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A n s w e r s
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swers
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Maths Quest 10/ Final Pages / 19/1/06
CHAPTER 1 Rational and irrational numbers
Are you ready?
1 a b c d
1
2 a
6
b
24
c
54
d
60
3 a b c d
4 a
2
b
2
c
2
d
1
5 a
0.375
b
0.1875
c
0.32
d
0.225
6 a b c d
7 a b c d
8 a
11
b
3
c
2
d
100
9 a
1.7
b
3.9
c
9.9
d
10.1
Exercise 1A — Operations with fractions
1 a b c d
e f g h
i j k l
2 a b c
1
d
1
e f g
2
h
2
i j k l
1
3 a b c d
e f g h
i
1
j
9
k
3
l
4
4 a b c d
2
e
1
f
1
g
1
h
1
i j
1
k
3
l
1
5 a
E
b
D
c
C
d
E
6
65 students
7
$9.80
8 i ii
Milly ate , was left.
iii
Dad ate , was left.
Maths Quest challenge (page 7)
1
−
50
2
(Other answers possible)
3
4
History of mathematics — Srinivasa Ramanujan (1887–1920)
1
More than 100 theorems
2
Number theory, elliptic functions, continued fractions, prime numbers
3
32
Exercise 1B — Finite and recurring decimals
1 a
0.75
b
0.4
c
0.9
d
0.625
e
0.66
f
0.275
g
0.9125
h
0.3125
i
0.52
j
0.45
k
0.57
l
0.08
2 a b c
d e f
g h
3 a b c
d e f
g h i
j k l4 a
B
b
D
c
C
d
E
e
D
5 a b c d e
f g h i j
k l
6 a b c d e
f g h i j
k l
7 a
C
b
A
c
C
d
D
10 Quick Questions 1
Exercise 1C — Irrational numbers
1 a
Surd
b
Not a surd
c
Not a surd
d
Surd
e
Surd
f
Surd
g
Not a surd
h
Not a surd
i
Surd
j
Not a surd
k
Surd
l
Not a surd
2 a
C
b
D
c
A
d
E
e
D
3 a
Rational
b
Irrational
c
Rational
d
Rational
e
Rational
f
Rational
g
Rational
h
Rational
i
Rational
j
Irrational
k
Rational
l
Irrational
4
Answers will vary.
5 a
8.185
b
9.055
c
12.124
d
2.285
e
2.627
f
0.868
g
50.339
h
44.294
i
24.653
j
94.526
k
3.473
l
18.846
6
Answers will vary.
7 a
2.844
b
−
9.637
c
4.019
d 5.983e 5.059 f 1.052 g 1.424 h −3.363i −2.431 j 0.9422
8 a 23 b 15 c 15 d −13 e 109 a A b C c D d E e B
10 a 56.37 b −9.48 c −1.05 d 0.51e 28.08 f 2.02
Answers
3
4---
1
4---
3
4---
1
5---
7
3---
15
4------
51
10------
39
8------
2
9---
1
3---
2
5---
6
11------
4.3̇ 5.428 571 13.83 19.6872
3
5---
3
4---
1
8---
1
40------
2
3---
2
5---
4
5---
16
25------
5
9---
8
15------
1
6---
1
4---
5
9---
8
9---
5
12------
3
5---
7
12------
5
6---
1
4---
1
10------
5
18------
1
4---
1
20------
7
24------
31
36------
3
4---
26
35------
13
20------
1
2---
16
63------
1
2---
9
55------
5
16------
7
24------
28
45------
3
5---
23
32------
7
12------
1
2---
5
6---
6
7---
5
6---
3
4---
3
4---
1
2---
7
8---
6
7---
5
8---
43
72------
3
4---
2
3---
3
4---
1
12------
2
3---
1
3---
1
3---
131
286---------
1
2---
2
3---
0.3̇ 0.16̇ 0.32
0.785̇ 0.594 0.125 125 1
0.375 46 0.814 35
0.6̇ 0.27 0.8̇
0.27̇ 0.83̇ 0.142 857
0.916̇ 0.06̇ 0.90
0.2916̇ 0.56̇ 0.259
4
5---
3
10------
7
50------
67
100---------
19
20------
3
4---
3
25------
7
8---
27
40------
357
1000------------
221
250---------
29
80------
5
9---
2
3---
28
33------
71
99------
7
15------
2
11------
17
90------
5
18------
109
300---------
379
990---------
616
999---------
725
999---------
1 2 4 3 48
11------
17
56------
1
15------
2
3---
5 1 6 0.325 7 85
6--- 0.16̇ 5
8---
9 107
9---
127
495---------
1A
1C
5_61_03274_MQV10 - A 1-15_tb Page 705 Thursday, January 19, 2006 7:42 PM
706 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
11 a Rali’s answer is approximate, Tig’s answer is exact, so the teacher is right.
b c
d Rali: 39600, Tig: 40 000e 400
f The difference between and 0.33 is only ,
but has much more significance if taken over a larger amount.
12 a cm or cm b 14.1 cm
13 or approximately 4.9 m
Maths Quest challenge (page 20)1 8, 18, 50 (other answers possible)2 a 177 768 889
b 399 960 001c 16
Exercise 1D — Simplifying surds1 a b c d 7 e
f g h i j
k l m n o
p q r s t
2 a b c 48 d e
f g h i j
3 a b c d e
f g h i j
4 a D b A c C d C e Df E g D h B
5 a b c d e
f g h
6 a m b m c ≈ 6.804 m
7 a cm b 224 cm
Exercise 1E — Addition and subtraction of surds1 a b c
d 0 e f
g h i
j k l
m n o
p q r 2
2 a b 0
c d
e f
g h
i j
k l
m n
3 a D b A c C d A
4 ( ) cm
5 a 67 m b m
c m d 10.39 m
Exercise 1F — Multiplication and division of surds1 a b 5 c −5 d
e f 8 g −10 h
i j k l 24
m −150 n o −168 p
q 1250 r s t
u
2 a E b C c A
3 a b c d
e f g h 4
i j k l
m n 15 o −8 p
q 1 r 1 s t
4 a B b C c C d D
5 a 1 b 1 c 1 d 1
6 a b c
d e f
g h i
j k l
m 12 n o
p −15 q r
7 a
b
c −1 d 2 e
f g h
i
8 a 42 b cm c cm
d cm2 e 42
9 a m b 18 m2
10 Quick Questions 2
33
100---------
1
300---------
1
3--- 0.003̇
200 10 2
2 6
2 5 2 2 3 2 30
5 2 2 7 6 3 12 2 4 3
10 5 9 2 2 13 55 2 21
7 2 11 3 7 7 78 4 10
4 2 15 3 35 2 20 6
10 3 4 42 72 2 27 5 132 2
12 175 108 80 384
90 32 720 600 338
15 3 19 5 16 7 17 2 a c
d2 b hk j f f
3 2 3 3 12 3 3–
60 14
2 2 4 5– 6 3–
5 11 2 7
10 2 7 3+ 8 5 6+ 8 10 7 3+
16 2 11 5– 2 6 5 5 15+
8 7 11– 11 2– 7 13–
3 6 4 3– 5 2 6 7–
2
5 3– 4 7–
5 6 6 5+ 2 3 3 5–
4 6 6 5 14 2–+ 29 5 22 3+
9 11 30– 28 2 39 5–
69 3 17 2– 18 10 42 2+
69 51 2– 41 5 6 3 6 2–+
40 5 2 2 13+ +
31 4 41+( )
36 4 41–( )
5 5 35
66– 2 15
2 14 6 2 15 2
24 5 6 2
108 2 1600 2 28 7
12 15
3 2 5 2
15 3 5
2 5– 9 3 2 2–2
2-------
3
4-------
1
6-------
4
45------
13
32------
3
4---
3 2
5----------
2
3---
1
6---
1
2---
3
4---
3 2 3 5+ 5 6 5 2– 6 5 6 11+
8 2 24+ 4 7 20– 10 2 2–
42 7 7+ 6 15+ 2 5 2 10+
42 8 14– 5 2 5+ 6 5 6–
60 18 5– 12 14 8 35+
50 15 2– 8 6 60–
2 10 15– 2 6 3–+
3 35 14– 3 10 2–+
7 10+
27 2 6+ 8 2 15– 5 2 6+
42 24 3–
4 2 168 2
504 2
3 2 2
1 2.25 2 3 40.45 7
11------ 99
5 4.05 6 2 7 83 10 16 2
9 1040 23
3-------
5_61_03274_MQV10 - A 1-15_tb Page 706 Thursday, January 19, 2006 7:42 PM
A n s w e r s 707
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swers
➔
Maths Quest 10/ Final Pages / 19/1/06
Exercise 1G — Writing surd fractions with a rational denominator
1 a b c d
e f g h
2 a b c d
e f g h
i j k l
3 a b c d
4 a b c d
5 a b
6 a b
c d
e f or 5 −
g h or 27 +
Summary
Chapter review1 a b − c d
2 126
3 a 0.08 b 0.8125 c d
4 a C b E
5 a b c d e
6 = 4
7 a 7.9 b 13.7 c 0.5 d 25.7
8 m or ≈ 8.06 m
9 a b c d
10 A
11 a b c d
12 a b c13 D
14 a cm b cm
15 a b c 13
d e 3 f
16 a 240 b
17 C18 B
19 a b c d
Chapter 2 Algebra and equationsAre you ready?1 a abc and 3acb b x2y and
c −2q2p and 2pq2
2 a −5 b 8 c 23 a 15 b −12 c 44 a −1 b 13 c 75 a −3x + 2 b −5a − 9 c −2p − 2q + 86 a 6 b 3ab c −4pq
7 a 1 b c
8 a b c 2
9 a b 1 c
Exercise 2A — Operations with pronumerals1 a 5k + 11c b 15m + 16f c 9d + 5c
d 4f + 8h e 12g + 13j f 15d + 13g 12n + 11 h 7h2 + 14y i 11nv + 10u
2 a 3m + 2c b 8a + 3f c 2k + 2d −4t + 2 e −r + 5 f −2v − 5g 4p − 14 h 4w + 4 i 7c − 17j −j + 3c k −7k + 4m l −4d + 3cm 7y2 − 4y n 2x3 − 6x4 o −c2 + 12
3 a C b B c A4 a x2 + 7x + 6 b d2 + 4d − 10
c v2 − 10v − 6 d a2 − 2ab + b2
e u2 + u − 12 f 5n4 − 6n2 − 255 a 40fh b 60abc c 28gm
d −54hnp e 84abst f −72ahmst6 a 3k b 2mn c −6g
d e − f
g h i
Exercise 2B — Substituting into expressions1 a 5 b 2 c 0 d 6 e −17
f 3 g 30 h 12 i −12 j 27k 30 l −5
2 a −11 b −1 c 1 d 30 e −24f 36 g −125 h 1 i 15
3 a b − c d 1 e
f 484 a 17 b 30 c 8 d 4 e 1.5
f 68 g 46 h 113.1 i 40 j 14.15 a D b C c B6 3.9 cm
3
3-------
5
5-------
6
6-------
7
7-------
10
5---------- 5
15
5----------
30
5----------
15
5----------
30
6----------
6
3-------
15
5----------
2 6
3----------
2 21
7-------------
3 10
5-------------
6
2-------
30 630
2----------
70
2----------
2 15
7-------------
2 42
3-------------
2 6
5----------
5
4-------
2
2-------
2 3
3----------
2
2-------
15
2----------
3 2
2---------- 2
5 2 3+( ) 2 1 2–( )–
4 5 2–( ) 3 3 7+( )
3 5 2+( ) 5 5 3–( ) 15
2 7 2+( ) 3 6 3 6 5 2+( )2
------------------------------------------- 15 3
1 decimal 2 repeater3 rational 4 recurring5 multiple 6 fractions7 surds 8 approximation9 perfect 10 factor
11 like 12 multiplied
7
12------
1
12------
1
12------
3
4---
0.285 714 0.5̇
4
5---
8
9---
83
100---------
5
6---
83
99------
16
65
3 11 5 7 24 2 12 10
150 180 605 18
4 6 7– 5 3 18 5 6 7–
12 4 2+( ) 56 2 5+( )
5 2 24 21
2 6– 1
2---
30
6----------
73 40 3–
7
14-------
5 6
6---------- 5 2– 30 3 3+
1
4--- yx2
5
12------
11
24------
5
12------
1
4---
1
9---
2
3---
5
6---
1
9---
2
3---
4a3c------
3g4h------
10a3b---------
x5---
8k2
3m--------–
3ac2bd2------------
7
12------
1
12------
1
12------
1
3---
1
576---------
1D
2B
5_61_03274_MQV10 - A 1-15_tb Page 707 Thursday, January 19, 2006 7:42 PM
708 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
7 65.45 cm3
8 361 m
9 a −1 In this case, addition is closed on integers.
b −1 In this case, subtraction is closed on integers.
c 2 In this case, multiplication is closed on integers.
d −1 In this case, division is closed on integers.
e −2 In this case, subtraction is closed on integers.
f − In this case, division is not closed on integers.
10 a 10 In this case, addition is closed on natural numbers.
b −4 In this case, subtraction is not closed on natural numbers.
c 12 In this case, multiplication is closed on natural numbers.
d In this case, division is not closed on natural numbers.
e −2 In this case, subtraction is not closed on natural numbers.
f 4 In this case, division is closed on natural numbers.
11 a (a + 2b) + 4c = a + (2b + 4c)
b (x × 3y) × 5c = x × (3y × 5c)
c 2p ÷ q ≠ q ÷ 2pd 5d + q = q + 5de 3z + 0 = 0 + 3z = 3z
f
g (4x ÷ 3y) ÷ 5z ≠ 4x ÷ (3y ÷ 5z)
h 3d − 4y ≠ 4y − 3d
Exercise 2C — Expanding 1 a 5k + 5 b 7m + 28 c 4y + 28 d 8d − 72
e 12h − 60 f 2k − 12 g 20m − 8 h 30t + 25
i 16k − 88 j 5m + 5n k 32y − 24f l 18v + 42wm bc − bd n ki + kef o 12pj − 18mp
2 a −3c − 3 b −5d − 10 c −6m − 66
d −8c − 8d e −12k + 8m f −14 + 21xg −50 + 10y h −k2 − 2k i −x2 + 3x
3 a 5c + 23 b 17k + 42 c 13m + 41
d −j − 16 e 2t − 11 f 13m + 39
g 16c − 14 h d − 8 i 4w − 112
j −18h + 22 k 10y − 19 l 9x + 27
m 7h − 5 n 6c + 38 o −5m + 22
4 a y2 − 4y − 12 b w2 − 4w − 12
c 2x2 − 13x + 20 d 3h2 + 8h + 5
e 3f 2 − 22f − 16 f 8a2 + 6a − 9
g −4x3 + 2x2 + 4x h 9x3 − 15x2 + 7xi 32p4 − 8p3 + 30p2
5 a D b B c E
6 a x2 + 5x + 6 b g2 − 6g + 8
c 6a2 + 7a − 20 d 12m2 − 25m + 12
e y2 + 10y + 25 f 4d2 − 12d + 9
7 a 3x2 units2 b (x2 + 5x) m2
10 Quick Questions 11 8r − 3t
2
3 −6g2hkm4 −6
5 a −1.3 b
6 a 4 In this case, multiplication is not closed on irrational numbers.
b In this case, multiplication is closed on irrational numbers.
7 6w − 12v8 −3pq + 15q2
9 20 + 31u10 −15r3 − 63r2 + 84r
Exercise 2D — Factorising using common factors 1 a 4(x + 3) b 6(y + 4) c 7(m + 7)
d 10(y + 11) e 2( f + 14h) f 3(a − 3)
g 5(b − 9) h 6(d − 1) i 8(e − 3)
j 6(l − 12) k 12(n − 3p) l 7( f − 14d)
2 a 2(3t + 5) b 3(3m + 2) c 4(3k + 7)
d 15(2m + 1) e 2(7m + 6n) f 5(2j − 5)
g 3(2c − 9) h 5(20h − 3) i 2(10m − 1)
j c(5 + d) k 6a(k − 5m) l bc(4a + d)
3 a −3(c − 5) b −7(m − 5) c −8(k − 3j)d −5( j + 4) e −4(h + 7j) f −6(p + 2s)
g −3(3k − 5) h −4a(4c − 3d) i −4b(3m + 5ac)
4 a m(m + 5) b d(d2 − 6) c 4x(x3 + 4)
d f(8 + f ) e y(y − 1) f 7p3(1 + 3p2)
g 2q2(2 − 5q6) h 5r4(3r − 1) i 3ab(4a + 5b)
j mn4(20m2 − n) k 2k2p(3kp + 4) l 11xy(x2 − y)
5 a A b B c E d D
Maths Quest challenge (page 56)1 24 units
2 4, 9, 25, 49 (Square numbers have an odd number of factors. Squares of prime numbers have 3 factors only.)
3 16, 81
4 64
5 60, 72 and 96 each have 12 factors.
Exercise 2E — Adding and subtracting algebraic fractions
1 a (1 ) b c 1
d e f
g h i
2 a b c d
e f g h
i j k l
1
2---
4
3---
2x 1
2x------× 1
2x------ 2x× 1= =
8
5e------
4
5---
6 5
15
A
B
26
21------ 5
21------
49
72------
17
99------
1
35------
6 5x–
30---------------
15x 4–
27------------------
15 16x–
40---------------------
15 2x–
3x------------------
5y12------
3y40------–
13x12
---------14x
9---------
3w28-------
y5---–
89y35
---------32x15
---------
7x 17+10
------------------7x 30+
12------------------
2x 11–
30------------------
19x 7+6
------------------
5_61_03274_MQV10 - A 1-15_tb Page 708 Thursday, January 19, 2006 7:42 PM
A n s w e r s 709
an
swers
➔
Maths Quest 10/ Final Pages / 19/1/06
3 a b c d
e f g h
i
4 a b
c d
e f
g h
i j
k l
Exercise 2F — Multiplying and dividing algebraic fractions
1 a b c d
e f g h
i j k l
2 a b c d
e f g
h i j
3 a b c d 3
e f or 5 g h
i j k l y2
4 a b
c d
e f
g h
Exercise 2G — Solving basic equations1 a a = 24 b k = 121 c g = 2.9 d r = 3
e h = 0.26 f i = −2 g t = 5 h q =
i x = 02 a f = 12 b i = −60 c z = −7 d v = 7
e w = −5 f k = 10 g a = 0.425 h m = 16
i y = 21
3 a t = 100 b y = ±17 c q = 6.25 d f = ±1.2
e h = f p = ± g g = h j = ±
i a = ±1
4 a a = 4 b b = 6 c i = 3 d f = 9
e q = 1 f r = 5 g s = 4 h t = 9
i a = −7
5 a f = 40 b g = 30 c r = −10 d m = 18e n = 28 f p = 62.4
6 a x = 1 b y = 9 c m = 4 d k = 1
e n = 5 f c = 1
7 a k = 25 b m = 16 c p = −11 d u = −4
e x = f v = 3
8 a B b E c C9 a x = −5 b d = −1 c p = 7 d x = −11
e h = −2 f t = 5 g v = −20 h r = −3i g = −0.8
10 a x = −1 b v = 1 c l = 2 d g = −2
e t = 3 f e = −23 g j = −3 h k = −36
i f = −12
11 a x = 2 b b = 5 c w = 2 d f = 7
e t = 3 f r = 2 g g = −1 h h = −2
i a = 0
12 a x = −1 b c = 2 c r = 2 d k = 1
e y = −1 f g = 7 g w = 1 h m =
i p = 1
13 a x = −15 b y = −4 c t = 21 d u = −2
e f = 12 f r = 7 g d = −6 h h = −12
i x = 114 a A b D c B
10 Quick Questions 2
Maths Quest challenge (page 69)1 51 2 9 3 17
Exercise 2H — Solving more complex equations1 a x = b x = 3 c x = d x = −7
e x = −2 f x = g x = −5 h x = −2
i x = 5 j x = 2 k x = −2 l x = −6
2 a x = 4 b x = 18 c x = d x =
e x = or x = −3 f x = g x = 3
h x = i x =
5
8x------
5
12x---------
38
21x---------
8
3x------
7
24x---------
9
20x---------
37
100x------------
51
10x---------
1
6x------–
3x2 14x 4–+x 4+( ) x 2–( )
----------------------------------2x2 3x 25+ +x 5+( ) x 1–( )
----------------------------------
2x2 6x 10–+2x 1+( ) x 2–( )
-------------------------------------4x2 17x– 3–
x 1+( ) 2x 7–( )-------------------------------------
7x2 x+x 7+( ) x 5–( )
----------------------------------2x2 6x 7+ +x 1+( ) x 4+( )
----------------------------------
–x2 7x 15+ +x 1+( ) x 2+( )
----------------------------------x 7–
x 3+( ) x 2–( )----------------------------------
x2 3x 9+ +x 2+( ) 3x 1–( )
-------------------------------------5 5x–
x 1–( ) 1 x–( )---------------------------------
5
x 1–-----------=
3x 7+x 1+( )2
-------------------3x 4–
x 1–( )2-------------------
4xy
------3xy
------4yx
------9x4y------
5– x4y
---------3w2x-------
6z7x------
2z7x------
3– x2y
---------5
24------
12zx
--------x–
6w-------
2
3x 2–---------------
5
x 3–-----------
9
2 x 6–( )--------------------
1
x 3+------------
2xx 1+( )2
-------------------x 1+
2 2x 3–( )-----------------------
a10 a 3+( )-----------------------
35d8 d 3–( )--------------------
9
32x2 x 2–( )----------------------------
3x10 x 1–( )-----------------------
3
5---
2
9---
1
3---
1
25------
35
6------
5
6---
4y2
7--------
2y2
25--------
8y2
9--------
32xy5
------------ 2
3---
9
3x 7–( ) x 3+( )-------------------------------------
1
x 2+( ) x 9–( )----------------------------------
4xx 1–( ) 2x 1+( )
-------------------------------------4 x 1–( )
x 1+( ) x 5–( )----------------------------------
1
2 x 1+( )--------------------
28
2x 3–( ) x 7–( )-------------------------------------
21 x 3–( )x 5+
-----------------------13
9 x 4–( ) x 1+( )-------------------------------------
1
6---
1
3---
5
8---
1
2---
16
49------
3
8---
225
484---------
14
31------
2
3---
1
8---
2
5---
5
6---
4
5---
1
2---
1
3---
2
5---
1
2---
2
3---
1
3---
3
7---
1
8---
8
11------
1
3---
3
8---
1
4---
1
3---
1
3---
1
5---
2
3---
1
8---
1
5---
2
3---
4
5---
5
7---
1
2---
1
2---
1 4(4 − 9y) 2 −4(8 − 7g)
3 48
15x---------
3x x 1+( )4
-----------------------
5 r = 12 6 t = −42
7 v = −3 8 y = 7
9 z = 2 10 b = − 1
2---
20
31------
5
8---
29
36------
8
11------
10
43------
3
4---
11
12------
2–
17------
3
2---
11–
3---------
2
3---
2
13------
5
7---
13
20------ 2C
2H
5_61_03274_MQV10 - A 1-15_tb Page 709 Thursday, January 19, 2006 7:42 PM
710 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
3 a x = b x = 15 c x = −6 d x = −e x = −1 f x = −192 g x = h x = 12
i x = 3 j x = 3 k x = 52 l x = 1
4 a x = b x = 1 c x = 4 d x = −3
e x = 5 f x = −1 g x = 1 h x = −4
i x = 1.5 j x = −4 k x = 3 l x = 1
Exercise 2I — Solving inequations1 a x > 1 b v < 10 c t ≤ −6 d p ≥ 1
e f ≥ −6 f h > 45 g k < 7.5 h j ≤ −0.16
i y > 1
2 a x < 3 b r ≥ 8 c y > 3 d e ≤ −3
e d ≥ −1 f k > −2
3 a h < 1 b d ≥ –1 c y ≤ 7 d p > 5
e u ≤ 5 f c > 6 g y ≥ −18 h k < 180
i x ≤ 20
4 a x > −4 b h < −7 c u ≥ −1 d x ≤ −2
e k > −4 f j ≤ −7 g f < −28 h w ≥ −4
i a < 16 j a > −9 k x ≤ −1 l a >
5 a i < −5
b u < 1
c g < 9
d c > −16
e y ≥ 4
f m > 15
g j ≥ 15
h x ≤ 1
i r < 3
j w > −
k x < 2
l y ≤ − 4
6 a C b A c A
Summary1 pronumeral 2 multiplying
3 Commutative, Associative, Identity, Inverse, Closure
4 substitute 5 brackets
6 Expanding 7 highest, Divide
8 lowest, common, single 9 numerators
10 reciprocal 11 numerical
12 isolate 13 inequality
14 negative
Chapter review1 a 7c − 13 b −7k + 3m c −5d − 5c d 7y2 − 5y
2 a −21mp b c d
3 B
4 35
5 D
6 a (a + 3b) + 6c = a + (3b + 6c)
b 12a − 3b ≠ 3b − 12a
c
d (x × 5y) × 7z = x × (5y × 7z)
e 12p + 0 = 0 + 12p = 12pf (3p ÷ 5q) ÷ 7r ≠ 3p ÷ (5q ÷ 7r)
g 9d + 11e = 11e + 9dh 4a ÷ b ≠ b ÷ 4a
7 a 96 In this case, multiplication is closed on natural numbers.
b In this case, division is not closed on natural numbers.
c −4 In this case, subtraction is not closed on natural numbers.
