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Answers
Chapter 1
Exercise 1A
1 a 57.5 b −1.2 c 24.25 d −16e −4 f 2.75 g 6 h 1.5i
10
7 j 3 k 7 l −4
m −14.4 n −5 o 9 p −1q −1.2 r 1.4 s 51 t 82u −11.5 v 12 w 8 x −2.8y 3.5 z 2.4
2 a − ba
be − d
cc
c
a− b d b
a − c
eab
a+
b f a + b g
b − d a
−c
hbd − c
a
3 a −18 b −78.2 c 16.75 d 28e 34 f 0.1154
Exercise 1B
1 a x ( x + 3) b x ( x − 5) c x ( x + 1)d x (3 x − 4) e x (5 x − 1) f 5 x ( x − 3)g 3 x (2 x − 5) h − x ( x + 5) i −4 x ( x − 4)
2 a ( x + 4) ( x + 6) b ( x + 1) ( x + 8)c ( x
−3) ( x
−8) d ( x
−6) ( x
+5)
e ( x − 4) ( x − 5) f ( x + 3) ( x − 40)g ( x + 2) ( x − 9) h ( x − 3) ( x − 16)i ( x − 7) ( x + 12) j (5 x + 3) ( x + 4)
k (3 x − 2) (2 x − 1) l (5 x − 4) ( x − 3)m ( x + 4) (6 x − 5) n (3 x + 2) (5 x − 7)o ( x + 1) (15 x − 14)
3 a ( x − 7) ( x + 7) b ( x − 4) ( x + 4)c (1− x) (1+ x) d (2 x − 9) (2 x + 9)e 2 (5 x − 7) (5 x + 7) f 4 ( x − 3) ( x + 3)
4 a 5 ( x − 4) ( x + 4) b 6 ( x − 3) ( x + 3)c 2 (2 x − 5)(2 x + 5) d 4 ( x + 2) ( x + 3)e 3 ( x
+2) ( x
+3) f 2 ( x
−2) ( x
−7)
g 5 ( y + 2) ( y − 6) h 3 ( x + 3) (2 x + 5)i 2 (2 x + 7)(3 x − 4) j x ( x − 6) ( x + 1)
k x (5 x
−6) ( x
−2) l 3 x ( x
−4)2
Exercise 1C
1 a −3 or 3 b −6 or 6c −8 or 8 d −0.5 or 0.5e 4 or −4 f −0.25 or 0.25g −4.98 or 4.98 h 0
2 a 2 or 4 b −3 or 11c −7 or 2 d −16 or 4e −1.5 or −1 f 0.5 or 1.5g −3 or 8 h −1.5 or −2
3i−
1.5 or 2
3 a −7.606 or −0.394 b 0.7085 or 11.29c −3.245 or 9.245 d −6.071 or 1.071e −5.804 or 0.804 f −12.75 or 2.746
4 a −1.5 or −1 b 0.134 or 1.866c −5.266 or 6.266 d −5.727 or 1.881e −13.57 or 1.572 f −1.954 or 8.954
5 a −0.65 or 4.65 b −0.41 or 2.41c −1.23 or 1.90
6 a −8 or −1 b −11 or 3c −4 or 7 d −2.243 or 6.243e −2.281 or −0.219 f −1.5 or −0.5
g −2 or 8 h 0.5 or 5
3
i −1.886 or 2.386 j 56
or 3
k −1.5 or 3 l 0.5 or 0.6m −0.75 or 2
3 n 0.5
o10
3 or 0
p 0 or 3
q −5 or −3 r 0.2 or 27 4 or 9
8 3
9 2 or 2.375
10 1 or −
m
4
563
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Exercise 1D
1 a
1 2
−2
−1
−1
1
2
3
4
x
y y = 3 x + 2
b
x
y
1 2
−1
1
2
3
4
5 y = 4 − 2 x
c
1 2 3 4 5 6
1
2
3
4
5
6
7
89
10
x
y
y = −1.5 x + 10
7
d
−2 −1 1 2
−4
−3
−2
−1
1
2
3
4
x
y y = x2 − 3
e
1 2 3 4 5
−2
−1
1
2
3
4
5
x
y
y = ( x − 1) ( x − 3)6
f
−1 1 2 3 4
−5
−4
−3
1
2
x
y
−2
−1
y = 3 x − x23
2 a
−3 −2 −1 1
−3
−2
−1
1
2
3
4
5
x
y
y = 0.5 x + 1
b
−3 −2 −1 1
−5
−4
−3
−2
−1
1
2
3
x
y
y = −2 − x
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c
−1 1 2 3−1
1
2
3
4
5
x
y
−2
y = −2.5 x + 56
d
−3 − −1 1
−4
−3
−2
−1
1
2
3
4
x
y
2
y = 3 x ( x + 2)
e
−4 −3 −2 −1
−4
−3
−2
−1
1
2
3
4
x
y y = ( x + 2)2 − 3
f
−2 −1 1 2
−3
−2
−1
1
2
3
4
5
x
y y = 5 − 2 x2
Exercise 1E
1 a
−3 −2 −1 1
−3
−2
−1
1
23
4
5
x
y y = 0 .75 x + 2
b
1 2 3 4
−5
−4
−3
−2
−1
1
2
x
y y = 0.5 x − 2
−6−6
c
−2 −1 1 2
−4
−3
−2
−1
1
2
3
4
x
y y = 2 x + 1
d
−2 −1 1 2
−5
−4
−3
−2
−1
1
2
x
y y = 3 x − 2
−6
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e
−1 1 2 3
−2
−1
1
2
3
4
5
6
x
y
y = −2 x + 4
f
−2 −1 1 2
−4
−3
−2
−1
1
2
3
4
x
y
y = −0.5 x
2 a
1 2 3 4
−3
−2
−1
1
2
3
4
5
x
y x + 2 y = 4
b y
−1 1 2 3
1
2
3
4
5
6
x
−1
73 x + y = 6
c
−3 −2 −1 1
1
2
3
4
5
6
7
x
y
−1
2 y − 3 x = 9
d
− 4 −3 −2 −1 1
−
−2
−1
1
2
3
4
5
x
y4 y − 2 x = 9
3
e
−4 −3 −2 −1
−3
−2
−1
1
2
3
4
x
y
4
x = 4 y − 4
−
f
−3 −2 −1 1−1
1
2
3
4
6
7
x
y
5
5 x − 2 y + 12 = 0
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3 a
−2 −1 1 2
1
2
3
4
5
7
x
y
−1
6
y = 6
b
−1 1 2 3
−6
−5
−4
−3
−2
−1
1
2
x
y
y = 4 x − 3
c
1 2 3 4
−5
−4
−3
−2
−1
1
2
x
y
−6
x − y = 4
d
−2 −1 1 2
−4
−3
−2
−1
1
2
3
4
x
y2 x − y = 0
e
−3 −2 −1
−3
−2
−1
1
2
3
4
x
y
−4
−4
x = −4
f
−2 −1 1 2
−4
−3
−2
−1
1
2
3
4
x
y2 y = 3 x
g
1 2−1−1
−2
−3
−2
1
2
3
4
x
y
−4
x = 0
h
1 2 3 4
−2
−1
1
2
3
4
5
x
y
−3
x + y = 3
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i
1 2−1−1
−2
−3
−2
1
2
3
4
x
y
−4
y = 0
4 a
x
y
–4
(2,2)
b
(−2,4)
x
y
–4
c
x
y
−
3
92
d
3
x
y
(1, –1)
e
–3.5
–7
x
y
f
–5
2
x
y
g
–2
8
x
y
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570 Queensland Mathematics B Year 11
d
−4 x
y
(−1,−3)
e
x
y
−1
−3
1.5
f
x
y
−2
8
2
2 a y = ( x + 1)2 + 4
x
y
5
(−1,4)
b y = ( x − 2)2 + 2
x
y
6
(2,2)
c y = ( x − 1.5)2 + 2.75
x
y
4
5
(1.5,2.75)
d y = ( x + 3)2 − 9
x
y
(−3,−9)
e y = 2( x − 1)2 + 7
x
y
(1,7)
9
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f y = 3( x + 1)2 − 4
x
y
(−1,−4)
−1
g y = ( x + 3)2
x
y
−3
9
h y = 3( x − 1.5)2 + 1.25
x
y
(1.5,1.25)
8
i y = 9− 2 x2
x
y
(1,7)
9
3 a
x
y
−2
8
2
b
x
y
−3
−12
2
(−0.5,−12.5)
c
x
y
11
(2,3)
d
x
y
8
(3,−1)42
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572 Queensland Mathematics B Year 11
e
x y
−10
(1.5,−12.25)
5−2
f
x
y
5
(2.5,−6.25)
g
x
y
−2
(−1,−2)
h
x
y
−12
(1.25,−15.125)
4−1.5
i
x
y
(0.5,0.75)
1
4 a
x
y
3 9,
8 16−
3
4
b
x
y
−8
8
3
8
3−
c
x
y
1
5 − 136
5 13
6
5 13,
6 12
–
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Exercise 1G
