iCE-Em Mathematics Year 10
April 5, 2017 | Author: Arah | Category: N/A
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Answers to exercises Chapter 1 answers Exercise 1A 1
a 0.56
b 0.082
c 0.96
d 0.18
e 0.12
f 0.0375
g 2.15
h 0.008
i 0.8825
j 0.002
k 1.508
l 0.0075
3
m 3.25 9 a 20 6 i 5 a 80%
f 44 49 %
g 166 23 %
h 1.75%
i 46%
j 20.5%
k 6%
l 2.5%
m 140%
n 205%
o 112.5%
p 0.075% 16 , 0.64 25 f 62.5%, 0.625
q 206%
r 217% c 16%,
s 775% 41 d , 0. 205 200 i 160%, 1.6
t 225%
b 60%, 0.6
d 523.2
e 96.77
2
16 b 25 3 j 2
n 0.006 17 c 20 11 k 8 b 87.5%
o 1.245 27 2 d e 40 3 7 29 l m 5 400 c 43.75%
p 0.00875 1 f 6 51 n 400 d 150%
33 h 400 1 o p 200 e 45%
7 125 39 500
g
5
a 9
b 116
4 25 11 h , 1.1 10 c 188.64
f 1011.6
g 2.77
h 7
i 1.1
6
a $26
b $254.10
c $48
d $1749
f $30
g $165
h $7.95
i $4176
7
a 6%
b 1.3%
c 4.7%
d 160%
e 16.7%
f 600%
8
a 4.80%
b 1.70%
c 0.23%
d 4.17%
e 4.11%
f 3.13%
g 3200%
h 0.14%
9 45
10 a 1.508 g
b 0.00728 g
12 a 16 500
b 31 500
4
a
g 104%,
26 25
11 61 minutes and 6 seconds
e 40%,
2 5
e $2.70
13 a $480
b 1600 g
c 240 mins
d 240 cm
e 208 ha
f $218
14 a $1900
b $3600
15 a 416.7 kg
b 800 mm
c 301 ha
d 373.3 mm
16 a $575
b $12 929
17 a 100% profit b 17.6% loss
18 a costs $518 519, total sales $546 519 19 a i $0 b i 0% c i $22 533.33
b costs $11 538 462, total sales $10 788 462 iii $4700 iii 12.37% iii $78 571.43
ii $450 ii 3% ii $34 640.00
iv $127 700 iv 31.93% iv $78 000.00
20 a $1600
b $8000
21 a $270
b $810
22 a $720
b $2025
c $5040
d $17 280
23 a 7.5% p.a.
b 6.8% p.a.
c 5.8% p.a.
24 a 50 years
25 a $5000
b $15 000
c $24 790.12
b 38 years
Exercise 1B 1
a $627
b $9786
c $483.36
d $3369.60
2
a $8100
b $4332
c $801.22
d $9139.20
3
a 35 840
b 171 360
c 278
4
a 434 mm
b 88 mm
c 786 mm
5
a 6% increase
b 88% increase
c 22% decrease
6
a 20% decrease
b 4.4% increase
c 44% decrease
7
a $630
b $5160
c $11 000
8
a 200
b 7299
c 68 142
9
a 4750 megalitres
b 11 340 megalitres c 1840 megalitres d 63 330 megalitres
10 a $52
b $31.20
c $442
11 a $3111.11
b $726.77
c $54.44
d $1300
d $1.04
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press
12 a $28 000 e $31 890.04
b $5992.56 f $421 741.12
c $13 806.10 g 4%
13 a i $280 000 b i $428 571 c i $803 333
ii $61 855 500 ii $6 625 714 286 ii $4 156 666 667
iii $2203 iii $1033 iii $1423
14 a i $3014 b i $3570 c i $4862
ii $147 400 ii $388 070 ii $3 818 254
iii $9.02 iii $2.90 iii $5.17
d $69 565.22 h 60.22%
i 17.16%
ii $13.83
15 a i $3102
ii $24.97
iii $373.27
b i $4061.90
16 a 50%
b 20%
c 81.8%
d 72.4%
17 a i 9.09% b i 11.11%
ii 15.25% ii 21.95%
iii 78.26% iii 400%
iv 3.94% iv 4.28%
iii $73.50
Exercise 1C 1
a $12 474
b $8042.64
c $22 425
d $61 412.50
e $233 436.82
2
$3.83
3 a $7706.79
b $96 119.13
c $128.24
d $3947.50
4
a 50.4%
b i $1503.63
ii $25 965.05
iii $324.84
6
a Decreased by 4% b Decreased by 64%
7
a $2.52
b $2.19
8
a i $370.74
ii $8897.76
9
5 25.75%
c Decreased by 1%
d Decreased by 9%
c $1.99 iii $4 199 930.76
b 62.9%
a 5158
b 51 589
10 a i 49.8°C
ii 24.8°C
iii 3.1°C
b i 20.2°C
11 a 18.6%
b 11.6%
12 a 90.9%
b 100.0%
c 61 895 ii 34.2°C
iii 68.6°C
c 100.0%
13 a Decrease of 25% b Decrease of 25% c As multiplication is commutative, if the percentage increases and decreases are the same, but in different orders, the overall effect is the same.
Exercise 1D 1
a $107 000
b $114 490
c $140 255.17
d 40.26%
e $40 255.17
f $35 000
2 a 59 750
b 76%
3
a i $1 630 910.87
ii 329%
b i $1 696 688.53
ii 346%
4
a $327 000
b $548 411.74
c 82.80%
d $248 411.74
5
a $83 521 265.73
b $153 336 556.81
c $12 000
6
a $1 024 290.31
b $1 131 996.29
c $252 000
7
a $33 542.00
b $13 542.00
8 a $58 349.04
b 71%
9
a $154 880.98
b 61%
10 a $43 297.57
b $12 335.56
11 a 74 488
b 69 356
c 55 985
d 39 178
12 a $226 057.28
b $212 924.56
c $177 929.33
d $131 911.86
13 a $41 734.65
b $36 508.33
c $26 530.27
d $19 290.79
14 a 12.68%
b 6.17%
c 10.95%
d 5.33%
15 a $16 282.30, $13 282.30
c $5 512 011.72, $2 512 011.72
e $2226.51, $7773.49
b $44 913.33, $34 913.33 d $552.75, $2447.25 f $1 632 797.69, $1 367 202.31
16 a $5400, $5724, $6067.44
b $5551.00, $5893.38, $6256.87
17 a $36 000, $39 240, $42 771.60
b $37 522.76, $41 042.65, $44 892.74
18 $1 202 680.28
20 a 3.72%
21 a 32.3%
19 $114 041 037.47 b 33.1%
e $189 000
c 33.8%
g The total percentage growth is approximately the same.
b 7.57% d 34.0%
e 34.4%
f 34.6%
22 87%
23 4 1%, 41%, they are the same as the percentages are the same just in a different order. As multiplication is commutative the total percentage increase will be the same. 24 $52 469.91
398
IC E - E M M at h emat i c s y ea r 1 0 B o o k 1
ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
Cambridge University Press
Exercise 1E 1
a $420 000
b $294 000 b $22 400
c $70 589.40
d 88.2%
e $88 235.10
c $7340.03
d 79%
e $3951.42
2
a $28 000
3
a $83 886.08, 73.8% b $507 799.78
5
Depreciated value is $15 126, Price obtained is better by $4874, to the nearest dollar
6
Price is worse than the depreciated price by $137 514.91
4 4902
7
$2 724 000, $1 634 400, $980 640, $588 384; $1 816 000, $1 089 600, $653 760, $392 256
8
$4650.00, $3487.50, $2615.63; $1550.00, $1162.50, $871.88
9
Sandra $116 731.38, Kevin $20 971.52 b Taxis $156 250, Other cars $2 166 757.03, difference = $2 010 507.03
10 a Taxis 98.4%, Other cars 78.3% 11 a $8387
b $10 822
c $83157
d 92.2%
e $7666
12 a $95 455
b $108 471
c $180 877
d 53.6%
e $16 146
c $83.2%
d $3964.07
e i $2480.80
15 a 2.6%
b 769.23 mL
13 a $15 523.20
b $33 348.49
14 a 59.31%
b $18 3094.15, $26 690
16 0.4096%
17 A gain of 58.86%
18 a i 27.75% ii 27.10% iv 26.49% v 26.26% iii 26.61% b The total depreciation is approximately the same, but decreases as the number of years increases.
ii $5941.76
vi 26.14%
19 92.09%
Review exercise 1
a $5400
b $2380
c $2700
2
a $5000
b $3333.33
c $2083.33
d $1500 d $4000
3
a 8%
b 12 12 %
c 10 23 %
d 11 19 %
4 16%
5 $85.50
6
a 66 g 115.5
b 125 h 225
c 177 i 390
d 36 j 294.84
e 64 7 $93.50
f 460 8 $67.82
9
a i 99
b No, they all end up close to 100.
ii 96
iii 99.84
iv 99.36
10 a 2250
b $2025
c $1822.50
d 27.1%
11 a $390
b 56%
12 7.58%
13 24.78%
14 a $4317.85
b $10 235.50
c $29 588.77
d $5320.45
15 a $19 467.20
b $16 477.04
c $10 859.71
d 25 000(0.92)n
16 a $7520
b $6245.99
c $5518.96
d $4308.92
17 a $13 721.03
b $7241.03
c 7 years
b 0.32% decrease
2 a Bank B
b $9653.19 d $53.03
Challenge exercise 1
a $37 036 $ 50 n
3
a
b $50.51
c $47.98
4
12.36%
5 0.8% increase
6 $69.32
7
80.4%
8 $369 814.11
9 r ≈ 4.88
Chapter 2 answers Exercise 2A 1
5
√60 √6 3
√30
√3 0
1
2
3
4
2
a 2 2
b 2 3
c 4 2
d 5 2
e 3 6
f 6 3
g 7 2
h 10 2
i 12 2
j 7 3
k 4 7
l 5 7
m 7 5
n 7 6
o 15
p 30
q 15 2
r 20 2
s 10 10 t 24 3
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press
3
4
a 10 3
b 20 5
c 18 11
d 15 6
e 24 5
f 35 5 k 35 3
g 24 2 l 12 11
h 35 2 m 8 13
i 66 3 n 25 11
j 560 o 40 3
a
8
b
45
c
147
d
216
e
300
f
160
g
605
h
2450
i
245
j
108
k
700
l
180
5
a
6
b
77
c
40 = 2 10
d
39
e
48 = 4 3
f
99 = 3 11
6
a
5
b
2
c
6
d
5
e
3
f
8=2 2
7
a
14
b
15
c 4
d 3 2
e
23
f 13 2
g 6 2
h
6
i 7
j 0
k 25
8
a
e
9
a
5 × 45 3
6 =
b
30
= 15
21 cm 2
10 a 3 2 2
11 a 112 cm
f
7 × 100 5
8 = =
c
56
g
20
1 30 cm 2 2 b 6 b
b 4 14 cm
12 × 91 7
3 =
36
= 13
c 17 cm2
d
c 3 5
d
15 ×
h
33 11
11 cm
d 2 2 2
=
3 =
l 3 45
3
i
5
e 2 6 cm
f 3 7 cm
e 5 2
f 3 5
52 = 2 13 cm
12 a 7 3 cm
b
d −17 5
e 3 2
i 36 7
j 11 10
Exercise 2B 1
a 13 2
b 25 3
c
f −27 5
g −2 3
h 22 5
2
a 12 2 + 2 3
b 10 7 − 10 5
c 13 11 − 3 10 d 10 2 − 4 3
e 6 15 − 4 7
3
a 6 2
b 5 3
c 5 5
d 15 2
e - 5 3
7
f −2 5
g 3 3 + 4 5
h
i 3 3 − 8 10
j −4 5 − 8 2
4
a 5 3
b 5 2
c 2 2
d −3 5
e 27 2 − 12 3 f 45 3
g 18 5
h
i 28 11
j −15 3
k 6 7 − 15 6
5
a
b 6 3
c 6 2
d
e 9 2
g 5 6
h 4 3
i 2 5
j 11 2
6
a 10 2 − 4 3
b 24 2 −
7
a x = 7 a 2 2 + 2 3, 6 , 5
b x = 5
8
2
7
11 + 8 5
f 11 5 − 2 2
3
k
c 6 3 + 15 10
5
5 5 2
l 181 3 9 f −7 5 l 15 2 − 10 5 d 24 3 − 19 2
c x = 6
b 2 3 + 2 5, 15, 2 2
Exercise 2C
400
1
a
15
b
55
c
112 = 4 7
d
65
e
12 = 2 3
f
54 = 3 6
2
a
5
b
2
c
6
d
5
e
3
f
8=2 2
3
a 12 10
b 77 15
c 54 35
d 32 6
e −30 6
f −28 22
g 12 77
h 18 14
4
a 2 3
b 2 10
j −10 21 1 d 3 2
k 12 14 5 5 e 3
l −42 33 2 6 f 3
h −2 3
5
−5 2 g 2 a 2
i −22 42 5 5 c 2
b 11
c 9
d 40
e 30
f 90
g -56
h -20
i 24
j 42
k -165
l -126
6
a
c 2 2
d 3 3
e 3 7
f 6 5
g 6 2
h 12 3
i 10 5
j 8 2
k 32 2
l 50 5
7
a 5 7
b 4 3
c −3 3
d −3 5
e 3 3
f 7 2
g 25 35
h 24 42
i 3 3
j 6 2
k 3 3
l 16 15
3
b
2
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8
a 16 6
b 43 15
c 7 10
d 3 30
e 6 6
f 6 10
g −17 30
h − 40 = −2 10 i 8 2
j − 4 5
k 2 2
l
9
a 8
b 12
c 80
d 45
e 98
f 150
g 63
h 44
i 108
j 250
k a2b
l m2n
b 2
c 2 2
d 4
e 4 2
f 8
k 192 3
l 40 5
e 2 15 + 2 6
f 3 14 + 3 10 l 60 2 − 32 5
10 a
2
g 3 3
h 5 5
i 16 2
j 54 2
m 625 5
n 56 7
o 512
p 2187 3
6 + 10
11 a
b
35 +
35 − 10
c
42
33 −
d
22
2
g 6 10 − 9 6
h 20 6 + 24 15 i 4 6 − 4 3
j 6 15 + 15
k 18 2 + 8 3
m 24 − 6 2
n 30 + 24 5
p 80 − 24 10
q 105 5 − 42 3 r 66 2 − 132 3
12 a 15 +
21 + 10 + 14
d 15 – 21 +
30 −
o 36 2 − 18 55 +
b
30 +
77 +
15 −
42 c
35 +
6 − 14
e 2 15 + 4 6 + 10 + 4
f 8 − 5 2
g 5 6
h 45 − 17 15
i 37 − 10 5
j 10 6 − 6 2 + 30 3 − 18
k −11 − 11 14
l 2 21 − 4 35 + 15 − 10
m 34 − 9 14
n 3 21 − 9 35 − 8 3 + 24 5
−3 2 13 a −18 2 b 2 b 6 −
14 a 4
42
c 66
3 c
d
11 108
e 24 3
3 + 10 d 2 3
f 1296 3 g
3 3 4
f 1
3
e 3 3 − 2
15 a 8 6 , 4 3 + 8 2
b 24, 12 3
c 29, 28
16 a
b 10 + 5 3 + 15
c 6 5 + 15
15
g
h -42 h 1
d 7 + 3 5, 6 + 4 5
b 79 + 20 3
17 a 8 + 20 3
Exercise 2D 1
a 7 + 2 10
b 10 + 2 21
c 5 − 2 6
d 13 − 2 42
e 13 + 4 3
f 22 − 12 2
g 43 + 30 2
h 25 − 4 6
i 14 + 4 6
j 59 − 24 6
k 83 + 12 35
l 98 − 24 10
m 34 − 24 2
n 79 − 20 3
o 37 − 20 3
p 13 − 4 a 13
q 45 − 9 6 r 103 − 5 3 8 4 4 c 2 d 5 e 1
2
3 b 3 l 34
m 1
n -3
f 3
o -1
p -1
g 11 1 q − 2
k 5
3
a 7 + 4 3
b 7 − 4 3
c 37 + 20 3
d 37 − 20 3
4
a 1
b 2 5
c 18
d 8 5
6
a 7 3 + 7
b 3 − 3 3
d 40 3 + 116
7
a
b 14 + 122
c 29 3 + 40 37 c 2
122
h 2 r 33 4
i 6 155 s − 9
75 2 e -42
j -1 7 t − 18
b 5 6
5 a
Exercise 2E 1
a
5 3 3
b 3 2
c
7
d
3 5 5
e
2 7 7
f
3
g
5 6 6
h
7 2 2
i
10 2
j
2
k
2 3
l
2 6 9
m
2 15 15
n
15 4
s
1 7 6 +6 2 12 5 + 10 t 30 30
(
(
)
o 1 8 3 + 3 6
(
)
p 1 3 2 − 4 10
(
)
q 1 3 + 2 6 3
r 1 8
(
10 − 2 6
)
)
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
401
Cambridge University Press
)
(
1 9 2+ 8 3 6
d
1 12 14 − 21 2 28
f 11 3 − 3 11 33
h
22 3 − 12 11 33
e 20 7 − 14 5 35
3
a
