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612 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
Answers
Chapter 1Start up
1 a 3 b −6 c 25 d −48e −8 f −10 g 0 h 9i 3 j −32 k 2 l −2
2 a b c d e 1
3 a 7400 b 158 c 0.3 d 0.0301e 0.535 f 0.0072
4 a 48% b 66 % c 175%
d 10.8% e 8.45%
5 a b 1 c d e 2
6 a $8.70 b $24 c 14 kg d 32 Le $15 f $28
7 a 1 : 1 b 2 : 1 c 1 : 2d 1 : 30 e 13 : 12 f 5 : 1g 1 : 10 h 1 : 2 i 1 000 000 : 1j 3 : 2
8 a 7 : 5 b 3 : 8 c 27 : 40d 18 : 13 e 3 : 2 f 3 : 1g 4 : 7 h 4 : 3
9 a 9 : 8 b 21 : 16 c 4 : 5d 5 : 6 e 25 : 1 f 1 : 4g 5 : 6 h 10 : 3
10 a 7 : 5 b 1 : 5 c 20 : 1 d 2 : 1511 a 4 : 3 : 6 b 6 : 3 : 8 c 17 : 5 : 1312 a 3 : 8 b 3 : 4 c 9 : 80
d 12 : 35 e 16 : 5 f 9 : 50
Exercise 1-011 a b − c d
e − f
2 a 2 b 1 c −4 d -1
e 1 f 2
3 a b = 1 c − = −1
d − e f −
4 a 29, 30, 31, 32, 33, 34b 33, 34, 35c 13, 14, 15, 16, 17d 49, 50, 51, 52, 53, 54, 55
5 a b c d e
6 a 1 b 3 c 6 d 4
e 2 f 1 g 2 h 6
7 a b c 2 d 6 e 22
f 3 g 7 h 17 i 8 j 8
8 a 6 b c 1 d 1 e
f g h 2 i j 2
9 a b c 4 d 9
e 12 f − g h 2
10 Any number greater than and less than
, such as
11 56 12 2 times
13 2 minutes or 2 minutes 11 seconds
(to the nearest second)
14 15 15 4 cans 16 $180
Skillbank 1A2 a 3.5 b 2.4 c 0.12 d 0.36
e 0.8 f 0.027 g 0.2 h 8.8i 0.24 j 0.012 k 1.8 l 0.028
4 a 66.3 b 6630 c 6.63 d 0.663e 6.63 f 663 g 0.663 h 663i 6630 j 66.3 k 0.663 l 0.0663
Exercise 1-021 a 56.37 b 0.1 c 636.0 d 3.765
e 19 f 2.47 g 8.00 h 244.00i 0.7 j 24.371 k 20 l 3.092
2 a 3.90 b 12.6 c 36.843d 0.6154 e 245.6615 f 7.37g 8.1 h 54.87 i 2.9861
3 a 56 000 b 19.71 c 1 600 000d 30 000 e 0.0079 f 1.00g 3.01 h 0.31 i 8 000 000j 0.0520 k 4 l 81
4 a 3 b 3 c 2 d 4 e 2f 2 g 4 h 5 i 2
5 a 4000 b 2.72 c 0.04729d 0.94 e 3 360 000 f 0.002
6 1 7 43 5008 a 6091.2 cents b $60.90
c i 2 ii 39 B 10 B 11 A
Exercise 1-031 a , recurring b 1.075, terminating
c 0.32, terminating d , recurring
e 1.375, terminating
f , recurring g 4.6, terminating
h , recurring i , recurring
j 0.85, terminating
2 a b c 5 d e
f 3 g h i 3 j
3 a 0.79 b 0.33 c 0.666 d
e f 0.55
4 a b c d e
f g h i j 1
Skillbank 1B2 a 0.5 b 90 c 8 d 0.9
e 90 f 0.7 g 8 h 30i 0.08 j 3.5 k 4 l 0.3
4 a 16 b 160 c 1.6 d 1.6e 1.6 f 16 g 160 h 0.16i 0.016 j 160 k 0.16 l 0.0016
Exercise 1-041 a 1 b 3 c d e 4
f g h i 1 j
2 a 0.37 b 0.08 c 0.042d 1.15 e 0.0004
3 a 32.5% b 1.5% c 66 %
d 87 % e 180% f 325%
g 133 % h 65% i 60%
j 68%4 a $24 b $161 c 36 kg
d $5100 e $35.70 f $2.985 a $50.40 b $1500 c 13.5 kg
d 1.827 cm e 97.2 kg f $1940.136 a $101.20 b $20.91 c 90.2 kg
d $921.38 e 12.9 L f 132 m7 a $1406 b $12588 $37.50 9 $70 157.50
10 $20.05 11 $831.25
12 a % b 8% c 76%
d 30% e % f 30%
g 7.5% h % i 7.1%13 28.6% 14 30% 15 10%16 40.6% 17 14.8%18 a 600 b 900 c 500
d e f 68
g h i 10219 a $75 b $75020 $16.32 21 19 44322 a $4347 b $24 15323 $275 24 80 25 $500 26 $560
Exercise 1-051 a 5.6 × 103 b 7.2 × 107
c 7 × 10−2 d 1 × 10−9
e 7.128 × 102 f 4 × 103
g 2.78 × 10−4 h 5 × 10−1
i 9 × 108
2 C 3 D4 a 37 000 b 0.0987
c 0.000 000 8 d 15 760 000e 0.3 f 80 700g 0.000 046 1 h 1 280 000i 0.030 61 j 99 100 000 000k 0.001 l 0.000 000 021
5 a 1.21 × 1013 b 1.44 × 10−15
c 1.76 × 1017 d 2.37 × 10−4
e 4.19 × 10−9 f 2.92 × 1015
g 8.23 × 1023 h 1.21 × 1012
i 1.96 × 10−5 j 9.13 × 10−4
6 a Maximum 2.06 × 108 km,
minimum 2.49 × 108 km
b 2.275 × 108 km7 a 1.5 × 104 b 7 × 106 c 3.5 × 105
d 7.61 × 107
8 a 1.9 × 10−3 kg b 1.66 × 10−24 g
c 1 × 10−6 m d 1.66 242 × 108 km2
e 3 × 108 f 2.817 × 10−15 m
37--- 3
5--- 17
40------ 9
16------ 2
5---
23---
725------ 9
20------ 21
25------ 1
40------ 21
50------
114
------ 138
------ 1003
--------- 214
------
52--- 15
8------
17--- 1
2--- 1
4--- 2
3---
19--- 4
7---
17--- 5
3--- 2
3--- 3
2--- 1
2---
18--- 4
11------ 7
10------
1518------ 12
15------ 3
5--- 4
3--- 8
11------
740------ 1
4--- 1
2--- 4
5---
920------ 5
6--- 23
24------ 1
10------
1528------ 1
5--- 2
5--- 1
2---
13--- 6
7--- 1
2--- 3
50------ 2
5---
23--- 9
10------ 3
4--- 19
21------ 5
48------
12--- 14
45------ 1
7--- 5
8--- 19
25------
1720------ 3
8--- 8
9--- 1
3---
35--- 7
10------ 39
100--------- 16
37------
1520------
1620------ 31
40------
47---
211------
23---
0.7̇
0.7̇2̇
0.3̇
0.46̇ 5.6̇
1750------ 13
20------ 3
10------ 1
250--------- 4
5---
625------ 41
200--------- 1
400--------- 5
8--- 7
20------
1340------
29---
79--- 38
99------ 875
999--------- 22
45------ 1
11------
17198--------- 35
198--------- 2
225--------- 541
990--------- 37
99------
720------ 9
20------ 7
25------ 1
200---------
33400--------- 1
3--- 3
8--- 1
2--- 11
200---------
23---
12---
13---
58.3̇
33.3̇
16.6̇
666.6̇ 545.4̇5̇ 47---
53.3̇ 32.6̇ 67---
ANSWERS 613
Exercise 1-061 a 5 : 6 b 7 : 6 c 9 : 13
d 2 : 11 e 16 : 11 f 21 : 55g 9 : 8 h 1 : 12 i 10 : 1j 3 : 2 k 5 : 6 l 50 : 1
m 50 : 3 n 32 : 15 o 20 : 1p 9 : 4
2 a 1 : 2 b 2 : 3 c 3 : 40d 1 : 10 e 1 : 7 f 1 : 10g 1 : 8 h 8 : 1 i 15 : 1j 15 : 1 k 500 : 3 l 3 : 8
m 1 : 50 000 n 9 : 4 o 1 : 3003 a 3 : 2 : 7 b 6 : 1 : 4
c 7 : 12 : 4 d 3 : 4 : 2e 8 : 1 : 6 f 9 : 8 : 12 : 12g 15 : 4 : 10 h 4 : 7 : 5i 12 : 7 : 10
4 a 48 b 48 c 5 d 7e 72 f 50 g 625 h 36i 9 j 55 k 39 l 4
m 4.8 n 28 o 37.5 p 175q 55 r 20
Exercise 1-071 10802 a 32 years b 38 : 18 = 19 : 93 $45 000 4 $146.705 a 8 cups b 9 cups c 10 cups
d 12 cups
6 a = 48% b 60 white, 40 red
7 a 43 : 1 : 6 b 2%c 45 kg plastic, 270 kg glassd 8.3%
8 a 1000b i 444.44 ii 750
9 972 tickets10 84 STD calls11 75 students12 1.24 kg of nickel and 2.17 kg of copper13 96 kg
14 30 m3 gravel, 22.5 m3 sand, 7.5 m3 cement15 a i 12 : 12 = 1 : 1 ii 18 : 6 = 3 : 1
b 18 g
Exercise 1-081 a 1 : 500 b 1 : 100 c 2000 : 1
d 1 : 50 000 e 1 : 10 f 1 : 5g 250 : 1 h 1000 : 1 i 1 : 200 000j 800 : 1 k 1 : 8 l 1 : 120 000
m 500 : 1 n 1 : 937.5 o 1 : 12 5002 a 1.5 m b 2.4 m c 1.15 m
d 2.75 m e 1.9 m3 C 4 825 m5 a 12 m b 4 m6 a 3.5 mm b 2.65 mm c 2.25 mm
d 1.4 mm e 0.4 mm7 A 8 D 9 B
10 1 : 2 000 000 11 1 : 25 000 00012 a i 20 km ii 37.5 km
b 9.7 km c 41 km d 2.5 km13 a 1 : 20 000 000 b 4000 km14 a 1 : 100 b 80 cm15 a 1 : 1000 b 1 : 2000
Exercise 1-091 a 90 km/h b $36/h
c 13 km/L d 80c/kge 18 mm/h f 76 cents/call
g $34/m h 1250 parts/hi $8.25/bottle j 25 km/day
k 42 words/min l 20 g/m2
m $0.90/min n 6.67 m/so 333.33 mL/s o 11 km/L
2 a $22.50/hour b $787.50c 120 hours
3 a $11.50/kg b $69 c 3.478 kg4 Sydney5 a 2000 L/h b 4000 L6 a 25 min b 4 h 35 min7 a 65 km b 162.5 km c 487.5 km8 85 km/h9 a 12 hours b 70.9 km/h
10 a 250 g b 1500 kg11 a i 255 km ii 495 km
b 5.5 hours c midnight
12 a 1 909 090.9 km2 b 930 769 23113 a $2360 NZ b $A4594
c i $76.18 US ii $129 Fiii 28 154 yen iv 298 200 baht
v 27 727.92 peso vi 610 000d $172e i $1.31 ii $2.38 iii $2.34
iv $5.79 v $819.67 vi $12.43
Exercise 1-101 a 14 L/100 k b 7.5 L/100 km
c 11 L/100 km d 8 L/100 kme 12.25 L/100 km f 12.8 L/100 kmg 12.5 L/100 km h 10 L/100 km
i 4 L/100 km
2 a 500 km b 50 km c 150 kmd 25 km e 47.9 km f 8.3 km
3 a 67.6 L b 39.52 L c 152.88 Ld 6.864 L e 0.26 L f 78 L
4 a 61.5 L b $67.595 a 1 birth/1000 people
b 2.5 births/1000 peoplec 13 births/1000 peopled 1.2 deaths/1000 peoplee 34 births/1000 peoplef 4.5 births/1000 people
6 a 240 000 b 130 000c 5.5 per 1000
7 John uses 324 L, Ken uses 144 L
Exercise 1-111 a 1167 m/min b 5000 m/min
c 22.2 m/s d 26.4 m/se 30.56 m/s f 1000 m/ming 2400 m/min h 1.8 km/hi 1.5 km/h j 252 km/hk 7.2 km/h l 36 km/h
2 a 50 g/cm b 125 kg/h
c 80 mL/g d 0.04 g/m2
e 8.64 kg/day f 54 L/h
g 11.52 t/day h 0.000 010 8 s/mm3
i 200 g/cm2 j 1.25 beats/sk 8 m/mL l 300 mL/s
3 a 4500 m b 9 km/h4 454.55 km 5 66.67 m6 2480 seconds7 a 15 m b 20 m c 27.5 m8 15 per 1000 9 Wonder Gal
10 240 km/h
Power plus1 a Teacher to check
b i 20 100 ii 500 500 iii (1 + N)
2 a i 1 ii 1 iii 2
iv 2 v 4 vi 49
b i + + = 1
ii + + + + + = 3
iii + + + + + + + …
+ = 7
iv 22 v 2475
3 3, 37, 1114 a 3 b 5 c 7 d 9 e 11
f 13 g 15 h 17 i 195 25196 a i 15.5% increase
ii 4% decreaseiii 8% decreaseiv 17.2% decrease
17 16 %
18 a , , , ,
b Depends on the values assigned to the pronumerals, e.g. if x = 3, y = 2, w = 1
then � ; if x = 6, y = 2, w = 1,
then �
9 10 M210 a Timmy b h
Chapter 1 Review1 a − b 2 c d 1
e 3 f − g −3 h 13
i 1 j 3 k 1 l 3
2 a $200 b 60 kg c 90 d
3 a 23.75 b 2.303 c 0.6d −0.8516 e 12.43 f 4.859
4 a 38 920 b 39 000 c 40 0005 9 461 000 000 000 km
6 a , recurring
b 0.575, terminating
c , recurring
d 0.95, terminating
e , recurring
7 a 0.84 b c
8 a $24.96 b $52.80 c 12 kgd 6.12 m e $132.60 f $10.50
9 a 62.5 b $84.80 c $432.32
10 a 1.27 × 105 b 7.01 × 10−2
c 7 × 10−6 d 3.7 × 103
11 a 0.000 000 43 b 87 530c 0.000 005
12 a 8.3 × 1011 b 2.0 × 1013
c 5.1 × 1012 d 1.1 × 10−10
13 a 2 : 5 b 4 : 3 c 1 : 10
1225------
23---
N2----
12---
12--- 1
2--- 1
2---
12--- 1
3--- 2
3--- 1
2---
12--- 1
3--- 2
3--- 1
4--- 2
4--- 3
4---
12--- 1
3--- 2
3--- 1
4--- 2
4--- 3
4--- 1
5---
56--- 1
2---
12---
23---
wx---- w
y---- y
x-- x
y-- x
w----
32--- 2
1---
62--- 2
1---
d720---------
115------ 3
10------ 9
35------ 19
20------
18--- 25
32------ 1
4--- 1
3---
23--- 2
5--- 3
8--- 3
4---
17120---------
0.2̇7̇
1.5̇71428̇
3.6̇2131------ 5
8---
614 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
d 9 : 2 e 3 : 8 f 1 : 3g 2 : 1 h 1 : 2 i 4 : 9
14 a 6 : 4 : 3 b 7 : 2 : 3 c 1 : 6 : 2d 5 : 9 : 6 e 3 : 8 : 6 f 1 : 2 : 4
15 a 4 : 5 = 12 : 15 = 32 : 40
b = =
16 a 180 b 30 c 48 d $16 250e i 240 : 360 ii 100 : 200 : 300
iii 120 : 60 : 240 : 18017 a 1 : 400 b 20 : 1
c 1 : 250 d 1 : 10 000 00018 a 93.33 km b 126.67 km
c 133.33 km d 266.67 km19 a i 36 km/h ii $5.98/kg
b i 36.60 ii 40 220 yeniii 15.25 iv $18.65
20 a i 12 L/100 km ii 17.2 L/100 kmb i 10.5 births/1000 people
ii 6.4 deaths/1000 people21 a 1.5 kg/min b 0.3456 kL/day
c 12.2 L/100 km d 10.53 km/Le 33.33 m/s f 18 km/h
Chapter 2Start up
1 a 115 b 15.7 c 37.7 d 69.9e 201 f 76.1 g 103 h 905i 48.5
2 a 1681 mm2 b 1650 mm2
c 1248 mm2 d 4750 mm2
e 2310 mm2 f 1800 mm2
g 182 mm2 h 770 mm2
i 680 mm2
3 a y = 29 b k = 29c 19
4 a 33 cm, 85 cm2 b 88 cm, 616 cm2
c 396 cm, 12 469 cm2
d 581 cm, 26 880 cm2
5 a 4.913 m; 1.508 m2
b 38.562 cm; 88.357 cm2
c 12.727 m; 8.727 m2
6 a 105 m3 b 480 m3 c 640 m3
Exercise 2-011 a 282 m2 b 204 m2 c 165 m2
2 a 1036 cm2 b 1020 mm2
c 204 m2 d 390 cm2
e 672 cm2 f 5672 mm2
3 a 57 m2 b 22 m c 269 m2
d 61 m2 e 20 m2 f 40 m2
4 a 657.1 cm2 b 2858.8 cm2
c 8314.2 cm2 d 2793.5 cm2
5 a 26.14 m2 b 29 m2
6 35 403 cm2
Exercise 2-021 a 275 m2 b 564 mm2 c 87.4 cm2
2 a 166.4 m2 b 3456 mm2 c 743.1 cm2
3 a 843 cm2 b 1592 cm2 c 3116 cm2
4 a 432 cm2 b 2150 cm2 c 173 cm2
5 a 144 m2 b 150 m2 c 137.8 m2
6 44 436 m2
7 a 1344 mm2 b 180 cm2 c 343.4 m2
8 a 42 m b A = 735 m2
c 35 m d 28 m
Exercise 2-031 a 101 cm2 b 628 cm2 c 2419 cm2
2 a 392.7 mm2 b 62.8 m2
c 192.4 cm2
3 a 90π b 224π c 577π4 a 942 cm2 b 393 cm2 c 1017 cm2
d 402 cm2
5 5525 cm2
6 a b 30.16 cm c 4.8 cm
d 8 cm e 193 cm2
7 a 17.3 cm b 16.5 cm
8 a 7.1 cm b 5.6 cm c 360 cm2
9 a 24.4 cm b 573.1 cm2
10 a 3.99 cm b 27.9 cm c 27.6 cm11 a 6.9 cm b 85.3 cm c 85 cm12 a r = 6.31, l = 11.2, h = 9.25
b r = 10.9, l = 12.8, h = 6.71c r = 13.1, l = 17.0, h = 10.9
Exercise 2-041 a 2827.43 mm3 b 380.13 m3
c 366.44 cm3
2 a 432π m2 b 192π m2 c 768π m2
3 a 628 m2 b 314 m2 c 236 m2
d 1257 m2
4 5.1 × 108 km2
5 a 716.3 m2 b 72 L6 a 6.91 cm b 7.98 cm c 9.77 cm7 a Teacher to check
b i 3800 cm2 ii 8.5 m2
c 6.5 cm
Exercise 2-051 a 857.7 cm2 b 270.9 cm2 c 1186.0 cm2
d 5969.0 cm2 e 250.6 cm2 f 628.3 cm2
g 282.7 cm2 h 652.9 cm2 i 501.6 cm2
j 3769.9 cm2 k 1148.8 cm2 l 3017.7 cm2
m 6615.4 cm2 n 3908.4 cm2 o 328 cm2
Exercise 2-061 a 64.8 m3 b 1534.5 m3
c 13.6 m3 d 135.7 m3
e 107.5 m3 f 146.3 m3
2 a 1539.4 m3 b 14 431.7 cm3
c 226.4 m3 d 4.7 m3
3 a 23 562 cm3 b 4825 cm3
c 5027 cm3 d 1989 cm3
e 536 cm3 f 12 900 cm3
g 33 117 cm3 h 2639 cm3
i 794 cm3
4 a 20 slices b 138 cm2 c 0.3 m2
5 Triangular prism 36.4 m2, half cylinder
28.8 m2.The triangular tent has the greater surface area.
6 2073.5 cm2 or 0.21 m2
7 a 1508 cm3 b 75.4 cm3
8 1028.3 cm3
9 a 29 040 cm3 b 24 Lc 0.0238 kL
10 4948 cm3 11 8.7 m2 12 10.7 cm
13 36 mm 14 3.2 m 15 3375 cm3
16 64 m2
Exercise 2-071 a 149.3 cm3 b 240 cm3
c 120 cm3 d 336 m3
e 1200 cm3 f 106.7 m3
2 a 950.7 cm3 b 41.1 m3
c 2400 mm3 d 840 cm3
e 14 842.7 mm3 f 21.9 m3
g 7883.3 cm3 h 8064 mm3
3 a i 24 cm ii 3200 cm3
iii 3200 mL
b i 20 m ii 19 200 m3
iii 19 200 kL
c i 40 mm ii 4320 mm3
iii 4.32 mL
d i 60 mm ii 28 160 mm3
iii 28.16 mL
e i 7.7 m ii 133.056 m3
iii 133.056 kL
f i 84 cm2 ii 564 480 cm3
iii 564 480 mL or 56.448 L
4 a Volume 343 cm3, capacity 343 mL
b Volume 240 cm3, capacity 240 mL
c Volume 1152 cm3, capacity 1152 mL
d Volume 6100 cm3, capacity 6100 mL
e Volume 6048 cm3, capacity 6048 mL
f Volume 2500 cm3, capacity 2500 mL
5 a 33.75 m3 b 18.98 tonnes
6 a 946 729 m3 b 0.411 m3
7 27 m 8 24 cm2 9 19 mm10 4.9 cm
Exercise 2-081 a 209 m3 b 872 cm3 c 1272 mm3
d 616 cm3 e 393 cm3 f 2545 mm3
2 a 12 566.4 cm3 b 3141.6 cm3
c 18 849.6 cm3 d 4712.4 cm3
3 a Perpendicular height 6.9 cm,
volume 115.6 cm3
b Perpendicular height 27.2 cm,
volume 13786.1 cm3
c Perpendicular height 12.4 m,
volume 378.6 cm3
d Perpendicular height 3.5 m,
volume 2.3 m3 e Perpendicular height 244.6 m,
volume 296 103.1 m3 f Perpendicular height 71.9 cm,
volume 129 674.2 cm3
4 a i Teacher to check. ii 40 mm
iii 3392.9 mm3
b i Teacher to check. ii 12 cm
iii 314.2 cm3
c i Teacher to check. ii 3 m
iii 8.0 m3
d i Teacher to check. ii 72 mm
iii 33 250.6 mm3
5 a 9 cm b 4 cm c 15 cm d 2 cm6 a 5.7 m b 9.7 cm c 9.2 cm
d 8.5 m7 2 cm 8 5.23 mm
23--- 12
18------ 14
21------
35---
ANSWERS 615
Exercise 2-091 a 14 137 mm3 b 697 m3
c 660 cm3 d 3619 m3
e 1072 cm3 f 8578 mm3
2 a 33 510.3 cm3 b 97.7 m3
c 179.6 cm3 d 0.9 m3
e 356.8 cm3 f 4.1 cm3
3 a 4 mm b 7 cm c 3 m
4 a 11 500 mm3 b 5150 cm3
c 409 000 mm3 d 1.07 m3
e 38.8 m3 f 51 000 mm3
5 1.1 × 1012 km3
Exercise 2-101 a cylinder and cone
b 58.6 m3 c 58.6 kL
2 a Volume 31 416 cm3, capacity 3.142 L
b Volume 616 cm3, capacity 0.616 L
c Volume 302 cm3, capacity 0.302 L
3 a Volume 1950 cm3, capacity 1.95 L
b Volume 22 988 cm3, capacity 22 988 L
c Volume 1309 cm3, capacity 1.309 L
d Volume 455 cm3, capacity 0.455 L
e Volume 25 656 cm3, capacity 25.656 L
f Volume 1527 cm3, capacity 1.527 L
4 a Tank A 28.27 m3, Tank B 56.55 m3
b 28.27 kL5 a 12 balls b 60 balls
c 31 416 cm3 d 48%
6 a 1963 cm3 b 0.55 cm3/min
7 a 250 m3 b 210 kL c $205.80
8 1.447 × 1015 km3
Exercise 2-111 a 1 : 16 b 9 : 16 c 25 : 4
d 4 : 492 a 9 : 1 b 9 : 25 c 81 : 25
d 4 : 93 a 3 : 5 b 1 : 10 c 8 : 5
d 4 : 94 a 7 : 6 b 7 : 95 a 1 : 11.56 b 1 : 39.3046 x = 3.5 7 a 1 : 11.56
b 1 : 39.3018 a i4 : 25ii 8 : 125
b i 1 : 4 ii 1 : 8c i 49 : 16 ii 343 : 64
9 a i 3 : 2 ii 9 : 4b i 5 : 3 ii 125 : 27c i Increased 8 times
ii of original length
10 a 162 cm2 b 154 mL
c 25 : 49 d 363.41 cm2
e Increases 9 times
f Decreases by
11 a V1 = 72 cm3 b V2 = 4V1 c V3 = 2V1
d V4 = 9V1 e V5 = V1
12 350% increase
Power plus1, 2, 3, 4, 5, 6 Teacher to check.
7 a b
Chapter review1 a 5236 cm2 b 277.7 m2
c 104.3 m2 d 14 294.4 cm2
e 5871.2 cm2 f 4427.8 cm2
2 a 960 cm2 b 7776 cm2 c 1356 cm2
3 a 704 mm2 b 3270 mm2
c 2490 mm2
4 a 452 m2 b 681 m2 c 5890 m2
5 a 3318 cm2 b 2592 cm2
c 3436 cm2 d 1268 cm2
e 16 416 cm2 f 3016 cm2
6 a 11 cm3 b 20 160 cm3
c 10 472 cm3
7 a 183 m3 b $21.96
8 a 323 m3 b 540 cm3 c 348 cm3
9 a i 1340.4 m3 ii 10 262.5 m3
iii 31.7 m3
b i 10.7 cm ii 6.9 cm
10 a i V = 180 000 mm3, C = 180 mL
ii V = 46.0 m3, C = 46.0 kLb i 12 mm ii 3 m
11 a 360 498 mm3 b 145 125 mm3
c 455 cm3 d 1152 m3
e 3054 cm3 f 12 667 m3
12 a 250 cm2
b i Increases 27 times
ii Decreases by
Chapter 3Start up
1 a m2 + 10m + 21 b k2 + 7k + 10
c y2 − 3y − 4 d w2 + 4w − 21
e n2 − 5n + 6 f 6d2 + 11d + 3
g 4 − 17p − 15p2 h 3a2 + 17af + 10f 2
i 3x2 − 7vx + 2v2 j ac + ad + bc+ bd
k 10e2 + 15eg + 6e + 9g
l 6h2 − 7h − 52 a i a2 + 2ab + b2 ii a2 − 2ab + b2
b i x2 + 8x + 16 ii y2 − 6y + 9iii 4k2 + 4k + 1 iv 9m2 − 24m + 16v 9k2 − 12kf + 4f 2
vi a2 + 4ab + 4b2
3 a a2 − b2
b i d2 − 9 ii 9a2 − 16iii 16w2 − y2 iv 25h2 − 9e2
v 1 − 9y2 vi 81d2 − 16w2
4 a 26 b 5−2 c 1010 d 70 e 3−5
f 157 g h 74 i 32
5 4 and 56 a 1.93 b 1.78 c 3.66
d 4.33 e 207.06 f 7.37g 5623.41 h 26.75
Exercise 3-011 a −1.8 b 0.7 c 0.4 d −3.5
e −2.5 f 2.6 g 1.6 h 1.9
2 a R b I c R d R e If R g R h R i I j Rk I l R m R n I o I
3 A, C, D
4 a , 5, b π, ,
5 a 1 , , b 2 , ,
6 b i 57 mmii Approximately 1.4 (= 56 ÷ 40)
7 a Teacher to check.b i Construct a right-angled triangle with
the two shorter sides equal to 10 units.