1 a x = −1 y = −1 b x = 5 y = 21c x = −1 y = 5 d x = 8 y = −2e x = 3 y = 4 f x = 7 y = 1/2
2 a x = 1 y = 5 b x = 2.5 y = −1c x = 2 y = −1 d s = −1 t = 4e p = 1 q = −1 f x = −1 y = 2.5g x = −1 y = 2
or
x = 3 y = 10
h x = −3 y = 8or
x = 1 y = 0i x = 1 y = 0 j x = −2 y = −4
or
x = 5 y = 17k x = 1 y = −1 l x = 0 y = 0
or
x = 1 y = 23 a x = 3 y = −2 b x = 7 y = 2
c x = 4 y = −3 d x = 7 y = 3e x = 4 y = −2 f x = 3 y = 1g x = 5 y = −2 h x = 3 y = −2i x = −2 y = −3
4 a x = 2 y = 4b x = −1 y = 7c x = 0 y = 8d x = −2.236 y = 0.528
or
x = 2.236 y = 9.472e x = −2.608 y = 3.804
or
x =
2.108 y =
1.446
f x = −0.781 y = 4.390or
x = 1.281 y = 3.360
Exercise 1H
1 a 20 b −12 c 16 d 9 e 412 a cross b neither c touch
d cross e neither f touch
3 a 2 b 0 c 1 d 2 e 1 f 0
4 a 1 rational root b 2 rational roots
c 2 irrational roots d 1 rational roote 2 irrational roots f no real roots
5 = m2 + 8m + 166 a m = 3 or −3 b −3 −1/4
Exercise 1J
1 a y = 3 x + 5 b y = −4 x + 6c y = 3 x − 4
2 a y = 3 x − 11 b y = −2 x + 93 a y = 2 x + 6 b y = 2.5 x − 24 a y = 4 x + 4 b y = −2
3 x
c y = − x − 2 d y = 12 x − 1
e y = 3.5 f x = −25 a 2 x + 3 y − 12 = 0 b 2 x + y + 6 = 0
c x + y − 8 = 0 d x + 2 y − 4 = 0e 2 x − 3 y − 2 = 0 f x = 3g 2 x − 2 y + 7 = 0 h y = 5i 2 x + 4 y − 1 = 0
6 yes
7 AB: 2 x − 3 y + 1 = 0 BC : 3 x + 2 y − 18 = 0 AC : x − 8 y − 6 = 0
Exercise 1K
1 a y = − 516
x2 + 5 b y = x 2
c y = 111
x ( x + 7) d y = ( x − 1) ( x − 3)
e y = −54
( x + 1)2 + 5 f y = ( x − 2)2 + 2
2 a y = 107
( x − 2) ( x + 4) b y = −4 x ( x − 4)
c y = 45
( x − 2) ( x − 5)
3 a y = 12
( x + 1)2 + 2
b y = −32
( x − 2)2 + 3
c y = 43 x2
4 a y = −2 x2 + x + 5b y = 1
4 x2 − 1
2 x + 3
c y = 4 x2 − 7 x5 y = 1
180( x − 90)2 + 30
Exercise 1L
1 a,b Check with your teacher
2 a C = 10, F = 50b −40◦ C = −40◦ F
3 a,b Check with your teacher
4 13
5 abc − ac − 1 b
2b
c − a
6
ab
a − b + c
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9 a
x
y
6
−3
b
x
y
3
9
c
x
y
7
2.8
10 a
x
y
2
−6
b
x
y
3
(1,1)
c
x
y
−10
6
11 a
x
y
(1,5)
3
b
x
y
29
(4,−3)
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c
x
y
6
−2 3
12 a y = ( x − 2)2 − 9
x
y
(2,−9)
−5
b y = −( x − 1.5)2 + 2.25
x
y
(1.5, 2.25)
c y = 2( x − 2)2 − 5
x
y
3
(2,−5)
13 a
x
y
1 6
6
(3.5, −6.25)
b
x
y
−25
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c
x
y
−5
13
12
−289
24,
1
3
5
2−
14 a x = −1 y = 2b x = −1 y = 4c x
=4 y
=1
15 a x = 2.5 y = −1b x = 4 y = −2c x = 1 y = −0.5
16 a x = 0.5 y = 6b x = −2.5 y = −3c x = 2.121 y = −0.257
or
x = −2.121 y = −8.74217 a x = 2 y = 4
b no solution
c x = 3.0769 y = −4.30718 a (−1, 1) and (3, 9)
b (−1.5, 4.5) and (4, 32)19 a 2 irrational roots b 1 rational root
c no real roots
20 Check with your teacher
21 a y = ( x + 4) ( x − 2)b y = 3 ( x − 1)2 − 3c y = −1
2( x − 1)2 + 2
22 a y = 43
( x + 1) ( x − 3)b y = −2 ( x − 3)2 + 2c y = 3 x2 + 4 x − 7
Chapter 2
Exercise 2A
1 a 4.10 b 0.87 c 2.94
d 4.08 e 33.69◦ f 11.92
240√
3cm
3 66.42◦, 66.42◦ and 47.16◦
4 23 m
5 a 9.59◦ b √ 35 m
6 a 60◦ b 17.32 m7 a 6.84 m b 6.15 m
8 12.51◦
9 182.7 m
10 1451 m
11 a 5√
2 cm b 45◦
12 3.07 cm
13 37.8 cm
14 31.24 m
15 4.38 m
16 57.74 m
Exercise 2B
1 a 8.15 b 3.98 c 11.75 d 9.46
2 a 56.32◦ b 36.22◦ c 49.54◦
3 a A = 48◦, b = 13.84 cm, c = 15.44 cmb a = 7.26, C = 56.45◦, c = 6.26c B = 19.8
◦
, b = 4.66, c = 8.27d C = 30◦, a = 5.41, c = 15.56
4 C = 26.69◦, A = 24.31◦, a = 4.185 554.26 m
6 35.64 m
7 1659.86 m
8 a 26.60 m b 75.12 m
Exercise 2C
1 5.93 cm
2 ∠ ABC
=97.90◦, ∠ AC B
=52.41◦
3 a 26 b 11.74 c 49.29◦ d 73e 68.70 f 47.22◦ g 7.59 h 38.05◦
4 2.626 km
5 3.23 km
6 a 8.23 cm b 3.77 cm
7 55.93 cm
8 a 7.326 cm b 5.53 cm
9 a 83.62◦ b 64.46◦
10 a 87.61 m b 67.7 m
Exercise 2D
1 400.10 m
2 34.77 m
3 575.18 m
4 109.90 m
5 16.51 m
6 027◦
7 056◦
8 a 034◦ b 214◦
9 a 3583.04 m b 353◦ or N7◦W
10 ∠ AS B = 113◦11 22.01◦
12 a ∠ B AC
=49◦ b 264.24 km
13 10.63 km
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Exercise 2E
1 a
3 b
4
5 c
4
3
d11
6 e
7
3 f
8
32 a 120◦ b 150◦ c 210◦ d 162◦
e 100◦ f 324◦ g 220◦ h 324◦
3 a 34.38◦ b 108.29◦ c 166.16◦ d 246.94◦
e 213.14◦ f 296.79◦ g 271.01◦ h 343.77◦
4 a 0.66 b 1.27 c 1.87 d 2.81
e 1.47 f 3.98 g 2.38 h 5.74
5 a 0.95 b 0.75 c −0.82 d 0.96e −0.50 f −0.03 g −0.86 h 0.61i −34.23 j 0.36
6 a 0, −1, 0 b −1, 0, undefined c 1, 0, undefined d −1, 0, undefined e −1, 0, undefined f 0, 1, 0g 0,
−1, 0 h 0,
−1, 0
Exercise 2F
1 a
√ 3
2 , −1
2, −
√ 3 b
1√ 2, − 1√
2, −1
c 0, −1, 0 d −√
3
2 , −1
2,√
3
e − 1√ 2,
1√ 2, −1 f 1
2,
√ 3
2 ,
1√ 3
g
√ 3
2 ,
1
2 , √ 3 h −1√ 2 , −
1√ 2 , 1
i
√ 3
2 ,
1
2,√
3 j −√
3
2 ,
1
2, −
√ 3
k −1, 0, undefined l 0, 1, 0
2 a
√ 3
2 b − 1√
2c 0 d −1
2
e 0 f √
3 g −√
3
2 h1√ 2
i − 1√ 3
j −1 k −1 l undefined
3 a −√
3
2 b − 1√
2c
1√ 3
d undefined
e 0 f − 1√ 2
g1√ 2
h −1
Exercise 2G
1 a 13 cm b 15.26 cm c 31.61◦ d 38.17◦
2 a 4 cm b 71.57◦ c 12.65 cm
d 13.27 cm e 72.45◦ f 266.39 cm2
3 17.58◦
4 1702.55 m