e
i 7
m 7 +
q 3 3 10 + 5 2 8
u −4 − 15
4
a
d −
5
a
13 3 3
b −
7
a
3− 3 2
b
8
a
4 19 + 6 2 289
d
1 63 2 − 13 34
9
i a
1 15 − 5 5 8
)
b
d
1 45 − 20
)
e
ii a 588 + 240 6
e
(
7 4+
b 3
13 5 −
(
) )
1 j 3
3
(
3
)
6
)
(
(
)
15 −
n 4 4 + 5
3
c
)
(
2
g 14 3 − 15 7 21
7− 6 43
f 6 2 +
6
(
)
(
)
)
(
)
(
a
5 + 2
2+ 2 2
(
)
h
k
21 − 14
l
o
2 2 6 −3 5
)
p
3 +
(
1 37 5 + 19 2 − 9 57
2 3 3 +2 5 17
e
1 6 3 −5 2 10
)
(
(
(
(
)
(
3 3
16 − 9 3 2
)
1 3 2 −1 2
)
e
1 915 2 − 137 578
)
)
c
)
(
1 115 − 17 5 40 1 b 5 3 −6 2 6
5
)
(
)
(
5 +
3
(
5 2 10 + 9
22
)
)
(
e
(
f
1 5795 − 1638 2 1156
(
6 10
(
)
2 +1
)
1 19 − 6 2 4
1 6+2 5 5
(
)
) )
(
1 123 − 9 5 40 1 c 147 − 60 6 36 5 f 1421 + 572 6 12 f
559 3 9
c
)
(
(
1 3528 + 1440 6 + 5 3 − 6 2 6
3
)
d 15 3 + 26
5
)
1 17 10 + 4 2 − 10 17
c 6 3 + 10
b
(
2
)
f -10
)
1 5+ 5
(
4
)
d 6 7
(
)
2 5+
(
c 14
(
6 −2
(
c
)
(
(
t
)
(
3
1 27 + 7 6 15 1 x 23 − 6 10 13
)
s 4 2 6 − 3 2 3 1 w 29 + 7 10 27
b
(
2
(
r 2 3 6 + 3 17 1 v 21 + 4 5 19
1 5+5 3 −4 2 4
d
g 2
)
(
(
1 b 1 15 7 − 14 5 c 5 6 +2 5 20 35
2
d
(
1 65 3 + 66 2 6
)
)
Review exercise
402
1
a 4 2
b
6
c 4 − 2 3
d 3 3 −
2
a 5 2
b
2
c −4 7
d 11 3
2
e −2 7
f 4 2 − 3 5
f 2 11
3
a 30 2
b 30 2
c 12 10
d 28 42
4
b 3 5 h 8 3
c 2 6 i 45 2
d 3 3
e 4 5
a 6 2 g 6 2
j 18 3
k 40 2
5
a
75
b
112
c
d
e
g
208
h
176
6
a 9 2
b 2 5 + 5 3
c 3 3
7
a 19 2
b 4 2
c 15 2 +
242
125
d 19 7 6
384
e 11 5
f
891
f 55 6
d 16 7 − 27 3
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8
a 6 3 + 6
b 30 − 10 2
c 24 − 16 21
d 30 5 − 50
e 12 7 − 21
9
a 10 −
b 32 − 9 3
c 2
d 17
e 50 j 14 − 4 6
2
f 8 + 2 15
g 8 − 2 15
h 2
i 14 + 4 6
k 10
l a - b
m 11 − 4 7
n 28 + 10 3
5 3 7 3 b 3 6 1 11 a − 5 + 7 2 12 a 2 + 1
)
(
3 +2
16 a 6
e 2 −
3
)
(
14
ii
21 20 2 23 14 5
6 4
f
42 7 42 h 7 12 1 3+ 5 d 2 d 3 + 2
g
)
(
)
(
1 3− 5 2 c 3 − 2 7 b 3 c
b 0
c 4 3 + 7
d 1
b 2 3 − 2
c 26 − 4 3
d 12 3 − 12
30 − 12 3 f 13
22 − 14 3 g 13
h 2 3 + 5
18 40 − 20 3
b 6
20 a i 3 + 5 2
5 3 9
e
b 4 3 + 7
17 a 4 3 + 8 19 a
3 2 7
14 a 5 5 + 2
)
(
d
1 10 + 3 5 11 b 2 − 3
b
13 p = 5, q = 2 15 a 2
2 2 3
c
10 a
f 45 − 12 6
c 5 −1 2
d
118
e
42
iii 1 + 5 2
iv 2 +
22 a −4 3
b -8
24 a 6
b -11
c -4 6 c − 11
30 − 10 3 3
f 20 2 − 10 6
v 2
5
d −7 − 4 3
Challenge exercise
( x)
2
+2 x y +
1
a
2
a a + b - a a +
d
3
9 a x = 4
b 4(1 +
a+b−
9
a
x2 )
b i
-1=3+ 2
e 2a + 2 a − b b x = 9
8
2 x
+
1 + 3 x
5
2 53
5
6
11 +
5
ii
11 −
5
iii 6 +
5
c 2x
2
c x = 2
4x2
15 −
d x = 9 5 − 3 +1 8
e x = 9 7
2 3 +3 2 − 12
f x = 8 30
ab
9
9 = 80
729 9 =9 80 80
4
4 = 15
64 4 =4 15 15
10 (2 6 + 5) cm 13 a 625
= x + y + 2 xy
b
x−2 x −
ii
5
b
b b =a c c b 2 b a+ =a c c b 2 a = (a − 1) c a b = 2 a −1 c a+
a Mixed numbers of the form a 2 . a −1 2 11 1 12 5
b 1799 2
2
2
4
14 a i
( y)
5 −1
iii
5 −1 2
iv
5 −1 2
5 − 1 5 − 1 b =1 + 2 2
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
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Chapter 3 answers Exercise 3A 1
a 8x
b 2x
c - 6x
d 4a + 7b
f -3x + 2y
g 4a + b
h -4a - 4b
i 3x - 4y
2
a 3ab
b 6x2y
c 12cd 3
d 9st2 + s2t
e 2mn + 4m2n
f 12x2y
g -6x2y2 - xy
3
a 3a + 6
b 7b + 21
c 8m + 16
d 11p - 77
e 4a - 10
f 24p + 108
g -12b - 27
h -20m - 25
i 2 - 6a
j 42 - 24b
k 35 - 10b
l 14 - 21b
m x + 3
o 4a + 2 1 x u − 5 10 c c2 - 5c
q 3x - 12
r 3 - 5l
4
n 3x + 6 5y 1 − t 2 5 b b2 + 7b
p 4t - 3
1 s 4 x + 2 a a2 + 4a
e
20h2 -
10i 2 +
12j 2 +
i l -
5
f
28h
g
14i
d 6g2 - 10g h 4k - 5k2
21j
j 6ac + 3bc
k 8c2 -
m 10m2 - 20mn
n 10x2y + 15x
o 6p - 15p2q
a 4a + 4b
b 5a + 5
c 8p - 56
g 30 - 6x
h 12a2
f
3l 2
6m2
+ 10m
l 10d 2 - 20de
4cd
d 15 - 3p
- 21a
i
12a2
e a2 + ab
+ 20a
j 6x2 - 21xy
6
a 8a + 23
b 3d + 3
c 2e + 5
d 3f - 18
e 13g + 10
f 3i - 30
g j - 6
h 8a2 + 13a
i 10b2 - 9b
j 15a2 + 6a
k 2b2 - 15b
l 10
7
34 x 15 + 21 7 a 5y + 14
f x2 + x - 6
d a + 1
g 2p2 + 7p + 5
h 10z
i 4y2 - 16y
q
p
x2
9 5 b 7a - 12
5x + 2 6 3x 9 r − 4 4 c 16b - 9
+ 10x + 21
n
m
b a2
+ 13a + 40
a2
7x 1 + 12 2
+ 12a + 27
o
19 x 3 − 15 5
e -5 j 15z2 - 10z d x2
- 6x + 5
8
a
e x2 - 5x + 6
f a2 - 11a + 28
g x2 - x - 42
h x2 + 8x - 33
i x2 - 5x - 24
j x2 - 3x - 54
k 6x2 + 17x + 12
l 15x2 + 22x + 8
m 5x2
q
2a2
u 12x2 + 13x - 14 2a2
+ 11x + 2
n 6x2
- 11a + 15 + 7ab +
r
c
+ 7x - 5 + 13a - 5
6a2
v 8x2 + 26x - 7
3b2
- 7mn -
b 6m2
o
20x2
- 11x - 3
p 6x2 - 13x + 5
s
8m2
+ 2m - 3
t 6p2 + 11p - 10
w 15x2 - 46x + 16
3n2
6p2
+ pq -
9
a
d 3x2 + 13ax - 10a2
e 6x2 + 13xy - 5y2
f 6c2 + 35cd - 6d2
g 8p2 + 2pq - 3q2
h 3b2 + 5ab - 2a2
i 15q2 - 4pq - 4p2
b 9x2 + 12x + 4
c 4a2 + 4ab + b2
10 a x2 + 2x + 1 + 24ab +
e
9a2
i
x2
- 2x + 1
11 a
x2
- 9
b a2
g 49 - 9y2
12 a
16b2
x2
f
4x2
j
4a2
+ 12xy +
9y2
g
- 12ab +
9b2
k 9a2
- 49
- 64
f 9 - 12x + 4x2
2
a a = 2 22 f y = 5 a x = -3
c
- 1
h a2 - 9b2
b b = 3
c x =
c x = -3
- 24ab +
- 4
4a2
9 2 h y = 6
d x = -2
d 4a2 + 12ab + 9b2
- 14x + 49
i 9x2 - 4y2
+ 4a + 1
g x = 4 b x = 2
y2
x2
g a2 + 6ab + 9b2
Exercise 3B 1
c
h 4m2 - p2 b 4a2
c
23 e x = 5
x 6x2 - 17x + 5
5q2
404
e 2t - s
h x2 - 2xy + y2
16b2
l 4x2 - 12xy + 9y2
- 1
e 4a2 - 9
d b2
f 25 - x2
j a2 - 9b2
k 64m2 - 25n2
d 9 -
e 9 + 12a +
4x2
l 49r2 - 4t2
4a2
i a2 - 6ab + 9b2
d y = -5
e x = 6
i a = 3 26 f a = 17
j b = -1 g y = -2
h x =
55 28
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3
a x > 3
c x ≤ -4
d x > -8
g x ≥ 1
h x < 8
i x >
b x ≤ 7
c x ≥ 13
d x < 11
e x ≤ -12
g x < 4
h x ≥ −
i x ≤ -20
j x < 5
5
4 5 11 a x > − 2 9 f x < 2 28
6 width 5m, length 13m
8
$90
9 4 km/h
4
f x ≤ −
e x > 7 2 2 j x ≤ 3
b x ≤ 2
29 4
1 2
7 17 and 25
10 $840 000
Exercise 3C 1
Giorgio 7 km/h, Fred 5 km/h
2 2 L
3
17 20-cent pieces, 10 $2 coins
4 $3600
6
1.5 L
5 12 12 hours, 7 hours
7 A to B 305 km, B to C 335 km
a $129 150 − x 11 a y ≥ 2 8
b $112, $142
9 26 hours
b i y ≥ 57.5
ii y ≥ 50.5
10 When sales exceed $1700
Exercise 3D 1
127
2 43.2
3 44.7
4
7
a 339 cm2
b 109 cm2
c 3695 cm2
8
a 465
b 1275
c 7260
9
88.36
10 99 cm
11 20 mm
6 5
5 2.45
d 5985
12 a y = z - x
b r =
A πs
c y =
ax t
d v =
2w m
m2 4
f m =
k2 4t
g p =
b2 a+b
h t =
x−1 x+1
e x =
i a = c 2 − b 2
j r = 2
m k = h − g t 2π
n a =
n−q v2
t l L = g 2π
k q = n - v3r
6 2.92
B2 + C 2 mbc o A = 2B bc − mc − mb
p s =
2
rc c − rv
Exercise 3E 1
a (b + 2)(b - 2)
b (y + 6)(y - 6)
c (3 + a)(3 - a)
e (3x + 4)(3x - 4)
f (5a + 1)(5a - 1)
g 16(y + 2)(y - 2)
h (a + 2b)(a - 2b)
i (3x + y)(3x - y)
j 2(2x + 1)(2x - 1)
k 3(2 + y)(2 - y)
l 5(m + 2n)(m - 2n)
m 3(3r + t)(3r - t)
n 7(a + 2b)(a - 2b)
o 5(x + 2y)(x - 2y)
p (1 + 3ab)(1 - 3ab)
2
a x + 3 x − 3
b x + 7 x − 7
3
( )( ) e ( x + 21 ) ( x − 21 ) a ( x + 2 5 ) ( x − 2 5 ) d ( x + 2 6 )( x − 2 6 ) (
)(
)
f
d (7 + x)(7 - x)
( )( ) c ( x + 13 )( x − 13 ) d ( x + 6 )( x − 6 ) ( x + 17 )( x − 17 ) g ( x + 19 )( x − 19 ) h ( x + 23 )( x − 23 ) c ( x + 3 3 ) ( x − 3 3 ) b ( x + 3 2 ) ( x − 3 2 ) f ( x + 2 7 ) ( x − 2 7 ) e ( x + 2 10 ) ( x − 2 10 ) (
)(
)
7 7 i x + x − 5 5 11 11 l x + x − 4 4
g x + 5 5 x − 5 5
h x + 10 2 x − 10 2
3 3 j x + x − 2 2
5 5 k x + x − 3 3
4
a (x + 2 + 2)(x + 2 - 2)
b (x - 1 + 3)(x - 1 - 3)
c (x - 5 + 5)(x - 5 - 5)
d (x + 3 + 7)(x + 3 - 7)
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
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2 2 e x − 4 + x − 4 − 2 2
g
5 5 i x + 3 + x + 3 − 3 3
k 2 x + 1 + 3 3
6 6 m 2 x − 3 + 2x − 3 − 3 3
o 12z(3z + 2)
5 6
3 3 f x − 1 + x − 1 − 3 3
(
( )( ) d 4 ( x + 7 ) ( x − 7 ) g 2 ( x + 1 + 5 ) ( x + 1 − 5 ) j 2 ( x + 1 + 2 3 ) ( x + 1 − 2 3 ) a ( a − b ) ( a + b )
)
h x + 3 + 2 2 x + 3 − 2 2 7 7 j x − 2 + x − 2 − 3 3
) (2x + 1 − 3 3 )
a 2 x + 7 x − 7
)(
(
( x − 1 + 2 3 )( x − 1 − 2 3 )
l
(3x + 1 + 2 3 ) (3x + 1 − 2 3 )
( )( ) e 3 ( x + 2 3 ) ( x − 2 3 ) h 3 ( x + 2 + 2 ) ( x + 2 − 2 ) k −2 ( x + 3 + 2 ) ( x + 3 − 2 ) b 3 x + 2 x − 2
b
15 4x − 1 − 5
n 4 x − 1 +
a +
15 5
( )( ) 2 ( x + 3 2 )( x − 3 2 ) 3( x − 2 + 2 2 )( x − 2 − 2 2 ) − 4 ( x − 2 + 5 )( x − 2 − 5 )
c 5 x + 3 x − 3 f i l
b
Exercise 3F 1
a (x + 1)(x + 8)
b (x + 1)(x + 6)
c (x + 2)(x + 6)
d (x + 2)(x + 4)
e (x + 3)(x + 9)
f (x + 6)2
g (x + 3)2
h (x + 3)(x + 8)
i (x + 3)(x - 2)
j (x + 6)(x - 5)
k (x + 8)(x - 5)
l (x + 10)(x - 6)
5)2
p (x - 2)(x - 16)
m (x - 1)(x - 6)
n (x - 3)(x - 4)
o (x -
q (x + 5)(x - 7)
r (x + 5)(x - 6)
s (x + 2)(x - 5)
t (x + 3)(x - 7)
u 3(x - 1)(x - 5)
v 2(x + 8)(x - 3)
w 4(x + 4)(x - 6)
x 5(x + 12)(x - 2)
y 3(x + 1)(x - 3)
z 2(x + 7)(x - 4)
2
a (m + 3)(x + 2)
b (x - 4)(x - 3)
c (a - 9)(b - 7)
d (2x + 5)(x - 1)
e (3a + 5)(b + 2)
f (2x + y)(3x - 2y)
g (2a - 5)(b - 6)
h (x + 2)2(x - 2)
i (4x - 5)(3 - 2y)
j (5x - 2y)(2x + 3y)
3
a (x – 11)(x + 7)
b (z – 5)(z + 5)
c 2x(x – 3)
g (x + 10)(x + 4)
h (x -
f (x + y)(x – y)
j
n –(x + 5)(x – 3)
o 2(x - 3)(x - 4)
s (x + y)(x + z)
t (3a – b)(2a – 3)
(
)(
)
x−6−3 2 x−6+3 2
7)2
d 5a2(4 – a) e (x + 3)(y + 2) i x − 11 x + 11
(
)(
)
k (11 – a)(7 + a)
l (x – 13)(x – 2)
m (z + 3)2
p 3(x – 8)(x – 1)
q 25(2 – x)(2 + x) r (8p – 9q)(8p + 9q)
Exercise 3G
406
1
a (2x + 1)(x + 1)
b (3x – 2)(x – 1)
c (5x – 2)(x – 1)
d (5x – 1)(3x + 1)
f (4x + 3)(2x – 1)
g (5x – 4)(4x – 3)
h (x + 11)(x – 2)
i 12(x – 3)(x + 2)
2
a (2x + 3)(x + 2)
b (2x + 1)(x + 5)
c (2x + 3)(x + 3)
d (3x + 2)(x + 3)
e (3x + 1)(x + 3)
f (3x + 2)(x + 4)
g (2x + 5)(x - 2)
h (2x + 1)(x - 7)
i (3x + 5)(x - 2)
j (5x + 4)(x - 1)
k (5x - 3)(x + 2)
l (7x + 1)(x - 3)
m (3x - 1)(x + 9)
n (7x + 10)(x - 1)
o (5x + 6)(x - 2)
p (3x - 2)(x + 2)
q (3x + 2)(x - 7)
r (4x - 1)(x + 3)
3
a (3x - 2)(2x + 1)
b (4x - 3)(2x + 1)
c (3x - 2)(2x + 3)
d (5x - 2)(3x - 2)
e (3x - 2)2
f (2x + 3)2
g (5x + 3)(2x - 3)
h (4x + 9)(3x - 2)
i (5x - 2)(2x - 3)
j (4x - 3)(3x - 2)
k (4x - 3)(3x - 1)
l (5x - 2)(3x - 5)
m 2(5x - 2)(2x - 5)
n 3(4x + 3)(2x - 1)
o 2(3x - 2)(2x - 3)
p 2(3x + 4)(2x + 3)
q 5(4x - 3)(2x + 1)
r 2(4x - 3)(2x - 7)
e (3x – 2)(2x – 1)
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4
a (2x + 1)(3x + 2)
b (2x – 3)(3x – 1)
c (2x + 3)(3x + 5)
d (3x – 4)(3x – 2)
e (a – 8)(a + 7)
f –(2x – 1)(3x + 4)
g –(x – 9)(x + 1)
h 4(2x + 1)(3x – 2)
i (x – 8)(3x – 7)
j 5 2 5 − x 2 5 + x k 2 x − 1 − 15 2 x − 1 + 15
l 3 10 −
5
a (x + y - 3)(x - y)
e (a + 2b)(a - 2b - 1)
f (3x + 2y)(3x - 2y - 1)
h (x + 3)(x - 9)
i (2x + 1)(2x + 4)
(
)(
(
)(
2 (a − 3) 3 10 +
2 (a − 3)
)(
) (
)
b (x - y - 5)(x + y)
c (x - 4)(x - 1)
)
d (x + 6)(x + 1) g (a + 2d)(a - 2d - 1)
j (x - 11)(x - 4)
Exercise 3H 4x 5
1
a
g –1
2
a 4 x+4 h − x−4 b
9x 20 9b f 20 4 b + 13 k 12 a
b
c 3 x−3 x+4 11x c 20 5− x h 12 10 − 5 x m 6 42 d x = 12 e 13 k x = 3 l x = 4
