Math AI SL

Answers 475

8 a x = −1.49 b x = 1.84 18 b 80
c ii Solutions to n2 + n − 4200 are not whole
9 a x = −4.89, −1.10 b No real solutions numbers.

10 8 cm 19 0.3
20 24
11 r = 2.5, −3.5 21 3.28
22 0.264 or −1.26
12 r = 2 23 x = 1, y = −1, z = 2
24 20
13 b x = −3.61, 1.11

14 a x3 − 3x − 1 = 0

b x = −1.53, −0.347, 1.88

15 b x = −1.42, 8.42

c x = 8.42 since length must be positive Chapter 13 Prior
Knowledge
16 a 5x2 + 2x − 19.8 = 0

b x = 1.8 m

17 x = 5 or 10 7 8
5 5
18 c r = 6.50 cm or r = 3.07 cm 1 y = x +

19 4 2 1, 0.264, −1.26

20 20 3 x = − 5 , y = 22 , z = 65
99 9
21 (4, 5) or (−5, −4)

22 a 1 n( n − 3) b 10 4 1.46
2

Chapter 12 Mixed Practice Exercise 13A

1 4 cm 1 a y = 2x − 6 b y = x + 3

2 b x = −3.19, 4.69 2 a y = −3x + 15 b y = −1.5x + 18

3 b x = −3.70, −3, 2.70 3 a y = 0.5x + 0.5 b y = 4x − 16

4 a x2 + x − 10 = 0 b 2.70, −3.70 4 a y = −2.4x + 9.4 b y = −2x + 4
5 a y = 5x − 11.5
5 1.65 cm b y = 7 x +1
3
6 a 8s + 6f = 142; s + f = 20
6 a y = −4x + 27.7 b y = −0.4x − 4.82
b 11
7 a i Each year the car will travel 8200 miles.
7 $2
ii The car had travelled 29 400 miles when it
8 39 was bought.

9 a x + y = 10 000 b 39 AUD b 70,400

c 12x + 5y = 108 800 d x = 8400, y = 1600 c Ninth

10 u1 = −11, d = 3 8 a C = 1.25d + 2.50
11 a A = x2 + 2x
b $6.50
b x = 9.5 
c No, the cost would be $10.63.
c 42 m
9 a i k = 100 ii m = −8

12 8 b 0  t  12.5

13 0.5 or −1.5 10 a C = −50d + 670 b £430 000

14 −3 11 a a = 0.225, b = 3.63

15 b x = 2.73 b i The change in weight per week

16 40 or 61 ii The weight of the baby at birth

17 a V = 2x3 + 7x2 + 6x b 1.5 × 3.5 × 6

476 Answers

12 a C = 00..1152ee++ 10 for 0  e  250 8 a x = −3
2.5 for e > 250
b 0 = 30.25a − 5.5b + c

b £47.50 0 = 0.25a − 0.5b + c

13 a i C = 7.5m + 80 ii C = 5m + 125 12.5 = 9a − 3b + c

b m > 18  miles c a = −2, b = −12, c = −5.5

14 a D(p) = 1125 − 15p d y

b S (p) = 12 p − 360

c p = $28.33

15 a 1.5 m b i a = 0.18

c 2.1 m ii b = −1.3 − 11 x
d 21.1 m 2
0  I  12.5
0 12.5 < I  50 − 1 11
50 < I  150 2 2
I > 150 −
00..24 II
16 a T = − 2.5
− 12.5

0.45I − 20

b i £4 500 ii £15 500 9 a 1350 = 2500a + 50b + c
c £155 556 5100 = 40000a + 200b + c

Exercise 13B 3100 = 160000a + 400b + c
b a = −0.1, b = 50, c = −900

1 a y = x2 + 5x − 3 b y = 2x2 − 3x + 5 c If the business produces no items it will make

2 a y = 3x2 − 2x − 4 b y = −x2 + 4x + 2 a loss of £900 .
3 a y = −2x2 + 6x + 5 b y = 0.5x2 − 3x + 2
4 a y = x2 + 3.2x − 5.4 b y = −0.3x2 − 2x + 4.1 d i 250 ii £5350
5 a y = 4x2 − 8x − 12 b y = −x2 + 3x + 10
6 a y = −2x2 + 12x − 8 b y = 3x2 + 12x − 5 10 10.8 cm