5 10.31◦ at B, 14.43◦ at A and C
6 45.04 m
7 a 24.78◦ b 65.22◦ c 20.44◦
8 42.40 m
9 1945.54 m
10 a 6.96 cm b 16.25 cm2
11 a 5 km b 215.65◦ c 6◦33
12 5 m
13 11.37 m
14 14 ≈ 401 km/h15 height = 8√
15= 2.07 m
Multiple-choice answers
1 D 2 C 3 C 4 D 5 B
6 C 7 A 8 C 9 B 10 E
Short-response answers
1 a7
6 b 0.51c c 2.06c
2 a 135◦ b 154◦42
3 a −12
b1√ 2
c −√
3
d 0 e −1
2 f undefined
g −12
h −12
i − 1√ 3
4 35.53 km
5√
91 ≈ 9.54 cm6 143◦
7 a 052.6◦
b ∠T QS = 33.14◦, bearing of T from Q is105.9◦
8 9.4 cm
9 a i 39◦ ii 9◦
b 1425.16 m c 1083.29 m
10 13.45 km11 a AC = 4.16km, BC = 2.4 km
b 57.6 km/h
12 804 m
13 a ∠ ACB= 12◦,∠CBO = 53◦,∠CBA= 127◦b 189.33 m c 113.94 m
14 a ∠TAB= 3◦,∠ ABT = 97◦,∠ ATB= 80◦b 2069.87 m
c 252.25 m
15 a 184.74 m b 199.71 m c 14.93 m
16 a 370.17 m b 287.94 m c 185.08 m
17 a 8√
2 cm b 10 cm c 10 cm d 68.90◦
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Chapter 3
Exercise 3A
1 a x5 b 8 x7 c x2 d 2 x3
e a6 f 26 g x2 y2 h x2 y6
i x3
y3 j x
6
y4 k 1 l 1
m3
2 n
23
52
2 a x9 b 216 c 317 d q8 p9
e a11b3 f 28 x18 g m11n12 p−2 h 2a5b−2
3 a x2 y3 b 8a8b3 c x5 y2 d9
2 x2 y3
4 a1
n4 p5 b
2 x8 z
y4 c
b5
a5 d
a3b
c
e an+2bn+1cn−1
5 a 1 b 317n c34n − 11
2
2
d 2n+133n−1 e 53n−2 f 23 x−3 × 3−4g 36−n × 2−5n h 33 = 27 i 6
j34n
2k
3n
5n l
1
2× 35n6 a 212 = 4096 b 55 = 3125 c 33 = 277 Check with your teacher
Exercise 3B
1 a 25 b 27 c1
9 d 16 e
1
2 f
1
4
g 125 h 16 i
110 000
j 1000
k 27 l3
5
2 a 253 b a
16 b
− 76 c a−6b
92 d 3
− 73 × 5−
76
e1
4f x6 y−8 g a
1415
3 a (2 x − 1)3/2 b ( x − 1)5/2 c ( x2 + 1)3/2d ( x − 1)4/3 e x√
x − 1f (5 x2 + 1)4/3
Exercise 3C
1 a 3 b 3 c 12
d3
4 e
1
3 f 4
g 2 h 3 i 3
2 a 1 b 2 c −32
d4
3 e −1 f 8
g 3 h −4 i 8 j 4 k 3
1
2 l 6
m 71
2
3 a4
5 b
3
2c 5
1
2
4 a 0 b 0,−2 c 1, 2 d 0, 15 a 2.32 b 1.29 c 1.26 d 1.75
6 a x > 2 b x >1
3 c x ≤ 1
2
d x 1
g x ≤ 37 a x ≤ 1.43 b x ≥ 0.77
c x > −1.89 d x > 2.718 0.5
9 x ≤ 210 a 2.5 b 2
Exercise 3D
1 y y = 2.4 x
y = 1.8 x
= 0.5 x
= 0.9 x
x
1
0
all pass through (0, 1)
base > 1, increasing
base >>
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A n s
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580 Queensland Mathematics B Year 11
5 a–f Check with your teacher
6 a–d Check with your teacher
Exercise 3E
1 a 3 b 4 c −7 d −3 e 4f −3 g 4 h −6 i −9 j −1
k 4 l −22 a log2(10a) b 1 c log2
9
4
d 1 e − log5 6 f −2g 3 log2 a h 9
3 a 2 b 7 c 9 d 1
e5
2f 5 log x a g 3 h 1
4 a 2 b 27 c1
125 d 8
e 30 f 2
3 g 8 h 64
i 4 j 10
5 a 5 b 32.5 c 22 d 20
e3±
√ 17
2 f 3 or 0
6 2+ 3a − 5c2
7 Check with your teacher
8 10
9 a 4 b6
5 c 3
d 10 e 9 f 2
Exercise 3F
1 a 2.81 b −1.32 c 2.40 d 0.79 e −2.58f −0.58 g −4.30 h −1.38 i 3.10 j −0.68
2 a x > 3 b x >>
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Answers 581
e y
x
3
2
1
−1−4 −3 −2 −1 1 2 3 4
−2
−3
−4
intercept: x = 1asymptote: x = 0
f y
x
3
21
−1−4 −3 −2 −1 1 2 3 4
−2
−3
−4
intercept: x = 1asymptote: x = 0
2 a i y = 2log10 x ii y = 13
log10 x
b i y = 1013 x
ii y = 13
1012 x
3 a y = log3( x − 2) b y = 2 x + 3
c y = log3
x − 24
d y = log5( x + 2)
e y = 13 × 2 x f y = 3× 2 x
g y = 2 x − 3 h y = log3
x + 25
4 a–f Check with your teacher
5 a 0.64 b 0.406 Check with your teacher
7 y = log10(√
x) = 12
log10 x for x ∈ (0, 10]
y
x0 1
8 Check with your teacher
9 a = 6
103
2/3 and k = 1
3 log10
10
3
10 a
f ( x)
y
x
g ( x) = f ( x) + 1
(1,0)
(1,1)
b y
x
h( x) = 2 f ( x)
(1,0)
f ( x)
11 Shift (translation) of cb
from y-axis parallel to
x-axis, followed by a 1
bstretch (dilation) also
from the y-axis parallel to the x-axis.
Exercise 3H
1 y = 1.5× 0.575 x2 p = 2.5× 1.35t 3 a
Total thickness,
Cuts, n Sheets T (mm)
0 1 0.2
1 2 0.4
2 4 0.8
3 8 1.6
4 16 3.2
5 32 6.4
6 64 12.8
7 128 25.6
8 256 51.2
9 512 102.4
10 1024 204.8
b T = 0.2(2)nc
200
T
150
100
500.2
0 2 4 6 8 10 n
d 214 748.4 m
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A n s
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582 Queensland Mathematics B Year 11
4 an 0 1 2 3 4
M 0 1 3 7 15
b M = 2n − 1
n 5 6 7
M 31 63 127
c M
n0
30
20
10
1 2 3 4 5
dThree discs 1 2 3
Times moved 4 2 1
Four discs 1 2 3 4
Times moved 8 4 2 1
5 n = 2
6 a 12
3n
b 12
5n−2c n =
3
7 a 729
1
4
nb 128
1
2
nc 4 times
8 11.21% (2009)
9 H = 1.26× 10−11 , given H >>
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An s
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Answers 583
e
0
y
x
f
y = 2(0,3)
0
y
x
7 x = 18 Check with your teacher 9 3
10 a k = 17
b q = 32
11 a a = 12
b y = −4 or y = 2012 a batch 1 = 15(0.95)n , batch 2 = 20(0.94)n
b 32 years
13 a
y = p(t )
p,q
y = q(t )
(millions)
1.7
1.2
0 t
b i t = 12.56 (i.e. mid 1962)ii t = 37.56 . . .(i.e. mid 1987)
14 a company X $1.82, company Y $1.51,
company Z $2.62
b company X $4.37, company Y $4.27,
company Z $3.47
c intersect at t = 21.784 and t = 2.090; hence,February 2006 until September 2007
d February 2007 until September 2007, approx.