9x + 3 20 5k − 3 l 42
g
a x = 6
b x = 8
c x = 3
h x = 108
i x = 9
j x = 3
e x – 6
2x − 5 x−4 16 x d 35 1 − 2z i 35 50 x − 3 n 30
k
f x = 2
g x = 20
m x = 3
n x = –2
Exercise 3I 1
a+b b x+4 g x−2
a
x+ y x− y a−2 h 2
a a x i x
b
c
2
a
2y + x 3(2 x + y )
e
( x − 2)(3 x − 1) 6( x + 2)
i
3
a
2 e ( x − 2)( x + 2)( x + 4)
h
2 2x + 3
i
4
a
−2( x + 4) x+3
b
a 2 + 3x 2 d 2ax
g
5
c
3x − 2 9
j 2x
k
x2 − x + 6 ( x − 2)( x + 2)
b
b
x 6( x − 3)
3 5 3a 2
c
d
(a − 3) (a + 2) a2 ( x + 4)( x − 3) ( x − 2)2
4x 2
x −9 x2 + 2 f ( x − 1)( x − 3)( x + 2)
2
x+7 x + 13 x−1 f 4x + 7
b
h
y y −1 − 3)
f 3(x - y) l
3x − y 4x − y
x ( x + 3)
d
h 2x l x − y x d
a (a − 4)(a + 1)(a − 2) 5 x + 49
e
x2 6y
49 − x 2 e ( x − 4)( x − 6)
−4(4 a − 1) (5a + 3)(2a − 1)(a + 4) + + + +
g
x+ x− 2x k 4( x
b+1 b x+2 j x+1
b b 3 7
(a + 1)2 a(a − 1) f a+1
2x 3x 4x e x a
3a 2
− + − +
f 2x + 9
x+2 x+4 23a e 20 13c + 1 j 15
j
i
x 15
3
d x – 3
2x + 5 ( x + 2)( x + 3)
g
x+5 ( x − 1)( x + 3)
j
4x (2 x − 3)(2 x + 3)
c
12 a2 − 1
f
1 (2 x + 1)( x − 1)(3 x − 2)
13a + 2 a2 − 9 x x+1 x+1 g x+5
c
d
a − 11 a−2
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
407
Cambridge University Press
1 5
6
a x = 2
b x =
e x = 7
f x = 4
7
a
28 5x
28 5 21 b x = 5 b x =
10 a x = 8
13 2 1 g x = − 7
d x = -17
8 x = -18
9 x = 2
c x =
c x =
h x = 7
54 13
d x = 6
e x = 2
f x =
2 5
Review exercise 1
a 13x
b 7p3q
c -5ab - ab2
d 20x3y - 10x2y
2
a -2a - 8
b -3b - 18
c -4m - 20
d -12b + 20
f 5a + 8
g 8p + 4
h 15q + 30
i -4p + 5
3
a
d2
- 9d
b 3e2
c
+ 6f
d -10m2 + 8m
f 6pq - 10pr
g
4
a 2c + 5
b 7h + 4
c 4 + 3q
d 4ab
f b - 22
g 12y2 - 11y - 15
h 6p2 - 6p - 4
i x2 + 7x + 12
k m2
5
a x = 3
b x = 5
f x = 4
g x =
- m - 20
6
a x > 4
1
c x ≥ -9
2
+ 8zy
3
4
–9
–8
–7
- 36x + 5
24 13
5
–1
7
12 a 7
9
a h =
19 x 10 − 12 3 j a2 - 14a + 45
e
d x = 15
e x = 10
i x = 2
7 b x < 2 0
6
1
2
3
4
–2
–1
0
5 d x ≥ − 2
–6
0
e -15n2 - 21n
8a2b
n 2m2 + 7mn + 3n2
h x = 20
–4
–1
i 6a +
c x = 2
–3
4 f x ≥ − 3
1
–2
7 g x ≤ 5
1 i x ≥ 8
–1
0
1
h x ≥ -15 0
1
–1
0
A − πr 2 πr
e (3x + 1)(3x - 1)
( )( ) d ( x − 4 2 ) ( x + 4 2 )
–16
2
–15
–14
–13
–12
1
b 3
10 a (7 + z)(7 - z)
+ 7x - 10
m 7x2
2 e x ≤ − 7 –2
- 15xy
–11 –10
l
6x2
h -6z2
0
408
+ e
-6x2
5f 2
e -20b + 35
b a =
8 a 52 2
A 2 − r c u = πr
b h =
v2 −
p −b 4
2E m
c -1
d x =
cb + by b(c + y) = c−a c−a
b (3 + 2a)(3 - 2a)
c (5 + 8x)(5 - 8x)
d (ab + 2cd)(ab - 2cd)
f (6p + 7q)(6p - 7q)
g (1 + 10b)(1 - 10b)
h (11a + 9b)(11a - 9b)
( )( ) ( x − 5 2 )( x + 5 2 )
(
)(
11 a x − 11 x + 11
b x − 15 x + 15
c x − 2 7 x + 2 7
e
f x − 3 5 x + 3 5
(
)(
)
)
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( x + 4 − 2 )( x + 4 + 2 ) ( x − 6 − 11 )( x − 6 + 11 ) (2x + 1 − 2 5 )(2x + 1 + 2 5 )
12 a
c
e
13 a (a + b + c)(a + b - c)
5 )( x − 3 + 5 ) ( d ( x − 7 − 7 ) ( x − 7 + 7 ) f ( 3 − 2 x − 10 ) ( 3 − 2 x + 10 )
b x − 3 −
b (a - b + c)(a - b - c)
14 a (a + 3)(2x -1)
b (a + 5)(b - 2)
c (x + y + 2z)(x + y - 2z)
c (2x + 3)(y + 5)
d (m - 7)(3x + 2)
15 a (x - 1)(x - 2)
b (c + 1)(c + 11)
c (x + 1)(x - 5)
f (z - 8)(z + 2)
g (c + 10)(c - 2)
h (p - 13) (p + 2) i (k - 14)(k + 3)
16 a 2(x - 6)(x + 11)
b 3(x -7)(x + 11)
d 3(x - 5)(x + 2)
17 a (2x + 3)(x + 4)
d (y - 1)(y + 10)
e (y - 7)(y + 2) j (x + 10)(x - 6)
c 3(x - 7)(x + 1)
e 3(x - 5)(x - 6)
f 6(x - 1)(x + 2)
b (3s + 2t)(2s - 5t)
c (3x + 1)(x + 1)
d (4x + 3)(3x - 4)
e (3x + 2)(x + 2)
f (x - 1)(2x + 3)
g (2x - 1)(2x + 3)
h (3x - 1)(x + 3)
i (3x - 2)(2x + 5)
j (3x - 2)(x + 1)
18 a x = 4
b x = 19
f x = 10
19 a
68 11
g x =
3x − 7 ( x − 2)( x − 1)( x − 3)
20 a 5
c x = 11 h x = − b
6 7 i x = 33 d x =
86 37
7 ( x − 1)( x − 2)
c
3 2
e x =
13 ( x − 1)( x – 2)( x − 3)
b 70 bottles of each
Challenge exercise 1
a (x + 10y)(5x + 4y)
b (8x + y)(2x + 3y)
2
a (a - x + 2b)(a - x - 2b)
b (3a - x + c)(3a + x - c)
c (b + d + a - c)(b + d - a + c)
d (y + b + a + 3x)( y + b - a - 3x)
a2
+
b2
+
c2
+ 2ab + 2bc + 2ac
3
a
4
a 4x2y(7x - 5)(x + 3)
5
x=6
6
a length 18 m, width 12 m cp am km km 8 p−q bp
7
b 343a6 - 27x3
(
)(
)
b ( p + 2q)( p − 2q) x + 2 2 y x − 2 2 y b length 15 m, width 12 m
Chapter 4 answers Exercise 4A 1
a
2
a
10 ≈ 3.16 5 ≈ 2.24
f
117 ≈ 10.82
b 4
c 7
b 5
c
h 5 2 ≈ 7.07
4
a (6, 5)
g 13 1 b , 2 b i 5
5
a u = 4
b v = -1 or 9
3
a (1, 4)
y −1 0 −1 −2
5
2
4
5
6
7
P(4, −2)
8
9
e
89 ≈ 9.43
i 10
ii 6 c w = -5 or 11
6 a 5
3
d 4 5 ≈ 8.94
1 e , 6 f (-5, 12) 2 c AC and BC are equal, so ABC is isosceles.
c (4, 6)
1
29 ≈ 5.39
d (-3, -7)
c i
2 12
(
b M 1 1 , 2 2
ii
2 12
)
iii 2 12
x
−3 −4 −5
(−1, −5)
(9, −5)
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7
b (22, -10)
a (-4, 1)
9 XY = XZ
8 PQ is perpendicular to PR.
10 AB = CD = DA = CA 11 (-2, -3)
12 AB = CD =
20 and BC = DA =
50
Exercise 4B 3 4
1 16 4 a = 8
5 1 d − e 4 3 11 d 8 5 b = -12
c 8
d 7
b 2
c -2
b -13
c
1
a
2
a 4
3
Both gradients = 1 b -4
6
a 4
7
Gradient of AB = 8, gradient of PQ = −
8
1 a − 6
b 2
9
a 2
b 2
10 a
y
3
D
2 1
0
12 a
2 c − 3 1 c − 2
A
C
B 1
2
2
3
4
5
b 1
d
f -1
e 1
f -2
1 e − 2
f 1
g 1
h 0
i −
13 9
1 8
5 4
g
5 7
b gradient of AB = gradient of DC = 0; gradient of AD = gradient of BC = 1 5 c , 1 ; Because the diagonals of a parellelogram bisect each other 2
x
c 1
13 gradient of AB = 2, gradient of BC = 2, points are collinear 1 7 14 a gradient of AB = , gradient of BC = , points not collinear 2 5 b both gradients = 3, points are collinear c both gradients = 3, points are collinear 11 13 d gradient of AB = , gradient of BC = , points not collinear 5 4 15 a 1 b 1 c (6, -3) d 5 12 , 3 12
(
)
Exercise 4C 2 b gradient = − , y-intercept is 5 3 2 4 d gradient = − , y-intercept is 3 11
1
a gradient = 4, y-intercept is 2
c gradient = -7, y-intercept is 10
2
a y = 8x + 3
b y = 11x + 5
c y = -6x - 7
3 d y = − x + 15 5
e y = -4x + 3
f y = -5x + 2
1 g y = − x + 5 2
3 2 h y = − x + 4 5
3
a
y
b
410
c
y
(0, 2)
(0, 1) 0
y
x
0
x
x
0 (0, −2)
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d
y
0
x
e
f
y
y
(4, 0)
(2, 0) x
0
x
0
(0, −3)
g
y
0
x
h
y
0
x
i
y
0
x
(−3, 0)
(−1, 0)
(0, −3)
j
y
k
0
x
l
y
(1, 0)
(−3, 0)
(0, 1) 0
y
x
a, b, c, d, i and j have gradient = 0.
x
0
e, f, g, h, k and l do not have a gradient.
4
a 2x + y - 6 = 0
b 2x + 3y - 33 = 0
c 9x + 15y + 10 = 0
d -24x + 42y - 7 = 0
e 2x + y + 8 = 0
f 3x + 4y - 40 = 0
g -x + 5y + 20 = 0
h 3x + 4y - 44 = 0
i 2x + 5y + 4 = 0
j -4x + 10y + 3 = 0
k 3x - y - 12 = 0
l 9x - 16y + 36 = 0
5
a y =
f
6
a
c
e
g
5 3 x − 3 b y = − x − 5 2 2 2 5 5 g y = − x − y = x + 4 3 4 2 3 gradient = , y-intercept is 6 2 3 gradient = − , y-intercept is -6 8 11 gradient = − , y-intercept is 11 4 3 gradient = , y-intercept is 6 7 gradient = -5, y-intercept is 20
i
7
a x-intercept is 5, y-intercept is -10
c x-intercept is -3, y-intercept is 15
5 5 e x-intercept is − , y-intercept is 12 4 g x-intercept is -16, y-intercept is 20
2 x − 4 3 3 h y = x − 6 2
c y =
1 x − 3 6 3 i y = x + 3 5 d y =
e y =
2 6 x− 15 5
b gradient = -4, y-intercept is -24 d gradient =
4 , y-intercept is 8 7
f gradient = 2, y-intercept is 4 h gradient =
2 , y-intercept is -2 7
11 , y-intercept is -11 3 d x-intercept is 16, y-intercept is 6 b x-intercept is
f x-intercept is -18, y-intercept is 12 h x-intercept is 14, y-intercept is -6
11 11 i x-intercept is − , y-intercept is 2 5
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8
a
y
b
(0, 12)
d
y
e
f
y 0
(0, 18)
(0, −6)
x
0
11 , 2
0
y
x 0
(–18, 0) 0
g
y
h
0
x
0
y
(7, 0)
(0, –4)
0, −
20 3
0, −
(0, –11)
x
11 , 7
x
(3, 0)
0
(12, 0) x
0
y
(0, 8)
(6, 0)
c
y
i
20 7
y (0, 5)
x
0
x
,0
5
− 2, 0
11 2
x
0
9
a x = 1
10 a
b y = 5
c x = - 4
y
b
(1, 2)
(3, 0)
y
e
0
5 2
f
y
x
b b = 10 b b = -8
13 34
19 14 − 5
412
y
9 4 9 c c = − 2 c c = −
x
0
x
0
3 2
d d = -14
e e = −
d d = 4
e e = -6
f f = 50 f f = -28
16 -6
15 12
Exercise 4D 1
a y = 6x - 24
b y = 2x + 10
c y = -4x + 14
e y = -2x + 10 1 a y = x + 8 23 3 5 5 a , 2 2 a y = -4x - 6
f y = -3x + 11
g y = 6
b y = 2x
3 a y = -2x
b 7
c y = 7x - 15
b y = -4x + 20
c y = 5x + 28
5
x
(4, 1)
12 a a = - 4
4
0
(–1, 2)
(–1, 4)
(0, 4)
2
x
0
d
11 a a = −
y
x
0
c
y
1 d y = x + 8 12 2 h x = -7 1 b y = − x 2 1 d y = − x + 2 67 7 d y = -2x - 12
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6
a y = x - 4
e x = 1
7
y = 5x - 4
9
a y =
d -1
b y = -3x - 9
c y = -x + 6
d y = 3
g x = 2
h y = 4
9 f y = x − 3 4 1 8 y = x − 4 12 2 b y = -x + 8
c y =
e 2 2
f 2 2
10 a i (3 1 , 3 1 ) 2 2
ii (3 12 , 3 12 )
b mAC = 1, mBD = -1
11 a mAB = 4
b (8, 11)
c (8, 11)
1 x+5 2
1 x+2 2
e y = 4x + 2
d 2 17
Exercise 4E 1
a
b
y
y 5
5 4
4
(1, 4)
3
3 1
1 −4 −3 −2 −1 0 −1
1
2
3
4
−4 −3 −2 −1 0 −1
x
1
−2
−2
y = 2x + 2
y = 3 − 2x
2
2
−3
y=x−3
−4
2
3
4
5
x
(2, −1)
−3 −4
y = 3x + 1
c
y 5
y = 5x + 3
4
y = 2x + 1
3
2
1
−3, −3
2 1
−4 −3 −2 −1 0 −1
1
2
3
4
5
x
−2 −3 −4
2
9 27 a x = − , y = − 7 7
e x = 4, y = 13
7 9 c x = − , y = − 5 5 4 20 g x = , y = 7 21 c x = 3, y = -2
13 8 ,y=− 5 15 5 3 h x = , y = 2 2 d x = -1, y = 2
3
a x = 4, y = 1
3 3 ,y= 2 4 1 9 f x = − , y = 7 7 b x = 1, y = 2
e x = 5, y = 2
f x = -3, y = 7
g x = 1, y = 2
h x = 1, y = -1
i x = 2, y = -1
j x = -4, y = 3
k x = 2, y = -1
l x =
m x =
4
60 10 ,y= 17 17
6 24 q x = − , y = 7 7 28 173 a x = − , y = − 33 33 63 30 d x = − , y = 19 19 385 19 ,y=− g x = 24 12 j x = 0, y = -6
b x =
n x = −
5 46 o x = 3, y = 5 ,y=− 13 13
30 121 ,y=− 19 19 864 35 ,y=− b x = 101 101 9 51 e x = , y = − 13 13 10 8 ,y= h x = 11 11 31 k x = 26, y = 3
d x =
14 9 ,y= 11 11
p x = 7, y = -4
r x = −
97 38 ,y= 6 3 552 60 ,y= f x = 59 59 168 24 i x = ,y=− 25 25 220 64 ,y= l x = − 17 17 c x = −
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5
y = 3x - 6
11 7 A(-1, - 1), B(3, 3), C , 5 1 7 9 a i y = x + 2 2 10 a i y = x 12 a y =
6 a y = 0 27 5
b x = 2 7 20 8 y = 2x + 2, x + y = 9, , 3 3
ii y = -2x + 16 b (5, 6) ii y = -x
26 43 b y = -2x + 19 c , 5 5
1 x − 1 2
14 5 ≈ 6.26 5 13 a = 4, b = -15
14 The point is (11, 5).
15 a y = 3x
b y = x
11 m = −
b (0, 0)
27 4
e 5 5 ≈ 11.18 f 70
d
18 a BD : 6y - x = 6; AC : y + 6x = 21
16 a a = 9
b b = -6
120 57 , b 37 37
c
9 37 37
21 26 17 , 5 5 d 9
Exercise 4F 1
45 and 67
3
The daughter is 8 and the father is 36.
4 58 $2 items, 43 $5-items
2 46 and 37
5
13 12 and 22 12
6 wallets $60.75, belts $65.25
7
Andrew is 36 and Brian is 14
8 10 metres and 14 metres
9
Standard: 5 kg, Jumbo: 7.5 kg
10 8000 $60 tickets sold and 2000 $80 tickets sold
11 1.875 L
12 14 10-cent coins, 13 20-cent coins
13 5 hours
14 3:20 p.m.