The hypotenuse is units.
ii Construct a right-angled triangle with the two shorter sides equal to a units.
Exercise 3-021 a 2 b 5 c 27 d 250
e 0.09 f 28 g 45 h 50
2 a 2 b 3 c 2 d 3
e 9 f 3 g 4 h 10
i 4 j 3 k 12 l 6
m 5 n 7 o 4 p 11
q 9 r 7 s 5 t 16
u 3 v 10 w 4 x 3
y 4
3 a 25 b 6 c 12
d 56 e f
g h 6 i 18
j k 3 l 3
m 40 n 15 o 14
p q 2 r 6
s 4 t 6 u 12
v 5 w 3 x 49
y 3
4 a F b F c T d T e T f F
Exercise 3-031 a b − c 4
d 6 e −5 f 45
g 15 h −10 i 140
j −30 k 36 l − 60
m −112 n 24 o 80
p 90 q −396 r 160
s 216 t −96 u 36
v − 60 w 252 x 72
2 a b − c 3
d 2 e − or −
f 21 g 1 h 8
1
23-------
2764------
34---
2 H4----
14---
4196
------
−4 −3 −2 −1 0 1 2 3 4
− 153 −145--- 2 5
9---
411------12 74%
π2--- 187%−
17 26 10 11
47--- π
2--- 2 7
9--- 203 2.6̇
10 2
2 3 6 6
3 5 3 2
6 7 2 3
3 3 2 2
2 5 5 2
35 7 7 11
10
2 2 3
2 10 3
73
------- 6 17
5 52
---------- 2 3
10 3 17
133
---------- 5 5
5 2 10
10 6 7
30
14 35 3
2
30 21
2 2
6
6 5
2 6 5
10 3 14
5 3 6
2 72
------- 12--- 7
616 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
i 5 j 5 k 4
l 2 m 2
n − or − o 1
p 10 q 4 r −21
s 12 t 2 u
3 a 2 b 4 c
d e 14 f 2
Exercise 3-041 a 7 b −5 c 7
d 4 e 0 f
g 8 h 0 i 2
j 10 k 6 l −14
2 a 5 − 8 b 13 + 3
c −9 + 5 d 7 + 8
e −3 − 2 f − − 8
g 13 − h − 6 + 11
i − 6 j −33 a C b D c B d A e C
4 a 6 b 3 c −2 d −
e 5 f 7 g − h 8
i 6 j 8 k 3 l 9
5 a 11 b −
c − 6 d 30
e 5 f 41
g 5 h 29
i −15 j 0
k 6 + 2 l 12 + 3
m 3 − 6 n −10
o 4 p −5
q 17 − 10 r −10 − 7
s 30 − 15 t − 4
u 8 v −13 + 10
w 9 − x −
Exercise 3-051 a − b 6 −
c −2 + 7 d 3 − 15
e 12 + 6 f − 4 +
g 42 − 8 h 5 − 75
i 24 − 3
2 a 10 + − 6 − 3
b 7 + 2 − − 2
c 28 + 21 + 8 + 2
d 20 + e 109 + 10
f 72 − 23 g 16 + 54
h −16 − i − − 1
j 45 − 5
3 a 8 − 2 b 9 + 2
c 21 − 4 d 49 + 12
e 77 + 30 f 179 − 20
g 110 + 60 h 35 + 20
i 159 − 30 j 73 − 12
4 a 5 b 22 c 67 d −57e 1 f 166 g 2 h −35
5 a 88 − 30 b −10 + 21
c 29 + 5 d 73 + 40
e 83 f 85 − 3
g 29 h 92 − 12
i − 6 j − 4
k 4 l −12 + 2
m −9 n 30 + 4
Exercise 3-061 a b c d 4
e f g h
i j k l
m = 1 − n
o or − 1
p or 2 −
q r 1 +
s t
Skillbank 32 a b c d
e f g h
i 5 : 9 j 5 : 9 k 9 : 20 l 4 : 5
m 9 : 7 n 4 : 3 o p
q r s t
3 a b c d e f
Exercise 3-071 a y8 b d4 c k15
d 1 e 9e f 12k9
g 4g3 h x3 i 2a4
j 20p3q7 k 3m2n l l14m13
m 48b17 n 4g2 o 1
p 3 q 6c9 r 36t5
s 5w2 t 5g4 u wt3
v −27e6 w fh3 xy 8
2 a 2 b 8 c −1 d −2e −4 f 3 g 15 h 24
3 a b c d e
f g h i j
4 a 12−1 b 5y−4 c (4k)−2
d h−4 or 3−1h−4 or e 2p−3
5 a 3 b 6 c 2 d
e f g h 7
i j k l
6 a 5 b 5 c 8 d 1 e 2f 10 g 3 h 25 i 6 j 3k 2 l 150 m 25 n 1 o 3
p 1 q r 27 s 4 t 8
u 1 v 1 w 1 x 24
7 a 3 b 3 c 4 d 2
e 2 f 4
Exercise 3-081 a 36 b (2d) c 8k
d (15w) e 5p f 74
g (9ab) h a
2 a b c
d e 14 f 4
g 7 h
3 a d b y c y
d h3 e p− f x−
g a h (5y4)
4 a b c 6 d 32
e f g h
5 a 125 b c 8
d e 27 f 100 000
g h 256 i
j 625 k 243 l
m 27 n 1000 o
6 a 1.7 b 0.23 c 2.4 d 0.0098e 0.046 f 0.30 g 15 h 2.3
Exercise 3-091 a y12 b k9 c m d w9
e a15 f m8 g d4 h x3
i p19 j c11 k 5k5 l 12q10
m 2 n 81e2 o 4mn6 p t8w7
q r 2x3 s 2d11 t 16a6b7
u 2c2h2 a 9 b 50 c 16 d 2
e 1 f 64 g 24 h 81
i 2 j 2 k −27 l
3 a 8 b (2m) c 20 d k
e y f (9k) g c h (5b)
4 a b c
3 2 3
6 14
24
------- 14--- 2
2
23---
6 30
245------ 3
7 2 5 7–
5 10
15 3
5 10 3
5 10 2
3 2 15 2
7 5 6 3
11 3 13 7
7 5
2 3 5 7
6 5 10 11
2 3 2 2
3 5
3 3
7 2
6 2
3
2 3 3 6
2 5 2
6 2
5 7 10 3
5 3 a
y n k
d 2a 2
15 10 3 6
21 10
6 11 55
7 5
6
10 5 2
7 21 3
6 2 3
10 77
6 10
35 15
21
15 14
5 10
6 7
2 3
6 35
7 2
35 3
77
5
30
15
15
33
------- 5 77
---------- 4 55
---------- 2
7 520------- 3 2
4---------- 10
8----------
146
---------- 5 66
---------- 772
---------- 4 155
-------------
2 2–2
---------------- 22
------- 5 2 6+4
-------------------------
35 5–5
------------------- 355
----------
22 11–11
---------------------- 1111
----------
2 3 2–2
------------------------ 2
1 2 7+2
-------------------- 6 5 3–3
------------------------
23--- 4
5--- 5
7--- 1
2---
14--- 1
6--- 5
6--- 2
5---
35--- 4
35------
14--- 4
9--- 5
32------ 1
4---
1740------ 2
3--- 16
25------ 1
4--- 5
24------ 2
25------
5 p2
3---------
142----- 1
65----- 1
8--- 1
53----- 1
27-----
1x2----- 1
5w------- 3
g4----- 4d2
y5--------- 15
n3------
13--- h 4–
3-------
14--- 10
27------ w
4----
9m6------ 125
k6--------- 5
a2b3----------- 1
9---
259d2--------- w4
m8 x12-------------- 16c10
a4------------- 64n3
27d6------------
14--- 1
16------
13--- 1
2---
12--- 1
16------
13-- 1
2-- 1
2--
13-- 1
3-- 1
2--
13-- 3
2--
5 603 2x3
8v a3 15w
t3 c8
------
14-- 5
3-- 1
2--
14-- 3
5--
72-- 1
3--
k5 1
d34---------- w25 p53
1
x4------- 1
8 a34------------- 1
64c35---------------- 10 k37
12---
18---
125------ 1
2---
1128---------
1216---------
2a3c------
12---
12-- 1
4-- 1
3-- 1
5--
43-- 3
5-- 7
4-- 8
3--
123----- 1
4--- 1
k2-----
ANSWERS 617
d e or
f g h
i 93 j k
l
5 a b c 2 d
e 2 f 4 g 4 h
i j 1 k 32 l
m 3 n 4 o p 27
Powerplus1 a Yes, since has been multiplied
by 1 .
b i ii Yes
2 a − 2 b
c d
3 a 1189 mm × 841 mm b m
4 s = or s = 5 =
6 (2 + 2 ) m 7
Chapter 3 Review1 a I b R c R d I e R
f R g R h R i N j I
2 a 6 b 7 c 5
d 8 e 15 f 14
g 48 h 15 i 28
j k l 2
m 4 n 6 o 6
p 3
3 a b 2 c 4
d e 2 f
g 4 h 1 i
j k
4 a −2 b 5
c 7 d 3 − 3
e 2 + 6 f 3
5 a 13 b 5 − 7
c 14 + 17 d 32 − 9
e 38 − 24 f 8
6 a −12 + 9 b − 10
c 7 − 27 d 23 − 8
e 77 + 10 f 43
g 9 h 70 + 38
7 a b c
d
8 a 30m3 b 3 c 4k6
d 4mn e 81y8 f 8
9 a d b d c w d y
10 a 32 b 10 000 cd e 4
Mixed Revision 11 202 a 17.0 b 14.72 c 2.0643 a 3 : 7 b 5 : 3 c 3 : 2
d 3 : 4 e 10 : 3 f 1 : 4g 12 : 1 h 4 : 1 i 1 : 180
4 40 320 km/h
5 a 6 b 6 c 2 d 12
6 a 673.9 b 0.0010
7 a 1.7 × 107 b 6 × 10−6 c 3.574 × 10−3
8 a 75 km/h b 70c/kgc $7.65/m d 85 words/min
9 a , recurring b 0.85, terminatingc 0.335, terminatingd 0.0019, terminatinge , recurring
10 a 95.2 L b 8.288 L c 154.56 L11 a $12 520 b $28 17012 a 3.6 m b 2.56 m
13 a 145.25 b $A204.82
14 a 7.18 × 1015 b 1.06 × 1020
15 20.6%
16 a b c 1
17 a 2145 mm3 b 8181 mm3
c 7069 mm3
18 a 6362 mm3 b 90 mm3
c 30 708 cm3
19 a 1294.43 m2 b 868 m2
c 1555.63 m2
20 a 15 927.9 cm3 b 8143.0 cm2
21 a 580 cm2 b 7.96 m c 7 cm
22 a 46 m3 b $49.22
23 a 22.4 m2 b 192.8 m2
c 303 m2
24 a 18 473 mm3 b 2413 mm3
c 1105 mm3
25 a 800 mm2 b 8.9 cm c 4904 cm2
26 a 445 cm2 b 61 m2
27 a 320 cm3 b 746.7 cm3
c 512 cm3
28 a i 1 : 5.76 ii 1 : 13.824b 64 : 125 c 1.86 md i 3.375 : 1 ii 45 cm
29 14 mm 30 4.89 m 31 10 cm
32 a 10 cm b 400 cm3
33 a 6 b 12 c 9 d 1
34 a 3 b 2 c −6d −12 e 2 f 13
g 14 h −2 i −18
35 a b c −10
d 60 e − f
g 15 h i
36 a 118 b y7 c 3k7m2
d 12 e f 35m4
37 a k b (3m5) c 2w
d 2y
38 a b c
d 27 e f
g h
39 a −2 b 15
c 9 d 7
e 0 f 32 − 6
g −14 h 9 − 6
i −940 a Irrational b Rational c Rational
d Neither e Rational f Irrational
41 a 1 b 2 c 12
42 a 5 b 24 − 12
c 4 + 10 d −7 − 7
e 33 + 4 f 54 + 22
g 8 − 2 h 163 + 8
i 5 j 259
k −20 l −18x − 6
43 a b c
d 3 e f
g h i
j 1 k l
44 a 5 − b 5
c 2 + d 7 − 2
e 5 − 2 f 2 − 7
45 (9 − ) cm2
Chapter 4Start up
1 a y = 5 b x = −3 c m = −24
d a = 7 e y = 2 f x = 1
2 a y < 50 b x � −13 c x > −8
d x < −1 e y < −10 f m � 10
3 a a2 + 13a + 30 b 2y2 − 3y + 1
c x2 + 9x + 20 d y2 − y − 6
e k2 + 2k − 15 f m2 − 4m + 4
g 25y2 + 30y + 9 h 9a2 − 24a + 26
i a2 + 14a + 514 a (4 − m)(4 + m) b (d − 11)(d + 11)
c 2y(7 − y) d 5p(2p + 5)e 5(x − 8)(x + 8) f 2(3w − 5)(3w + 5)
5 a (k + 1)(k + 4) b (y − 8)(y − 2)c (m − 8)(m + 7) d (u + 13)(u − 5)e (w − 7)(w − 3) f (x − 6)(x + 4)
6 a (y + 2)2 b (d − 3)2
c (n − 6)2 d (p + 9)2
e (2w − 3)2 f (8q + 5)2
7 a (3a + 1)(a + 3) b (5x + 2)(x − 3)c (2y − 5)(3y + 8) d (3t − 1)(5t + 4)e (5v + 3)(v − 7) f (2y + 5)(4y + 7)g (3h − 4)(5h − 1) h (4p − 3)(3p + 5)
i (4d + 5)2
3a5----- 1
25g2------------ 1
5g( )2--------------
p3
h5----- 1
d2 f 3------------ 7a2
x4---------
25m4------ 9
h3-----
27k6
y12-----------
19--- 1
4--- 1
15------
12--- 1
36------
45--- 1
2--- 1
216---------
38--- 17
27------ 1
32------
1
7 2–---------------------
7 2+
7 2+--------------------- 1=⎝ ⎠
⎛ ⎞
7 2+5
---------------------
10 2 15 3+17
-----------------------
13 5 5+11
----------------------- 8 3 6–2
--------------------
3 22
----------
D
3------- 3D
3------------ 1
2------- 2
2-------
3 43--- 10 3
3-------------
2 2 11
2 6 7
2 5 3
6 503
------ 5 2
11 2
6
21 10 3
55 6 14
6 66
-------
3 22
---------- 23---
3 2
5 7 2
5 7
2 2 5
2 3 5 7
2 3 11
2 10 5
35 7
6
5
3 1010
------------- 2 2 1+3
-------------------- 34
-------
5 66
----------
15-- 2
3-- 7
4-- 3
2--
12---
1343---------
25--- 3
8--- 3
4---
0.5̇4̇
0.64̇
6599------ 7
55------ 313
990---------
2 5 2
5 5 2
7 3 10
2 6 43--- 14
2 2 18---
22
------- 3 32
----------
56---
75-- 1
2-- 5
4--
13--
y23 32m53 24
p3 454 1025
1
3125 c5---------------------- 4n( )56
2 5
3 2
5 3
6 3 10
y
916------ 5
6---
2
7 2
14 10
15 30
6 wx
55
------- 7 33
---------- 10
5 115
---------- 2 23
----------
302
---------- 6 2 355
-------------
3 14 7–21
--------------------------- 6 15+3
--------------------
10
5 2 10
10 10
12--- 5
34---
12---
618 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
8 a 5.6 b 13.54
Exercise 4-011 a y = 2 b a = 5 c x = 8
d a = 1 e a = −5 f a = −2g w = 6 h x = 7.5 i y = −2
2 a y = 10 b a = −2.5 c y = 5d a = −3 e y = 8 f x = 1g y = −1 h x = −10 i m = 1.5
3 a x = 16 b m = 6 c y = 4d y = 8 e y = −7 f x = 11g m = −8 h a = −1 i y = −5
j a = k a = l a =
4 a y = 15 b a = 9 c m = 7d k = 57 e n = −35 f y = −7g x = 31 h y = 46 i m = 18j x = −29 k x = 24 l m = 10
5 a m = 2 b x = 4 c w =
d x = 3 e y = 3 f a = −8
g p = 9 h y = 3 i y =
j w = 10 k w = 50 l w = 9
m a = n y = 60 o a = 1
6 a x = b y = 15 c m = 19
d x = 10 e x = 10 f y = 10
g a = -41 h a = 7 i y = -14
j a = 5 k w = 11 l y = 2
Exercise 4-021 $7.50, $182 18 mm, 36 mm, 36 mm3 57 mm, 19 mm 4 29 cm, 13 cm5 61, 62, 636 Vatha is 22, Chris is 14.7 213, 214, 215, 216 8 269 Scott is 11 and his mother is 34
10 85 10-cent coins, 117 20-cent coins11 612 The son is 3 and the father is 27.13 25°, 50°, 105° 14 72 L15 8 teachers, 120 students
Skillbank 4A2 a 160 b 70 c 240 d 900
e 2600 f 900 g 220 h 300i 180 j 770 k 18 l 34
m 46 n 26 o 18 p 12q 40 r 8 s 104 t 24u 44 v 135 w 600 x 80
Exercise 4-031 a 52 b 172 a i 36 km/h ii 86.4 km/h
iii 180 km/hb m/s
3 434 a I = 300 b P = 781.255 a 27°C b 0°C c 100°C
d 39°C6 a 11.2 b 9 c 17.3
7 a 15.1 m b 31.8 cm8 a 21.0 b 105.84 kg
9 a 137.3 cm3 b 4.9 m10 a 80 km/h b km/h
c 325 km d 1 hours
11 a $97.50 b 620 km
12 a 410.0 cm2 b 73.9 m2
Exercise 4-041 a y = b y = c y =
d y = e y = f y =
g y = h y =
i y = − 3 or y =
j y = ± k y = ±
l y = kx2 m y = ±
n y = T 2c − k o y =
2 a r = b b = ±
c a = d h =
e r = f s =
g R = ± h l =
i n = j T =
Exercise 4-051 a x � 9
b x � 5
c y � − 4
d m � −8
e a �
f w � −10
g a � 5
h y � 20
i a � 3
j a � −1
k w � −3
l a � − 6
2 a x � 1 b m � 6 c y � -8
d r � e a � 3 f w � 0
g a � 4 h a � 6 i m � 3
j m � 3 k d � 2 l m � −2
3 a x � 3 b y � −8 c k � −11d m � 0 e p � −6 f p � −4g x � −3 h k � −12 i y � 4
j t � −2 k x � 12 l h � −18
m w � −11 n m � 29 o h � −9p x � −8 q p � −4 r y � −6
4 a w � −1
b y � −2
c x � −7
d x � 4
e a � 1
f x � −2
g k � − 4
h m � 3
i d � −5
5 a m � −10 b y � −2
c k � d x �
6 a 22, 23, 24b length = 23.2 cm, width = 5.8 cmc 12 cm, 12 cm, 21 cm d 3, 5 and 17
Exercise 4-061 a x = 3, y = −1 b x = 1, y = 3
c x = 7, y = 3 d x = 2, y = 3e x = 1, y = 4 f x = 5, y = 4
1113------ 4
3--- 9
10------
25--- −3
4-----
35---
23--- −5
6-----
35---
611------ 11
13------
715------ 4
7---
67--- 2
9--- 3
5---
914------ 1
5---
514------ 1
2---
30.5̇
55.5̇12---
5 x–2
------------ k m–p
------------- 5 2x–d
----------------
P 8–k
------------- 5m3
------- K D–M
---------------
4d 5–8
---------------- 2c k+a
---------------
20m3
---------- 20m 9–3
--------------------
t2 d2– w 5–x
-------------
a2 p–m
---------------
5n d–n
----------------
IPn------- c2 a2–
2 s ut–( )t2
---------------------- 2 Aa b+------------
3V4π-------3
v2 u2–2a
-----------------
A πr2+π
------------------- A πr2–πr
-------------------
s 360+180
------------------ DS----
0 3 6 9 12 15
0 1 2 3 4 5 6
−6 −4 −2 0 2 4
−12 −10 −8 −6 −4 −2 0
12---
−1 0 1 2 3 4
−12 −10 −8 −6 −4 −2 0 2
0 1 2 3 4 5 6 7
0 5 10 15 20 25
0 1 2 3 4 5
−3 −2 −1 0 1 2
−4 −3 −2 −1 0 1 2
−10 −8 −6 −4 −2 0
12--- 2
5---
12---
25--- 3
5--- 1
2---
25---
−4 −3 −2 −1 0 1 2
−5 −4 −3 −2 −1 0 1
−10 −9 −8 −7 −6 −5 −4
−2 0 2 4 6 8 10
−2 −1 0 1 2 3 4
14---
−3 −2 −1
−10 −8 −6 −4 −2 0 2
14---
2 3 4
12---
−8 −7 −6 −5 −4 −312---
13--- 11
3------
ANSWERS 619
2 a x = 2, y = 2 b x = 1, y = 4c x = 2, y = 1
3 a x = −2, y = 2 b x = 5, y = 10c x = 1, y = 4 d x = 1, y = 3e x = 2, y = 8 f x = 1, y = 1
Exercise 4-071 a x = −2, y = −3 b x = 0, y = 1
c x = 4, y = −32 a x = 1, y = 2 b x = −3, y = −3
c x = 3, y = 2 d x = 0, y = 1e x = 3, y = −1 f x = −2, y = 9g x = 5, y = − 4 h x = 3, y = 2
3 b The lines are parallel, so they do not intersect.
Exercise 4-081 a x = −2, y = −5 b x = 8, y = 2
c x = 2, y = 2 d x = 24, y = 8
e x = , y = 2 f x = , y = 2
2 a x = 1, y = 3 b x = 5, y = 2c x = 1, y = 6 d x = 4, y = 1e x = 4, y = 1 f x = 3, y = 2g x = 9, y = 3 h x = -13, y = 6
i x = -8, y = -3 j x = 3 , y = −1
Exercise 4-091 a a = 2, c = −3 b e = 4, g = − 4
c x = 1, y = −3 d n = 1 , p = −2
e x = 3, y = 2 f w = −1, y = −2g x = 5, y = −10 h x = 5, y = −5
i c = 1 , e = 1
2 a x = 6, y = 1 b a = 3, c = 2c h = −2, p = 3 d x = − 4, y = 6e x = 3, w = 6.5 f p = 1, w = 5
g x = 3, y = −3 h x = −3, y =
i x = 6, y =
3 a x = 10, y = 3 b a = 1, c = 3c x = 5, w = 5 d a = 3, g = 4e x = 6, y = −7 f x = 2, y = −2g p = 2, q = 0 h m = −1, n = 1i x = 4, y = 7 j x = 1, y = − 4k a = 2, c = −8 l h = −2, d = −2
Skillbank 4B3 a 13 b 2460 c 0.13 d 24.6
e 24.6 f 1300 g 1.3 h 2460i 2.46 j 13 k 130 l 246
6 a 14 076 b 1407.6 c 140.76d 1407.6 e 140.76 f 140.76g 140 760 h 14.076 i 1407.6j 14 076 k 14 076 l 1.4076
Exercise 4-101 120 women 2 280 children3 b 110 adults4 210 children 5 22 inkjet, 38 laser6 10 400 adults, 4600 children7 4 videos 8 11 Supreme pizzas9 Jenni is 19 years old
10 Adam is 12 years old11 a Pie = $2.20 b Hotdog = $1.8012 b 115 × 50-cent coins
Exercise 4-111 a m = ±12 b x = ±20 c y = ±15
d k = ±13 e y = ±1 f w = ±4g x = ±2 h t = ±4 i a = ±4j k = ±6 k w = ±10 l d = ±12
m k = ±1 n w = ±5 o x = ±
p m = ±6 q y = ±1 r p = ±3s k = ±2 t y = ±10 u x = ±9
2 a m = +2 b a = ±9 c m = ±5.29d m = ±1.94 e k = ±0.58 f x = ±7.58g k = ±9.80 h k = ±9.49 i y = ±0.35j w = ±7.07 k a = ±9.24 l y = ±6.20