8 months
15 a 13.81 years b 7.38 years
16 a Temperature = 87.065 × 0.94t b i 87.1◦ C ii 18.6◦ C
c Temperature = 85.724 × 0.94t d i 85.7◦ C ii 40.8◦ C
e 28.2 min
17 a a = 0.2 and b = 5b i z = x log10 b
ii a = 0.2 and k = log10 518 a y = 2× 1.585 x b y = 2× 100.2 x
c x = 5log10 y
2
Chapter 4
Exercise 4A
1 a
x
y
−1 1 2 3−1
−2
−3
1
2
3
b
x
y
−2 −1 1 2 3−1
−2
−3
1
2
3
c
x
y
−2 −1 1 2 3−1
−2
−3
1
2
3
d
x
y
−2−3 −1 1 2 3
1
2
3
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A n s
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584 Queensland Mathematics B Year 11
2 a
x
y = −5
y
−2−3 −1 1 2 3−1
−2
−3
−4
1
2
3
4
b
x
x = −3
y
−2−3 −1 1 2 3 4 5−1
−2
−3
12
3
c
x
y = 1
y
−2−3 −1 1 2 3
1
2
3
4
5
6
d
x
y x = 2
1 2 3 4 5 6−1
−2
7 8 9 10
1
2
3 a
x
y
−2−4
−1 1 2 4 5 6−1
−2
−3
−4
−5
1
2
3
4
−3 3
b
x
y
−2−3 −1 1 2 3−1
−2
−3
1
2
3
c
x
y
−2−3−4 −1 1 2 3 4−1
−2
−3
−4
−5
1
2
3
4
5
−5
5
d
x
y
−2−4 −1 1 2 4 5−1
−2
−4
−5
−5
12
4
5
−3 3
3
−3
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Answers 585
4 a
x
x = 1
y
−2−3−4 −1 1 2 3 4 5 6−1
−2
−3
−4
−5
1
2
3
4
5
b
x
x = 2
y
−2−3−4 −1 1 2 3 4 5 6 7−1
−2
−3
−4
−5
1
2
3
4
5
c
x
x = 1
x = −
1
y
−2−3−4 −1 1 2 3 4 5−1
−2
−3
−4
−5
−5
1
2
3
4
d
x
y
−2−3−4 −1 1 2 3 4 5−1
−5
1
2
3
4
Exercise 4B
1
1 2 3 4 5 6
12
11
10
9
8
7
5
4
3
2
1
6
Cost
No. cartons7
2 y
1 2 3 4 5 6 x
7
5
4
3
2
1
6
3 a y
-781 2 3 4 5 6 7 x
8
7
5
4
3
2
1
6
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A n s
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586 Queensland Mathematics B Year 11
b y
-781 2 3 4 5 6 7 x
8
7
5
4
3
2
1
6
4 y
-781 2 3 4 5 6 7 x
8
7
5
4
3
2
1
6
(4, 8)
5V
98 10
10.5
111 2 3 4 5 6 7
1200
1100
(4,1144)
(6,972)
(2,884)
(8,560)
(10,100)
Length of cut
1000
900
800
700
500
400
300
200
100
600
6 a y
5 x
5
b y
-7 9 1081 2 3 4 5 6 7
9
8
7
5
4
3
2
1
6
x
(5,4)
(7,6.3..)
(9,8.4..)
7
2
1
Cost ($)
50 100 150 200 250 500
Weight (g)
8Cost ($)
1 2 3 4 15 24
Time (h)2
456
8
10
12
14
36
Exercise 4C
1 a Independent: Number of cartons
Dependent: Cost
b Domain: {1, 2, 3, 4, 5}Range: {2, 4, 6, 8, 10}
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Answers 587
c
1
2
3
4
5
2
4
6
8
10
2 a Independent: Number on uppermost face
Dependent: Number on the table
b Domain: {1, 2, 3, 4, 5, 6}Range: {1, 2, 3, 4, 5, 6}
c
1
2
3
4
5
6
1
2
3
4
5
6
3 a i Independent: Height
Dependent: Base
ii Domain: 0
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A n s
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588 Queensland Mathematics B Year 11
2 a i 7 ii 15 iii −3 iv 10b i 2a + 5 ii a2 + 2a + 7
iii −6b − 1 iv b2 − 6b + 15c i 9.5 ii ±
√ 14
3 a i −5 ii 5 iii 4 iv 15b i 7− 3b ii 2a2 + 8a + 5
iii −
6a−
5 iv 8b2
+24b
+15
c i 21
3 ii ±54 a i −24 ii 4
b i 5a + 5h − a2 − 2ah − h2 ii 4c i no solution ii 1 or 4
5 a i
x
y
1−4
−4
ii
x
y
4
4−1
b i Domain: All real numbers
Range: y ≥ −6 14
ii Domain: All real numbers
Range: y ≤ 6 14
6 a i
x
y
−1 3
3
ii
x
y
3
(1,2)
b i Domain: All real numbers
Range: y ≤ 4ii Domain: All real numbers
Range: y ≥ 2Exercise 4F
1 a Domain: All real numbers
Range: All real numbers
b Domain: x ≥ 0Range: y ≥ 0
c Domain: All real numbers
Range: y ≥ 1d Domain: −3 ≤ x ≤ 3
Range: −3 ≤ y ≤ 0e Domain: x > 0
Range: y > 0
f Domain: All real numbers
Range: y ≤ 3g Domain: x ≥ 2
Range: y
≥0
h Domain: x ≥ −1Range: All real numbers
i Domain: x ≤ 1 12
Range: y ≥ 0 j Domain: x = −2
Range: y = 0k Domain: x > 5
Range: All real numbers
l Domain: x = 12
Range: y = 0m Domain: x
= −2
Range: y = −4n Domain: x > 3
Range: y > 0
o Domain: x > −1Range: y > 0
2 a
x
y
(1,1)
−2
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Answers 589
b
x
y
(1,2)
−1
c
x
y
3
(−1,5)
d
x
y
(−2,4) (3,4)
e
x
y
(−2, 6)
4
2
f
x
y
−2
2
g
x
y
(1,3)
2
h
x
y
1
(10,1)
Exercise 4G
1 a y
x0
Range = [0, ∞)b y
x0
1
1
Range = [0, ∞)c y
x0
Range = (−∞, 0]d y
x0
Range = [1, ∞)e y
x
2
0
(1,1)
Range = [1, ∞)
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590 Queensland Mathematics B Year 11
2 a y
x
4
3
2
1
1 2 30
b Range = (−∞, 4]3
y
x
1
2
1 2 3 54
–3 –2 –1
–1
–2
–3
–4
–5
0
4 a y
x
(0,1)
0
b Range = [1,∞)5 a y
x –3 3
0
–9
b Range = R6 a y
x
(1,1)
0
b Range = (−∞, 1]
7
f ( x) =
x + 3, −3 ≤ x ≤ −1− x + 1, −1
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Answers 591
12 f ( x) and h ( x) are even
g ( x) is odd
13 Offer A is cheaper if average number of calls is
>
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A n s
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592 Queensland Mathematics B Year 11
b i 2a + 11ii b2 − 4b + 6
c i 7.5
ii x = ±√
22
d
x
y
7
(−1,6)
6 a Domain: x ≥ −2Range: y
≥0
b Domain: x = 12
Range: y = 0c Domain: x > 8
Range: All real numbers
d Domain : All real numbers
Range: All real numbers
7 a
x
y
(3,8)
−1
b
x
y
−2 2
4
c
x
y
−1
(2,−3)
(2,−1)
4
1
8 a {(3, 1), (4, 3), (6, 4), (8, 7)}
b f −1 ( x) = 12 x − 3
2
c g −1 ( x) = 34 − 1
4 x, x ≤ 3
9 a f −1 ( x) = √ x + 3 or f −1 ( x) = −√ x + 3
b f −1 ( x) =√
x2 + 9, x ≥ 0 or f −1 ( x) = −
√ x2 + 9, x ≥ 0
10 A ( x) = 25 − x2, 0 < x <√
50
11 V ( x) = x (30 − 2 x) (21 − 2 x) , 0 < x >>
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Answers 593
4 a
P e r c e n t a g e o f s t u d e n t s
25
50
W a t c h
T V R e a d
L i s t e n
t o m u s i c
W a t c h
a v i d
e o
P h o n
e f r i e
n d s
O t h e
r
Leisure activity
b watching TV
Exercise 5C
1 Number 0 1 2 3 4 5 6
Frequency 4 4 4 4 3 2 1
2 a 4 b 2 c 5 d 28
3 a 0 b 48 c 60–69 d 334 a, b Temperatures Relative
(◦C) Frequency frequency
0 – 1 0.03
5 – 0 0
10 – 1 0.03
15 – 9 0.28
20 – 4 0.13
25 – 5 0.16
30 – 7 0.22
35 – 4 0.13
40 – 0 0
45 – 1 0.03
c
0
2
4
6
8
10
5 10 15 20 25 30 35 40 45
Temperature
N o . o f c i t i e s
d 47%
5 a
0
2
4
6
N o . o f b o o k s
Price
5 10 15 20 25 30 35 40 45
b $5.00–$5.99
cPrices ($) Cumulative frequency
less than 5 3
less than 10 9
less than 15 12
less than 20 15
less than 25 19
less than 30 19less than 35 20
less than 40 20
less than 45 21
Price
C u m u l a t i v e f r e q u e n c y
50
5
10
15
20
10 15 20 25 30 35 40 45
6 a
0 28 29 30 31 32 33 34
2
4
6
8
10
Measurement
F r e q u e n c y
b
0
28
28 29 30 31 32 33 34Measurement
C u m u l a
t i v e f r e q u e n c y
c The students’ estimates ranged from
28.9 cm to 33.3 cm, with most students (89%)
overestimating the 30 cm measure.