15 50 tonnes of the 52% iron alloy, 150 tonnes of the 36% iron alloy 16 480 km/h and 640 km/h 2 18 15
17 model horse $30.00; model cow $10.00
19 30 km
Review exercise 1
a 2 17
2
3 a , 1 2
3
c 12
d
b − 1, 4 12
1 c − , 0 2
9 d , 2
a 4
b 3
c 3
d 1
5 h − 2
2 i − 5
j -3
4
5 g − 2 - 4
5
a gradient = 2, y-intercept is 4
b gradient = 3, y-intercept is 5
c gradient = 1, y-intercept is -4 1 e gradient = , y-intercept is 1 2 g gradient = -2, y-intercept is 5
d gradient = 2, y-intercept is -1 2 f gradient = , y-intercept is 2 3 h gradient = -1, y-intercept is 6
i gradient = -1, y-intercept is 2
j gradient = -3, y-intercept is 4
k gradient = -3, y-intercept is 4 2 m gradient = , y-intercept is -2 3 o gradient = 2, y-intercept is 0
l gradient = -2, y-intercept is 1 3 n gradient = − , y-intercept is 3 4 p gradient = 3, y-intercept is 0
q gradient = -3, y-intercept is 0 3 s gradient = , y-intercept is 0 2
r gradient = -4, y-intercept is 0 3 t gradient = − , y-intercept is 3 2
414
b 6
(
)
26 5
e
34
3 13 e − , 2 2 1 2 k 0 e
f 4 5 9 f , −3 2 2 3 l 0
f
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6
a
b
y (0, 4)
(0, 2)
(3, 0) x
0
d
y
(0, 3) (4, 0)
c
y
x
0
y
e
(1, 0) x
0
y (0, 4)
(0, 2) (6, 0)
(6, 0) x
0
f
y
x
0
g
y
(0, 3) 0
(4, 0) x
0
h
(0, −4)
i
y
x
(2, 0)
j
y
y 0
(3, 0) 0
x
(0, −1)
0
(2, 0)
(2, 0)
x
x
(0, −3) (0, −8)
k
y
l
y
(6, 0)
(8, 0) x
0
(0, −12)
7
a
x
0
(0, −12)
y
b
y 2 3
0 (0, −3)
3 2
,0
x
0
c
y (0, 2)
,0 x
(0, −2)
(4, 0) 0
x
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Cambridge University Press
d
y
e
f
y
y
(5, 0) (0, −1)
0
1 2
x
,0
(0, 6) x
0 (0, −2)
(−2, 0) x
0
g
h
y
y 0,
(1, 0) 0, −
2 3
(0, 4) x
0
1 2
y
(−2, 0)
x
0
i
(4, 0) x
0
j
y
k
y
(0, 1)
(2, 0) 1 3
,0
m
y
(−1, 0) x
0
x
0
x
0
l
y
n
y
o
y
(0, 3) (0, 2) (−1, 0) x
0
x
0
0
x
(0, −2)
p
y
q
y
r
y (4, 0)
0
1 3
,0
(0, 3)
x
(0, −1)
0
s
y 3 2
x
t
y (0, 6)
,0 x
0 0, −
416
x
(2, 0) 0
(0, −1)
2 3
x
0 0, −
3 2
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8
a 6x + y = 11
b 2x - 3y + 14 = 0
9
a 4x + 9y = 74
b 4x + 5y = 32
c 4x - 3y + 26 = 0
10 a x = 4 18 , y = −1 87
b x = 1 87 , y = 1 87
11 a x = -3, y = -7
b x = -2, y = 8
d x = 1, y = 1
e x =
5 139 12 a i y = − x + 6 12 57 14 13 , − 13 13
c x = 5, y = 0
156 42 ,y= 11 11
ii y = −
f x =
138 3 ,y=− 65 13
701 273 b , 74 74
3 69 x+ 11 11
14 y = x + 3; y = x - 2; 3x + 2y = 21; 3x + 2y = 6
15 The gradient of AC = -1 and the gradient of AB = 1. 16 b 12 2 , 4 2 5 19 a p = 2
17 113 and 25
18 Stool $30; Chair $100
b p = - 23
20 4x + 3y + 5
21 The point is (4, 3)
Challenge exercise 1
a
b i After 1 hour 52 12 minutes, 150 km from Mildura
d
300
Steve
ii After 2 12 hours, 200 km from Mildura
Tom
200
iii 20 km
100 0
1
2
3
t
4
2
1 1 a ( x1 + 3 x2 ) , ( y1 + 3 y2 ) 4 4
3
mx2 + nx1 my2 + ny1 m + n , m + n
4
c b a , 2 2
5
a i y =
a + c b b i , 2 2
6
a i
10 a
b x a+c
b a+c
p p ;− m
b i ii y =
1 1 b (2 x1 + 3 x2 ) , (2 y1 + 3 y2 ) 5 5
c2 + b2 2
ii
b2 + c2 2
iii
c2 + b2 2
b ( x − c) a−c
a + c b ii , 2 2 b a−c m m b y = − x + p p
c a 2 + b 2
ii
c y =
1 m x+ p p
Chapter 5 answers Exercise 5A 1
a 0, -3
b 0, 7
c 0, -5
d 3, -6
e -7, -9
2
a 0, 5
b 0, -7
c 0, -8
d 0, 25
e 0, -18
3
a -1, -8
b -1, -6
c -2, -6
d -2, -4
e -3, -9
f 10, 7 1 f 0, 2 f -6
g 2, 4
h -3, -8
i 2, -3
j 5, -6
k 5, -8
l 6, -10
m 1, 6
n 3, 4
o 5
p 2, 16
q -5, 7
r -5, 6
s -2, 5
t -3, 7
u 10
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Cambridge University Press
4
a 0, 8
b 1, 16
c 1, 2
d no solution
e 3, 5
f no solution
g 0, -3
h 7, 3
i -5, 4
j no solution
k 8, 2
l 8, 3
m -10, 1
n 7, 1
o -1, 1
Exercise 5B 1
3 a 0, − 2
b 0, 7
c 0,
7 f − , − 4 2
g 3 , − 4 2 3
h
a -1, -2 3 a − , − 4 2
b 3, 2 1 b − , − 4 3
3 4 g − , − 2 5
h
3 4 , 2 3
1 m − , −1 2
2 n − , −1 3
o
s −
1 , −15 15
u
4
a
1 4 , 2 3
1 5 b − , 6 4
c 1,
3 4
d −2,
g
5 2 , 2 5
1 3 h − , − 4 5
i 2,
1 2
j 3, −
2 3
5 ,3 12
t
Exercise 5C 1
a 9, -2
b 8, -4
g -3, 4
h -3, -1
2
4 3 m − , 3 2 a -7, 2
3 7 n , − 2 5 b 5, -3
g -6, 5
h -5, 4
5 7 m − , − 4 6
7 n − , 3 3
3
a
50 x
b
5 2
5 , −4 2
d 0, −
4 5
3 2 e − , 4 3
2 3 i − , − 5 7
j 2 , 4 3 3
c -7, 5 3 c − , −2 2
1 d − , 2 2
e
3 ,3 2
f
4 ,2 3
1 5 i − , 2 4
2 3 j − , 3 4
1 2 k − , 4 3
l
2 5 ,− 3 2
1 , −8 2
2 p − , −7 3
2 q − , 7 3
2 r − , −3 5
9 8 , 8 9
2 3 v − , 3 2
w
10 7 , 3 4
x
3 2
e
2 5 , 3 2
1 4 f − , 2 3
5 4
2 1 k − , 7 3
l
d 4, -2
e -5, 5
f no solution
j 5, 1
k -5, 1
l 8, -8
3 c − , 4 2 i 0, -5 o 4, 8 c -3, 2 1 i − , −4 2
7 5 , 2 4
5 , −7 4
19 p − , 1 2 d 2, -1
e 6, 1
f 16, 2
j 4
k -13, 11
1 l − , 3 2
q 2, -4
50 = x + 5, x = 5 x
Exercise 5D 1
2 12 cm
11 sides
3 16
4 5
5
9 and -7 or 7 and -9
6 8 cm
7 4 cm, 32 cm, 42 cm
8
5 and 7
9 6 km/h
10 a 20 - x cm
11 4 m
b 14 cm, 6 cm
12 8 cm, 12 cm, 4 13 cm 14 20 cm and 25 cm
13 25 km/h
Exercise 5E
418
1
a -1
b -2
7 g − 2
h
5 2
c -3 i −
1 2
d 1 j
3 2
e -4
f 5
2 5
l −
k
3 4
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2 d − 7
5 g 5 f − 5 7 2 3 b parts ii, iv, vii, x, xii
2 5
a 1 b − 1 3 5 a parts i, v, ix
c 2 5
5
a 4 1 g 4 a (x + 3)2 + 1
b 16 9 h 4 b (x + 4)2 - 21
c 25 25 i 4 c (x + 6)2 - 46
d 36 49 j 4 d (x + 5)2 - 30
e 49 25 k 4 e (x + 1)2 - 16
f 100 121 l 4 f (x + 4)2 + 7
g (x - 4)2 - 10
h (x - 3)2 - 21
i (x - 6)2 - 50
j (x - 5)2 - 19
k (x - 3)2 - 17
l (x - 10)2 - 95
6
3 17 a x + − 2 4
3 19 e x + + 2 4
13 189 i x − − 2 4
2 3 4
e
2
2
1 3 b x + + 2 4
2
11 117 f x − − 2 4
2
2
1 41 d x − − 2 4
5 1 c x − − 2 4
2
2
2
9 141 h x + − 2 4
11 125 g x − − 2 4
2
2
h −
2
3 25 j x − − 2 4
2
1 49 k x + − 2 4
1 55 l x + + 2 4
Exercise 5F 1
a ± 5
b ± 11
c ± 13
d ±2 3
f ± 2
g ± 10
h ± 5
i ± 7
2
a −1 ±
c 6 ± 13
d −3 ±
f −4 ± 14
g −4 ± 2 5
h −4 ± 15
i −5 ± 2 6
j −2 ±
k −6 ±
41
l 5 ± 5 3
m −10 ±
n 10 ± 105
o 50 ± 2 645
p 25 ±
615
q 7 ±
r 14 ± 197
3
a
−1 ± 5 2
b
g
−5 ± 41 2
h
m
9 ± 101 2
n
2
b −2 ±
51
3± 5 2 3±
29
1±
21
b 4 ±
g
5
a −3 ± 17
b 5, -2
1 g −2, − 2
h 1, −
6
5 2 3 4 s − , 2 3 m ±
a −1 ±
h
5
−5 ± 13 2
n ±
6
2
a 3 ±
−1 ± 5 2
c
2
4
7
3
i o
1 3
−3 ± 17 2
e
3 ± 17 2
j
−7 ± 53 2
k
1±
401
2 7±
2 29
2
i −1 ±
p
7±
37
6
5 1 t − , − 3 4
u -3
b 1 ±
c −2 ± 11
2
q
2
−5 ± 21 2 7±
449 2
3± 5 2
f l r
−9 ± 89 2 3±
33 2
5 ± 13 2
e 2 ± 2 2
f 5 ±
5
k −25 ± 2 157
l 5 ±
34
e 2, -3
f
j −3 ± 14
k no solution
l no solution
p -3, 4
q ±
d −3 ± 11 j
−3 ± 37 2
d 2 ±
o -7, 5
2
e 4 ± 17
d
29 2
c 1, -7
1 3
2
5±
c −5 ± i
95
e ± 3
7
d −3 ±
5
e
5 2
1± 5 2
1 ± 13 2
r no solution
f 2 ± 2 3
Exercise 5G 1
a 4 ± 15
f
−5 ± 17 2
b -2, 4 g
−9 ± 69 2
3 ± 13 2
d
h 4 ± 14
i
c
13 ± 3 21 2 1±
21 2
e -2, 6 j
1± 5 2
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k 1 ± 2 2
2
a
3
−1 ± 22 3 3 2
g −1,
l
−1 ± 29 2
m
−5 ± 5 2
n −1 ±
b
−3 ± 29 10
c
3± 5 4
d 1,
5 2
i
−11 ± 161 10
j
h −2,
a 1.53, -0.13
b 7.87, 0.13
2 7
1 ± 29 14
l
2 ± 19 3
f -0.41, 3.41
j 0.15, 3.35
k 0.13, -1.53
l -1.46, 1.71
h -0.24, 0.84
i -1.37, 1.70
ii one solution
iii two solutions
b i 2
ii 2
iii 1
iv 1
v 0
vi 0
vii 2
viii 2
5
a
c
3+
33 2
6 30 km/h
8 50
5+5 7 −5 + 5 7 5−5 7 −5 − 5 7 and or and 2 2 2 2
10 4.47
11 5( 201 + 1) km/h ≈ 75.89 km/h 2095 and 50 +
10 ± 106 3
f
e 2, 3
g -0.35, 2.85
13 50 −
2 5
d -0.20, -14.80
a i no solutions
9
k
33
c -2, 5
7
o −6 ± e −1,
−6 ± 38 2
4
1+ 5 b 3.52 cm 2 AP =12 + 4 5 cm , BP = 4 + 4 5 cm
5
−1 − 401 −1 + 401 or 2 2
12 x =
2095
14 a x = 4 or x = 7
b x = 6 − 2 10 or x = 6 + 2 10
c x=
−1 − 19 −1 + 19 or x = 3 3
Review exercise a 6, -3
b −2,
g -2, 4
h -1, -3
1 m − , 3 2
n
2
a 3, 5
b 5, 6
f 10 ± 107
g
k 5, -4
l -6, 4
3
a 6, -4
b −2,
4
a 1 ±
5
a
e
b
2
1 , 3 3
4
21 3 − ,
3 1 d − , 2 3
i
5 , −1 2
j 1, −
o
1 3
p 3
h −3 ± 19 m
3±
37 2
5 7
7 3
d −1 ±
2
i 6 ±
30
n 3, -1
1 , 5 2
q
7 7 ,− 3 2
e
−1 ± 13 2
j 5 ± 2 7
c
−1 ± 5 2
d
−5 ± 17 4
e
−5 ± 57 4
c
1 ± 17 8
d
1 ± 17 2
e 1, −
−7 − 73 −7 + 73 , 4 4
c
f
9 + 113 9 − 113 , 4 4
g
a -7 and 9 or 7 and -9
b length 16 cm, width 9 cm
c 3, 5 or -3, -5
7 x = 2 5
Train 28 km/h, car 49 km/h 28 10 6 cm2 or cm2 27
4
33 5 − ,
f
1 ± 13 6
7 6
f
1 , −1 3
33 4
33
−1 ± 7 2
1 + 51 1 − 51 , 10 10 5+
1 l − , −5 2
k
o −1 ±
b
1 f − , −4 2
3 e − , 4 2
1 2
21 4
1 2
c 2 ±
3 ± 101 2
7 + 13 7 − 13 , 2 2 3+
c −1,
6
8
420
1 3
1
d
4+ 6 4− 6 , 5 5
h
−2 + 10 −2 − 10 , 3 3
9 (10 + 5 2) cm and (10 - 5 2) cm 11 AP = 4 cm
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Challenge exercise 1
±2, ±3
2 -1, -2, 3, 4
6
7
7 10 + 2 21
8
a AA
b
−1 + 5 2
d
2(5 + 4
9
a 2x2 + 6x - 3 = 0
b
−3 + 15 2
c
−1 + 5 2
7 17 5 − , 2 2
4 x2 - 4x + 1 = 0
5)
10 0, 5a
Chapter 6 answers Exercise 6A 1
a 318 cm2
b 448 cm2 f 54 + 36π
m2
e 432
h 100 + 37.5π
m2
≈ 217.8
2
a 3 3 ≈ 5.1962 cm
c 760 cm2
d 192π cm2 ≈ 603.2 cm2
cm2
g 4140 cm2
i 1000
l 144 + 40 2 ≈ 200.6
k 194
6
≈ 167.1
m2
cm2
5
cm2
cm2
j 596 cm2
cm2
3 9 m b 216 + 18 3 ≈ 247.1769 cm2 350 cm ≈ 37.1 cm a 1425π cm2 ≈ 4476.8 cm2 b 3π b 216 m3 c 144 m3 d 28 000 mm3 a 280 cm3 cm3
f 1215π
i 3240 cm3
≈ 3817.0
cm3
g 23 040
j 1080 cm3 cm3
7
a 5.29 cm
b 761.98
9
a 4 cm
b 7.4 cm3
cm3
k 480 cm3
e 300π cm3 ≈ 942.5 cm3
1178.1 cm3
l 360 cm3
8 a 19.9 cm
b 6.31 cm
10 16.25 m2
11 4 cm
1500
12 180 + 24π cm3 ≈ 255.4 cm3
13
14 a 1099.56 cm3
c 300.44 cm3
b 1400 cm3
h 375π
cm3 ≈
4 4.5 cm
2+3 2
cm ≈ 240.28 cm
Exercise 6B 1
60 cm3
2 a 80 cm3
cm3
e 216
3
a 400 cm3
f 180
b 120 cm3 m3
4
2 547 758.6
6
a 5 cm
5 a 800
7
a 64 + 32 13 ≈ 179.378
8
a 61 cm ≈ 7.810 cm
9
a
b i 10 cm
cm3
b 13 cm
VDA,
VAB,
cm2
c 56 cm3
d 144 cm3
c 28 cm3
d 53 13 cm3
e 80 cm3
b 2333.33
cm3
c 65 cm2
c 733.33
f 96 cm3
m3
d 360 cm2
b 36 + 12 34 ≈ 105.971 cm 2 b 2 13 cm ≈ 7.211 cm
VCB,
b 100 cm3
cm3
c (80 + 8 61 + 20 13 ) cm ≈ 214.593 cm 2
VDC: all are right-angled triangles.
ii 2 41 cm ≈ 12.806 cm
iii 10 cm
d 1099.56 cm2
c 192 cm2
Exercise 6C 1
a 138.23 cm2
b 235.62 cm2
2
a 24π cm2
b (16 π + 16 π 10 ) cm 2
3
a 10 cm
b 9.1652 cm b i 10 cm
cm2
5
a 104.72
6
a i 5 5 cm ≈ 11.18 cm
360 ° ≈ 255° b 2
c 395.84 cm2
c 70π cm2
49 7 449 2 d + π cm 4 4
4 a 685.84 cm2
b 21.83 cm
c 19.41 cm
ii 3.33 cm
iii 9.43 cm
ii 25π 5 cm 2 ≈ 175.62 cm2
360 ° ≈ 161° iii 5
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7 8
a 100.53 cm3
b 335.10 cm3 9 a 648π cm
127 mm
c 235.62 cm3
3
b 270π cm
10 a (1800 + 36 21) π cm 3
d 513.13 cm3
e 376.99 cm3
f 339.29 cm3
2
b 726π cm3
11 $19 635
c 615.75 cm2
d 1385.44 cm2
Exercise 6D 1
a 804.25 cm2
2
a 64π cm ≈ 201.06 cm
3
6.3078 cm
4 a 2.11 m
b 1.72 m
5
a 195π cm2
b 115π cm2
c 512π cm2
d (440 + 16π) cm2
6
a 2144.66
c 904.78
d 1436.76
e 452.39 cm3
7 a 6.20 cm
b 179 mm
c 9.8 cm
c 575.96 cm3
d 329.87 cm3
9 8249 cm3
2
b 2827.43 cm2
b 128π cm ≈ 402.12 cm2
2
b 4188.79 3
3
g 883.57 cm
h 3619.11 cm
8
a 167.55 cm3
b 368.61 cm3
Exercise 6E 1
a i 9
ii 9
b i
9 4
9 ii 4
c i
2
a i 4
ii 8
b i 9
ii 27
9 c i 4
3
a 42 875 cm3 cm2
c 192π cm2 ≈ 603.19 cm2
2
36 25
ii
36 25
d i
9 4
ii
9 4
ii
27 8
d i
36 25
ii
216 125
b 7350 cm2
4 960 cm2
b 325.5 cm3
7
a 172 800
8
a 192 m3
9
a The area of the triangle increases by a factor of 7.
b 36
5 8
e i
25 4
f 3619.11 cm3 10 545 131.78 mm3
ii
125 8
6 2 2
c (96 + 16 13 ) m 2 ≈ 153.7 m 2 d 614.8 cm2
b 1536 cm3 c 9
d 81
Review exercise 1
a i 1000 cm3 c i 480
cm3
e i 90π m3
2
a i 550 m3
3 4 5 6
ii 600 cm2 ii 528
b i 600 cm3
cm2
d i 168π
ii 460 cm2
)
(
ii 36 π + 12 58 π cm 2
cm3
2048 π m3 ii 78π m2 f i 3 ii 375 + 25 17 cm 2 ≈ 478.1 cm 2
ii 256π m2
)
(
340 π + 120 cm 2 ≈ 476.0 cm 2 b i 200π cm3 ≈ 628.3 cm3 ii 3 2 cm 250 π cm 3 ≈ 261.8 cm 3 a 1 500 000 m3 b 80 cm3 c 3 8 cm 848 π cm 3 ≈ 888.02 m 3 ii 168π cm2 ≈ 527.8 cm2 a i 3 2π 3 3 b i 4 + ii (16 + π) m2 ≈ 19.1 m2 m ≈ 6.09 m 3
d 156π cm3 ≈ 490.1 cm3
7
1.2 m
8 a 662.5 cm2
b 937.5 cm3
9
a 80 m3
2 2 b 63 + 8 34 + 3 89 m ≈ 137.95 m
10 a V =
250 000 12 500 π mm 3 , S = 20 000 + π mm 2 3 3
b V = 2250 cm3, S = 675π m2
875 d V = π m 3 , S = 150 π m 2 3
)
(
(
)
c V = 4 π m 3 , S = 6 + 13 π m 2
Challenge exercise
422
1
4 cm
3 6 cm
7
b 7 : 8
9 a Use similar triangles.