8 a 6x − 18 b −8 + 4xc −10p2 + 21p + 12 d 11x3 + 16x2 − 4xe 2k2 − 10k − 48 f 25d2 − 60d + 36
9 C
10 B
11 a 4a(p − 3g) b −4(h + 18)
c 3p4(4p2 + 5) d 6p2q(1 − 4pq2)
12 B
13 a b c
d
14 a b c
d e
f
15 a p = 88 b s = 3.01 c b = 16
d r = −35 e x = 144 f x = −
g y = 60 h a = ±6 i k = 12
16 a b = 4 b t = 2 c p = −2
17 a x = b x = 6 c x = −
d x = 1 e x = 12 f x = 1
5
17------
2
9---
10
19------
1
2---
4
7---
1
4---
5
8---
5
19------
31
58------
11
14------
15
17------
20
43------
10
13------
2
61------
9
26------
1
3---
3
4---
2
5---
8
9---
5
6---
5
8---
4
5---
7
9---
1
2---
25
9------
–6–7–8–9 –5 –4 –3 –2 –1 i
–3 –2 –1 0 1 2 3 4 5 u
5 6 7 8 9 10 11 12 13g
–20 –19 –18 –17 –16 –15 –14 –13 –12 c
1
2---
1 2 3 4 5 6 7 8 9 y
11 12 13 14 15 16 17 18 19 m
11 12 13 14 15 16 17 18 19 j
–3 –2 –1 0 1 2 3 4 5 x
–1 0 1 2 3 4 5 6 7 r
1
2---
–4 –3 –2 –1 0 1 2 3 4 w
–2 –1 0 1 2 3 4 5 6 x
1
4---
–8 –7 –6 –5 –4 –3 –2 –1 0 y
a20------
4cd25 f---------
8a3
------
7 p 1
7 p------× 1
7 p------ 7 p× 1= =
1
3---
7y6
------7x 18+
10------------------
22
15x---------
3x2 6x 9–+x 3+( ) x 2+( )
----------------------------------
8yx
------25z4x--------
5
x 3+------------
5
6---
y2
50------
2xx 1–( ) 9x 1+( )
-------------------------------------
13
2------
1
2---
1
5---
3
14------
2
9---
1
6---
5_61_03274_MQV10 - A 1-15_tb Page 710 Thursday, January 19, 2006 7:42 PM
A n s w e r s 711
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swers
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18 a x = b x = 22 c x = 2
d x = 5 e x = 3 f x = −
19 a x > 4 b x > −7 c e ≤ −12
d y ≤ −30 e u ≥ −1 f x ≥ −14
20
Chapter 3 Linear graphsAre you ready?1 a −6 b 4 c 3
2 a i y = 2 ii x = 3
b i y = −3 ii x = 9
c i ii x = 2
3 a y = −2x + 4 b y = 4x − 5 c
4 a 1 b 2 c −1
5 a b c
6 a
b
c
Exercise 3A — Plotting linear graphs1 a
b
c
d
2 a
x −10 −8 −6 −4 −2 0 2 4 6 8 10
y −28 −22 −16 −10 −4 2 8 14 20 26 32
6
7---
1
2---
3
8---
16
21------
–6 –5 –4 –3 –2 –1 0 1 2 n
y 3
2---–=
y 2
3---x–
5
3---–=
5
2---
7
3---
1
4---–
–5
4
0 x
y
5y – 4x = 20
0 x
y4y – 2x = 5
–21–2
11–4
3y + 4x = –12
y
–3
–4
x0
10
20
30
–10
–20
–30
5 10–10 0–5
y
x
y = 3x + 2
x −6 −4 −2 0 2 4 6
y 20 14 8 2 −4 −10 −16
x −3 −2 −1 0 1 2 3
y 6 5 4 3 2 1 0
x −6 −4 −2 0 2 4 6
y 15 11 7 3 −1 −5 −9
x −5 −4 −3 −2 −1 0 1
y −25 −15 −5 5 15 25 35
0 5 10
5
–5
–10
–15
–20
–5–10 x
y
y = –3x + 2
10
15
20
02 x
y
y = –x + 3
31–1–2–3
1
2
3
4
5
6
0 5 10
5
–5
–10
–5–10 x
y
y = –2x + 3
20
10
15
1 x
y = 10x + 25
2–1–2–3
–4–5
35
30
25
15
10
5
–5
–10
–15
–20
–25
y
20
2I
3A
5_61_03274_MQV10 - A 1-15_tb Page 711 Thursday, January 19, 2006 7:42 PM
712 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
b
c
d
e
f
Maths Quest challenge (page 85)1 From top to bottom, the three numbers are −31, 25, 102 a 67 + 31 + 4 + 5 = 107, 46 + 24 + 19 + 17 = 106
b Total of the eight numbers is an odd number.
Exercise 3B — Sketching linear graphs1 a
b
c
d
e
x −1 0 1 2 3 4
y −17 −12 −7 −2 3 8
x −6 −4 −2 0 2 4
y 13 12 11 10 9 8
x 0 1 2 3 4 5
y −240 −140 −40 60 160 260
x −3 −2 −1 0 1 2
y 18 13 8 3 −2 −7
1 x
y = 5x – 12
2 3 4 5–1–2
10
5
–5
–10
–15
–20
y
x
y = –0.5x + 10
2 4 6–2 0–4–6
2
4
6
8
10
12
14y
x
y = 100x – 240
1 2 3 4 50
–250
–200
–150
–100
–50
50
100
150
200
250
300y
x
y = –5x + 3
1–1–2–3 20
–10
–5
5
10
15
20y
x −3 −2 −1 0 1 2
y 19 15 11 7 3 −1
x
y = 7 – 4x
1–1–2–3 2 30–5
5
10
15
20y
x
y = 4x + 1
(1, 5)
1
1
5
y
0
x
y = 3x – 7
(1, –4)
1
4
y
0
7
x
y = –2x + 3
(1, 1)
1
1
y
0
3
x
y = –5x – 4
(1, –9)
1
–4
y
0
–9
x
y = x – 21 – 2
(2, –1)2
–1–2
y
0
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swers
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Maths Quest 10/ Final Pages / 19/1/06
f
g
h
i
2 a b
c d
e f
g h
i j
k l
m
n
o
3 a b
c d
e f
x
y = – x + 32 – 7
(7, 1)
7
1
3
y
0
x
y = 0.6x + 0.5
(5, 3.5)
5
1.5
3.5
y
0
x
y = 8x
(1, 8)
1
8
y
0
x
y = x – 7
(1, –6)
1
–7–6
y
0
2
4
–2
–4
2 40–2
y
x
5x – 3y = 10
2
4
–22 40–2
y
x
5x + 3y = 10
2
4
–22 4–4 0–2
y
x
–5x + 3y = 10
2
4
–2
–4
2 4–4 0–2
y
x
–5x – 3y = 10
5
–55 100–5–10
y
x
2x – 8y = 20
5
10
–55 100–5
y
x
4x + 4y = 40
10
20
–1050–100 0–50
y
x
–x + 6y = 120 –2x + 8y = –205
–55 10–10 0–5
y
x
5
–5
–10
5 10–15 0–10 –5
y
x
10x + 30y = –150
5
10
–5
–10
10 20–30 0–20 –10
y
x5x + 30y = –150
–9x + 4y = 36
5
10
–55 10–10 0–5
y
x
6x – 4y = –245
10
–55 10–10 0–5
y
x
x
y = 2x – 10
5
–10
y
0
x
y = –5x + 20
4
20
y
0
x
y = – x – 41 – 2
–8
–4
y
0
y = 2x
2
10
y
x
y = 5x5
10
y
x
y = –3x
–3
10
y
x
y = x
10
y
x
–21
–21
y = x
3
2
0
y
x
–32
10
y
x
y = – x –25
–25– 3B
3B
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714 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
4 a b
c d
e f
g h
i
5 a x-intercept: −0.5; y-intercept: 0.4
b x-intercept: 0.5; y-intercept: −0.4
c x-intercept: 0; y-intercept: 0
d x-intercept: −3; y-intercept: 12
e x-intercept: −4; y-intercept: −4
f x-intercept: −1; y-intercept: −0.5
g x-intercept: 2.75; y-intercept: 2.2
h x-intercept: 7; y-intercept: 3.5
i x-intercept: 9.75; y-intercept: −3.9
j x-intercept: ≈ 1.77; y-intercept: 4.6
10 Quick Questions 11 y = −3x + 1
2 y = 4x − 2
3
4 x-intercept = or 1 , y-intercept = −7
5 x-intercept = 10, y-intercept = 5
6
7
8
9
10 y-intercept = −5 , x-intercept = 8
Exercise 3C — Finding linear equations 1 a y = 2x + 4 b y = −3x + 12 c y = −x + 5
d y = 2x − 8 e y = x + 3 f y = − x − 4
g y = 7x − 5 h y = −3x − 15
2 a y = 2x b y = −3x c y = x d y = − x3 a y = 3x + 3 b y = −3x + 4 c y = −4x + 2
d y = 4x + 2 e y = −x − 4 f y = 0.5x − 4
x −4 −3 −2 −1 0 1 2
y 13 10 7 4 1 −2 −5
y = 10
5
10
–55 10–10 0–5
y
x
y = –10
5
–5
–10
5 10–10 0–5
y
x
x = 10
5
10
–5
–10
5 100–5
y
x
x = –10
5
10
–5
–10
5–10 0–5
y
x
y = 100
50
100
–505 10–10 0–5
y
x
y = 05
–55 10–10 0–5
y
x
x = 05
10
–5
–10
50–5
y
x
x = –100
5
10
–5
–10
50–100 0–50
y
x
x
y = –12–12
y
0
23
13------
1 2 3 4 5
12963
–3–6
–4–3–2–1 0 x
yy = –3x + 1
x −6 −4 −2 0 2 4 6
y −26 −18 −10 −2 6 14 22
–6 –4 –2 2 4 60 x
y
y = 4x – 230
20
10
–10
–20
–30
•
•
•
•
•
•
•
−2 −1 1 2 3 4 5 x
y
y = x − 3
2
1
−1
−2
−3
−4
0
•
•
3–5
7
6---
1
6---
x
y
3y = –2x – 161
–4
2
–1
–2
–3
–5
–2–4–6–8–10 0 2 4
•
•
–5–4–3–2–1 1 2 3 4 50 x
y
y = –2
54321
–1–2–3–4–5
–8–10 –6–4–2 2 4 6 100 x
yx = 85
4321
–1–2–3–4–5
8
y 2
3---x 17
3------–=
2
3---
1
2---
1
2---
1
4---
1
2---
3
4---
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A n s w e r s 715
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➔
Maths Quest 10/ Final Pages / 19/1/06
g y = 5x + 2.5 h y = −6x + 3 i y = −2.5x + 1.5
j y = 3.5x + 6.5
4 a y = 5x − 19 b y = −5x + 31
c y = −4x − 1 d y = 4x − 34
e y = 3x − 35 f y = −3x + 6
g y = −2x + 30 h y = 2x − 4.5
i y = 0.5x − 19 j y = −0.5x + 5.5
5 a y = x + 3 b y = 2x − 1 c y = − x +
d y = x + e y = −2x − 2 f y = −x − 8
Exercise 3D — Linear modelling1 a
b Pay = $8.50 × number of hours worked
2 a
b Cost = $3 × number of rides + $10
3 a Price = 40 × number of books b $3560
4 6 assistants
5 a Cost = 0.33 × number of brochures + 166.66
b $216.67
6 a Time = 0.005 × number of people + 0.5
b 2.75 hours
7 a Price = 0.1 × number of pages − 5 b $21.40
8 a Charge = 160 × time b $2240
9 30 days
10 a Shipping cost = 2.5 × number of CDs + 2.50
b $502.50
11 39 minutes
12 a Cost = 0.8 × distance + 3.25
b $15.49 c 17.69 km
Maths Quest challenge (page 101)1 5 moves
2 There are 6 pairs (a, b): (1, 16), (3, 13), (5, 10), (7, 7), (9, 4), (11, 1).
3 When the pairs of (a, b) coordinates are plotted, the points lie on a straight line.
10 Quick Questions 21 y = x + 1 2 y = 1 − x
3 y = 4x + 2 4 y = 3x − 10
5 y = 1 − 2x 6
7 y = x + 8 y = 2x
9 C = 2 + 3.5k, where C is the cost to travel k kilometres.
10 8 km
Exercise 3E — Sketching linear inequations1 a b
c d
e f
g h
i j
k l
2 a b
c d
e f
Number of hours 0 2 4 6 8 10
Pay $0 $17 $34 $51 $68 $85
Rides 0 2 4 6 8 10
Cost $10 $16 $22 $28 $34 $40
1
2---
7
2---
1
2---
1
2---
1
3---
1
2---
1
2---
5
2---
0
–1
1
Region required
x
y
y ≥ x – 1
y = x – 1
0
–1
Region required
x
y
y < 2x – 1
y = 2x – 1
–21
0
–2
–2
Region required
x
y
y > –x – 2
y = –x – 2 0
6
Region required
x
y
y < 6 – x
y = 6 – x
6
0
–3
Region required
x
y
y > x – 3
y = x – 3
30
5
Region required
x
y
y < 5
y = 5
0–3
Region required
x
y
x ≥ –3
x = –3
0
–8
Region required
x
y
y ≤ x – 8
y = x – 8
8
0
–1
Region required
x
y
y ≥ –1
y = –10–4
Region required
x
y
y < x + 4
y = x + 4
4
0 7
Region required
x
y
x < 7
x = 7
0
2
1
Region required
x
y
y ≤ 2x
y = 2x
0
Region required
x
y
x + y – 5 ≥ 0
5
5
0–2
Region required
x
y
x – y + 2 < 02
0
Region required
x
y
3
1
y < x–31
0
Region required
x
y
–4
y ≤ 3x – 4
–34
0
Region required
x
y
4
2
y ≥ 4 – 2x
0
Region required
x
y
6
y > 6 – 4x
–23
3C
3E
5_61_03274_MQV10 - A 1-15_tb Page 715 Thursday, January 19, 2006 7:42 PM
716 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
g h
i j
k l
4 a C b B c C
5 a i ii
b
6
Summary
Chapter review1
2 a b
c d
3 a x-intercept = , y-intercept c = 6
b x-intercept = (13 ), y-intercept c = −5
c x-intercept = (1 ), y-intercept c = −
d x-intercept = −5.6, y-intercept c = 2.84 a b
c d
5 a b
c d
0
Region required
x
y
3
5y < 2x +15—2
15 – 0
Region required
x
y
–6 3x – 2y ≥ 12
4
0
Region required
x
y9
6x + y > 9
–23
0
3
2
Regionrequired
x
y
2y ≤ 3x
0
–3
Region required
x
y
y = –3
y + 3 < 0
0 2
Regionrequired
x
y x = 2
x – 2 ≥ 0
y ≤ x + 2
Region required
0 x
y
2
–2
y ≥ 4 – 2x
Region required
x
y
4
20
Region required
x
y
4
20
2
–2
Regionrequired
x
y
2
1 20
1
–1
x – 2y = 0
2x + y = 0
1 Cartesian plane, coordinates
2 infinite 3 points
4 y = mx + c 5 positive, negative
6 intercept 7 parallel
8 undefined 9 substituting
10 modelling 11 inequality
12 half plane 13 unwanted
14 broken 15 reverse
x −10 −8 −6 −4 −2 0 2 4 6 8 10
y 65 55 45 35 25 15 5 −5 −15 −25 −35
y = –5x + 15
25
50
–25
–50
5 10–10 0–5
y
x
x
y = 3x – 2
(1, –1)
1
1
–2
y
0
x
y = –5x + 15
(1, 10)
1
10
15
0
y
x
y = x + 1
(3, –1)3–1
1
–2 – 3
0
y
x
y = x – 3
5
–3
4
7 – 5
0
y
6
7---
40
3------
1
3---
21
16------
5
16------
3
4---
3
–2
0 x
y2x – 3y = 6 3
–1 0 x
y
y = –3x
–3
– 0 x
y
5x + y = –3
–53
–3
–3 0 x
y
x + y + 3 = 0
x
y = x
1
(1, )
1 – 2
1 – 2
1 – 2
0
y
0 x
y
1
–4y = –4x
0 x
y
–2
x = –2
0 x
y
7 y = 7
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A n s w e r s 717
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Maths Quest 10/ Final Pages / 19/1/06
6
7 a y = 2x − 2 b y = −x − 4 c y = − x + 2
d y = 4x e y = − f x = 5
8 a y = 3x − 4 b y = −2x − 5
c y = x + 5 d y = 6
9 a y = 7x − 13 b y = −3x + 4
c y = x + 6 d y = x −
10 a y = −x + 8 b y = x + 12
c y = x +
11 a
b Pay = $13.50 × number of hours worked
12 a
b Cost = $2.50 × number of rides + $12.50
13 a
b Cost = 22.5 × time + 160 c $435.63
14 a b
c d
e f
g h
i
Chapter 4 Quadratic equationsAre you ready?1 a 12x + 20 b 10x2 − 15x
c −12x + 8x2
2 a x2 − 4 b 4x2 − 12x + 9
c 6x2 − 11x − 10
3 a 4x(x + 2) b −3x(5x + 3)
c x(6x − 1)
4 a (x + 2)(3x + 4) b (x − 1)(4x − 1)
c −(x + 3)(2x + 1)
5 a 2 and 3 b −2 and +2 c −3 and −1
6 a b 2 c 4
7 a b 9 c
8 a x + 3 b c
9 a b c
Exercise 4A — Expanding algebraic expressions1 a 2x + 6 b 4x − 20 c 21 − 3x
d −x − 3 e x2 + 2x f 2x2 − 8xg 15x2 − 6x h 10x − 15x2 i 8x2 + 2xj 4x3 − 6x2 k 6x3 − 3x2 l 15x3 + 20x2
2 a x2 − x − 12 b x2 − 2x − 3 c x2 − 5x − 14
d x2 − 6x + 5 e −x2 − x + 6 f x2 − 6x + 8
g 2x2 − 17x + 21 h 3x2 − x − 2 i 6x2 − 17x + 5
j 21 − 17x + 2x2 k 15 + 14x − 8x2
l 110 + 47x − 21x2
3 a 2x2 − 4x − 6 b 8x2 − 28x − 16
c −2x2 + 12x + 14 d 2x3 − 2xe 3x3 − 75x f 6x3 − 54xg 2x3 − 12x2 + 18x h 5x3 − 30x2 + 40xi −6x3 − 6x2 + 120x
4 a x3 + 2x2 − x − 2 b x3 − 2x2 − 5x + 6
c x3 − 5x2 − x + 5 d x3 − 6x2 + 11x − 6
e 2x3 − 7x2 − 5x + 4 f 6x3 − 7x2 + 1
5 a x2 − x − 2 b −2x2 + 4x + 10
c 5x2 − 6x − 5 d 19x − 23
e −5x − 1 f −2x + 6
g x2 − 2x − 3 + x
h
6 a A b C
Number of hours 0 2 4 6 8 10
Pay($) 0 27 54 81 108 135
Rides 0 2 4 6 8 10
Cost($) 12.50 17.50 22.50 27.50 32.50 37.50
–27 0 x
y
–
7
3(y – 5) = 6(x + 1)
(0, 7)
1
3---
3
4---
1
2---
1
2---
3
5---
18
15------
3–
2------
2
5---
27
5------
8Time
Cost
0
100
200
300
400
500
0 2 4 6 10 12
•
•
•
0 x
y
1
–1
y ≤ x + 1
Region required
0 x
y
10
–5
y ≥ 2x + 10
Region required
0 x
y
–12
4
y > 3x – 12
Region required
0 x
y
1
y < 5x
y = 5x5
Regionrequired
0 x
y
7x ≥ 7
x = 7
Regionrequired
0 x
y
–2
1
Regionrequired
y ≤ x + 1 –21
0 x
y
9
Regionrequired
2x + y ≥ 9
–29 0 x
y
–16
12
4x − 3y ≥ 48
5
Regionrequired
•
•
0 x
y
–12
Regionrequired
y > –12
1
12------
1
2---
1
2---
2
3---
1
x 3–( ) x 7+( )----------------------------------
x 2+2 x 3+( )--------------------
2 6 6 3 36 3
3
6 2 2x 3 3x– 6x2– 5x–+ 4A
4A
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swers
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7 a x2 − 2x + 1 b x2 + 4x + 4c x2 + 10x + 25 d 16 + 8x + x2
e 49 − 14x + x2 f 144 − 24x + x2
g 9x2 − 6x + 1 h 144x2 − 72x + 9i 25x2 + 20x + 4 j 4 − 12x + 9x2
k 25 − 40x + 16x2 l 1 − 10x + 25x2
8 a 2x2 − 12x + 18 b 4x2 − 56x + 196c 3x2 + 6x + 3 d −4x2 − 12x − 9e −49x2 + 14x − 1 f 8x2 − 24x + 18 g −12 + 108x − 243x2 h −45 + 330x − 605x2
i −16x2 − 16x − 49 a x2 − 49 b x2 − 81 c x2 − 25
d x2 − 1 e 4x2 − 9 f 9x2 − 1g 49 − x2 h 64 − x2 i 9 − 4x2
10 a (x + 1)(x − 3) b x2 − 2x − 3 c 6 cm, 2 cm, 12 cm2
11 a b
c (x + 1)(x + 2) d x2 + 3x + 2e 4 m2, 12 m2
12 a (x + 2)2 b 5(x + 2)2 c 5x2 + 20x + 20d 500 cm3 e 100 cm2, 100 tiles
Exercise 4B — Factorising expressions with two or four terms
1 a x(x + 3) b x(x − 4) c 3x(x − 2)d 4x(x + 4) e 3x(3x − 1) f 8x(1 − x)g 3x(4 − x) h 4x(2 − 3x) i x(8x − 11)
2 a (x − 2)(3x + 2) b (x + 3)(5 − 2x)c (x − 1)(x + 5) d (x + 1)(x − 1)e (x + 4)(x − 2) f (x − 3)(4 − x)
3 a (x + 1)(x − 1) b (x + 3)(x − 3)c (x + 5)(x − 5) d (x + 10)(x − 10)e (y + k)(y − k) f (2x + 3y)(2x − 3y) g (4a + 7)(4a − 7) h (5p + 6q)(5p − 6q)i (1 + 10d)(1 − 10d)
4 a 4(x + 1)(x − 1) b 5(x + 4)(x − 4)c a(x + 3)(x − 3) d 2(B + 2D)(B − 2D)e 100(x + 4)(x − 4) f 3a(x + 7)(x − 7)g 4p(x + 8)(x − 8) h 4(3x + 2)(3x − 2)i 3(6 + x)(6 − x)
5 a C b B c B
6 a (x + )(x − ) b (x + )(x − )
c (x + )(x − ) d (2x + )(2x − )
e (3x + )(3x − )f 3(x + )(x − )
g 5(x + )(x − ) h 2(x + )(x − )
i 12(x + )(x − )
7 a (x − 3)(x + 1) b (x − 4)(x + 6)c (x − 5)(x + 1) d (x − 1)(x + 7)e (6 − x)(x + 8) f (10 − x)(x + 2)g 8(x − 3) h (7 − x)(5x + 1)i (x − 22)(9x + 2)
8 a (x − 5)(x + 5) b (x − 5) cm, (x + 5) cmc 2 cm, 12 cm d 24 cm2 e 120 cm2 or 6 times bigger
9 a r metres b (r + 1) mc A1 = πr2 m2 d A2 = π(r + 1)2 m2
e A = π (r + 1)2 − πr2 = π(2r + 1) m2
f 34.56 m2
10 a (x − 2y)(1 + a) b (x + y)(2 + a)c (x − y)(a + b) d (x + y)(4 + z)e ( f − 2)(e + 3) f (n − 7)(m + 1)g 3(2r − s)(t + u) h 7(m − 3)(n + 5)i 2(8 − j)(4 + k) j a(3 − b)(a + c)k x(5 + y)(x + 2) l m(m + n)(2 − n)
11 a (y + 7)(x − 2) b (m + 2)(n − 3)c (q + 5)(p − 3) d (s + 3)(s − 4t)e (b + d)(a2 − c) f (1 + 5z)(xy − z)
12 a (a − b)(a + b + 4) b (p − q)(p + q − 3)c (m + n)(m − n + l) d (x + y)(7 + x − y)e (1 − 2q)(5p + 1 + 2q) f (7g + 6h)(7g − 6h − 4)
13 a (x + 7 + y)(x + 7 − y)b (x + 10 + y)(x + 10 − y)c (a − 11 + b)(a − 11 − b)d (3a + 2 + b)(3a + 2 − b)e (5p − 4t + 3t)(5p − 4t − 3t)
f
14 a E b A c D
Exercise 4C — Factorising expressions with three terms1 a (x + 2)(x + 1) b (x + 3)(x + 1)
c (x + 8)(x + 2) d (x + 4)2
e (x − 3)(x + 1) f (x − 4)(x + 1)g (x − 12)(x + 1) h (x − 6)(x + 2)i (x + 4)(x − 1) j (x + 5)(x − 1)k (x + 7)(x − 1) l (x + 5)(x − 2)m (x − 3)(x − 1) n (x − 4)(x − 5)o (x + 14)(x − 5)
2 a −2(x + 9)(x + 1) b −3(x + 2)(x + 1) c −(x + 2)(x + 1) d −(x + 10)(x + 1)e −(x + 2)(x + 5) f −(x + 12)(x + 1)g −(x + 3)(x + 4) h −(x + 2)(x + 6)i 2(x + 2)(x + 5) j 3(x + 1)(x + 10)k 5(x + 20)(x + 1) l 5(x + 4)(x + 5)
3 a (a − 7)(a + 1) b (t − 4)(t − 2)c (b + 4)(b + 1) d (m + 5)(m − 3)e (p − 16)(p + 3) f (c + 16)(c − 3)g (k + 19)(k + 3) h (s − 19)(s + 3)i (g + 8)(g − 9) j (v − 25)(v − 3)k (x + 16)(x − 2) l (x − 15)(x − 4)
4 a C b B5 a (2x + 1)(x + 2) b (2x − 1)(x − 1)
c (4x + 3)(x − 5) d (2x − 1)(2x + 3)e (x − 7)(2x + 5) f (3x + 1)(x + 3)g (3x − 7)(2x − 1) h (4x − 7)(3x + 2)i (5x + 3)(2x − 3) j (4x − 1)(5x + 2)k (3x + 2)(4x − 1) l (3x − 1)(5x + 2)
6 a 2(x − 1)(2x + 3) b 3(3x + 1)(x − 7)c 12(2x + 1)(3x − 1) d −3(3x + 1)(2x − 1)e −30(2x − 3)(x − 1) f 3a(4x − 7)(2x + 5)g −2(4x − 3)(x − 2) h −(2x − 7)(5x + 2)i −(8x − 1)(3x − 4) j −2(3x − y)(2x + y)k −5(2x − 7y)(3x + 2y) l −12(5x + 3y)(10x + 7y)
7 a w2 + 5w − 6 b (w + 6)(w − 1)c (x + 5)(x − 2)
8 a x(x + 5) b x(x + 5)c (x − 1)2 d (x + 9)(x + 5)e (x − 15)(x − 6) f (x − 10)(x − 3)
x m (x + 2) m
(x + 1) m
11 11 7 7
15 15 13 13
19 19 22 22
3 3 2 2
3 3
6t 1– 5v+( ) 6t 1– 5v–( )
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swers
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9 a (x − 5)(x + 1) b x − 5
c x = 15 d 160 cm2
e 3000(x − 5)(x + 1) cm2
or (3000x2 − 12 000x − 15 000) cm2
Maths Quest challenge (page 127)1 a 173
b The pattern is based on the difference of two squares rule:
a2 − b2 = (a + b)(a − b). The factor (a − b) is 1 in each case, so a2 − b2 = a + b.
2 Numbers that are 2 or 4 more than a multiple of 6 are even and so cannot be prime. Any number which is 3 more than a multiple of 6 will be divisible by 3 and hence not prime. So that only leaves 1 more or 1 less than a multiple of 6 for the prime numbers. The two exceptions are 2 and 3.