7 a
0 10 20 30 40 50 60 70 80 90 100
2
4
6
8
Marks
N o . o f s t u d e n t s
b
0 10 20 30 40 50 60 70 80 90 100
5
10
15
20
25
30
Marks
C u m u l a t i v e f r e q u e n c y
c The students’ marks ranged from 21 to 99,
with most students (over 70%) scoring more
than 50% on the test.
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594 Queensland Mathematics B Year 11
8 a
Length of hole
N o . o f h o l e s
0 240 260 280 300 320 340 360 380 400 420
2
4
6
8
10
12
14
b
Length of hole
0240 260 280 300 320 340 360 380 400 420
10
20
30
40
50
C u m u l a t i v e f r e q u e n c y
c i below 300 m = 425
ii no. of holes ≥ 360 m = 17, proportion = 17
50
iii approx. 280 m
Exercise 5D
1 a centre b neither c both
2 a positively skewed
b negatively skewed c symmetrical
3 symmetrical 4 symmetrical
5 approximately symmetrical
Exercise 5E
1 a 4 0 2 4 5 7 | 7 represents 7.7 days5 8 9
6 5 8
7 7
8 4 8
9 4
b four months
2 a 0 4
1
1 6 8 9 2 | 5 represents 25 hours2 1 1 3 (truncated)
2 5 5 5 6 7 7 9 9
3 1 1 2 3 3
3 6 9
4 1
4 6
b nine batteries
3 a 0 0
1 0 0 4 5 5 6 9
2 0 0 1 3 7 8 9
3 3 7 9
4 6 4 | 6 represents 46 minutes5
6 3 7
7 0
b three students
c positively skewed
4 a 2 5 8
3 5 6 9
4 5 6 9
5 2
6 8
7 5 5 6 8 9
8 2 4
9 5
10 16 | 4 represents $16411 (truncated)
12
13
14 9
15
16 4
17
18
19
20
21 0
b approximately symmetrical
5 a Father's age Mother's age
3 7 8 8 9
4 4 4 3 3 3 1 1 0 4 0 0 0 1 2 3 3 3 3 3 4 4
9 8 8 8 8 7 7 6 6 6 5 4 5 6 7 8 9 9 9
4 2 1 1 0 5 0 0 0
5 5
0 | 4 represents 40 years 4 | 0 represents 40 yearsb Both distributions are approximately
symmetrical. Fathers, with ages centred in the
late forties, tend to be older than mothers, with
ages centred in the early forties. The spread is
similar for both distributions.
6 a Class B Class A
3 2 1 9
2 2
3 9
4 5 7 8
5 5 8
9 6 5 8
6 4 3 3 2 2 1 0 0 7 1 6 7 9 9
8 8 4 4 3 2 1 1 0 0 8 0 1 2 2 5 5 9
8 1 9 1 9
9 | 6 represents 69 marks 7 | 1 represents 71 marksb six students in class A and two in class B
c Class B performed better as more students
scored in the higher values of 70s to 90s.
Exercise 5F
1 a 800 b 400 c 14 d 9
2 a 56 000 b 1988–89 c 1995–96 d 72%
e 1996–97
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3 a 51% b Jan 2005 c Aug 2007 d 128 000
e Wivenhoe
4 a 47 000 000 b 100 000 000
c 2004/05 d 21%
e Check with your teacher
5 a $8 billion b 4.7%
6 a 0.029 b 0.26 c 53
Exercise 5G
1 a mean = 18.36,median = 14b mean = 9.19,median = 10c mean = 7.41,median = 7.65d mean = 1.62,median = 1.15
2 a mean = 3.24,median = 3b mean
= −0.38,median
=0
3 mean = $193 386,median = $140000; themedian is a better measure of centre as it is
typical of more house prices.
4 mean = 4.06,median = 4; both are reasonablemeasures of centre in this example.
5 a range = 602, IQR = 455b range = 5.3, IQR = 3.2c range = 0.57, IQR = 0.21d range = 7, IQR = 3.5
6 a 145 b 42
7 a 2.4 kg b 1.0 kg
8 a
0 C u m u l a t i v e r e l a t i v e
f r e q u e n c y
Age
17 22 27 32 37 42 47
0.2
0.4
0.6
0.8
1.0
median = 18, IQR = 2b mean = 20.97, s = 7.37 c 92%
9 a 12.39 b 1.33 c 281.24 d 3.04
1 0 a i mean = 17.61, s = 15.96ii mean
=195.3, s
=52.9
b i 94% ii 100%
1 1 a i mean = 6.79,median = 6.75ii IQR = 1.8, s = 0.93
b i mean = 13.54,median = 7.35ii IQR = 1.81, s = 18.79
c The error does not affect the median or
interquartile range very much. It doubles the
mean and increases the standard deviation by
a factor of 20.
12 Approximately 95% of share prices lie between
$44 and $56.
13 About 95% of days lie in this interval.
Exercise 5H
1 a m = 154, Q1 = 141.5, Q3 = 161.5,min = 123,max = 180
b
120 130 140 150 160 170 180
c The distribution of heights is slightly
negatively skewed, centred at 154 cm, with the
middle 50% of heights ranging from 141.5 cm
to 161.5 cm.
2 a m = 3, Q1 = 0, Q3 = 13,min = 0,max = 52
b 38, 52
c * *
0 10 20 30 40 50
d The distribution of number of books borrowed
is positively skewed, centred at 3. While 75%of people borrowed 13 books or less, one
person borrowed 38 books and another
borrowed 52.
3 a
0 105 15 20 25 30 35 40 million
b The distribution of winnings is positively
skewed with a median value of
$5.3 million. The middle 50% of players won
between $2.9 million and $10.0 million. Roger
Federer is the outlier, winning
$39 million.4 a
0 5 10 15 20
b The distribution is symmetrical, centred at
$10.00. The middle 50% of students earn
between $8.15 and $11.85 per hour.
5 a
0 100 200 300 400 500 600 ,
000
*
d The distribution is approximately
symmetrical, centred at about 210 000, withan outlier at 570 000. The middle 50% of
papers have circulations from about 88 000 to
270 000.
Exercise 5I
1 a
0 10 20 30 40 50
*
*
*After
Before
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596 Queensland Mathematics B Year 11
b The distribution for the number of sit-ups is
negatively skewed before the course, centred
at 26. After the course, the distribution is more
symmetric, centred at 30, indicating that the
course has been effective. The distribution
after the course is more variable than before
the course, showing the course has not had the
same effect on all participants. There is one
outlier in the before group, who can achieve
46 sit-ups, and two in the after group,
recording 50 and 54 sit-ups, respectively.
2
0 1 3 5 72 4 6 8 9
Year 12
Year 8
a Year 12 b Year 12
3 a
15 20 30 40 5025 35 45
1990
1970
b The distributions of ages in both groups are
slightly positively skewed, with the mothers in
1970 (median = 24.5) generally younger thanthe mothers in 1990 (median = 28). Thevariability in both groups is the same
(IQR = 10 for both groups).