6 c Substitute the result of part b into that of part a. c Show s2 = h2 + (R - r)2
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10 a Use Pythagoras’ theorem
c cross-section area = area of cylinder cross-section - area of cone cross-section 1 2 d π r 3 − π r 3 = π r 3 3 3 e Volume of a sphere is twice the result in part d.
Chapter 7 answers Exercise 7A 1
a x = 4, (4, 0)
b x = 1, (1, 2)
c x = 2, (2, 6)
f x = -7, (-7, -2)
g x = 3, (3, -4)
h x = -2, (-2, -3) i x = 6, (6, 6)
2
a -3
b 46
c 5
3
a
y
b
d x = -3, (-3, 7)
e x = -2, (-2, 3)
d -65 c
y
y
(0, 25)
(0, 7) 0
0
x
(0, −2)
d
x
(5, 0)
(−2, 3)
(1, −3)
0
e
y
y
f
(0, 13) (0, 7)
x
(4, −4) x
0
4
a y = (x - 3)2
b y = (x + b)2
c y = x2 - 6
5
a y = (x +
b y = (x -
c y = (x -
4)2
+ 3
- 5
a y =
y
(−3 + 2√2, 0)
6)2
x
0 (1, 6)
(4, −3)
y
(0, 12)
0
6
x
a)2
d y = x2 + c + b
x2
d y = (x + d)2 - c b y = (x + 5)2 - 11
c y = (x + 3 - a)2 - 8 + b
(0, 1) x
0 (−3 − 2√2, 0) (−3, −8)
7
a a = 0
8
a
b a = 2
y 0 (0, −√2)
( 2, 0) √ x
c a = -1
b
d a = -8
y 0 x
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Exercise 7B 1
a i 3
ii x = 0, (0, 3)
iii
y
iv no x-intercepts
iv two x-intercepts
(0, 3) x
0
b i -7
ii x = 0, (0, -7)
iii
y x
0 (0, −7)
c i 8
ii x = 2, (2, 4)
iii
y
iv no x-intercepts
(0, 8)
(2, 4) x
0
d i 2
ii x = 3, (3, -7)
iii
iv two x-intercepts
y
(0, 2) x
0 (3, −7)
e i 25
ii x = -5, (-5, 0)
iii
iv one x-intercept
y (0, 25) 0 (−5, 0)
f i 48
ii x = 7, (7, -1)
iii
x
iv two x-intercepts
y (0, 48)
0 (7, −1)
2
a i 0
v two x-intercepts
ii y = (x - 3)2 - 9
iii x = 3, (3, -9)
x
iv
y
x
0 (3, −9)
424
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b i 0
v two x-intercepts
ii y = (x + 3)2 - 9
iii x = -3, (-3, -9)
iv
y
0
x
(−3, −9) 2
c i 0
v two x-intercepts
7 49 7 7 49 ii y = x − − iii x = , , − iv 2 4 2 2 4
y
x
0 7 , 2
− 49 4
d i -4
v two x-intercepts
ii y = (x + 1)2 - 5
iii x = -1, (-1, -5)
iv
y
0 (0, −4)
x
(−1, −5)
e i -1
v two x-intercepts
ii y = (x + 2)2 - 5
iii x = -2, (-2, -5)
iv
y
0 (0, −1)
x
(−2, −5)
f i -3
v two x-intercepts
ii y = (x + 3)2 - 12
iii x = -3, (-3, -12)
iv
y 0 (0, −3)
x
(0, −2)
x
(0, −5)
x
(−3, −12)
g i -2
v two x-intercepts
ii y = (x + 4)2 - 18
iii x = -4, (-4, -18)
iv
y 0
(−4, −18)
h i -5
v two x-intercepts
ii y = (x + 5)2 - 30
iii x = -5, (-5, -30)
iv
y 0
(−5, −30)
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ii y = (x + 6)2 - 40
i i -4
v two x-intercepts
iii x = -6, (-6, -40)
iv
y 0 x
(0, −4)
(−6, −40) 2
3 11 3 3 11 ii y = x − + iii x = , , 2 4 2 2 4
j i 5
v no x-intercepts
iv
y
(0, 5) 3 11 2, 4
0
x
2
5 17 5 5 17 ii y = x − − iii x = , , − iv 2 4 2 2 4
k i 2
v two x-intercepts
y
(0, 2) 0
x 5 2,
−
17 4
2
7 49 7 7 49 iii x = − , − , − iv ii y = x + − 2 4 2 2 4
l i 0
v two x-intercepts
y
(0, 0) x
7
− 2, −
3
a -3, 1
b -5, 1
d 3, 5
e −1 −
g 4 −
7, 4 +
c -2, 8 2 , −1 +
f 3 −
2
5, 3 +
5
h −3 − 11, −3 + 11
i 1, -3
j 2 + 2 2 , 2 − 2 2
k 5 + 3 2 , 5 − 3 2
l 3 + 5 2 , 3 − 5 2
m −3 + 2 5, −3 − 2 5
n −2 + 4 2 , −2 − 4 2
o 3, -5
4
a −1 − 5, −1 +
b 3 −
c 4 −
d −2 − 2 2 , −2 + 2 2
5
a i -5
ii y = (x + 2)2 - 9
iii x = -2, (-2, -9)
iv -5, 1
7
5
49 4
2, 3 +
2
e -11, 1
3, 4 +
3
f 10 − 5 6 , 10 + 5 6 v
y (−5, 0)
(1, 0) 0
x (0, −5)
(−2, −9)
426
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b i 5
ii y = (x + 2)2 + 1
iii x = -2, (-2, 1)
iv no x-intercepts
v
y
(0, 5)
(−2, 1) x
0
c i 9
ii y = (x + 3)2
iii x = -3, (-3, 0)
iv -3
v
y
(0, 9)
d i -7
ii y = (x - 3)2 - 16 iii x = 3, (3, -16)
iv -1, 7
v
x
0
(−3, 0)
y
(−1, 0)
(7, 0)
0
x
(0, −7)
(3, −16)
ii y = (x + 4)2 - 19
e i -3
iv −4 − 19 , −4 + 19
iii x = -4, (-4, -19) v
y
(−4 − √19, 0)
(−4 + √19, 0) 0
x
(0, −3)
(−4, −19) 2
f i -7
iv
3 3 37 3 37 ii y = x + − iii x = − , − , − 2 2 4 2 4
−3 − 37 −3 + 37 , 2 2
v −3 − √37 2
y
−3 + √37 2
,0
,0 x
0 (0, −7)
3 37 − ,− 2 4
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2
g i 11
iv no x-intercepts
5 19 ii y = x + + 2 4
5 5 19 iii x = − , − , 2 2 4 v y
(0, 11)
5
− 2,
19 4
x
0
2
h i -2
iv -1, 2
1 9 ii y = x − − 2 4
iii x =
1 1 9 , ,− 2 2 4
v
y
(−1, 0)
(2, 0)
0
x
(0, −2) 1 2,
2
i i 10
iv no x-intercepts
5 15 ii y = x + + 2 4
−
9 4
5 5 15 iii x = − , − , 2 2 4 v y
(0, 10)
− 52 , 15 4 0
2
j i -3
iv
7 61 ii y = x + − 2 4
−7 + 61 −7 − 61 , 2 2
x
7 7 61 iii x = − , − , − 2 2 4 v
y
−7 − √61 2
−7 + √61 2
x
0 (0, −3)
7
− 2, −
428
61 4
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ii y = (x - 1)2 + 3
k i 4
iv no x-intercepts
iii x = 1, (1, 3)
v
y
(0, 4) (1, 3) x
0
2
l i -2
iv
11 129 ii y = x − − 2 4
11 + 129 11 − 129 , 2 2
iii x =
11 11 129 , ,− 2 2 4
v
y 11 − √129 2
11 + √129 2
0
x
(0, −2)
11 2 ,
129
− 4
Exercise 7C 1
a i -7
iv no x-intercepts
ii x = 0, (0, -7)
iii
0
b i 7
iv two x-intercepts
ii x = 0, (0, 7)
y
iii
(0, −7)
y (0, 7) x
0
c i -4
iv two x-intercepts
x
ii x = 3, (3, 5)
iii
y (3, 5) 0 x (0, −4)
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ii x = 3, (3, -7)
d i -16
iv no x-intercepts
iii
y 0 (3, −7)
x
(0, −16)
ii x = -4, (-4, 0)
e i -16
iv one x-intercept
iii
y (−4, 0) x
0
(0, −16)
ii x = 6, (6, 0)
f i -36
iv one x-intercept
iii
y (6, 0) x
0 (0, −36)
2
a Reflect y = x2 in the x-axis then translate 9 units down.
b Reflect y = x2 in the x-axis then translate 3 units right.
c Reflect y = x2 in the x-axis then translate 4 units right and 7 units down.
d Reflect y = x2 in the x-axis then translate 4 units left and 3 units down.
e Reflect y = x2 in the x-axis then translate 3 units left and 11 units up.
f Reflect y = x2 in the x-axis then translate 1 unit right and 6 units up.
3
a
y
b
c
y
(0, 3)
(−2, 9)
(−2, 0) x
0
(− √3, 0)
(0, −4)
(−5, 0)
d
y
e
(−1, 1) (−2, 0)
x
x
y
(3, 22)
(0, 13) 0
x
(3 + √22, 0)
(3 − √22, 0) 0
430
( √3, 0) 0
(0, 5) (1, 0) 0
y
x
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f
y
(4, 9)
g
(4, 23)
(7, 0)
(1, 0)
x
0
(4 + √23, 0)
(0, 7) (4 − √23, 0)
(0, −7)
y
h
y
x
0
i
y
7 85 2, 4
(2, 7) (0, 3) (2 − √7, 0)
(2 + √7, 0) x
0
(0, 9) 7 − √85 2
7 + √85 2
,0 0
j
y
11 201 2, 4
k
,0
x
(6, 42)
y
(0, 20) (0, 6) 11 − √201 , 2
11 + √201 , 2
0
l
(6 − √42, 0)
(6 + √42, 0) x
0
x
0
0
y 1 5 2, 4
(0, 1) 1 − √5 2
1 + √5 2
,0
,0 x
0
Exercise 7D 1
a
y = 3x2
y
b
y 1
0
(1, 3)
(1,
(1, − 3)
y = x2
(1, 1) 0
y = 13 x2 1 3)
x
c
y y = 2x2
(1, 3)
(1, −1) x
y = −x2
y = 3x2 (1, 2) (1, 1)
y = − 13x2 (1, −3)
y = x2
0
x
y = −3x2
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
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2
y = 2x2 + 1
y
a
b
c
y
y = 6x2 − 1
(1, 3)
y = 4x2 −1
(0, 1) 0
x
1
−2
vertex: (0, 1) y-intercept: 1
d
y
vertex: (0, 8) y-intercept: 8 x-intercepts: −2, 2
8
0
1 2
(0, −1) vertex: (0, −1) y-intercept: −1 1 x-intercepts: − 2,
e
y 9
0
2
0 1 1 x − √6 √6 (0, −1) vertex: (0, −1) y-intercept: −1 x-intercepts: − 1 ,
x
1 2
√6
vertex: (0, 9) y-intercept: 9 3√2 x-intercepts: − 2 ,
x
−3√2 2
0
3√2 2
3√2 2
x
3
a, d, e, g are congruent; b, f are congruent; c, h are congruent
4
a y = 3(x + 1)2
7 25 d y = 2 x + − 4 8
5
a i 3
ii y = 2(x + 1)2 + 1
iii x = -1, (-1, 1)
iv no x-intercepts
v
2
2
3 41 b y = 3 x − + 2 4 2
1 √6
y = −2x2 + 9
y = −2x2 + 8 −2
y
5 51 c y = 2 x + − 2 2
2
5 59 e y = 3 x + + 6 12
f y = -5(x - 2)2 + 57
y
(0, 3) (−1, 1) x
0
b i 7
ii y = -2(x + 1)2 + 9
iii x = -1, (-1, 9)
−2 + 3 2 −2 −3 2 iv , 2 2
v
y (−1, 9) (0, 7) −2 − 3√2 , 2
432
0
0
−2 + 3√2 , 2
0
x
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ii y = 3(x + 2)2 - 31
c i -19
−6 + 93 −6 − 93 iv , 3 3
iii x = -2, (-2, -31) v
y −6 − √93 , 3
−6 + √93 , 3
0
0 x
0 (0, −19)
(−2, −31) 2
d i -2
iv
4 22 ii y = 3 x + − 3 3
−4 + 22 −4 − 22 , 3 3
4 4 22 iii x = − , − , − 3 3 3 v
y −4 − √22 , 3
−4 + √22 , 3
0
x
0 4
− 3, − 2
e i -15
iv −3,
5 2
(0, −2)
22 3
1 121 ii y = 2 x + − 4 8
1 1 121 iii x = − , − , − 4 4 8
v
0
y
5 2,
(−3, 0 )
0
0
x (0, −15)
1
− 4, − 2
f i 5
iv no x-intercepts
1 39 ii y = 2 x − + 4 8
iii x =
121 8
1 1 39 , , 4 4 8
v
y
(0, 5)
1 39 4, 8
0
1 1 11 , , , no x-intercepts 2 2 4
6
a x =
d x = −
7
a 6
b 8
c -8
9
a ±1
b ±3
5 5 181 , has x-intercepts b x = − , − , − 6 6 12
5 5 59 3 3 131 , no x-intercepts e x = , , − , has no x-intercepts , − , 6 6 12 10 10 20
10 y = (x - 1)2 - 2
x
or y = x2 - 2x - 1
d -2
8 a 3 b 1 2 c ± 2 11 y = 3(x + 2)2 - 1
1 c 2
1 1 7 c x = − , − , , no x-intercepts 4 4 8 3 3 17 f x = , , , has x-intercepts 2 2 4
d -3
d ± 3
or y = 3x2 + 12x + 11
12 y = -2(x - 1)2 + 6 or y = -2x2 + 4x + 4 Answers to exercises
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Cambridge University Press
13 a
y = 2(x − 1)2 + 3
y
y
b
(1, 8)
5
6
y = −2(x − 1)2 + 8
0
3
(1, 3) 0
−1
x vertex: (1, 3) y-intercept: 5 axis: x = 1
c
vertex: (1, 8) y-intercept: 6 x-intercepts: −1, 3 axis: x = 1
d
y
(−3, 12) y
(2, 12) vertex: (2, 12) y-intercept: −4 x-intercepts: 2 − √3, 2 + √3 axis: x = 2
2
y = −4(x − 2) + 12 2 − √3 0
x 2 + √3 −24
e
y
f
y
y = 4(x + 2)2 − 16 −4
0
(−2, −16)
vertex: (−3, 12) y-intercept: −24 x-intercepts: −3 + √3, −3 − √3 axis: x = −3 −3 + √3 x 0
−3 − √3
−4
x
y = −4(x + 3)2 + 12
y = 2(x + 2)2 − 12 vertex: (−2, −12) x-intercepts: −2 − √6, −2 + √6 y-intercept: −4 axis: x = −2
x vertex: (−2, −16) x-intercepts: 0, −4 y-intercept: 0 axis: x = −2
0
−2 − √6
−2 + √6
x
−4
(−2, −12)
g
y (3, 4)
0
3 − √2
vertex: (3, 4) x-intercepts: 3 − √2, 3 + √2 y-intercepts: −14 axis: x = 3 3 + √2
x
h
y y = 3(x + 1)2 − 15
−12
−14 y = 4 − 2(x − 3)2
434
0
−1 − √5
(−1, −15)
−1 + √5 x
vertex: (−1, −15) x-intercepts: −1 + √5, −1 − √5 y-intercept: −12 axis: x = −1
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i
y
81
vertex: (−4, 1) y-intercept: 81 axis: x = −4
(−4, 1)
x
0
Exercise 7E 1
4
3
a
2 1
y
b
c
y
y
(0, 6) (4, 0)
(−1, 0)
x
0
(3, 0) (2, 0) 0 1 1 2 2, − 4
(2, −4)
d
e
y
(0, −15)
x
(1, −20)
y
(2, 18)
x
(3, 0)
0
f
y
(0, 60)
(0, 10) (−2, 0)
(5, 0)
(−1, 0)
x
0
(5, 0) 0
g
y (−2, 0)
x
1
1 −2,
1 , 2
3 −64
i
y (0, 72)
(0, −6)
x
1 2
1 73 2
y (2, 0)
(−5, 0)
x
0
x
(0, −40)
j
(6, 0)
5 2, −
h
(4, 0) 0
(1, 0) 0
(−6, 0) 0
(−1, −100)
(4, 0) x (0, −96)
k
1
x
0
y
(0, −70)
(4, 0)
(−3, 0)
(1, −45)
l
y (−3, 0)
3
−12 , −85 4 y (−1, 0)
(1, 0)
x (0, −7)
(0, −21) (−1, −28)
,0
0
x
0
1 2
1 −4
,
63 −8
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press
g -4, 4
5
a
b -4, -3 1 1 h − , 2 2 b
y
c -3, 6
d -5, 3
e −12, −
i 0, 3
j -4, 0
k -4, 0
y
x=3
3
−1
y
d −6
0
e
y
x=1
x = −2
x
y
f
−3
−12 0
3
(1, −4)
x
2
0
−3
x
0 1 3 (2, −1)
x
2 4 (3, −1)
0
1 f − , 10 2 4 l 0, 5
y
x=2
8
c
1 2
x=1
a -5, -1
x = − 32
4
−3, −9 2 4
x
2 (1, −1)
x
0
(−2, −16)
1 2 1 x
k
436
0
−3
l
n
3 2 x
3
y
3
x
6 a y = x2 - 4x b y = 9 - x2
1 25 , 4 8
−1
2
0
c y = -3x2 + 6x + 9
3 2 0
−2
x
y
7 1 ,− 12 24
4
0
x
0
x
0
x=
1
3 9 , 4 8
3
m y
x=4
x
y
8
1 3 − ,− 8 8
1 2 2 3
9
−1 4
−1 2
0
7 1 − ,− 4 8
y
2
1 4
j x = −3
y
−3 2
−2
3 1 ,− 4 8
i
6
x = − 74
x= 1
0
y
7 x = 12
h
x=0
y
x=0
g
3 4
3x 2 9x − +6 5 5 1 5x e y = x 2 − −2 3 3 7x2 21x f y = − − +7 4 4
d y = − x
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7
y = x2 - x - 2
8 y =
1 2 x x − − 3 2 2
9 y = 6x2 - 15x + 6
Exercise 7F 1
80 m, 130 m
2 60 cm
3 25, 26
4 14, 16
5
13, 15
6 3
7 8 years
8 $6
9
8 cm, 15 cm, 17 cm
10 20 cm, 21 cm, 29 cm 11 5 cm, 30 cm, 40 cm 12 44 sides 5 14 1.84 seconds 15 -7 16 4 73 d 46 b -2 c − 12 19 75 m by 150 m 20 31.888 m, 5.102 seconds
13 16 17 a 4 18 25 cm by 25 cm, 17cm
22 3 83 square units
21 62.5 m by 80 m, or 160 m by 31.25 m
Exercise 7G 1
a -3 < x < 5
b x < -4 or x > 1
c x = -2
d x ≤ -1 or x ≥ 2
e all values of x
f x < −
2
a x < -2 or x > 3
b -4 ≤ x ≤ -1
c x ≤ 2 or x ≥ 5
3 3 or x > 2 2 d -3 < x < 0
3
a x < -10 or x > 7
b -3 < x < 8
c x ≤ -5 or x ≥ -4
d 3 ≤ x ≤ 4
4
a -8 < x < 5
b -3 ≤ x ≤ 8
c x ≤ 5 or x ≥ 7
d x < 0 or x > 11
Review exercise 1
a 2
b 24
c 0
d 0
2
a 0
b ±4
3
a -4, 1
1 b −6, − 2
c ±3 3 3 c − , 4 2
d ±2 1 5 d − , 2 2
4
a −2 −
d 2 −
5, 2 +
g 3 +
35 ,3− 5
5
a − 2 − 6 , −2 +
6
5, −2 +
b 3 −
5 5 35 5
5 19 5 − ,− + 2 2 2 a Minimum c −
2, 3 +
2
e −2 −
6 6 , −2 + 2 2
h 2 +
2, 2 −
c −1 − f 1 −
e 4
f -16
e -7, 7
f 0, 5
5 , −1 + 2, 1 +
5
2
2 b 3 − 2 2 , 3 + 2 2
6 19 2
26 , −2 + 2 c Maximum d −2 −
b Maximum
26 2 d Minimum
7
y = x2 is congruent to y = x2 - x, y = -2x2 is congruent to y = 3 - 2x2, y = 3x2 is congruent to y = 3x2 + 1, y = 2 + 3x - 4x2 is congruent to y = 1 + 4x2
8
a Translate 1 unit down
d Reflect in the x-axis then translate 1 unit up
9
a Translate 2 units left
d Translate 1 unit left and 3 units down
b Translate 2 units up c Reflect in the x-axis then translate 4 units up (There are other correct answers to this question.)