11 a c = 1.45 b
2a
b i 1.4 = −

7 a (−5, 0) 8.41a + 2.9b + 1.45 = 0

b b = 3, c = −10 y ii a = −5, b = 14
c
c t = 0.281, 2.52s
12 a 3.88m2

b Assumes that R is always growing at the same

x rate, e.g. ignoring temperature.
2
−5 c The algae will eventually fill the pond, so its

surface area will no longer increase, but in the

model it continues to grow indefinitely.

−10

x = −23

Answers y b 477
y
Exercise 13C

1 a

8

y=3 3 x
−ln 4 y = −1
x
2 a y b y

1 ln 3 8
2 y
x
y = −2 y=1 x
3 a
y b

6

y=4

−1 x
−1.5
y = −2.5

x

478 Answers
4 a
y 14 a c = 0.0462 b 99.7 hours
y=2 1 ln 2
15 50 m s−1

16 a 66.6 °C

b i No change

ii Increase but stay less than 1

x Exercise 13D

1 a y = 4x2 b y = 0.5x4

2 a y = 33 x b y = 1.5 x

3 a y = 5 b y = 2
x x3

4 a y = 15 b y = 12
x 4x

b y 5 a y = x3 + 2x2 − 5x + 4
y=5
b y = 3x3 − 4x2 − x + 2
6 a y = 2x3 + 4x2 − 3x − 6

b y = −x3 − 3x2 + 2

2 7 a y = −2x3 + 6x2 + 1
b y = 0.5x3 − 3x2 + 0.3x + 10

− log2 ( 5 ( x 8 a y = −1.2x3 + 4x − 2.5
3 b y = 3x3 + 2x2 − 4.7x − 3.8

9 a F = 2.4x b i 9.6 N

ii 8.33 cm

10 a T = 0.2 l b i 1.55 s

ii 16 cm

11 a 5.63 × 106 Watts b 17.0 m s−1

12 a P = 900 000 b i 7200 Pa
V ii 11.25cm3
5 a y = 2 × 3x − 1 b y = 4x + 1.5

6 a y = 10 × 5−x + 6 b y = 6 × 2−x− 5 13 29
14 a −1 = −8a + 4b − 2c + d
7 a y = 1.2x + 3 b y = 4 × 0.5x + 2
−8 = −a + b − c + d
8 a y = 5e−x − 2 b y = ex + 4 2= a+b+c+d
−5 = 8a + 4b + 2c + d
9 a 4.68 billion

b 2% b a = −2, b = 0, c = 7, d = −3
c 5.27 billion 15 a T = −0.00342d3 + 0.251d2 + 1.04d − 0.265

10 a 1.5mg l−1 b i 11.4 ii 3.88%
b 0.556 mg l−1
c 9 hours and 2 minutes c 252 years

11 a k = 2.5, r = 0.206 b 1 205 000 3

16 x = kz2

12 a k = 45.1, a = 1.15 b 27.2 s 17 17.4%

13 a V = 8500 × 0.8t 18 f (5) = 38

b $912.68 19 7 days
c c = 8000, d = 500
20 f (x) = kx(x − 1)(x − 2) + x

Answers 479

Exercise 13E Chapter 13 Mixed Practice

1 a i 3 ii 90 iii y = 2 1 a 16% b 52%

iv 5 v 1 vi (0, 2) c 70 minutes b 6 = − b
2 a c = 3 2a
b i 2 ii 720 iii y = −5
12 = 36a + 6b
iv −3 v −7 vi (0, −5)
b y = 3.5
2 a i 0.5 ii 1080 iii y = −2 c a = − 1 , b = 4
3
iv −1.5 v −2.5 vi (0, −1.5)
3 a p = 1.5, q = 3.5
b i 5 ii 4 iii y = 3
4 a V = 0.4T b V = 117 cm3
iv 8 v −2 vi (0, 8)
5 a m = 90
3 a 3 b 8 c 2
b Change per year in the number of apartments
1
4 a 4 b 2 c −1 c 60
5 a 50 m c 5 m
b 1.8 m d Number of apartments initially