Exercise 5J
1 a
1.0 2.0 3.0 4.0 5.0 6.00
10203040506070
Drug dose (mg)
R e s p o n s e t i m e ( m i n )
b negative association c no outliers
2 a
100 200 300 400 500 600
5
10
15
0 B u s i n e s s ( $ ’ 0 0 0 )
Advertising ($)
b positive association c no outliers
3 a
100 200 300 400
400
500
600
700
800
No. of seats
A i r s p e e d ( k m / h )
b positive association
c (122, 378) is an outlier
4 a
2 4 6
5
10
15
0 8 10 12
P r i c e
( $ ’ 0 0 0 )
Age (years)
b negative association
c (10, 8700) is an outlier
Exercise 5K
1 a no correlation
b weak negative correlation
c strong negative correlation
d weak positive correlation
e strong positive correlation
f strong negative correlation
g strong positive correlation
h no correlation
i strong negative correlation
j weak positive correlation
k strong positive correlation
l moderate negative correlation
2 a 0.71 b 0.78 c 0.82 d 0.92
3 a −0.6 b moderate negative correlation4 a 0.67 b moderate positive correlation
5 a 1 b strong positive correlation
6 a −0.43 b weak negative correlation
Exercise 5L
1 a no linear relationship
b weak negative linear relationship
c strong negative linear relationship
d weak positive linear relationship
e strong positive linear relationship
f strong negative linear relationship
g strong positive linear relationship
h no linear relationship
i moderate negative linear relationship
j weak positive linear relationship
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Answers 597
k perfect positive linear relationship
l perfect negative linear relationship
2 a 0.8 b 0.8 c 0.7
d 0.8 e −0.7 f −0.23 a −0.86
b strong negative linear relationship
4 a 0.95b strong positive linear relationship
5 a 0.77
b strong positive linear relationship
6 a −0.77b strong negative linear relationship
7
50
50
55
55
60
60
65
65
70
70
40 45Attempt 1
A t t e m p t 2
a a strong positive relationship
b Yes, the data are numerical and the
relationship is linear. There are no outliers.
8
210230250270290310330
200 240220 260280 300 320 3400
T e s t 2
Test 1
a There is a strong positive linear relationship
between the scores on Test 1 and Test 2.
b Yes, the data are numerical and the
relationship is linear.
c q = 0.71, r = 0.87d q: moderate positive linear relationship
r : strong positive linear relationship
e i q = 0.43, r = −0.004ii The error in the data has a much greater
effect on Pearson’s correlation coefficient.
Exercise 5M
Note: Answers will vary for lines drawn by eye.
1
x
y
1 2 3 4 5 6 7 8
5
10
15
0
y = 1 + 2 x
2
x
y
0 1 2 3 4
5
–5 –1 –2 –3
–20
–15
–10
y = – 4.5 – 3.75 x
3 a
2000
2000
4000
4000
6000
6000
K B A
J D
E CG H
I
F
Year 1
Y e a r 2
0
b y = 424 + 0.794 xc The positive slope indicates that districts with
high rates in Year 1 also had high rates in
Year 2.
4 a
85
90
95
36 40 44 48 52 56 60
H e i g
h t ( c
m )
Age (months)
b y = 72+ 0.4 xc The intercept (72 cm) is the predicted height
at age 0. The slope predicts an increase of 0.4 cm in height each month.
d i 89 cm ii 158 cm
e Part i is reasonable as it is a value close to the
data. Part ii is not reliable as the relationship
may no longer be linear here.
5 a
150
150
160
160
170
170
180
180
Mother
D a u g h t e r
b y = 18.3+ 0.91 x c 173 cm6 a
160 180100
2.5
3.0
3.5
4.0
200140120
C o s t ( $ ’ 0 0 0 )
Number of MP3 players
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598 Queensland Mathematics B Year 11
b y = 1300 + 13 x c ≈ $1300d ≈ $13
7 a y = 70− 14 xb The intercept is the predicted time taken to
experience pain relief if no drug is given.
From the slope we predict a reduction of
14 minutes in time taken to experience pain
relief for each mg of drug administered.
c −14 min, which is not a realistic answer 8 a y = 18.2 x + 1000
b Intercept predicts $1000 of sales if nothing is
spent on advertising. The slope means that, on
average, each $1 spent on advertising is
associated with an increase of $18.20 in sales.
c i $19 200 ii $1000
Exercise 5N
1 a y = 68.2+ 0.46 xb The y-intercept is the predicted height at birth.
From the slope, we predict an increase in
height of 0.46 cm each month.
c i 88 cm ii 168 cm
d The height at 42 months is reliable since this
is within the range of data given. The height at
18 years is less reliable since this is outside the
range of data given.
2 y = 487.6+ 0.77 x3 a y = 50.2+ 0.72 x
b An increase of 1 cm in the mother’s height is
associated with an increase of 0.72 cm in thedaughter’s height, on average.
c 172 cm (to the nearest cm)
4 a y = 1330 + 12 xb $1330
c $12
5 a response time = 57.0− 10.2× drug doseb The intercept of 57.0 minutes is the predicted
time for pain relief when no drug is given.
From the slope, we predict a 10.2 minute
decrease in response time for each 1 mg of
drug given.
c −4.2 min, which is not a realistic answer.6 a business = 1123.8+ 18.9× advertising
b Intercept is the volume of business with no
advertising. From the slope we predict an
increase in business of $18.90 for every dollar
spent on advertising.
c i $20 044 (to the nearest dollar)
ii $1124 (to the nearest dollar)
Exercise 5O
1 72.667
2 8.5 years
3 $34 000
4 $12 000
5 $21 000
6 9% p.a.
7 a–d Check with your teacher
8 Check with your teacher
9 102
10 Check with your teacher
11 Check with your teacher
12 a Method 1
Method 2
Method 3
50 60 70 80 90 100
b The distributions of scores are negatively
skewed for methods 1 and 3 and symmetrical
for method 2. The scores for method 1 are
higher than for methods 2 and 3 (90, 79 &70, respectively), and are also less variable
than method 2. They show similar variation
to the scores for method 3.
c Thus training method 1 would be
recommended, as it consistently produces
higher scores.
13 a First-born
Second-born
Third-born
10 20 30 40 50
b The distribution for the first-born is
symmetrical, while for the second and
third-born the distributions are positively
skewed. The centre for the first born is higher
than for the second, which is higher than the
third (35, 21 & 12, respectively), whilst the
variability is most for the first-born, followed
by the second-born and then the third-born.
14 a yes
b 60; this is reliable because it is an
interpolation within the data.
15 4750; this is not reliable as it is an extrapolationfar beyond the data.
Multiple-choice answers1 D 2 B 3 D 4 C 5 D
6 C 7 A 8 D 9 C 10 A
11 D 12 E 13 A 14 B 15 B
16 E 17 C 18 C 19 A 20 D
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Answers 599
Short-response answers1 a continuous b ordinal
c discreate d nominal
2 a numerical b categorical
3 a Class composition by gender
Girls
Boys
b Responses to survey
0
2
4
6
8
10
12
14
16
18
Agree Neutral Disagree
F r e q u e n c y
4
0
F r e q u e n c y
0
2
4
68
10
12
10 20
No. of cigarettes smoked
30 40
5 a 2 2
3 9 4 | 7 represents 47 minutes 4 3 4 5 7 9
5 0 1 1 2 2 4 5 6 6 7 9
6 5 8 9 7 2
b m = 52, Q1 = 47, Q3 = 576 x = $283.57,m = $267.507 a 92.9%
b yes, it is close to 95%
8
0 5 10 15 20 25
9
0 10 20 30 40 50
*
10 a numerical
b 0 0 5 6 6 8 9
1 4 4 5 8 9 3 | 2 represents 32 %2 5 6 7 8
3 2 2
4 4
5 3
c positively skewed
d 21.1%
e x = 20.05,m = 18f
0 10 20 30 40 50 60
0
1
2
3
4
5
6
Divorce rate
F r e q u e n c y
i positively skewed
ii 5
g
10 20 30 40 50 600
0.0
20.0
40.0
60.0
80.0
100.0
Divorce rate
C u m
u l a t i v e f r e q u e n c y %
i 58%
ii ≈ 17%
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600 Queensland Mathematics B Year 11
11 a
600
2
4
6
8
10
12
70 80 90
Travel time
F r e q u e n c y
i 21.4% ii positively skewed
iii 38.1%
b x = 69.60, s = 9.26,min = 57,Q1 = 62,m = 68, Q3 = 76,max = 90
c i 69.60 ii 68
iii 33, 14 iv 76
v 9.26 vi 51.08, 88.12
d
50
Hillside
Met
60 70 80 90 100
e The distributions of travel times are both
positively skewed. The travel times for the
Met (median = 70) tend to be longer thanthe travel times for Hillside trains
(median = 68). The spread of times is alsolonger for the Met (IQR
=24) than the travel
times for Hillside trains (IQR = 14).12 a
30 40
40
50 60
60
70
80
100
120
140
Inside 50
S c o r e ( p o i n t s )
b positive c 0
13 0.927
14 weight ≈ −200 + 2× height15 errors = 14.9− 0.533 × time16 a Intercept: no sensible interpretation. Slope:
For each additional second taken to complete
the task, on average, the number of errors is
reduced by about 1
2.