Exercise 4D — Factorising by completing the square1 a x2 + 10x + 25 = (x + 5)2
b x2 + 6x + 9 = (x + 3)2
c x2 − 4x + 4 = (x − 2)2
d x2 + 16x + 64 = (x + 8)2
e x2 − 20x + 100 = (x − 10)2
f x2 + 8x + 16 = (x + 4)2
g x2 − 14x + 49 = (x − 7)2
h x2 + 50x + 625 = (x + 25)2
i x2 − 2x + 1 = (x − 1)2
2 a (x − 2 + )(x − 2 − )
b (x + 1 + )(x + 1 − )
c (x − 5 + )(x − 5 − )
d (x + 3 + )(x + 3 − )
e (x + 8 + )(x + 8 − )
f (x − 7 + )(x − 7 − )
g (x + 4 + )(x + 4 − )
h (x − 2 + )(x − 2 − )
i (x − 6 + )(x − 6 − )
3 a 2(x + 1 + ) (x + 1 − )
b 4(x − 1 + ) (x − 1 − )
c 5(x + 3 + 2 ) (x + 3 − 2 )
d 3(x − 2 + ) (x − 2 − )
e 5(x − 3 + ) (x − 3 − )
f 6(x + 2 + ) (x + 2 − )
g 3(x + 5 + 2 ) (x + 5 − 2 )
h 2(x − 2 + ) (x − 2 − )
i 6(x + 3 + ) (x + 3 − )
4 a (x − + )(x − − )
b (x − + )(x − − )
c (x + + )(x + − )
d (x + + )(x + − )
e (x + + )(x + − )
f (x + + )(x + − )
g (x − + )(x − − )
h (x − + )(x − − )
i (x − + )(x − − )
5 a B b E
10 Quick Questions 11 8x2 − 20x + 12
2 −7x2 − 42x − 63
3 4x2 − 49
4 6x(4x2 − 3)
5 2(7x − 6y)(7x + 6y)
6 (x − 2)(4x + y)
7 (x − 11)(x + 2)
8 (3x + 5)(2x + 3)
9 2(2x + 1)(x − 7)
10 (x + 3 + 2 )(x + 3 − 2 )
Exercise 4E — Mixed factorisation
46 a
b
c
47 a b
c d
e f
g h
i j
11 11
3 3
13 13
19 19
65 65
6 6
7 7
17 17
11 11
3 3
6 6
2 2
17 17
7 7
5 5
3 3
11 11
14 14
1
2---
5
4-------
1
2---
5
4-------
3
2---
21
2----------
3
2---
21
2----------
1
2---
21
2----------
1
2---
21
2----------
3
2---
13
2----------
3
2---
13
2----------
5
2---
17
2----------
5
2---
17
2----------
5
2---
33
2----------
5
2---
33
2----------
7
2---
53
2----------
7
2---
53
2----------
9
2---
29
2----------
9
2---
29
2----------
1
2---
13
2----------
1
2---
13
2----------
7 7
1 3(x + 3) 2 (x +2 +3y)(x + 2 − 3y)3 (x + 6)(x − 6) 4 (x + 7)(x − 7)5 (5x + 1)(x − 2) 6 5(3x − 4y)7 (c + e)(5 + d) 8 5(x + 4)(x − 4)9 −(x + 5)(x + 1) 10 (x + 4)(x − 3)
11 (m + 1)(n + 1) 12 (x + )(x − )7 713 4x(4x − 1) 14 5(x + 10)(x + 2)15 3(3 − y)(x + 2) 16 (x − 4 + y)(x − 4 − y)17 4(x2 + 2) 18 (g + h)( f + 2)19 (x + )(x − ) 20 5(n + 1)(2m − 1)5 521 (x + 5)(x + 1) 22 (x + 1)(x − 11)23 (x + 2)(x − 2) 24 (a + b)(c − 5)25 (y + 1)(x − 1) 26 (3x + 2)(x + 1)27 7(x + 2)(x − 2) 28 −4(x + 6)(x + 1)29 (2 + r)(p − s) 30 3(x + 3)(x − 3)31 (u + v)(t − 3) 32 (x + )(x − )11 1133 (4x − 1)(3x − 1) 34 (x + 1)(x − 3)35 (x + 6)(x − 2) 36 4(x − 1)(x + 4)37 3(x + 2)(x + 8) 38 (3 + x)(7 − x)39 4(3 − x + 2y)(3 − x − 2y)40 3(y + x)(y − x) 41 4(x + 2)42 (3x − 4y)(x − 2y) 43 (x + 7)(x + 4)44 (x + 2)(x − 5) 45 2(2x + 3)(x + 2)
x 5+( ) x 2–( )x 2+( ) x 2–( )
----------------------------------x 2+( ) x 2+( )x 4–( ) x 2+( )
----------------------------------×
x 5+( ) x 2–( )x 2+( ) x 2–( )
----------------------------------x 2+( ) x 2+( )x 4–( ) x 2+( )
----------------------------------×
x 5+x 4–------------
x 1–
x 6–-----------
x 1+2x 3+---------------
18
x x 5–( )-------------------
2x 1–
x 4+---------------
x 2+x 5+------------
x 6–
x 3+------------
4 b 2+( )5
--------------------p p 7+( )
p 3+( ) p 2–( )-----------------------------------
5 m 2 n+ +( )2 2m 5–( )
-------------------------------5 d 3– 5e+( )
4 4d 3+( )-------------------------------- 4B
4E
5_61_03274_MQV10 - A 1-15_tb Page 719 Thursday, January 19, 2006 7:42 PM
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Exercise 4F — Solving quadratic equations
1 a −7, 9 b −2, 3 c 2, 3 d 0, 3
e 0, 1 f −5, 0 g 0, 3 h −2, 0
i − , j −1.2, −0.5
k 0.1, 0.75 l − ,
2 a , 1 b −2, − c , 7 d − , 1
e , f − , g 0, , 3
h 0, , − i 0, −3,
3 a 0, 2 b −5, 0 c 0, 7 d − , 0
e 0, 1 f 0, g 0, h − , 0
i 0, 1
4 a −2, 2 b −5, 5 c −2, 2 d −7, 7
e −1 , 1 f −2 , 2 g − , h − ,
i − , j −4, 4 k − ,
l − ,
5 a −2, 3 b −4, −2 c −1, 7 d 3, 5
e 1 f −1, 4 g 5 h −2, 5
i 2, 6 j −3, 7 k −5, 6 l 3, 4
6 a − , 3 b , −1 c −2, d , 1
e − , 1 f , g −1 , 2
h −1 , −1 i − , j 1 , 2 k − ,
l 3, 4
7 a 2 + , 2 − b −1 + , −1 −
c −3 + , −3 − d 4 + , 4 −
e 5 + , 5 − f 1 + , 1 −
g −1 + , −1 − h −2 + ,−2 −
i −2 + , −2 −
8 a + , − b − + , − −
c + , − d + , −
e + , − f − + , − −
g − + , − − h + , −
i + , −
9 a −3, 1 b −4.24, 0.24 c −1, 3
d −0.73, 2.73 e 0.38, 2.62 f −0.30, 3.30
g −1.19, 4.19 h −2.30, 1.30 i −2.22, 0.22
10 No real solutions — when we complete the square we get the sum of two squares, not the difference of two squares and we cannot factorise the expression.
11 8 and 9 or −8 and −9
12 6 and 8, −6 and −8
13 9 or −10
14 2 or −2
15 8 or −10
16 6 seconds
17 a l = 2xb
c x2 + (2x)2 = 452, 5x2 = 2025
d Length 40 cm, width 20 cm
18 8 m, 6 m
19 a
b (2 + x) m, (4 + x) m c (2 + x)(4 + x) = 24
d x = 2, 4 m wide, 6 m long
20 a (l − 4) cm b l − 8, l − 4
c (l − 8)(l − 4) = 620 d 31 cm
e 836 cm2
Exercise 4G — Using the quadratic formula1 a a = 3, b = −4, c = 1 b a = 7, b = −12, c = 2
c a = 8, b = −1, c = −3 d a = 1, b = −5, c = 7
e a = 5, b = −5, c = −1 f a = 4, b = −9, c = −3
g a = 12, b = −29, c = 103
h a = 43, b = −81, c = −24
i a = 6, b = −15, c = 1
2 a −1 b c
d e f
g h i
j k l
3 a −0.54, 1.87 b −1.20, 1.45 c −4.11, 0.61
d −0.61, 0.47 e 0.14, 1.46 f 0.16, 6.34
g −1.23, 1.90 h −1, 1.14 i −0.83, 0.91
j −0.64, 1.31 k −0.35, 0.26 l −1.45, 1.20
m 0.08, 5.92 n −0.68, 0.88 o −0.33, 2
4 C
5 a 0.5, 3 b 0, 5 c −1, 3
d 0.382, 2.618 e 0.298, 6.702 f 2, 4
g No real solution h −1, 8
i −4.162, 2.162 j −2, 1 k −7, 1.5
l No real solution m 2, 7
n − , o No real solution
6 a 2πr2 + 14πr − 231 = 0 b 3.5 cm
c 154 cm2
10 Quick Questions 2
1
2---
1
2---
2 31
2---
2
3---
1
4---
6
7---
1
2---
3
5---
2
3---
5
8---
2
3---
1
2---
1
2---
2
5---
2
5---
2
3---
1
2---
1
3---
7
2-------
3
3-------
1
4---
1
3---
1
3---
1
2---
1
2---
2
3---
2
3---
1
2---
1
2---
1
5---
1
5--- 5 5
11
3----------
11
3----------
1
2---
2
3---
1
5---
1
3---
1
2---
3
14------
1
4---
1
3---
1
3---
1
2---
3
4---
1
3---
2
5---
1
2---
1
2---
2
3---
2
5---
1
6---
2 2 3 3
10 10 2 3 2 3
2 6 2 6 3 3
6 6 10 10
15 15
3
2---
5
2-------
3
2---
5
2-------
5
2---
29
2----------
5
2---
29
2----------
7
2---
33
2----------
7
2---
33
2----------
1
2---
21
2----------
1
2---
21
2----------
11
2------
117
2-------------
11
2------
117
2-------------
1
2---
5
2-------
1
2---
5
2-------
3
2---
37
2----------
3
2---
37
2----------
5
2---
37
2----------
5
2---
37
2----------
9
2---
65
2----------
9
2---
65
2----------
2
3---
1
2---
2x cm
x cm45 cm
4 m
2 m
−3 13±2
-----------------------5 17±
2-------------------
2 13 ± 1– 2 3± 7 45±2
-------------------
9 73±2
------------------- 3 2 3± 4– 31±
1 21±2
-------------------5 33±
2------------------- 1– 4 2±
1
2---
1
3---
1 −7, −2 2 12, −3
3 0, −1 4 ±112
3---
5 − , 1 63
4---
2
3--- 3 2±
7 82 10± 1 13±2
-------------------
9 1 , 10 2.193, −3.1931
2---
1
2---
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Exercise 4H — Finding solutions to quadratic equations by inspecting graphs
1 a
x
=
−
2,
x
=
3
b
x
=
1,
x
=
10
c
x
=
−
5,
x
=
5
d
x
=
2
e
x
=
−
1, x = 4 f x ≈ −1.4, x ≈ 4.4g x = −25, x = 10 h x = 0i x ≈ −2.3, x ≈ 1.3 j x ≈ −1.5, x = 1
2 a–j: Confirm by substitution of above values into quadratic equations.
3 150 m4 7 m
Exercise 4I — Using the discriminant1 a −11 b 0 c 169 d 0 e 37
f 0 g 52 h −7 i −4 j 109k 129 l 1
2 a No real solutions b 1 rational solutionc 2 rational solutions d 1 rational solutione 2 irrational solutions f 1 rational solutiong 2 irrational solutions h No real solutionsi No real solutions j 2 irrational solutionsk 2 irrational solutions l 2 rational solutions
3 a No real solutions b 2
c −11, 2 d −
e ≈ −4.541, 1.541
f g ≈ −0.869, 1.535
h No real solutions i No real solutions
j ≈ −2.573, 0.907
k ≈ −4.589, 1.089 l 5, 6
4 a a = 3, b = 2, c = 7 b −80c No real solutions
5 a a = −6, b = 1, c = 3 b 73
c 2 real solutions d
6 A
Maths Quest challenge (page 148)
Summary
Chapter review1 a 3x2 − 12x b −21x2 − 7x
c x2 − 6x − 7 d 2x2 − 11x + 15e 12x2 − 23x + 5 f 6x2 − 3x − 84g 2x3 + 15x2 − 8x − 105 h 3x2 − 5x + 65i 5x2 + 12x − 3
2 a x2 − 14x + 49 b 4 − 4x + x2
c 9x2 + 6x + 1 d −18x2 + 24x − 8e −28x2 − 140x − 175 f −160x2 + 400x − 250
g x2 − 81 h 9x2 − 1i 25 − 4x2
3 a 2x(x − 4) b −4x(x − 3)c ax(3 − 2x) d (x + 1)(x + 2)e 2(2x − 5)(4 − x) f (x − 4)(x + 1)
4 a (x + 4)(x − 4) b (x + 5)(x − 5)c 2(x + 6)(x − 6) d 3(x + 3y)(x − 3y)e 4a(x + 2y)(x − 2y) f (x − 1)(x − 7)
5 a (x − y)(a + b) b (x + y)(7 + a)c (x + 2)(y + 5) d (1 + 2q)(mn − q)e (5r + 1)(pq − r) f (v − 1)(u + 9)g (a − b)(a + b + 5) h (d − 2c)(d + 2c − 3)i (1 + m)(3 − m)
6 a (2x + 3 + y)(2x + 3 − y)b (7a − 2 + 2b)(7a − 2 − 2b)
c7 a (x + 9)(x + 1) b (x − 9)(x − 2)
c (x − 7)(x + 3) d (x + 7)(x − 4)e −(x − 3)2 f 3(x + 13)(x − 2)g −2(x − 5)(x + 1) h −3(x − 6)(x − 2)i (4x − 1)(2x + 1) j (3x − 1)(2x + 1)k 4(2x + 3)(x − 1) l 5(7x − 3)(3x + 1)m −2(3x − 5)(2x − 7) n −3(3x − 1)(5x + 2)o −30(2x + 3)(x + 3)
8 (3x + 4) m
9 a (x + 3 + 2 )(x + 3 − 2 )
b (x − 5 + 2 )(x − 5 − 2 )
c (x + 2 + )(x + 2 − )
d (x − + )(x − − )
e (x + + )(x + − )
f 2(x + + )(x + − )
10 a 3x(x − 4)b (x + 3 + )(x + 3 − )c (2x + 5)(2x − 5) d (2x + 5)(x + 2)e (a + 2)(2x + 3) f −3(x − 2)(x + 3)
11 a b c
12 a −5, −3 b −6, −1 c −8, −3 d 2, −6e 5, −2 f 4, −7 g 3, 1 h 5, 6i 7, −5
13 a −2, −6 b −2, −1 c , −3 d 2, −7
e − , 4 f − , 2 g 2, 1 h ,
i −7,
14 a −4 ± b −1 ± c −1, 15 416 a −0.651, 1.151 b −0.760, 0.188
c −0.441, 0.566
1
2---
2
3---
−3 37±2
-----------------------
1
5---
1 13±3
-------------------
−5 109±6
--------------------------
−7 129±4
--------------------------
1 73±12
-------------------
1 220 2 1323 10, 11, 13, 18, 35 4 6, 225 If x = a2 + b2, then 2x = 2a2 + 2b2.
This can be written as (a2 − 2ab + b2) +(a2 + 2ab + b2) or (a − b)2 + (a + b)2.
1 expansion 2 FOIL3 perfect 4 Factorisation5 common factor 6 quadratic7 difference8 coefficient, factor pair, grouping9 square
10 Null Factor, quadratic formula11 discriminant 12 x-intercepts
8s 1– 3t+( ) 8s 1– 3t–( )
2 2
7 7
6 6
5
2---
17
2----------
5
2---
17
2----------
7
2---
53
2----------
7
2---
53
2----------
9
2---
85
2----------
9
2---
85
2----------
7 7
2 x 4+( )5 x 1+( )-------------------- 7
8---
x 2–( ) x 1–( )x x 4–( )
---------------------------------
1
2---
1
2---
2
3---
5
3---
5
2---
1
2---
17 6 1
4---
4F
4I
5_61_03274_MQV10 - A 1-15_tb Page 721 Thursday, January 19, 2006 7:42 PM
722 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
17 a −0.571, 0.682 b −0.216, 3.836c −0.632, 0.632
18 −3, 719 −3, 120 a 2 irrational solutions b 2 rational solutions
c No real solutions
Chapter 5 Quadratic graphsAre you ready?1 a 0 b −16 c −382 a x = −2 b x = 3 c x = or x = 1.5
3 a (x + 1)2 + 1 b c 2(x − 1)2 + 4
4 a b c
5 a x = −2 or x = −3 b x = 1 or x = −2c x = 2 or x = −2
6 a x = − or x = −2 b x = 2 or x = −
c x = or x =
Exercise 5A — Plotting parabolas1 x = 0, (0, 0)
2 a b
x = 0, (0, 0) x = 0, (0, 0)3 Placing a number greater than 1 in front of x2 makes
the graph thinner. Placing a number greater than 0 but less than 1 in front of x2 makes the graph wider.
5 Adding a number raises the graph of y = x2 vertically that number of units. Subtracting a number lowers the graph of y = x2 vertically that number of units.
7 Adding a number moves the graph of y = x2 horizontally to the left by that number of units. Subtracting a number moves the graph of y = x2 horizontally to the right by that number of units.
9 The negative sign inverts the graph of y = x2.The graphs with the same turning points are:
y = x2 + 1 and y = −x2 + 1; y = (x − 1)2 and
y = −(x − 1)2; y = (x + 2) and y = −(x + 2)2;
y = x2 − 3 and y = −x2 − 3.
They differ in that the first graph is upright while the second graph is inverted.
10 a x = 5, (5, 1), min, 26
3
2---
x 3
2---–⎝ ⎠
⎛ ⎞ 2 7
4---+
–1 5±2
-------------------2 2±
2----------------
1 7±3–
----------------–1 7+−
3-------------------=
1
2---
1
3---
3
2---
2
3---
x
y
1
y = x2
2 3–3–2–1
2
–2
4
6
8
10
–4
(0, 0)
x
y y = 3x2
30252015105
0–1 1 2 3–2–3 x
y y = x2
1
0–1
1 – 4
1 2 3–2–3
2
4 a
x = 0, (0, 1), 1
c
x = 0, (0, −3), −3
x
y
0
y = x2 + 1
2
4
6
8
10
–3–2–1 1 2 3
x
y
0
y = x2 – 3
–3–2–1 1 2 3
2
–2
4
6
b
x = 0, (0, 3), 3
d
x = 0, (0, −1), −1
x
y
0
y = x2 + 3
(0, 3)
–3–2–1 1 2 3
2
4
6
8
10
12
x
y
(0, –1)
y = x2 – 1
–3–2–1 1 2 3
2
–2
4
6
8
6 a
x = −1, (−1, 0), 1
c
x = 2, (2, 0), 4
x
y
0–1 1 2
(1, 4)
(–5, 16) y = (x + 1)2
–2–3–4–5–6
4
8
12
16
20
x
y
0 2
4
y = (x – 2)2
2
6
8
10
1 4 53
b
x = −2, (−2, 0), 4
d
x = 1, (1, 0), 1
x
y
0–2
y = (x + 2)2
2–4–6
4
8
12
16
x
y
0 1
y = (x – 1)2
2
4
6
8
10
2 3 4 5
8 a
x = 0, (0, 1), 1
x
y
01 2 3 4–1–2–3
1
–2–3–4–5–6–7–8 y = –x2 + 1
b
x = 1, (1, 0), −1
x
y
0
1 2 3 4 5–1–2
y = –(x – 1)2
–3–4–5–6–7–8–9
c
x = −2, (−2, 0), −4
x
y
0
–4
–6
–8
–2–4–6
y = –(x + 2)2
1–2
d
x = 0, (0, −3), −3
x
y
0
y = –x2 – 3
1 2 3 4 5–2
–4
–6
–8
–10
–12
x
y
0
26
15
y = (x – 5)2 + 1
64321
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b x = −2, (−2, −3), min, 5
c x = 3, (3, 4), max, −5
d x = 1, (1, 2), max, −1
e x = −2, (−2, −9), min, −5
f x = −1, (−1, 16), max, 15
g x = −1, (−1, 27), max, 24
h x = 2, (2, 1) min, 5
11 a If the x2 term is positive, the parabola has a minimum turning point. If the x2 term is negative, the parabola has a maximum turning point.
b If the equation is of the form y = a(x − b)2 + c, the turning point has coordinates (b, c).
c The equation of the axis of symmetry can be found from the x-coordinate of the turning point. That is, x = b.
12 C 13 B 14 C 15 A
16 a
b i 16 m ii 8 s
17 a
b i 18 m ii Yes, by 3 m iii 1.5 s iv 3 s
Exercise 5B — Sketching parabolas using the basic graph of y = x2 1 a Narrower, TP (0, 0) b Wider, (0, 0)
c Narrower, TP (0, 0) d Narrower, TP (0, 0)
e Wider, TP (0, 0) f Wider, TP (0, 0)
g Narrower, TP (0, 0) h Narrower, TP (0, 0)
2 a Vertical 3 up, TP (0, 3)
b Vertical 1 down, TP (0, −1)
c Vertical 7 down, TP (0, −7)
d Vertical up, TP (0, )
e Vertical down, TP (0, − )
f Vertical 0.14 down, TP (0, −0.14)
g Vertical 2.37 up, TP (0, 2.37)
h Vertical up, TP (0, )
3 a Horizontal 1 right, (1, 0)
b Horizontal 2 right, (2, 0)
c Horizontal 10 left, (−10, 0)
d Horizontal 4 left, (−4, 0)
e Horizontal right, ( , 0)
f Horizontal left, (− , 0)
g Horizontal 0.25 left, (−0.25, 0)
h Horizontal left, (− , 0)
4 a (0, 1), max b (0, −3), min c (−2, 0), max
d (0, 0), min e (0, 4), max f (0, 0), max
g (5, 0), min h (0, 1) min
5 a Narrower, min b Narrower, max
c Wider, min d Wider, max
e Narrower, max f Wider, min
g Narrower, min h Wider, max
i Narrower, min j Narrower, max
k Narrower, min l Narrower, max
6 a i Horizontal translation 1 left
ii (−1, 0) iii
–2–4–6–8–4
4
8
12
16
x
y
0
y = 2(x + 2)2 – 3
1 2 3 4 5 6
–5–4–3–2
1234
x
y
0
y = –(x – 3)2 + 4
x
y
0
5
–2 42–5
–10–15
–20
–25
y = –3(x – 1)2 + 2
–2–4–6
5
10
–5
–10
x
y
0
y = x2 + 4x – 5
–2–4–6 2 4
10
–20
–10
20
30
40
x
y
0
y = –x2 – 2x + 15
–4 –2 2–6
25
2015105
–5–10–15–20–25
x
y
0
y = –3x2 – 6x + 24
–2 2 4
4
8
12
16
20
x
y
0
y = (x – 2)2 + 1
68
1012141618
42
t
h
0 1 2 3 4 5 6 7 8
h = –(t – 4)2 + 16
68
1012141618
42
t
h
0 1 2 3
1
4---
1
4---
1
2---
1
2---
3 3
1
2---
1
2---
1
5---
1
5---
3 3
x
y
0(–1, 0)
y = x2
y = (x + 1)2
5A
5B
5_61_03274_MQV10 - A 1-15_tb Page 723 Thursday, January 19, 2006 7:42 PM
724 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
b i Reflected, narrower (dilation)
ii (0, 0) iii
c i Vertical translation 1 up
ii (0, 1) iii
d i Wider (dilation)
ii (0, 0) iii
e i Vertical translation 3 down
ii (0, −3) iii
f i Horizontal translation 4 right
ii (4, 0) iii
g i Reflected, wider (dilation)
ii (0, 0) iii
h i Narrower (dilation)
ii (0, 0) iii
i i Reflected, vertical translation 2 up ii (0, 2) iii
j i Reflected, horizontal translation 6 rightii (6, 0) iii
k i Reflected, vertical translation 4 downii (0, −4) iii
l i Reflected, horizontal translation 1 left
ii (−1, 0) iii
m i Narrower (dilation), horizontal translation 1 left, vertical translation 4 down
ii (−1, −4) iii
x
y
y = –3x2
y = x2
0
x
y
0
(0, 1)
y = x2
y = x2 + 1
x
y
(0, 0)
1–3
y = x2
y = x2
x
y
0
(0, –3)
y = x2
y = x2 – 3
x
y
0 (4, 0)
y = x2 y = (x – 4)2
x
y
y = – x22–5
y = x2
(0, 0)
x
y
(0, 0)
y = x2
y = 5x2
x
y
0
y = x2
y = –x2 + 2
(0, 2)
x
y
0
(6, 0)
y = –(x – 6)2
y = x2
x
y
0
y = x2
y = –x2 – 4
x
y
0
(–1, 0)
y = x2
y = –(x + 1)2
x
y
y = 2(x + 1)2 – 4
y = x2
0
(–1, –4)
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n i Wider (dilation), horizontal translation 3 right, vertical translation 2 up
ii (3, 2) iii
o i Wider (dilation), reflected, horizontal translation 2 left, vertical translation up
ii (−2, ) iii
p i Narrower (dilation), reflected, horizontal translation 1 right, vertical translation down
ii (1, ) iii
7 a 10 cm b 5 cm c 5 cm d y = (x − 5)2
Exercise 5C — Sketching parabolas in turning point form1 a (1, 2), min b (−2, −1), min
c (−1, 1), min d (2, 3), max
e (5, 3), max f (−2, −6), min
g (2, 8), max h (3, −2), min
i (−8, 2), max j (− , − ), min
k ( , ), min l (−0.3, −0.4), min
m (1.6, 2.7), min n (−2, 5) max
o (7, 2), max
2 a i (−3, −5) ii Min iii Narrower
b i (1, 1) ii Max iii Same
c i (−2, −4) ii Min iii Narrower
d i (3, 2) ii Min iii Wider
e i (−1, 7) ii Max iii Wider
f i (− , − ) ii Min iii Wider
3 A b y = −(x − 2)2 + 3 B e y = −x2 + 1
C f y = (x + 1)2 − 3 D d y = −(x + 2)2 + 3
E c y = x2 − 1 F a y = (x − 1)2 − 3
4 a A b C c B d C e B
5 a i −3 ii −3, 1
b i 12 ii 2
c i −18 ii No x-intercepts
d i −5 ii −1, 5
e i 4 ii No x-intercepts
f i 4 ii −3 − , −3 + (approx. −5.24, −0.76)
6 a i (4, 2) ii Min iii Same width
iv 18 v No x-intercepts
vi
b i (3, −4) ii Min iii Same width
iv 5 v 1, 5
vi
c i (−1, 2) ii Min iii Same width
iv 3 v No x-intercepts
vi
d i (−2, 3) ii Min iii Same width
iv 7 v No x-intercepts
vi
e i (−5, −3) ii Min iii Same width
iv 22
v −5 − , −5 + (approx. −6.73, −3.27)
x
y
y = (x – 3)2 +2y = x2
0
(3, 2)1 – 2
1
4---
1
4---
x
y
y = (x + 2)2 + 4
y = x2
0
(–2, )14
13
–
3
2---
3
2---–
x
y
y = (x − 1)2 −
y = x2
0 (1, )74
−32 3
2
−
1
2---
3
4---
1
3---
2
3---
1
5---
1
2---
5 5
x
y
0
18
1 2 3 4
(4, 2)
y = (x – 4)2 + 2
x
y
0
5
1 2 3 4 5
(3, –4)
y = (x – 3)2 – 4
–4
x
y
0–1
(–1, 2)
y = (x + 1)2 + 2
32
1
x
y
0–1–2
(–2, 3)
y = (x + 2)2 + 3
3
7
21
3 3 5C
5C
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726 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
vi
f i (−1, 1) ii Min iii Same width
iv 2 v No x-intercepts
vi
g i (1, 2) ii Max iii Same width
iv 1
v 1 − , 1 + (approx. −0.41, 2.41)
vi
h i (−2, −3) ii Max iii Same width
iv −7 v No x-intercepts
vi
i i (−3, −2) ii Max iii Same width
iv −11 v No x-intercepts
vi
j i (1, 3) ii Min iii Narrower
iv 5 v No x-intercepts
vi
k i (−2, 1) ii Max iii Narrower
iv −11
v −2 − , −2 + (approx. −2.58, −1.42)
vi
7 a $1.90 b $1 c 3 pm
d $1.40 e
10 Quick Questions 11
2
3
4 5
6 (−2, −4)7 Minimum 8 09 −4, 0
10
x
y
0(–5, –3)
–5 – 3
–5 + 3
y = (x + 5)2 – 3
22
x
y
0
y = (x + 1)2 + 1
2
1
–1
(–1, 1)
2 2
x
y
0
1
–1 1
(1, 2)21 – 2 1 + 2
y = –(x – 1)2 + 2
xy
0
–3
–7
(–2, –3)
–2
y = –(x + 2)2 – 3
xy
0–2
–3
(–3, –2)
–2 –1
–11
y = –(x + 3)2 – 2
x
y
0
5
(1, 3)
y = 2(x – 1)2 + 3
1
3-------
1
3-------
x
y
0
–11
(–2, 1)
y = –3(x + 2)2 + 1
–2 – 1—3
–2 + 1—3
t (Hoursafter 12 pm.)
p ($)
0
1.0
3 5
1.4
1.9
x
y
0
y = x2
x
y
y = x2 + 4
0
4
x
y
0
y = (x – 1)2
1
1
x
y
0
y = x21–4
x
y
0
y = –x2
x
y
0
y = (x + 2)2 – 4
–2
–4
–2–4
(–2, –4)
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swers
➔
Maths Quest 10/ Final Pages / 19/1/06
Maths Quest challenge (page 175)1 There is more than one correct answer. One possible
answer is shown.
