New Century Maths 11
b
45
100%  62.5%
72
c
4
9
100%  7 %
52
13
d
16
1
100%  33 %
48
3
Mathematics General
Preliminary course
(Pathway 2)
Worked Solutions
5 a 52
b6
Chapter 5
c 12
SkillCheck
d 26
7
1a
25
b
e4
f5
1
11
6a
c
d
2a
b
1
4
b
6
100%  30%
20
c
12 3

20 5
2
3
27
40
4 1

20 5
7a
3
25
c 1
1 7

8 8
d 1
4 5

9 9
3 a 17  20  0.85
b 9  36  0.25
c 25  40  0.625
d 4  15  0.26
4a
10
100%  40%
25
b
Total  4  16  11  1  32
4 1
 
32 8
1
c 0%
d
28
100%  87.5%
32
9
1
9
d
7 Total = 15 cards
e1
a P(2, 4, 6,8,10,12,14) 
8 a Teacher to check.
b P (1,3,5, 7,9,11,13,15) 
b AC
c VU
e VU
8
15
1
15
c
d AC
d P (1,10,11,12,13,14,15) 
f Teacher to check.
g VU
7
15
7
15
8 Total = 10 letters
h Teacher to check.
a
2 1

10 5
b
3
10
iU
jL
2 1

10 5
c
Exercise 5-01
d
9
5 1

10 2
1
3
10 Total = 24 coins
6 Total = 6  3  9
a
8
1
100%  33 %
24
3
b
1
1
 100%  4 %
24
6
a
6 2

9 3
c
19
1
 100%  79 %
24
6
b
3 1

9 3
d
18
100%  75%
24
c
0
0
9
2
e
11
6
100%  25%
24
4
2
 100%  66 %
6
3
16 Each of the 16 teams is not equally
likely to win the competition, the levels of
ability/success are different.
17 a P (9to 20) 
12
100%  60%
20
12 Total = 6 equal parts
a
2 1

6 3
1
b
6
b P(1, 4,9,16) 
4
100%  20%
20
c P(3, 6,9,12,15,18) 
6
100%  30%
20
18 Total = 7
3 1
c 
6 2
a
2
7
5
6
b
4
7
1
2
c
1
7
6
3

52 26
d
5
7
4
1

52 13
e
5
7
d
36 9

52 13
f 0
e
2
1

52 26
f
1
52
g
2
1

52 26
h
13 1

52 4
d
13 a
b
c
14
8
1

1200 150
2 1
15
  0.16
12 6
Exercise 5-02
3
b 12
ci
3
100%  25%
12
ii
6
 100%  50%
12
iii
6
 100%  50%
12
iv
3
100%  25%
12
c 12
d i P( H 2, H 4, H 6) 
ii P(T 1, T 2) 
3 1

12 4
2 1

12 6
iii P( H 2, H 3, H 4, H 5, H 6) 
3a
5
12
b8
ci
3
 0.375
8
ii
4
 0.5
8
iii
1
 0.125
8
4
4
 0.5
8
iv
5a2
ii
6 1

36 6
ei7
b4
ii 2,12
c 16
7a
d 1024
6a
b 12
ci
6 1

12 2
ii
2 1

12 6
iii
6 1

12 2
iv
1
12
b 36
ci
3
1

36 12
ii
5
36
iii
3
1

36 12
8a
iv 0
v
vi
di
18 1

36 2
bi
1
 0.05
20
8 2

36 9
ii
8
 0.4
20
1
36
iii
4
 0.2
20
5
c9
2
 0.1
20
iv
1
9
di
9a
ii
3 1

9 3
iii
8
9
iv
5
9
v
2
9
bA
1
ci
8
ii
3
8
iii
7
8
Exercise 5-03
1 a 2  6  6  72
bi
iv
4 1

8 2
1
72
ii
v
2 1

8 4
P(T 26, T 62, H 26, H 62) 
vi
4 1

8 2

10 a
4
72
1
18
2
2  2  6  24
B
3a
bi
2
100%  25%
8
b Sat and Sunday
 2 stages
ii
3
1
 100%  37 %
8
2
A
6
iii
7
100%  87 12 %
8
bi
1
36
iv
1
100%  12 12 %
8
ii
1
36
v
4
100%  50%
8
iii
26 10 65