b Translate 1 unit right c Reflect in the x-axis then translate 1 unit left e Translate 2 units right and 3 units down
f Reflect in the x-axis then translate 3 units right and 1 unit up
(There are other correct answers to this question.)
10 a y = (x - 3)2
b y = (x + 2)2
11 a y = (x - 3)2 + 1
b y = (x - 5)2 - 2
12 a y = 3(x +
b y = 3(x -
3)2
+ 2
13 a y = -(x - 1)2 14 a (1, 2)
d (5, 11)
15 y = x2 - 2x - 1
3)2
c y = (x + 3)2 - 2
-2
b y = -(x + 2)2
c y = -(x + 1)2 - 2
b (-2, 3)
c (-4, -2)
e (-2, -1)
f (3, 4)
16 y = x2 + 2x - 15
17 y = -x2 - 6x - 8
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Cambridge University Press
5 x
−2
0
6
x
(2, −16)
−4
4
−1 − √2
0
x
axis of symmetry x = −1
b
x=3
13
1 1 , 6 4
0
x=0
x
y
axis of symmetry
x = −5 2 0
axis of symmetry
y
f
(0, 16)
5 25 − ,− 2 4
19 a
y
−1 + √2
0
(3, 4)
c
y
−3 − √5
x
−3
y
5 − √2 0 −69
−3 + √5 0
x
−4
(5, 6) 5 + √2 x
x=5
d
x
axis of symmetry
x
y
(−3, 5)
(−1, −6) 0
1 3
x=1 6
e
x
3 9 ,− 2 4
axis of symmetry
y
axis of symmetry
−5
axis of symmetry
(3,−4)
d
3
0
−12
y
2
c
x = −3
0 1
x=3
y
axis of symmetry
b
x=2 axis of symmetry
5
x=3
y
axis of symmetry
18 a
20 y = 11(x - 2)2 - 4 21 h = 1 or h = -1
438
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c
y 2
−1 + 6 x
0
y
0 3− 7
0 3+ 7 x
(−2, −3)
−5 (−1, −6)
d
(3, −7)
y
e
axis of symmetry
(4, 3)
4+ 3
4− 3
x
axis of symmetry
0
−13
y
f
−7
y 3 23 , 2 2 7 3 − 23 2 0
5 (−1, 3) 0
23 The side lengths are 5 cm, 12 cm and 13 cm
x
x
axis of symmetry
b
axis of symmetry
−1 − 6
y
axis of symmetry
axis of symmetry
22 a
3 + 23 2 x
24 52 cm, 42 cm and 4 cm
25 125 m by 250 m 26 a -3 < x < 5
b -5 ≤ x ≤ -2
c x < 2 or x > 5
d x < 0 or x > 2
Challenge exercise 1
a -5
b -5
2
a -21
b 4
3 4 5
c c > 4
1+ 5 1− 5 1− 5 1+ 5 x= ,y= ;x= ,y= ; x = 1, y = 1; x = −2, y = −2 2 2 2 2 3 4 d x = 80, t = ; x = 90, t = 2 3 40 km/h 6 45 cattle 7 32 km/h, 40 km/h
8
9 4.844 m/s, 5.844 m/s
75 runs 24 11 a = 5 2
10 40 km/h, 60 km/h
2
5 7 25 12 a x − + y − = 2 2 2
b Let A(0, 1) and B(g, h) be the given points and P(x, y) be any point. P is on the circle with diameter AB if ∠APB = 90° (angle in
B(5, 6)
semicircle). If gradient of AP × gradient of y−1 y−h BP = -1, then × = 1. x x−g
5 7 , 2 2 A(0, 1) 2
3
x
So (y - 1)(y - h) + x(x - g) = 0 is the equation of the circle. This meets the x-axis when y = 0. That is when x2 - gx + h = 0.
2500 100 π x2 (100 − x )2 cm 2 when x = + b The minimum area is π+4 π+4 4π 16 14 d If (x, y) goes to (x, 5y), then (a, a2) goes to (a, 5a2). The graph of y = x2 goes to the graph of y = 5x2. (0, 0) goes to (0, 0), (1, 0) goes to (1, 0), and (0, 1) goes to (0, 5). It is not a similarity transformation since some distances are unchanged and others multiplied by 5. Note that 5 can be replaced by any a > 0.
13 a
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Cambridge University Press
2
a a2 a2 a 1 e T he enlargement with centre at (0, 0) and enlargement factor maps A(a, a2) to B , . Since = 5 , B lies on 5 5 5 5 5 y = 5x2. That is y = x2 is similar to y = 5x2. f By c y = ± ax2 + bx + c is congruent to y = ax2, a > 0. By d y = ax2 is similar to y = x2. That is, all parabolas are similar. 15 a
y (−8, 17)
17
b
y y = 2x2 + 5x − 3
y = x2 + 8x + 17
0 (− 4, 1)
0
x
− 5 ,− 4
y
c
y = 3x2 + 11x − 12
d
y = dx2 + ex − f x
0
−12
3
49 8
y
x
0
− 11 , −12
x −3
− 5 ,−3 2
e − , −f d
− 11 , − 265 6 12
−
−f
e2 e , − −f 4d 2d
Chapter 8 answers Exercise 8A 1
a y = 6, b = 45° d a = 90°, b = 60°
2
a q = 75° (isosceles
b x = 5, a = 30° e x = 4, a = 45° ABC), a = 30° (angle sum of
b a = b = g = 60° (equilateral c 3a = 180° (angle sum of d q = 20° (angle sum of
ABC)
RST) FGH), a = 60°, x = y = 6 m (equilateral
WXY), x = 12 cm (isosceles
e ∠RQP = 60° (base angle of isosceles f a = 60° (angle sum of
c a = g = 60°, b = 120°, x = 6 f a = 60°, x = 5
WXY)
PQR), q = 60° (angle sum of
IJK), x = 3 cm (equilateral
g y = 5, a = 60° (equilateral
FGH) PQR), y = 7 cm (equilateral
RPQ)
IJK)
LNQ), a = 2b (exterior angle of isosceles
LQM), b = 30°
h ∠ GCB = 70° (base angle of isosceles BCG), q = 70° (vertically opposite angles at C), a = 40° (angle sum of b = 140° (straight angle at G), ∠BGF = 70° (alternate angles, FG || AD), g = 110° (straight angle at G)
BCG),
ABD = 55° (straight angle at B), ∠BAD = 55° (base angle of isosceles ABD), ∠ADB = 70° (angle sum of ABD), i ∠ q = 70° (vertically opposite angles at D), ∠AGE = 55° (alternate angles, AC || HE), a = 125° (straight angle at G) j ∠DCG = 55° (vertically opposite angles at C), ∠CDG = 55° (base angle of isosceles CDG), a = 70° (angle sum of CDG), ∠AGE = 55° (corresponding angles, BD || HE), ∠EGD = 55° + 70° = 125° b = 125° (vertically opposite angles at G) k ∠ DAB = 50° (complementary angles), a = 50° (alternate angles, AB || CE), ∠AFD = 40° (base angle of isosceles a + ∠FDC = 100° (angle sum of ADF), ∠FDC = 50°, q = 130° (straight angle at D)
440
l a = 60° (equilateral
KLM), b = 120° (exterior angle of
m a = 60° (equilateral
ABC, vertically opposite angles at A), b = g = 120° (exterior angles of
ADF),
KLM) ABC)
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n y = 2 m (radii of circle), b = 25° (base angle of isosceles BCO)
ABO), a = 60° (base angle of isosceles
BCO, angle sum of
o 3a + b = 180° (co-interior angles, AD || BC), a + 2b = 180° (angle sum of isosceles triangle), a = 36°, b = 72° 3α p ∠ CBO = 2a ( BOC isosceles), ∠OBA = 90 (angle sum of isosceles triangle), ∠OBC + ∠ABO = 95° (alternate 2 angles, AB || CD), a = 10° q a = 120° (co-interior angles, AD || BC), 3b = 180° (angle sum of quadrilateral), b = 60° (co-interior angles, AD || BC) r a = 130° (angle sum of quadrilateral) s a = 108° (angle sum of regular pentagon) t a = 120° (co-interior angles, AB || DC), b = 60°(co-interior angles, BC || AD) 3
4 36°
6 sides
Exercise 8B BAC ≡ ABC ≡
b e
ABC ≡ PQR ≡
a d
2
a ∠BCA = 40°, ∠FED = 100°, ∠FDE = 40°, AB = BC = 3 cm, FD = 4.6 cm
YXZ (SAS) XYZ (ASA)
XZY (ASA) XYZ (ASA)
c
ABC ≡
1
PQR (SSS)
b ∠ABC = 10°, ∠EFD = 130°, CB = 4.2 cm, AC = 1.1 cm, ED = 5 cm c ∠ABC = 28°, ∠FED = 28°, ∠FDE = 62°, AC = 8 mm, FE = 15 mm, DE = 17 mm d ∠FED = 32°, ∠CAB = 111°, ∠BCA = 37°, AB = 9.1 cm, AC = 8 cm, FE = 14.1 cm e ∠FED = 67°, ∠CAB = 67°, ∠ABC = 67°, ∠BCA = 46°, CB = 38.3 mm, AC = 38.3 mm, FE = 38.3 mm, ED = 30 mm 3
a AD = BC (equal side lengths of square), DE = CE (given), ∠ADE =∠BCE
b AE = BE (matching sides in congruent triangles)
4
a PR = RS (equal side lengths of square), PX = ST (half side lengths of square),
(given),
5
ADE ≡
BCE (SAS)
b
∠RPX =∠RST = 90°, RPX ≡ RST (SAS), RX = RT (matching sides in congruent triangles), RX = RT (from a), VR = VR (common side), VT = VX (half side lengths of square)
X
RXV ≡ RTV (SSS), ∠TRV =∠XRV (matching angles of congruent triangles)
V
T
S
a Diagonals of a rectangle are equal and bisect each other. b i ∠AOD = 56°
6
R
P
ii ∠AOB = 124°
iii ∠OBC = 62°
iv ∠ABO = 28°
∠YDO = ∠OBX (alternate angles, AB || CD), ∠BOX = ∠DOY (vertically opposite), OB = OD (diagonals of a parallelogram bisect each other), BOX ≡ DOY (AAS), OX = OY (matching sides of congruent triangles) Q
7
P
B
A
8
Y
B P = PC (P is the midpoint of BC), ∠QPB = ∠CPD (vertically opposite), ∠PQB = ∠CDP (alternate angles, AQ || DC), BQP ≡ CDP (AAS), BQ = CD (matching sides of congruent triangles), CD = AB (opposite sides of parallelogram), AQ = 2AB
C
D D
C
A
B
C = AB = YX (opposite sides of parallelogram), DC || AB || YX (opposite sides of D parallelogram) Therefore, DCXY is a parallelogram (opposite sides equal and parallel).
X
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9
AKO = ∠AOK (isosceles triangle AKO), ∠ ∠KAO = 45° (diagonals of a square bisect the vertex angles), ∠AKO = ∠AOK = 67 12 ° (angle sum of triangle), ∠BOA = 90° (Diagonals of a square intersect at right angles), ∠KBO = 45° (Diagonals of a square bisect the vertex angles), ∠BOK = 22 12 °, ∠AOK = 3∠BOK
C
B K O
D
A
10 a
B
DBC ≡
DBA (AAS)
AB = CB (matching sides of congruent triangles)
DA = DC (matching sides of congruent triangles)
A
C
D
b
B
AXB ≡ CXB (SAS) So AB = CB (matching sides of congruent triangles) CDX ≡ ADX (SAS) AD = CD (matching sides of congruent triangles)
A
C
D
11
442
C
B
β α
∠DAC = ∠BCA (Alternate angles, AD || BC) ∠ACD = ∠CAB (Alternate angles, AB || CD) ABC =
CD = BA (matching sides of congruent triangles)
AD = BC (matching sides of congruent triangles)
A
∠ABC = ∠CDA (matching angles of congruent triangles)
∠BAD = a + b = ∠OCB Thus, opposite angles are equal.
α
β
D
CDA (AAS)
Thus, opposite sides are equal.
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12
C
B
A
α
β
Let X be the point of intersection of diagonals BD and AC.
β α
We use the results of Question 11. AD = CB
∠DAC = ∠BCA (alternate angles, BC || AD)
∠BXC = ∠DXA (vertically opposite)
BXC ≡
D
DAX (AAS)
CX = AX (matching sides of congruent triangles)
DX = BX (matching sides of congruent triangles)
Thus, the diagonals of a parallelogram bisect each other.
13
C
B
A
A rhombus is a parallelogram.
β α
BC = CD (all sides of a rhombus are equal) BX = DX (diagonals of a parallelogram bisect each other) BXC ≡
α
β
∠BXC + ∠DXC = 180°
D
So ∠BXC = ∠DXC = 90°
Thus, diagonals of a rhombus are perpendicular. C BAD ≡
14 B
DCB (SAS) BD = AC (matching sides of congruent triangles)
A
15
DXC (SSS)
∠BXC = ∠DXC (matching angles of congruent triangles)
D B 2α
Let ∠ABK = ∠DBK = a
Then ∠CBD = 2a (diagonals of rhombus bisect the vertex angles)
∠BKA = 180° – (180° – 4a) - a (angle sum of triangle) = 3a = 3∠ABK A
16
α α
C
∠BAK = 180° - 4a (co-interior angles AD || BC)
K
D C
B
X
60°
∠CBX = ∠CDY = 90° + 60° = 150° ∠XAY = 360° – (90° + 60°+ 60°) = 150°
XBC ≡
CDY ≡
XAY (SAS congruence test)
ence, XC = CY = YX (matching sides of congruent H triangles)
60° A
60°
60°
D
Y
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17
B β
18
α
A
C
α
2a + 2b = 360° (angle sum of quadrilateral ABCD) a + b = 180° so ∠ABC + ∠BAD = 180° so ∠BCD + ∠ADC = 180° Thus, AB || OC and BC || AD (co-interior angles are supplementary).