6 a i 1.68 m ii 0.52 m 6 a −8 p + 4q − 2r = −8

b 300° s−1 p + q + r = −2

c 0.81 m 8 p + 4q + 2r = 0

7 a 152 cm b 0.4 s c 140 cm b p = 1, q = −1, r = −2

8 a p = 40, q = 18, r = 0 7 a i 3 ii 180°

b 10.8 s b i a = 2 ii b = 1 iii c = −1

9 13.7 °C c 5

10 3.30 m 8 a $1.00 b $2.00

c 350  w  500

Exercise 13F 9 a a = 4, b = 5 b 2

1 a F y = 2 × 3−x + 1 c c = − 1
b F y = 4 × 3x − 2 2
2 a C y = x3 − 3x2 + x + 1
b C y = −x3 − x2 + 4x + 2 10 36%
3 a E y = 3cos(180x) − 2
b E y = −2cos(18x) + 1 11 a i 40 °C ii 20 °C iii 10 °C
4 a D y = −sin(0.5x) + 3
b D y = 10sin(90x) + 4 c y
5 a P = 67 × 1.01n
b Population growth rate may change over 100

the longer term due to, e.g. changes in 80
immigration, birth rate, death rate, etc.
There might be large errors by the time we get 60
to 2100.
6 a 0  t  55 40
b An exponential model so that the rate of decrease
slows down and the pressure tends to zero. 20
7 a 0 °C
b T = 80 × 0.87t + 20 1 2345 6 t

d i 94 = p + q ii 54 = p + q
e p = 80, q = 14 2
g i r = −0.878 f y = 14
h 36.7
ii y = 71.6 − 11.7t

i 52.8%

480 Answers

12 a 1.5 b C = 2.5 15 6.85 cm
c 3 hours 19 minutes 16 1.32 cm, 13.7 cm
17 0.219 cm
13 a i 100 m ii 50 m 18 per = 16.6 cm, area = 3.98 cm2
19 x = 3
b ii 2.4 20 r = 23.4, θ = 123
21 per = 33.5 cm, area = 78.6 cm
c h

100

80 Exercise 14B

60 1 a x + 2 y = 2 b x − 3y = 39

40 2 a 3x − 4 y = −60 b 5x + 2 y = 60
3 a 5x − 4 y = 6 b 4x + 3y = −13
4 a 12x + 3y = 11 b 12x − 4 y = −41

20 5 a y = −2x + 22 b y = −3x + 40

6 a y = 3 x + 5 b y = − 2 x + 2
2 4 5

10 20 30 40 t 7 a y = 2 x + 1 b y = − 1 x + 53
d a = −50, b = 18, c = 50 3 12 2 10

8 a y = −x − 26 b y = 2x − 16
3 3
Chapter 14 Prior
Knowledge 9 a i A, B, C, D, E ii F, G, H

iii [FG], [GH] iv No finite cells

b i A, B, C, D, E, F ii G, H, I, J, K, L

1 a 10.4 b (−0.5, −1) iii [GH], [HI], [IJ], [JG], [JL], [LK], [KI]
2 2x + 5y = 27
iv GHIJ, IJLK