36  36 324
vi
4
100%  50%
8
4 1 26  26  676
5 a 4  5  4  80
b i 4  5 1  20
14
6  6  6  6  6  7776
C
15 a 12
{25,26,29,52,56,59,62,65,69,92,95,96}
b6
ii 4  4  4  64
iii 3 3 4  36
c
3 1
  0.25
12 4
6
d 4  3 2  24
26  26  26 10 10 10
e 4  4  16
 17576000
D
7 a 110 10 10  1000
16 9  8  72
17 5  4  3 2 1  120
18 a i 2
b 5-digit postcodes
8 8  8  8  8  4096
9 a 5  2  3 3  90
b i 5 1 3 2  30
ii 3 2 1 3  18
ii 2  2  4
iii 2  2  2  8
iv 2  2  2  2  16
b 2  4  8  16  30
iii 11 3 3  9
10 8  5  4  160
11 2  2  2  2  2  2  2  2  256
12
1110 10 10 10 10 10  1000
 000
13 a
36  36  36  36  36  36  2176
 782336
7
Exercise 5-04
b
1  P(two tails)
1
 1
4
3

4
1
c
2 1

4 2
d
2a
1  P(6)
1
 1
6
5

6
1  P(one tail)
1
 1
2
1

2
e
b
1  P(less than 3)
2
 1
6
4

6
2

3
f
1  P(no tails)
1
 1
4
3

4
4
c
1  P(multiple of 3)
2
 1
6
4

6
2

3
3
a
1
4
1
4
a
3
8
b
1  P(two tails)
3
 1
8
5

8
c
3
8
8
1  P (contains 7)
d
1  P(one tail)
3
 1
8
5

8
e
1
8
f
1  P(no tails)
1
 1
8
7

8
 1  P (7,17)
2
 1
20
18

20
 0.9
e
1  P(divisible by 5)
 1  P(5,10,15, 20)
4
 1
20
16

20
 0.8
f
5a
P (10,11,..., 20)
11

20
 0.55
b
1  P(divisible by 3)
 1  P(3, 6,9,12,15,18)
6
 1
20
4

20
 0.7
c
P (1, 2,...,9)
9

20
 0.45
d
1  P(factor of 20)
 1  P(1, 2, 4,5,10, 20)
6
 1
20
14

20
 0.7
6
1  P(win prize)
8
 1
1000
992

1000
124

125
A
7a
1  P( P)
1
 1
11
10

11
9
b
1  P( B)
2
 1
11
9

11
c
1  P(vowel)
4
 1
11
7

11
d
1  P(A,B)
3
 1
11
8

11
8
1  P (sat,sun)
2
 1
7
5

7
9 total = 9
a
1  P(green)
4
 1
9
5

9
b
3 1

9 3
1  P(white)
2
 1
9
7

9
d
1  P(yellow)
1
 1
3
2

3
e
1  P (red)
 1 0
1
f
1  P(green,yellow)
7
 1
9
2

9
10
Total possibilities = 36
P (2,12)
2

36
1

18
P (not losing)
 1  P(2,12)
1
 1
18
17

18
B
c
10
11
1  P (rain)
 1  0.23
 0.77
 77%
f
1  P(black 7,8,9,10)
8
 1
52
11

13
12 a
1  P(10)
4
 1
52
12

13
13 Total = 164 seconds
a
63
 0.38414...
164
 0.384
b
1  P(picture card)
12
 1
52
10

13
c
1  P(club)
1
 1
4
3

4
b
1  P(red)
 1  0.384
 0.616
c
1  P(amber)
2
 1
164
81

82
 0.9878...
 0.988
d
1  P(black)
1
 1
2
1

2
e
1  P (red ace)
2
 1
52
25

26
d
1  P(green)
99
 1
164
65

164
 0.3963...
 0.396
e
11
1  P(red,amber)
65
 1
164
99

164
 0.6036...
 0.604
f
18 Total = 9 parts
a
1  P(win holiday)
1
 1
9
8

9
b
1  P(red,green,amber)
164
164
 11
 1
0
14
1  P(born 3rd June)
1
 1
365
364

365
1  P(cash or computer)
3
 1
9
2

3
Exercise 5-05
1 a 2  5  70  56  6 1  150
bi
56 28

150 75
ii
15
1  P(end 0,9)
2
 1
10
8

10
8
  100%  80%
10
16 a draw
b 100  42  51  7%
P  51,52,  52 
63
150
21

50

iii P(48, 49) 
2 a no
b yes, biased towards 6
17
1  P (win)
13
 1
20
7

20
17
150
c
22
100%  27.5%
80
d 27.5  28
3a1
b Total = 40
12
i
14
 0.35
40
ii
5
 0.125
40
iii
1  P(no children)
 1  0.125
 0.875
iv
P (2,3, 4,5)
25

40
 0.625
v
P(3, 4,5)
11

40
 0.275
vi
P (0,1, 2,3)
36

40
 0.9
c
bi
ii
1  P(cloudy)
 1

23
30
 50.65%
b female as greater than 50%
c
males  19855288  105638
 9799250
9799250

 22400000
19855288
 11055150.65...
 