3 Because the square of a positive number or a negative number is always a positive number.
4 a, c and f
5 a x = 1 or x = −3 b a = 0 or a = −5
c m = −5 or m = 6 d k = or k = −1
e p = −7.5 or p = 4.5 f x = 21 or x = −19
g m = −2 or m = 14 h y = or y = 2
i x = −32 or x = 33
Exercise 4-121 a m = −7 or −3 b d = 3 or 7
c y = −5 or 3 d k = 0 or 3e t = −7 or 0 f p = 0 or 3
g w = 0 or h n = − or 3
i a = or j x = − or −1
k e = l f =
m c = − or − n h = −1 or
o e = or 1
2 a m = 0 or −2 b y = 0 or 3c f = −5 or 5 d p = 4 or −4e x = 3 or −3 f g = −1 or −2g t = −3 or 6 h u = −2 or −24i n = 7 j w = −11 or 6k p = 4 or 6 l k = 3 or 4
m d = 6 or −3 n y = 5 or −3o k = −9 or 3 p a = 3 or −2q c = −5 or 3 r r = 11 or −3s y = 2 t d = −6u m = 1
3 a k = −3 or − b g = −1 or −1
c d = −1 or d t = −2 or −1
e m = or f y = −1 or 3
g x = or −2 h a = 2
i u = or 3 j q = −4 or 1
k w = −1 or 1 l c = −4 or 3
4 a x = −2 or 3 b t = −2 or
c u = − or −5 d m = or 1
e p = −4 or 7 f e = −1 or 5
g t = or 5 h d = − or
i h = ±5 j f = 0 or
k w = or 3 l a = −2 or
5 8
Exercise 4-131 a x2 + 2x + 1 = (x + 1)2
b p2 − 6p + 9 = (p − 3)2
c m2 − 8m + 16 = (m − 4)2
d k2 + 4k + 4 = (k + 2)2
e y2 + 7y + = (3 + )2
f w2 − 3w + = (w − )2
g x2 + x + = (x + )2
h h2 − 5h + = (h − )2
i a2 + a + = (a + )2
j v2 − v + = (v − )2
2 a −3 + , −3 −
b 5 + , 5 −
c −1 + , −1 −
d 1 + , 1 −
e + , − or ,
f ,
g ,
h ,
i ,
j ,
k 2 + , 2 −
l ,
3 a h = −1 ± b r = 1 ±
c m = −3 ± d w = 2 ±
e a = 5 + f x =
g p = h c =
i f = j y =
k x = l e =
m k = n u =
o b = −2 ±
4 a x = −0.88 or −5.12 b m = 4.40 or −0.40c g = 0.72 or −1.39 d h = 1.27 or −2.77e w = −1.27 or −0.47 f y = 1.14 or −1.47g p = −2 or 1.33 h e = 1.13 or −0.88i n = 1 or −2.5
Exercise 4-141 a x = 36, −3 b n = −2 , 1
c k = , 1 d p = , −2
e y = , 2 f x = −3 ±
g a = h m =
i c = j n =
34--- 3
4--- 2
3--- 1
3---
13---
12---
12---
12---
12---
12---
23---
23--- 1
3---
23---
23--- 1
2---
12--- 3
5--- 1
3--- 1
2---
52--- 1
2---
13--- 1
4--- 1
2---
57---
25--- 1
2---
−23----- 1
5---
56--- −2
3----- 1
4---
13--- 1
2--- 1
2---
−45----- 1
3---
25---
12--- 1
2--- 1
2---
18--- 1
7---
32--- 7
3--- 1
2---
12---
16--- 1
3---
494
------ 72---
94--- 3
2---
14--- 1
2---
254
------ 52---
72--- 49
16------ 7
4---
53--- 25
36------ 5
6---
7 7
5 5
10 10
2 2
12--- 5 1
2--- 5 1 2 5+
2-------------------- 1 2 5–
2--------------------
−2 3 3+3
------------------------- −2 3 3–3
------------------------
−2 42+2
------------------------ −2 42–2
------------------------
6 82+2
-------------------- 6 82–2
-------------------
−2 7+3
--------------------- −2 7–3
---------------------
3 71+2
-------------------- 3 71–2
-------------------
5 5
3 4 2+4
-------------------- 3 4 4–4
--------------------
6 2
7 3
30 7 61±2
--------------------
−1 21±2
----------------------- 9 73±2
--------------------
−5 17±2
----------------------- 3 17±2
--------------------
3 5±2
----------------- −5 17±2
-----------------------
−7 61±2
----------------------- 1 21±2
--------------------
2
12---
56--- −2
3-----
−15----- 7
1 17±4
-------------------- −1 22±3
-----------------------
3 41±8
-------------------- 3 21±6
--------------------
620 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
k d = -2 ± l h =
m g = n w =
o v = p
2 a 8.89, 0.11 b 1.41, −1.41c 5.37, 0.37 d 3.19, 0.31e 0.85, −2.35 f 0.30, −1.13g 2.39, 0.28 h 3.83, −1.83i 1.62, −0.62 j 4, 9k 1.48, −1.48 l 2.31, 0.69m 8.09, −3.09 n 4.27, 7.73o 3.31, −1.81
Exercise 4-151 a −5, −9 b 20, −7 c 5 ±
d −2 ± e −4, 2 f no solution
g 2, −2 h i
j k 3 , −5 l
m 1 , −2 n o
p 3.5 q r no solution
2 0, 1 or 2 solutions
Exercise 4-161 a 58 m × 38 m b 6.5 m × 13 m
c 17 cm × 12 cm d x = 62 a 24 and 25 b 42 and 44
c 15 and 273 a 1275 b 3240 c 1965
d 36 e 444 15 and 175 a 1800 m b 1080 m
c 19 s (approximately) d 13.4 s6 a $9200 b $5550
c 471 (truncate)
Exercise 4-171 a 16a2p2 + 7 b 2at − 4a2t2
c 11 + 6h + h2 d 3k2 − 6k − 4
e u2 − 3u + 4 f k2 − 4k + 5
2 a −1, b − , 1 c − , −1
d , 2 e −1 , 1
3 a x = ±2 or ±3 b x = ±2 or ±2
c m = ± or ±2 d y = ± or ±3
e k = ± or ±2
4 a y = ±3 b m = ±4c no solutions
5 a m = 1 or 2 b m = −1 or 10c x = ±2.8 or ±1.2 d x = ±0.9 or ±1.4e x = −0.894 or 1.26
Exercise 4-181 a x = 3, y = 9 and x = 2, y = 4
b x = 0, y = 0 and x = 4, y = 32c x = 4, y = 26 and x = −2, y = 14
d x = −1, y = 5 and x = 1 , y = 7
e x = −2, y = 18 and x = , y = 11
f x = −2, y = −22 and x = 5, y = 20g x = 7, y = 39 h no solution
i no solution j
Power plus1 a x = b x =
c x = −8 d x = ±2 x � 10 3 Teacher to check.4 p = 4, q = 3 5 b x = 1
6 Intersects twice
Chapter 4 review1 a w = −3 b y = −2 c p = 0
d m = 2.5
2 a m = 4 b w = 13 c m =
d m = 8
3 a y = b m = 1
4 a 92, 93, 94, 95b Grace is 13, Jane is 16
5 a i 160 mm3 ii 300 m2
iii 1.5 cmb 120 m
6 a = − 6 or a =
7 a y � 5
b y � −5
c y � −1
d m � −2
8
∴ x = −2, y = 69 a x = −3, y = −11 b x = 3, y = 3
10 a x = 2, y = 11 b x = −4, y = 12c x = 1, y = 1 d x = −1, y = 1
11 a m = 5, c = −9.5 b x = −5, y = −2c x = 9, y = −4 d k = −3, w = −10
12 a 900 childrenb 17 DVDs and 13 videos
13 a y = ±11 b m = ±5c m = 6 or −2 d y = 1.5 or -4.5
14 a m = −2 or 2 b x = 0 or 10c p = ±6 d m = -6 or 8
e m = − or 2 f x = −3 or 5
15 a y = -4 ± b x = 3 ±
c y = d k =
16 a m = −5.5 or 0.54 b y = −1.2 or 4.7c k = 2.8 or −2.5 d m = 1.8 or −2.2e y = 2.0 or 4.0 f m = −1.9 or 2.4
17 a 32 and 34 b 14 m × 22 m
18 a y = or b m = ± or ±2
c y = ±219 a x = 4, y = −20 and x = −3, y = 29
b x = 0, y = 0 and x = 3, y = 6
Chapter 5Start up
1 a m = 63 (vertically opposite angles)b x = 51 (angles on a straight line)c y = 57 (angles at a point)d k = 318 (angles at a point)e w = 89 (vertically opposite angles)f a = 19 (angles at a point)
2 a d = 127 (alternate angles)b 4m = 88 (corresponding angles)
m = 22c 2h + 66 = 180 (co-interior angles)
2h = 114h = 57
d a = 72 (alternate angles, corresponding angles)
e k = 104 (vertically opposite angles)m = 86 (angles on a straight line, corresponding angles)
f p = 81 (corresponding angles)w = 99 (angles on a straight lines, alternate angles)
3 a h = 46 (angle sum of a triangle)b m + 35 = 124 (exterior angle of a
triangle)m = 89
c 10h = 180 (angle sum of a triangle)h = 18
d 5y + 137 + 83 = 360 (angle sum of a quadrilateral)5y = 140y = 28
e 3k + 105 = 180 (co-interior angles)3k = 75k = 25
f a + 27 + 197 + 34 = 360° (angle sum of a quadrilateral)a = 102
4 a No, corresponding angles are not equal.b Yes, co-interior angles have a sum of 180°.c No, alternate angles (or corresponding
angles) are not equal.
Exercise 5-011 a y = 50 (angle sum of an isosceles
triangle)b m = 108 (angle sum of an isosceles
triangle)c h = 120 (exterior angle of an equilateral
triangle)d x = 45 (angle sum of a right-angled
isosceles triangle)e a = 135 (exterior angle of a right-angled
isosceles triangle)f 5d = 180 (angle sum of an isosceles
triangle)d = 36
g y + 70 + 70 = 180 (angle sum of an isosceles triangle)y = 40
5 9 17±4
--------------------
−1 6±5
-------------------- 1 7±3
-----------------
−2 14±5
----------------------- 2 7±3
-----------------
7
11
−1 97±6
----------------------- 2 2 10±3
-----------------------
3 37±2
-------------------- 12--- -3 11±
2----------------------
13--- 4 46±
3-------------------- 5 3 5±
4--------------------
−7 33±4
-----------------------
13--- 4
5--- 2
5--- 1
2--- 1
2---
45--- 1
3--- 1
2---
2
1
3------- 1
2---
2
5-------
15--- 1
5---
13---
3 41±2
--------------------
1113------ 1
3---
k2 2y–
35--- 1
11------ −1
9-----
13---
57--- 1
2---
2 Ah
------- 2 A bh–h
--------------------
0 1 2 3 4 5
−5 −4 −3 −2 −1 0
−2 −1 0 1 2 315---
−3 −2 −1
x −3 −2 −1 0 1
y 7 6 5 4 3
x −3 −2 −1 0 1
y 4 6 8 10 12
53---
26 29
−3 3 3±2
----------------------- 6 6±3
-----------------
−12----- 1
2--- 2
ANSWERS 621
h 2p = 130 (exterior angle of an isosceles triangle)p = 65
i h = 84 (exterior angle of an isosceles triangle)
2 a y = 53 (rhombus, co-interior angles)b k = 27 (opposite angles of a
parallelogram are equal)c w = 35 (angles in a square)d h = 109 (co-interior angles in a
trapezium)e f = 25 (angle sum of a right-angled
triangle in a rhombus)f a = 46 (opposite angles of a
parallelogram are equal, angles on a straight line)
3 a t = 70 (angle sum of an isosceles triangle)
b m = 58 (alternate angles)y = 61 (angle sum of an isosceles triangle)
c d + 74 = 120 (exterior angle equals sum of interior opposite angles)d = 46
d m = 115 (corresponding angles)k = 115 (vertically opposite angles)w = 65 (co-interior angles)h = 65 (corresponding angles)
e c = 94 (angle sum of isosceles triangle, and vertically opposite angles)
f w = 24 (alternate angles and angle sum of an isosceles triangle)a = 102 (angle sum of straight line or alternate angles)
g ∠DXA = 90° (diagonals of a rhombus meet at right angles)∴ p = 56 (angle sum of ∆DXA)
h h = 70 (corresponding angles, angle sum of a triangle)
i w = 93 (isosceles triangle, and angle sum of a quadrilateral)
4 a q = 70 (alternate angles)p = 70 (angle sum of a straight line)w = 40 (angle sum of ∆XYZ)
b ∆XYZ is isosceles since p = q = 70.5 a T b T c F d F e T
f F g T h T i F j Fk T l T m F n T
6 ∠LMK = ∠LKM = 45° (angle sum of a right-angled isosceles triangle)∠PMN = 135° (angles on a straight line)2x + 135 = 180 (angle sum of isosceles triangle MNP)∴ x = 22.5
7 Let ∠XYP = x°, ∠TWP = y°∠PYW = x° (YP bisects ∠XWY)and ∠PWY = y° (WP bisects ∠TWY)2x + 2y = 180 (co-interior angles, YX || WT)∴ x + y = 90But x° + y° + ∠YPW = 180° (angle sum of ∆YPW)90° + ∠YPW = 180°∴ ∠YPW = 90°
8 ∠DBF = ∠DCE (corresponding angles, BF || CE)∠BFD = ∠DEC (corresponding angles, BF || CE)But ∠DBF = ∠BFD (equal angles of isosceles ∆BDF)∴ ∠DCE = ∠DEC
∴ ∆CDE is an isosceles triangle (two angles equal)
9 ∠NKL + 93° = 147° (exterior angle of ∆NKL)∠NKL = 54∠NKH = 54 (NK bisects ∠NHK)∠HKL = 108°∠KHL + 108° = 147 (exterior angle of ∆HKL)∴ ∠KHL = 39°
10 a ∠CED = ∠CDE = 42° (equal angles in isosceles ∆CDE)∠DCE = 96 (angle sum of ∆CDE)∴ k = 96
b ∠BCE = ∠CED = 42° (alternate angles, CB || DE)∠BEC = 42° (equal angles in isosceles ∆BCE)∠ABE = ∠BCE + ∠BEC (exterior angle of ∆BCE)= 84°∴ m = 84
11 a y° = 60 (angle in equilateral triangle)x = 120 (co-interior angle in a rhombus)
b ∠ACB = 60° (angle in equilateral triangle)x + x = 60 (exterior angle in isosceles triangle)∴ x = 30∠B = 60° (angle in equivalent triangle)∴ y = 90 (angle sum of ∆ABD)
c 2x + 40 = 180 (angle sum of isosceles triangle)∴ x = 70y + y = 40 (exterior angle of isosceles triangle)∴ y = 20
d ∠HJK = ∠JHK = 36 (equal angles in isosceles triangle)∠IJK = 72° (co-interior angles on parallel lines IJ, HK)∴ x = 72 − 36 = 36∴ y = 180 − x − x (equal angles in isosceles triangle)y = 108°
12 ∠CDB = ∠CBD (equal angles of isosceles ∆CDB)∠EAB = ∠CBD (corresponding angles, AE || BD)∠EBD = ∠CDB (alternate angles, BE||CD)and ∠EBD = ∠AEB (alternate angles, AE || BD)∴ ∠CBD = ∠AEBBut ∠CDB = ∠CBD = ∠EAB∴ ∠AEB = ∠EAB∴ ∆ABE is an isosceles triangle (two equal angles)
13 ∠LMN = ∠LNM (equal angles of isosceles ∆LMN)∠KLN = ∠LMN + ∠LNM (exterior angle of triangle)= 2 × ∠LMN
∠KLP = × (2 × ∠KMN)
(LP bisects ∠KLN)∠KLP = ∠LMN∴ LP || MN (corresponding angles are equal)
14 Let ∠B = ∠A = x° (equal angles of isosceles ∆ABC)
∠ACB = 180 − 2x° (angle sum of ∆ABC)∠DCE = 180 − 2x° (vertically opposite angles)
∠D = ∠E = [180 − (180 − 2x)] (angle
sum of isosceles ∆DCE)∠D = ∠E = x°∠B = ∠D∴ AB || DE (alternate angles are equal)
15 ∠YUX = ∠UYX = ∠UXY = 60° (angles in equilateral ∆UXY)∠UXW = 120° (angles on a straight line)∠XWU + ∠WUX + 120° = 180° (angle sum of ∆WXU)∠XUW = ∠XWU = 30° (∆WXU is isosceles)∴ ∠WUY = ∠XUW + ∠YUX= 30° + 60°= 90°
16 ∠WTP = ∠P and ∠YTQ = ∠Q (alternate angles, WY || PQ)∴ Angle sum ∆PQT = ∠P + ∠PTQ + ∠Q= ∠WTP + ∠PTQ + ∠YTQ (angles on a straight line)= 180°
17 ∠BAD + ∠DAH + ∠ BAC + ∠CAF = 180° (angles on a straight line)But ∠BAD = ∠DAH (AD bisects ∠HAB)and ∠BAC = ∠CAF (AC bisects ∠FAB)2 ∠BAD + 2 ∠BAC = 180°∠BAD + ∠BAC = 90°∴ ∠CAD = 90°
Exercise 5-021 a 3240° b 1440° c 2340°
d 3960° e 5040° f 1800°2 a 28 b 12 c 7 d 40 e 14 f 93 a 108° b 140° c 150°
4 a 168° b 172° c 128 ° d 174°
5 a 18 b 24 c 8 d 45 e 126 a 60° b 30° c 12°7 a 6 b 24 c 45 d 8 e 158 24
Skillbank 5A2 a 70% b 66% c 45% d 88%
e 75% f 75% g 80% h 80%i 55% j 35% k 30% l 80%m 90% n 45% o 52%
4 a 25% b 68% c 17%d 60% e 10% f 33.3%g 59% h 70.2% i 84%j 70% k 42.8% l 5.5%
m 91% n 78.25% o 31.4%
Exercise 5-031 a No b Yes, SAS c No
d Yes, SSS e No f Yes, RHSg Yes, AAS h Yes, SAS i Yes, AAS
2 a d = 31 b k = 25c y = 12, w = 25 d a = 33, p = 9e p = 109 f k = 11
3 a AB = CB (given)EB = CB (given)∠ABE = ∠CBD (vertically opposite angles)∴ ∆ABE ≡ ∆CBD (SAS)
12---
12---
47---
622 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
b QW = PT (given)TW is common.∠QTW = ∠PWT = 90° (given)∴ ∆QTW ≡ ∆PWT (RHS)
c ∠HCD = ∠GEF (corresponding angles, CH || EG)∠HDC = ∠GHE (corresponding angles, DH || FG)CH = EG (given)∴ ∆CDH ≡ ∆EFG (AAS)
d UX = WX (given)XY is common.∠UXY = ∠WXY (YX bisects ∠UXW)∴ ∆UXY ≡ ∆WXY (SAS)
e AB = CB (equal sides of a square)AY = CX (given)∠A = ∠C = 90° (angles in a square)∴ ∆ABY ≡ ∆CBX (SAS)
f LM = NP (given)LP = NM (given)PM is common.∴ ∆LMP ≡ ∆NPM (SSS)
g OA = OC (equal radii of small circle)OB = OD (equal radii of large circle)∠AOB = ∠COD (vertically opposite angles)∴ ∆AOB ≡ ∆COD (SAS)
h FH = FG (given)∠F is common.∠HNF = ∠GMF = 90° (HN ⊥ FG, GM ⊥ FH)∴ ∆FHN ≡ ∆FGM (AAS)
4 a i PQ = PR (∆PQR is isosceles)QA = RB (given)∠Q = ∠R (equal angles of an isosceles triangle)∴ ∆PQA ≡ ∆PRB (SAS)
ii PA = PB (matching sides of congruent triangles)∴ ∆PAB is isosceles (two sides of the triangle are equal)
b i TP = XP (given)AP = CP (given)∠TPA = ∠XPC (vertically opposite angles)∴ ∆TAP ≡ ∆XCP (SAS)
ii ∠T = ∠X (matching angles of congruent triangles)∴ TA || XC (alternate angles proved equal)
c i ∠ADB = ∠CBD (alternate angles, AD||CB)∠ABD = ∠CDB (alternate angles, AB ||| CD)BD is common.∴ ∆ABD ≡ ∆CDB (AAS)
ii ∴ AD = CB and AB = CD (matching sides of congruent triangles)
d i AD = AC (equal radii of circle, centre A)BD = BC (equal radii of circle, centre B)AB is common.∴ ∆ADB ≡ ∆ACB (SSS)
ii ∠DAC = ∠CAB (matching angles of congruent triangles)∴ AB bisects ∠DAC
e i ∠HEF = ∠GFE (given)EH = FG (given)EF is common.
∴ ∆HEF ≡ ∆GFE (SAS)ii ∠EHF = ∠FGE (matching angles of
congruent triangles)f i OT = ON (equal radii)
OL = OM (equal radii)LT = MN (given)∴ ∆LOT ≡ ∆MON (SSS)
ii ∠LOT = ∠MON (matching angles of congruent triangles)
g i OC = OE (equal radii)OD is common.∠ODC = ∠ODE = 90° (OD⊥CE)∴ ∆OCD ≡ ∆OED (RHS)
ii CD = ED (matching sides of congruent triangles)∴ OD bisects CE
h i AB = AD (given)CB = CD (given)AC is common.∴ ∆ABC ≡ ∆ADC (SSS)
ii ∴ ∠BCA = ∠DCA (matching angles of congruent triangles)
iii Since ∠BCA = ∠BCYand ∠DCA = ∠DCY,∠BCY = ∠DCY (∠BCA = ∠DCA, proved in ii)CB = CD (given)CY is common.∴ ∆BCY ≡ ∆DCY (SAS)
iv BY = DY (matching angles of congruent triangles)
Exercise 5-041 a TX = TY (given)
XW = WY (given)TW is common.∴ ∆TXW ≡ ∆TYW (SSS)
b ∠X = ∠Y (matching angles of congruent triangles)
2 a PL = ML (equal sides of a rhombus)PN = MN (equal sides of a rhombus)NL is common.∴ ∆LMN ≡ ∆LPN (SSS)
b MN = ML (equal sides of a rhombus)PN = LP (equal sides of a rhombus)PM is common.∴ ∆PMN ≡ ∆PML (SSS)
c ∠PNL = ∠MNL and ∠PLN = ∠MLN (matching angles of isosceles triangles LMN and NPL)∴ LN bisects ∠PNM and ∠PLM.Similarly, by considering matching angles in ∆PMN and ∆MLP, PM bisects ∠NPL and ∠NML.∴ Angles of a rhombus bisected by the diagonals.
3 a ∠ABD = ∠CDB (alternate angles, AB || CD)∠ADB = ∠CBD (alternate angles, AD || CB)BD is common.∴ ∆ABD ≡ ∆CDB (AAS)
b AB = CD (matching sides of congruent triangles)and AD = CB∴ Opposite sides of a parallelogram are equal.
4 a i AC = BC (equal sides of an equilateral triangle)AX = BX (given)
CX is common.∴ ∆AXC ≡ ∆BXC (SSS)
ii ∠A = ∠B (matching angles of congruent triangles)
b i AB = CB (equal sides of an equilateral triangle)AY = CY (given)BY is common.∴ ∆AYB ≡ ∆CYB (SSS)
ii ∠A = ∠C (matching angles of congruent triangles)
c ∴ ∠A = ∠B = ∠CBut ∠A + ∠B + ∠C = 180°∴ ∠A = ∠B = ∠C = 60°
5 a ∠XWY = ∠VYW (alternate angles, WX || VY)∠XYW = ∠VWY (alternate angles, WV || XY)WY is common.∴ ∆WXY ≡ ∆YVW (AAS)
b ∠WXV = ∠YVX (alternate angles, WX || VY)∠WVX = ∠YXV (alternate angles, WV || XY)XV is common.∴ ∆VWX ≡ ∆XYV (AAS)
c ∠V = ∠X (matching angles in congruent triangles WXY and YVW)∠W = ∠Y (matching angles in congruent triangles VWX and XYV)∴ Opposite angles of a parallelogram are equal.
6 a ∠XDE = ∠XFG (alternate angles, DE || FG)∠XED = ∠XGF (alternate angles, DE || FG)DE = FG (opposite sides of a parallelogram are equal)∴ ∆DEX ≡ ∆FGX (AAS)
b DX = FX and EX = GX (matching sides of congruent triangles)∴ Diagonals of a parallelogram bisect each other (X is the midpoint of both diagonals).
7 a XW = XY (given)∠XTW = ∠XTY = 90° (XT⊥WY)XT is common.∴ ∆WXT ≡ ∆XYT (RHS)
b ∴ WT = YT (matching sides of congruent triangles)
8 a AB = AC (given)AX is common.BX = CX (AX bisects BC)∴ ∆AXB ≡ ∆AXC (SSS)
b ∴ ∠AXB = ∠AXC (matching angles of congruent triangles)
c ∠AXB + ∠AXC = 180° (angles on a line)∠AXB = ∠AXC = 90°∴ AX ⊥ BC
9 a ∠P = ∠R (given)∠TXP = ∠TXR = 90° (TX⊥PR)TX is common.∴ ∆PXT ≡ ∆RXT (AAS)
b TP = TR (matching sides of congruent triangles)∴ Sides opposite the equal angles are equal.
10 a FC = FE (equal sides of a rhombus)FB is common.
ANSWERS 623
∠CFB = ∠EFB (diagonals of a rhombus bisect the angles)∴ ∆CBF ≡ ∆EBF (SAS)
b CB = EB (matching sides of congruent triangles)∴ Diagonal CE is bisected at B.
c Prove ∆FBC ≡ ∆DBC (SAS)FB = DB (matching sides of congruent triangles)∴ Diagonal DF is bisected at B.
d i Matching angles of congruent triangles CBF and EBF.
ii ∠CBF = ∠EBFBut ∠CBF + ∠EBF = 180° (angles on a straight line)∠CBF = ∠EFB = 90°∴ FB ⊥ CEand FD ⊥ CE
e The diagonals of a rhombus bisect each other at right angles.
11 a In ∆LMN and ∆LPN,LM = LP (given)NM = NP (given)LN is common.∴ ∆LMN ≡ ∆LPN (SSS)i ∴ ∠LMN = ∠LPN (matching angles
of congruent triangles)ii ∠MLN = ∠PLN and ∠MNL = ∠PNL
(matching angles of congruent triangles)∴ ∠MLP and ∠MNP are bisected by the diagonal LN.
b LM = LP (given)LT is common.∠MLT = ∠PLT (LN bisects ∠MLP (proved in a)∆LMT ≡ ∆LPT (SAS)MT = PT (matching sides of congruent triangles)∴ Diagonal MP is bisected.Also, ∠LTM = ∠LTP (matching angles of congruent triangles)and ∠LTM + ∠LTP = 180° (angles on a straight line)∠LTM = ∠LTP = 90°LT ⊥ MPand LN ⊥ MP∴ Diagonal MP is bisected at right angles by diagonal LN.
Exercise 5-051 a ∠A = ∠C and ∠B = ∠D
Now ∠A + ∠C + ∠B + ∠D = 360° (angle sum of a quadrilateral)∴ 2∠A + 2∠B = 360° (∠C = ∠A, ∠D = ∠B)∴ ∠A + ∠B = 180°Note: A pair of co-interior angles have a sum of 180°∴ AD || BCAlso, from ∠A + ∠B + ∠C + ∠D = 360°2∠A + 2∠D = 360° (∠C = ∠A, ∠D = ∠B)∠A + ∠D = 180°AB || DC (co-interior angles have a sum of 180°)Opposite sides are parallel.∴ ABCD is a parallelogram.
b Draw the diagonal PM.In ∠LMP and ∆NPM,LM = NP (given)PM is common.∠LMP = ∠NPM (alternate angles, LM || NP)∆LMP ≡ ∆NPM (SAS)∠LPM = ∠NMP (matching angles of congruent triangles)LP || NM (alternate angles proved equal)∴ LMNP is a parallelogram (opposite sides are parallel).
c In ∆LMF and ∆GMH,LM = GM (given)FM = HM (given)∠LMF = ∠GMH (vertically opposite angles)∆LMF ≡ ∆GMH (SAS)∠FLM = ∠HGM (matching angles of congruent triangles)∴ FL || GH (alternate angles proved equal).Similarly, ∠FGM = ∠HLM (matching angles of congruent triangles FGM and HLM)FG || HLOpposite sides are parallel∴ FGHL is a parallelogram.
d Join the diagonal PR.In ∆PQR and ∆RTP,PQ = RT (given)QR = TP (given)PR is common.∆PQR ≡ ∆RTP (SSS)∠QPR = ∠TRP and ∠QRP = ∠TPR (matching angles of congruent triangles)PQ || TR and PT || QR (alternate angles proved equal)∴ PQRT is a parallelogram.However, since the sides of PQRT are equal, PQRT is a rhombus.
e ∆FHC ≡ ∆FHE ≡ ∆DHE ≡ ∆DHC (SAS)FC = FE = DE = DC (matching sides of congruent triangles)Also, ∠CFH = ∠EDH and ∠CDH = ∠EFH (matching angles of congruent triangles)CF || DE and CD || FE (alternate angles equal)∴ CDEF is a rhombus (opposite sides parallel and all sides equal).
f Since WY = XV, and diagonals bisect each other, TW = TV = TY = TXThen ∆TWV ≡ ∆TXY (SAS), and ∆TVY ≡ ∆TWX (SAS)∠VWT = ∠XYT and ∠TVY = ∠TWX (matching angles of congruent triangles)WV || XY and VY || WX (alternate angles equal)∴ WXYV is a parallelogram.Also ∆WXV ≡ ∆YVX ≡ ∆WVY ≡ ∆WXY (AAS)∠W = ∠X = ∠Y = ∠V (matching angles of congruent triangles)Since the angle sum of WXYV = 360°∠W = ∠X = ∠Y = ∠V = 90°.∴ WXYV is a rectangle.
g Since ∠A + ∠D = 180° and ∠A + ∠B = 180°AB || DC and AD || BC (co-interior angles have a sum of 180°)ABCD is a parallelogram with right angles∴ ABCD is a rectangle.
h Since the sides are equal, TWNE is a rhombus (proved in part d).Since ∠M = 90°, TWME is a square (a square is a rhombus with a right angle).
i Since the angles of the quadrilateral are right angles, GHKL is a rectangle (proved in part g).If GL = GH,GL = GH = LK = KH (opposite sides of a rectangle are equal)∴ GHKL is a square (all sides are equal, all angles are 90°).
j The diagonals bisect each other at right angles, so MNPT is a rhombus.∴ MN = NP = PT = MT∆MNT ≡ ∆NPT ≡ ∆PTM ≡ ∆MNP (SSS, since TN = PM)∠M = ∠N = ∠P = ∠T = 90° (angle sum of a quadrilateral and matching angles of congruent triangles)∴ MNPT is a square.
2 a i BX = DY (given)∠B = ∠D (opposite angles of a parallelogram)AB = CD (opposite sides of a parallelogram)∴ ∆ABX ≡ ∆CDY (SAS)
ii AX = CY∴ XC = AY as BX = YD and BC = AD (opposite sides of a parallelogram)∴ AXCY is a parallelogram as pairs of opposite sides are equal
b i AE = EB (given)∠DAE = ∠CEB (corresponding angles, AD || EC)AD = EC (sides of a rhombus)∴ ∆DAE ≡ ∆CEB (SAS)
ii ED = BC (matching sides in congruent triangles above)AE = DC and AE = EB∴ DC = EB∴ BCDE is a parallelogram because opposite pairs of sides are equal
c i AP = CR (given)∠A = ∠C (opposite angles of a parallelogram)AS = CQ (given)∴ ∆APS ≡ ∆CRQ (SAS)∴ PS = RQPB = RD (given AP = CR and opposite sides of a parallelogram)∠B = ∠D (opposite angles of a parallelogram)BQ = DS (given CQ = AS)∴ ∆PBQ ≡ ∆RDS (SAS)∴ PQ = RS
ii PQRS is a parallelogram because pairs of opposite sides are equal
d AC and DB are the diagonals.DO = BO (equal radii small circle)AO = CO (equal radii large circle)∴ ABCD is a parallelogram because the diagonals bisect each other.
624 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
e SQ and PR are the diagonals.PO = RO (equal radii small circle)SO = QO (equal radii large circle)PR ⊥ SQ (given)∴ PQRS is a rhombus because the diagonals bisect each other at right angles.
f Since WD = WE = GY = YF, and DZ = ZG = FX = EX, WZ = WX = YX = ZY (by Pythagoras’ theorem)∴ i WXYZ is a parallelogram (opposite sides equal).∴ ii WXYZ is a rhombus (all sides equal).
