

x +

3

Exercise 16I

1a5

4 and

7

4b

4and7

4

2 a0.93 and 2.21 b4.30 and 1.98

c3.50 and 5.93 d0.41 and 2.73e2.35 and 3.94 f1.77 and 4.51

3 a150 and 210 b30 and 150 c120 and 240d120 and 240 e60 and 120 f45 and 135

4 a0.64, 2.498, 6.93, 8.781

b5

4 ,

7

4 ,

13

4 ,

15

4

c

3,

2

3 ,

7

3 ,

8

3

5a 3

4 ,3

4 b

3,

2

3 c 2

3 ,2

3

6

x

y

,

0

1

1

3

52

1

,3

42

1

, 3

22

1,

3

22

1

,3

42

1

,3

52

1,

3

2

1,

3

2

1

7a7

12,

11

12 ,

19

12 ,

23

12

b

12,

11

12 ,

13

12 ,

23

12

c

12,

5

12,

13

12 ,

17

12

d5

12,

7

12,

13

12 ,

15

12 ,

21

12 ,

23

12

e5

12,

7

12,

17

12 ,

19

12

f5

8 ,

7

8 ,

13

8 ,

15

8

8 a2.034, 2.678, 5.176, 5.820b1.892, 2.820, 5.034, 5.961

c0.580, 2.562, 3.721, 5.704

d0.309, 1.785, 2.403, 3.880, 4.498, 5.974

Exercise 16J

1 a

x

y

3

1

01

6

7

6

11


49/67

Answers


b

x

y

2 3

2 3

3

0

6

73

6

3

4

c y

x

(0, 1 + 2)

1 2

0

24

34

5

dy

x0

2

4

4

4

5

ey

x

1 + 2

1 20 (2, 0)

2

3

2 ay

22

x

2

4

0(2, 2)

(2, 2)

6

116

76

52

32

6

6

2

b

x2

(2, 1.414) (2, 1.414)

22

0

2

y

12

234

12

125

129

1213

1217

1221

12

712

1112

1512

19

c

x

y

2 20

1

3

5

(2, 3)(2, 3)

d

x

(2, 3)

2

(2, 3)

2

y

3

01

1

3

5

3

4

3

2

3

2

3

5

3

4

3

3

e

(2, 2)

2

(2, 2)

2

y

x0

2

3

2

3

6

5

6

116

7

2

3

2

6

2

f

2

1 + 3

(2, 1 + 3)

10

3

x2

(2, 1 + 3)

y

12

19

12

7

45

125

47

1217

43

4

3 a

10

3

x

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

1 + 3

y

127

4

125

43 b

10

3

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

y

4

12

12

11

4

3


50/67

Ans

wers

Answers 757

c

0

3

x

(, 3) (, 3)

3 2

2 + 3

1 + 3

y

6

6

53

2

Exercise 16K

1 a 0.6 b0.6 c0.7 d0.3 e0.3f

10

7(1.49) g0.3 h0.6 i0.6 j0.3

2 a

3 b

3 c

5

12d

14

3sinx= 45

and tanx= 43

4cosx= 1213

and tanx= 512

5sinx= 2

6

5 and tanx= 2

6

Exercise 16L

1a

4 b

3

2 c

2

2 ay

0x

2

34

2

x =34

x =

4x =

4

x =

b

0 x

y

56

x =5

6

23

23

x =

3

2

6

x =

3

x =6

x =2

x =

c y

0 x

23

56

x =5

6x =

2

6

x = x =6

x =2

x =

23

3

3

3 a7

8 ,

38

,

8,

5

8

b17

18 ,

1118

,5

18 ,

18,7

18,

13

18

c 5

6 ,

3 ,

6,2

3

d13

18 ,

718

,

18,

5

18,

11

18 ,

17

18

4 a

y

0x

6

5

6

2x=

2x =

(, 3)3

(, 3)

by

0

2x

4

34

2

x=

2x=

(, 2)(, 2)

cy

0

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

x2

4

34

Exercise 16M

1 a0.74 b0.51

c0.82 or0.82 d0 or 0.882y= asin (b + c) + d

aa= 1.993 b = 2.998 c = 0.003d= 0.993

ba= 3.136 b = 3.051 c = 0.044d= 0.140

ca= 4.971 b = 3.010 c = 3.136d= 4.971

Exercise16N

1 ax= (12n + 1)6

orx= (12n + 5)6

bx= (12n 1)18

cx= (3n + 2)3

2 ax= 6

orx= 56

bx= 18

orx= 1118

cx=2

3 orx=5

3


51/67

Answers


3x= norx= (4n 1)4

;

x= 54 ,,

4,0,

3

4 , or

7

4

4x= n3

;x= , 23 ,

3,0

5x= 6n 112

orx= 3n + 26

;

x= 23, 7

12, 1

6, 1

12,

1

3,

5

12,

5

6,

11

12

Exercise16O

1 a

0 3 6 12 18 24 t

D

13

10

7

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

c12.9 m

2 a p= 5,q= 2b

0 6 12 t

D

7

5

3

cA ship can enter 2 hours after low tide.

3 a 5 b1ct= 0.524 s, 2.618 s, 4.712 sdt= 0 s, 1.047 s, 2.094 seParticle oscillates about the pointx= 3 from

x= 1 tox= 5.


1C 2D 3E 4C 5E

6D 7E 8E 9C 10B

Short-answer questions(technology-free)

1 a11

6 b

9

2c6 d

23

4 e

3

4

f9

4g

13

6 h

7

3 i

4

92 a 150 b315 c495 d45 e1350

f135 g 45 h 495 i1035

3a12

b12

c12

d

3

2

e

3

2

f

1

2

g1

2

h

1

2

4Amplitude Period

a 2 4

b 3

2

c1

2

2

3

d 3

e 4 6

f2

3 3

5 a

2

0

2

y = 2sin2x

y

x

b y

x0

3

3

mathsmethods1/2

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