Exercise 14A 10 a i A ii F

b i B ii C

11 a i B and C ii x = 4

1 a 3.49 cm, 8.73 cm2 b 9.07 cm, 36.3 cm2 b i A and B ii y = 3 − x

2 a 6.98 cm, 14.0 cm2 b 14.7 cm, 51.3 cm2 c i D and C ii 6 y + 4x = 25

3 a 33.5 cm, 134 cm2 b 25.3 cm, 63.3 cm2 12 a

4 a 20.3 cm2, 28.4 cm b 4.32 cm2 , 15.5 cm

5 a 43.3 cm2 , 34.2 cm b 15.4 cm2, 19.1 cm C

6 a 176 cm2 , 57.1 cm b 8.81 cm2 , 13.3 cm

7 per = 20.9 cm, area = 19.5 cm2

8 per = 27.5 cm, area = 47.0 cm2

9 10.1 cm D
A
10 a 80.2° b 17.5 cm2 B

11 40.1°

12 15.5 cm

13 a 11.6 cm b 38.2 cm

14 area = 57.1 cm2 , per = 30.3 cm

Answers 14 a 481
b C B

C BD
D E
A
A

13 a b

C

C D D E
A B A
B

b 15 y = 2x − 2.5
16 a x = 5.5
C c 1025 mbar b 22 °C
D 17 a i F, G d 31 mm

B ii A, B, G, I
A
b B, C, D, H, I

18 C; highest mean contamination at sites in that
region

19 a i D ii C iii D

b Contains all stores served by distribution

centre A

482 Answers

c y d y

13 12
12
11 11
10
10 9B
8
9 E 7C
8

7
6
5 D 6 E
5
4
4
3
3
2 D
1
−−11 AB x 2 A
1 2 3 4 5 6 7 8 9 101112131415 1

−2 1 2 3 4 5 6 7 8 9 10 11 12 x
−3 e 412 m

d (14, 1) to B 22 a 3x − 2 y = 1 b D

20 a i y = 2.5 − 0.5x c (9, 13)

ii x = 3 23 a D b (3, 6)

iii y = 4 − x c 2400 m2 d y = −x + 11

b (3, 1) e y
d y

5 10
C

4 CB 8

3 6 M
D
2 4
x A
1
A 2
12345

e B 2 4 B 10 x
21 b (7.83, 5.67) 68
c (7, 5)
24 a (25, 5) b 25.5 km

c e.g. the towns have no size − they are just a

point

25 a (31, 29) to be at the vertex furthest from all

towns on a Voronoi diagram

b Moves further away from B (i.e. larger x and

y coordinates)

Chapter 14 Mixed Practice

1 a 27.9 cm2 b 25.6 cm
2 8
3 1.18 m3

Answers 483

4 a y = −7.42x + 1440 b 142 km e y

5 a 5390 m b 5 6
c 4x + 10 y = 55 2 5

d It is the same distance from A, C and D. 4
3
6 a 7 km2, 316 people per km2 2C B

b 0.194 D
1
c B − cell has the highest probability of cholera
A
7 a (15, 25) b 15.8 km 1234567 x

8 a 6x + 4 y = 29 −1

b y f 16.7%
11 a i 23 °C
7D b y ii 28 °C iii 25 °C

6

5 9
4C
8
3E B B
2 7A
1A 6

123456789 x 5D
−1 4 EC
−2 3

9 25 600 2F G
10 a i y = 1
1
ii x = 3
iii y = 10 − 3x 123456789 x
b (3, 1)
c y c (6, 3), to 24 °C

12 6 cm

13 4.0cm and 3.4cm

5 14 a 44.0° b 1.18cm2

4 15 b 58.3° c 23.9 cm

16 a 82.3° b 111 cm

3 17 a 130° b 36.1%
2C B
c 18.2 °C

1 18 (3.1, 1.9) (although (1.25, 3.75) is also locally
A stable)

−1 1 23 45 67 x

−2

−3
d (3, 1)

484 Answers

19 a y 5 a Sufficient evidence that the mean is not 2.6
5
b Insufficient evidence that the mean is not 8.5
4 B
C3 6 a Sufficient evidence that the data does not
come from the distribution N(12, 3.42 )
2
1A D b Insufficient evidence that the data does not
come from the distribution N(53, 82 )
−2 −1 1 2 34 x
−1 7 a i Expected: 25, 50, 25, χ 2 = 7.26

−2 ii 2

b 8 km iii p = 0.0265, reject H0
b i Expected: 50, 30, 20, χ 2 = 1.00
20 a a2 + b2 = 1 b (1+ a,b)
ii 2
( ) c y− b = − 1 + a x − 1 + a
2 b 2 iii p = 0.607, do not reject H0
8 a i Expected: 10, 10, 10, 10, 10, χ 2 = 8.00
Chapter 15 Prior
Knowledge ii 4