Skillbank 5B2 a b c d e
f g h i j
k l m n o
4 a b c d e
f g h i j
k l m n o
Exercise 5-061 a Yes b No c Yes d Yes
e Yes f Yes2 a 16.2 b or 6
c 10.5 d 10e 5.625 f or 5
3 a 18 b c 31.5
d 7 e 30 f
g h 16.5
4 a A and C,
b A and B, . Also A and D, 1 . Also B
and D, 2.4.5 15.8 6 6.3 7 21.35 m8 a T b T c T d T
e F f T g T h F
Exercise 5-071 a RHS, b AA c SAS,
d SSS, e SAS, f AA
g SAS h SSS, i RHS,
2 a ∠X is common.∠XLM = ∠XYW (corresponding angles)∴ ∆XLM ||| ∆XYW (AA)
b = =
= =
∴ =
and ∠DCE = ∠BCA (vertically opposite)∴ ∆ABC ||| ∆EDC (SAS)
c ∠P is common.∠PLM = ∠PTW = 90° (given)∴ ∆PLM ||| ∆PTW (AA)
d = =
= =
∴ =
and ∠K = ∠T = 90°∴ ∆KGX ||| ∆TYX (RHS)
e ∠GMK = ∠LHK (given)∠K is common.∴ ∆GMK ||| ∆LHK (AA)
f ∠Y is common.∠WZY = ∠ZXY = 90° (given)∴ ∆WYZ ||| ∆ZYX (AA)
3 a i ∠K is common.∠FGK = ∠MHK (corresponding angles)∴ ∆GKF ||| ∆HKM (AA)
ii m = 6 or 6.86
b i ∠F = ∠C (alternate angles)∠G = ∠D (alternate angles)∴ ∆CDE ||| ∆FGE
ii k = 11.25c i ∠F = ∠E (given)
∠FCB = ∠ECD (vertically opposite angles)∴ ∆BCF ||| ∆DCE (AA)
ii x = 7.5d i ∠A is common.
∠AXM = ∠AYN = 90° (given)∴ ∆AXM ||| ∆AYN (AA)
ii AX = 12e i ∠T is common.
∠TFR = ∠TPD = 90° (given)∴ ∆TFR ||| ∆TPD (AA)
ii TR = 14.4f i ∠E is common.
∠ECF = ∠EBA (corresponding angles, FC || AB)∴ ∆EFC ||| ∆EAB (AA)
ii ∠AEB = ∠FAD (alternate angles, AD || BE)∠B = ∠D (opposite angles of a parallelogram)∴ ∆EAB || ∆AFD (AA)
iii ∠ECF = ∠ADF (alternate angles, AD || EC)∠EFC = ∠AFD (vertically opposite angles)∴ ∆EFC ||| ∆AFD (AA)
iv AB = 14.4 cm4 a ∠T = 90 − ∠P (in ∆PWT)
∴ ∠TWN = 180 − [90 + (90 − ∠P)] (angle sum of ∆TWN)= ∠PIn ∆PWN and ∆WTN,∠WPN = ∠P = ∠TWN (proved)∠PNW = ∠WNT = 90° (WN⊥PT)∴ ∆PWN ||| ∆WTN (AA)
b WN = 6, WP = 2 , WT = 3
5 a ∠FEG = ∠EHF∠F is common.∴ ∆EFG ||| ∆HFE (AA)
b FH = 25 cm
Exercise 5-081 ∠ABC = 180° − (∠CAB + ∠ACB)
(angle sum of ∆ABC)
∠CBD = 180° − ∠ABC (angle on a straight line)∴ ∠CBD = 180° − [180° − (∠CAB + ∠ACB)]= 180° − 180° + (∠CAB + ∠ACB)∴ ∠CBD = ∠CAB + ∠ACB
2 a XA = YA (equal radii of circle, centre A)XB = YB (equal radii of circle, centre B)AB is common.∴ ∆AXB ≡ ∆AYB (SSS)∴ ∠XAC = ∠YAC (matching angles of congruent triangles)
b XA = YA (equal radii)AC is common.∠XAC = ∠YAC (proved in part a)∴ ∆XAC ≡ ∆YAC (SAS)
c ∴ XC = YC (matching sides of congruent triangles)Also, ∠XCA + ∠YCA = 180° (angles on a straight line)and since ∠XCA = ∠YCA (matching angles of congruent triangles)∠XCA = ∠YCA = 90°∴ XY ⊥ AB
3 a ∠P is common.∠PQR = ∠PMN (corresponding angles, QR || MN)∆PQR ||| ∆PMN (AA)
b = (ratio of matching sides)
Since Q is the midpoint of PM. =
=
∴ PR = RN4 a ∠A is common
∠ADC = ∠ACB = 90° (given)∆ADC ||| ∆ACB (AA)
= (matching sides are in the
same ratio)
∴ AC2 = AB × ADb ∠B is common.
∠BDC = ∠BCA = 90° (given)∆BDC ||| ∆BCA (AA)
= (matching sides in the same
ratio)
∴ BC2 = AB × DB
c AC2 + BC2 = AB × AD + AB × DB= AB × (AD + DB)= AB × AB
= AB2
5 In ∆DEG: DE2 = EG2 + DG2
DG2 = DE2 − EG2
In ∆DFG: DG2 = DF2 − GF2
DE2 − EG2 = DF2 - GF2
∴ DE2 + GF2 = DF2 + EG2
6 a no b yes c yes7 a 53 cm b 12 m
c 5 cm d 96 m
8 a In ∆AXB: AB2 = AX2 + BX2 (1)∆AXD: AD2 = AX2 + DX2 (2)∆CXD: CD2 = CX2 + DX2 (3)∆BXC: BC2 = BX2 + CX2 (4)Adding (1) and (3): AB2 + CD2
= AX2 + BX2 + CX2 + DX2
= (AX2 + DX2) + (BX2 + CX2)
= AD2 + BC2 (using (2) and (4))∴ AB2 + CD2 = BC2 + AD2
34--- 7
25------ 3
10------ 7
50------ 3
50------
1720------ 8
25------ 49
100--------- 14
25------ 9
10------
1825------ 13
20------ 1
5--- 6
25------ 53
100---------
1925------ 1
10------ 4
5--- 9
20------ 22
25------
1425------ 3
4--- 31
100--------- 17
25------ 1
20------
35--- 27
50------ 3
50------ 49
100--------- 41
50------
6.6̇ 23---
5.0̇9̇ 111------
7.3̇17--- 22.53̇
10.26̇58---
58--- 1
2---
32--- 3
2---
45--- 8
15------
45--- 3
5---
ACEC-------- 10
7.5------- 4
3---
BCDC--------- 8
6--- 4
3---
ACEC-------- BC
DC---------
KXTX-------- 7.5
18------- 0.416̇
GXYX-------- 12
28.8---------- 0.416̇
KXTX-------- GX
YX--------
67---
13 13
PRPN-------- PQ
PM---------
PQPM--------- 1
2---
PRPN-------- 1
2---
ACAB-------- AD
AC--------
BCAB-------- DB
BC--------
23
ANSWERS 625
9 a Since 52 = 32 + 42,∠D is a right angle.In ∆ABC and ∆EDC:∠B = ∠D = 90°∠ACB = ∠ECD (vertically opposite angles)∴ ∆ABC ||| ∆EDC (AA)
b m = 16 or
Power plus1 Teacher to check proofs.2 X and Y are midpoints of BC and AY.
Medians AX and BY meet at B.Draw CP to T, so that CP = PT.Prove ∆CYP ||| ∆CAT (SAS)∴ YP || AT∴ PB || ATSimilarly, prove PA || BT∴ APBT is a parallelogram∴ W is the midpoint of AB (the diagonals of a parallelogram bisect each other).
Chapter review1 a ∠BCE = 110° (corresponding angles,
BD || CE)∴ c = 110 (corresponding angles, BC || DE)
b 4w = 114 + w (exterior angles of a triangle)∴ w = 38
c 2y + 36 = 180 (angle sum of an isosceles triangle)y = 72
2 a Let ∠PTR = ∠PRT = x (equal angles of an isosceles ∆PRT)∠TRM = 180 − x (angles on a straight line)∠MTR = ∠RMT = [180 − (180 − x)] ÷ 2 (angle sum of isosceles ∆MRT)
∠MTR = x
∠MTR = × ∠PTR
∴ ∠PTR = 2 × ∠MTRb ∠DBA = 120° (exterior angle of
equilateral ∆BCD)∴ ∠ADB = ∠BAD = 30° (angle sum of isosceles ∆ABD)∴ ∠ADC = ∠ADB + ∠BDC= 30° + 60°= 90°∴ AD ⊥ CD
3 a Teacher to check b 724 a SAS b AAS c SAS
5 a No, the angle in the second triangle is not included.
b Yes, RHSc No, sides are not opposite the same
angle.6 a i KM = PN (sides of a square)
MX = NX (sides of an equilateral triangle)∠KMX = ∠PNX = 30° (angle of a square and angle of an equilateral triangle)∴ ∆KMX ≡ ∆PNX (SAS)
ii XK = XP (matching sides of congruent triangles)∴ ∆KPX is isosceles (two equal sides)
b i DF = EG (given)EF = DG (given)DE is common.∴ ∆DEF ≡ ∆EDG (SSS)
ii ∴ ∠DEG = ∠EDF (matching angles of congruent triangles)∴ ∆DEY is isosceles (two equal angles)
iii DY = EY (equal sides of isosceles ∆DEY)and DF = EG (given)By subtraction: YF = YG∴ ∆FGY is isosceles.
iv Let ∠DYE = ∠GYF = x (vertically opposite angles)
∠EDY = (180° − x°) (angle sum of
isosceles ∆DEY)
and ∠GFY = (180° − x°) (angle
sum of isosceles ∆FGY)∠EDY = ∠GFY∴ DE || GF (alternate angles proved equal).
7 a Opposite angles of a parallelogram are equal.
b BC = AD (opposite sides of a parallelogram are equal)BC = DX (given)∴ AD = DX
c ∠DAX = ∠BCY (opposite angles in parallelogram)Since ∆DAX, ∆BCY are isosceles:∠DXA = ∠BYC (equal to ∠DAX, ∠BCY)DX = BY∴ ∆DAX = ∆BCY (AAS)
d AX = CY (matching sides of congruent triangles)AB = CD (opposite sides of parallelogram)AB − AX = CD − CY∴ BX = DYAlso, BY = DX∴ BXDY is a parallelogram (opposite sides are equal).
8
a 2α + 2θ + 2β + 2φ = 360° (angle sum of quadrilateral)α + θ + β + φ = 180° (1)
b In ∆ABD, β + α + 2φ = 180°φ = 180 − (α + β + φ) (2)From (1): θ = 180 − (α + β + θ)∴ θ = φ
c Similarly, α = βABCD is a parallelogram (opposite angles equal)Also, ∆ABC is isosceles (θ = φ)AB = BC
d ∴ ABCD is a rhombus (a parallelogram with two adjacent sides equal).
9 a 12 b 6 c 4.2 d 5.5
10 10.6 11 11
12 In ∆ACD and ∆ABC:
= = 0.4
and = = 0.4
Also ∠DAC = ∠CAB (included angles equal)∴ ∆ACD ||| ∆ABC (SAS)
13 a ∠CDA = ∠BAD = 90° (angles in a square)∠WAB = 90° and ∠YDC = 90° (angles on a straight line)Let ∠AWX = ∠AWB = x°∠WYX = ∠DYC = (90 − x°) (angle sum of ∆WYX)∠YCD = x° (angle sum of ∆CYD)In ∆WBA and ∆CYD:∠WAB = ∠YDC (proved above)∠AWB = ∠YCD (proved)∴ ∆WBA ||| ∆CYD (AA)
b = (matching pairs of sides in
similar triangles)WA × DY = AB × CDBut AB = CD = AD (equal sides of a square)∴ WA × DY = AD × AD
and AD2 = WA × DY
14 a
JK2 = JX2 + KX2
= ( JL)2 + ( KM)2
JK2 = JL2 + KM2
∴ JK2 + KM2 = 4JK2
23--- 16.6̇
C
X
BT
A
Y
P
W
12---
12---
12---
12---
A
B
D Cθθ
φφ
β
αα
β
67---
37---
ADAC-------- 3.2
8------- } (matching
pairs of sides in same ratio)
ACAB-------- 8
20------
WACD--------- AB
DY--------
M L
J K
X
12--- 1
2---
14--- 1
4---
626 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
b Prove ∆NMA = ∆PQB (SAS)Hence show AC = BC (∆ABC is isosceles)Prove NC = PC∴ ∆NPC is isosceles.
Chapter 6Start up
1 a $100 b $3.25 c $42d $1425 e $100 000 f $0.38
2 a $2.52 b $13.96 c $81.50d $8.91 e $215.63 f $0.31
3 a 0.07 b 0.11 c 0.2d 0.085 e 0.105 f 0.02g 0.045 h 0.0675 i 0.1225j 0.032
4 a $60 b $189 c $119.10d $58.21 e $4.20 f $179.29
5 a $73.5 b $4060.80 c $963.20d $4795.50 e $564.08 f $1027.04
6 a $837 b $1610 c $18 619d $538.13 e $35.52 f $2229.98
7 a 8.3% b 7.1% c 0.3%8 a 1152 b 83.2 c 600 d 0.069 a i 24 ii 60 iii 120
b i 52 ii 156 iii 26010 a 10 b 20 c 2 d 411 a 596.26 b 2388.10 c 3425.78
d 357.99
Exercise 6-011 a $120 b $1600 c $7500
d $8 e $3645 f $602 a $78.75 b $3750 c $117 000
d $156 0003 a $11 200 b $1575 c $22 800
d $7400 e $9690 f $9943.75g $618 h $70 000
4 a $1.25 b $80 c $53.10d $0.85 e $1.15 f $4.40g $322.75 h $2.05
5 a $106 600 b $429 c $25.70d $131.73 e $640.15 f $3856.44
6 a 2 years b 39 weeks c 137 daysd 5 years e 4.5% p.a.f i $3150 ii 13.125% p.a.
g 11 years 156 days h 2 years
i 1.9% p.a. j 9.75% p.a.
Skillbank 6A2 a 0.75 b 0.7 c 0.6
d 0.07 e 0.45 f 0.98g 0.4 h 0.85 i 0.264j 0.15 k 0.648 l 0.44
m 0.725 n 0.54 o 0.2624 a 0.61 b 0.04 c 0.29
d 0.9 e 0.085 f 0.271g 0.87 h 0.124 i 0.05j 0.1825 k 0.2 l 0.401
m 0.007 n 0.0628 o 0.3005
Exercise 6-021 a $84 b $1284 c $89.88
d $1373.88 e $173.882 a $270 b $282.15 c $6552.15
d $294.84 e $6846.99 f $846.993 a i $563.08 ii $63.08
b i $2823.33 ii $1233.33c i $3846.79 ii $446.79d i $32 756.04 ii $4756.04e i $7956.75 ii $456.75f i $861.13 ii $21.13g i $255 256.31 ii $55 256.31h i $20 528.77 ii $2278.77i i $41 787.70 ii $3337.70j i $11 099.89 ii $2199.89
4 a $47.29 b $223.73 c $90.83d $1314.50 e $1939.39 f $199.78
Exercise 6-031 a $7919.61 b $12 298.74
c $13 433.47 d $1592.04e $4032.59 f $19 233.59g $30 387.65 h $5418.33i $68 135.36 j $185 196.18
2 a $42.31 b $275.56 c $1337.90d $2039.04 e $4311.43 f $10 324.28g $100 902.81 h $25 625.00 i $25 464.10j $217 461.14
3 a $849.27, $49.27b $13 488.50, $3488.50c $52 751.13, $17 251.13d $53 366.91, $11 366.91e $19 473.43, $2973.43f $21 832.76, $9832.76g $4448.55, $948.55h $1901.76, $101.76
4 a $2910.78 b $2934.27 c $2946.20d $2954.22 e $2957.32
Exercise 6-041 a $2148.79 b $9586.40 c $3411.52
d $935.92 e $1616.28 f $12 750.402 a $307.60 b $18.12 c $5597.21
d $104.09 e $8922.06 f $7155.04g $480.45 h $777.97
Skillbank 6B2 a $88 b $800 c 900
d $60 e $24 000 f $800g $140 h 48 i $550 000j 45 k $22 500 l $75, $66
Exercise 6-051 $1589.25 2 $17 977.643 a i $8215.83 ii 60.0%
b i $4416.17 ii 52.2%c i $16 120 ii 58.6%d i $470.21 ii 45.2%e i $403.03 ii 46.3%
4 a i 90% ii 72.9% iii 53.1%iv 47.8%
b 6–7 years5 a i $7120 ii $5696 iii $2916.35
b 32.77%6 a $11 984.47 b $10 666.18
c $8448.687 a i $10 540 ii $8959
iii $6472.88b 8–9 yearsc 23.2%
8 Yes, since the depreciated value is 0.48 (or 48%) of the original value.
Exercise 6-061 $731 2 $409 3 $78 4 $1255 $162.706 a i $299.80 ii $1199.20
iii $139.11 iv $1338.31v $55.76
b i $119.25 ii $675.75iii $54.06 iv $729.81v $28.07
c i $247.50 ii $1402.50iii $561 iv $1963.50v $40.91
d i $1281 ii $2989iii $807.03 iv $3796.03v $105.45
e i $64 ii $576iii $69.12 iv $645.12v $12.41
7 a $217.50 b $1232.50 c $295.80d $1528.30 e $63.68 f $1745.80
8 a $1100 b $1440 c $340 d 30.9%9 a $2080 b $8320 c $13 200
d $4880 e 14.7%10 a $990 b $190 c 23.75% p.a.11 a $2599 b $3576 c $677
d 10.4% p.a.12 $140.45
Exercise 6-071 a $22 080 b $9580 c 19.2%2 a $9120 b $3120 c 26%3 a $33 280 b $8280 c 6.6%4 a i $209.60 ii $12 576 iii $4576
b i $322.20 ii $38 664 iii $20 664c i $1096 ii $263 040 iii $183 040d i $580.31 ii $6963.72 iii $713.72e i $40.96 ii $983.04 iii $183.04
5 a $33.20 b $29 100 c $68 400d $3540
Exercise 6-081 a Teacher to check.
b $2000 + $1800 + $1580 = $5380c $6000 d $620
2 a i $14 700 ii $14 376b $2376 c $2400d $24
3 a 0.005 b $1500c $297 500 d $1487.50e $294 987.50 f $2987.50g $3000
4 a $40 b $3940c i $39.40 ii $3879.40d i $38.79 ii $3818.19e $118.19 f $120 g $1.81
Exercise 6-091 a $10 000 b $6485.13
c i $729.75 ii $47.58d i 16.99% p.a. ii 16.99% p.a.e July 8, 2004f The first $34.90 is a purchase charge, the
second $34.90 is a credit (or refund)2 a i $896.60 ii $44.83
b i $926 ii $46.30c i $320.60 ii $16.03d i $708 ii $35.40e i $1649 ii $82.45f i $250 ii $12.50
12---
ANSWERS 627
g i $3738 ii $186.90h i $711.50 ii $35.58
3 a $10 b $2.57 c $81.25d $8.06 e $0.43 f $10.85g $56.85 h $22.25
Power plus1 4 years, 61 days 2 $4444.443 a $541.57, $41.57
b $12 838.71, $2838.714 $63 367.49 5 $5839.78 6 3 924 8727 a 17.67 years b 17.67 years
c No (when compounded annually it doesn’t double till the end of the 18th year)
Chapter 6 review1 $14402 a $45 b $208.33 c $495.19
d $67.593 6.25% p.a. 4 1.85 years5 a $2249.73 b $9274.196 a $215.87 b $2063.297 a $2497.04 b $391.048 $16 638.94 9 65.61% 10 $440
11 a i $320 ii $88.80b i $176 ii $48.84
12 23.8%13 a $2600 b $3200 c $600 d 9.23%14 a $15 120 b $5120 c 10.2% 15 a $8.04 b $36.50
Mixed Revision 21 a 8 b 1 c −9 d 2
e −1 f −1
2 a m = 2, n = 1 b k = 3, h = −2
c y = 1 d =
3 20 mL4 a x = ±4.5 b m = ±2.2 c a = ±1.75 87, 88, 89 6 $357 a 0, 3 b no solutions c 1, 3
d , −2 e , 5 f −1, 2
g 7, 1 h − , i −2, 6
8 a b c
d
9 a x = 4, y = 16 and x = −1, y = 1
b x = 2 + , y = 10 + 2 ;
and x = 2 − , y = 10 − 2
10 −2 + , −2 − 11 r = ±
12 a 1.9 or −1.4 b −0.4, −2.6
13 a 4w2 − 5w − 1071 = 0b 17 m × 63 m
14 a x = 1, y = 4 b x = 2, y = 1
c x = − , y = −3
15 a x � 1
b x � 6
c x � −3
16 x = ±2, x = ±317 a ii x = 3, y = −3 b ii x = 4, y = 218 a x = 1, y = 2 b x = 1, y = 5
c x = −1, y = 619 a m = 35 b d = 25 c w = −3020 a w = 64 (angle sum of an isosceles
triangle)b m = 42 (opposite angles of a
parallelogram, angles on a straight line)c x = 48 (angles at centre is 90° since the
shape is a rhombus, angle sum of a triangle)
d h = 23 (the shape is a rectangle, angle sum of right-angled triangle)
e a = 108 (equal angles of an isosceles triangle, angles on a straight line)
f y = 52 (angle sum of an isosceles triangle)
21 160°22 a ∠CDW = ∠ABW (alternate angles,
CD || AB)∠DWC = ∠BWA (vertically opposite angles)∴ ∆CDW ||| ∆ABW (AA)
b 4 ≈ 4.7
23 a 12 b 6
24 10.6 25 165°26 a ∠LPM = ∠NMP, ∠PLN = ∠MNL
b Opposite sides of a rectangle are equalc AASd Matching sides of congruent triangles
LPT and MTNe Rectangle, bisect
27 a PT = RQ (given)RT = PQ (given)PR is common.∆PRT ≡ ∆RPQ (SSS)
b ∠PRT = ∠RPQ (matching angles of congruent triangles)TR || QP (alternate angles proved equal in part b)
c PQRT is a parallelogam because it has one pair of opposite sides equal and parallel.
28 CO = DO (equal radii of circle)∠CEO = ∠DFO (CE⊥AB, DF⊥AB)∠COE = ∠DOF (vertically opposite angles)∴ ∆CEO ≡ ∆DFO (AAS)
29 a ∠P is common.∠PTW = ∠PRQ (corresponding angles, TW || RQ)∴ ∆PWT ||| ∆PQR (AA)
b = (matching sides of similar
triangles)But T is the midpoint of PR.
=
=
∴ TW = PQ
30 a m = 12.375 b 10
31 a No b Yes, SAS c No32 a $1400 b $264 c $1.56 d $7.5433 a $7941.60 b $903.86 c $21 216
d $6080 e $206034 a i $20 400 ii $17 340
iii $14 739b 61.4%
35 a $15 900 b $3900 c 6.5% p.a.36 D 37 B 38 45 months39 $232.33 40 $10.1241 a $8.70 b $878.7042 $6843 a $373.50 b $2116.50 c $465.63
d $2582.13 e $107.59 f $2955.66
Chapter 7Start up
1 a i (3, 2) ii (6, 1)iii (−1, − 6) iv (−5, −4)
b i 6 units ii 2 unitsiii 6 units
c 4.2 units, isoscelesd (−2, 2) e (6, 6)
f i ii −
2 a
b
c
d
3 a N b X c P d P e N f X4 a −2 b 8 c −1 d 5 e 64 f 64
5 a − = − b − = − c 8.25
6 y = 4 7 x = −28 a x = 7 b y = −3 c (7, −3)
Exercise 7-011 a i (3 , 1 ) ii iii
b i (8, ) ii iii
c i (−2 , 3 ) ii iii −
d i (2 , −2 ) ii iii
e i ( , −7) ii iii −
f i (2 , 1) ii iii −
g i (−1 , − 6 ) ii iii 1
35--- 1
4---
23--- 1
2---
3941------ −36
41--------
34--- 1
3--- 1
2---
12--- 4
5---
−1 5±2
-------------------- 1 21±5
-------------------- −5 17±4
-----------------------
3 3±3
-----------------
10 10
10 10
2 2 πR2 A–π
--------------------
23---
−2 −1 0 1 2
−1 0 1 32 54 7614---
−5 −4 −3 −2 −1 0
57---
67---
TWRQ--------- PT
PR--------
PTPR-------- 1
2---
TWRQ--------- 1
2---
12---
522------
13--- 2
3---
x 0 1 2 3
y −3 −2 −1 0
x −2 −1 0 1
y −4 −1 2 5
x −1 0 1 2
y 3 1 −1 −3
x −2 0 2 4
y −4 −1 2 5
28--- 1
4--- 2
8--- 1
4---
12--- 1
2--- 26 1
5---
12--- 41 5
4---
12--- 1
2--- 34 3
5---
12--- 1
2--- 50 1
7---
12--- 29 2
5---
12--- 85 2
9---
12--- 1
2--- 2
628 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
h i (4, −3) ii iii −2
i i (1, 2) ii iii −4
2 a AB = , BC = 1, AC = ; scalene
b XW = , WV = , XV = ;
scalene
c PQ = , QR = , PR = ;
scalene
d MN = 5, NT = , MT = 5; isosceles
e AF = 13, FL = , AL = 13; isosceles
f DX = 10, XP = , DP = 10; isosceles
3 a AB = , BC = , CD = , AD
=
b 25.5 units
c mAB = , mBC = − , mCD = , mAD = −
d A parallelogram, because opposite sides are equal
4 a LM = , MN = , NP = ,
LP =
b mLM = − , mMN = , mNP = − , mLP =
c LN = = MP
d A rectangle, because opposite sides are equal and diagonals are equal
e 68 units2
5 a WX = , XY = , WY =
b i 104 ii 104c They are the same.d A right-angled triangle (using
Pythagoras’ theorem)
6 a The length of each side is units.
b DF = EG =
c A square, because all sides are equal and the diagonals are also equal
7 a The length of each side is units.
b mHJ = , mJK = , mKL = , mHL =
c HK = , JL =
d A rhombus, because all sides are equal and the diagonals are not equal
Skillbank 7A2 a y = 3x + 4 b y = 2x + 10
c b = 5a − 2 d n = 8m − 6e p = 10k − 3 f z = r + 9g h = 4 − 2d h w = 9t − 9
Exercise 7-021 a No b No c Yes d Yes e Yes
f Yes g No h No i Yes j Yes2 b (2, 2)3 They intersect at one point because (1, 2)
lies on each of the three lines.