 11055151
 
7–11 Teacher to check.
Exercise 5-06
1a
bi
ii
4
5 a No
7
30
6a
10056038
100%
19855288
 50.646...%
3
 150
40
 11.25
 11
12
 100%  42 76 %
28
B
10 1

30 3
3
1
 100%  37 %
8
2
4
100%  50%
8
ci
37 12 % of 88
37 12
 88
100
 33

13
ii
50% of 88
50

 88
100
 44
iii Teacher to check
iv P (2, 4, 6) 
b
d Total = 88
3
 0.5
6
1
150  25
6
c Teacher to check
i
30
 100%  34.090...%
88
 34.09%
3a
ii
47
100%  53.409...%
88
 53.41%
e They are close.
fi
37 12 % of 200
37 1
 2  200
100
 75
ii
50% of 200
50

 200
100
 100
bi
3
 0.8333...
36
 0.0833
ii
2
 0.05555...
36
 0.0556
iii
g Total = 200
i
80
100%  40%
200
97
ii
100%  48.5%
200
8
 0.22222...
36
 0.2222
c Teacher to check
di
h yes
2 a i Teacher to check.
ii
2
 120  6 23  7
36
iii
8
120  26 23  27
36
ii
1
 0.1666...
6
 0.167
3
 120  10
36
14
e Teacher to check
4ai
12
 0.2307...
52
 0.231
vi
21
 0.4375
48
 0.438
ci
ii
20
 0.2564...
78
 0.256
b
12
 78  18
52
A
ii
3
 48  12
12
iii
5
 48  20
12
6 Each outcome has an equal chance.
7ai
c Teacher to check
bi
1
 0.08333...
12
 0.083
iii
iv
84
100%  42%
200
26
100%  41.85...%
iv 540
 41.9%
ii
iii
P( H 2, H 4, H 6)
3

12
 0.25
23
100%  46%
50
55
 100%  39.28...%
ii 140
 39.3%
5 a 12
5
 0.1041...
48
 0.104
1
 48  4
12
b Yes, the coin seems to be coming up
heads around 42% of the time instead of
50%.
Sample HSC problem
a
12
 0.25
48
v
P (T 1, T 2, T 3, T 5, T 6)
5

12
 0.4166...
 0.417
15
b There are ten 1s  1
ci
8 2

36 9
ii
18 1

36 2
iii
1  P(5)
2
1
36
17

18
1 1 7
 
3 4 12
7
1
  100%  58 %
12
3
d
3 Being married and being single are not
equally likely outcomes.
4a
Chapter Review
1a
1
26
bi
1
8
ii
3
8
iii
4 1

8 2
b
P(a, e, i, o, u, y )
6

26
3

13
iv
c
1  P( w, x, y, z )
4
 1
26
11

13
1  P(no girls)
1
 1
8
7

8
v
2 a 25%
b
1 1 1
P(black)  1   
3 4 4
1

6
1
2
  100%  16 %
6
3
c
1  P(0,1 girls)
4

8
1

2
vi
4 1

8 2
5a
16
P(11, 22,33, 44,55, 66)
6

36
1

6
11 Total = 140
a
30
3

140 14
b
49
7

140 20
b
P(26, 62,35,53, 44)
5

36
6 2  2  2  2  2  2  2  128
7
6  6  6  216
C
8 a 10 10 10 10  10000
bi
1
100%  0.01%
10000
c
1  P (bicycle)
 1

18
140
61
70
77
100%  96.25%
80
12 a
b
3
 5000  187.5  188
80
13 a Teacher to check.
b Teacher to check.
10
ii
100%  0.1%
10000
c
1
6
d
1
 48  8
6
iii
5  5  5  5  625
625

100%  6.25%
10000
9
1  P(miss)
 100%  68%
 32%
10
1  P (win)
54
 1
240
31

40
 0.775
D
17