2 There is more than one correct answer. One possible answer is shown.
3 21%
Exercise 5D — Sketching parabolas of the form y = ax2 + bx + c1 a y = (x + 2)2 − 6, (−2, −6)
b y = (x + 6)2 − 40, (−6, −40)
c y = (x − 4)2 − 10, (4, −10)
d y = (x − 1)2 + 11, (1, 11)
e y = (x − 2)2 − 2, (2, −2)
f y = (x − 2)2 − 6, (2, −6)
g y = (x + )2 − , (− , − )
h y = (x + )2 − , (− , − )
i y = (x + )2 − , (− , − )
j y = 2(x + 1)2 + 6, (−1, 6)
k y = 3(x − 2)2 − 6, (2, −6)
l y = 5(x − 3)2 − 20, (3, −20)
2 a y = (x + 1)2 − 6, x-intercepts are −1 ± (≈ −3.4,
1.4)
b y = (x − )2 + 4 , no x-intercepts
c y = (x + )2 − 3 , x-intercepts are
(≈ −2.3, 1.3)
d y = (x − )2 − 5 , x-intercepts are
(≈ 0.2, 4.8)
e y = −(x + )2 + 7 , x-intercepts are
(≈ −5.2, 0.2)
f y = −(x − )2 − 2 , no x-intercepts
g y = −(x + )2 + 10 , x-intercepts are
(≈ −4.7, 1.7)
h y = −(x + 1)2 − 10, no x-intercepts
i y = 2(x + 1)2 − 20, x-intercepts are −1 ±
(≈ −4.2, 2.2)
3
2---
5
4---
3
2---
5
4---
1
2---
9
4---
1
2---
9
4---
7
2---
41
4------
7
2---
41
4------
6
x
y
0–1–1 – 6
–5(–1, –6)
y = x2 + 2x – 5
–6
–1 + 6
3
2---
3
4---
x
y
0
7
y = x2 – 3x + 7
(1 , 4 )1–2
3–4
1
2---
1
4---
−1 13±2
-----------------------
x
y
0
–3( , –3 )
y = x2 + x – 3
1–4
1–2
–
1–2
–
5
2---
1
4---
5 21±2
-------------------
x
y
0
y = x2 –5x + 1
1–2
1–2
1–4
(2 , –5 )
2
1
5
2---
1
4---
−5 29±2
-----------------------
x
y
01
y = –x2 –5x + 1
1–2
1–2
1–4
(–2 , 7 )1–4
7
–2
1
2---
3
4---
xy
0
–3
y = –x2 + x – 3
1–2
1–2
3–4
( , –2 )
3
2---
1
4---
−3 41±2
-----------------------
x
y
0
8
–1
y = –x2 – 3x + 8
1–2
(–1 , 10 )1–2
1–4
x
y
0
(–1, –10)
–1
y = –x2 – 2x – 11
–11
10
x
y
0–1
–18(–1, –20)
y = 2x2 + 4x – 18
5D
5D
5_61_03274_MQV10 - A 1-15_tb Page 727 Thursday, January 19, 2006 7:42 PM
728 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
j y = 3(x + )2 − 12 , x-intercepts are
(≈ −2.6, 1.6)
k y = −5(x − 1)2 − 30, no x-intercepts
l y = −7(x + )2 + 50 , x-intercepts are
(≈ −3.2, 2.2)
3 a b
c d
e f
g h
i
4 a b
c d
e f
g h
i
5 a B b C6 a iv b vii c vi d iii
e i f viii g ii h v 7 a b h = 0
c 2500 m d 25 s after launchinge 50 s
1
2---
3
4---
−1 17±2
-----------------------
x
y
0–
–12
1–2
(– , –12 )1–2
3–4
y = 3x2 + 3x – 12
xy0
–30
–35
1
(1, –30)
y = –5x2 + 10x – 35
1
2---
3
4---
−1 29±2
-----------------------
x
y
0
49
–
(– , 50 )
1–2
1–2
3–4
y = –7x2 – 7x + 49
x
y
0
–12(– , –12 )
–4
1–2
1–4
3
y = x2 + x – 12
x
y
0 4 8
y = x2 – 12x + 32
32
(6, –4)
x
y
0 9–1
y = x2 – 8x – 9
(4, –25)
x
y
0
–8
–4
(–3, 1)
y = –x2 – 6x – 8
–2
x
y
0
27
(–3, 36)
y = –x2 – 6x + 27–9 3 x
y
0
35(1, 36)
y = –x2 + 2x + 35–5 7
x
y
0
2( , 1 )1–2
3–4
y = x2 – x + 2
x
y
0
–8
–2
(–3, 1)
–4
y = –x2 – 6x – 8
x
y
0
–5
1
(–2, –9)
–5
y = x2 + 4x – 5
x
y
0–9
(4 , –45 )
–1–2
1–4
1–8
9
y = 2x2 – 17x – 9
x
y
0
14
(3 , –30 )5–6
1—12
72–3
y = 3x2 – 23x + 14
x
y
0
10
–5 –
(–2 , –26 )7—10
9—20
2–5
y = 5x2 + 27x + 10
x
y
0–
(– , –5 )7—12
1—24
3–2
1–3
y = 6x2 + 7x – 3
–3
x
y
0– 4
(1 , 10 )3–4
1–8
1–2
y = –2x2 + 7x + 4
4
x
y
0 7– 3–2
(2 , 36 )3–4
1–8
y = –2x2 + 11x + 21
21
x
y
0–
( , 7 )5—12
2–3
3–2
1—24
y = –6x2 + 5x + 6
6
x
y
0
(1 , 48 )31—36
7–2
2–9
25—72
y = –18x2 + 67x – 14–14
x
y
0
(1 , 1 )3–4
7–8
y = 2x2 – 7x + 8
8
t
h
0
2500
50
(25, 2500)
5_61_03274_MQV10 - A 1-15_tb Page 728 Thursday, January 19, 2006 7:42 PM
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Maths Quest 10/ Final Pages / 19/1/06
8 a A = xy m2 b 2x + y = 40 m
c y = (40 − 2x) m d A = 2x(20 − x) m2
e (10, 200) f
g Maximum area
is 200 m2, paddock is 10 m wide and 20 m long.
9 a b 2 s
c 0.15 s d 17.11 m
10 a A = 2x(150 − x) m2
b
c 11 250 m2, 75 m and 150 m
10 Quick Questions 21 (0, −5)
2 (−7, 0)
3 Increasing the value of a makes the graph narrower.
4 The graph becomes inverted.
5 y = (x + 1)2 − 36
6 y = 2(x − )2 − 4
Exercise 5E — Solving quadratic inequations using sketch graphs1 a x < −3 and x > 4 b 1 < x < 5
c −3 ≤ x ≤ 3 d x ≤ −3 and x ≥ −e − < x < 4 f x < 0 and x > 7
2 a −3 < x < −1 b x < 1 and x > 7
c x ≤ −3 and x ≥ 1 d −5 ≤ x ≤ 2e x < −2 and x > 1 f − < x < 1
3 a −5 < x < 4 b x < 5 and x > 7
c −1 ≤ x ≤ 2 d −2 < x < 1
e x ≤ 0 and x ≥ 4 f x < −4 and x > 4
4 a C b D c E
5 a −6t(t − 1.6) b 0, 1.6
c d 0.8 s
e 3.84 m f 1.6 s
g From 0 to 1.6 s
6 a
b 5 s
c 1 s
d 2 s
Summary1 plotting
2 axis of symmetry
3 turning point
4 vertically
5 horizontally
6 thinner, wider
7 upright, minimum, inverted, maximum
8 turning point form, (b, c)
9 x = 0
10 y = 0
11 halfway, divide, equation
12 sketch, above, below
x
y
0
200
20
(10, 200)
t
h
0–1.72 2.02
h = –4.9t2 + 1.5t + 1717
x
A
0
11 250
150
(75, 11 250)
1
2---
1
2---
7 8
x
y
0
3
(–1, 2)
y = (x + 1)2 +2
x
y
0
−7
y = −(x − 2)2 − 3
(2, −3)−3
2
9 10
x
y
0
–10
–5 2
y = x2 + 3x – 10
(–1 , –12 )1–2
1–4
x
y
0
–3
–3 1–2
y = 2x2 + 5x – 3
(–1 , –6 )1–4
1–8
1
2---
1
3---
1
2---
1
3---
1
2---
1
2---
t
h
0
3.84
1.6
h = –6t2 + 9.6t
t (s)
h (m)
0
31.25
2.5 5
h = –5t2 + 25t
t (s)
h (m)
0
–30
1 2 3 4 5
h = –5t2 + 25t – 30
t (s)
h (m)
0
–20
1 2 3 4 5
h = –5t2 + 25t – 20
5E
5E
5_61_03274_MQV10 - A 1-15_tb Page 729 Thursday, January 19, 2006 7:42 PM
730 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
Chapter review1 a x-intercepts are −1 and 3.
b x-intercepts are −7 and −1.
c x-intercepts are 1 and 3.
2 (−2, −3)
3 a i Graph upright, vertical translation 3 units down
ii (0, −3) iii
b i Graph upright, horizontal translation 2 units to the left
ii (−2, 0) iii
c i Graph inverted, dilation of factor 5 makes the graph narrower.
ii (0, 0) iii
d i Graph upright, dilation of factor 2 makes the graph narrower, horizontal translation 4 units to the left
ii (−4, 0) iii
e i Graph inverted, dilation of factor of makes the graph wider, horizontal translation 1 unit to the right, vertical translation of 3 units down
ii (1, −3) iii
f i Graph upright, dilation of factor makes the
graph narrower, horizontal translation 3 units
to the left, vertical translation of 1 unit up
ii (−3, 1) iii
4 a TP (3, 1), no x-intercepts, y-intercept is 10.
b TP (−1, −5), x-intercepts are −1 − and −1 + ,
y-intercept is −3.
c TP (4, 1), x-intercepts are 3 and 5, y-intercept is−15.
x
y
0–3
3
(1, –4)
y = x2 – 2x – 3
–4
–1
x
y
0–1
7
(–4, –9)
y = x2 + 8x + 7
–9
–9 –7
x
y
0
–3
y = –x2 + 4x – 3
2
1 3
(2, 1)
x
y
0
y = –(x + 2)2 – 3
–7
–3
–2(–2, –3)
x
y
0
y = x2 – 3
y = x2
–3
x
y
0–2
y = (x + 2)2
y = x2
x
y
0
y = –5 x2
y = x2
x
y
0–4
y = 2(x + 4)2
y = x2
1
2---
x
y
0(1, –3)
y = x2
y = (x – 1)2 – 3–
1 – 2
5
2---
x
y
0(–3, 1)
y = x2y = (x + 3)2 + 15 – 2
x
y
0
10
y = (x – 3)2 + 1
(3, 1)
10
2----------
10
2----------
x
y
0
–3
–3 1
y = 2(x + 1)2 – 5
(–1, –5)
x
y
0
–15
1
3 4 5
(4, 1)
y = –(x – 4)2 + 1
5_61_03274_MQV10 - A 1-15_tb Page 730 Thursday, January 19, 2006 7:42 PM
A n s w e r s 731
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Maths Quest 10/ Final Pages / 19/1/06
d TP (5, 3), x-intercepts are and ,
y-intercept is −9 .
5 a (4, −15) b (−2, −9)
c d
6 a b
c d
7 a b
c
8 a i −3 ii 1 − and 1 + iii (1, −5)b
9 a b 4 m
c 2 s d 4 s
10 a b 25 m
c 2 m d 7 m
11 a x < −6 and x > 1 b −4 ≤ x ≤ 1c 3 < x < 4
12 a b 4 s c 2 s
d The ball is never above a height of 20 m.
Chapter 6 VariationAre you ready?1 a 4 b −5 c 0 d − or −5.2
2 a 5.0 b 3.83 c 428.672 d 58
3 a 2 b − c 1 d −2
4 a 4 b 2 c 768 d 2.25
5 a b c d
6 a 2 b c 0.2 d 17.5
Exercise 6A — Direct variation
1
2 a b 22 500
5 6– 5 6+1
2---
x
y
0
–9
(5, 3)
5– 6 5+ 6
y = (x – 5)2 + 3–1 – 2
1 – 2
11
2---– 3
4---–,⎝ ⎠
⎛ ⎞ 11
4---– 61
8---–,⎝ ⎠
⎛ ⎞
x
y
0 4 + 154 – 15
y = x2 – 8x + 1
(4, –15)
1x
y
0
y = x2 + 4x – 5
(–2, –9)
–5
–5
1
x
y
0
y = 3x2 + 9x + 6
6
–1(–1 , – )
–21–2
3–4
x
y
0
–3
–3
(–1 , –6 )
y = 2x2 + 5x – 3
1 – 4
1 – 2
1 – 8
x
y
0
y = x2 + 6x + 8
8
–2–4(–3, –1)
–3
–1
x
y
0
y = –x2 + 6x – 5
–5
1 5
4(3, 4)
x
y
0
y = –x2 – 2x + 15
–5 3
15
(–1, 16)
10
2----------
10
2----------
x
y
0
y = 2x2 – 4x – 3
–1 31
–5(1, –5)
t
h
0
h = 4t – t2
4
4
s 10 20 30 40 50 60 70 80 90 100
P 60 120 180 240 300 360 420 480 540 600
x
h
0
21
25
2 7
h = –x2 + 4x + 21
1
2---
t
h
0
h = –5t2 + 20t
4
20
26
5------
1
2---
1
2---
1
3---
1
3---
1
2---
2
3---
5
6---
5
12------
3
4---
s
P
0
100
200
300
400
500
0 20 40 60 80 100
a
F
0
5 000
10 000
15 000
20 000
25 000
0 5 10 15 20 25 6A
6A
5_61_03274_MQV10 - A 1-15_tb Page 731 Thursday, January 19, 2006 7:42 PM
732 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
3 a b 24 c a = 2b
4 a b $520 c y = 0.8x
5 a Distance travelled is directly proportional to time travelled.When time = 0, distance = 0, therefore direct variation.
b 80 minutes c d = 2t
6 a d = 3000t b 16.67 minutes
7 a t = n
b
c 45 questions
8 a D b B
9 0.67 10 0.5
11 1.53
12 20
13 20
14 0.75
15 a 0.5 b 7 kg
16 2.4 hours
17 $17.77
18 30 000
19 a 6000 lines b c 36 000 lines
20 a E b B
10 Quick Questions 11
2
3 c = 0.15A4 c = $48
5 d = 75t6 10 hrs 40 min
7
8 4
9 2 hrs 37 min
10 $18.67
Exercise 6B — Direct variation and ratio (rate)1 a i 148.8 cm ii 68.5 kg b 0.0248
2 a h = w b 85.7 cm c 70 cm
3 20 teeth
4 a i 12 (graduates) ii 8 (professionals)
b 16 professionals
5 a 1818 m2 b $35 750
6 a 624 km b 38.5 hours c 47.5 hours
7 a 570.15 L b Nissan Pulsar
8 a 81.9 L (4 WD), 38.4 L (small car)
b 2131.6 km
9 a 48 L/100 km b 2917 km
Exercise 6C — Partial variation1 y = 2x + 2
2 y = 25x + 150
3 a y = 4x + 2 b 46.8 c 3.475
4 a b = 0.25a + 38 b 51.75 c 248
5 a c = 0.9d + 2.20 b $13.36 c 22 km
6 a c = 40t + 300 b $480 c 10 hours
7 a c = 4m + 100 b $612
s 1 2 3 4 5
P 9 18 27 36 45
b
a
0
5
10
15
20
25
0 5 10 15 20 25
Bricks
y
x
Cost
($)
0
200
400
600
800
1000
0 200 400 600 800 1000
Time
Dis
tance
0
10
20
30
40
50
0 5 10 15 20 25
8
3---
Questions
Tim
e
0
20
40
60
80
100
0 10 20 30 40 50
1
60------
A 100 200 300 400 500
c 15 30 45 60 75
s
P
0
9
18
27
36
45
0 1 2
(2, 18)
(1, 9)
(3, 27)
(4, 36)
(5, 45)
3 4 5
Area (m2)
Cost
($)
0
15
30
45
60
75
0 100 200
(100, 15)
(200, 30)
(300, 45)
(400, 60)
(500, 75)
300 400 500
2
3---
7
12------
1
2---
10
7------
10
22
0 104
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Maths Quest 10/ Final Pages / 19/1/06
8 a 2.5 b c = 2.5s + 50 c $125
9 a B b D
Exercise 6D — Inverse variation
1 a k = 1000, y = b
2 a k = 48, p = b
3 a k = 42, y = b
4 a 4000 b a =
c 20 m/s2 d 4 m/s2
5 a 500 b 2500 pencils
c 1000 pencils
6 a 337.5 b 3.97 hours = 3 h 58 min
c 96.4 km/hr d 337.5 km
7 a 17 500 b $233
c 70 people d $17 500
8 a 200 b 1 amp c 13.3 ohms
9 a 200 b 2 minutes c 600 mHz
10 a 10 500 b 210 days c 105 workers
11 a C b E
Maths Quest challenge (page 218)1 1 2 6 3 12 4 8 5 0
Exercise 6E — Other forms of direct and inverse variation1 a b E = x2
c
d e 9 units
2 a 15 b d = 15t2
c d 1500 m
3 a 11.25 b c = 11.25h2
c $115.20 d 4.22 m
4 a 4 096 000 000 b F =
c 25 units d 74.8 units
5 a 20 b b =
c 0.8 units d 4.5 m
6 a 10−12 b F = c 10−2
7 a 1 000 000 b C =
c $1 d C =
e Inverse variation
8 a y = 0.5x2 b y =
c y = 1.022x2 d y =
9 a 1250 b 2
c 2555 d 0.003 312
10 a 16 b y = 16x3
c 128 d 628.9
11 a E b A
10 Quick Questions 2
Exercise 6F — Identifying the type of variation1 a Partial, k = 4 b Inverse, k = 20
c Square, k = 2 d Direct, k = 2
e None f Inverse, k = 5000
g Inverse square, k = 32 h Partial, k = 8
i Direct, k = 3 j Partial, k = 100
Exercise 6G — Joint variation1 a 1.0475 b V = 1.0475hr2 c 100.56 cm3
2 a 0.333 b V = 0.333ah c 12 000 cm3
3 a 0.2 b Number =
c 10 000
4 170.7 days
5 250 units
6 432 m
7 0.32 units
8 a C b B
x 0 4 8 12 16 20
E 0 1 4 9 16 25
1000
x------------
x
y
0
200
400
600
800
1000
0 5 10 15 20 25
48
q------
q
p
0
10
20
30
40
50
0 2 4 6 8 10
42
x------
x
y
0
10
20
30
40
50
0 2 4 6 8 10
4000
m------------
1
16------
1
16------
Speed
Ener
gy
0
5
10
15
20
25
0 5 10 15 20 25
Time
Dis
tance
0
1000
2000
3000
4000
5000
0 5 10 15 20 25
4 096 000 000
d2---------------------------------
20
d2------
10 12–
d2------------
1 000 000
n2-----------------------
1 000 000
n-----------------------
5000
x2------------
82.8
x2----------
1 130 m2 2 C = 50 + 15h3 $102.50 4 6 hours5 64 Mb 6 2 hours 48 min7 1 hour 20 min 8 $37.509 125 m 10 27 m
0.2 Budget×Price
-------------------------------
6B
6G
5_61_03274_MQV10 - A 1-15_tb Page 733 Thursday, January 19, 2006 7:42 PM
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swers
Maths Quest 10/ Final Pages / 19/1/06
Maths Quest challenge (page 231)1 60 2 12 minutes
3 30 breakfasts
Summary1 direct 2 partial
3 inverse, inverse square 4 variation symbol
5 gradient 6 direct square
7 constant 8 joint
9 origin 10 k
Chapter review 1 a
b 20.8 c y = 4x2 a
b 105 m c d = 2.1r3 1.54 a 150 b $9305 12 teeth6 a 300 km b 3.5 hours7 a C = 50t + 400 b $450 c 9.5 hours8 a C = 40n + 1000 b $3400 c 75 people
9 a k = 44 b y =
c d 220
10 a 360 b t = c 4 hours
11 a 26 b L = 26w2 c 2.4 m12 31.25 km13 a Direct b Inverse square c Partial
14 a 96 b y = c 24
Chapter 7 Simultaneous equationsAre you ready?1 a y = −3 b x = 1 c x = 1
2 a i y = −3 ii x = 2
b i y = 5 ii x = 1
c i y = 3 ii x = −7
3 a x-intercept = 3
y-intercept = 2
b x-intercept = −3
y-intercept = 9
c x-intercept = −3
y-intercept = 4
4 a or 1 b or 1 c or 2
5 a False b False c True
Exercise 7A — Graphical solution of simultaneous equations 1 a (2, 1) b (1, 1) c (0, 4) d (2, −1)
e (−2, −4) f (−0.5, 1.5)
2 a No b Yes c Yes d No
e Yes f No g No h Yes
i No j Yes
3 a (3, 2) b (4, 3) c (−3, 4) d (−2, 2)
e (2, 0) f (3, 0) g (−2, 4) h (3, 8)
i (− , 1 ) j (2, 5) k (5, 3) l (2, )
4 a (3, 5) b (−2, 4) c (5, 7) d (−2, −5)
e (5, 1) f (6, −2) g (−4, 7) h (3, 4)
5 a No solution b (2, −1) c No solution
d (1, 9) e (3, 1) f No solution
g No solution h (2, 1)
6 a b
c 2 d (−1, 1) and (2, 4)
7 a (−2, 4) and (3, 9)
b (−2, 5) c No solution
x
y
0
10
20
30
40
50
0 5 10 15 20 25
r
d
63
0 30
44
x------
x
y
0
10
20
30
40
50
0 1 2 3 4 5
360
s---------
96x2
w2-----------
1
3---
1
4---
1
2---
0 1
1
2
2 3
2x +3y =
6
x
y
2
4
6
7
8
9
1
3
5
021 3–2 –1–3
x
y=
3x
+9
y
0–1
1
2
3
4
–2–3
4x –
3y +
12 =
0
x
y
4
3---
1
3---
3
2---
1
2---
5
2---
1
2---
1
2---
1
2---
2
3---
x
y
0
y = x2
x
y
0
y = x2
y = x + 2
2(–1, 1)
(2, 4)
5_61_03274_MQV10 - A 1-15_tb Page 734 Thursday, January 19, 2006 7:42 PM
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Maths Quest 10/ Final Pages / 19/1/06
Exercise 7B — Algebraic solutions of simultaneous equations — substitution method1 a (2, 3) b (2, −1) c (3, −2) d (7, 6)
e (3, 6) f (2, 1) g (−1, −2) h (2, −2)
i (−1, −2) j (6, −2) k (3, 1 ) l (−3, −5)
2 a (−6, −23) b (5, 23) c (2, −6)
d e (1, −7) f (− , −4)
g h i (−3, −1.5)
j (−4, −2.8) k l (1, −1)
10 Quick Questions 11 Not a solution 2 Is a solution
3 (4, 8) 4 (5, 1)
5
Exercise 7C — Algebraic solutions of simultaneous equations — elimination method1 a (3, 1) b (−2, 3) c (−2, 6)
2 a (5, −1) b (2, 3) c (−3, 1)
3 a (6, 3) b (−3, −7) c (2, −5) d (−3, 5)
e (−5, −8) f (2, −2) g (1 , 3 ) h (2, 1 )
i (1, 1)
4 a (2, 1) b (3, 5) c (3, 3)
d (1, 3) e (2, 4) f (5, 2)
g (4, 2) h (−3, 4) i (−3, −1 )
j (−6, −5) k (−3, 5) l (2, 1.8)
5 a (5, 2) b (3, 3) c (−2, 6)
d (5, −1) e (7, 0) f (3, 1)
g (6, 3) h (2, −2) i (1, 3)
j (−1.5, −3) k (−8, 18) l (−3, 5)
6 a (1, 3) b (4, 0) c (−3, 5)
d (4, 3) e (8, 5) f
Maths Quest challenge (page 251)1 Rollercoaster ride $6, Ferris wheel ride $4, Gravitron
ride $8
2 89 246
Exercise 7D — Problem solving using simultaneous equations1 Maths mark = 97, English mark = 66
2 8 and 3
3 9 and 7
4 6 and 5
5 Length = 12 m and width = 8 m
6 18 nuts, 12 bolts
7 Lemons cost 55c and oranges 25c
8 Length 60 m and width 20 m
9 Eight 20 cent coins and three 50 cent coins
10 Twelve $1 coins and nine $2 coins
11 Paddlepops costs $1.20 and a Magnum costs $2.10.
12 Cost of the Golden rough = 35c and cost of the Redskin = 25c
13 Fixed costs = $87, cost per person = $23.50
14 PE mark is 83 and science mark is 71
15 Mozzarella costs $6.20, Swiss cheese costs $5.80
16 x = 3 and y = 4
17 Fixed costs = $60, cost per person = $25
18 $4 each for CDs and $24 each for zip disks
10 Quick Questions 2
Maths Quest challenge (page 262)1 Length 11 m, width 8 m
2 12 minutes
3 24 cm2 ≈ 41.6 cm2
Exercise 7E — Solving a quadratic equation and a linear equation simultaneously1 (−4, 1) and (1, 6)
2 a (−4, 12) and (−3, 10)
b (−2, −5) and (6, 35)
c (3, −2) and (5, 0)
3 (2, 4)
4 Δ = −8
5 a ( −2, 4) and (5, 18)
b (−2, −9) and (−1, −8)
c (4, 10)
d (−7, 18) and (−1, 6)
e (1, 1) and (3, 9)
f (1, 4) and (10, 22)
6 (−3, 1) and (−2, 1)
7 a (1, −5)
b No, but the straight line is vertical and intersects at one point only.