b 10
17 a IV = Exam scoreDV
= Number of new clients
b
7.00
70.00
5.00
8.00
80.00
6.00
60.00
9.00
10.00
11.00
65.00 75.00 85.00
Exam score
N u m b e r o f n e w c l i e n t s
c positive d 1, strong positive
e 0.748, moderate positive
f number of new clients=−4.00+ 0.173 × examscore
g Intercept: no sensible interpretation. Slope:
On average, each extra 1 mark in the final
exam is associated with an increase of 0.173
clients.
h 13
i Not very reliable as it is outside the range of
the data.
Chapter 6
Exercise 6A
1 a k = 2.6 b k = 92 a k = 1.3 b k = 4.83 $55.44 4 $6.40/kg 5 k
=38.88
6 a w (ox ) = 45
ox b 60 moles c 10.8 moles
7 3.75 barrels
8 1+ 2 ≈ 2.57
9 Check with your teacher
10 k = 1411 Check with your teacher
12 a A = klb, k = 1 b V = kr 2h, k = 3
c A = kd 2, k = 8
13 a F = km1m2d 2
b new force = old force ×4
9
Exercise 6B
1 a y
x0
(1,1)
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b y
x
(1,2)
0
c y
x0
1
21,
d y
x0
(1,–3)
e y
x
2
−12
0
f y
x0
–3
13
g y
x0
–4
12
h y
x
5
0
110
i y
x0
–1 1
j y
x –2
0
– 12
k y
x
–1 0
3
4
43
–
l y
x0 3
–4
– 31
352
2 a y = 0, x = 0 b y = 0, x = 0c y = 0, x = 0 d y = 0, x = 0e y = 2, x = 0 f y = −3, x = 0g y = −4, x = 0 h y = 5, x = 0i y = 0, x = 1 j y = 0, x = −2
k y = 3, x = −1 l y = −4, x = 3
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602 Queensland Mathematics B Year 11
Exercise 6C
1 a
x
f ( x) = 2 x − 3
y
1.5
−3
b
x
y
5
5
f ( x) = 5 − x
c
x
y
3
f ( x) = 3
d
x
y
(3,2)
f ( x) =2 x3
e
−2
x
y
f ( x) = −2
f
x
y4
(4,1)
f ( x) = 4 − 3 x4
163
2 a
x
y
−1 2
−2
b
x
y
−1 3
−3
c
x
y
−2−3
6
d
x
y
2 3
−6
e
x
y
−1 1 22
−6
f
x
y
(1,3)
212
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Answers 603
3 a
x
y
4
(1, −3)
b
x
y
−2 2
−4
c
x
y
1 3
3
d
x
y
−2 4
−8
e
x
y
42
8
f
x
y
−2 −1
4
g
x
y
−1.5 1.5
9
h
x
y
2
−10
56
−
Exercise 6D
1 a x2 + 2 x + 3 x − 1 b 2 x
2 − x − 3+ 6 x + 1
c 3 x2 − 10 x + 22− 43 x + 2
d x2 − x + 4− 8 x + 1
e 2 x2 + 3 x + 10 + 28 x − 3
f 2 x2 − 5 x + 37 −133
x + 4 g x2 + x +2
x + 3
2 a1
2 x2 + 7
4 x − 3
8 + 103
8(2 x + 5)b x2 + 2 x − 3− 2
2 x + 1c
1
3 x2 − 8
9 x − 8
27 + 19
27(3 x − 1)d x2 − x + 4+ 13
x − 2e x2 + 2 x − 15
f
1
2 x
2
+3
4 x −3
8 −5
8(2 x + 1)
Exercise 6E
1 a ( x − 1)( x + 1)(2 x + 1) b ( x + 1)3c ( x − 1)(6 x2 − 7 x + 6)d ( x − 1)( x + 5)( x − 4)e ( x + 1)2(2 x − 1) f ( x + 1)( x − 1)2g ( x − 2)(4 x2 + 8 x + 19)h ( x + 2)(2 x + 1)(2 x − 3)
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604 Queensland Mathematics B Year 11
2 a ( x − 1)( x2 + x + 1)b ( x + 4)( x2 − 4 x + 16)c (3 x − 1)(9 x2 + 3 x + 1)d (4 x − 5)(16 x2 + 20 x + 25)e (1− 5 x)(1+ 5 x + 25 x2)f (3 x
+2)(9 x2
−6 x +
4)
g (4m − 3n)(16m2 + 12mn + 9n2)h (3b + 2a)(9b2 − 6ab + 4a2)
3 a ( x + 2)( x2 − x + 1)b (3 x + 2)( x − 1)( x − 2)c ( x − 3)( x + 1)( x − 2)d (3 x + 1)( x + 3)(2 x − 1)
4 a = 3, b = −3, P ( x) = ( x − 1)( x + 3)( x + 1)
Exercise 6F
1 a
x
y
1 2 30
b
–2 –2
–1 1
y
x
c
1 2 3
−6
0
y
x
d
–3 –2 –1
612
y
x1 2
e
–3 –2 –1 1 2 3
y
x
f y
x –1 0 1
g y
x321
–30
h y
x32
18
1 –1 –2
i y
x210 –2 –1
–3
– √3 √3
j y
x –1
01 2 3
–6
– – 23
k
–1
1
0 1 – – 12
1
3 –
y
x
2 a y
x
(1.02, 6.01)1.5
0.5
(–3.02, –126.01)
0
–5
–15
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Answers 605
b y
x
(1.26, 0.94)1.5
1
(–1.26, –30.94)
0
–2.5
–15
c y
x
(1.35, 0.35)
1.5
(0, –18)
(–1.22, –33.70)
0
–2.5 1.2
d y
x
(–0.91, –6.05)
–2.5 –3 0
(–2.76, 0.34)
e y
x0
(–2, 8)
–3
f y
x0
(–2, 14)
–3.28
6
3 ( x + 1)( x + 1)( x − 3) = 0,∴ graph just touches the x-axis at x = −1 and cuts it at x = 3.