1 18.4 iii p = 0.0916, do not reject H0
2 0.0280 b i Expected: 15, 15, 15, 15, χ 2 = 10.5
3 0.388
4 a 0.662 ii 3
b Significant positive (linear) correlation
iii p = 0.0145, reject H0
Exercise 15A 9 a i Expected: 66.7, 50, 40, 43.4, χ 2 = 18.7

1 a H0: µ = 23.4, H1: µ  23.4, one-tail ii 3
b H0: µ = 8.3, H1: µ  8.3, one-tail
2 a H0: µ = 300, H1: µ ≠ 300, two-tail iii p = 0.000320 , reject H0
b H0: µ = 163, H1: µ ≠ 163, two-tail b i Expected: 53.3, 26.7, 32, 48, χ 2 = 7.38
3 a H0: There is no correlation between
ii 3
temperature and the number of people
H1: There is a negative correlation between iii p = 0.0606, do not reject H0
the two; one-tail 10 a χ 2 = 6.81, do not reject H0
b χ 2 = 6.43, do not reject H0
b H0: There is no correlation between rainfall 11 a χ 2 = 10.1, reject H0
and temperature b χ 2 = 1.69, do not reject H0
H1: There is correlation between the two; 12 a χ 2 = 9.13, reject H0
two-tail b χ 2 = 5.12, do not reject H0
13 a χ 2 = 5.51, in sufficient evidence that they are
4 a Insufficient evidence that the mean is greater
than 13.4 not independent

b Insufficient evidence that the mean is smaller b χ 2 = 33.2, sufficient evidence that they are
than 26 not independent

14 a χ 2 = 0.695, insufficient evidence that they are
not independent

b χ 2 = 11.2, sufficient evidence that they are
not independent

15 a χ 2 = 42.8, sufficient evidence that they are
not independent

b χ 2 = 5.19, insufficient evidence that they are
not independent

16 χ 2 = 12.14, p = 0.00231, sufficient evidence that
the age and food choices are not independent

Answers 485

17 χ 2 = 4.41, p = 0.110, insufficient evidence that 27 χ 2 = 5.46, p = 0.243, insufficient evidence that
the model is not appropriate
type of insect depends on location
28 a χ 2 = 14.0, p = 0.007 33, sufficient evidence
18 a 10, 20, 30; 2 b 1.93 that diet and age are dependent

c Insufficient evidence that the ratio is different b χ 2 = 15.9, p = 0.0141, insufficient evidence
that diet and age are dependent
from 1:2:3
29 a χ 2 = 3.37, p = 0.185, insufficient evidence
19 a H0: Each course is equally likely, that favourite science depends on gender
H1: Each course is not equally likely
b χ 2 = 10.1, p = 0.00636, sufficient evidence
b 20, 20, 20, 20; 3 that favourite science depends on gender

c p = 0.001 19, sufficient evidence that each Exercise 15B

course is not equally likely 1 a t = 0.278, p = 0.395

20 p = 8.55 × 10−23, sufficient evidence that city and Do not reject H0
b t = 1.95, p = 0.0492
mode of transport are dependent
Reject H0
21 χ 2 = 11.6, p = 0.0407, insufficient evidence that 2 a t = −1.63, p = 0.0823

the dice is not fair Reject H0
b t = −1.19, p = 0.143
22 a H0: The data comes from the distribution
B(3, 0.7) Do not reject H0
3 a t = −2.78, p = 0.0239
H1: The data does not come from the
distribution B(3, 0.7) Do not reject H0
b t = 3.03, p = 0.0163
b 5.40, 37.8, 88.2, 68.6; 3
Reject H0
c 3.57 4 a t = 10, p = 2.46 × 10−10