8 (−2, 0) and (2, 0)
Exercise 7F — Solving simultaneous inequations1 a True b False c False d True e True
f False g True h False i False j False
1
2---
3
2---
15
2------⎠
⎞–,⎝⎛ 1
2---
3
2---
1
2---⎠
⎞–,–⎝⎛ 1
5---
4
5---⎠
⎞,–⎝⎛
4
5---
4
5---⎠
⎞,–⎝⎛
6 (1, −3)7 (3, −1)
8 ( , 1 )
9 (4, 24)
10 , −
1
2---
1
2---
9
10------⎝
⎛ 4
5---⎠
⎞0
3
–3
–6
–9
6
9
y
x21–1–2–3 3
1
2---
1
2---
4
5---
1
2---
1
3---
1
3---–,⎝ ⎠
⎛ ⎞
1 (−5, 3) 2 (−1, −4)
3 (1 , −4 ) 4 (−7, −1)1
9---
1
9---
5 (− , −1) 6 19 and −61
3---
7 48 m × 160 m
8 36 five cent coins and 68 ten cent coins
9 Tomatoes cost 25c and a lettuce costs 55c.
10 Entry fee is $18.50 for adults and $8.50 for children.
3
7A
7F
5_61_03274_MQV10 - A 1-15_tb Page 735 Thursday, January 19, 2006 7:42 PM
736 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
2 Note: The shaded region is the region not required.
a x + y > 3
b x + 2y ≤ 6
c 3x − 2y > 12 d 4x + y ≥ −8
e y ≥ x + 4 f y < 3 − 3x
g y − 3x < 9
h 2x + y ≥ 8
3 a A b C c B d E
4 Note: The shaded region is the region not required.
a
b
c
d
e
f
2
6
4
2 4 60
–2–2–4–6
–4
y
x
21
2 4 60
–2–1–2–4
3
y
x
21
2 40
–2–1–2–4
–4–3
–6–7
–5
y
x
21
20
–2–1–2–4
–4–3
–6–7–8–9
–5
y
x
2
2 40
–2
–2–4–6
4
6
–4
y
x
1
1 2 30
–1
–1
2
3
–2
y
x
2
1 20–1–2–3
1
43
56789
10
y
x
2
1 20–1
1
43
56789
10
y
x3 4
2
6
4
1 2 30
–2–1
–4
–6
y
x4 5
x + y < 3
2x − y ≥ 4
2
6
4
2 4 60
–2
–2
–4
–6
y
x
8 10x + 5y ≤ 10
3x + 2y > 12
1
3
2
1 2 30
–1
–1
–2
–3
y
x4 5
y < 3 − x
2y > x − 2
2
6
4
1 2 30
–2
–1–2–3
–4
–6
y
x
y < 4 − 2x
y > 2x + 4
2
6
4
2 4 60
–2
–2–4–6
–4
–6
y
x
x + y > 4
y − 2x ≤ 5
42
6
101214161820
8
1 2 3 4 5 60
–4–2
y
x
52–3
3x + y > 17
y < 8
5_61_03274_MQV10 - A 1-15_tb Page 736 Thursday, January 19, 2006 7:42 PM
A n s w e r s 737
an
swers
➔
Maths Quest 10/ Final Pages / 19/1/06
g
h
i
j
k
l
m
n
5 a a ≥ 18, where a is the age of a person.
b w ≤ 2, where w is the number of litres of water.
c x + y ≤ 800 where x is the number of reserved tickets and y is the number of general admission tickets.
d 2c + 3p ≤ 20, where c is the number of Christmas cards and p is the number of sheets of wrapping paper.
6 a r + x ≤ 2000 b r ≤ 600
c r ≥ 0, x ≥ 0. Amount of money cannot be negative.
d
e Answers will vary.
7 a 100a + 75b ≥ 450 b 50a + 75b ≥ 300
c
d Answers will vary.
Summary1 accurate
2 linear
3 gradient
4 parallel
5 elimination
6 define, number, simultaneously, original
7 Cartesian
5
15
10
50
–5
–10
–15
y
x10
3x + y > 15
x + 2y ≥ 10
21
3
56
7
4
1 2 3 4 50
–2–1
–4–5
–3
y
x
x < 5
y > 2x − 3
42
6
81012141618
2 4 6 8 10 12 14 160–2–3–6
–4–6
y
x
3y − 2x < 6
y ≥ 2x − 2
21
3
56
4
2 4 60
–2–1–2–4–6
–4–3
–6
–5
y
x
2x + 3y ≤ 6
y − x > 4
21
3
56
4
1 2 30
–2–1–1–2
–4–3
–6
–5
y
y < 2x
y + 2x > 3
x1–2
1
42
6
108
2 4 60
–4–2–2–4–6
–8–6
–10
y
x
y − 2x ≥ 9
x + y ≤ 4
42
6
108
2 4 60
—4—2—2
—8—6
—10
y
x8
2x − 3y ≥ 18
x + y > 7
21
3
56
4
1 2 30
–2–1–1–2
–4–3
–6–5
y
x4
y ≥ 2x
y > 4
0
2000
x
r2000600
Regionrequired
0
6
2
4
b
a
Regionrequired
654321
7F
7F
5_61_03274_MQV10 - A 1-15_tb Page 737 Thursday, January 19, 2006 7:42 PM
738 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
8 two, intersect, tangent9 graphical
10 intersection11 unwanted
Chapter review1 a (3, 1)
b (2, 3)2 D3 a No
b Yes4 a (−2, 1) b (0, −2) c (5, 2)5 a (2, 7) b (−5, −3) c (−2, 2)
d e (−14, −53) f ( , −7)
6 a (5, 2) b (−2, 3) c (−3, −1)d (1, 3) e (2, −2) f (4, 2)
7 a (0, 3) b (−3, −3) c (2, 1)8 a Numbers are 9 and 14.
b Length = 11 metres, width = 6 metresc Chupa-chups cost 45c and Whizz fizzes cost 55c
9 Milk $1.75, bread $2.3510 a (−8, 22) and (2, 2)
b (5, 10)c No solution
11 a
b
c
Chapter 8 Exponential functionsAre you ready?1 a b 1 c 1
2 a − b 1 c −
3 a b c 1
4 a 0.25 b 0.025 c 1.0255 a $22.50 b $21 c $26
Exercise 8A — Index laws1 a a7 b a6 c b8 d a4b7
e m5n13 f a5b7c3 g m6n4p5 h 6a2bi 10a4b9 j 36m8n7 k 12x6y6 l 4x8y6
2 a a b a5 c b3 d a4
e 3b4 f 4m5 g m3n h y2
i x3y j 7b3 k m2p2 l xy2
3 a 1 b 1 c 1 d 3e 4 f −3 g 3 h −7i 4
4 a a6 b 16a20 c m8 d n8
e a6b3 f 9a6b4 g 16m12n20 h m6n3
i j k l
5 a D b D6 a 64 b 72 c 625 d 48
e 1600 f g 20 h 1
i 47 a x3yz b ab c manb
d e n3 − pm2 − q f amp + np
Exercise 8B — Negative indices
1 a b c d
e f g h
i j k l
2 a b c d
e f g h
i j k l
m n o
3 a b c d
e f g 48 h
i = 1 j 4 k 125 l
4 a 0.001 371 742 b 0.000 048 225c 0.000 059 499d 256e 7.491 540 923f 5 419 228.099
5 a B b D c C d E
7
3---–
7
3---,⎝ ⎠
⎛ ⎞ 5
2---
2
6
4
2 4 60
–2
–2–4–6
–4
–6
y
x
y ≥ 3
y ≤ x + 4
21
3
56
4
2 4 60
–2–1–2–4–6
7
–4–3
y
x
2y − 3x ≥ 12
y + 3x > 0
21
3
56
4
4 86 12 14100
–2–1–2–4–6–8
789
10
y
x162
5x + y < 10
x + 2y < 11
5
6---
1
12------
7
40------
1
4---
1
6---
3
4---
3
8---
1
2---
1
2---
4
3---
1
2---
3
4---
5
4---
1
2---
1
81------
4
9---
27
64------
a4
b6-----
625m12
n8------------------
343x3
8y15--------------
81a4
625b12-----------------
27
125---------
a2x
b3x-------
1
x5-----
1
y4-----
2
a9-----
4
5a3--------
3x2
y3--------
1
4m3n4---------------
6a3
bc5-------- a6
2a4
3-------- 2ab2 7b3
2a4--------
2m3a2
3b4n5---------------
1
a2b3-----------
6
x6y--------
3
n8-----
4
a2b5-----------
2y3x------
5y6x3--------
3
m2n2------------
4y12
x5----------
1
3m3n3---------------
1
32a15m20-----------------------
4q8
p14--------
3
a8b12-------------
27q9
8 p6-----------
b6
4a8--------
1
8a6b6--------------
1
8---
1
36------
1
81------
8
9---
1
16------
5
36------
32
27------
27
25------
2
25------
3
4---
5_61_03274_MQV10 - A 1-15_tb Page 738 Thursday, January 19, 2006 7:42 PM
A n s w e r s 739
an
swers
➔
Maths Quest 10/ Final Pages / 19/1/06
10 Quick Questions 1
Maths Quest challenge (page 287)1 a 100 × 10 000, 10 × 100 000
b 64 × 15 625 as 106 = (2 × 5)6 = 26 × 56
2 20
3 320 is the larger number since 230 = (23)10 = 810 and 320 = (32)10 = 910
Exercise 8C — Fractional indices1 a 4 b 5 c 9 d 2
e 4 f 3 g 2 h 125
i 216 j 10 000 000 k 8 l 9
2 a 1.44 b 2.24 c 1.48 d 1.26
e 2.54 f 0.66 g 0.54 h 0.81
i 0.86
3 a b c d
e f g h
i
4 a b c d
e f
5 a b c d
e f g h
i
6 a b c d
e f
7 a b c d
e f g h
i
8 a b c
d e f
g h i
9 a E b B
10 a b c d
e f g h
i
Exercise 8D — Further use of index laws
1 a b c d
e f g h
i j
2 a b c d
e f g h
i
3 a b c d
e f g h
i
4 a b c d
e f g h
i
5 a b c d
e f g h
6 a b 1
7 1
8 a b y = 4
9 E
1 240a2b3c8 3d3e2 f 4
7--------------------
3 4729g21h3 25 j6
k6-----------
5 9 6 1
4---
7 83
n4-----
q6 p4---------
9 10sr8----
216t6
u21-------------
4
4
5---
2
1
2---
a5
6---
x23
20------
10m8
15------
2b5
7---
4y20
9------
– 0.02a9
8---
5x7
2---
ab3
2---
x4
5---
y5
9---
6a8
5---
b17
15------
2m19
28------
n2
5---
x19
6------
y5
6---
z5
6---
8a2
5---
b8
9---
c
3
1
6---
5
5
12------
12
1
2---
a3
7---
x5
4---
m11
45------ 1
2---x
3
20------ 1
3---n
2
3---
5
4---b
7
20------
x5
3---
y7
5---
a7
45------
b4
15------ 1
3---m
3
8---
n11
56------
2x2
15------
y3
4---
1
4---a
11
20------
b7
20------ 1
7--- p
5
24------
q1
12------
2
9
20------
5
1
6---
7
6
5---
a3
10------
m1
6---
2
1
3---
b1
6---
4 p2
5---
xmp----
3
bc---
aac---
a1
4---
b1
6---
a3b3
4---
x6
5---
y7
4---
3
1
3---
a1
9---
b1
5---
c1
4---
5x1
4---
y1
3---
z1
5--- a
1
2---
b2
3---
-----
m8
5---
n7
4---
------b
2
5---
c8
27------
-------2
1
2---
x7
2---
y3
8---
-----------
a4 b3 m4 4x2
2y3 2x2y3 3m3n5 2 pq2
6a2b6
54a10b9 48a5b16 2n13
m9----------- 500 p8q18
36a20b10 15b5
c26----------- 12x
7
8---
y11
15------
8m15
4------
n15
4------
6
p7
12------
-------- 8 p7
45------
q5
18------
5
8a7--------
x4y6--------
27
128m29n26--------------------------
64y36
x24-------------
24a24b7 27h12
8g6-------------- p
35
3------
q1
2--- 625
81b20c28---------------------
x
5
3---
y
1
8---
z
3
2---
3a2
2-------- 8n2 m2n4
3------------
4x5
3y8--------
36x6
y-----------
y2
x4-----
b7
3a4--------
75q5
2 p11-----------
x17
10------
y7
10------
2
5a4b7--------------
4a3b3
15--------------
n9
4m9----------
4m5
9n15-----------
4
81x2y14------------------- 48x11y6 3 p4
5q9---------
2b1
12------
3a17
24------
-----------
4x1
12------
3y21
20------
-----------
5
2a13-----------
56a11b6
81-------------------
1024b2
81a-----------------
25
128x23y4----------------------
4y36
27x16------------- 6m19n19 16m
11
12------
n3
------------------4b
11
2------
3
1
2---
c7
30------
-------------
125
8---------
5y 1–
8A
8D
5_61_03274_MQV10 - A 1-15_tb Page 739 Thursday, January 19, 2006 7:42 PM
740 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
Exercise 8E — Exponential functions and their graphs
1
2 a b
c
3
4 Increasing the value of a increases the steepness of the graph where x is positive and flattens the graph where x is negative.
5 a b 2 c y = 0
6
7 The coefficient, k, affects the steepness of the graph: the larger the value of k, the steeper the graph.
8
9
10 The negative index reflects the graph in the y-axis.
11 a
b = (2−1)x = 2−x
12
13 a b 10 c y = 0
x −4 −3 −2 −1 0 1 2 3 4
y 1 10 100 1000 10 000
x
y
0
y = 10x
4321–1–2–3–4
1000
1
10 000
1
10 000----------------
1
1000------------
1
100---------
1
10------
x
y
0
y = 4x
321–1–2–3–4 4
20
40
60
80
100
x
y
0
y = 5x
321–1–2–3–4 4
20
40
60
80
100
x
y
0
y = 6x
321–1–2–3–4 4
20
40
60
80
100
x
y
0
y = 4x y = 3xy = 2x
321–1–2–3–4 4
20
40
60
80
100
x
y
0
y = 2 × 3x
321–1–2–3–4 4
10
20
30
40
50
60
x
y
0
y = 3 × 2x
y = 2x
y = × 2x1–5
321–1–2–3
10
8
6
4
2
x −3 −2 −1 0 1 2 3
2x −0.125 −0.25 −0.5 1 2 4 8
3 × 2x −0.375 −0.75 −1.5 3 6 12 24
× 2x −0.025 −0.05 −0.1 0.2 0.4 0.8 1.6
x −3 −2 −1 0 1.0 2.00 3.000
y −8 −4 −2 1 0.5 0.25 0.125
1
5---
y-intercept at (0,1).Equation of horizontal asymptote is y = 0.
x
y
0
y = 2–x
321–1–2–3
10
8
6
4
2
x
y
0
y = 3–x y = 3x10
8
6
4
2
x
y
0
y = ( )x1–2
321–1–2–3
10
8
6
4
2
1
2---⎝ ⎠
⎛ ⎞ x
x
y
0
y = (1.8)x
y = (1.5)x
y = (1.2)x
1
x
y
0
y = 10 × (1.3)x
10
5_61_03274_MQV10 - A 1-15_tb Page 740 Thursday, January 19, 2006 7:42 PM
A n s w e r s 741
an
swers
➔
Maths Quest 10/ Final Pages / 19/1/06
14 a
b c $1331
15 a
b
c As n increases, the value of the car decreases.
d $17 748
10 Quick Questions 2
1 2 3 4 5
10
Maths Quest challenge (page 304)1 15 minutes
2
3 a = 2, b = 4 or a = 4, b = 2
4 b = 2 and c = 2. (a can take any integer value)
Exercise 8F — Modelling exponential growth and decay1 a 2000 b 486 000
c d 1.26 h
2 a $5000 b $7717
c d 10 years
3 a C b D
4 a $883.50 b $821.66
c V = 950 × (0.97)n
d $659.15
5 a 102 b 86.7
c A = 120 × (0.85)t d 83.927
e f Approximately 210 years
6 a i 96.04% ii 90.39%
b C = 100(0.98)w c
d 8 washings
7 a 118 (million) b a = 1.02; P = 118 × (1.02)n
c
Calculated population is less accurate after 10 years.
d 288 (million)
n 0 1 2 3 4 5 6
A 1000 1100 1210 1331 1464.10 1610.51 1771.56
n 0 1 2 3 4 5
V 40 000 34 000 28 900 24 565 20 880 17 748
n
A
0
A = 1000 × (1.1)n
1000
1 2 3 4 5 6
n
V
0
V = 40 000 × (0.85)n
40 000
15 000
1 2 3 4 5
b13
2------
5a2-------- 12e2d
3
4--- 1
108 j9k7
3---
--------------------h2
2i1
2---
------- 1
3---
6 7
x
y
0
y = 5x
3 421–1–2–3–4
20
40
60
80
120
140
100
x
y
0
y = 10x
321–1–2–3
50
100
150
200
250
300
350
400
8 9
x
y
0
y = 5 × 2x
321–1–2–3
105
20
30
40
50
x
y
0
y = × 3x1–4
1–4
321–1–2–3
1
2
3
4
x
y
0
y = 2–x
321–1–2–3
2
1
4
6
8
10
Year 1970 1975 1980 1985 1990
Population 118 130 144 159 175
x
N
0 3 421 5
2000
4000
6000
8000
10 000
12 000 N = 2000 × 3x
A
2000
4000
6000
8000
10 000
12 000
14 000A = 5000 × (1.075)n
n0 2 4 6 8 10
t
A
0
20406080
100120140
A = 120 × (0.85)t
w
C
0
20
40
60
80
100 C = 100 × (0.98)w
5 10 15 20
8E
8F
5_61_03274_MQV10 - A 1-15_tb Page 741 Thursday, January 19, 2006 7:42 PM
742 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
8 a 32 b 0.98 c T = 32 × (0.98)t
d 26.1, 21.4, 17.5, 14.3Values are close except for t = 40.
Summary
Chapter review
1 a b c
d
2 a 16 b
3 a b c
4 a 8 b c 0
5 a b c
6 a 1 b 4
7 a b
8 a b c
9 a 46 b −
10
11 a
b
12 13
14 a
15 a
16 a
17 a 3.5 g b 2 g
c d 17 days
18 a A = A0 × (1.065)n b $7769.93
c d 11
Chapter 9 MeasurementAre you ready?1 a 3.6 × 106 mm2 b 2 × 10–6 km2
c 5.2 × 10–4 m2
2 a 24 m2 b 30 cm2 c 4.9 cm2
3 a 150 cm2 b 232 cm2 c 1.22 m2
4 a 3.4 × 106 cm3 b 2.5 × 10–4 m3
c 6.5 × 103 mm3
5 a 125 cm3 b 160 cm3 c 0.03 m3
x −3 −2 −1 0 1 2 3
y 0.008 0.04 0.2 1 5 25 125
1 added 2 subtracted3 multiply 4 zero5 factor 6 reciprocal7 surds 8 power9 exponential 10 steeper
11 reflected 12 increases, larger13 decreases, smaller 14 asymptote15 a initial amount b growth c decay
9x10y10 13ab3c2
6--------------------
1000m15n6
27---------------------------
16 p28
81q12--------------
3
2---–
8
a11b2-------------
y2
5x17----------
m12
16n8-----------
3
2---
30a41
20------
b33
20------ 4
x1
20------
y2
9---
-------------2a
1
6---
b3
2---
---------
−2a3 2a2b1
2---
+ 6xy2
2a13
5b2-----------
9y4
32x15------------- 2
4
3---
m
1
18------
1
36------
x
y
0
y = 5x
321–1–2–3–4 4
20
40
60
80
100
120
140
160
x
y
0
y = 10 × 3x
321–1–2–3
100150200250300350400450
50
x
y
0
y = 10–x
321–1–2–3
20
40
60
80
100
120
140
x
y
0 321–1–2–3
2
4
6
8
10
y = (1.5)x
y = (1.2)x
b Increasing the value of a makes the graph steeper for positive x-values and flatter for negative x-values.
x
y
0 321–1–2–3
4
12
8
16
20
24
28
32
36
y = 5 × 3x
y = 2 × 3x
1–2
y = × 3x
b Increasing the value of k makes the graph steeper.
x
y = (2.5)xy = (2.5)–x y
0 321–1–2–3
5
10
15
20
25
30
35
40
45
b Changing the sign of the index reflects the graph in the y-axis.
t
m
0 2 4 6 8 10 12 14 16
1
2
3
4 m = 3.5 × 2–0.2t
n
A
0 2 4 6 8 10
2000
4000
6000
8000
10 000
12 000
14 000 A = 5000 × (1.065)n
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Exercise 9A — Errors in calculations1 a 44.5 m and 45.5 m b 18.45 cm and 18.55 cm
c 475 km and 485 km d 29.355 cm and 29.365 cm2 a 60 ± 4
b Largest 64, smallest 56. The sum is between 56 and 64.
c 20 ± 4d Largest 24, smallest 16. The difference is between
16 and 24.3 a Length 6.25%, width 8.33%
b 48 cm2
c Max. area 55.25 cm2, min. area = 41.25 cm2
d 15.10%4 15.1%5 30.5 m/min, 35.6 m/min, 26.4 m/min, 16.7%
Max. percentage error is 16.7%.6 a 5% b 13 cm ± 0.65 cm
Exercise 9B — Perimeter1 a 26 cm b 40 mm c 18π or 56.55 cm
d 24 cm e 15 cm f 31 cmg 10 cm h 30 cm
2 a 64 cm b 90 mm c 36 cmd 25π or 78.5 cm e 44 cm
3 a i (30 + 15π) cm ii 77.12 cm
b i cm ii 74.99 cm
c i cm ii 43.56 cm
d i (74 + 7π) cm ii 95.99 cme i (200 + 20π) cm ii 262.83 mf i (28 + 14π) cm ii 71.98 cm
4 a 42.43 cm b 174.55 cm c 163.98 cm d 148.5 cm e 47.14 cm f 54.27 cm
5 650 m6 B7 E8 27 m × 13.5 m9 35 m
10 13 cm11 406.28 m, 412.57 m, 418.85 m12 12 cm13 Azi’s, Robyn’s and Simon’s suggestions are correct
as their figures have a perimeter of 64 mm. Lauren is incorrect as her suggested figure has a perimeter of 32 mm.
14 658.95 m15 Approx. 2250 m
Maths Quest challenge (page 326)1 2 km 2 m ≈ 5.83 m
Exercise 9C — Area1 a 16 cm2 b 48 cm2 c 75 cm2
d 120 cm2 e 706.86 cm2 f 73.5 cm2
g 254.47 cm2 h 21 cm2 i 75 cm2
2 e 225π cm2 g 81π cm2
3 a 20.7 cm2 b 7.64 cm2
4 a 113.1 cm2 b 188.5 cm2
5 a i 12π cm2 ii 37.70 cm2
b i mm2 ii 108.38 mm2
c i 261π cm2 ii 819.96 cm2
6 E
7 D
8 a 123.29 cm2 b 1427.88 m2 c 52 cm2
d 30.4 m2 e 78 cm2 f 2015.5 cm2
9 a 125.66 cm2 b 102.87 cm2 c 13.73 m2
d 153.59 m2 e 13.85 m2 f 37.5 m2
10 11 707.92 cm2
11 21 m2
12 60
13 $840
10 Quick Questions 11 0.5 cm 2 6.15 m to 6.25 m
3 28.2 m 4 43 m
5 (16 + 4π) cm
6 i cm ii 26.09 cm
7 16π cm2 8 48.515 cm2
9 88.4 cm2 10 86 m2
Maths Quest challenge (page 335)1 Approximately 56%
2 The parallelogram has twice the area of the triangle.
3 a 50 cm2
b Perimeter of EFGH = × perimeter of ABCD
Exercise 9D — Total surface area1 a 600 cm2 b 384 cm2 c 1440 cm2 d 27 m2
2 a 113.1 m2 b 6729.3 cm2 c 8.2 m2
d 452.4 cm2
3 a 1495.4 cm2 b 502.7 cm2
4 a 506.0 cm2 b 9.4 cm2 c 340.4 cm2
d 224.1 cm2
5 a 13.5 m2 b 90 m2 c 11 309.7 cm2
d 9852.0 mm2 e 125.6 cm2 f 1531.4 cm2
6 a 880 cm2 b 3072.8 cm2 c 75 cm2
d 70.4 cm2 e 193.5 cm2 f 1547.2 cm2
7 B
8 63
9 3.6 m2
10 11 216 cm2
11 a 70.0 m2 b $455
12 a 3063.1 cm2 b $168.47
Exercise 9E — Volume1 a 27 cm3 b 74.088 m3 c 3600 cm3
d 94.5 cm3
2 a 6333.5 cm3 b 19.1 m3 c 280 cm3
d 288 mm3
3 a Vnew = 27l3, the volume will be 27 times as large as the original volume.
b Vnew = l3, the volume will be of the original volume.
c Vnew = 2πr2h, the volume will be twice as large as the original volume.
d Vnew = πr2h, the volume will remain the same.
e V = 3lwh, the volume will be 3 times as large as the original value.
4 a 7.2 m3 b 14 137.2 cm3 c 1436.8 mm3
d 523 598.8 cm3
5 a 169.6 cm3 b 3539.5 mm3
4221π
2---------+⎝ ⎠
⎛ ⎞
2015π
2---------+⎝ ⎠
⎛ ⎞
34
69π2
---------
242π3
------+⎝ ⎠⎛ ⎞
2
2-------
1
8---
1
8---
9A
9E
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Maths Quest 10/ Final Pages / 19/1/06
6 a 784 cm3 b 8960 cm3 c 5.34 m3
d 540 cm3
7 a 33.50 m3 b 64 000 cm3 c 3.7 m3
d 1385.44 mm3
8 a 630 mm3 b 420 cm3 c 3152.7 mm3
d 1319.5 mm3
9 E
10 a 12 800 cm3 b 268.08 cm3 c 7438.34 cm3
11 C
12 6809.4 L
13 2
14 4775.22 cm3
15 84 823 cm3
16 10 215.05 cm3
10 Quick Questions 21 138.24 cm2 2 273.9 cm2
3 27 m2 4 226.2 cm2
5 277.6 cm2 6 768 cm3
7 6 m3 8 1436.76 cm3
9 136 cm3 10 32.7 m3
Exercise 9F — Time, speed, density and concentration
1 64 km/h
2 a 120 km/h b 30 km/h c 1692.3 km/h
3 a 30 km b 30 min c 1 h 30 min
d 45 min e 2.00 pm f 53.33 km/h
4 a 150 km, 200 km and 250 km
b 50 km
c Both stop for 1 hour.
d Brian — 300 km, 3.5 h; Margaret — 300 km, 4 h
e Brian — 85.7 km/h; Margaret — 75 km/h
5 a 2.5 hb 20 kmc
6 a 0750 h b 1430 h c 9.30 pm
d 6.55 pm
7 B
8 D
9 280 km
10 a 6.30 am b 50 min c 2 h 10 min
d 2 h 15 min
11 a 1 h 30 min b 4.25 am c 5 h 15 min
d 8.45 pm e 2 h 20 min
12 4 g/cm3
13 a 5 g/cm3 b 20 kg/m3 c 39 g/mm3
14 a 3.5 b 25 c 570
15 D
16 600 g/L
17 30 g/L
Maths Quest challenge (page 364)1 Side length of 6 units2 a Volume is 8 times larger
b Surface area is 4 times greater
Summary1 error
2 × 100%
3 perimeter4 2πr 5 area6 hectares7 surface area
8 4πr2; πr3
9 volume10 AH11 rate
12
13
14 Concentration
Chapter review1 a 2 cm b 1.3%2 a 55 ± 3 b 52 and 58 c 2 and 8
d 750, 10.93% e 1.2, 0.133 a 128 cm b 112 cm
c 800 cm2, max. error is 124 cm2, percentage error is 15.5%
4 a 56 cm b 30 cm c 69.12 cmd 25.66 cm e 32.38 cm f 87.96 cm
5 c 22π cm e cm f 28π cm
6 39 cm by 13 cm7 12.6 m8 a 81 cm2 b 300 cm2 c 84 cm2
d 100 cm2 e 452.39 cm2 f 6.5 cm2
g 56.52 cm2 h 60 cm2 i 244.35 cm2
9 e 144π cm2 i cm2
10 a 60 cm2 b 300 cm2 c 224.25 cm2 d 160 cm2 e 23.56 cm2 f 80.19 cm2
11 a 40.04 cm2 b 129.53 cm2 c 499.86 cm2
d 44.54 cm2 e 85.84 cm2 f 128.76 cm2
12 a 59.6 m2 b $268213 a 2400 cm2 b 700 cm2 c 18 692.48 cm2
d 1495.14 cm2 e 804.25 cm2 f 642 cm2
g 8444.6 mm2 h 873.36 mm2 i 760 cm2
14 a 3606.55 cm2 b $180.3315 a 343 cm3 b 672 cm3 c 153938.04 cm3
d 1.45 m3 e 1800 cm3 f 1256.64 cm3
g 297 cm3 h 8400 cm3 i 7238.23 mm3
16 , the volume will be 1.5 times as large
as the original volume.17 V = 3lwh, the volume will be 3 times as large as (or
triple) the original volume.