Exercise 7-031 a − b = c =
d − e =
2 a L b N c N d L e N f P
3 a − b − c
4 a − b c − d −
e − f g − h m
5 a Yes b Yes c A parallelogram
6 a mPQ = mRT = b mPT = mQR = −
c PT = , QP =
d No, because − × ≠ −1
e A parallelogram
Exercise 7-041 a
b
c
d
e
f
g
h
i
j
k
l
20
272
17 20
41 5 52
26 50 16
20
208
80
41 40 41
40
45--- 1
3--- 4
5--- 1
3---
136 34 136
34
53--- 3
5--- 5
3--- 3
5---
170
32 72 104
40
80
58
37--- 7
3--- 3
7--- 7
3---
200 32
15--- 10
3------ 31
3--- 7
5--- 12
5---
23--- 20
7------ 26
7---
65--- 6
5--- 5
6---
16--- 4
5--- 20
17------ 1
k---
ba--- n
3--- 10
21------
211------ 8
3---
73 73
83--- 2
11------
20
4
y
x
y = 4 − 2x
−4 0
2
y
2x = 4y − 8
x
−6 0
6y
x
y − x = 6
40
−6
y
x
3x − 2y = 12
2.50
2.5
y
x
2x + 2y = 5
60
3
y
x
6 − x = 2y
−2 0
4
y
x
y = 4 + 2x
30
5
y
x
5x + 3y − 15 = 0
20
−6
y
x
3x − y = 6
100
−4
y
x
2x − 5y − 20 = 0
20
4
y
x
4x + 2y − 8 = 0
20−0.5
x − 4y − 2 = 0
y
x
ANSWERS 629
Exercise 7-051 a m = 6, b = −5 b m = −3, b = 8
c m = − , b = 9 d m = , b = 6
e m = − , b = 2 f m = 2, b = −
g m = 2, b = −7 h m = − 4, b = 3
i m = , b = 2 j m = , b = −
2 a y = 3x + 7 b y = −5x − 2
c y = + 8 d y =
e y = + 7 f y = − 2
g y = h y = − 4x + 2
i y = − + 1 j m = 4x +
3 a m = , b = 2, y = + 2
b m = − , b = 5, y = − + 5
c m = , b = 4, y = + 4
d m = −1, b = −5, y = −x − 5
4 a y = + 2 b y = − + 3
c y = − 2 d y = −x − 4
5 a C b B c B d C, D e D6 a B, C b A c B, D
d C, D7 a y = 5x + 6 b y = − 4x − 1
c y = − + 4 d y = − + 2
e y = − − 5 f y = − − 2
Exercise 7-061 a i m = 1, b = − 4
ii
b i m = 2, b = 5
ii
c i m = −2, b = 3
ii
d i m = − , b = 1
ii
e i m = − , b = 7
ii
f i m = 6, b = 1ii
g i m = 2, b = − 4
ii
h i m = − , b = − 4
ii
i i m = − , b =
ii
Exercise 7-071 a x + y − 2 = 0 b 3x − y + 2 = 0
c 5x − y + 8 = 0 d x − 2y + 3 = 0e x − 5y + 1 = 0 f x − 2y − 6 = 0g 8x − y + 2 = 0 h 6x − y − 3 = 0i x + y − 10 = 0 j 2x − y + 4 = 0k 8x − 2y − 1 = 0 l x − 2y − 6 = 0
m 3x − 5y + 10 = 0 n x + 2y + 10 = 0o 15x − 3y − 2 = 0 p 3x − 24y − 8 = 0q 4x − 3y − 2 = 0 r 4x − 3y + 15 = 0s 3x − 2y − 12 = 0 t 2x + 5y + 3 = 0
2 a y = −2x − 5, m = −2, b = −5b y = −x − 6, m = −1, b = − 6c y = −2x + 1, m = −2, b = 1
d y = + 2, m = b = 2
e y = 4x − 5, m = 4, b = −5f y = 2x − 6, m = 2, b = − 6
g y = − + 4, m = − b = 4
h y = − , m = b = − (or −3 )
12--- 5
4---
23--- 1
2---
54--- 1
2--- 3
4--- 1
4---
x3--- x 4+
5------------
5x2
------ 4x3
------
3x10------
5x3
------ 34---
23--- 2x
3------
27--- 2x
7------
12--- x
2---
x2--- x
3---
3x2
------
x2--- 4x
3------
x3--- 2x
3------
−2 2
11
4 6 8−4
2
4
6
−2
−8
0
−6
−4
y
y = x − 4
x
−2 2
1
2
4 6−4
2
4
6
−20
−6
−4
8
y
y = 2x + 5
x
−2 2
12
4 6−4
2
4
6
−20
−6
−4
8
y
x
12---
−2 2
−12
4 6−4
2
−20
−4
y
x
23---
−2 2
−23
4 6 8−4
2
0
4
6
8
y
x
−2 2
6
1
4 6 8−4
2
0
4
6
8
y
x
−2 24
2
4
2
−20
−4
−6
−8
4
6
y
x
32---
−2 2
−32
4−20
−4
−6
−8
y
x
54--- 1
2---
−2−4−6 2−5
4
4−2
2
4
6
8
0
−4
−6
−8
y
x
3x2
------ 32--- ,
4x3
------ 43--- ,
3x2
------ 72--- 3
2--- , 7
2--- 1
2---
630 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
i y = − − 2, m = − b = −2
j y = + 4, m = b = 4
k y = + , m = b =
l y = 2x − , m = 2, b = −
Exercise 7-081 a 2x − y + 1 = 0 b x + y + 2 = 0
c 4x − y − 20 = 0 d x − 2y − 10 = 0e 2x − 3y − 4 = 0 f x + 5y + 38 = 0g 3x + y − 4 = 0 h 4x + y + 1 = 0i 3x − 4y + 10 = 0 j 2x + y + 10 = 0
2 i and ii
3 a k: x + 2y − 7 = 0, l: 3x − y + 7 = 0b 4x + y − 20 = 0 c 5x − 7y − 30 = 0d 2x − 3y + 18 = 0 e 3x + 5y − 30 = 0
4 a x − y − 4 = 0 b 4x − 5y + 18 = 0c 5x − 6y + 23 = 0 d 8x + 3y − 10 = 0e 3x + 2y − 6 = 0 f 5x − 3y − 1 = 0g 6x + 11y + 38 = 0 h 13x − 10y + 24 = 0i x + y − 3 = 0 f 4x − 3y − 11 = 0
Exercise 7-091 a 2x − y + 4 = 0 b x + 2y − 11 = 0
c 3x + y − 13 = 0 d 2x − y − 13 = 0e 5x + 4y + 37 = 0
2 a 2x + y + 2 = 0 b x + 3y − 4 = 0c 2x − 3y − 19 = 0 d 2x + y − 2 = 0e 2x + 9y − 8 = 0
3 a x + y − 9 = 0 b 3x − 4y + 22 = 0c 2x + y − 11 = 0 d x + 2y − 16 = 0
4 a x − 3y + 3 = 0 b 3x + y − 11 = 05 a 4x + 5y − 40 = 0 b A(10, 0)
c 5x − 4y − 50 = 0 d B(0, −12.5)
6 a i m = − ii (6, 6)
iii 3x − 4y + 6 = 0
b i m = − ii (1, 2)
iii 7x − 3y − 1 = 0c i m = −2 ii (−2, 1)
iii x − 2y + 4 = 0
d i m = − ii (−2, −3)
iii 3x − y + 3 = 0
e i m = − ii (3, −1)
iii 2x − y − 7 = 0
f i m = − ii (2, 1)
iii 8x − y − 15 = 0
7 a x − 4y + 10 = 0 b x + 2y + 4 = 0c x − 4y − 2 = 0 d x − y + 1 = 0e (6, 1)
8 a i x − 2y + 4 = 0 ii (− 4, 0)
iii 2x + y − 7 = 0 iv A(0, 7), B(3 , 0)
b Area ∆AOB = 12.25 units2
Area ∆RQB = 14.25 units2
Area ∆RQB − Area ∆AOB = 2 units2
Skillbank 7B2 a 19 b $7.50 c 87.5
d $20.20 e $3.76 f 40g $0.93 h 89.6 i $270j 9.7c (10c) k $152.76 l $8.26m 31.54 n $1.01 o 42.6p $2431.76
4 a 10 b 124 c $490d $1.72 e 160.4 f $255g 79.4 h $0.76 i 8.8c (9c)j $1.45 k $768 l 64
6 a 100 b $0.60 c 2.5 d $1.35e $1.84 f $4.23 g 4c h 6.6i $0.48 j $6.95 k $0.40 l 42.9
Exercise 7-101 a i 5x + 2y − 18 = 0
ii 3x − 4y − 16 = 0
iii (0, 9) iv (0, − 4) v 26 units2
b i x − 5y + 20 = 0ii x + 2y + 6 = 0
iii (0, 4) iv (0, −3) v 35 units2
c i 3x − y − 46 = 0ii 7x + 15y + 66 = 0
iii (15 , 0) iv (−9 , 0)
v 123 units2
2 a 5x − 2y − 25 = 0 b 5x + 7y − 25 = 0
c 5 d t = −10
3 a DE = EF = FG = DG = 5 unitsb For DE and GF, m = 0
For DG and EF, m = −
c Diagonal DF, m = −
Diagonal EG, m = 2
Since − × 2 = −1 it is true that DF ⊥ EG
d Midpoint of DF = (0, 0)Midpoint of EG = (0, 0)The diagonals bisect each other because their midpoints are the same.
e Opposite sides are equal and parallel, adjacent sides are equal, diagonals bisect each other at right angles
4 a units b units c (−1, −1)
d (−1, −1) e No, since mPR × mQS ≠ −1f Rectangle, diagonals are equal and
bisect each other but not at right angles
5 a CE = units, DF = units
b and
c mCE = , mPR = −
∴ CE ⊥ DF because × − = −1
d Square, diagonals are equal and bisect each other at right angles
6 a BC = DE = units,
CD = BE = units
b mBC = , mCD = − , mDE = , mBE = −
c Midpoint of BD = ,
Midpoint of CE =
d Parallelogram, opposite sides are parallel and equal
7 a AC = BD = units
b Midpoint of AC = (1, 2), midpoint of BD = (1, 2)
c mAC = −5, mBD = , ∴ AC ⊥ BD
d The diagonals are equal and bisect each other at right angles.
8 Midpoint of KM = Midpoint of LN
=
mKM × mLN = 1 × −1 = −19 Teacher to check.
10 a mJK = − , mLM = − , mKL = − , mJM = −
b JK = LM = units, KL = JM
= units∴ JKLM is a parallelogram because opposite sides are parallel.
11 Teacher to check. 12 Trapezium
13 ST = WX = units
TW = SX = units
XS ⊥ ST because mXS = − , mST = 6∴ STWX is a rectangle because opposite sides are equal and angles are right angles.
14 a Midpoint of TU = A(4, −1)Midpoint of UV = B(0, 3)Midpoint of SV = C(−5, −1)Midpoint of ST = D(−1, −5)
b Gradient of AB = −1 = gradient of CD
Gradient of AD = = gradient of BC
AC = 9 units, BD = units∴ ABCD is a parallelogram.
15 a X(3, ) Y(−1 , 3 )
b mXY = − , mCB = −
∴ XY || CB
c XY = , CB =
∴ CB || 2.XY
16 a i mLM = ii mLM = iii mMN =
b L, M and N are collinear points.
Exercise 7-111 a C b A c B d C e D2 a B b A c C d D e C
3 a
x2--- 1
2--- ,
x3--- 1
3--- ,
x10------ 7
10------ 1
10------ , 7
10------
92--- 9
2---
0
y
xP
a
d
b
c
4x + y − 10 = 0
x − y − 5 = 0
x − 5y − 13 = 0
x + 3y + 3 = 0
(3, −2)
43---
37---
13---
12---
18---
12---
13--- 3
7---
1721------
25---
43---
12---
12---
6 5 6 5
130 130
12--- 1
2---,⎝ ⎠
⎛ ⎞ 12--- 1
2---,⎝ ⎠
⎛ ⎞
113
------ 311------
113
------ 311------
61
6556--- 4
7--- 5
6--- 4
7---
112--- 21
2---,⎝ ⎠
⎛ ⎞
112--- 21
2---,⎝ ⎠
⎛ ⎞
104
15---
212---
12---,⎝ ⎠
⎛ ⎞
13--- 1
3--- 5
2--- 5
2---
40
29
37
2 3716---
45---
65
12--- 1
2--- 1
2---
23--- 2
3---
12--- 117 117
13--- 1
3--- 1
3---
0
2
y
x
ANSWERS 631
b
c
d
e
f
g
h
i
4 a
b
c
d
e
f
g
h
i
5 a i y � −x and y �
ii y � −x and y �
b i y = − − 1 ii y � − − 1 and y < 2
c i y = 3 − 3x ii y = x + 3iii A: y > x + 3 and y � 3 − 3x
B: y < x + 3 and y � 3 − 3xC: y < x + 3 and y � 3 − 3xD: y > x + 3 and y � 3 − 3x
d i y = ii p: x = 3, q: x = − 4
iii y � and x � − 4 and x � 3 and
y � 0iv 14 units2
Exercise 7-121 a
0
1
y
x
0 3
y
x
0−4
y
x
0
−2
y
x
0 4−1
y
x
1
5
0
y
x
2
0
y
x
−2
4
0
y
x
0
yy = 2x
x
0
7
7
yy = 7 − x
x
0
1
1
y
y = 1 − x
x
0
2
(1, 6)
yy = 4x + 2
x
0
2
2
yy = 2 − x
x
0(−1, −1)
−4
yy = 3x − 4
x
0
−66
y
x − y = 6
x
0−4
2
y
x = 2y − 4
x
0
−6
12
y
12 = x − 2y
x
x2---
x2---
12--- x x
2---
13 2x–7
-------------------
13 2x–7
-------------------
x −3 −2 −1 0 1 2 3
y 9 4 1 0 1 4 9
632 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
b
2 a
b
3 a
b
4 a
b
5 a
b
6 a
b
Power plus1 D(−3, 5)
2 a y = x − 2
b C(9, 4), 4 is the y-coordinate3 a k = 5 b k = −2
4 a − b 3x + 2y + 2 = 0 or y = − − 1
5 B(2, −1)6 x + 5y + 9 = 0, x − y − 1 = 0, 2x + y + 3 = 0.
Point of intersection is
7
Area = (15 + 5)
= 5 × 20
= 100 units2
Chapter 7 review1 a (4, 2) b units c m = 4
2 a T(−1, 6) b Z(5, 10)
c mTZ = d mAC =
e Length of AC = units
= units
f Length of TZ = units
∴ AC || TZ and AC = 2.TZ3 a No b Yes c No
d Yes e Yes f Yes4 a Neither b Parallel
c Perpendicular d Parallele Neither f Parallel
5 a i m = −2 ii m =
b i m = 3 ii m = −
c i m = − ii m = 3
d i m = 2 ii m = −
e i m = − ii m = 4
f i m = ii m = −
6 a
b
−2−4 2 4
2
4
6
8
10
0
y
x
y = x2
x −3 −2 −1 0 1 2 3 4 5
y 15 8 3 0 −1 0 3 8 15
−2−4 2 4 6
2
4
6
8
10
0
12
14
16
y
x
x −2 −1 0 1 2
y −8 −1 0 1 8
−2−4 2 4
2
−2
−4
−6
−8
4
6
8
0
y
x
x −3 −2 −1 0 1 2 3
y 1 2 4 818--- 1
4--- 1
2---
−2−4 2 4
2
4
6
8
0
y
x
x −3 −2 −1 0 1 2 3
y 8 4 2 1 12--- 1
4--- 1
8---
−2−4 2 4
2
4
6
8
0
y
x
x −3 −2 −1 0 1 2 3
y 1 2 1411------ 2
3--- 1
3--- 1
3--- 2
3--- 4
11------
−2−4 2 40
y
x−2
−4
2
4
23---
32--- 3x
2------
− 23--- − 5
3---,⎝ ⎠
⎛ ⎞
−2−4−6−8 2 4 6 80
y
x
x = 4x = −6
−2
−4
−6
−8
2
4
6
8
y = − 3x_2
y = 6 − x_2
102
------
68
23--- 2
3---
208
2 52
52
12---
13---
13---
12---
14---
32--- 2
3---
40
y
x
x − y = 44
−4
160
y
x
x + 2y = 168
ANSWERS 633
c
d
7 a m = −3, b = 5 b m = 6, b = 10
c m = , b = − 4 d m = , b = −2
e m = − , b = f m = 2, b = −5
8 a y = 2x − 4 b y = − + 5
9 a m = −3, b = 5
b m = , b = 3
c m = , b = −3
10 m = − , b = 5
11 a 3x + y − 9 = 0 b 5x − 4y + 4 = 0c 2x + 7y − 3 = 0 d 3x − 2y − 36 = 0e x + 4y − 10 = 0 f x − 6y − 12 = 0
12 a y = + 2, m = , b = 2
b y = + , m = , b =
c , m = − , b = 6
13 a 3x − y − 10 = 0 b 2x − 3y + 26 = 0c x + 5y + 33 = 0 d x − 4y − 10 = 0
14 a 3x − 5y − 20 = 0 b x + y + 3 = 0c 2x − 5y − 16 = 0 d 8x − 5y + 22 = 0
15 a 4x − y − 19 = 0 b 3x − 2y − 18 = 0c x − 2y − 10 = 0 d 5x − 2y − 26 = 0
16 8x + 3y − 95 = 0 17 5x − 4y + 4 = 018 a x + y − 6 = 0 b Q(0, 6)
c 5x − 8y − 30 = 0 d P(0, −3.75)
e 29.25 units2
19 PN = LM = units,
PN = PL = units,
mPN = , mPL = − , ∴ PN ⊥ PL
∴ LMNP is a square because all sides are equal and PN ⊥ PL.
20 A only
21 a
b
c
22 a
b
c
Chapter 8Start up
1 a i 17.6 ii 5.0 iii 70.5b i 65° ii 70° iii 28°
2 a 37.2 m b 164.5 cm c 270.3 mm3 a 29°46′ b 55°7′ c 34°53′
Exercise 8-011 a 64.7 cm b 14.2 cm c 54.5 cm
d 18.5 cm e 5.1 cm f 17.4 cmg 48.8 cm h 59 cm i 17.5 cm
2 a 39° b 56° c 43°3 a 52°57′ b 64°37′ c 45°1′4 a 73° b 5.7 m5 10° 6 65 m 7 60° 8 3.60 m9 a 11.6 m b 11.2 m
40
y
x
3x + 4y − 12 = 0
3
80
y
x
5x − 2y = 40
−20
13--- 2
3---
52--- 1
2---
x2---
−1−2−3−4 1 2 3−1
2
3
4
5
0
1
−2
−3
−4
yy = 5 − 3x
x
−1−2−3 1
1
2
2 3−1
1
2
3
4
0
−2
−3
y
x
y = + 3x_2
12---
32---
−2−4−6 23
2
4 6 8−2
2
4
6
8
0
−4
−6
−8
y 3x − 2y = 6
x
−2−4−6 2
5
−2
4 6 8
2
4
6
8
0
y
y = − x + 5
x
2_5
25---
2x5
------ 25---
3x5
------ 65--- 3
5--- 6
5---
9x2
------ 6+ 92---
34
34
35--- 5
3---
0
6
y
x
0 2
y
x
0 1
3
y
x
0−1.5
3
y
x
0 3
3
y
x
0 3.5−2
y
x1_3
634 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
10 33.64 m 11 11 m 12 79°13 480 m 14 48 m15 a 052° b 206° c 300°
d 125° e 238° f 338°16 a 68 km b 015°17 a 58 km b 301°18 2757 km 19 367 km20 a 14.1 m b 53.0 m21 389 m
Exercise 8-021 a 43° b 16° c 87.45°
d 34.8° e 51°43′ f 72°22′2 a 0.6 b 0.75
c cos β = , cos α = , sin β =
d sin F = , sin E = , cos F =
e cos Y = , sin Y = , sin X =
f cos φ = , sin φ = , cos =
3 a 0 b 1 c 1
4 a sin = , cos = , tan = , =
so = tan
b sin = , cos = ,
tan = , =
so = tan
c sin = , cos = ,
tan = ,
= so = tan
5 a tan A = b tan Y = c tan X =
d tan P = e tan Q =
6 a sin X = b cos X = c sin X =
d cos X =
7
8 a 45° b 30° c 30°
9 a 8 b 12 c
10 11 Teacher to check.
Exercise 8-031 a P b P c N d N e P f N2 a 10° b 70° c 50° d 83° e 65° f 12°3 a − 0.89 b 0.95 c −1.17 d − 0.11
e − 0.19 f 0.26 g 0.05 h − 0.60i − 0.43
4 a −cos 38° b −tan 59.5° c sin 79°25′d −tan 22° e −cos 27.3° f −cos 59°35′g sin 85° h −tan 9.2° i −cos 45°j −tan 39°50′ k sin 4.5° l sin 75°
5 a b −1 c d − e 0
f g − h − i 0 j −
6 a 123° b 143° c 110° d 130° e 173°f 135° g 100° h 155° i 114° j 147°k 105° l 118°
7 a 34°51′, 145°9′ b 48°3′, 131°57′c 20°34′, 159°26′ d 6°12′, 173°48′e 27°2′, 152°58′ f 64°9′, 115°51′
8 a 53°8′ b 126°52′c 16°42′ d 163°18′e 53°8′, 126°52′ f 25°23′, 154°37′g 136°39′ h 136°28′i 61°3′, 118°57′ j 69°18′k 41°49′, 138°11′ l 143°8′
9 a 30.5 b 23.1 c 44.3d 11.5 e 0.61 f 69.6
10 Teacher to check. 11 Teacher to check.
Exercise 8-041 a 18.4 b 21.1 c 105.02 a a = 20.51 b b = 11.91 c c = 12.58
d d = 4.10 e e = 30.85 f f = 3.553 a 27° b 37° c 54°4 a 44.5° b 46.6° c 32.0°
d 67.3° e 18.8° f 31.8°5 a 149°7′ b 129° c 142°8′
d 135°29′ e 129°29′ f 162°13′6 a k = 6.1 cm b w = 28.7 m
c p = 8.3 m7 a 46° or 134° b 48°
c 55° or 125° d 44° or 136°e 51°
8 a 75° or 21° b 41° c 84°9 79 m 10 106°31′
11 a 113° b 1042 cm12 a 110° b 131.6 m13 235° or 205°14 d 124.7 m15 b 595 m
Skillbank 8The following are the exact answers. Check
how close your estimates were.2 a 331 b 157 c 1587 d 255
e 421 f 203 g 413 h 734i 6723 j 15 744 k 276l 72.43 (to two decimal places)
4 a 177.4967 b 416.752 c 16.6957d 5.0237 e 38.36 f 4.4066g 5.8065 h 9097.3444 i 8.1998
Exercise 8-051 a 5.6 b 13.1 c 35.82 a a = 8.30 b c = 54.52 c e = 88.41
d b = 16.33 e d = 19.44 f f = 40.723 a 70° b 33° c 109° d 131°4 a 111.8° b 108.0° c 121.2°
d 23.0° e 60.0° f 82.8°5 20.8° 6 64°40′ 7 99° 8 47 km
Exercise 8-061 a 413.4 m2 b 463.1 cm2
c 326.9 mm2 d 132.9 mm2
e 320.4 cm2 f 0.1 m2
2 a 97.4 m2 b 463.6 m2 c 246.2 m2
d 227.6 m2 e 93.5 m2 f 152.2 m2
3 a 46 cm2 b 20 cm2 c 294 cm2
d 321 cm2
4 a 225 m b 2770 m2
5 a 130° b 82 m c 766.7 m2
6 852.7 m2
7 a 1256.6 cm2 b 418.9 cm2
c 173.2 cm2 d 245.7 cm2
8 a 112° b 37 cm2 c 740 cm3
Exercise 8-071 a 10.2 m b 16.1 mm c 17.1 cm
d 13.1 m e 3.9 m f 18.2 m2 a 32° b 142° (or 38°)
c 34° d 55° e 37° f 125°3 105 m4 a 32°–23° = 55° (exterior angle of a
triangle)b 108.45 m c 89 m
5 a 15.4 b 15.4c Results are the same. When using the
sine rule:
=
becomes:
d = (since sin 90° = 1)
which is the same result when using the sine ratio.
6 7.5 km 7 486 km
Power plus1 a x = 12 b y = 16 c g = 10
2 a b
3 a 45° b 60° c 30°d 150° e 120° f 135°
4 a 0 b 1c, d and e Teacher to check.
5 Teacher to check. 6 Teacher to check.7 a 67°1′ or 112°59′ b 30° or 150°
c 20.7° or 159.3°
Chapter 8 Review1 a 10.9 m b 4.4 m c 11.5 cm2 a 64°59′ b 48°59′ c 57°12′3 a 12° b 0.114 km or 114 m4 a 27° b 38°48′ c 61.63°
5 a cos α = b tan A =
6 a 30° b 45° c 60°
7 a 24 b 48 c =
8 a −cos 17°13′ b sin 25.6°c −tan 67.19° d sin 84°43′
9 a 23°19′ b 64°28′c 25°32′ or 154°28′ d 117°2′e 114°27′ f 27°2′ or 152°58′
10 a 0.4 m b 14.8 cm c 136.4 mm11 a 81°54′ b 77°24′ or 102°36′
c 49°37′12 a 6.8 m b 112.1 mm c 7.6 cm13 a 95.7° b 55.9° c 125.1°14 a 165 cm2 b 286 m2 c 30 mm2
513------ 12
13------ 12
13------
4041------ 9
41------ 9
41------
53
------- 23--- 5
3-------
54
------- 114
---------- 114
----------
35--- 4
5--- 3
4---
35---
45---
34---
sincos------------
513------ 12
13------
512------
513------
1213------
512------
sincos------------
5 1–
2 3---------------- 5 1+
2 3-----------------
5 1–
5 1+-----------------
5 1–
2 3----------------
5 1+
2 3-----------------
5 1–
5 1+----------------- sin
cos------------
6091------ 1.3̇ 2
3---
940------ 40
3----------
1161------ 7
25------ 2
3---
1
3-------
2
3 2
35 3 m
12--- 1
2------- 1
2---
32
------- 32
------- 3 1
2-------
d90°sin
------------------ 12.856°sin
------------------
12.856°sin
------------------
3 1– 12--- 3 1–( )
6061------ 39
80------
3 48
2------- 24 2
ANSWERS 635
Chapter 9Start up
1 a quantitative, continuousb quantitative, discretec quantitative, continuousd quantitative, continuouse categoricalf quantitative, discreteg categoricalh quantitative, continuousi quantitative, discretej categorical
2 a 34 b 8 c 4 d 44%3 a i 26% ii 7% iii 5%
b i 312 or 313 ii 187 or 188iii 75
4 a
b
c 4 d 115 a 30 b 2 c 27%6 a i 5 ii 5.3 iii 5 iv 5
b i 9 ii 16.4 iii 15 iv 15c i 9 ii 8.7 iii 8.5 iv 7
Exercise 9-011 a i 5 ii 16.125 iii 15.5
iv 15b i 6 ii 3.1 iii 3 iv 2c i 6 ii 6.75 iii 6.5 iv 6, 8d i 1.6 ii 7.6 iii 7.5 iv 7.5
2 a range = 5, mean = 17.325, median = 17, mode = 17
b range = 52, mean = 28.5, median = 29, mode = 28, 29
c range = 6, mean = 5.4, median = 5, mode = 5
d range = 5, mean = 12.4, median = 13, mode = 13
3 a
= 7.42
b i 7 ii 8 iii 8
4 a
b i 36 ii 70.3 iii 70 iv 685 a i 21 ii 74 iii 74.5 iv 76
b 69, 78, 75, 90, 81, 81c i 21 ii 79 iii 79.5 iv 81d Range, no change; mean, median and
mode have increased by 56 a i 26 ii 20.5 iii 17.5 iv 16
b i 26 ii 30.5 iii 27.5 iv 267 a 120 b 154 c 348 85%9 11 or 42
10 a 12, 12; 11, 13; 10, 14; 9, 15b 17, 17c Any three scores who have a sum of 62,
such as 20, 20, 22.11 a 342 b 38
Exercise 9-021 a i no outliers
ii some clustering near 5 to 18iii symmetrical
b i outlier at 24ii clustered near 15 to 18
iii positively skewedc i no outliers
ii clustered near 13, 16iii not symmetrical, not skewed
d i no outliersii clustered near 3, 4 and 7, 8
iii symmetricale i no outliers ii no clustering
iii symmetricalf i no outliers ii clustered near 7
iii negatively skewedg i one outlier at 25
ii clustered near 18, 19iii positively skewed
h i two outliers ii clustered near 4, 6iii symmetrical
i i no outliers ii clustered near 12iii positively skewed
j i one outlier (24)ii clustered near 5, 9
iii symmetrical (if outlier is ignored)k i no outliers ii clustered near 51
iii symmetricall i no outliers
ii clustered near 20, 26iii symmetrical
2 a
b no outliersc i clustered near 69–70 and 72–74
ii not symmetrical, not skewed3 a
b no outliersc i clustered around 2–3
ii positively skewed4 a 26
b Clustering in 30s, 40s and 50sc Skewed towards lower marksd The test was difficult, or most students
did not study.5 a Yes, 7
The opposition team in this game was missing some of its better players, possibly through injury.
b Clustered around 2c Symmetrical (especially without the
outlier). The team is consistent in the number of goals it scores.
6 a No, because results of 3, 2, 1 are not far from the main body of data which begins at 5.
b Clustering occurs around 8–9.c Skewed towards the higher marksd The students practised spelling the
words before the test, or the words were very easy to spell.
Exercise 9-031 a Richard 167, James 45
b Richard 46, James 41.8c Richard 7, James 46.5d James because he is the more consistent
batsman. However, Richard who frequently scores 0, has had two very good scores.
2 a i $95 250 ii $91 500b The median because the value of the
mean has been affected by $170 000 which is much higher than the other salaries.