4 a y = −18
( x + 2)3 b y − 2 = −14
( x − 3)3
5 y = 2 x( x − 2)26 y = −2 x( x + 4)27 a y = ( x − 3)3 + 2
b y = 2318
x3 + 6718
x2 c y = 5 x3
Exercise 6G
1 a x = 0 or x = 3b x = 2 or x = −1 or x = 5 or x = −3c x = 0 or x = −2 d x = 0 or x = 6e x = 0 or x = 3 or x = −3f x = 3 or x = −3g x = 0 or x = 4 or x = −4h x = 0 or x = 4 or x = 3i x = 0 or x = 4 or x = 5
j x = 2 or x = −2 or x = 3 or x = −3k x = 4 l x = −4 or x = 2
2 a
5
y
x0
(3.15, –295.24)
b y
x –4 0 5 6
(5.53, –22.62)
480 (0.72, 503.46)
c y
x
(–1.89, –38.27)
0 –3
d y
(3, –27)
4 x
0
e y
(3.54, –156.25)(–3.54, –156.25)
5 –5 x
0
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606 Queensland Mathematics B Year 11
f y
2
–2
0
16
x
g y
–9 9
(–6.36, –1640.25) (6.36, –1640.25)
x
0
h y
4
0 3
(3.57, –3.12)
(1.68, 8.64)
x
i y
5
0 4
(4.55, –5.12)
(2.20, 24.39)
x
j y
–5 –4 0 4 5
400
(–4.53, –20.25) (4.53, –20.25)
x
k y
0 2 x
–20
l y
(1.61, –163.71)
(–5.61, 23.74)
50
–4
–14
–7 x
Exercise 6H
1 a
x
y
−2−3−4 −1 1 2−1
1
2
3
4(2,4)
shift left 2
b
x
y
−2 2
−2
−4
2
4
4−4
shift down 2
c
x
y
−2−4 2
(3,3)
−1
11
2
3
4
shift left 3
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Answers 607
d
x
y
1 2
(1,1)
(1,−3)
3 4 5 6−1
−2
−3
−4
1
2
shift down 4
e
x
y
2
(1,0)(1,0)
(10,−1)(10,−1)
(10,1)(10,1)
4 6 8−1
−2
−3
−4
10
1
(1,−2)(1,−2)
shift down 2
f
x
y
−2−3−4 −1 1
(1,2)(1,2)(−1,1)(−1,1)
2−1
1
22
3
4
5
shift left 12 a
x
y
−2−3 −1 1
(1,1)
(−1,−3)
−1
−2
−3
1
2
shift left 1, shift down 3
b
x
y
1 2 3
(3,4)
(3,1)
(2,3)
(1,1)
4−1
5 6
1
2
3
4
shift right 2, shift up 3
c
(2,1)(0,1)
(0,0)
2 4−4− 6 −2
2
−2
−4
x
y
shift left 2, shift down 1
d
x
y
−2 1 2 4−1
1
2
3
4(1,4)
(−1,−1)
3−1
−3
−2
shift right 1, shift up 4
e
x
y
1 2 4−1
−2
−3
−4
6 8 10
1
(1,0)
(2,−2)
shift right 1, shift down 2
f
x
y
−2−3 −1 1 2 3−1
1
2
3
(3,4)
(1,3) (3,3)
4
5
(2,2)
(2,1)
shift right 2, shift up 1
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608 Queensland Mathematics B Year 11
3
x
y
−1 1 2 3
(2,−3)
(−1,1)
(2,1)
(3,−4)
4 5
−1
−2
−3
−4
1
2
3
shift right 3, shift down 4
4
x
y
−2−4−1
−2
−6
1
2
(−3,1)
(−4,−1) (−1,−1)
3
4
(−3,0)
shift left 3, shift up 1
5
x
y
−2−3−4 −1 1 2
(1,−1)(−1,−1)
(−2,−3)
(−1,−4)
3
−2
−4
−6
2
shift left 2, shift down 3
6
x
y
−2 2 4
−4
6
(3,−1)(0,−1)
(1,1) (3,1) (4,1)
(4,3)
2
4
−2
shift right 3, shift up 2
Exercise 6I
1 a
x
y
−2 −1 1
(1,−1)
(1,1)
2−1
−2
−3
−4
12
3
4
reflect about the x-axis
b
x
y
2
(2,2)
(1,1)
4
2
4
−2−4
−2
−4
dilate by 4 from the x-axisc
x
y
−2−3 112
2
1
2
3
4
−1−13
4,( ) 1
214
,( )
dilate by 3 from the x-axis
d
x
y
1 2
34 , 2
(4 ,2)
3 4−1
5 6
1
2
3
4
( )
dilate by 13 from the y-axis
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Answers 609
e
x
y
2 4
(4,2)
(4,−2)
−1
−2
−3
−4
6 8
1
2
3
4
reflect about the x-axis
f
x
y
−2−3−4 −1 1 2−1
12
2(−1, 3)
3
4
5
2(−1, 1)
dilate by 3 from the x-axis
2 a
x
y
−2 −1 1
(1,−3)
(1,1)
(1,3)
2−1
−2
−3
−4
1
2
3
4
dilate by 3 from the x-axis, reflect about the
x-axis
b
x
y
1 2 3
(4, 2)
4
−2
−4
5 6
2
4
3( 4 ), −2
3( 4 ), −2
dilate by 13 from the y-axis, reflect about the
x-axis
c
x
y
2 4
−2
2
4
(2,2)
(2,1)
(2,−1)
−2−4−4
−4
dilate by 12 from the y-axis, reflect about the
x-axis
d
x
y
−2−3 2 3
(2,−4)
(2,−2)
−1
−2
−3
−4
−5
−6
1
−1 1
dilate by 12 from the x-axis
e
x
y
21
12
4−1
−2
−3
−4
6 8 10
1
2
3
4
dilate by 3 from the x-axis, dilate by 12 from the
y-axis
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610 Queensland Mathematics B Year 11
f
x
y
−2−3 −1 1 2 3
(1,2)
−1
−2
−3
−4
−5
−6
2
4
6
3( 1 ), 2
3( 1 ),−2
dilate by 13 from the y-axis, reflect about the
x-axis3
x
y
−2−3 1 2 3−1
−3−4
1
2
3
4
−1
−2
dilate by 3 from the x-axis, reflect about the
x-axis
4
x
y
−2−3−4−5 −1 1 2−1
−2
−3
−4
1
2
3
4
2( ),−1 1 81 1
2( ),−1 1 4−212( ),−1 1 8−11
dilate by 12 from the x-axis, reflect about the
x-axis
5
−5
x
y
−2 −1 1 2
(1, 2)
(2, −2)
3 4−1
−2
−3
−4
1
2
3
4
(1, −2)
dilate by 12 from the y-axis, reflect about the
x-axis
6
x
y
−1 1 2 3
(4,3)
(4,2)
4−1
−1
−2
−3
−4
5 6
1
2
3
4
12
dilate by 11
2 from the x-axis
Exercise 6J
1 a
x
y
−2−3 −1 1
(1,1)
(−1,3)
2−1
1
2
3
4
5
6
shift left 1, dilate by 2 from the x-axis, shift
up 3
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b
x
y
1 2 3 4
−2
−4
−6
−8
5
2
4
(4,−4)
(4,2)
(4,4)
(4,−8)
dilate by 2 from the x-axis, reflect about the x-axis, shift down 4
c
3 5
2
x
y
21 4
−2
4
(2, −1)
−2
−4
shift right 2, reflect about the x-axis,
shift up 3
d
x
y
−2 −1 1
(−1,1)
2 3−1
−2
−3
−4
1
2
3
4
shift right 1, dilate by 3 from the x-axis, reflect
about the x-axis
e
x
y
2 4−1
−2
−3
−4
−5
6 8 10
1(5,1)
(10,1)
(5,−2)
dilate by 12 from the y-axis, shift down 3
f
x
y
−2−3 −1 1−1
−2
−3
−4
2
4
2( 1 ), −1
(1,3)2( 1 ),3
dilate by 12 from the y-axis, shift down 4
2
x
y
−2−3 −1 1
(1,1)
2−1
−2
(−2,−4)
(−1,−1)
−3
−4
2
4
shift left 2, dilate by 3 from the x-axis, shift
down 4
3
x
y
2 4
(2,3)
−2
−4
6 8 10
2
4
shift right 1, dilate by 2 from the x-axis, reflect
about the x-axis, shift up 3
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612 Queensland Mathematics B Year 11
4
x
y
−2 −1 1 2 3
12
(1,1)
4−1
−2
−3
−1
−4
−5
1
2
( ),
dilate by 12 from the y-axis, dilate by 3 from the
x-axis, shift down 4
5 a
x
y
−2−3 −1 1
(−2,4)(−1,4)
2
1
2
3
4
5
(1,3)
6
dilate by 12 from the y-axis, shift right 1, shift
up 3
b
x
y
2 4
(2,1)
6 8 10−1
−2
−3
−4
−2
1
2
(5,1)(10,1)
dilate by 12 from the y-axis, shift left 3, shift
down 1
c
x
y
−2−3−4 −1 1 2
6
8
2
(1,3)3
4 12
,3( )
dilate by 12 from the y-axis, shift left 1
2, dilate
by 2 from the x-axis
6 a Dilate from y-axis by a factor of 13; translate
right 2 units; dilate from x-axis by a factor of
4; translate upwards 1 unit
b Dilate from y-axis by a factor of 12; translate
left 5 units; dilate from x-axis by a factor of 8;
reflect about the x-axis; translate downwards
7 units
c Dilate from y-axis by a factor of 16; translate
right 1 unit; dilate from x-axis by a factor of 2;
translate downwards 5 units
Exercise 6K
1 a 7 b 6 c 5
d 6 e 11 f −42 a
x −2 −1 0 1 2 y 6 3 0 3 6
6
5
4
3
2
1
−4 −3 −2 −1
−1
1 2 3 4
y = 3| x|
y
x
b x −2 −1 0 1 2 y 1 2 3 2 1
6
5
4
3
2
1
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