7.30 8.00 8.30 9.00 9.30 10.00
20
15
10
5
0
Time (am)
Dis
tanc
e fr
om h
ome
(km
)
error
actual value----------------------------
4
3---
distance
time-------------------
mass
volume------------------
248π3
------+⎝ ⎠⎛ ⎞
700π9
------------
V 3
2---πr2h=
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18 785.4 m3
19 303.478 m3
20 a 90 km/h b 3 h21 a 20 km b 15 min c 40 km
d 30 min e 30 minf i 80 km/h ii 40 km/h iii 80 km/h
22 a 5.30 pm b 6.45 am c 10.10 pm23 a 7.30 am b 1 h 20 min c 10.20 am
d 2 h 20 min24
25 a 0.14 g/mL b 0.07 g/mL
Chapter 10 Circle geometryAre you ready?1 a a = 70° b b = 75°
c c = 100°
2 ∠ABC, ∠BCA, ∠CAB, ∠ACD
3 a d = 60° b e = 50° c f = 145°
4
5 8.7 cm
Exercise 10A — Intersecting chords, secants and tangents1 a m = 3 b m = 3 c m = 6 2 a n = 1 b m = 2 c n = 133 a w = 6 b x = 10 c y = 84 a x = 5 b m = 7 c x = 2.5, y = 3.15 a x = 2.8 b x = 3.3 c x = 5.6 d m = 90°6 A7 ST = 3 cm8 Check with your teacher.9 Check with your teacher.
10 Check with your teacher.
History of mathematics — They couldn’t do it!1 20°2 It was said to be created by the Oracle at Delphi.3 Anaxagoras about 440BC, Archimedes about 300BC,
Hindu mathematicians from about 800BC to 500BC.4 Pierre Wantzel5 a x3 − 3x − 1 = 0
b the square root of πc the cube root of 2
Exercise 10B — Angles in a circle1 a x = 30° b x = 25°, y = 25°
c x = 32° d x = 40°, y = 40° e x = 60° f x = 40° g x = 84° h x = 50°, y = 100°i x = 56°
2 a s = 90°, r = 90° b t = 90°, u = 90° c m = 90°, n = 90° d x = 52°e x = 90° f x = 90°, y = 15°
3 a x = 90°, y = 20°, w = 20°, z = 90°b s = 90°, r = 90° , t = 140°c x = 20°, y = 70°, z = 70°d x = 70°, y = 90°, z = 20°, r = 20°, s = 90°e x = 70°, y = 20°, z = 20°f x = 75°, y = 75°, z = 75°
4 D
5 D
6 a Base angles of an isosceles triangleb r + s = 90°, s = 45° ⇒ r = 45°c u is the third angle in ΔABD. d m is the third angle in ΔOCD.e ∠AOC and ∠AFC stand on the same arc with
∠AOC at the centre and ∠AFC at the circumference.
7 OR = OP (radii of the circle)∠OPR = x (equal angles lie opposite equal sides)∠SOP = 2x (exterior angle equals the sum of the two
interior opposite angles)OR = OQ (radii of the circle)∠OQR = y (equal angles lie opposite equal sides)∠SOQ = 2y (exterior angle equals the sum of the
two interior opposite angles)Now ∠PRQ = x + y and ∠POQ = 2x + 2y = 2(x + y).Therefore ∠POQ = 2 × ∠PRQ.
8 Check with your teacher.
9 Check with your teacher.
10 Check with your teacher.
Maths Quest challenge (page 387)1 904.8 cm or approx. 9 m
2 ( )% or approx. 47.64%
3 m2 or approx. 23.4 m2 (regular hexagon)
Exercise 10C — Cyclic quadrilaterals1 a x = 85°, y = 88° b m = 115°
c n = 25° d x = 130° e x = y = 90° f x = 45°, y = 95°
2 a x = 85°, y = 80° b x = 110°, y = 115° c x = 105° d x = 150° e x = 90°, y = 120° f m = 120°, n = 130°
3 D
4 a A b D
5 a 2x b 180 − 2x c 90 − x d 180°
6 Check with your teacher.
10 Quick Questions 1
Mass Volume Density
500 g 20 cm3 25 g/cm3
1500 g 30 cm3 50 g/cm3
2040 g 120 cm3 17 g/cm3
10050π
3---------–
27 3
2-------------
1 a = 8 cm 2 b = 8.5 cm 3 c = 6 cm
4 d = 7 cm 5 e = 56° 6 f = 130°7 g = 90° 8 h = 38° 9 i = 101°
10 j = 85° 9F
10C
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Maths Quest 10/ Final Pages / 19/1/06
Maths Quest challenge (page 393)1
2 One possible solution:
Exercise 10D — Great circles 1 a 14 074 km b 32 393 km c 12 175 km
d 18 877 km2 a 3574 km b 11 170 km c 7372 km
d 10 947 km3 a 8042.48 km b 1675.52 km
c 1340.41 km d 9829.69 km4 a 10 053 km b 3351 km
c 4468 km d 6702 km
10 Quick Questions 2
Maths Quest challenge (page 401)1 50 cm2
2 a 2π m ≈ 6.28 mb Height above the equator needs to be the same as
the radius of the Earth.
Exercise 10E — Locus1 a x = 4 and x = −4
b y = 2 and y = −2c x = 11 and x = −9
2 a x2 + y2 = 9b x2 + y2 = 49 c x2 + y2 = 1
3 a (x − 2)2 + (y − 1)2 = 9 b (x + 3)2 + (y + 2)2 = 36c (x − 3)2 + (y + 1)2 = 16
4
These are two possible locations the marker could be.5 D6 D7 a x2 + y2 = 36
b (x + 2)2 + (y + 1)2 = 16c (x − )2 + (y − )2 = 5
Summary
Chapter review1 a m = 3 b m = 12 c m = 9 d m = 6
2 a x = 5 b k = 12 c m = 6, n = 6
d x = 7 e b = 4, a = 2 f w = 3, x = 5
3 C
4 D
5 CE × ED = AE × EB (theorem)
AE = CE (given)
∴ ED = EB
6 ∠AYC = ∠AXC (same segment)
∠BXD = ∠BYD (same segment)
But ∠AXC = ∠BXD (vertically opposite)
⇒ ∠AYC = ∠BYD
7 a x = 50° b x = 48°, y = 25°
c x = y = 28°, z = 56° d x = 90°
e y = 90° f x = 70°
g x = 55° h x = 125°
i x = 70° j x = 100°
k m = 40° l x = 90°, y = 60°, z = 50°8 a x = 90° b x = 20°
c x = 55° d x = 125°9 ∠PQT & ∠PST, ∠PTS & ∠RQS, ∠TPQ & ∠QSR,
∠QPS & ∠QTS, ∠TPS & TQS, ∠PQS & ∠PTS, ∠PUT & ∠QUS, ∠PUQ & TUS
10 a x = 80°, y = 95° b x = 99°
c x = 78°, y = 92° d x = 97°, y = 92°11 D
12 8042.48 km
13 a 8378 km b 17 984 km
14 a 5027 km b 11 952 km
15 a 3128 km b 1117 km
16 335.10 km
17 x2 + y2 = 64
18 a x2 + y2 = 36 b (x − 1)2 + (y + 2)2 = 9
19 D
Chapter 11 Further geometry
Are you ready?1 a
b
A
B EF
C
D
1 2 3 10
4 5 6
8 9
7
5 C 6 A 7 1340 km8 3016 km 9 2.5° 10 34.3°S
1 k = 6 cm (theorem 5) 2 l = 40° (theorem 5)3 m = 8 cm (theorem 6) 4 n = 5 cm (theorem 6)5 p = 31° (theorem 8) 6 q = 22° (theorem 11)7 4245 km 8 1452 km9 58° 10 6500 km
A B
Abdul Joe
Possible position
of marker
Possible position
of marker
100 m
1
2---
1
3---
1 circumference 2 chord3 segment 4 sector 5 secant 6 bisects 7 tangent 8 equal in size 9 twice 10 right angles
11 supplementary 12 exterior 13 great circles 14 longitude15 latitude 16 arc length17 6400 km 18 locus 19 x2 + y2 = r2 20 (x − h)2 + (y − k)2 = r2
23°
130°
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c
2 a 2 : 3 b 3 : 1 c 5 : 3
3 a 2 b 1.5 c 2.5
4 a ΔPTQ b PR c SR, PR
5 a = 50° (vertically opposite angles), b = 90° (sum of angles in triangle), c = 90° (alternate angles), d = 40° (alternate angles)
6 F6
7 G4
8 G4
Exercise 11A — Review of 2-dimensional and 3-dimensional drawing1 2
3 a i Draw a line segment of length 7 cm. Call it AB.
ii Using the same radius draw an angle of 70° at A, with the other arm of length 3 cm. Label the end of this line segment D.
iii Draw a line segment of length 7 cm from D. Call the end point C.
iv Join points C and B by a line.
b i Draw a circle of radius r cm.
ii Place the pair of compasses at any point on the circumference and draw an arc from one point on the circumference to another.
iii Place the pair of compasses where the arc ends and draw another arc.
iv Repeat iii until the pattern is complete.
4 a
c
5 a
b
c
6 a Front Top
b Side Top and bottom
c Front Side Top
7 a C b A c B
8 a
250°
12 cm
8 cmA B
C
28°R
10.8 cm
7.8 cmP
Q
40°
10 cm
12 cm
2 cm 30°10 cm12 cm 2 cm
3030° °
b
45° 45°
10 cm
12 cm
••VP2 VP1
HL
HL•
HL•
Side
Front
20 m
12 m
10D
11A
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Maths Quest 10/ Final Pages / 19/1/06
b
Exercise 11B — Cross-sectional view of objects1 a b
c d
e f
g
h
2 C
3 B
Exercise 11C — Similarity1 a Yes, scale factor = 2 b No
c Yes, scale factor = 3 d Yes, scale factor = 1.5
e No
2 a ΔABC ∼ ΔPQR; x = 14, y = 10
b ΔRPT ∼ ΔRQS; x = 50°, y = 3, z = 7
c ΔAEC ∼ ΔBDC; y = 12, x = 3
d ΔABC ∼ ΔEDC; x = 75° y = 70° z = 16.8
e ΔLMN ∼ ΔRST; x = 15, y = 12, z = 53.1°
3 a ΔPQT ∼ ΔRST; x = 30°b ABCD ∼ PQRS; x = 89°, y = 85°, z = 6
c ABCD ∼ PQRS; x = 50°, y = 8 cm
4 5 m
5 a 26.67 cm b 133.33 cm
6 150 cm × 125 cm
7 D8 C9 100 cm and 150 cm
10 a Yes, scale factor = 10, all sides in the equal ratiob No. The length increases 2 times but the width
remains the same.11 8 m12 m = 4 cm, n = 1.5 cm13 8 cm14 a Show two sides with corresponding ratios equal
and the included angle equal.b Show two sides with corresponding ratios equal
and the included angle equal.
Maths Quest challenge (page 433)1 20 cm by 30 cm2 20 cm from the torch.
History of mathematics — Thales of Miletus1 He proposed the first natural cosmology and
pioneered the scientific method.2 A solar eclipse.3 Any two of the following:
Any circle is bisected by its diameter.The base angles of an isosceles triangle are equal. The opposite angles of intersecting straight lines are the same. The angle subtended by the diameter of a circle touching the circle is a right angle.Two triangles are congruent if they have the same base and base angles.
10 Quick Questions 11 2
3
4 5 6
7 a = 12
8 b ≈ 14.2
20 m
10 m
1 mfence
9 cm
7 cm
3 cm
Top
Front Side
VP1 VP2• •
5m
7m
12m
6m
6m
2
3---
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9 c = 3 , d = 7
10 e = 6 , f = 13
Exercise 11D — Congruence1 a I and III, SAS b I and II, AAS
c II and III, RHS d I and II, SSS
2 a x = 7 cm b x = 78°
c x = 75°, y = 35°, z = 70° d x = 28°, y = 10 cm
e x = 50°, y = 40°, z = 40°, m = 90°, n = 90°3 a Use SAS. b Use SAS. c Use ASA.
d Use ASA. e Use SSS.
4 a I and II b II and IV c I and II
5 B
6 a x = 110°, y = 110°, z = 4 cm, w = 7 cm
b x = 70° c x = 30°, y = 65°7 The triangles will only be similar. They may have
different side lengths.
8 The third sides are not necessarily the same.
9 Corresponding sides are not the same.
10 Use RHS.
Exercise 11E — Nets, polyhedra construction and Euler’s rule1
2 a
b
3 Two possible nets are:
or
4 A possible net is:
5–7 Check with your teacher.
8 a E = 12, V = 8, F = 6, 8 + 6 − 2 = 12
b E = 12, V = 6, F = 8, 6 + 8 − 2 = 12
9 a E = 18, V = 12, F = 8, 8 + 12 − 2 = 18
b and c E = 12, V = 8, F = 6, 6 + 8 − 2 = 12
10 No. Edges, vertices and faces can not be identified on a solid with curves.
11 No
12 For each shape, E = 9, V = 6 and F = 5 where 5 + 6 − 2 = 9.
Maths Quest challenge (page 450)1
2 13.61 cm in this case. In general, the position of X can be found by construction as shown in the diagram.
10 Quick Questions 21 1.16 m
2 5 m
3 13 m
4 17 m
5 1.2 m
6 a = 70, b = 12, c = 12, d = 70
7 e = 19, f = 35
1
3---
6
11------
4
9---
5 cm
5 cm
10 cm
c
or
A
B
B'
Wall
X
1
3---
11B
11E
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Maths Quest 10/ Final Pages / 19/1/06
8
9 F = 5, V = 5, E = 8, 5 + 5 − 2 = 8
10 F = 7, V = 10, E = 15; 7 + 10 − 2 = 15
Exercise 11F — Transformation of points and figures
1 a–d
2 a
b
3 a Centre of rotation: C, angle of rotation: 180°. C is an invariant point.
b Centre of rotation: B, angle of rotation: 90°. B is an invariant point.
c Centre of rotation: G, angle of rotation: 180°. There are no invariant points.
4
5
6 a
b
c
d
7 a P′(−2, −2), Q′(−1, 2), R′(2, −1), S′(2, 3)
b P′(2, 2), Q′(1, −2), R′(−2, 1), S′(−2, −3)
c P′(2, −2), Q′(−2, −1), R′(1, 2), S′(−3, 2)
d P′(4, 2), Q′(3, −2), R′(0, 1), S′(0, −3)
e P′(−2, −4), Q′(−1, 0), R′(2, −3), S′(2, 1)
8 a x b y9 a b
c d
e
10 a 3 b 12 cm c 3.5 cm d 9
4
2
1
–1
3
421 3–4 –2 –1–3
R'''R''
R'
R''''
R
0
DA'
CB' C'B
A D'x
y
2
2
1
1 3–1
–2
–1–2
B C
DA
D'
C'B'
A'
x
y
2
2
1
1–1
–2
–1–2
B
CA
C'
B'
A'
x
y
2 3
2
3
1
1–1
–1–2
B C
DA
C'
B'
D'
A'
x
y
2
2
1
1 3–1
–2
–1–2–3
e
•
A B
CDA' B'
C'D'
•
A B
C
A' B'
C'
•A
B
C
A'
B'
C'•
A B
CD
A' B'
C'D'
•AB
C
A'B'
C'
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Maths Quest 10/ Final Pages / 19/1/06
11 a b cm2
12 a 120 cm3 b 3 cm by 9 cm by 15 cmc 405 cm3 d 3.375
13 216 cm3
14
15 B
16 D
17 a i (−1, 2) ii (−2, 3) iii (−1, 4)b x- and y-coordinates are swapped, the sign of the
x-coordinate is changed (that is, x′ = −y, y′ = x).c i (−y, x) ii (−n, m) iii (8, −2) iv (5, 3)
v (−2, −7)d i (2, −4) ii (−5, 2) iii (2, −5)
18
19 a Side lengths and anglesb Anglesc Side lengths and anglesd Side lengths and angles
20 B
21 P′(4, 0), Q′(4, 2), R′(5, 0)
22
23 a 200 b i 6 m × 6 m ii 5 m × 4.8 mc 37.44 m2
Summary
Chapter review1
2 a i Draw a circle of radius 2 cm.
ii Using the same centre, draw a circle of radius 1 cm.
iii Construct a set of radii, 60° apart at the centre.
iv Join consecutive points of intersection of the radii and circumference by straight lines (chords) in both circles.
v Shade in the minor segments in the outside circle.
b i Draw a rectangle ABCD, with AB = CD = 6 cm and AC = BD = 2 cm.
ii Mark a set of points at intervals of 1 cm on CD.
iii Taking the first point on CD as centre draw a semicircle with radius 1 cm towards the outside.
iv Repeat iii but use the 5th point as the centre.
v Using the third point as the centre, draw a semicircle towards the inside of the rectangle.
vi Rub off the side CD and shade in the enclosed area.
3 a b c
4
5 Top Left Right Front and back
6 a b c
7 a Similar, scale factor = 1.5
b Not similar c Similar, scale factor = 2
8 a x = 48°, y = 4.5 cm
b x = 86°, y = 50°, z = 12 cm
c x = 60°, y = 15 cm, z = 12 cm
9 a ΔABC ∼ ΔDEC b ΔABC ∼ ΔDEC
c ΔABC ∼ ΔPQR
10 a Show 3 pairs of equal angles (AAA).
b ΔABC ∼ ΔEDC
c d d = 5 cm, e = 4.5 cm
11 10 m
12 a I and III, ASA b I and II, RHS
13 a x = 8 cm b x = 70°c x = 30°, y = 60°, z = 90°
14 a Use SAS. b Use ASA.
1
2---
P
4---
y
x21–2–1
654321
M''
M'
y
x642
–6
–4
–2–6 –4 –2
6
4
2
•
•
•
••
••
•
K
L
M
N K'
L'
M'
N'
y
x654321
3
2
1
Q
RP
Q'
R'P'
• •
Mirror line
1 Construction 2 sketch 3 isometric 4 oblique5 planometric 6 perspective7 Orthogonal 8 Similar
910 Congruentimage length
object length-------------------------------
11 net 12 polyhedron13 Euler’s 14 translation15 centre of rotation 16 size, angles 17 image 18 area, volume
50°11.1 cm
6.7 cm
B
A
C
5 cm
3 cm
6 cm
30°30°
5 cm
3 cm
6 cm
30°
60°30°
6 cm
3 cm
5 cm
HLVP
•
AB
ED--------
AC
EC--------
BC
DC--------= =
11F
11F
5_61_03274_MQV10 - A 1-15_tb Page 751 Thursday, January 19, 2006 7:42 PM
752 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
15 Corresponding sides are not the same.
16 a b
17 Check with your teacher.
18 a F = 6, V = 8, E = 12, 6 + 8 − 2 = 12
b F = 5, V = 6, E = 9, 5 + 6 − 2 = 9
c F = 5, V = 5, E = 8, 5 + 5 − 2 = 8
19 No. Edges, vertices and faces are not clearly defined.
20 a (4, −2) b (2, −3) c (4, −3)
21 a b c
d e f
Q is an invariant point in part b; R is an invariant
point in parts b−f.
22 B
23
24 a 3 b 10 cm c 15 cm d 81 cm2
25 a 2.1 cm3 b 1 cm by 2.4 cm by 7 cm
c 16.8 cm3 d 8
26 216 cm3
27
28 a (−1, 0) b (−2, 0) c (0, −1) d (−1, −1)
Chapter 12 Trigonometry
Are you ready?1 a 0.685 b 1.400 c 0.749
2 a i 15°33′ ii 15°32′41″b i 63°16′ ii 63°15′32″c i 27°10′ ii 27°10′16″
3 a b c
4 a x = 30 × tan 15° b x = c x = 5.3 × tan 64°
5 a b
Exercise 12A — Angles and the calculator1 a 0.5000 b 0.7071 c 0.4663 d 0.8387
e 8.1443 f 0.71932 a 0.6944 b 0.5885 c 0.5220 d −1.5013
e 0.9990 f 0.6709 g 0.8120 h 0.5253i −0.8031 j 0.4063 k 0.9880 l −0.9613m 1.7321 n −0.5736 o 0.1320
3 a 50° b 24° c 53° d 71°e 86° f 41°
4 a 54°29′ b 6°19′ c 0°52′ d 72°47′e 44°48′ f 26°45′
5 a 26°33′54′′ b 64°1′25′′ c 64°46′58′′d 48°5′22′′ e 36°52′12′′ f 88°41′27′′
6 a 2.824 b 71.014 c 20.361 d 2.828e 226.735 f 1.192 g 7.232 h 32.259i 4909.913 j 0.063 k 0.904 l 14.814
7 E8 D
Exercise 12B — Using trigonometric ratios to find side lengths1 a 8.660 b 7.250 c 8.4122 a 0.792 b 4.721 c 101.3823 a 33.45 m b 74.89 m c 44.82 m
d 7.76 mm e 80.82 km f 9.04 cm4 a x = 31.58 cm b y = 17.67 m
c z = 14.87 m d p = 67.00 me p = 21.38 km, q = 42.29 km f a = 0.70 km, b = 0.21 km
5 a 6.0 m b 6.7 m6 D7 B8 B9 1.05 m
Exercise 12C — Using trigonometric ratios to find angles1 a 67° b 47° c 69°2 a 54°47′ b 33°45′ c 33°33′3 a 75°31′21′′ b 36°52′12′′ c 37°38′51′′4 a 41° b 30° c 49°
d 65° e 48° f 37°5 a a = 25°47′, b = 64°13′ b d = 25°23′, e = 64°37′
c x = 66°12′, y = 23°48′
3 cm
7 cm
–1–2 1 2
1
2
–2
–1
y
x
P Q
S R
P' Q'
S' R'
–1–2 1 2
1
2
–2
–1
y
x
PQ
S R
P'Q'
S'R' –1–2 1 2
1
2
–2
–1
y
x
P Q
S R
P' Q'S' R'
–1–2 1 2
1
2
–2
–1
y
x
P Q
S R
P'
Q'
S'
R' –1–2 1 2
1
2
–2
–1
y
x
P Q
S R
P'Q'
S'R' –1–2 1 2
1
2
–2
–1
y
x
P Q
S R
P' Q'
S' R'
P
RS
QP'
R' S'
Q'
–1–2 1 2
1
2
–2
–1
y
x
A
B
CA'
C'B'
H
A
Oθ H
A
O
θ
HA
O
θ
4.2
tan 28°-----------------
7.5km
CA
B
N
N
5km
120°25°
N
180
km
70 km
S
20°
60°
5_61_03274_MQV10 - A 1-15_tb Page 752 Thursday, January 19, 2006 7:42 PM
A n s w e r s 753
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Maths Quest 10/ Final Pages / 19/1/06
6 B7 D8 C
10 Quick Questions 1
Maths Quest challenge (page 492)1 25 cm from the base of the container2 12, 13, 27, 37
Exercise 12D — Applications1 a 36°52′ b 53°8′ c 2.4 m
2 a 14°29′ b 31 cm
3 8.74 m
4 687.7 m
5 a 176.42 m b 152.42 m
6 23.88 cm
7 a 56.83 cm b 62.50 cm
8 65°46′9 6.09 m
10 19°28′11 62°33′12 16.04 m
13 a h = x tan 47°12′ m h = (x + 38) tan 35°50′ mb x = 76.69 m c 84.62 m
14 a h = x tan 43°35′m h = (x + 75) tan 32°18′mb 148.37 m c 143.1 m
15 a 11°32′ b 4°25′16 a iii 35.36 cm ii 51.48 cm iii 51.48 cm
iv 57.23 cm v 29°3′ vi 25°54′b iii 25.74 cm ii 12.5 cm iii 25°54′
iv 28.61 cm
17 a 77° b 71°56′ c 27.35 cm
18 a 7.05 cm b 60°15′ c 8.12 cm
19 a 28.74 cm b 40.64 cm c 66°37′20 a 26.88 cm b 11.07 cm
Maths Quest challenge (page 499)1 a b c
2 Any multiple of 3 (extension of part b) or power of 2 (extension of parts a and c).
Exercise 12E — Bearings1 a 020°T b 340°T c 215°T d 152°T
e 034°T f 222°T
2 a N49°E b S48°E c S87°W d N30°We N86°E f S54°W
3 a 3 km 325°T b 2.5 km 112°T c 8 km 235°T d 4 km 090°T, then 2.5 km 030°T e 12 km 115°T, then 7 km 050°T f 300 m 310°T, then 500 m 220°T
4 a b
c d
e
5 a iii 13.38 km ii 14.86 km iii 222°T
b iii
iii 51.42 km iii 61.28 km
iv 310°T
c iii
iii 38.97 km iii 22.5 km
iv 030°T
6 215°T 7 A 8 1.732 km
9 a 9.135 km b 2.305 km c 104° 10′T10 684.86 km
11 B
12 a 60°43′T b 69°27′T c 204°27′T
Exercise 12F — The unit circle — quadrant 11 a 0.6499 b 0.4931 c 0.3919
d 0.7454 e 0.9428 f 0.9165
2 a 0.5235 b 0.9882 c 0.8660
d 0.9165 e 0.7042 f 0.9448
1 3.2472 2 78° 3 74°58′4 10.6 5 13.5 m 6 41°7 422 m 8 35°53′ 9 7.34 cm
10 26°33′54′′
40°
100°
N
N30 km
40 k
m
240°
135°N
N
230 km140 km
260°120°
32°
N
N
N
0.8 km
1.3 km
2.1
km
50°
40°30°
S
N
N
8 km
5 km
7 km
30°
70°
20°N
N
S
180 km
320
km
220 km
42°
130°N
N
80 km
20 k
m
A
B
C
42°
130°
210°
N
N
N
80 km
20 k
m
45 k
m
A
B
D
C
12A
12F
5_61_03274_MQV10 - A 1-15_tb Page 753 Thursday, January 19, 2006 7:42 PM
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swers
Maths Quest 10/ Final Pages / 19/1/06
3 a 0.577 b 1.333 c 4.000d 0.224 e 1.397 f 2.582
4 a i 0.9766 ii 0.2201 b i 0.1262 ii 0.1273c i 0.9491 ii 3.0130 d i 0.4823 ii 1.8163e i 0.6 ii 0.75 f i 0.9035 ii 0.474
5 a 0.7915 b 0.7722 c 37°41′d
6 a 0.9766 b 4.5423 c 75°35′d
7 a B b A c D d D
10 Quick Questions 21 30.2 2 19.9 3 57°51′ 4 19°20′ 5 89°2′6 623 m 7 338 m 8 243°T 9 0.454 10 0.510
Exercise 12G — Circular functions1 a 1st b 2nd c 4th d 3rd
e 2nd f 3rd g 4th h 4th2 A3 D4 a 0.35 b 0.95 c −0.17 d 0.99
e −0.64 f 0.77 g −0.57 h −0.825 a 1 b 0 c 0 d −1
e −1 f 0 g 0 h 16 a 0.87 b 0.507 a 30° b −0.87 c cos 150° = −cos 30°
d 0.5 e sin 150° = sin 30°8 a 30° b −0.87 c cos 210° = −cos 30°
d −0.50 e sin 210° = −sin 30°9 a 30° b 0.87 c cos 330° = cos 30°
d −0.50 e sin 210° = −sin 30°10 a 0.34 b 0.94 c 0.36
d 0.36 e They are equal.11 a 0.71 b −0.71 c −1 d −1
e They are equal. f tan 135° = −tan 45°12 a −0.64 b −0.77 c 0.84 d 0.83
e They are approx. equal. f tan 220° = tan 40°13 a −0.87 b 0.5 c −1.73 d −1.74
e They are approx. equal. f tan 300° = −tan 60°14 D
15 a b c
16 a b c
d e f
17 a 45° b 60° c 270°d 120° e 36° f 315°
Exercise 12H — Graphs of y = sin x and y = cos x1
2
3 360°, after which the graph repeats itself.4 a 0.7 b 0.8 c 0.35 d −0.35
e 0 f 0.9 g −0.2 h −0.95 a 64°, 116°, 424°, 476°
b 244°, 296°, 604°, 656°c 44°, 136°, 404°, 496°d 210°, 330°, 570°, 690°e 233°, 307°, 593°, 667°f 24°, 156°, 384°, 516°
6
7
8 The graph would continue with the cycle.9 It is a very similar graph with the same shape;
however, the sine graph starts at (0, 0), whereas the cosine graph starts at (0, 1).