β D
B
C
BC = AD and AB = DC BCD ≡
Hence,
∠ABD = ∠CDB (matching angles of congruent triangles)
∠ADB = ∠CBD (matching angles of congruent triangles)
A
D
DAB (SSS)
Hence, AB || CD (alternate angles are equal) and BC || AD (alternate angles are equal)
Exercise 8C 1
a
ABC is similar to
RQP (SSS)
b
LMN is similar to
QRP (AAA)
c
STU is similar to
XZY (AAA)
d
LNM is similar to
TSU (SAS)
e
BCA is similar to
ECD (AAA)
f
GHF is similar to
JHI (SAS)
g
OMK is similar to
LMN (AAA)
h
STR is similar to
SUQ (AAA)
EDC is similar to
ADB (SAS)
j
IFG is similar to
IHF is similar to
i
2
a AAA
4
b a = 49° a = b, b = 131 - b EC ED DC c x = 2 DBA (SAS) b a DEC is similar to = = BA DB DA a ∠ACB = ∠DCE, ∠CAB = ∠CED, ∠ABC = ∠EDC b 3.75 cm c 7.2 cm
b x = 3.5
a
JKL is similar to
IGH (SSS)
7
a
ADE is similar to
ACB (AAA)
8
a AAA
5 6
FHG (AAA)
b y = 14.4
3 a SAS
b 8 47 cm
b LM = 16 23 cm 10 130 m
12 33 m 11 26 23 m 13 ∠BAE = ∠BDC (alternate angles, AE || DC), ∠BEA = ∠BCD (alternate angles, AE || DC), ∠ABE = ∠CBD (vertically opposite), ABE is similar to DBC (AAA)
9
18.75 m
2 CD DB = = , DB = 2AB 1 AE AB
14 ∠DOC = ∠AOB (vertically opposite), ∠ABO = ∠CDO (alternate angles, AB || CD),
AOB is similar to
15 a ∠BAD = ∠BAC (common angle), ∠ADB = ∠ABC = 90°,
ABD is similar to
ACB (AAA)
b ∠BCD = ∠BCA (common angle), ∠ABC = ∠BDC = 90°,
BCD is similar to
ACB (AAA)
16 a ∠BAP = ∠DCP = 90° and ∠APB = ∠CPD (given),
BAP is similar to
DCP (AAA)
AB AP x b qx = , = ,y= CD CP y q b
b
17 a ∠AEP = ∠CBP (alternate angles, CB || AD), ∠EPA = ∠BPC (vertically opposite),
COD (AAA)
OB OA = = 2 , OB = 2OD OD OC
APE is similar to
CPB (AAA)
AP AE AE (ratios of matching sides of similar triangles) = = PC AD CB AD AE AE 1 = = = , and AD = CB (opposite sides of a parallelogram) PC CD CB 2
b
AP = 2PC and AC = 3AP 18 a ∠EDB = 180° - ∠ACB (given), ∠ADE = ∠ACB (supplementary), ∠BAC = ∠DAE (common), ACB (AAA)
444
b
ADE is similar to
AE AD (equal ratios of matching sides of similar triangles) = AB AC
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19 a ∠PDB = 90° (given), ∠AFB = 90° (given) ∠DBP = ∠ABF (common). PBD is similar to ABF ≡
ABF (AAA)
ACF (RHS) So
PDB is similar to
ACF
FC AC (matching sides of similar triangles) = DB PB
b
1 1 AC and NM = AB. But AC = AB. Therefore, LM = NM. 2 2 21 a ∠ABC = ∠ADB = 90°, ∠BAC = ∠BAD (common), ABC is similar to b a c = = b . Therefore, a(a - y) = c2 (1) x c a− y
20 LM =
c ∠ABC = ∠BDC = 90°, ∠BCA = ∠BCD (common), c b a d = = , so ay = b2 (2) x y b
e From (1)
From (2)
ABC is similar to
ADB (AAA)
BDC (AAA)
a(a - y) = c2 and a2 = c2 + ay ay = b2 a2 = b2 + c2
Exercise 8D 1
B
Let N be the midpoint of BC.
1 1 AB and NB = CB. 2 2 MB 1 NB 1 = and = Hence, AB 2 CB 2
2
M
A
Let M be the midpoint of AB.
MB =
N
MBN is similar to
Hence, ∠BAC = ∠BMN (matching angles of similar triangles)
C
MN || AC (corresponding angles equal) 1 1 Also, MN = AC (similarity factor of ) 2 2
B
∠BMN = ∠BAC (corresponding angles, MN || AC) BMN is similar to BAC (AAA) BM = 1 BA 2 BM BN 1 = = (matching sides of similar angles) BA BC 2 Thus, N is the midpoint of BC.
3
A
N
C
B
M is the midpoint of AB. N is the midpoint of BC.
γ
M
Let M be the midpoint of AB and MN || AC.
M
ABC (SAS)
P is the midpoint of AC. β
α γ β
α
From Question 1, 1 1 1 MN = AC , MP = BC , PN = AB 2 2 2 Therefore,
N
A
MP = BN = NC
Hence, all the triangles are congruent by the SSS test.
All triangles have angles a, b, and g.
herefore, each of the small triangles is similar to the large T triangle.
α
β
γ P
α
γ β
C
NP = MB = AM MN = AP = PC
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4
5
C N B Q M A
Join vertices B and D.
In
CBD, NQ || BD (Question 1)
In
ABD, MP || BD (Question 1)
Therefore, NQ || MP.
In a similar way, MN || PQ.
Thus, MNQP is a parallelogram.
D
P
A 12
In PAQ and BAC ∠PAQ = ∠BAC (Common) BA AC 7 PA = AQ = 4
4
P
PAQ is similar to BAC (SAS) ∠APQ = ∠ABC (matching angles of similar triangles PQ || BC (corresponding angles are equal)
Q
9
3
B
6
Let M, N, Q and P be the midpoints of AB, BC, CD and DA respectively.
C
QB = 10 and PQ : BC = 3 : 8
A 6 Q
9 P
B
15 C
7
a
B
A
S
C
SAC is similar to
SBD
SA SC = SA + AB SC + CD
SA × (SC + CD) = SC(SA + AB) D
SA × SC + SA × CD = SC × SA + SC × AB
SA × CD = SC × AB SA = SC AB CD SA : AB = SC : CD
b
SAC is similar to
SBD
SB SD = SB − AB SD − CD SB × (SD - CD) = SD × (SB – AB)
SB × SD - SB × CD = SD × SB – SD × AB SB SD = . That is, SB : AB = SD : CD. so SB × CD = SD × AB, AB CD SA SC = c or SAC is similar to SBD. SB SD SA : SB = SC : SD.
446
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8
SA SC = AB CD
A
SA × SC + SA × CD = AB × SC + SC × SA
SA × (SC + CD) = SC(AB + SA)
S
C
B
SA × CD = AB × SC
SA × SD = SC × SB SA SC = SB SD
D
Therefore, SAC is similar to Hence, AC || BD.
SBD (SAS).
Review exercise 1
a = 37°, b = 80°, g = 20°
2
a
3
a In
b i 2
4
a a = 112°
b a = 78°, b = 58°
c a = 20°
d b = 104°, a = 76°, g = 52°
e a = 36°, b = 144°
f a = 130°, b = 50°, g = 130°
PCQ is similar to BEC and
to
5
b x = 5
ACB
FED, ∠CBE = ∠DFE (alternate angles, BC || AD) ∠CEB = ∠FED (vertically opposite),
BEC is similar
FED (AAA) ii 4
a ∠BAC =
70°,
∠ACB =
55°
b ∠FIC = 55° (corresponding angles, FI || AB). ∠FCI = 55°. So c ∠DIB = 55° (corresponding angles, AC || DI).
FIC is isosceles.
DBI is isosceles (equal base angles), DB =DI
d ∠GIF = 53°, ∠EFG = 17° ONM ≡
6
a
7
a 6x
8
b
ZYX (AAS)
ABC ≡
b x =
AOB is similar to
9
RTS (RHS)
3 2
COD (AAA). Ratio of matching sides gives:
B
F
BO AO = OD OC
In CFA and BEA, ∠CFA = ∠BEA = 90° (altitudes), ∠FAC = ∠EAB (common), CFA is similar to BEA (AAA) BE AB (ratios of matching sides) = CF AC
A
E
C
b x =
10 a x = 10 11 a x = 13, EF =
102 11
c x =
21 4
d x =
9 2
119 12 20 cm 13
Challenge exercise Only outlines of proofs are given in the challenge exercise b 50 55 cm
c 16 55 cm
Z
1
a 272 cm
2
Produce AY to Z. Let ∠BAY = a ∠AZC = a (alternate angles, BA || ZC) Then ∠YAC = a (angle bisector), and ZCA is isosceles So ZC = CA = 2OA Next, AOX is similar to ZCY (AAA) CY ZC = = 2 or CY = 2OX OX OA
d approximately 67 cm Y
B
C
X α A
O D
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3
PQ = BC = AB = CD = x and BP = CQ = 2x
AXB ≡ QXP (AAS) ∴ PX = XB = x = QY = YC
PQYX is a rhombus and XQ and PY are diagonals of a rhombus. Hence, AQ ⊥ DP.
X
A
4
Show that ∠BAD = ∠BDA = ∠ACF = 54°. Hence, ABD is isosceles with AB = BD.
Draw the perpendicular from B to AD. Let X be the point where the perpendicular meets AD. X is the midpoint of AD. AFC ≡
Q
P
Y
B
A F
X
BXD (AAS). Hence, FC = XD and AD = 2FC. B
5
6
448
C
SXY is similar to SPR. 1 1 SX = PS , SY = SR Q 2 2 1 area of SPR Therefore, area of SXY = 4 1 Therefore, area of SPR = area of parallelogram PQRS 2 P X 1 area of parallelogram PQRS Therefore, area of XYS = 8 a ∠BAD = ∠BEC (corresponding angles, AD || EC) ∠DAC = ∠ACE (alternate angles, AD || EC) BAD is similar to BEC (AAA) AEC is isosceles; AC = AE (Base angles of AEC) BD BA = BD + DC BA + AE BD BA BD + DC = BA + AC ( AC = AE )
R Y S
A
B
BD (BA + AC) = BA (BD + DC) BD × BA + BD × AC = BA × BD + BA × DC BD × AC = BA × DC BD BA DC = AC BC BE BA BE b From a, = and = . CD ED AD ED BC BA = CD AD AD CD Therefore, = AB CB
D
E
D
C
D
C
C
B
Therefore,
E
AF AC = c From a, FB BC
BD = BA DC CA
AE = AB EC BC
BD CE AF BA BC AC × × = × × =1 DC EA FB CA AB BC
A
D B
F
A
D
E
C
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7
a B’
C’
8
c
A”, B’
ca
cb
ab
b B”
C”
A’
2
b
cb
ca
a2
a2
A”
ab
ab
B”
d ∠B" A" A' is a right angle. The new triangle is similar to triangle ABC. 1 1 AC = WZ and YZ = BD = XW. 2 2 Therefore XYZW is a parallelogram. Diagonals bisect each other. XY =
A’
b2
C” , C’
f a2 + b2 = c2
e c
B Y X
C
A
Z W
D
Chapter 9 answers Exercise 9A 1
a 16
b 125
c 64
d 27
e 10 000
f 216
2
a 23
b 26
c 34
d 25
e 54
f 35
3
a 13-1
b 7-2
c 13-3
d 2-10
e 3-4
f 11-10
4
a a15 f m9n6
c m15 h 15x5y4
d p11 i 15x8y9
e a6b7
5
a
c a4m3
d
c a5b10c15d 20
5 xy 4 d 4a6b2
e
6
x y a a6b12
b a11 g 8a5b5 3 xy 2 b 2 b x21y35
3b e 9a4b8
4 x3 y f 64a9b6
7
a 18m12
b p 4
c a6b3
d m5n
e a5b2
f p12q
8
1 2 1 f 25
b
1 4
g
1 343
9
a
k
11 a
14 15 1 b 81
1 7 f 6
g
1 2
2x y 1 m 1 4 9
a b 7 x m 2 y
b b
1 4 4
9x y 8 4 3
a b
h m7n6
d
i
1 1000 h 1
d
c c i
y3 64 x
3
15 p3 q
5
p8 n2 m
4
1 9
e
5 3 27 n 8
h
c
g 4
1 3
1 1000 25 m 9
l
a
10 a
8 7
c
d
d
j
y6 8x
6
3 p11 2q
6
a2 8 2
b c
2
f
1 27
11 4 1331 o 125 j
4 3 i 1
2c 2
e
125 8
e
x6 y6 27
f 4x10y10
e
y 2x6
f
k
4 ab 5 3
l
x4 16 1 m 9 n17
n a3b11
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Cambridge University Press
12 a
1 27
b
x y
b
15 a −
1 64 1 x2
−
c − 1 y2
c
1 64
7 208
13
3x 2 y 2 x+ y
14
2 2 d x y + xy 2 2 x + y
e
13 3 x4 1 + 2x2 y + x4 y2
f
x2 y x2 + y
Exercise 9B 1
a 6.3 × 101
e 2.1 ×
b 4 × 10-1
c 6.2 × 10-1
-4
-5
d 7.4 × 103
-2
f 2.6 × 10
107
g -8.6 × 10
h 2 × 1012
-3
l 1.00051 × 10-1
i 9.1345 × 10
j 5.732 × 10
k 3.012 × 10
2
a 2.25 × 107
b $1.5585 × 108
c 6.700 × 10-7 m
3
a 6400
b 92 000
c 0.048
e 7 412 000
f -402
g -0.004 657
h 47.26
4
a 8 × 105
b 1 × 106
c 1.26 × 108
d 2.04 × 106
e 2 × 10-4
f 4 × 10-6
g 1.21 × 10-16
h 2 × 10-21
i 2.5 ×
k 5 ×
106
l 5 × 100
5
a 1.026 ×
c 6 ×
10-7
d 4.34 × 109
6
5.8473 × 109 kg = 5.8473 × 106 tonnes
7
a 2.18 × 1018 km
8
1840 electrons
9
a 5.766 × 102
b 4.73 × 102
c 4.7 × 102
d 5 × 102
e 5.124 × 10-2
f 5.12 × 10-2
g 5.1 × 10-2
h 5 × 10-2
i 1.603 × 103
j 1.60 × 103
k 1.6 × 103
l 2 × 103
m 2.994 ×
n 2.99 ×
o 3.0 ×
p 3 ×
q 5.73 ×
r 4.32 × 105
j 3 ×
101
1027
e 0.003 51
11 2.93 × 10-8 m3
101
e 1.2 × 102
d 0.0087
f 1.6 × 102
b 1.23 × 1023 km
1027
c approximately 8 minutes 18 seconds
1027
1027
b 538
c 9670
d 732 000
f 0.0142
g 372
h 478 000
12 1.82 × 1016
13 a 4.14 × 104
b 2.97 × 109
14 a 20
d 1.5 × 1011 m
10-2
b 5.83 ×
109
10 a 5.60
4
e 0.007
b 500
c 420
d 34 000
f 0.0492
g 475.235
h 0.006 84
105
15 a between 14.5 cm and 15.5 cm
b between 1.995 × 103 kg and 2.005 × 103 kg
d between 4.874 45 × 107 ml and 4.874 55 × 107 ml
c between 18.665 m and 18.675 m
16 a 2820
e 1.24 × 1012
b 42 900
c 3.22
d 11.3
f 1.44 × 105
g 1.93
h 1.13
Exercise 9C 1
a 2
b 7
c 5
d 11
e 3
f 3
g 2
h 10
i 5
k 5
l 7
2
1 a 2
j 2 1 d 2
3
f 100 000
k
8 125
l
4
a
3
b
5
2
a
1 43
f
3 72
1 5
1 100 c 27 16 h 81
h 1 10 b 16 1 g 4
g 1 5 a 9
450
b
1 c 4 i
125 216 4
3
b
1 13 7
g
3 115
e 243
i 1331
j 14 641
1 1000
n
7 10
c
5
d
4
c
1 54
h
2 10 7
1 8
j 1 10 d 8
m
20
e
d
10
2 117
e e
5
f
1 2
9
2 53
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4
3
a 5 2
b 5 3
c 6 4
7
a
1 3
b
1 2
c
8
a
1 8
b
1 1000
c 4
9
a 2
f 10 12
17
7
h
1 1 2 20
e
1 1000
f
1 11
d
1 625
e
1 100
f
1 8
i
e 10 2
1 3
510
9
2
3
d 312
1
10 a 2 5
b 7
c 2 28
d 7 6
11 a 3.9811
b 8.3203
c 12.748
d 3.7477
e 9.3152
f 2.4150
g 2202.7
h 2.4721
11 m12
11 a 15
12 a
b
c
8
5 b 21
g
m 4 m 2
h 4 m 5
5
i
3
n
f 112
1 10
7
c 7 5
g 315
e 5 3
d
3
b 315 11
d 7 5
1 11
5
7
6
5
6
3 11 x 4 y 15
d
1
j
3 27m 2
e 2
10 5 a 11 b 6
f
1
e a 2
f
a2 625
k
7 m12
c 1.130
d 0.9547
e 0.010 46
b m6
c p3.6
d b1.25
e 4p2.6
b1.6 2a
0.7
h
2 n8 3m
4.2
i
b 5.1 a
0.8
1 13
80
14 a a4.8
g
1 m6
13 a 8.586
f 64p6.3
2
10 7
8m3 b 458.2
1
j
l m 6
f 0.001406
1 2.1 5.8
m n
Exercise 9D 1
a
x 2x
-2 0.25
-1 0.5
0 1
1 2
2 4
y
y = 2x
8 7 6 5 4 3 2 1 –2
b
x 2-x
-2 4
-1 2
0 1
1 0.5
–1
1
2
1
2
x
y
y = 2−x
2 0.25
0
8 7 6 5 4 3 2 1 –2
c
x 4x
-2 -1 0.0625 0.25
0 1
1 4
2 16
–1
0
y
y = 4x
x
30 20 10
–2
–1
0
1
2
x
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d
x
-2
-1
0
1
2
5-x
25
5
1
0.2
0.04
y
40
y = 5−x
30 20 10 –2
2
y = 3 × 4x y = 2 × 4x y 5
y=
4
3
0
–1
1
y = 2 × 2x y = 2x y
x
2
4
y = 5−x y=
4x
y
3−x
y = 2−x
3 2
2
1
y=
1
x
0
1 2
× 2x
1 2
−2
−1
0
1
1
x
2
x
0
Exercise 9E 1
a 3
b 9
c 5
d 2
e 3
f 2
g 3
h 2
i 5
j 3
k 6
l 4
m 9
n 12
o 5
p 3
2
a -4
b -4
c 0
d -3
g -5 1 a 2
h -10
i -11
3
5 3
4
a
5
a 5
b
3 2
c
3 2
d
2 3
e
3 2
j -6 2 f 3
b
7 2
c
5 7
d
3 4
e
1 3
2 f − 3
1 g − 3
b -2
c
4 3
d
2 15
e
3 2
f 6
g −
g
2 3
e -5
f -3
k -4 3 h 4
l -5
h −
5 4
h
3 2
3 2
6
a 4 and 5
b 2 and 3
c 5 and 6
d 2 and 3
e 4 and 5
f 2 and 3
g 1 and 2
h 2 and 3
i 3 and 4
j -1 and 0
k 0 and 1
l 1 and 2
Exercise 9F 1
a
t y
0 200
1 400
2 3 4 5 800 1600 3200 6400
b y 7000 6000 5000
4000
c i 303.14 ii 918.96 iii 2262.74
3000 1000 200
0
2
a
t y
0 200
1 100
2 50
3 25
4 5 12.5 6.25
1
2
3
4
5
t
5
t
b y
c i 131.95 ii 21.76 iii 8.25
200
y = 200 × 100
0
452
y = 200 × 2t
2000
1
2
3
()
4
1 t 2
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3
a i 60
4
a 8.2007 × 109
ii 3840
iii 10 861 b i 1000 b 1.360 × 1011
ii 100
iii 1
iv 0.1
5
a 63.2°
b 42.0°
6 a 80 000
b 86635
7
a 1000
b i 250
ii 125
iii 31
Exercise 9G 1
a log2 8 = 3
d 34 = 81 is equivalent to log3 81 = 4
e 53 = 125 is equivalent to log5 125 = 3
f 73 = 343 is equivalent to log7 343 = 3
g 25 = 32 is equivalent to log2 32 = 5
h 104 = 10 000 is equivalent to log10 10 000 = 4 2-1
b log10 100 = 2
c log7 49 = 2
i 10-3 = 0.001 is equivalent to log10 0.001 = -3
= 0.5 is equivalent to log2 0.5 = -1
j
2
a 2
b 6
c 7
d 12
e 0
f 8
g 3
h 2
3
a 3
b 4
c 3
d 2
e 3
f 0
g 4
h 3
4
a -2
b -1
c -2
d -2
e -3
f -5
g - 4
h -3
c 3
d 5
e 100
f -2
g -7
h -13
5
a 1
b 0
6
1
7 0
8
a 3
b 1.6021
c 2.8971
d -3.5229 e 7.8573
9
a 6
b 5
c -4
d -3
e 27
i 2
j 4
k 5
l 4
m 2
f -3.9085 g -2.2680 h 56.8028 1 1 h f 25 g 5 8
Review exercise 1 2
a a17 1 a 16
3
a
4
a
g
5 6
1 a8 1 52
b 1296m16 b 6
b
b
1 3
c a6b2 1 c 10 000
h
14 a3 1 34
c
c
1 5a
4
4 a2 1 a6
81 16
d
1
4a6 1 d 3 b
i 2x2
d
e
j 4x3
k
2 x4
4b3 a
f
2
l
2x 1 a 36
1 b 64
1 c 64
1 d 16
1 e 5
1 f 100
1 g 49
h
1 64
a 1
b 5
c 1
d 7
e 1
f 5
g 1
h
1 3
7
a 8ab
b ab5
8
22n - 4
9 63x
−
1 6
c 2b
11
1 d 32a
7
e
9 a2
5 x3 5m3 6
i 1
3
b a 20
c 2 3
d 2 5
11 a 4.2 × 103
b 6.2 × 10-3
c 7.4 × 108
d 2 × 10-7
12 a 5400 e 0.18
b 112 000 f 0.000 064
c 0.068 g 7 410 000
d 0.0097 h 402
13 a 63 000 000
b 1 400 000 000
c 0.000 08 m
d 0.003 cm
14 a 20 f 0.0492
b 500 g 480
c 420 h 600
d 34 000 i 0.09
e 0.0068 j 0.00684
15 a 3 g -2
b 4 h -3
c -2 i -4
d 0 j -2
e 2 k -3
16 a 1
b 2
c 3
d 4
e 5
f 6
h 15
i -1
j -2
k -3
l -6
10 a 2
g 10
j 4
a4 f 2
f 3 l -2
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press
5 17 a − 3
7 b − 3
18 a 12
b
19 a
37 b 2 3a
4
20 a 32 5 5 21 a 8400
f
1 2
1 c − 5
d -2
c 4
d
37 a 8 27 b 9
e -5
1 128
b 1 x10
c
b 37 = 2187
c 1
d 72 = 49
g 5
h 3
i 10
e
1 2
e
1 10
3
1 6
5 7
c 8000 × (1.05)n
b 9261
x 16 a 2306, 2308, 2312, 2320, 2336, 2368, 2432, 2560 1 c x = 5, y = 3 a x = 3, y = 2 b x = , y = 2 2 a 120x
1 a3
+
b 128
1 b3
+
1 c3
a
9
a 3a − 2a 3 b 2 − a 3 b 1
1
1 1
c
b x + x −1 +
8
−
1
f 7 2
d ab4
Challenge exercise 1
f 6
1
−
1 2
1
10 3 3 > 2 2 > 5 5
2 -8
4 -1
b n = 12 d x = 3, y = 7
1 32
2
−
2 3
1
− 6a 3 b + 4 a 3 b 2 + 2b 2 b a - b2 1 11 xy 12 5
13 18 063
Chapter 10 answers 10A Review Chapter 1: Consumer arithmetic 1
$4500
2 $5100
3 3 years
4
3 years
5 8.25% p.a.