Length of queue Frequency
1 5
2 4
3 7
4 9
5 7
6 3
7 1
2
0
4
6
8
10
1 2 3 4 5 6 7
Frequency histogramand frequency polygon
Length of queue
Freq
uenc
y
x f xf
4 4 16
5 3 15
6 6 36
7 10 70
8 16 128
9 6 54
10 3 30
11 2 22
50 371
x
Stem Leaf
5 5
6 0 2 2 4 4 7 8 8 8
7 2 2 3 3 4 5 7
8 0 1
9 1
67 68 69 70 71 72 73 74 75
2
0
4
6
8
10
1 2 3 4 5 6 7 8 9
Frequency histogram
Number of children
Freq
uenc
y
636 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
3 a i 1.5 ii 0.5 iii 0b A combination of the mean and mode; the average number of
accidents is 1.5 per workplace although many of the workplaces had no accidents.
4 a Toyotab The data is categorical.c The range, mean and median can only be found for numerical data.
5 a Jade: range 39, mean 66, median 69Amy: range 62, mean 64.6, median 73
b Amy because she is more consistent, apart from the score of 23, which could be due to illness. Amy’s mean of 64.6 has been affected by the mark of 23.
6 a i 64 ii 63b i 70–79 (the 70s) ii 60–69 (the 60s)
Exercise 9-041 a
Σf = 40b
2 a
b 17 c 80%
d
e Median = 15
3 a
b
Number of calls f cf
0 5 5
1 6 11
2 6 17
3 6 23
4 9 32
5 6 38
6 1 39
7 1 40
5
0
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7
Cumulative frequency histogramand polygon
Number of calls/minute
Freq
uenc
y
Median = 3
Ages f cf
13 1 1
14 6 7
15 10 17
16 7 24
17 5 29
18 1 30
5
0
10
15
20
25
30
13 14 15 16 17 18
Cumulative frequency histogramand polygon
Ages
Cum
ulat
ive
freq
uenc
y
10
0
20
30
40
50
60
70
17 18 19 20 21 22
Cumulative frequency histogramand polygram
Score
Cum
ulat
ive
freq
uenc
y
Median = 20
5
0
10
15
20
25
30
0 1 2 3 4 5
Cumulative frequency histogramand polygram
Score
Cum
ulat
ive
freq
uenc
y
Median = 2.5
ANSWERS 637
4 a
Σf = 45
b
c Median = 8
Exercise 9-051 a
b
c 50–54
d
e 55.1
2 a
b i 145.4 ii 140–144
c
3 a
b Mean = 0.121c Modal class = 0.05–0.09
Number of seedlings f cf
4 4 4
5 2 6
6 4 10
7 8 18
8 10 28
9 9 37
10 4 41
11 3 44
12 1 45
10
0
20
30
40
50
4 5 6 7 8 9 10 11 12
Cumulative frequency histogramand polygon
Number of seedlings/punnet
Cum
ulat
ive
freq
uenc
y
Classinterval
Classcentre Frequency fx cf
35–39 37 5 185 5
40–44 42 4 168 9
45–49 47 6 282 15
50–54 52 2 624 27
55–59 57 8 45 35
60–64 62 3 186 38
65–69 67 5 335 43
70–74 72 6 452 49
75–79 77 0 0 49
80–84 82 0 0 49
85–89 87 1 87 50
50 2755
6
4
2
0
8
10
12
37 42 47 52 57 62 67 72 77 82 87
Frequency histogram and polygon
Marks in maths exam
Freq
uenc
y
10
0
20
30
40
50
37 42 47 52 57 62 67 72 77 82 87
Cumulative frequency histogramand polygon
Marks (class centres)
Cum
ulat
ive
freq
uenc
y
Median ≈ 54
Heights ofstudents (cm)
Classcentre, x f fx cf
130–134 132 3 396 3
135–139 137 7 959 10
140–144 142 10 1420 20
145–149 147 8 1176 28
150–154 152 6 912 34
155–159 157 4 628 38
160–164 162 2 324 40
Σf = 40 5815
10
0
20
30
40
132 137 142 147 152 157 162
Cumulative frequency histogram and polygon
Heights (class centres)
Cum
ulat
ive
freq
uenc
y
Median ≈ 144.5
Classinterval x f cf fx
0.00–0.04 0.02 10 10 0.2
0.05–0.09 0.07 12 22 0.84
0.10–0.14 0.12 9 31 1.08
0.15–0.19 0.17 9 40 1.53
0.20–0.24 0.22 6 46 1.32
0.25–0.29 0.27 4 50 1.08
Σf = 50 Σfx = 6.05
638 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
d
e Median ≈ 0.12f The mean and median are the most useful measures of location.
The mean and median are approximately equal; both are much larger than the mode.
4 a
b 105–109
c
d ≈ 99.9 km/h
5 a
b 100–109
c
d ≈ 99.7
e Both methods of grouping give similar results. Grouping of 70–79, … condenses data more than grouping of 75–79, … and so some information about actual speeds of vehicles may be lost.
Skillbank 92 a $408 b 132 c 200
d $408 e $672 f 81g $517 h $1170 i 525j 364 k $262.50 l $954.60
4 a 330 b $240 c 1600 d $470e $175 f 370 g $51 h $105i 68 j $388 k $108 l $237.50
Exercise 9-061 a Q1 = 4, Q2 = 8, Q3 = 16
b Q1 = 18.5, Q2 = 21, Q3 = 22.5
c Q1 = 6.5, Q2 = 9, Q3 = 13
d Q1 = 55, Q2 = 71, Q3 = 78
2 a 9 b 3 c 15 d 2 e 3 f 22.53 a 9 − 6 = 3 b 50 − 48 = 2 c 6 − 3 = 3
d 24 − 22 = 24 a Range for X is 10, and for Y is 10.
b Median for X is 5, and for Y is 9.c Interquartile range for X is 5 and for Y is 4.d Technician Y’s spread of calls is less, since the interquartile range
is 4 compared with 5 for X.
5 a
b Median speed for A is 73 km/h, and for B is 71 km/h.c Range for A is 30 km/h and for B is 24 km/h.d Interquartile range for A is 15 km/h and for B is 8 km/h.e Suburb B is much safer for driving because the speeds are more
clustered (interquartile range of B < half that of A). Also the median speed is lower, while the range is significantly lower.
10
0
20
30
40
50
0.020.07
0.120.17
0.220.27
Blood alcohol levels (class centres)
Cum
ulat
ive
freq
uenc
y
Classinterval
Classcentre, x
f fx cf
75–79 77 2 154 2
80–84 82 4 328 6
85–89 87 5 435 11
90–94 92 3 276 14
95–99 97 7 679 21
100–104 102 7 810 28
105–109 107 11 1177 40
110–114 112 9 1008 49
115–119 117 1 117 50
Σf = 50 4995
10
0
20
30
40
50
77 82 87 92 97 102 107 112 117
Cumulative frequency histogram and polygon
Speed (km/h) (class centres)
Cum
ulat
ive
freq
uenc
y
Median ≈ 102 km/h
x
Classinterval
Classcentre f fx cf
70–79 74.5 2 149 2
80–89 84.5 9 760.5 11
90–99 94.5 10 945 21
100–109 104.5 19 1985.5 40
110–119 114.5 10 1145 50
Σf = 50 4985
10
0
20
30
40
50
74.584.5
94.5104.5
114.5
Cumulative frequency histogram and polygon
Speeds (class centres)
Cum
ulat
ive
freq
uenc
y
Median ≈ 102 km/h
x
B A
8 5
9 8 8 7 4 3 3 3 2 0 6 0 0 1 2 3 5 5 7 8 9
9 9 6 5 5 4 4 3 3 2 2 1 1 0 0 0 7 0 0 2 2 3 3 5 5 5 6 6
2 0 0 8 0 2 3 4 5 5 5 8
9 0
ANSWERS 639
6 a
b The range for Day 1 is 36, and for Day 2 is 44.c The interquartile range for Day 1 is 13, and for Day 2 is 20.d Day 2, because the interquartile range and the range are
significantly larger than for Day 1.
7 a
b
c Median ≈ 37Interquartile range ≈ 45–21
= 23
Exercise 9-071 a 7 b 5.5 c 8 d 2.5
e
2 a
b Team 1 is more consistent because the interquartile range (the length of the box) is smaller than that of team 2.
3 a 21 hours b 16 hours c 24 hours d 8 hourse i 75% ii 25%
4 a i 9 ii 10b The median mark for Class 1 is 6.5, and for Class 2 is 5.5.c The interquartile range for Class 1 is 3, and for Class 2 is 4.d Class 1 is more consistent.e i 75% ii less than 50%
5 C 6 C7 a
b The interquartile range for the Bushrangers is 60, and for the Ghosts is 73.
c Bushrangers are more consistent. Their interquartile range is lower and the range of the Bushrangers is 152 compared to 190 for the Ghosts.
Exercise 9-081 a ≈ 6.4, σ ≈ 2.7 b ≈ 23.6, σ ≈ 2.6
2 a ≈ 13.71, σ ≈ 2.12 b ≈ 7.8, σ ≈ 1.83
c ≈ 22.73, σ ≈ 0.90 d ≈ 12.29, σ ≈ 1.19
e ≈ 4.07, σ ≈ 1.33
3 a i 4 ii 4 iii 1.09b i 4 ii 4 iii 1.41c i 4 ii 4 iii 1.63
4 a ≈ 7.0, σ ≈ 2.0 b ≈ 46.4, σ ≈ 13.4 c ≈ 49.3, σ ≈ 1.4
Exercise 9-091 a Men: ≈ 71.4, σ ≈ 6.8 Women: ≈ 77.5, σ ≈ 7.0
b Yes, the mean of women’s pulse rates is much higher, which may be due to the stresses involved in shopping (and looking after children at the same time).
2 a i ≈ 150.1, σ ≈ 9.5 ii ≈ 165.9, σ ≈ 11.2
iii ≈ 158, σ ≈ 13.0
b The boys are much taller than the girls, since their mean is much larger than the girls’ mean. The mean of the class is the average of the girls’ and boys’ means.
3 The first set, since the standard deviation is less than that of the second set.
4 a Team A: ≈ 122.9, σ ≈ 27.0
Team B: ≈ 120.9, σ ≈ 23.6
b Team B is slightly more consistent as its standard deviation is 23.6, compared with 27.0 for Team A.
5 a Vatha: ≈ 13.8, σ ≈ 0.5 Ana: ≈ 14.1, σ ≈ 0.7
b Vatha is more consistent as the standard deviation for her times is significantly lower than the standard deviation for Ana’s times.
6 a Test 1: ≈ 58.3, σ ≈ 14.5 Test 2: ≈ 67.7, σ ≈ 13.3
b The mean for Test 2 is higher than for Test 1 and the standard deviation for Test 2 is lower than the Test 1. This means the results are higher and not as spread, so the class has benefitted from the remedial work.
Exercise 9-101 a i The range for science is 47, and for English is 44%.
ii The interquartile range for science is 22, and for English is 14.iii The standard deviation for science is 14.7, and for English is 10.0.
b Mean mark for science is 63.5, and for English is 65.9.c The interquartile range, range and standard deviation for English
are all lower than those for science. Also the mean is higher for English, so student marks for English are better than those for science.
2 a Male 53, Female 46 b Range: Male 52, Female 45
Day 1 Day 2
0 8
9 8 7 5 5 5 4 2 2 1 1 2 3 4 5 5 6 6 6 7 8 8 9
9 8 7 6 4 2 2 1 1 0 2 2 4 7 8 8
8 8 5 1 0 0 3 1 2 3 5 6 7 7
6 7 0 4 0 2 3 8
5 2
Class interval Frequency Cumulative frequency
0–9 1 1
10–19 8 9
20–29 5 14
30–39 9 23
40–49 11 34
50–59 4 38
60–69 2 40
0–9
10–1
9
20–2
9
30–3
9
40–4
9
50–5
9
60–6
9
5
0
Number of SMS communications
Cumulative frequencyhistogram and polygon
Cum
ulat
ive
freq
uenc
y
10
15
20
25
30
35
40
Med
ian
Q1
Q3
4 5 6 7Number of hours
8 9 10
22 24 26 28 30 32 34 36Number of goals
38 40 42 44 46 48 50 52 54
Team 1
Team 2
40 60 80 100 120 140 160 180 200 220 240 260
Bushrangers
Ghosts
x x
x x
x x
x
x x x
x x
x x
x
x
x
x x
x x
640 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
Interquartile range: Male 22, Female 16.5Standard deviation: Male 13.84, Female 10.48
c Males spend more than females at the restaurant. Also females as a group are more consistent in their spending because the range, interquartile range and standard deviation are all significantly lower than those of the males.
3 a A 75.25, B 74.06b A 38, 12.5, 9.64
B 42, 17.5, 11.39c Group A has less spread since the range, interquartile range and
standard deviation are all less than Group B’s.4 a i Team 1 51, Team 2 36 ii Team 1 22, Team 2 23
iii Team 1 14.6, Team 2 11.9b Team 2 appears to be more consistent since its range and standard
deviation are lower than Team 1’s. However Team 1’s range and standard deviation have been affected by the score of 73.
Power plus1 a Maths, since it is 1 standard deviation above the mean while 65 for
English is less than 1 standard deviation above the mean.b Maths, since it is less than 1 standard deviation below the mean
while 58 for English is more than 1 standard deviation below the mean.
c Neither, since they are both 2 standard deviations below the mean.d Neither, since they are both 3 standard deviations above the mean.e English, since it is 4.5 standard deviations above the mean while
maths is only 4 standard deviations above the mean.
2 ≈ 8, σ ≈ 1.63
3 a 12 11 13 15 14 13 10 14 15 b ≈ 13, σ ≈ 1.63
c Mean increases by 5, σ is unchanged.
4 a 14 12 16 20 18 16 10 18 20 b ≈ 16, σ ≈ 3.27
c is doubled and so is σ.
5 a 1.94b i 2.67 ii 2.63
6 a Because it is a measure of the ‘distance’ of a score from the mean.b 0
Chapter 9 Review1 a Range = 10, mean = 17, median = 17.5, mode = 18
b 4, 9 , 9, 9
c 6, 13.45, 13.5, 14d 40, 103.7, 107, 94, 108 and 112
2 Teacher to check.
3
b Centre A: = 23.08, median = 23.5, range = 37
Centre B: = 16.63, median = 15.5, range = 28
c Yes. The mean of the waiting times of Centre B is much lower than that of Centre A. The waiting times for Centre B also have less spread than the waiting times for Centre A.
4 a
b
c 7
5 a
b Modal class 105–109
c
d 107 km/h
e = 105
6 a 6 b 2.5 c 12.57 a X 81, Y 73.5
b X 9, Y 12
x
x
x
x
13---
Centre A Centre B
9 7 5 4 0 5 7 8 8 8 9
8 6 6 5 2 1 1 2 2 3 3 4 5 6 7 7 8
6 5 5 2 1 2 0 1 2 4 5
9 8 8 5 1 0 0 3 0 2 3
1 0 4
x
x
x f cf
5 5 5
6 8 13
7 15 28
8 13 41
9 8 49
10 1 50
10
0
20
30
40
50
5 6 7 8 9 10
Cumulative frequency histogram and polygon
Score
Cum
ulat
ive
freq
uenc
yClass
intervalClass
centre x f cf
85–89 87 3 3
90–94 92 2 5
95–99 97 7 12
100–104 102 7 18
105–109 107 15 33
110–114 112 13 27
115–119 117 3 50
50
10
0
20
30
40
50
87 92 97 102 107 112 117
Cumulative frequency histogram and polygon
Speeds (km/h)
Cum
ulat
ive
freq
uenc
y
x
ANSWERS 641
c
d Group Y had greater spread since its range and interquartile range were both larger than those for group X.
8 a Girls: ≈ 55, σ ≈ 19.6
Boys: ≈ 55.8, σ ≈ 23.8
b The mean mark for boys was just higher than the mean for girls, but the spread of marks for boys was much greater.
9 a
b
c Box plot, as it shows the difference in spread between men and women and the positions of the medians
d Women: ≈ 41.65, σ ≈ 6.94
Men: ≈ 37.2, σ ≈ 8.97
e Yes. The mean of reaction times of men is much lower than that of women, but the reaction times for men have more spread than the women’s reaction times.
Mixed revision 31 a Neither b Parallel
c Parallel d Perpendicular
2
3 a i m = , b = −2 ii
b i m = −3, b = 1 ii
c i m = − , b = 2 ii
4
5 3x − y − 5 = 0
6 y = − x + , m = − , b =
7 a y = − x + 5 b y = 3x − 2 c y = − x + 6
8 a b
c
9 3 × (−1) − 3 + 6 = 0
10 AB = AC = , BC =
11 All sides have the same length of units, and diagonals AC and
BD both have length units.
12 y = −4x + 3
13 a y = x − 3 b y = − x + 1 c y = 5
14 a i (−1, − ) ii iii
b i (3 , ) ii iii −
c i (−1, ) ii iii −
15 a y = − x + 1 b y = −x − 4 c y = x − 2 d y = 2x + 2
16 C17 a 2x − y − 3 = 0 b 3x + 4y − 6 = 018 6x + 8y + 7 = 019 a x + y − 1 = 0 b 4x − y − 13 = 020 a 2x + 3y − 6 = 0 b 3x − 2y − 5 = 0
62 64 66 68 70 72 74 76 78 80 82 84 86 88
Group X
Group Y
Pulse rates
x
x
Women Men
2 6 8 9 9 9
9 9 8 7 7 6 5 5 4 2 3 0 1 1 2 2 3 7 8
8 5 3 2 0 0 4 3 4 7 8 9
7 4 1 1 5 3 5
25 30 35 40 45 50 55 60
Women
Men
x
x
23---
12---
−2 2 4
2
−20
4
y
x
−2 2 4
2
−4
−20
4
y
x
45---
−2 2 4
2
−4
−20
4
y
x
−2−4−6 2 4 6
2
−4
−20
−6
4
6a
b
c
d
y
x
12--- 3
2--- 1
2--- 3
2---
34--- 1
2---
−2−4 2 4
2
−4
−20
4
y
y = 2x + 1
x −2−4 2 4
2
−4
−20
4
y
2x + y = 3
x
−2−4 2 4
2
−4
−20
4
y
x + 2y − 4 = 0
x
40 32
29
58
12--- 3
4---
12--- 97 9
4---
12--- 1
2--- 34 5
3---
12--- 125 1
2---
12--- 2
5---
642 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
c d −2
c 4 square units (≈4.67)
21 a 3x − 2y + 4 = 0 b 5x − 2y − 17 = 022 a 39.3° b 51.9° c 44.4°
23 a b c
d e f 1
24 72 m 25 C 26 50 m 27 13°
28 a −1 b − c −
d e f −
g 1 h 0 i 029 a 20.6° or 159.4° b 118° c 95.4°30 a 5.4 m b 32.5 m c 5.5 m
31 a 41 m2 b 340 mm2
32 a 43°54′ b 95°12′33 52°, 128°34 a 116° b 43° c 41°35 019°36 a i 8 ii 6.7 iii 7 iv 7
b i 5 ii 5.0 iii 5 iv 4c i 6 ii 18.1 iii 18 iv 18d i 39 ii 117.7 iii 119 iv 134e i 4 ii 22.3 iii 22 iv 23
37 a 8 − 5 = 3 b 6 − 4 = 2c 19 − 17 = 2 d 128 − 102.5 = 25.5e 23 − 21.5 = 1.5
38 a 10 b i 6 ii 11.5 c 5.5
39 a A: ≈ 396.7, σ ≈ 34.2
B: ≈ 369.7, σ ≈ 27.8
b Brand A is the longer-lasting brand because the mean for Brand A is nearly 30 hours above that of Brand B.
40 a i No ii Clustered around 3 and 4iii Nearly symmetrical
b i No ii Clustered around 15–17iii Negatively skewed
c i Yes, 125 ii Clustered in 70–80siii Skewed
d i No ii No iii Symmetrical41 a i Range for Runner A is 3.3. Range for Runner B is 0.8.
ii Interquartile range for Runner A is 0.5 and for B is 0.55.iii The standard deviation for Runner A is 1.03 and for B is 0.29.
b A 12.55, B 12.6c Runner B is more consistent because the range and standard
deviation are much lower than for Runner A and the interquartile ranges are nearly the same.
d Although Runner B is more consistent, Runner A is no better. Runner A’s results have been affected by the outlier of 15.2 (which may be an error in timing or recording).
42 a ≈ 55.2, σ ≈ 13.6 b ≈ 50.3, σ ≈ 1.7
43 a
b and c
44 = 7.2, median = 6.5, mode = 2. The median best represents the data
because the mean is affected by the score of 20 and the mode is the lowest score.
45 a
b
c 40–49
2713------ 8
13------,⎝ ⎠
⎛ ⎞ 12---
3552------
32
------- 1
2------- 1
3-------
32
------- 1
2-------
32
------- 1
2-------
1
2------- 3
2------- 1
3-------
x
x
x x
Number of DVDs f cf
0 6 6
1 10 16
2 12 28
3 8 36
4 7 43
5 4 47
6 2 49
7 1 50
Σf = 50
0
10
20
30
40
50
0 1 2 3 4 5 6 7Number of DVDs
Median = 2
Cumulative frequency histogramand polygon
Cum
ulat
ive
freq
uenc
y
x
Marks in test x f cf fx
10–19 14.5 1 1 14.5
20–29 24.5 10 11 24.5
30–39 34.5 11 22 379.5
40–49 44.5 20 42 890
50–59 54.5 8 50 436
Σf = 50 Σfx = 1965
0
4
8
12
16
20
10–1
9
20–2
9
30–3
9
40–4
9
50–5
9
Marks in test
Frequency histogramand polygon
Freq
uenc
y
ANSWERS 643
d
e 42 f ≈ 39.8
46 4
Chapter 10Start up
1 a even chanceb likely or unlikely (depending on today’s
weather)c even chance d very unlikelye unlikely f very unlikelyg even chance h likelyi certain j impossible
2 a 1, 2, 3, 4, 5 or 6b Team A wins, Team B wins, or it is a
drawc Player A wins, Player B winsd It lands heads, or it lands tailse HH, HT, TH, TTf 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
3 a A b B c B d B4 a C b C c A d D5 a C b D
Exercise 10-011 a 26 b Yes2 a 5
b No, since the coins are not identical in size.
3 a 5 b c i ii 0 iii d
4 a i 2, 4 ii 1, 1, 3, 5 iii 2, 3, 4, 5iv 1, 1, 2, 3
b i or ii or
5 a {Jack of hearts, Jack of diamonds, Jack of clubs, Jack of spades}
b {4 of clubs, 5 of clubs, 4 of spades, 5 of spades, 4 of hearts, 5 of hearts, 4 of diamonds, 5 of diamonds}
c Jack of hearts, Queen of hearts, King of hearts, Jack of diamonds, Queen of diamonds, King of diamonds}
6 a = b = c =
d =
7 a Drawing a blue marble (or any colour other than green or red)
b 18 a 0 b 1
9 a i ii iii 0
iv = 1
b + = 1
10 a Obtaining an odd numberb Obtaining a tailc Selecting a diamond, club or spaded Selecting a vowele Missing the bullseyef Selecting a red disc
11 a {1, 2, 3, 4, 5} b
c Obtaining a number other than 1
d {2, 3, 4, 5} e f + = 1
12 a i ii 0.01 iii 1%
b = 0.99 or 99%
13 0.714 a 40% b 6015 a Missing the bullseye b 0.4
Skillbank 102 a Mark $100, Jenni $50
b Simon $1200, Sunil $900c Lisa $160, Bree $560d William $500, Adriana $1500e Ed $2700, Pete $1800f Sharanya $900, Asam $2100g Cindy $3000, Carmen $600h Nancy $600, John $1000i Carol $550, Louis $440j Yvette $800, Andre $3200k Arden $2100, Ivan $2800l Tan $2000, Mai $1200
Exercise 10-021 a i ii iii or
b 1c i 102 or 103 ii 52 or 53
2 a 10 b to e Teacher to check.3 a Teacher to check. b 1
c and d Teacher to check.4 a Teacher to check. b 1
c and d Teacher to check.
5 a i = ii =
b 16 a 40
b i ii
iii iv =
v = vi =
c 1
7 a b = c
d
8 a b c
9 a b = c
d e
10 a b = c =
d =
11 a 1000b i 0.17 ii 0.155 iii 0.505
iv 0.486 v 0.845 vi 0.66912 a 0.2525 b 0.2325 c 0.515
d 0.7675
Exercise 10-031 a Each outcome to appear 5 times
b P(R) = P(G) = P(Y) = P(B) =
c to f Teacher to check.2 a to f Teacher to check.
g i
ii
iii Teacher to check.iv A 67 B 222 C 1111v Teacher to check.
3 a 1, 2, 3, 4, 5, 6
b i ii iii
4 a 1, 2, 3, 4, 5, 6, 7, 8
b i ii
c Yes d 3, 4, 5, 6 e or
f i 0.125 ii 0.5 iii 0.375iv 1 v 0.5 vi 0
5 a A, E, I b or 0.3
c i or 0.1 ii or 0.7
0
10
20
30
40
5010
–19
20–2
9
30–3
9
40–4
9
50–5
9
Marks in test
Cumulative frequency histogramand polygon
Cum
ulat
ive
freq
uenc
y
x
15--- 1
5--- 1
5--- 2
5---
46--- 2
3--- 2
6--- 1
3---
2652------ 1
2--- 12
52------ 3
13------ 5
52------ 1
13------
852------ 2
13------
37--- 4
7---
66---
36--- 3
6---
15---
45--- 1
5--- 4
5---
1100---------
99100---------
41100--------- 21
100--------- 38
100--------- 19
50------
55100--------- 11
20------ 45
100--------- 9
20------
340------ 11
40------
740------ 6
40------ 3
20------
540------ 1
8--- 8
40------ 1
5---
19100--------- 15
100--------- 3
20------ 57
100---------
97100---------
731------ 10
31------ 12
31------
320------ 2
20------ 1
10------ 1
20------
1120------ 3
20------
1160------ 21
60------ 7
20------ 8
60------ 2
15------
4260------ 7
10------
14---
Dice 1Dice 2 1 2 3 4 5 6
1 0 1 2 3 4 5
2 1 0 1 2 3 4
3 2 1 0 1 2 3
4 3 2 1 0 1 2
5 4 3 2 1 0 1
6 5 4 3 2 1 0
Difference Frequency Probability
0 6 =
1 10 =
2 8 =
3 6 =
4 4 =
5 2 =
Total 36 1
636------ 1
6---
1036------ 5
18------
836------ 2
9---
636------ 1
6---
436------ 1
9---
236------ 1
18------
16--- 1
2--- 5
6---
18--- 1
8---
48--- 1
2---
310------
110------ 7
10------
644 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
d + = 1
6 a = b or
7 6 red marbles
8 a b c d
e f g h
9 a b c
10
11 a b c d
e f g 0 h
12 a b c d
13 a b c d
14 a b c d
15 a 0.62 or or or 62%
b 0.07 or or 7%
16 D17 a 1, 2, 3, 4, 5, 6, 6
b i ii iii iv v
18 a b c 0 d
19 C
20 a b 50
21 27 or 28
Exercise 10-041 2
3 a Independent b Dependentc Dependent d Independente Dependent
4 a i ii b i ii
c i ii d i ii
5 Independent, because the outcome for the coin will not affect the outcome for the die.
6 Dependent, snce the number of balls from which the draw is made is decreasing.
Exercise 10-051 BA AB RB MB
BR AR RA MABC AC RC MRBM AM RM MC
2
b 36 c 11
3 a
b
4 a
b
5 a b No
6 i with replacement
a b c d e =
ii without replacement
a = b = c =
d = e =
7
b i = ii
iii = iv =
8
b i = ii =
iii = iv
v = vi
vii viii =
9
a = b =
c = d =
10 a i ii
b i = ii =
Exercise 10-061
310------ 7
10------
26--- 1
3--- 4
6--- 2
3---
14--- 1
2--- 1
13------ 2
13------
113------ 3
13------ 1
52------ 3
13------
511------ 4
11------ 9
11------
37---
13--- 1
2--- 1
2--- 1
3---
310------ 1
5--- 2
5---
13--- 1
3--- 1
6--- 5
6---
111------ 4
11------ 7
11------ 3
11------
12--- 1
2--- 1
5--- 1
10------
62100--------- 31
50------
7100---------
27--- 4
7--- 4
7--- 3
7--- 3
7---
12--- 2
3--- 5
6---
14---
12--- 1
6---
58--- 4
7--- 3
8--- 5
7---
58--- 3
7--- 3
8--- 2
7---
a Red die
1 2 3 4 5 6
1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6
2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6
3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6
4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6
5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6
6 6, 1 6, 2 6,3 6, 4 6, 5 6, 6
Blu
e d
ieB BB
B B BBR BRB BB
B B BBR BRB RB
R B RBR RR
B BBB R BR
B BBB R BR
B RBR B RB
3 334 34
36 367 373 434 44
46 467 473 63
64 646 667 673 73
74 746 767 77
4 343 6 36
7 373 43
4 6 467 473 63
6 4 647 673 73
7 4 746 76
H HHH
T HT
H THT
T TT
1sttoss
2ndtoss
425------ 4
25------ 4
25------ 16
25------ 5
25------ 1
5---
220------ 1
10------ 4
20------ 1
5--- 2
20------ 1
10------
1220------ 3
5--- 4
20------ 1
5---
a 2nd die
1 2 3 4 5 6
1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6
2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6
3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6
4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6
5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6
6 6, 1 6, 2 6,3 6, 4 6, 5 6, 6
1st
die
636------ 1
6--- 11
36------
1636------ 4
9--- 2
36------ 1
18------
a 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
336------ 1
12------ 6
36------ 1
6---
336------ 1
10------ 35
36------
1036------ 5
18------ 7
36------
136------ 15
36------ 5
12------
K spadesK clubs Q clubs
Q spadesK clubs
K spades Q spadesQ clubsK spades
Q clubs K clubsQ spadesK spades
Q spades K clubsQ clubs
212------ 1
6--- 2
12------ 1
6---
412------ 1
3--- 4
12------ 1
3---
925------ 4
25------
620------ 3
10------ 2
20------ 1
10------
PeopleCar J M L S
C CJ CM CL CS
F FJ FM FL FS
M MJ MM ML MS
H HJ HM HL HS
ANSWERS 645
2
3 a
b i ii iii iv v vi
4 a
b 2, 3, 4, 5, 6, 7, 8, 9, 10
c i ii iii iv
v vi vii viii
5
a b c d e f
6 a 50 b 300 c 750
7 a Teacher to check.
b i ii iii iv
8 a 0.277 b 0.54 c 0.4659 a 640 b 260 c i 0.333 ii 0.173 d 0.45
Power plus1 Teacher to check.