10 a 0.7 b −0.99 c −1 d 0.9 e −0.5 f −0.8 g 0.8 h −0.96
11 a 120°, 240°, 480°, 600°b 37°, 323°, 397°, 683°c 46°, 314°, 406°, 674°d 127°, 233°, 487°, 593°e 26°, 334°, 386°, 694°f 154°, 206°, 514°, 566°
Summary
1 , , tan θ, hypotenuse 2 trigonometric ratios
3 right-angled 4 elevation 5 depression 6 north–south line7 north, east or west 8 clockwise9 radius 10 Pythagorean
37° 41'
cos 37° 41'
tan
37° 4
1'
x
ysin 37° 41'
75° 35'
cos 75° 35'
tan
35° 7
5'
x
y
sin
75° 3
5'
π2---
c π3----
c 4π3
-------c
π6----
c 2π5
-------c 5π
4-------
c
10π9
---------c π
2---
c 4π15-------
c
x 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
sin x 0 0.5 0.87 1 0.87 0.5 0 −0.5 −0.87 −1 −0.87 −0.5 0
x 390° 420° 450° 480° 510° 540° 570° 600° 630° 660° 690° 720°
sin x 0.5 0.87 1 0.87 0.5 0 −0.5 −0.87 −1 −0.87 −0.5 0
x 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
cos x 1 0.87 0.5 0 −0.5 −0.87 −1 −0.87 −0.5 0 0.5 0.87 1
x 390° 420° 450° 480° 510° 540° 570° 600° 630° 660° 690° 720°
cos x 0.87 0.5 0 −0.5 −0.87 −1 −0.87 −0.5 0 0.5 0.87 1
1
0
–1
y
x
90°
18
0°
27
0°
36
0°
45
0°
54
0°
63
0°
72
0°
y = sinx
1
0
–1
y
x
90
°1
80°
27
0°
36
0°
45
0°
54
0°
63
0°
72
0°
y = cosx
O
H----
A
H----
5_61_03274_MQV10 - A 1-15_tb Page 754 Thursday, January 19, 2006 7:42 PM
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Maths Quest 10/ Final Pages / 19/1/06
11 rearranged 12 tangent
13 unit circle, 90° and 180°, 180° and 270°14 cos θ, sin θ 15 radians, degrees
16 17
Chapter review1 a 0.276 b 0.511 c −24.898 d 0
2 a 56°49′ b 63° c 42°13′3 a 0.848 b 1623.10 c 0.8168 d 3.368
4 D
5 B
6 D
7 a tan θ = b sin θ = c cos θ =
8 a 17.48 m b 53°35′ c 51°20′ d 15.02
9 40°32′10 17.6 m
11 26.86 m
12 a 11.04 cm b 15.6 cm c 59°2′13 a h = x tan 47°48′ m
h = (x + 64) tan 36°24′ mb 129.07 m c 144.29 m
14 a 346°T b 156°T c 217°T d 048°T
15 a S52°E b N66°W c N34°E d S38°W
16 67.98 km
17 4.16 km
18 a 0.951 b 0.527
19 a 0.888 b 0.613
20 a 0.467 b 14.114
21 cos θ = 0.929, tan θ = 0.397
22 a 0.8 b −0.64 c 0.5 d −0.34
23 a b c
24 a b c
25 a 72° b 300° c 216°26
27
Chapter 13 ProbabilityAre you ready?1 a Set A: 3, Set B: 4, Set C: 4
b 4 c 2
2 a b c
3 a Not drawing an ace
b Drawing a red card
c Obtaining a 4 or a 5
4 a b c
5 a b c
Exercise 13A — Probability revision1 a b
2
3 a b c
4 a i ii iii iv
b i ii iii
5 A
6
7 a b c
8 a b c d
e 0 f
9 a b c d 0
10 a No. Pr(Azi rolls a 5) = and
Pr(Robyn rolls a 5) =
b Yes. Pr(Azi wins) = and Pr(Robyn wins) =
11 a C b D c E
12 Yes. Both have a probability of .
13 a
b i = ii = iii =
iv = v =
14 a Pr(A ∩ B) b Pr(X′ ∩ Y)
c Pr(A′ ∩ B′) d Pr(A ∩ C ∩ B′)
π c
180---------
180°
π-----------
O
A----
O
H----
O
A----
3π2
------c 5π
3------
c 2π9
------c
π8----
c 4π5
------c π
5---
c
1
0
–1
y
x
90°
18
0°
27
0°
36
0°
45
0°
y = sinx
1
0
–1
y
x
90
°
18
0°
27
0°
36
0°
45
0°
y = cosx
1
4---
1
9---
2
3---
x f Relative frequency
1 2 or 0.1
2 5 or 0.25
3 6 or 0.3
4 3 or 0.15
5 4 or 0.2
Σ f = 20 1.00
2
3---
133
156---------
7
13------
1
12------
5
312---------
1
8---
9
10------
1
10------
5
6---
17
30------
1
10------
4
5---
3
20------
1
5---
9
80------
7
40------
1
16------
17
80------
1
16------
1
10------
1
4---
3
10------
3
20------
1
5---
1
5---
1
10------
3
10------
1
13------
1
4---
1
2---
12
13------
1
2---
1
5---
7
20------
11
20------
1
8---
1
6---
1
2---
1
2---
1
2---
A B
1
9
35
117 13
1517
19122
6 8 10 18
14
204
16
x
10
20------
1
2---
4
20------
1
5---
2
20------
1
10------
12
20------
3
5---
8
20------
2
5---
12G
13A
5_61_03274_MQV10 - A 1-15_tb Page 755 Thursday, January 19, 2006 7:42 PM
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swers
Maths Quest 10/ Final Pages / 19/1/06
15 a i
ii
iii
b 96
c i ii =
d i = ii
16 a
b i ii iii iv v
c i ii
17 a
b i 25 ii 23
c i ii iii iv
d i ii iii
18 a b $50 c 9–4
19 4–3
20 a b c
21 A
Exercise 13B — Complementary events1 a b
2 20% or
3 a i ii b Yes
4
5 C
6 a b c
7 D
8 a b c d
9 a No. There are many other foods one could have.
b No. There are other means of transport, for example, catching a bus.
c No. There are other possible leisure activities.
d No. The number 5 can be rolled too.
e Yes. There are only two possible outcomes; passing or failing.
10 a b
11 No, getting 1 Tail is possible too.
12 a b
10 Quick Questions 11
2
3
4
5 5–4
6 Getting a number greater than 2
7 90% or
8
9
10
Maths Quest challenge (page 549)1 39
2 a 36 b
3 17 576 000
Exercise 13C — Mutually exclusive events1
2 a b c
3 a b c
4 C
5 No. The number 2 is common to both events.
6 a b
7 a or b or c or
Volleyball Walking
Tennis
10
8
6
2 17
3815
x
Volleyball Walking
Tennis
10
8
6
2 17
3815
x
Volleyball Walking
Tennis
10
8
6
2 17
3815
x
35
96------
8
96------
1
12------
63
96------
21
32------
23
96------
Volleyball Soccer
ξ
Tennis
= 30
5
7
2 4
17 4
1
2---
1
6---
1
30------
2
5---
7
15------
1
2---
8
15------
Calculator Graph book
ξ
5
= 35
187 5
18
35------
5
7---
1
7---
12
35------
1
5---
6
7---
1
7---
3
10------
1
6---
4
17------
1
8---
1
13------
12
13------
1
5---
1
2---
1
2---
16
25------
1
10------
9
10------
47
50------
9
50------
41
50------
12
25------
13
25------
15
16------
7
9---
1
8---
7
8---
1
26------
1
18------
1
9---
4
5---
9
10------
4
5---
7
18------
4
7---
1
36------
2
3---
8
25------
19
25------
9
25------
7
26------
2
13------
3
13------
5
9---
4
7---
8
14------
4
7---
2
14------
1
7---
10
14------
5
7---
5_61_03274_MQV10 - A 1-15_tb Page 756 Thursday, January 19, 2006 7:42 PM
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Maths Quest 10/ Final Pages / 19/1/06
8 a b c
9 a Yes b
10 a i T ii F iii T iv F
v F vi F
b i ii iii iv
c i ii iii
Maths Quest challenge (page 554)
Area inside the smaller circle ( cm2) is larger than
the area between the circles ( cm2).
Exercise 13D — Two-way tables and tree diagrams1 a
b
2 a
b c
3 a
b
c
4
a b c
d e f
5 a
b {(R, R), (R, G), (R, B)}
c
d
1
13------
1
4---
4
13------
1
2---
3
16------
1
8---
5
8---
3
16------
5
16------
3
16------
3
4---
25π4
---------
24π4
---------
Die 1 outcomes
Die
2 o
utco
mes
1
1
2
3
4
5
6
2 3 4 5 6
(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)
1
12------
Die outcomes
Coi
n ou
tcom
es
H
T
1 2 3 4 5 6
(H, 1) (H, 2) (H, 3) (H, 4) (H, 5) (H, 6)
(T, 1) (T, 2) (T, 3) (T, 4) (T, 5) (T, 6)
7
(H, 7)
(T, 7)
8
(H, 8)
(T, 8)
9
(H, 9)
(T, 9)
10
(H, 10)
(T, 10)
1
5---
1
4---
Green octahedron outcomes
Yello
w o
ctah
edro
n ou
tcom
es
1
1
2
3
4
5
6
7
8
2 3 4 5 6
(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)
(7, 1) (7, 2) (7, 3) (7, 4) (7, 5) (7, 6)
(8, 1) (8, 2) (8, 3) (8, 4) (8, 5) (8, 6)
7 8
(1, 7) (1, 8)
(2, 7) (2, 8)
(3, 7) (3, 8)
(4, 7) (4, 8)
(5, 7) (5, 8)
(6, 7) (6, 8)
(7, 7) (7, 8)
(8, 7) (8, 8)
Green octahedron outcomes
Yello
w o
ctah
edro
n ou
tcom
es
1
1
2
3
4
5
6
7
8
2 3 4 5 6
(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)
(7, 1) (7, 2) (7, 3) (7, 4) (7, 5) (7, 6)
(8, 1) (8, 2) (8, 3) (8, 4) (8, 5) (8, 6)
7 8
(1, 7) (1, 8)
(2, 7) (2, 8)
(3, 7) (3, 8)
(4, 7) (4, 8)
(5, 7) (5, 8)
(6, 7) (6, 8)
(7, 7) (7, 8)
(8, 7) (8, 8)
1
8---
R
B
R
B1–2
1–2
1–2
1–2
R
B
1–2
1–2
R
B
1–2
1–2
R
B
1–2
1–2
R
B
1–2
1–2
R
B
1–2
1–2
1 2 3 Outcomes
RRR
RRB
RBR
RBB
BRR
BRB
BBR
BBB
Probability1–8
1–8
1–8
1–8
1–81–8
1–8
1–8—1
1
8---
3
8---
3
8---
1
8---
7
8---
1
2---
G
R
B
Outcomes
RR
Probability1–9
1—18
1–6
RG
RB
GR1—18
1—36
1—12
GG
GB
BR1—6
1—12
1–4
BG
BB—1
1–3
1–6
1–2
R
B
G
1–3
1–6
1–2
R
B
G
1–3
1–6
1–2
R
B
G
1–3
1–6
1–2
1 2
1
3---
7
18------ 13B
13D
5_61_03274_MQV10 - A 1-15_tb Page 757 Thursday, January 19, 2006 7:42 PM
758 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
6 a
b 3
c
d They are equally likely.
e
7 a
b No
c i ii iii iv
8 a
b c d
9 E
10
11 a
Sample space = {SS, SS′, S′S, S′S′}b c d
12
a b c 1
13 a
b c d e No
14 a
Sample space = {ACF, ACG, . . . , BEG, BEH}
b c
15 a
B
G
B
G1–2
1–2
1–2
1–2
B
G
1–2
1–2
B
G
1–2
1–2
B
G
1–2
1–2
B
G
1–2
1–2
B
G
1–2
1–2
1 2 3 Outcomes
BBB
BBG
BGB
BGG
GBB
GBG
GGB
GGG
Probability1–8
1–8
1–8
1–8
1–81–8
1–8
1–8—1
3
8---
7
8---
2
1
3
Outcomes
1 1
Probability1–4
1–8
1–8
1 2
1 3
2 11–8
1—16
1—16
2 2
2 3
3 11–8
1—16
1—16
3 2
3 3—1
1–2
1–4
1–4
1
3
2
1–2
1–4
1–4
1
3
2
1–2
1–4
1–4
1
3
2
1–2
1–4
1–4
1 2
1
4---
1
2---
3
8---
9
16------
R
R'
R
R'1–4
3–4
1–4
3–4
R
R'
1–4
3–4 1–
4
3–4
1–4
3–4 1–
4
3–4
R
R'
R
R'
R
R'
R
R'
1–4
3–4
1 2 3 Outcomes
RRR
RRR'
RR'R
RR'R'
R'RR
R'RR'
R'R'R
R'R'R'
Probability1—64
3—64
3—64
9—64
3—649—64
9—64
27—64—1
1
64------
27
64------
27
64------
t
t'
t
t'1–6
5–6
1–6
5–6
t
t'
1–6
5–6
1 2 Outcomes
tt
tt'
t't
t't'
Probability1—36
5—36
5—36
25—36—1
a
b
c
d
1
36------
5
18------
25
36------
11
36------
S
S'
S
S'1–4
3–4
1–4
3–4
S
S'
1–4
3–4
1 2 Outcomes
SS
SS'
S'S
S'S'
Probability1—16
3—16
3—16
9—16—1
1
16------
9
16------
3
8---
W
Y
X
Z
Outcomes
XY
Probability1—121—121—12
1—121—121—12
1—121—121—12
1—121—121—12
XW
XZ
YX
YW
YZ
WX
WY
WZ
ZX
ZY
ZW—1
1–4
1–4
1–4
1–4
1–3 1–
3
1–3
1–3 1–
3
1–3
1–3
1–3
1–3
1–3
1–3
1–3
Y
Z
W
X
Z
W
X
Z
Y
X
W
Y
1 2
1
4---
1
4---
B
G
B
G6–9
4–9
5–9
3–9
4—10
6—10
B
G
1 2 Outcomes
BB
BG
GB
GG
Probability
2—15
4—15
4—15
1–3—1
2
15------
1
3---
8
15------
D
C
E
Outcomes
ACF
Probability
1—181—181—18
ACG
ACH
ADF 1—181—181—18
ADG
ADH
AEF 1—181—181—18
AEG
AEH
BCF 1—181—181—18
BCG
BCH
BDF 1—181—181—18
BDG
BDH
BEF 1—181—181—18
BEG
BEH—1
1–3
1–3
1–3
D
C
E
1–3
1–3
1–3
A
B
1–2
1–2
F
H
G
1–3 1–
3
1–3
F
H
G
1–3 1–
3
1–3
F
H
G
1–3 1–
3
1–3
F
H
G
1–3 1–
3
1–3
F
H
G
1–3 1–
3
1–3
F
H
G
1–3
1–3
1–3
1 2 3
1
2---
1
6---
b
c
1
2---
1
2---
R
G
R
G1–2
1–2
1–2
1–2
1–2
1–2 R
G
Outcomes
RR
RG
GR
GG
Probability
1–4
1–4
1–4
1–4—1
5_61_03274_MQV10 - A 1-15_tb Page 758 Thursday, January 19, 2006 7:42 PM
A n s w e r s 759
an
swers
➔
Maths Quest 10/ Final Pages / 19/1/06
16 a
Results obtained differ to those in question 15 as the first counter is not replaced, altering the probability of drawing the second counter as displayed in the tree diagram.
Maths Quest challenge (page 563)(36 − 18 ) cm2 ≈ 4.8 cm2
Exercise 13E — Independent and dependent events1 a Yes b i ii c
2
3
4 a b c d
5 a 0.28 b 0.12 c 0.42 d 0.186 a C b D
7 a b c d
8 a b c
9 0.9
10
11 a b c d
12 a b c
13 a b c
Exercise 13F — Karnaugh maps1 a
b i ii iii =
2 a
b c d
3 a
b c = d =
4 a
b = c
5 a
b 0.01 c 0.99 d 0.91
6 a
b 0.46
7 a
b 0.70
Red Green Blue Yellow Probability
Heads =
Tails =
Probability = = 1
Spinner 1
ProbabilityRed Blue Green Yellow
Red =
Spinner 2 Blue =
Green =
Probability = = = = 1
b
c
1
3---
2
3---
R
G
R
G2–3
2–3
1–3
1–3
2–4
2–4 R
G
Outcomes
RR
RG
GR
GG
Probability1–6
1–3
1–3
1–6—1
3
1
2---
1
6---
1
12------
1
40------
5
36------
16
25------
64
125---------
1
25------
4
25------
3
77------
48
77------
8
77------
18
77------
1
37------
1
1369------------
73
1369------------
1
14------
1
5---
1
5---
1
10------
1
3---
1
17------
1
221---------
25
102---------
26
145---------
136
435---------
221
435---------
1
20------
5
20------
2
20------
2
20------
10
20------
1
2---
2
20------
4
20------
2
20------
2
20------
10
20------
1
2---
3
20------
9
20------
4
20------
1
5---
4
20------
1
5---
1
2---
13
20------
4
20------
1
5---
1
12------
1
12------
1
12------
1
12------
4
12------
1
3---
1
12------
1
12------
1
12------
1
12------
4
12------
1
3---
1
12------
1
12------
1
12------
1
12------
4
12------
1
3---
3
12------
1
4---
3
12------
1
4---
3
12------
1
4---
3
12------
1
4---
1
12------
11
12------
1
2---
Die outcomes
Proba-bility1 2 3 4 5 6
Dieout-
comes
1 =
2 =
3 =
4 =
5 =
6 =
Probability = = = = = = 1
Bag 1
ProbabilityR B
Bag 2
R =
B =
Probability = = 1
Dart 1
ProbabilityA B C D
Dart 2
A 0.16 0.12 0.08 0.04 0.40
B 0.12 0.09 0.06 0.03 0.30
C 0.08 0.06 0.04 0.02 0.20
D 0.04 0.03 0.02 0.01 0.10
Probability 0.40 0.30 0.20 0.10 1
Drawer 1
ProbabilityG Y
Drawer 2
G 0.18 0.12 0.30
Y 0.42 0.28 0.70
Probability 0.60 0.40 1
Weather
ProbabilityS S′
Result
W 0.42 0.28 0.70
W′ 0.18 0.12 0.30
Probability 0.60 0.40 1
1
36------
1
36------
1
36------
1
36------
1
36------
1
36------
6
36------
1
6---
1
36------
1
36------
1
36------
1
36------
1
36------
1
36------
6
36------
1
6---
1
36------
1
36------
1
36------
1
36------
1
36------
1
36------
6
36------
1
6---
1
36------
1
36------
1
36------
1
36------
1
36------
1
36------
6
36------
1
6---
1
36------
1
36------
1
36------
1
36------
1
36------
1
36------
6
36------
1
6---
1
36------
1
36------
1
36------
1
36------
1
36------
1
36------
6
36------
1
6---
6
36------
1
6---
6
36------
1
6---
6
36------
1
6---
6
36------
1
6---
6
36------
1
6---
6
36------
1
6---
11
36------
2
36------
1
18------
20
36------
5
9---
35
120---------
40
120---------
75
120---------
5
8---
21
120---------
24
120---------
45
120---------
3
8---
56
120---------
7
15------
64
120---------
8
15------
35
120---------
7
24------
61
120---------
13D
13F
5_61_03274_MQV10 - A 1-15_tb Page 759 Thursday, January 19, 2006 7:42 PM
760 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
8 a 20 b i ii
c Events A and B are not independent.
d
9 a
b i = ii
c Events A and B are not independent since Pr(A ∩ B) ≠ Pr(A) × Pr(B).
10 a
b i = ii =
c Events A and B are independent since Pr(A ∩ B) = Pr(A) × Pr(B).
11 a
b
12 a
b
c 0.84
10 Quick Questions 21 2 3 4 5
6 7 8 9 10
Exercise 13G — Subjective probability1 a The outcome depends upon whether it is a Test
match or a one-day game and how effective the bowlers and batsmen are; not forgetting the pitch usually favours spin bowling.
b The outcome depends on which team is best on the day and which team can adjust to the conditions.
c No. The third one has an equal chance of being a girl or a boy.
d This is not necessarily true. Current position and form of both teams should be used as a gauge.
e It does not mean it will rain again on Friday.
f There is no certainty about that. It depends upon the condition of your house.
g Cricket games are not won or lost by the attractiveness of the uniform.
h It is possible to get 6 Heads in a row on a normal coin.
i They will have a good chance but there is no certainty. The country with the better competitors on the day of each event will win.
j This is dependent on the person’s country or state of origin.
2 a You still have a chance.
b No horse is certain to win. Lots of problems can occur on the track.
c This is not true. Even though Heads and Tails have equal chances, it does not mean half the results will show Heads.
d Favourites do not always win.
e Sometimes outsiders pay well, if you back the right one! You can lose more money than you win.
3 a There is a contradiction. The job was never hers. She had to do well to win the position.
b The team may have had a lead but a match is won when finished.
c No horse is sure to win.
4 Answers will vary. Class discussion required as there are many factors to consider.
Maths Quest challenge (page 581)
1 a b 2 or 0.0166
Summary1 chance, likelihood
2 zero, one
3 event
4 equally-likely
5
6
7
A A′ Probability
B
B′
Probability = = 1
A A′ Probability
B
B′
Probability = = 1
A A′ Probability
B =
B′ =
Probability = = 1
Outcome Probability
A ∩ A 0.36
A ∩ B 0.24
B ∩ A 0.24
B ∩ B 0.16
A B Probability
A 0.36 0.24 0.60
B 0.24 0.16 0.40
Probability 0.60 0.40 1
7
10------
9
20------
6
20------
3
20------
9
20------
8
20------
3
20------
11
20------
14
20------
7
10------
6
20------
3
10------
11
40------
6
40------
17
40------
17
40------
6
40------
23
40------
28
40------
7
10------
12
40------
3
10------
28
40------
7
10------
17
40------
121
400---------
99
400---------
220
400---------
11
20------
99
400---------
81
400---------
180
400---------
9
20------
220
400---------
11
20------
12
40------
3
10------
220
400---------
11
20------
220
400---------
11
20------
A B
43
1
2
x
9
10------
2
3---
1
4---
15
52------
17
18------
4
13------
3
8---
5
9---
1
12------
1
6---
1
3---
1
20------
1
20------
1950
117 453-------------------
number of times an event has occurred
total number of trials---------------------------------------------------------------------------------------------
frequency of the score
total sum of frequencies---------------------------------------------------------
n E( )n S( )------------
5_61_03274_MQV10 - A 1-15_tb Page 760 Thursday, January 19, 2006 7:42 PM
A n s w e r s 761
an
swers
➔
Maths Quest 10/ Final Pages / 19/1/06
8 odds
9 ,
10 complementary
11 Pr(A) + Pr(A′) = 1
12 mutually exclusive
13 Pr(A ∪ B) = Pr(A) + Pr(B)
14 Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)
15 two-way tables, tree diagrams, Venn diagrams
16 independent
17 Pr(A ∩ B) = Pr(A) × Pr(B)
18 Karnaugh map
19 Subjective
Chapter review1 a and d are equally likely; b and c are not.
2 or 0.8
3 a b c d
4 a
b
c
5 a i 50 ii 7 iii 25 iv 8
b i ii iii
c i ii
6 a b c $28
7 a b c
8 a 4–7 b 1–6 c 8–25
9 A
10 a b
11 a b
12 C
13 B
14 a Yes b Pr(A) = and Pr(B) =
c
15 a No b Pr(A) = , Pr(B) = , Pr(A ∩ B) =
c
16 a
b 6
c No. Frequency of numbers is different.
d
e i ii iii
f i ii iii
g 50
17 a
b No
c 0 and 6
d 3
e 0 and 6, 1 and 5, 2 and 4
18 a
b i ii iii iv
ba b+------------
aa b+------------
4
5---
1
13------
1
4---
2
13------
3
4---
A B ξ
A B ξ
A
C
Bξ
1
2---
3
50------
6
25------
Friedrice
Chickenwings
ξ
Dim sims
= 50
2
11
6 5
410 12
1
25------
3
7---
4
7---
7
10------
2
7---
5
17------
1
7776------------
7775
7776------------
Sum 2 3 4 5 6 7 8 9 10 11 12
Frequency 1 2 3 4 5 6 5 4 3 2 1
3
8---
5
8---
1
2---
1
6---
2
3---
1
4---
1
13------
1
52------
4
13------
Die 1 outcomes
Die
2 o
utc
om
es
1
1
2
3
4
5
6
2 3 4 5 6
(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)
1
36------
1
6---
1
18------
1
36------
1
6---
1
18------
Die 1 outcomes
Die
2 o
utc
om
es
1
0
0
1
2
3
2 3
(1, 1) (1, 2) (1, 3)
(0, 1) (0, 2) (0, 3)
(2, 1) (2, 2) (2, 3)
(3, 1)
(1, 0)
(0, 0)
(2, 0)
(3, 0) (3, 2) (3, 3)
F
F'
F
F'1–8
7–8
1–8
7–8
F
F'
1–8
7–8 1–
8
7–8
1–8
7–8 1–
8
7–8
F
F'
F
F'
F
F'
F
F'
1–8
7–8
Outcomes
FFF
FFF'
FF'F
FF'F'
F'FF
F'FF'
F'F'F
F'F'F'
Probability1–8
1–8
1——512× =1–
8×1–8
1–8
7——512× =7–
8×1–8
7–8
7——512× =1–
8×1–8
7–8
49——512× =7–
8×7–8
1–8
7——512× =1–
8×7–8
1–8
49——512× =7–
8×7–8
7–8
49——512× =1–
8×7–8
7–8
343——512× =7–
8×
1
512---------
343
512---------
21
512---------
11
256--------- 13F
13G
5_61_03274_MQV10 - A 1-15_tb Page 761 Thursday, January 19, 2006 7:42 PM
762 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
19 a
b
c
d
20 B
21 a b c
22 a b
23 a
b i = ii iii
24 a
b i = ii =
c Events A and B are not independent sincePr(A ∩ B) ≠ Pr(A) × Pr(B).