d Insufficient evidence that the data does not Reject H0
b t = 2, p = 0.0262, reject H0
come from the distribution B(3, 0.7) 5 a t = −0.232, p = 0.409, do not reject H0.
b t = −0.861, p = 0.202, do not reject H0.
23 a Binomial, n = 6, p = 0.5 6 a t = 0.253, p = 0.806, do not reject H0.
b t = −3.35, p = 0.0285, reject H0
b 9.38, 56.3, 141, 188, 141, 56.3, 9.38 7 a t = 1.55, p = 0.0781, do not reject H0
b t = 15.49, p < 10−10, reject H0
c χ 2 = 7.82; insufficient evidence that the coins 8 a t = −0.885, p = 0.189, do not reject H0
b t = −2.30, p = 0.0118, reject H0
are biased 9 a t = −2.97, p = 0.00331, reject H0
b t = 1.34, p = 0.196 , do not reject H0
24 a X ~ B(4, 0.5)
10 t = 0.579, p = 0.290, do not reject H0
b H0: The data comes from the distribution
B(4, 0.5) 11 t = −1.49, p = 0.152, do not reject H0

H1: The data does not come from this 12 t = −1.229, p = 0.1251, do not reject H0
distribution
13 H0: µ = 14.3. H1: µ > 14.3
c 4 t = 1.91, p = 0.0330, reject H0
14 a t = 0.314, p = 0.381, do not reject H0
d χ 2 = 13.4, p = 0.00948, sufficient evidence b The times are normally distributed.

that the data does not come from B(4, 0.5)

25 a 0.440, 0.334, 0.0668

b 3

c HN0(:5T.8h,e0.d8a2t)a comes from the distribution

Hdi1s:tTrihbeutdioantaNd(o5e.s8,n0o.t8c2o)me from the

d χ 2 = 2.82, p = 0.420, insufficient evidence to

reject H0
26 a 14.1, 7.09, 7.62, 7.09, 14.1

b 4

c 30.7

d Sufficient evidence that the times do not
follow the distribution N(23, 2.62 )

486 Answers

15 a The amounts are normally distributed. 12 a y
100
b t = −3.37, p = 0.00213, reject H0
80
16 t = 0.673, p = 0.514 , do not reject H0
17 a The times of two groups are normally 60

distributed and have a common variance. 40

b H0: µ1 = µ2, H1: µ1  µ2 where µ1 and µ2 are 20
population mean times of the two groups
00 2 4 6 8 10 x
c t = −2.20, p = 0.0162
b Appears non-linear
i Reject H0
ii Do not reject H0 c −0.994, strong negative association
18 a H0 : µ = 0, H1 : µ > 0 where µ is the mean
13 H0: There is no correlation, H1: There is positive
change in share price correlation, rs = 0.303. Do not reject H0

b The changes in price are normally distributed. 14 0.332; no significant evidence of correlation

c t = 0.582, p = 0.288, do not reject H0 between time playing video games and sleep
19 a equal variances
15 a −0.952 b No
b t = −0.9, p = 0.197, do not reject H0
20 a t = 1.90, p = 0.0365, reject H0 16 a 0.8857
b −8, 4, −12, −11, −11, −15, 0, 2
b Insufficient evidence of correlation,
c H0: µ = 0, H1: µ < 0, where µ is the mean
change in blood pressure rs  0.886

d t = 2.68, p = 0.0125 < 0.05, reject H0 17 Insufficient evidence of correlation, rs = −0.505
e The differences are distributed normally.

f The first test required that the observations
are independent, which is not true.

Exercise 15C 18 a 0.806 b No c Yes

19 a 0.738

1 a 0.857 b 0.762 b Sufficient evidence of positive correlation

2 a −0.617 b −0.983 20 a Increases

3 a 0.286 b −0.833 b No change

4 a −0.0357 b −0.357 c No change

5 a −0.891 b −0.873 Chapter 15 Mixed Practice

6 a −0.593 b −0.593

7 a 0.795 b −0.807 1 a 0.946 b 1

8 a 0.343 b 0 c As c increases V increases, but not in a
perfectly linear pattern
9 a B b C

c A 2 a 0.0668

10 a i 0.879 b Sufficient evidence to support claim
( p = 0.00186)
ii 0.886

b Strong positive correlation 3 Insufficient evidence to reject H0 ( p = 0.681)
4 a H0: The dice is unbiased (all outcomes have
11 a 0.929
equal probability), H1: The dice is biased
b As temperature increases, it appears that the b all = 50 c 5

number of people visiting increases.