2 a b
3 a p3 b (1 − p)p2 = p2 − p3
c (1 − p)3 d 3p2 − 2p2q
4 a ≈ 0.191 b ≈ 0.417 c ≈ 0.139
Chapter 10 review1 a 5 b Yes c i ii iii
2 a 9b No, as there are two 3s, 3 is twice as likely.
c i ii iii iv
3 a Red, yellow, blue, green
b c Yellow, blue, green d e + = 1
4 a 70%5 a 0 is 0.117, 1 is 0.383, 2 is 0.41, 3 is 0.09
b i 0.383 ii 0.41 iii 0.883
6 a b c
7 a b c
8 a b c
9 a b c d e f
10
11 a i ii =
b i = ii =
c i ii =
d i = ii =
12 a b c d
13 a 0.1 or b 0.2 or c 0.2 or
14
a b i ii
1 H12 H2
H3 H34 H45 H56 H61 T12 T2
T3 T34 T45 T56 T6
Coin Die
1st 2nd 3rd Samplecoin coin coin space
H HHHH
T HHTH
H HTHT
T HTTH THH
HT THT
TH TTH
TT TTT
18--- 3
8--- 1
8--- 1
2--- 1
8--- 1
2---
Normaldie
Tetra-hedral die
1 2 3 4 5 6
1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6
2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6
3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6
4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6
124------ 1
6--- 1
6--- 1
6---
14--- 1
8--- 23
24------ 23
24------
H HHHHH
T HHHTH
H HHTHT
T HHT TH
H HTHHH
T HTHTT
H HT THT
T HT T TH THHH
HT THHT
HH THTH
TT THT T
TH T THH
HT T THT
TH T T TH
TT T T T T
1stcoin
2ndcoin
3rdcoin
4thcoin
Samplespace
116------ 1
4--- 3
8--- 1
4--- 1
16------ 15
16------
18--- 1
8--- 3
8--- 1
8---
16--- 25
216---------
22115--------- 48
115--------- 16
115---------
15--- 1
5--- 2
5---
19--- 2
9--- 1
9--- 5
9---
14--- 3
4--- 1
4--- 3
4---
23--- 1
3--- 5
6---
712------ 1
4--- 5
6---
25--- 3
5--- 3
5---
152------ 2
13------ 12
13------ 1
26------ 1
13------ 3
4---
19---
2564------ 20
56------ 5
14------
1064------ 5
32------ 10
56------ 5
28------
964------ 6
56------ 3
28------
1064------ 5
32------ 10
56------ 5
28------
14--- 1
6--- 1
12------ 1
6---
110------ 1
5--- 1
5---
B B B B BB
G B B BGB
B B BGBG
G B BGGB
B BGB BB
G BGBGG
B BGGBG
G BGGGB GB B B
BG GB BG
BB GBGB
GG GBGG
GB GGB B
BG GGBG
GB GGGB
GG GGGG
12--- 3
8--- 15
16------
646 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
Chapter 11Start up
1 a i −2 ii 13 iii 23 iv −12b i −1 ii 95 iii −3 iv 29
c i 4 ii −2 iii iv −8
d i −2 ii −10 iii 5 iv −10
e i 1 ii iii 81 iv
2 a
b
c
d
3 a m = 2, b = 2, y = 2x + 2b m = −2, b = 2, y = −2x + 2
c m = , b = −1, y = x − 1
Exercise 11-011 a i 160 m/min ii 100 m/min
b i 160 ii 100c The steepness of the graph.
d Zaid 11 min, Nooreen 9 min
2 a No, because the slope of the graph changes.
b During the 3rd hour (from t = 2 to t = 3), 0 km/h
3 a i 12.5 km/h ii 25 km/hb i 12.5 ii −25c i Yes ii No, opposite in signd Car A is moving away from the starting
point (positive gradient) while car B is moving towards the starting point (negative gradient).
4 a The cyclist leaves the starting point, travels at a speed of 20 km/h for 1 h; stops for 1 h; and continues for another hour at a speed of 10 km/h. At D, the
cyclist stops for h, then cycles back
towards the starting point at a speed of 30 km/h for 1 h.
b No, since the gradients of the intervals are all different.
c The cyclist is moving back towards the starting point.
d i 10 km/h ii From C to D.5 a Kate 133 m/min, Colleen 114 m/min
b Kate, in 9 minutes c 1.5 mind 690 m e 110 mf The graph shows the distance they move
down the slope and this increases as more time passes.
Exercise 11-021 a The person starts the journey fast (the
graph is steep), then slows down (the graph becomes less steep) before increasing speed again (the graph becomes steeper).
b The person’s speed is fast initially, then slows down and stops (the graph is horizontal).
c The person starts the journey at a high speed and then gradually slows down to a stop.
2 a H b D c A–B d F e E f C
3
4 a
b
c
d
5 a i C ii B iii A
b i Is the steepest (has the greatest gradient) and must be the fastest (Jade).
ii Is the least steep (the smallest gradient) and must be the slowest (Cameron).
iii The slope of this graph is between the other two (Kiet).
c Jade stopped to talk to a friend. (Other answers possible.)
d This person speeds up slightly and maintains speed for a while, slowing down gradually to a stop.
6 No, because it does not account for variable speeds when leaving and arriving at home.
7 a C b D c E d F e B f A
Skillbank 11A2 a 12:25pm b 1:10am
c 10:50am d 10:55pme 0610 hours f 0010 hoursg 9:10am h 3:15ami 1100 hours j 2305 hoursk 12:20am l 11:35am
4 a 6:05pm b 6:40amc 12:10pm d 2:50ame 1245 hours f 0355 hoursg 10:50pm h 12:15pmi 1545 hours j 0400 hoursk 1:35pm l 7:20am
Exercise 11-031 a i Independent variable is time.
Dependent variable is temperature.ii Teacher to check.
b i Independent variable is time. Dependent variable is height of tide.
ii Teacher to check.c i Independent variable is distance.
Dependent variable is volume of petrol.
ii Teacher to check.d i Independent variable is age.
Dependent variable is height.ii Teacher to check.
2 a i Independent variable is number of persons. Dependent variable is cost.
ii Cost per personiii 1
b i Independent variable is time. Dependent variable is speed.
ii Rate of change of speed (acceleration)
iii −3c i Independent variable is quantity.
Dependent variable is profit.ii Profit/item
iii
d i Independent variable is time. Dependent variable is water level.
ii cm/min
iii −3
3 Teacher to check.
85---
13--- 1
9---
n −2 −1 0 1 4
c 1 2 1614--- 1
2---
B 1 4 10 32 80
L 40 10 4 1 14--- 1
2---
x −3 −1 0 1 2
y −27 −1 0 1 8
x −3 0 1 4 9
y 9 0 1 16 81
12--- 1
2---
14---
12---
40
20
0
60
80
100
120
2 4 6 8 10 12
Time
Dis
tanc
e
Time
Dis
tanc
e
Time
Dis
tanc
e
Time
Dis
tanc
e
13---
19---
ANSWERS 647
4 a
b
c
d
e
f
5 a
b
c
d
e
f
g
h Same as g
i
j
k
l
Exercise 11-041 a viii b i c iii d vi e ix
f iv g ii h v i vii2 a C b B c A d B e A
f B g A3 a B b A c C d B e A4 a B b H c A d F e C5 a C b E c F d D e A f B6 a i C ii A iii E iv F
b i A ii D iii F iv C7 a E b F c B d C e A f D
8 a
b
H
t
H
t
H
t
H
t
H
t
H
t
Time
Dis
tanc
e
Time
Dis
tanc
e
Time
Dis
tanc
e
Time
Dis
tanc
e
Time
Litr
es
Distance
Litr
es
Time
Dis
tanc
e
Time
Wat
er le
vel
Time
Hei
ght
Time
Dis
tanc
e fr
om s
hop
Time
Soun
d le
vel
Time
Spee
d
Time
Spee
d
648 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
c d
e f
9
10
Skillbank 11B2 a 8 h 30 min b 5 h 40 min c 3 h 25 min
d 8 h 25 min e 11 h 25 min f 1 h 40 ming 5 h 10 min h 5 h 45 min i 7 h 55 minj 7 h 25 min
Exercise 11-051 a b
c d
e f
Exercise 11-061 a y = 3x2 b y = 3x2 c y = 0.5x2
d y = −2x2 e y = −3x2 f y = −10x2
2 a i
b i
c i
d i
e i
Time
Spee
d
Time
Spee
d
Time
Spee
d
Time
Spee
d
Time
Spee
d
Time
Spee
d
Time
Spee
d
a b c
Distance from home
Spe
ed
30
−6
y
x
40
12
y
x
20
−8
y
x
0
3
y
x−11_
2
0
y
x1_3
1_2
0 5
y
x−21_
2
0
y
x
y = 4x2
y = x2
ii (0, 0)iii y = 36
y = −2x2
y
x
y = x2
0
ii (0, 0)iii y = −18
0
y
x
y = x2
y = x2
1_3
ii (0, 0)iii y = 3
y = 6x2
0
y
x
y = x2
ii (0, 0)iii y = 54
y
x
y = x2
0
y = − x2 1_2
ii (0, 0)
iii y = −4 12---
ANSWERS 649
f i
g i
h i
i i
Exercise 11-071 a b
c d
e f
g h
i j
k l
Axis of symmetry is the y-axis (or x = 0) for all parabolas.2 a vi b ix c i d xi e x f iii
g ii h xii i viii j v k vii l iv
3 a y = −x2 b y = x2 c y = −x2 +
d y = x2 + 3 e y = −x2 + 9 f y = x2 − 84 a i narrower ii moved down iii −1
b i wider ii moved up iii 3c i wider ii moved up iii 4d i wider ii moved down iii −4e i the same ii moved down iii −3
f i narrower ii moved down iii −
5 a ii b vii c iv d x e v f ixg i h xi i viii j xii k vi l iii
6 a ii y = x2
b ii y = x2 − 1
c ii y = −x2
d ii y = −x2 + 37 b 80 m c 43.2 m
d Approximately 4.1 seconds.8 a to c Teacher to check.
d i parabola ii 100 m2 iii 10 m × 10 m
y = 2x2
0
y
x
y = x2
ii (0, 0)iii y = 18
y
x
y = x2
0
y = −3x2
ii (0, 0)iii y = −27
0
y
x
y = 3x2
y = x2
ii (0, 0)iii y = 27
0
y
x
y = x2
y = x2
1_4
ii (0, 0)
iii y = 2 14---
y
x
V = (0, 10)10
0
V = (0, 10)y
x0
10
y
x0
−10V = (0, −10)
y
x0
V = (0, −10)−10
y
x
V = (0, 6)6
0 V = (0, −6)
y
x
−6
0
V = (0, 6)y
x0
6
y
x0
−6 V = (0, −6)
V = (0, 10)y
x0
10
V = (0, −3)
y
x
−3
0
y
x0
−5 V = (0, −5)
y
x
V = (0, 2)2
0
12---
12---
650 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
Exercise 11-08(Diagrams not to scale.)
1 a
b
c
d
e
f
g
h
i
Exercise 11-091 a x = 3 b x = 0 c x = 2
d x = −1 e x = 1 f x = 3
2 a i x = 3 ii (3, −1)b i x = 3 ii (3, 0)c i x = 3 ii (3, 1)d i x = 4 ii (4, 25)e i x = 4 ii (4, −9)f i x = −4 ii (−4, −80)g i x = 4 ii (4, 80)
h i x = − ii (− , − )
j i x = − ii (− , )
3 a (−2, −3) b (1, 6)
c (− , − ) d (3, −9)
e (1, 1) f (− , )
4 a i −4, 10ii −40
iii x = 3iv (3, −49)v Up
b i 0, 3ii 0
iii x =
iv ( , − )
v Up
c i Noneii 4
iii x = −
iv (− , − )
v Up
d i Approx. (−0.7, 6.7)ii 5
iii x = 3iv (3, 14)v Down
e i Approx. (−4.2, 1.2)ii 21
iii x = −
iv (− , 30)
v Down
f ii 3iii x = 4iv (4, −13)v Up
g i Noneii 4
iii x = −0.7iv (−0.7, 1.55)v Up
h i 0, 4ii 0
iii x = 2iv (2, 8)v Down
i i , 2
ii −6
iii x =
iv ( , )
v Down
y
x0 4
y
x0−2
−3
−15
−5 x
y
0
y
x0
10
2 1_22
− 2_3
y
x0−1
−2
−4 −2
8
x
y
0
x
y
0 5
5
−1
1_31−5
−20
x
y
0
1_2−2
0 x
y
2
12---
12---
14--- 1
4--- 11
4---
16--- 1
6--- 11
4---
212--- 101
4---
112--- 31
4---
0 10
−40
−4 x
y
(3, −49)
0 x
y
31_4
1_2(1 , −2 )
112---
112--- 21
4---
0
4_78
_34(− , 2 )
x
y
34---
34--- 27
8---
x
y
0
5
(3, 14)
0
21
1_2(−1 , 30)
x
y11
2---
112---
3
(4, −13)
0 x
y
4
(−0.7, 1.55)
0 x
y
4
(2, 8)
0 x
y
2
−6
0 x
y
1_21
1_8
3_4(1 , )
112---
134---
134--- 1
8---
ANSWERS 651
Exercise 11-101 a y = 3x3 is narrower than y = x3
b y = 3x3 is narrower than y = 2x3
c y = 0.5x3 is narrower than y = x3
d y = −2x3 is narrower than y = −x3
e y = −3x3 is narrower than y = − x3
f y = −10x3 is narrower than y = − x3
2 a i
ii y = 108
b i
ii y = −54
c i
ii y = 9
d i
ii y = 162
e i
ii y = −13.5
f i
ii y = 54
g i
ii y = −81
h i
ii y = 81
i i
ii y = 6.753 a i narrower ii moved down 1
b i wider ii moved up 3c i wider ii moved up 4d i wider ii moved down 4e i same ii moved down 3
f i narrower ii moved down
2 a i b v c vii d iii e vif iv g viii h ix i ii
Exercise 11-111 a
b
c
d
e
f
g
h
14---
15---
110------
0 x
y
0 x
y
0 x
y
0 x
y
0 x
y
0 x
y
0 x
y
0 x
y
0 x
y
12---
x
y
0
(1, 4)
x
y
0
(1, −2)
x
y
0
(4, 5)
x
y
0
(1, 5)
x
y
0(2, −4)
x
y
0
(−4, 4)
x
y
0
(3, 4)
x
y
0
(−1, 1)
652 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
Exercise 11-121 a
b 1
c For y = ax where a = 2, 3 or 5, as a increases the graph increases more rapidly as x becomes larger.
2 a
b i y = 3−x ii y = a−x
3 a
b
c y = −2x and y = −3x are reflections of
y = 2x and y = 3x in the x-axis.
d y = a−x
4 a increasing b decreasingc increasing d decreasing
5 a
b
c
d
e
f
g
h
Exercise 11-131 a Centre (0, 0), r = 3
b Centre (0, 0), r = 10c Centre (0, 0), r = 7d Centre (0, 0), r = e Centre (0, 0), r = 3f Centre (0, 0), r = 16
g Centre (0, 0), r =
h Centre (0, 0), r = i Centre (0, 0), r = 5
2 a x2 + y2 = 144 b x2 + y2 = 25
c x2 + y2 = d x2 + y2 = 7
3 (Diagrams not to scale.)
a
b
c
d
e
0 x
y
4
3
2
1
−2 −1 1 2
y = 2xy
= 5
x
y =
3x
0 x
y
4
3
2
1
−2 −1 1 2
y = 3xy = 3−x
0 x
y
2
1
−1
−2
−2−3 −1 1
y = 2x
y = −2x
0 x
y
2
1
−1
−2
−2−3 −1 1
y = 3x
y = −3x
x
y
(1, 2)
1
x
y
(1, −4)−1
x
y
(−1, 3)1
x
y
(−1, 2) −1
x
y
(−1, 4)
1
x
y
(−1, −5)
−1
x
y
(−1, 2)
1
x
y
(−1, −10)
−1
20
12---
6
19---
x
y
−4
4
4
−4
x
y
1
1
−1
−1
x
y
− 1–2
− 1–2
1–2
1–2
x
y8
8
− 8
− 8
x
y
5
5
−5
−5
ANSWERS 653
f
Exercise 11-141 a H b P c L d E e L
f L g H h L i P j Lk P l E m Q n L o C
2 a vii b xii c x d xi e viiif i g iv h ii i vi j iiik v l ix
3 a
b
c
d
e
f
g
h
i
Power plus1 i
ii For y = 5x + 1 asymptote is y = 1.
For y = 5x asymptote is y = 0.
For y = 5x − 2 asymptote is y = −2.2 a Centre (1, −2), r = 6
b Centre (−4, −3), r =
3 i
ii The asymptote for each graph is y = 0, the x-axis.
4 a
b
5 a i
ii
iii
x
y
11
11
−11
−11
x
y
(2, 5)
−30
x
y
(1, −1)0
x
y
(2, 25)
1
0
x
y
(−2, 11)
0
3
x
y(5, 3)
−47
0
x
y
(2, 3)
0
x
y
(5, 3)
53
0
x
y
4
20
x
y
12
12
−12
−12
−1
0
1
2
x
y
y = 1
y = −2
y = 0
y = 5x + 1y = 5x
y = 5x − 2
2
0
y = 5 × 2x
y = 2x
1
0.5
5
x
y
y = × 2x1–2
x
y
−5 2 3 40
40
x
y
30 5−11–2
−45
x
y
0−4
−4
−2−2
2
2
4
4
x
y
0−4
−4
−2−2
2
2
4
4
x
y
0−4−2
−2 2
2
4
4
6
654 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
b i x = 0, y = 1 ii x = 0, y = −2iii x = 0, y = 3
6 a
b
c
7 a i
ii
iii
b i x = 1, y = 0 ii x = −1, y = 0iii x = −3, y = 0
8 a b
c
9 a
b
c
Chapter 11 review1 a Ben, his graph is steeper
b Ben 20 km/h, Anita 16 km/hc 1 h 22.5 min
2
3 a i Time ii Distanceb Teacher to check.
4 a ii b iii c i
5 a i
ii
iii
6
x
y
40
4
−4
Semi-circleCentre (0, 0)
Radius 4
x
y
5
4
−5 0
−5
Semi-circleCentre (0, 0)
Radius 5
x
y
0 2
Semi-circleCentre (0, 0)
Radius 2
2
− 2
x
y
0−2
−2
2
2
x
y
0−2
−2
2
2
x
y
0−2
−2
2
−4
x
y
2
2
0
−2
x
y
1
1
0−1
x
y
0
−21–2
−21–2
21–2
x
y
(1, 1)
0
x
y
−2
2
20
x
y
−3
−3
3
0
Time
Dis
tanc
e
40
−8
y
x
40
8
y
x
4
0−8
y
x
y
x
y = x2
y = 4x2
0
y = − x2 1__10
0
ac
ANSWERS 655
7 a
Both have the same vertex (0, −1) and axis of symmetry x = 0. y = 4x2 − 1 is narrower than y = x2 − 1. Both are concave up.
b
Both have the same vertex (0, 3) and axis of symmetry x = 0. y = 3 − 2x2 is narrower than y = 3 − x2. Both are concave down.
8 a
x-intercepts: −1, 1y-intercept: 1
b
x-intercepts: −2, 10y-intercept: −20
c
x-intercepts: 0, 2y-intercept: 0
9 a i x = 1 ii (1, −7)
b i x = 1 ii (1 , 14 )
c i x = −4 ii (−4, −40)
10 a
b
11 a
b
c
12 a
b
c
13 a
b
c
d
14 a Centre (0, 0), radius 10 b Centre
(0, 0), radius
c Centre (0, 0), radius 715 a ix b vii c vi d iieif
ivg viii h iii i v j xikxl
xii
Mixed Revision 41 Teacher to check.
2 a = b = c =
y = 4x2 − 1
−1
y
x
y = x2 − 11–4
14---
0
3
y = 3 − x21–2
y = 3 − 2x2
x
y
12---
−1 1
1
x
y
−2
−20
10 x
y
x
y
21–2
12---
12--- 1
2--- 3
4---
x
yy = 2x3
y = x3
x
y
y = x3
y = − x31–2
x
y
(2, 23)
−10
x
y
(2, −38)
−1
2
0
x
y
(2, −1)
−5
0
x
y
0
(1, 3)
x
y
0
(−2, 4)
x
y
0
(5, 8)
x
y
(1, 4)
1
0
x
y
(−1, 4)1
0
x
y
(1, −4)
−10
x
y
(−1, −4)
−10
5
46--- 2
3--- 3
6--- 1
2--- 3
6--- 1
2---
656 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
3 a b i ii
4 a
b The theoretical probability is the expected probability, whereas the probabilities in the table are based on an experiment. The experimental probability will approach 0.5 as the number of trials is increased.
5 a b
6 0.8
7 a b c
8 B
9 a b c d e
f g h i j
10 a b c d
11 a i ii
b
c i ii iii iv
12
13 a b c
14 B
15 a
b
c
16 C
17
18 a Naim; Naim’s graph is steeper than Jenna’s.b Naim is stationary. c 6 d 6 km/h
19 a C
20 a i
b i
c i
d i
Numberof trials
10 20 30 50 80 100 200 400 1000
Numberof heads
6 10 15 29 41 51 109 214 511
P(H) 0.6 0.5 0.5 0.58 0.5125 0.51 0.545 0.535 0.511
45--- 2
3--- 1
3---
112------ 1
6---
126------ 3
13------ 12
13------
136------ 4
9--- 5
6--- 5
36------ 35
36------
14--- 1
2--- 1
4--- 11
36------ 5
6---
28121--------- 56
121--------- 49
121--------- 105
121---------
38--- 5
8---
2564------
27512--------- 135
512--------- 125
512--------- 387
512---------
12---
18--- 3
8--- 7
8---
−2 2 51 3−3
2
1
3
−2
−4
0
−5
−3
y
x4
−2 21 3−3
2
1
3
−2
−4
0
−5
−3
y
x
−1 2 51 3
2
1
3
4
5
−2
0
−3
y
x
4
1−1−2−3−4 2 3 4
2
−1−2−3−4
4
1
3
0
y
x
y
x
(1, 4)30
ii (0, 3)
y
x−20
ii (0, −2)
y
x
2
0
ii (0, 2)
y
x
11−1
0
ii (0, 1)
ANSWERS 657
e i
f i
21 y = x2 + 3 is the graph of y = x2 moved up 3 units (along the y-axis)
22 Speed on the journey away from home increases then decreases to a stop. The journey begins again towards home, increasing before decreasing and finally stopping without reaching home.
23 x-intercepts are −2, 7.y-intercept is −14.
24 a i x = − ii (− , − )
b i x = ii
c i x = 0 ii (0, 10)
d i x = ii
25 a
b
26 a i ii
iii 9
b i ii − , 5
iii −5
27 a
b
28 a Independent variable is distance, dependent variable is amount of fuel.
b Independent variable is air leaking out, dependent variable is diameter.
c Independent variable is number of people, dependent variable is cost.
d Independent variable is temperature, dependent variable is height.
29 a vii b vi c v d xii e iiif x g ii h iv i xi j ik ix l viii
30 a Centre (0, 0), r = 6 unitsb Centre (0, 0), r = 1 unit
c Centre (0, 0), r = unit
General revision1 a k = b y = 10 c m =
d k = e x = ±3 f y = ±5
g x = −11, 2 h y = −1 or y =
i d = − or d = 1
2 C3 a 4 b 2.5 c 27.54 a $5618 b $935.88 c $1597.98
5 a
b
c
6 a 2x − y + 7 = 0 b 2x + 3y + 5 = 0c 3x + 7y − 2 = 0 d x + 2y − 9 = 0
7 a i $851.20 ii $4742.40b $1200 c 69%
8 a 7.2 m3 b 78.5 cm3
c 5747.0 cm3
9 a x = 2, y = 2 b m = 1, p = −1
c a = , c = −
10 a = 63.1, σ = 10.3
b = 3.6, σ = 1.6
c = 50.3, σ = 12.2
11 a b
c
12 a i ii iii
b i −172 ii
iii
13 a 270b i 0.15 ii 0.19 iii 0.47
iv 0.47 v 0.7214 a y ≥ 1,
b x < ,
c x < −3,
15 a 19.1 m b 10.6 m c 4.8 cm16 a In ∆ABD and ∆ACD:
AB = AC (given)AD is common∠ADB = ∠ADC = 90° (AD ⊥ BC)∴ ∆ABD ≡ ∆ACD (RHS)
b i ∴ BD = CD (matching sides of congruent triangles)∴ AD bisects BC
ii ∴ ∠BAD = ∠CAD (matching sides of congruent triangles)∴ AD bisects ∠BAC
17 a 396 mm2 b 476 mm2 c 792 mm2
d 3750 mm2 e 785 mm2 f 6792 mm2
18 a x = b x =
19 a b c d
20 a 81°45′ b 37°45′ c 142°49′21 a 360 498 mm3, 25 700 mm2
b 145 125 mm3, 17 604 mm2
22 a i , m = ii , m = −
y
x
(2, 6)4
0
ii (0, 4)
y
x
(1, −7)
−50
ii (0, −5)
x
y
7−2−14
32--- 3
2--- 1
4---
34---
34--- 11
8---,⎝ ⎠
⎛ ⎞
54--- 11
4--- −11
8---,⎝ ⎠
⎛ ⎞
y
x0−1
−4
11_3
( , -4 )1_6
1_12
y
x021_
2
(1 , 3 )1_4
1_8
1 12--- 0,⎝ ⎠
⎛ ⎞ 112---
2 13--- −211
3---,⎝ ⎠
⎛ ⎞ 13---
x
y(1, 3)
1
0
x
y
(−1, −6)
−10
13---
612--- 71
3---
337---
23--- 1
2---
25--- 1
3---
x
y
110
−33
−3
(4, −49)
x
y
(2, −2)0
x
y
2
4
0
11823------ 111
23------
x
x
x
9 2 7 2
7 3 15 2–
3 55
---------- 3 2+2
----------------- 2 5–3
----------------
1 9 3+
98 24 10+
−2 −1 0 1 2 3
112---
−3 −1−2 0 1 2 3
−5 −3−4 −2 −1 0 1
-7 41±2
---------------------- -1 34±3
----------------------
14--- 5
32------ 169
256--------- 169
256---------
13 23--- 40 1
3---
658 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
iii , m = iv , m = −
b i 5, (2 , 1) ii 9, (2 , 1)
c A parallelogram.