25 a Whether it rains or not on Thursday is not affected by what happened on Monday, Tuesday or Wednesday. It can still rain on Thursday.
b The team’s win or loss depends upon how other players bat and bowl or how the other team plays.
c There is an equal chance of having a boy or a girl.
26 a If you were defeated, the opponent was the winner.
b The slowest motocross rider could not win the race. He/she must have been fastest.
c The person elected was the most popular choice for the position.
Chapter 14 StatisticsAre you ready?1 a Suitable b Not suitable (irrelevant)
c Suitable
2 Junior school:
Middle school:
Senior school:
3 a Numerical, continuousb Categorical, nominalc Categorical, ordinal
4 a Most popular: cartoonsleast popular: documentaries and lifestyle programs
b 50 c 405 Junior 180° middle 109° senior 71°6 a 20% b 42.9% c 28.6%7 a Number of kilograms: independent
total cost: dependentb Temperature: independent
number of swimmers: dependentc Age: independent
height: dependent
Exercise 14A — Collecting data1 By observation: a, e, f, i.
By questioning: b, c, d, g, h.2 Check with your teacher.3 a Suitable, as it requires yes/no answers
b Suitable, as it requires yes/no answersc Suitable, as the answers can be easily put into
categories — main course, dessert, entree and so on.
d Suitable, as the answers can be easily put into categories — daily, once a week, and so on.
e Not suitable, as it requires a lengthy responsef Not suitable; it is an ambiguous question; the
answer will probably depend on who the ‘somebody’ is.
g Suitable, as the answers can be easily put into categories — excellent, good, fair, poor
h Not suitable, because it is irrelevant4 a 14
b Assign the numbers 1–200 to each member; put 200 numbered pieces of paper in a container and mix; select 15 pieces and match selected numbers with the corresponding names on the list. Another method could involve the use of a graphics calculator.
Die
Probability1 2 3 4
Coin
H =
T =
Probability = = = = 1
DVD
ProbabilityS S′
Video
S =
S′ =
Probability = = 1
A A′ Probability
B =
B′ =
Probability = = 1
Die outcomes
Coin
ou
tcom
esH
T
1 2 3 4
(H, 1) (H, 2) (H, 3) (H, 4)
(T, 1) (T, 2) (T, 3) (T, 4)
H
T
H1
H2
H3
H4
T1
T2
T3
T4
1–2
1–2
1
4
2
3
1–4
1–4
1–4
1–4
1
4
2
3
1–4
1–4
1–4
1–4
Outcomes Probability
—1
1–2
1–8=1–
4×1–2
1–8=1–
4×1–2
1–8=1–
4×1–2
1–8=1–
4×
1–2
1–8=1–
4×1–2
1–8=1–
4×1–2
1–8=1–
4×1–2
1–8=1–
4×
1
4---
1
8---
1
8---
1
8---
1
8---
4
8---
1
2---
1
8---
1
8---
1
8---
1
8---
4
8---
1
2---
2
8---
1
4---
2
8---
1
4---
2
8---
1
4---
2
8---
1
4---
1
19------
21
38------
15
38------
1
169---------
1
221---------
15
80------
9
80------
24
80------
3
10------
35
80------
21
80------
56
80------
7
10------
50
80------
5
8---
30
80------
3
8---
15
80------
3
16------
59
80------
21
80------
16
50------
14
50------
30
50------
3
5---
12
50------
8
50------
20
50------
2
5---
28
50------
14
25------
22
50------
11
25------
28
50------
14
25------
30
50------
3
5---
1
2---
13
43------
17
86------
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swers
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5 a B
b B
6 Answers will vary for each example. One possible solution has been provided.
a By the year the student is enrolled in
b By gender
c By the course they are enrolled in
d By the university they graduated from
7 9 customers in total: 3 customers with standard phone line connection and 6 with broadband
8 D
Exercise 14B — Presenting categorical and discrete data1 Categorical: a, d, f, h, k Numerical: b, c, e, g, i, j, l
2 Discrete: g, j Continuous: b, c, e, i, l
3 Nominal: a, d, f, h, k Ordinal: none
4
5 a 1°C
b July
c April and October
d 21°e A country in the Northern Hemisphere where
winter is in December–February and summer is in June–August.
6 a C b C c D
d C e D
7
8
9 a 2 b 1 c 9
10
11
12 a
b 50% c 8.33%
13 Key: 1| 2 = 12 years of age
14 a Key: 4 | 2 = 42 seconds
b Key: 4 | 2 = 42 seconds
4* | 7 = 47 seconds
c Graph in part b. The smaller categories better show the spread of the data.
Maths Quest challenge (page 605)Piece D
The Inc
redibl
es
The Fant
astic F
our
Charlie
and th
e Choc
olate F
actory Shr
ek
Bend it
Like Beck
ham
121086420
f Favourite movies
302826242220181614121086420 2003 2004 2005 2006
Model AModel BModel C
f
Year
MonTue
WedThuFriSat
Sun
= 10 ‘Happy meal deals’
Happy meal deals
Stem Leaf12345678
2 7 8 90 1 3 4 6 7 8 92 4 55 6 81 91 2 4 6 7 93 50 2
LeafBoys
Stem LeafGirls
9 89 8 7 6 5 3 2 2
1 0 0
345
0 2 4 6 7 8 90 1 2 6 8 9
LeafBoys
Stem LeafGirls
9 83 2 2
9 8 7 6 51 0 0
3*4*4*5*5*
0 2 4 6 7 8 90 1 26 8 9
A+ A B+ B C+Marks
Marks obtained in a maths test
TV commercials 45%
Beauty salonspromotion 3%
Radio commercials 10%
Newspaper ads 5%
Women’smagazine ads 25%
Promotionsin major shops 12%
Promotional advertising budget
Sleep
Shopping
Work
Watching TVHouseworkEating
CookingTravel
Maya’s day
14A
14B
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Exercise 14C — Representing data grouped into class intervals
1
2 a and b
3 a
b
4 E
5
6
7 a cf: 2, 6, 13, 23, 35, 43, 47, 50%cf: 4, 12, 26, 46, 70, 86, 94, 100
b
8 a 35 b 25 c 50 d 22
9 a
b and c
d and f
e i 13 ii 9 g i 18 ii 12
h 50% of scores lie below 18 and 30% of scores lie below 12
10 a
Class interval Frequency
120–129 6
130–139 6
140–149 5
150–159 3
160–169 7
170–179 3
Total 30
Size of house (m2)
20181614121086420
f
100 150 200 250 300 350 400
Data
300250200150100500
f
10 20 30 40 50 60 70 80 90
Data
20151050
f
110 120 130 140 150 160 170 180 190
Age
60
50
40
30
20
10
015 20 25 30 35 40 45 50 55
Cum
ula
tive
freq
uen
cy
Weekly spending ($)
1009080706050403020100
10 20 30 40 50 60
Cum
ulat
ive
freq
uenc
y (%
)
Class interval Frequency
0–9 8
10–19 9
20–29 6
30–39 5
40–49 2
Total 30
Score Frequency
0–4 3
5–9 5
10–14 4
15–19 5
20–24 3
25–29 3
30–34 4
35–39 1
40–44 2
Total 30
Data
50
40
30
20
10
0
100
80
60
40
20
030 35 40 45 50 55 60 65 70
Cum
ula
tive
freq
uen
cy
Cum
ula
tive
freq
uen
cy (
%)
Number of books
9876543210
f
100 20 30 40 50
Number of books
30
25
20
15
10
5
100
75
50
25
010 20 30 40 50
Cum
ula
tive
freq
uen
cy
Cum
ula
tive
freq
uen
cy (
%)
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b
c There is more detail shown about the categories; however, there is more information to take in.
11 D
Maths Quest challenge (page 618)j, f, g, b, i, k, h, c, e, a, d
10 Quick Questions 11 Questioning2 703 25294 445 Numerical — continuous6
7
8 Key: 13 | 5 = 135 cm
9
10 16 minutes
Exercise 14D — Measures of central tendency1 a i 7 ii 8 iii 8
b i 6.875 ii 7 iii 4, 7c i 39.125 ii 44.5 iii no mode
d i 4.857 ii 4.8 iii 4.8e i 12 ii 12.625 iii 13.5
2 Science: mean = 57.6, median = 57, mode = 42, 51 Maths: mean = 69.12, median = 73, mode = 84
3 a i 5.83 ii 6 iii 6b i 14.425 ii 15 iii 15
4 a Mean = 2.5, median = 2.5b Mean = 4.09, median = 3c Median
5 a 72 b 72
c 70–<806 a i 124.83 ii 120 − <129
b i 66.33 ii 66 − <707 a B b B c C d E8 a Mean = $32.93, median = $30
b
Mean = $32.50
c ; median = $30
d The mean is slightly underestimated; the median is exact. The estimate is good enough as it provides a guide only to the amount that may be spent by future customers.
9 a 3b 4, 5, 5, 5, 6 (one possible solution)c One possible solution is to exchange 15 with 20.
LeafGirls
Stem LeafBoys
98 8 3 2
2 2 19
13141516171819
2 31 1 2 9
1 51
Fre
quen
cy
Number of books
5
4
3
2
1
00 5 10 15 20 25 30 35 40 45
Fre
quen
cy
20
15
10
5
0M E H G
Favourite subjectsKeyM = MathematicsE = EnglishH = HistoryG = Geography
Maths
English
History
Geography
Favourite subjects
15 30 45 60 75
70
60
50
40
30
20
10
0
Cum
ula
tive
freq
uen
cy
Number of sit-ups
Class interval FrequencyCumulative frequency
0–9 5 5
10–19 5 10
20–29 5 15
30–39 3 18
40–49 5 23
50–59 3 26
60–69 3 29
70–79 1 30
Total 30
2
3---
Data
30
25
20
15
10
5
040 50 60 70 80 90 100
Cum
ula
tive
freq
uen
cy
Amount spent ($)
30
25
20
15
10
5
010 20 30 40 50 60 8070
Cum
ula
tive
freq
uen
cy
14C
14D
5_61_03274_MQV10 - A 1-15_tb Page 765 Thursday, January 19, 2006 7:42 PM
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Exercise 14E — Measures of spread1 a 15 b 77.1 c 9
2 a 7 b 7 c 8.5 d 39
3 a 3.3 kg b 1.5 kg
4 a
b i 62 ii QL = 58, QU = 67
iii 9 iv 14 v 6
5 ; IQR = 24
6 a
i Range = 23 ii IQR = 13.5
b
i Range = 45 ii IQR = 27.5
c
i Range = 49 ii IQR = 20
7 a
b Range = 17, IQR = 68 a C b C
9 a
b Outliers 1, 3 and 8 due to poor play, injury, tough opposition. Outlier 40 due to team playing well, improved skill, weaker opposition
10 a
b Men: mean = 32.3; median = 32.5; range = 38; IQR = 14 Women: mean = 29.13; median = 27.5; range = 36; IQR = 13
c No outliers
d Typically, women marry younger than men although the spread of ages is similar.
Exercise 14F — Bivariate data1 Independent Dependent
a Number of hours Test results
b Rainfall Attendance
c Hours in gym Visits to the doctor
d Lengths of essay Memory taken
e Cost of care Attendance
f Age of property Cost of property
g Number of applicants Cut-off ENTER score
h Running speed Heart rate
2
3 a Perfectly linear, positive
b No correlation
c Non-linear, negative, moderate
d Strong, positive, linear
e No correlation
f Non-linear, positive, strong
g Perfectly linear, negative
h Moderate, negative, linear
i Weak, negative, linear
j Non-linear, moderate, positive
k Positive, moderate, linear
l Non-linear, strong, negative
m Strong, negative, linear
n Weak, positive, linear
o Non-linear, moderate, positive
4 a
Battery life (h)
60
40
35
30
25
20
15
10
5
050 55 65 70 75 80
Cum
ula
tive
freq
uen
cy
Class interval
140
55
50
45
40
35
30
25
20
15
10
5
0120 130 150 160 170 180 190 200
Cum
ula
tive
freq
uen
cy
2 4 6
610.5 16.5 24
29
8 10 12 14 16 18 20 22 24 26 28 30
0
28 19 35.5
47
10 20 30 40 50
90
87 136100 111 120
95 100 105 110 115 120 125 130 135 140
8
9 26181412
10 12 14 16 18 20 22 24 26
0
1 3 8 15 4039302824
5 10 15 20 25 30 35 40
×× ××
15 20 25 30 35 40 45 50 55 60 Age
Men
Women
4.64.44.24.03.83.63.43.23.02.82.62.42.22.01.81.61.4
Cos
t ($1
000)
30 40 50 60 70 80 90 100 110 120Number of guests
121110987654321
Num
ber o
f bag
s so
ld
30 35 40 45 50 55 60 65 70 75 80Cost ($)
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b Negative, linear, moderate. The price of the bag appears to affect the number sold; the more expensive the bag, the fewer sold.
5 a
b Moderate positive linear correlation. There is evidence to show that the larger the number of bedrooms, the higher the price of the house.
c Various answers; location, age, number of people interested in the house, and so on.
6 a
b Strong, positive, linear correlation
c Various answers — some students are of different ability levels.
7 a
b Weak, negative, linear relation
c Various answers, such as some drivers are better than others, live in lower traffic areas, traffic conditions and so on.
8 a T b F c T d F e T
9 B 10 C 11 D
10 Quick Questions 21 Mean = 30.3, median = 29, mode = 18
2 Mean = 31.8, median = 38, mode = 46
3 1.1
4 $2.63
5 $2.25
6 50
7 2
8 48
9 26
10
Exercise 14G — Lines of best fitNote: Answers may vary depending on the line of best fit drawn.
1 a and b
c Using (23, 3) and (56, 8), the equation is
P = d −
2 a and b
c Using (8, 47) and (12, 74), the equation isE = 6.75h − 7
d On average, students were paid $6.75 per hour.
3 a 38 b 18
4 a i 460 ii 290 iii 130
b i 39 ii 24 iii 6
c y = −11.71x + 548.57
d y-values: i 466.60 ii 290.95 iii 127.01
x-values: i 36.60 ii 24.64 iii 5.86
5 a
b Using (1, 75) and (5, 150), the equation isC = 18.75x + 56.25
c On average, weekly cost of food increases by $18.75 for every extra person.
d i $206.25 ii $225.00 iii $243.75
420
400
380
360
340
320
300
280
260
240
220
200
180
160
140
Pri
ce (
$1000)
1 2 3 4 5 6 7
Number of bedrooms
100
90
80
70
60
50
40
30
20
10
0
Tota
l sc
ore
(%
)
1 2 3 4 5 6 7 8 9 10
Number of questions completed
6
5
4
3
2
1
0Num
ber
of
acci
den
ts
5 10 15 20 25 30 35 40
Number of lessons
0 10 20 30 40 50 60 Goalsscored
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Pet
rol use
d (
L)
10 20 30 40 50 60 70 80 90 100
Distance travelled (km)
5
33------
16
33------
130
120
110
100
90
80
70
60
50
40
30
20
10
0
Ear
nin
gs
($)
2 4 6 8 10 12 14 16 18
Hours worked
165
160
155
150
145
140
135
130
125
120
115
110
105
100
95
90
85
80
75
70
Cost
of
food (
$)
1 2 3 4 5 6 7
Number of people
14E
14G
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swers
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6 a
7 a
b M = 0.973d + 1.285
c Each day Rachel’s crystal gains 0.973 g in mass.
d 7.123 g; 8.096 g; 13.934 g; 14.907 g; interpolation (within the given range of 1–16)
e 17.826 g; 18.799 g; predictions are not reliable, since they were obtained using extrapolation
8 a D b C 9 E
Exercise 14H — Time series1 a Linear, downward b Non-linear, upward
c Non-linear, stationary in the mean
d Linear, upward e Non-linear, downward
f Non-linear, stationary in the mean
g Non-linear, stationary in the mean
h Linear, upward
2 a
b Linear downward trend
3 a
b Sheepskin products more popular in winter — discount sales, increase in sales, and so on.
c No trend
4 a
b General upward trend with peaks around December and troughs around April.
c Peaks around Christmas where people have lots of parties, troughs around April where weather gets colder and people less inclined to go out
d Yes. Peaks in December, troughs in April
5 a Peaks around Christmas holidays and a minor peak at Easter. No camping in colder months.
b Check with your teacher.
6 a
Upward linear
b In 2011 expected amount = 122
7 a
b Positive, strong, linear correlation; M = 0.247t − 6.408
c With every week of gestation the mass of the baby increases by 247 g.
d 3.719 kg; 3.966 kge 1.002 kgf 36 weeks
3.63.43.23.02.82.62.42.22.01.81.61.41.21.0
Mas
s (k
g)
31 32 33 34 35 36 37 38 39 40Weeks
181716151413121110987654321
Mas
s (g
)
1 2 3 4 5 6 7 8 9 10111213141516Day
Tem
per
ature
(°C
)
Day
May temperature
14.014.214.414.614.815.015.215.415.615.816.016.216.416.616.817.017.217.417.617.818.0
302928272625242322212019181716151413121110987654321
130
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
Sal
es (
× $1000)
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 QuarterYear2002 2003 2004 2005
Rev
enue
($1000)
100
95
90
85
80
75
70
65
60
55
50
45
40
121110987654321 121110987654321 1211109876543212004 2005 2006
Month
Year
120
110
100
90
80
70
60
50
40
30
20
10
01997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
Year
Enro
lmen
t
14
13
12
11
10
9
8
7
6
June July Aug Sep Oct Nov Dec AprJan Feb Mar MayMonth
Num
ber o
f chi
ldre
n
(1, 7)
(8, 11)
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b Yes, the graph shows an upward trend.
c y = x +
d i 15 ii 18 (The assumption made was that business will continue on a linear upward trend.)
8 The trend is non-linear, therefore unable to forecast future sales.
Maths Quest challenge (page 668)1 One possible answer is 2, 3, 10, 12, 13.
2 When the two sets each contain the same number of data or when both sets have the same mean.
3 5, 5, 4, 1, 0
Summary1 questioning 2 random, target
3 chance
4 strata, randomly, proportional, population
5 square root
6 categorical, measured, counted, categorised
7 discrete 8 stem-and-leaf
9 frequency 10 symbols
11 observation 12 sector
13 increasing 14 continuous, class intervals
15 polygons 16 gaps
17 midpoints 18 ogive
19 total 20 percentile
21 quantile 22 central tendency
23 average 24 ,
25 26 50th percentile
27 highest 28 modal
29 spread 30 lowest
31 upper, lower 32 QL, QU33 five-number 34 bivariate
35 scatterplot 36 independent, dependent
37 correlation, positive 38 linear, best fit
39 interpolation, extrapolation
40 reliable 41 time series
42 general trends
Chapter review1 a i Suitable ii Yes, no
b i Suitable
ii Very difficult; difficult; moderate; not difficult; easy
c i Suitable ii Under 5, 5–10, 10–15 and so on.
d i Suitable ii Mathematical methods, English, Chemistry and so on.
e i Not suitable — requires lengthy answer
f i Not suitable — ambiguous
g i Suitable ii Yes, no, unsure
h i Suitable ii Just empty box for the actual score
2 a Assign numbers to every employee (from 1 to 290); select 17 numbers randomly and match with the names of employees.
b Select 5 programmers and 12 technicians at random.
3 a i Categorical
b i Numerical ii Continuous
c i Categorical
d i Numerical ii Discrete
e i Numerical ii Continuous
f i Categorical
4 a
b
c
5 Key: 1 | 6 = 16 calculators
6 a
4
7---
45
7------
x Σxn
------= x Σfxn
--------=n 1+
2------------
LeafGraphics calculator
Stem LeafScientific calculator
9 6 3 29 8 7 5 2 0
9 6 4 4 06 3 0
01234
6 7 80 1 5 80 3 6 80 2 5 8 91 3
Class interval Frequency
20–<30 1
30–<40 4
40–<50 5
50–<60 3
60–<70 5
70–<80 4
80–<90 5
90–<100 3
Total 30
Sachertorte
Des
sert
Chocolatemud cake
Lemontang cake
Chocolatemousse
Pavlova
1 2 3 4 5 6 7 8 9 101112131415
Chocolatemousse
Sachertorte
Lemontangcake
Chocolatemudcake
Pavlova
Pavlov
a
Choco
late
mou
sse
Lemon
tang ca
ke
Choco
late
mud
cake
Sach
er to
rte
Dessert
14H
14H
5_61_03274_MQV10 - A 1-15_tb Page 769 Thursday, January 19, 2006 7:42 PM
770 A n s w e r san
swers
Maths Quest 10/ Final Pages / 19/1/06
b and c
d and f
e i 8 ii 14using raw data:
i 8 ii 14
g i 45 ii 70
7 a Mean = 11.55; median = 10; mode = 8
b Mean = 36; median = 36; mode = 33, 41
c Mean = 72.18; median = 72; mode = 72
8 a Mean = 32.03; median = 29.5
b
c Mean = 31.83
d
e Median = 30
f Estimates from part c and e were fairly accurate.
g Yes, they were fairly close to the mean and median of the raw data.
9 a
b Year 7: mean = 26.83; median = 27; range = 39;
IQR = 19; sd = 11.45
Year 12: mean = 40.7; median = 39.5; range = 46;
IQR = 20; sd = 12.98
c The typing speed of Year 12 students is about 13
to 14 wpm faster than that of Year 7 students. The
spread of data in Year 7 is slightly less than in
Year 12.
10 iii Calculate % cf
ii Construct an ogive
iii Find the 25th and the 75th percentiles from ogive
iv Calculate IQR
11 a Number of questions — independent; mark on a
test — dependent
b
c Strong, positive, linear correlation; the larger the
number of completed revision questions, the
higher the mark on the test.
d Different abilities of the students
12 a i 12.5 ii 49
b i 12 ii 22.5
c y = x −
d i 12.33 ii 49 and i 11.82 ii 22.05
13 a and b
L = 1.062n + 19.814
c 25.124 cm; 27.248 cm; 29.372 cm; 31.496 cm;
32.558 cm; 35.774 cm; 36.806 cm; 38.93 cm;
39.992 cm
d Interpolation (within the given range of 1–20)
e 42.116 cm; 43.178 cm; 44.24 cm
f Not reliable, because extrapolation has been used.
Class interval Frequency
0–9 2
10–19 7
20–29 6
30–39 6
40–49 3
50–59 3
60–69 3
Total 30
5
4
3
2
1
0
f
Number of students attending20 30 40 50 60 70 80 90 100
Attendance
30
25
20
15
10
5
0
100
75
50
25
020 30 40 50 60 70 80 90100
Cum
ula
tive
freq
uen
cy
Cum
ula
tive
freq
uen
cy (
%)
Age
30
25
20
15
10
5
010 20 30 40 50 60 70
Cum
ula
tive
freq
uen
cy0 10 20 30 40 50 60 70
Year 12
Year 7
100
90
80
70
60
50
40
30
20
10
0
Number of questions
Tes
t re
sult
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5_61_03274_MQV10 - A 1-15_tb Page 770 Thursday, January 19, 2006 7:42 PM
A n s w e r s 771
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Maths Quest 10/ Final Pages / 19/1/06
14 a Linear downward b The trend is linear.c
About 65 occupantsd Assumes that the current trend will continue.
Strategies for investigation and problem solvingCreate a table, then look for a pattern or a result1 a 3.2 b 2.4 c 3.1 d 3.2
e 2.9 f 1.2 g 2.7 h 4.52 s = 4 units3 Radius = 3 units4 x = 25 x = 5 and x = −2
Draw a diagram, then look for a pattern or a result1 48 2 243 3 108 4 545 294 6 32 7 40 8 405
Set up equations and find a solution, making use of technology such as a computer spreadsheet1 a 3.214 466 b 2.434 838 c 3.115 163
d 3.229 708 e 2.942 268 f 1.165 427g 2.724 775 h 4.526 176
2 x = 1.584 9733 The population is predicted to reach 1000 during the
year 2012.4 x = 11.185 13.5 minutes
Work backwards from the answer1 $1240
2 $48 900
3 $75
4 Original price = $177.95
5 Overall discount = 33.5%
Use a process of elimination1 4 motorboats
2 4 orchestral CDs
3 7 kitchens
4 8 tractors
5 7 tractors
Look at similar but simpler problems1 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
2 9 × 8 × 7 × 6
3 23 × 22 × 21 × 20
4 30 × 29
5 50 × 49 × 48 × 47 × 46
Create a mathematical model of a complicated situation1 Approximately 12.30 pm on day 5.
2 Approximately 1.10 pm on day 2.
3 Approximately 4 pm on day 3.
4 Approximately 9.55 am on day 2.
5 Approximately 8.15 am on day 3.
Communicating, reasoning and reflecting1 x = −7 and x = −1
2 640
3 x = 0.314 980
4 35.875%
5 2 overlockers
6 10 × 9
7 About 4 pm on day 2
8 A spreadsheet could be set up with a column for the independent variable, x, a column with the formula for the first function and a column for the second function. Finding the same value in both columns 2 and 3 would help solve the problem. Communicating and reasoning involves stating those values of x for which the second column had a lesser value than those in the third column. Reflecting on this method, it might be decided that finding the x-value was difficult and that a better way to find it might be with a fourth column of differences so that when the differences change from positive to negative the required x-value is close.
9 A spreadsheet of interest returns throughout the year will indicate that Soonju is ahead in interest during the year but will ultimately end up with the same interest at the end of the year. Communicating and reasoning is in terms of the spreadsheet and formulae used. On reflection, it might be reasonable to ponder the effect of weekly or even daily interest calculations (which some financial organisations offer).
10 Communication and reasoning would involve mention of all measures of central tendency, leading to the most appropriate in this case. Reflection might involve other scenarios for which the same measure is again appropriate.
11 Mario really needs a first estimate of the value of x, so that the spreadsheet he sets up will not be enormous. He will also then be able to use smaller increments to find x accurately. On reflection, it might be apparent how necessary it is to quickly find a rough estimate of the solution in order to narrow down the options and quickly find the better approximation of the answer.
12 A sketch using all information will ensure that an appropriate trigonometry equation is set up. If no sketch was made, and the solution was unnecessarily lengthy and/or erroneous, reflection might lead us to recommend a sketch in a similar sort of problem in the future.
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5_61_03274_MQV10 - A 1-15_tb Page 771 Thursday, January 19, 2006 7:42 PM
Maths Quest 10/ Final Pages / 19/1/065_61_03274_MQV10 - A 1-15_tb Page 772 Thursday, January 19, 2006 7:42 PM