d Insufficient evidence that the dice is biased,
p = 0.123  0.10

Answers 487

5 a 0.857 16 a H0 : All the coins are fair (P(tails = 0.5))
b H0: There is no correlation between hours H1 : (Some of) the coins are biased

of sunshine and temperature, H1: There is a b 6.25, 43.75, 131.25, 218.75, 218.75, 131.75,
positive correlation 43.75, 6.25
c There is sufficient evidence of positive
correlation. c p = 0.0401, χ 2 = 14.7
d Insufficient evidence that the coins are not fair
6 Sufficient evidence that the mean time is greater 17 a Mean is 1, which is consistent with the
than 17.5 minutes ( p = 0.0852  0.10)
claimed B(5,0.2) distribution but does not
7 a H0: µ = 16, H1: µ  16 prove it must be that
b Insufficient evidence of decrease b i H0: data is drawn from a B(5,0.2)

( p = 0.0538  0.05) distribution
8 a H0: The grades are independent of the school H1: data is not drawn from a B(5,0.2)

H1: The grades depend on the school distribution
b p = 0.198, insufficient evidence that grades ii χ 2 = 0.588
iii p = 0.899
depend on school iv There is no significant evidence to doubt

9 H0: The times are the same after training, the manufacturer’s claim.
H1: The times have decreased after training. 18 a t = −1.70, p = 0.0754
Insufficient evidence that times have improved No: there is insufficient evidence to reject H0
( p = 0.277) b i H0: There is no correlation

10 H0: The data comes from the given distribution H1: There is negative correlation
H1: The data does not come from this distribution ii −0.857
iii Reject H0 (as age increases there is a tendency
Insufficient evidence to reject claim ( p = 0.465)
11 a 791 for 100 m time to decrease)
b H0: The average weight of a loaf is 800 g,     

H1: The average weight of a loaf is less than 19 a t = −1.19, p = 0.118  0.05
800 g. t = −2.63, p = 0.0137. Sufficient b χ 2 = 9.79, p = 0.0440  0.05
evidence to reject H0 and say the average c They could come from a normal distribution
weight of a loaf is less than 800 g.
12 a not random (observations may not be with a different mean (or different variance).
independent) d Scores are discrete, normal distribution is
b H0: µ1 = µ2, H1: µ1 ≠ µ2
c The masses of the eggs of both chickens are continuous.
distributed normally with equal variance. 20 a t = 2.51, p = 0.0129. Reject H0 (significant
d p = 0.666, insufficient evidence that the means
of egg masses for the two chickens are different change from 3.3)
13 a −0.511 b rS = 0.696, do not reject H0 (no significant
b H0: no correlation, H1: some correlation
c Insufficient evidence of correlation evidence of increasing frequency with score)
(0.511  0.564) c χ 2 = 1.93, p = 0.858, do not reject H0 (no
14 a t = 1.689, p = 0.103, do not reject H0
b The lifetimes follow a normal distribution significant evidence of non-random pattern)
with equal variances.
15 a B(5, 0.45) Chapter 16 Prior
b 5.03, 20.6, 33.7, 27.6, 13.1 Knowledge
c 4
d p = 0.0249, sufficient evidence that Roy’s 1 f′(x) = 2x 2 3 3 5.75
belief is incorrect
Exercise 16A

1 a 8 b 10

488 Answers

2 a −6 b − 9 4 3.15 hours b 100 cm2
2 5 a 40 cm b  25.9cm
6 a 42 cm2
3 a 1 b 1
18 4
7 108 cm3
4 a −12 b −16 72
8 a P = 2x + x b 24cm
5 a 0, 2.67 b −1.73, 1.73 9 253 m2