23
24 a 32 b 4 c d
Chapter 12Start up
1 a SSS b SAS c RHSd AAS e SAS f AAS
2 a SSS b AA c RHSd SSS e SAS or AA f AA
Exercise 12-01 1–5 Teacher to check.6 a UC = 4.5 m
(line from the centre is the perpendicular bisector of the chord and the chords are equal because they are the same distance from centre)
b DE = 12 m(sides opposite equal angles)
c ∠UVO = 58°(angle sum of isosceles ∆; chords of equal length subtend equal angles)
d PQ = 30 mm(Pythagoras: the line from the centre is the perpendicular bisector of the chord)
e OM = 21 cm(Pythagoras: the line from the centre is the perpendicular bisector of the chord)
f OD =
(Pythagoras: the line from the centre is the perpendicular bisector of the chord)
7 a i 52 cm ii 96 cmb i 58 cm ii 8 cmc 18.4 km
8 a 77 cm b 34 cm, 20 cm
c i AB = 30 cm ii area = 540 cm2
d i 52 cm ii area = 1920 cm2
Exercise 12-02 1–3 Teacher to check.4 a 45
b 112c 120d 232 e 40f 74g 63h 104i 90 (angle in semi-circle theorem)j 48k 36l 30
m 23n 9o 45p 63 (opposite angles of a cyclic
quadrilateral are supplementary)q 75r 88
5 a x = 75 (angle at centre theorem)y = 33 (angles at circumference
theorem)z = 72 (angle sum triangle)
b x = 108 (angle at centre theorem)y = 126 (opposite angles of a cyclic
quadrilateral theorem)z = 252 (angle sum at a point or angle at
centre)c x = 70 (straight line)
y = 110 (exterior angle cyclic quadrilateral theorem)
z = 70 (straight line)d x = 96 (angle at centre theorem)
y = 42 (base angles of isosceles ∆)z = 264 (angle sum at a point)
e x = 140 (base angles of isosceles ∆)y = 70 (angle at centre theorem)z = 35 (angle sum of isosceles ∆, and
by subtraction)f x = 62 (angle in semi-circle theorem)
y = 118 (opposite angles of cyclic quadrilateral theorem)
z = 31 (base angles of isosceles ∆)6 a iii WXYZ is a cyclic quadrilateral
because ∠W + ∠Y = 180° and∠X = ∠Z = 180°i.e. opposite angles are supplementary
b Draw the circumcircle of ∆BCD. We show that A lies on the circle. Join OB, OD and let A′ be a point on the circumference.
If ∠A = x, then∠BCD = 180° − x (given)
∴ Reflex ∠BOD = 2 × (180° − x)= 360° − 2x
∴ ∠BOD = 2x∴ ∠A′ = x (angle at centre and
circumference of a circle)∴ ∠A = ∠A′ (both equal x)
∴ A lies on the circle∴ A, B, C, D are concyclic.
7 In ∆PBC:
sin P =
but ∠P = ∠A (angles at the circumference standing on the same arc are equal)
∴ sin A =
∴ = 2R
Similarly, construction diameters from A and B,
= 2R and = 2R
∴ = = = 2R
where R is the radius of the circumcircle of ∆ABC
Exercise 12-03 1–3 Teacher to check.4 a a = 56 (the angle between the radius
and the tangent is a right angle)
b b = 21 (radius is perpendicular to a tangent, and Pythagoras’ theorem)
c c = 134 (a tangent is perpendicular to the radius; angle sum of a quadrilateral)
d g = 67 (alternate segment theorem)5 a 15 b 5 c 9 d 7 e 20 f 46 a x = 7 cm
b i XP = 10 cm ii AB = 24 cm
Exercise 12-041 ∠R = ∠Q
∠P = ∠S
∠RYP = ∠QYS (vertically opposite angles)
∴ ∆PYR ||| ∆SYQ
∴ = (ratio of sides opposite equal angles)
∴ PY × YQ = RY × YS2 ∠ADC = ∠BEC (opposite angles of a
parallelogram equal)∠ADC = ∠CBE (exterior angle of
cyclic quadrilateral)∴ ∆CBE is isosceles as ∠BEC = ∠CBE (base angles equal)
3 ∠TZY = ∠X (alternate segment theorem)
∠X = ∠Y (base angles of isosceles ∆ZXY (XZ = YZ) )
∴ ∠TZY = ∠Ynow XY || ST as alternate angles ∠TZY and ∠Y are equal
4 Construction: Draw a perpendicular from O to meet DG at P. Since the perpendicular from the centre to a chord bisects the chord:
DP = GP andEP = FP
∴ DE = DP − ED= GP − FP= FG
5 ∠THJ = ∠HIJ (alternate segment theorem)
∠THJ = ∠HPI (equal alternate angles)∴ ∠HIJ = ∠HPIIn ∆HIP and ∆HJI
∠HIJ = ∠HPI (proved above)∠IHJ = ∠IHP (common angles)
∴ ∆HIP ||| ∆HJI (equiangular)∴ ∠HIP = ∠HJI (third pair of equal
angles in similar triangles)
6 In ∆UVX and ∆UWX,∠UXV = 90° = ∠UXW (angle in a semi-
circle, straight line)UV = UW (given)
13 23--- 40 1
3---
12--- 1
2---
x
y
20
−11–3
19--- 1
32------
18 2
⎭⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎫
(the angle at the centre; twice the angle at the circumference standing on the same arc)
⎭⎪⎬⎪⎫ (angles at the circumference of
a circle standing on the same arc are equal)
⎭⎪⎬⎪⎫ (angle in semi-circle,
angle sum triangle)
⎭⎬⎫ (exterior angle of cyclic
quadrilateral theorem)
B
C
D
O
A
A′
a2R-------
a2R-------
aAsin
-------------
bBsin
------------- cCsin
-------------
aAsin
------------- bBsin
------------- cCsin
-------------
⎭⎪⎪⎬⎪⎪⎫ (angles at the
circumference standing on the same arc are equal)
PYYS-------- RY
QY--------
ANSWERS 659
UX is common∴ ∆UVX ≡ ∆UWX (RHS)∴ VX = VW (matching sides in congruent
triangles)∴ circle bisects base of triangle
7 Let ∠QRP = x (alternate segment theorem)
∴ ∠SRP = x (PR bisects ∠QRS)∴ ∠PSQ = x = ∠QRP (angles at the
circumference standing on the same arc)
∠PQS = x = ∠SRP (angles at the circumference standing on the same arc)
∴ ∠PQS = ∠PSQ = x∴ ∆SPQ is isosceles because base angles PQS and PSQ are equal
8 Now AP2 = YP × PX (intersecting tangent and secant theorem)
and BP2 = YP × PX (intersecting tangent and secant theorem)
∴ AP2 = ΒP2
∴ AP = ΒP 9 Join APX and BQX
In ∆XPQ and ∆XAB∠X is common
∠XPQ = ∠XAB (corresponding angles)∠XQP = ∠XBA (corresponding angles)
∴ ∆XPQ ||| ∆XAB (AAA)
∴ = (ratio of sides opposite equal angles in similar ∆s)
but XP = XA (radius half diameter)
∴ =
∴ =
∴ PQ = AB
10 Let ∠PTX = a∴ ∠R = a (alternate segment
theorem)now ∠QTY = a (vertically opposite
angles)∴ ∠S = a (alternate segment
theorem)∴ ∠R = ∠S = a
∴ PR || SQ because alternate angles R and S are equal.
Chapter 13Start up
1 a 13 b 3 c 122 a (x − 4)(x + 4) b x(x − 4)(x + 4)
c 3(x − 3)(x + 3) d 3x(x − 3)(x + 3)e (x − 5)(x + 3) f (x + 8)(x − 3)g (x − 2)(2x + 5) h x(x − 10)(x + 7)i (x − 5)(x + 5)(x − 2)(x + 2)
3 a
b
c
d
e
f
4 a x = − b x = − or x = 2
c x = 0 or x = 10 d x = 0 or x =
e x = −1 or x = −5 f x = −10 or x = 12
g x = −2 or x = − h x = − or x = 2
i x = −3 or x =
Exercise 13-011 a
b
c
d
e
f
g
XPXA-------- PQ
AB--------
12---
12--- XA
XA----------- PQ
AB--------
12--- PQ
AB--------
12---
0
y
x
y = x + 2
2
−2
0
y
x
y = x2
0
y
xy = −x2
0
y
x
y = x3
0
y
x
(2, −4)
y = −x3
0
y
x
x2 + y2 = 11
1
−1
−1
52--- 5
2---
35---
32--- 5
4---
32---
0
y
x
(−1, −3)−2
0
y
x
1−1
0
y
x
(1, 2)
0
y
x
(2, −5)
3
0
y
x
(1, −3)
0
y
x
(2, −6)
2
0
y
x
(2, 11)
−5
660 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
h
i
j
k
l
m
n
o
2 a i −3, −2, 2 ii −12
iii
b i −1, 0, 2 ii 0
iii
c i −1, 1, 3 ii 3
iii
d i −1, 4, 5 ii −20
iii
e i −3, −2, 1 ii 6
iii
f i 0, 3, 6 ii 0iii
g i −3, 2, 5 ii −30
iii
h i −2, 1, 3 ii −12
iii
3 a i −2, 1, 2 ii 4
iii
b i −1, 0, 3 ii 0
iii
c i −1, 1, 3 ii −3
iii
d i −6, 0, 1 ii 0
iii
e i − 4, −1, 1 ii 4
0
y
x
(1, 1)
4
0
y
x
(2, 8)
4
0
y
x
(−3, 7)
−2
0
y
x
(2, 1)
3
(−1, 3)
0
y
x−1
−40
y
x
(3, 5)
0
y
x
4
(−2, 8)
0
y
x(1, −1)
−3
1−1−2−3 2 30
−12
y
x
1−1 20
y
x
1−1 2 30
3
y
x
1 2 3 4 5 60
−20
y
x
1−1−2−3 20
6
y
x
1−1 2 3 4 5 60
y
x
0
y
x−3
−30
2 5
0
y
x−2
−12
1 3
0
y
x−2 1 2
4
0
y
x−1
3
0
y
x−1 31
0
y
x−61
ANSWERS 661
iii
f i −2, 0, 3 ii 0
iii
g i −6, −1, 2 ii − 4
iii
h i − 4, 1, 4 ii 48
iii
4 a x-intercepts are −2, 1 and y-intercept is 6
b x-intercept is −1y-intercept is 1
c x-intercepts are −2 and 3y-intercept is 18
d x-intercepts are , −2
y-intercept is 4
Exercise 13-021 a v b viii c ix d iii e vii
f i g vi h iv i ii
2 a
b
c
d
e
f
13 a Move up 4 unitsb Move right 5 unitsc Move left 3 unitsd Move up 4 unitse Move right 3 unitsf Move left 2 units
Exercise 13-031 a i (−3, 4) ii 5
b i (5, −12) ii 13c i (−2, 4) ii 5d i (3, 1) ii 1e i (9, 12) ii 15f i (1, −3) ii 2g i (−6, −1) ii 1h i (−5, −8) ii 4i i (0, 0) ii 11j i (−2, 1) ii 1k i (2, 0) ii 8
l i (4, −3) ii
m i (3, 4) ii 5
n i (0, −1) ii
2 a x2 + y2 = 16
b (x − 1)2 + (y + 2)2 = 9
c x2 + (y + 11)2 = 4
d (x + 3)2 + (y − 2)2 = 100
e (x + 1)2 + (y + 1)2 = 1
f (x − 7)2 + y2 = 81
g (x + 6)2 + (y − 2)2 = 5
h (x + 1)2 + y2 = 8
3 a
0
y
x−4−1 1
4
0
y
x−2 3
y
x−6
−4−1 1
y
x−4 4
48
1
112---
y
x−2
6
1 2
y
x−1
1
y
x3−2
12---
y
x−2
4
1
0
y
x
4
2
y = (x − 2)2
y = x2
y = 3x2 + 1
0
y
x
y = 3x21
0
y
x
2y = −x3
y = −x3 + 2
0
y
x
y = −2x3y = −2(x + 2)4
−2
(−1, −1)
0
1
1
y
x
y = x4 y = (x − 1)4
0
y
xy = −x5 y = −x5 − 2
−2
212---
13---
0
1
2y
x
(1, 0)
662 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
b
c
d
e
f
4 a (−3, 1), r = 5 b (4, 2), r = 7c (−2, 5), r = 6 d (−10, 6), r = 1e (2, − 4), r = 5 f (6, −3), r = 4
g (−3, 10), r = 9 h (4, −1), r =
Exercise 13-041 a (0, 0) and (1, 1) b (1, 1) (3, 9)
c (3, −3), (2, −4) d (2, 0) (−1, 3)e (3, 0), (0, 3) f (1, 3), (−1, −3)g (1, 3), (3, 1) h (− 4, −3), (3, 4)i (−8, −1), (1, 8) j (−3, 9), (1, 1)k (2, 3), (−6, −1) l (12, 5), (−5, −12)
2 a ( , − )
b The algebraic method is more accurate.
Exercise 13-051 a Yes, not monic b No
c Yes, monic d Yes, not monice No f Yes, monic
g Yes, not monic h Noi Yes, monic j Nok Yes, monic l Yes, not monic
2 a i 5 ii 9 iii 1b i 5 ii −6 iii 3c i 2 ii 11 iii −10d i 1 ii −6 iii 0e i 5 ii 7 iii 3f i 0 ii 9 iii 9g i 6 ii 1 iii −11
h i 1 ii − iii 22
i i 3 ii iii 0
j i 4 ii iii 8
3 a 17 b −1 c −7
d e −
4 a 1 b −3 c 0
d e 14 f
g 56 h −7
Exercise 13-061 a 9x3 + 8x2 + 6x − 2
b 3x4 + 2x3 − 3x2 −2x
c − 4x2 − 3x − 2
d −x4 − 3x3 − 5x2 + 4
e x6 + 2x5 + 11x4 + 10x3 + 25x2
f 6x4 + x3 − 2x2 + 2x + 11
g 7x6 + x5 − 3x4 − x3 + 2x2
h 7x4 − 16x3 − 48x2 + 6x + 4
i 6x3 − 8x2 + 4x + 6
j − 4x4 − 8x3 + 5x2 + x − 2
2 a x2 + 11x − 1 b −x2 − 3x − 5
c x2 + 3x + 5 d 2x2 + 26x − 5
e 4x3 + 25x2 − 13x − 6
3 a −x2 + 7x + 3 b x2 + 4x − 15
c −3x2 + 11x + 6 d 3x2 − 15x + 3
e 3x2 − 7x − 15 f 2x2 − 15x + 9
g − 4x3 + 9x2 + 24x − 54
h 8x3 − 62x2 + 99x
4 a x2 − 7x + 6 b
c x = 6, x = 1
5 a x3 + 4x2 − 5x − 20
b
c x = −5, x = 0, x = 1
Exercise 13-071 a x2 + 7x + 4 = (x + 2)(x + 5) − 6
b x2 − 6x + 2 = (x − 3)(x − 3) − 7
c 4x2 + 3x + 10 = (x − 1)(4x + 7) + 17
d 8x2 + 9x + 11 = (2x + 1)(4x + ) +
e x3 + 6x2 + 5x − 4
= (x − 3)(x2 + 9x + 32) + 92
f 4x3 + 2x2 + x
= (x + 4)(4x2 − 14x + 57) − 228
g 2x3 − x2 + 5x + 3
= (x + 6)(2x2 − 13x + 83) − 495
h 3x3 − x2 + 11
= (x + 2)(3x2 − 7x + 14) − 17
i x5 − x4 + 8x3 + 2x2 − x − 1
= (x + 1)(x4 − 2x3 + 10x2 − 8x + 7) − 8
j x4 − x2 − 10
= (x + 3)(x3 − 3x2 + 8x − 24) + 62
2 a (3x − 1), R = 3 b (x + 7), R = 14
c (3x3 + 14x2 − 2x + 21), R = 42d (4x + 6), R = 17
3 a (2x − 1)(3x + 2)
b (2x − 1)(x2 + x + 1)c (2x − 1)(4x + 7)
d (2x − 1)(3x2 + 2x + 1)
e (2x − 1)(x3 − 3x2 − 4x + 2)
f (2x − 1)(x3 − x + 3)
g (2x − 1)(3x2 + 1)
h (2x − 1)(4 − 3x − x5)
Exercise 13-081 a 5 b −181 c −1 d 179
e −7 f 1709 g −16 h −12 a 54 b 2 c 14 d −2 e −12
f 174 g 0 h 6 i −1 j 115
Exercise 13-091 a B, C b B, C c A
d A, B, C e A, B2 Teacher to check.3 a x(x + 2)(x + 4)
b x(x − 2)(x + 1)c (x − 1)(x + 1)(x + 2)d (x − 2)(2x − 1)(x + 4)e (x − 1)(x − 2)(x − 3)f (x − 2)(x + 8)(x − 5)g (x − 6)(x + 1)(3x − 1)h (x − 2)(3x + 1)(2x − 1)
i x2(2x − 1)(x − 2)j (x − 3)(2x + 1)(2x + 3)
4 a x = − 4, , 3 b x = − 4
c x = −2, , 3 d x = ±5
e x = ±4, −3, 2 f x = −7, 0, 2g x = −3, −2, 3 h x = −2, −1, 4
i x = − 4, −1, 5 j x = − , 1, 3
k x = , , 1 l x = 3
Exercise 13-101 a
b
20
3
y
x
(2, 3)
2
0
y
x
(−1, −1)2
5
0
y
x
(3, 4)
4−3 50
y
x(1, 0)
−20
−2
y
x
2
(−2, −2)
2 3
225--- 31
5---
54---
13---
2 2
7 2 5– 58---
13 5 3– 4764------
24 21 2–
2 3 8–
212--- 81
2---
12---
52---
12---
14--- 2
3---
y
x−4
−2
y
x−2
−2
−1 1
ANSWERS 663
c
d
e
f
g
h
2 a
b
c
d
e
f
g
h
i
j
3 a
b
c
d
e
f
y
x2
8
−4 1_2
y
x1
20
y
x
−20
−4−51_2
y
x−2 1 6
−12
y
x0−2 1 4
y
x−3
−36
−2 2 3
−4
−32
2 x
y
0
−18
3−2x
y
0
−1
−4
2x
y
0
−1 3 x
y
0
−8
−2 1 2 x
y
0
4x
y
0
−4
−1 2x
y
0
−64−1 4 x
y
0
−1−2 1 2 x
y
0
4
−64
−2 2 4 x
y
0
−24
−2 34 x
y
0
1
−3
−3x
y
0
x
y
0−5 3
x
y
0
4
1 2−1−2
y
x−1
−64
4
y
x−2
−48
664 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
Exercise 13-111 a i
ii
iii
iv
b i
ii
iii
iv
c i
ii
iii
iv
d i
ii
iii
iv
2 Teacher to check.3 Teacher to check.
Chapter 14Start-up
1 a
b
c
0
y
x
2
−2
y = P(x)
y = −P(x)
0
y
x
2
y = P(−x)y = P(x)
0
y
x
2
y = P(x) − 3
y = P(x)
0
y
x
2
4 y = P(x)
y = 2P(x)
0
y
x
y = P(x)
y = −P(x)
−2−2
2
−1 1
0
y
x
y = P(−x) y = P(x)
−2−2
−1 1
0
y
x
y = P(x)
y = P(x) − 3
−2−2
−5
−1 1
0
y
x
y = P(x)
y = 2P(x)
−2−2
−4
−1 1
0
y
x
y = −P(x)
y = P(x)
−1 2
0
y
x
y = P(−x) y = P(x)
−1−2 1 2
0
y
x
y = P(x)
y = P(x − 3)
−1
−3
2
0
y
x
y = 2P(x)
y = P(x)−1 2
0
y
x
y = −P(x)
y = P(x)
31
9
−9
0
y
x
y = P(−x) y = P(x)
31
−9−1−3
−12
0
y
x
y = P(x) − 3
y = P(x)
31
−9
y = 2P(x)
0
y
x
y = P(x)
31
−9
−18
0
y
x
0
y
x
−3
0
y
x−3
ANSWERS 665
2 a
b
c
3 a
b
c
4 a
b
c
5 a x = b x =
c x =
6 a 2.47 b 3.83 c 1.58 d 3.31
Exercise 14-011 a Yes b No c No d Yes
e No f Yes g No h Yes2 a Yes b Yes c No d No
e No f Yes g No h Yesi Yes j No k Yes l Yes
Exercise 14-021 a i 6 ii −2 iii 0
b i −2 ii 2 iii 1c i 12 ii 0 iii 0
d i ii −1 iii 1
e i 27 ii iii 1
f i 3 ii 1 iii
g i ii iii −1
h i −2 ii 6 iii 12 a i 16 ii −14 iii 6x + 16 iv x = 2
b i 4 ii − iii −2 iv x =
c i 5 ii 0 iii
iv t = −2 or t =
3 a i 11 ii 15 iii 10b 9 − 2x c −2 d x = 16 e x = 3
4 a i 11 ii 4 iii 20
b Teacher to check. c x = 1 ±
5 a b x = 7.5
c has no value. d x = −
6 a i − 4 ii 4b −8 c x = 1 or x = 3
d Teacher to check. e has no value.
7 a 8 b 11 c 40
d 3k4 − 2k2 + 3
8 a −26 b −7 c 1 + y3 d −2y3
Exercise 14-031 a i −3 � x � 3 ii 0 � y � 3
b i − 4 � x � 0 ii 0 � y � 4c i x � 0 ii y � 5d i x � −1 ii y � 0e i x � −1 ii y � −3f i No restrictions ii y = −3g i No restrictions ii y > 0h i No restrictions ii y = 3i i No restrictions ii y > −2j i No restrictions ii −1 � y � 1k i No restrictions ii y � − 4l i No restrictions ii 0 < y � 4
2 a
i All x ii All y
b
i All x ii y ≤ 1
c
i All x ii y > 0
d
i x ≠ 0 ii y ≠ 0
e
i All x ii All y
0
y
x
(2, 8)
0
y
x
(1, 4)
0
y
x−11
0
y
x
(1, 1)
0
y
x−1
1
0
y
x−2
1_2
0
y
x
(1, 2)1
0
y
x
(−1, 3)
1
0
y
x
(1, −4)
−1
12--- y 1
2---+ 3
y--- 1–
y 3–±
17---
13---
3
72--- 1
2---
12--- 5
2---
12---
32---
3
2
−1 12---
30---
0
y
x
(1, 4)
1
0
y
x
1
1−1
0
y
x
1
(1, 2)
0
y
x
(1, 2)
0
y
x
1
1–2
666 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3
f
i All x ii All y
Exercise 14-041 a
b i
ii
iii
iv
c i y = f(x) is moved up 1 unitii y = f(x) is moved down 3 units
iii y = f(x) is moved 2 units to the rightiv y = f(x) is moved 1 unit to the left
2 a
b i ii
iii iv
v vi
3 a
b i ii
iii
4 a b
c
5 a b
c
Exercise 14-051 b, c, h
2 a i f −1(x) = 3x
ii
b i y = 8 − 4x
ii
c i f −1(x) =
ii
d i f −1(x) =
ii
0
y
x
(2, 8)
0
y
x
(1, 2)
0
1
y
x
(1, 3)
0
−3
y
x(1, −1)
20
−4
y
x
−1 0
2
y
x
0
y
x
0
y
x
−3 0
2
y
x
−10
y
x3
0
y
x
0
y
x−10
y
x(−2, −1)
3
0
y
x
0
y
x1
0
y
x
2
0
y
x
(−2, 1)
−3
−2
y
x0
y
x0
1
y
x0
−3
3
y
x0 −2
y
x0
0
y
x
y
x0
y = 3x
y = x_3
y
x0
y = 2 − x_4
y = 8 − 4x
x3
y
x0
1
1−1
−1
y = x3
y = x3
x 5+2
------------
y
x0
y = x + 5__2
y = 2x − 5
ANSWERS 667
e i f −1(x) =
ii
f i y =
ii
3 a Teacher to check.b The graph of f(x) = 2 − x is itself
symmetrical about the line y = x.
4 a b No.
c f −1(x) =
d f −1(x) = −
5 a
b f −1(x) = y = −
c x � 2
Exercise 14-061 a 2 b 3 c 2 d 4 e 5
f 3 g 3 h 2 i 6 j 8k 6 l 3
2 a log5 25 = 2 b log4 64 = 3
c log10 10 000 = 4 d log25 5 =
e log2 = − 4 f log3 = −2
g log8 4 = h log10 0.01 = −2
i log4 = j log16 4 =
k log9 27 = l log6 = −
3 a 125 = 53 b 10 = 101
c 27 = d = 23.5
e 64 = 26 f = 3−4
g = 5−3 h =
i 10 = j =
k 2 = l = 100−1
Exercise 14-071 a 7 b 3 c 2 d −1
e f −2 g − 4 h − 4
2 a 1 b 2 c 1 d 3
e 3 f 2 g −1 h
i 3 j 2 k −1 l − 43 a logx 30 b logx 5 c logx 8
d logx 2 e logx 40 f logx 10
g logx h logx i logx 12
j logx 2
4 a 1.2042 b 2.6021 c 3.6021d 0.30105 e −0.3979 f 2.2042g 0.3979 h 0.80105
5 a a + b b a + b + cc 2b + c d c − (a + b)e 2c + a + b f −a
g h −(a + c)
i (a + c) − 2b j (b − c)
6 a logm x + logm y + logm z
b logm a +logm b − logm c
c logm a − (logm b + logm c) or
logm a − logm b − logm c
d logm a + logm (x + y)
e (logm a + logm x)
f −(logm x + logm y) or −logm x − logmy
g 2 logm x − logm y
h logm x − 3 logm y
i − (logm x − logm y) or
logm y − logm x
Exercise 14-081–5 Teacher to check.
6 a i 1.3010 ii 2.7973 iii 3.7345iv 0.9138 v 0.3979 vi −0.1192
b, c, d Teacher to check.
Exercise 14-091 a k = 10 b m = 8 c d = 10
d x = 2.5 e y = −4.5 f a = 3.5g k = 1.5 h n = −1.5 i d = 2.75
2 a x = 1.425 b x = 2.227c x = 2.519 d x = −0.943e x = 0.428 f x = −0.661g x = 7.555 h x = −0.107i x = −1.121 j x = 1.011k y = −0.975 l k = −2.069
3 a x = 2 b x = c x =
d x = − e x = f −
g x = h x = −2
4 a x = 8 b x = 1000 c x =
d x = e x = f x = 32
g x = h x = i x =
j x = k x = 128 l x =
5 a x = 2 b x = c x =
d x = 0.1 e x = 256 f x = 2g x = 3.915 h x = 23.04 i x = 3.846j x = 6.687 k x = 1.682 l x = 19.180
6 a 11.89 ≈ 12 years b 22.43 ≈ 23 months7 a A ≈ 106 g b t = 20 days
c t ≈ 58 days d Teacher to check.
2 2x–3
----------------
y
x0
y = 2 − 2x_____3
y = 1 − 3x__2
2x 3+2
----------------
y
x0
y = 2x + 3_____2
y = 2x − 3_____2
0
y
x
−3
x 3+
−3
−3
0
y
x
y = x2 − 3, x � 0
(x + 3)y =
x 3+
0
y
x
−3
−3
y = x2 − 3, x � 0
(x + 3)y = −
0
y
x
2
2 x–
0
y
x2
y = − (2 − x)
12---
116------ 1
9---
23---
2 14--- 1
2---
32--- 1
6------- 1
2---
36
8 2
181------
1125--------- 2 8
16---
10012---
5 5 532---
813---
1100---------
12---
12---
14--- 1
5---
b c+2
------------
13---
12---
12---
12---
12--- 1
2---
53--- 5
4---
12--- 7
2--- 13
6------
54---
125------
164------ 81
16------
11000------------ 16 2 1
10----------
18--- 1
25------
15--- 1
2---