6 8.7% p.a.
7
a $239.40
b $72
8 a $30.60
b $102
9
20%
10 30%
11 a $24
b $120
b 103.5
c 36
d 20.8
f 150% increase
g 253.75
b 15.5% increase
c 4.32% decrease
d 19% decrease
15 a 20% increase
b 5% increase
c 20% decrease
d 15% decrease
16 $56 314.12
17 a $19 042.49
b $19 301.25
c $19 362.03
12 a 108
e 25% decrease
13 a 4% decrease 14 11.384% increase
Chapter 2: Review of surds
454
1
a 5 2 + 2 3
b 7 5 + 4
2
a 3 2
b 8 2
c 24 2
d 9 3
3
a − 2
b 26 3
c 16 3
d 8 7
4
a 4 3 − 4
b 2 2 − 13
c 13
d 5 − 2 6
5
a 8
b a - b
c a + 2 ab + b
6
a
7
a
6 7 + 2 10 3
b b
3
c
3+5 3 3
13 + 5 6 19
c
4 3+4 2 + 2
2 +
d
15 − 5 5
6 +2
e 7 5 + 18 2
f 2 a
e 2 15 + 2 2
f
d
14 2 − 9
2 2 −1 3
5
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Chapter 3: Algebra review 1
a 9mn2 - 3m
b 0
2
a 5a + 4
b 2b + 21
c
e 2x2 + 11x + 5
f 6a2 + 17a - 14
g y2 + 5y + 9
3
a
6x2
+ 33x + 45
b 12a2
4
a
5 a + 1 4
b
e
2x2 7x 1 + + 3 6 4
f −
5
a x =
17 2 10 f y = − 3
c 7ab2 + a2
- 10a - 8
7 7 b+ 6 3
c c
4 y 2 15 y 73 + + 9 4 12 10 b x = − 3 g x = 16
6x2
+ 7y - 14
2
2
11 3 79 h x = 7
a (x - 5) metres
b length 17.5 m, width 12.5 m 8 4 litres
9
a x > 4
12 a g =
d −
y 7y2 − 10 10
38 3 65 x = j 2 e x =
i y = -9
c x ≥
d 7b2 + 37b + 1
d y = -4
11 6 g x > 3
7 4 f x ≤ -5 b x ≤ −
t2
h b2 + 23b - 1
c x = −
24 km/h
2(s − ut )
d 17y - 5y2
11x 31x + 12 12
6
b a =
- 2x
10y2
7
17 e x ≤ − 7 11 a s = 42.225
d 10p2q2 - 5p3
d x < 37 10 9.61 × 106 watts 2
4π p T2
b g = 9.8
v 2 − u2 13 a a = 2x c−b 14 a x = a yc f x = y−c
b x =
c x =
d x =
e x = ya2
15 a x2 - 25
b x2 - 4
d 9x2 - 1 1 2 4 2 x − y i 16 9 d (2a - 3)(2a + 3)
e 4b2 - 49
b (x - 4)(x + 4)
c a2 - 9 1 2 h a − 1 4 c (a - 8)(a + 8)
f (9b - 1)(9b + 1)
g (x - 2y)(x + 2y)
h (3x - 2y)(3x + 2y)
b 3x(x - 6)
c 4(a - 4)(a + 4)
d 9(a - 2)(a + 2)
f 3(2b - 3)(2b + 3)
g 5(x - 2y)(x + 2y)
h 7(2x - 3y)(2x + 3y)
f x2 - y2
16 a (x - 6)(x + 6)
e (3b - 5)(3b + 5)
17 a x(x - 18)
e 2(3b - 5)(3b + 5)
b v = 7.9 c − ab a n− p g x = m2
g 25x2 - 4y2
cd − b a − (a + b) h x = a
c−b r −t 3y − 1 i x = 2
1 1 1 1 i 6(3a - 2b)(3a + 2b) j x − y x + y k 2 x − 2 y x + 2 y 3 3 2 2 2 1 2 1 l 3 x − y x + y 2 5 2 5
18 a (x + 2)(x + 3)
b (x + 1)(x + 6)
c (x + 2)(x + 6)
d (x + 1)(x + 12)
e (x - 1)(x - 2)
f (x - 2)(x - 5)
g (x - 1)(x - 5)
h (x - 3)(x - 6)
i (x - 6)(x + 1)
j (x - 5)(x + 2)
k (x - 4)(x + 2)
l (x - 7)(x + 3)
19 a (x + 2)(2x + 3)
b (3x + 1)(x + 6)
c (5x + 4)(x + 3)
d (3x + 2)(x + 6)
e (2x - 1)(x - 2)
f (3x - 10)(x - 1)
g (5x - 1)(x - 5)
h (7x - 9)(x - 2)
i (3x + 2)(x - 3)
j (2x - 5)(x + 2)
k (5x + 4)(x - 2)
l (2x + 3)(x - 7)
20 a (2x + 1)(2x + 3)
b (3x + 2)(2x + 3)
c (4x + 3)(x + 4)
d (2x + 1)(4x + 3)
e (2x - 5)(2x - 3)
f (2x - 5)(3x - 2)
g (2x - 5)(4x - 1)
h (5x - 6)(2x - 3)
i (2x - 5)(2x + 3)
j (2x - 5)(3x + 2)
k (4x + 5)(x - 2)
l (4x + 5)(2x - 3)
b 5(x + 2)(x + 3)
c 4(x + 1)(x - 4)
d 3(x - 1)(x + 5)
21 a 2(x + 3)(x - 7)
e 3(2x + 5)(x - 1)
f 4(2x + 1)(x + 2)
g 3(2x + 3)(3x - 1)
h 2(2x - 3)(4x + 3)
i 3(x + 7)(x - 5)
j 5(2x + 1)(x - 6)
k (4 - x)(x - 6)
l -2(3x + 4)(x + 1)
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press
22 a
d
23 a
d
34 x + 13 35
b
5x + 7 ( x + 2)( x + 1)
c
−3 x − 11 ( x + 2)( x − 3)
4x + 5 x ( x + 2)(2 x + 3)
e
x−4 x( x − 1)( x + 1)
f
5x + 9 ( x − 3) ( x + 3)( x + 1)
x+2 x+4
b
( x + 1)( x + 6) ( x − 1)( x + 1)
c
x+5 x−5
( x + 1)( x − 1) ( x + 2)( x + 4)
e
1 x
f
− x ( x + 3) 3( x + 1)
Chapter 4: Lines and linear equations 1
34
c
b 2 2
a 2 13
d
2
a (3, 2)
b (4, 3)
3
2 a 3
b 1
1 7 c − , 2 2 3 c 5
4
a m = 2, c = -1
b m = 1, c = 3
c m = -1, c = 7
d m = -1, c = 4
f m =
5
a i 3
2 1 e m = − , c = 3 3 1 ii − 3
c i
6
a
1 2
y
b
0
x
y
e
0
3
0
y
x
f
y
x (1, −2)
y
h
x
(1, 4) 0
−3
0
x
(−1, 2)
4
g
y
2
3 2
0
c
3
− 12
d
1 2 3 ii 2 ii
2 d i − 3
y
1
3 1 ,c=− 4 2
b i -2
ii -2
34
5 1 d − , − 2 2 5 d 3
0
x
y
x 0
3
x
−4
7
a y = 3x + 2
c y = -x - 3
8
a 3x + 2y = 6
b 2y - x = 2
c y = 3
d x = 5
e y = 2x - 1
f y = 1 - x
b x = 1, y = 1
c x = 6, y = -5 1 44 f x = , y = 13 13
a x = 2, y = 3 1 2 d x = , y = 2 3 10 a = 3, b = 2 9
456
b y = 2x + 1
e x = 6, y = -5
d x + 2y = 8
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b i a - 2
11 a 3
c i
2 a
ii
d i
2 5
ii y =
e i a 2 + 4 12 a 31 000 L
2 4−a 2 x + 1 5
5 iii , 2 2
ii 3 5 b 40 L/h
c V = 31 000 - 40t
13 a i
ii (a - 2, 1)
d after 775 hours
A started in Melbourne, B started 100 km from Melbourne.
ii A finished 160 km from Melbourne, B finished in Melbourne.
iii A took 2 hours 20 minutes, B took 1 hour 36 minutes.
iv A averaged 68 47 km/h, B averaged 62.5 km/h. 480 b A: d = t, B: d = -62.5t + 100. 7 c They passed after 46 minutes of travelling.
Chapter 5: Quadratic equations 1
a -4, 4
f -2, 2
2
3 a 0, 2
b -8, 8 1 1 g , − 2 2 1 b 0, − 3
g 0, -5
h 0, 7
a a = 4 or -3
b t = -3 or -5 1 11 b , 3 3
3
5
5 a , 7 2 1 e − , 1 3 a 3
6
a ( x +
d ( x − 4 +
f 2( x − 3 – 5 )( x − 3 +
5)
7
a y = −1 ±
b a = 2 ±
e y = −
8
a d = ± 2
b y = ±
d y = 4 ± 13
e m =
9
a (x + 7) metres, (x + 8) metres
4
b 2( x − 6 )( x +
5)
7 )( x − 4 −
5
1 2 ± 4 4
10 a 381.6 m
f − 3 , 5 2 6 c -6
b -5
5 )( x −
c -2, 2 5 5 h , − 2 2 3 c 0, 5 3 i 0, 4 c m = -7 or 3 1 c , −8 3 3 g , −4 2 d 7 e -1
7)
f x =
12 a
13 a b = ±4 14 a a =
4 3
d 0, -3
1 2 1 k 0, 2 e x = 4 e 0,
j 0, 12 d m = 4 or -1
f 0,
2 7
1 4 f b = 9 or -3 l 0, −
7 3 d − , − 6 2 h 3 f 4
6)
3 g − 2
h 5
c 6( x − 2 3 )( x + 2 3 )
e ( x + 2 − 2 2 )( x + 2 + 2 2 )
6
1±
33 4
30 2 1 5 ± 2 2
41 2
c x = 1 ± g n =
5±
41 2
d x = h x =
7±
41 2
−1 + 2 4
c x = 10 ± 2 5 f n =
3±
21 2
b 5 cm, 12 cm, 13 cm b 102 seconds
11 a length = 24 - 2x cm, width = 17 - 2x cm x2 m2
5 5 e − , 2 2
d -7, 7 3 3 i , − 2 2
b (8x + 16)
m2
b x = 2.5
5x 2 c 8 x + 16 = 4 c -4 < b < 4
b b < -4 or b > 4 4 4 c a > b a < 3 3
15 a c = 4
c 19 cm × 12 cm × 2.5 cm d 8 m × 8 m b c < 4
c c > 4
Answers to exercises ISBN 978-1-107-64844-9 © The University of Melbourne / AMSI 2011 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press
Chapter 6: Surface area and volume 1
a 2200 cm2 cm2
b 6000 cm3
2 5 cm
b 728 cm2
c 880 cm3
5 a 4712 litres
b i 96π cm2
ii 96π cm3
8 a 4 cm
b 8 6 cm
c 328 cm3
c 2, 8
d 6, 216
e 5, 25
4
a 44
6
a i 360 cm2
ii 400 cm3
7
a 254.6 m2
9
a 18π m2
b 160 m3 45 b π m3 4 b 2.25, 3.375
10 a 9, 27
3 5 cm b 64 cm
f 9, 81
Chapter 7: The parabola 1 b − , 6 2 a y = (x + 3)2 - 6, (-3, -6) 2 3 7 3 7 c y = 2 x + − , − , − 2 2 2 2
c −4 +
3, −4 − 3
b y = (x -
2)2
3
a y = -2x2 + 6x + 8
b y = x2 - 8x + 15
4
a y =
2x2
5
a y =
3x2
6
a
1 2
a -1, -3
- 12x + 16 - 12x + 12
- 2, (2, -2)
b y = -3x2 + 12x - 7 -x2
- 2x + 2
b
c y = x2 - 10x + 21
y
5
5
2
2
1 25 −2, 4
0 1
2, 2 −
4 10 4 10 d y = 3 x + − , − ,− 3 3 3 3
b y =
y
d 2 +
c
y 4
6
−2
x −3
2
0
2
0
x
x
(3, −4)
d
y
e
y
f
y
9
0 −1
−3
g
0
3
−√2
x
−3
x
√2
0
x
−2 (1, −4)
y
h
y
1
−1 − √2
y −5−√37 2
(−1, 2)
i
x
0
−1 + √2
0
−5+√37 2
x
−3
11 0
5
− 2, −
(3, 2)
37 4
x
458
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Cambridge University Press
j
11
361 24
− 12 ,
y
k
y
l
y
1 + √57 4
6 1 – √57 4
10 7
−
5 2
− 8,
2 3
−7
47 16
0
x
0
7
1 4 a , − 3 3
1 b − , 3
9
a 12 - 2x
b A = x(12 - 2x)
10 a (-1, -8)
0 , (1, 0)
1 4,
x
)(
)
b (0, −7), −1 − 2 2 , 0 , −1 + 2 2 , 0
d −2 2 − 1 ≤ x ≤ 2 2 − 1
57 8
c 3 m by 6 m c
−
8 5 cm × 10 cm × 20 cm
(
x
0
y −2√2 − 1
2√2 − 1 x
0
−7 (−1, −8)
11 a
y
1 − √3 2 1 0
3 3 < x
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