6 a −0.779, 0.178, 0.601 b 0.607 10 b 821 cm3

7 a 1.10 b −0.632, 0.632 11 a h = 460 , S = 2x2 + 1840
x2 x
8 a i max (−0.577, 1.77), min (0.577, 0.23)

ii 1.77 b 9.73
b i max (0, 5), min (2.67, −4.48)
12 a 4x2 + 675
ii 5 x
9 a i max (−0.167, −2.99), min (0, −3) b 4.39cm × 8.77 cm × 5.85cm

ii −2.25 13 b r = 4.30, h = 8.60
b i max (0.667, −0.852), min (0, −1)
c e.g. doesn’t take into account the neck of the
ii 1
10 a i max (0, 1), min (1.58, −5.25) bottle

ii 1 14 b 1210 cm3
b i max (−0.707, 1.25), (0.707, 1.25), min (0, 1)
15 23
ii 1.25
16 b 39

17 a $7

b It does not predict negative sales when x  10

11 a i min (0.630, 1.53) c $9

ii 4 18 a 800 − 200 b 0.5 kg
m m2
b i local maximum (0,2), local minimum
(0.750, 1.90) 19 102 000 cm3

ii 4 20 12.9 cm3

12 b = −2, c = 3 Exercise 16C

13 b = −4, c = 1 b 0.909 1 a 290 b 478
14 a −0.329, 1.54 2 a 0.921 b 0.337
3 a 3.65 b 2.16
15 (0, 0), (3, 27) 4 a 1.48 b 1.08
5 a 14.6 b 16.4
16 (−1, −5) b −32 6 a 1.71 b 5.44
17 a (0.180, 7.81) 7 a 3.57 b 3.47
8 5.15
18 108.9 9 5.82 b Larger (the tunnel
10 13.2 is curved)
19 13.5  f(x)  30.7 11 a 11.1 m2
b Underestimate
20 f(x)  −30 12 4%
13 137 m2
21 a = 2, b = 5 14 2250 m2
15 a 1.84
22 a = 3, b = 6

23 (−1.49, 20.9)

Exercise 16B

1 $2.5 million
2 $1726
3 77.1 km h−1

Answers 489

Chapter 16 Mixed Practice 6 a y

( )1 a 2 , − 4 3 , 81
3 3 √2 4

b y

x
3

4 x
3
A ( ) 3 , 81 , mimima in the interval
maximum at 2 4

2 , − 4 at the end points (0, 0) and (3, 0)
3 3
b 31.3

2 −16 y 7 11.4
3 a 8 a h = 44 − w
b w = 22, h = 22
c 484cm2

9 65.8 km h−1 d 2.67 m
10 c 48x − 18x2 f 5.33 m, 4 m
e 56.9 m3
g 7

11 (−1.14, 3.93)

− 5 5 12 a = 3, b = 213 b 0.982, 8.20%
12 12 x 13 (−1, 9) b 6

14 a 1.06
15 a (0.423, 0.385)

16 −256

17 cylinder, r = 3.63 cm, h = 7.26 cm

18 a V = 20lw
d S

b ± 5 500
12

4 −0.855 400

5 (0.5, 7)

300

200

100

w
5 10 15 20

490 Answers

e 4 − 300 b 3.48 years
w2
f 8.66 c Continues to spread, tending to an upper limit

of 10 000 (which is possibly the full population

g 17.3 of this species) [6]

h 109 cm 9 a 51.8 cm

19 b i 1200 ii A = 2πx2 + 1200 b 26.8 cm
x x
c 5980 cm3 [7]

c dA = 4πx − 1200 d 4.57 10 a H0: µA = µB
dx x2 H1: µA ≠ µB
b p = 0.0338
e 394 cm2

20 (−2.67, 107) c 0.0338  0.05 so reject H0. There is sufficient
evidence at the 10% level that the mean length
21 a T = −0.022t3 + 0.56t2 − 2.0t + 5.9

b −21.5 °C; it predicts a much lower temperature of perch in the two rivers is different.
at midnight on the next day than the previous
day d Both populations are normally distributed.

22 6.25 The two populations have equal variance. [8]

23 a 33 units2 b 20 625 m2 11 a 6x + 8y = 59

b C as that point is in the Voronoi cell

Applications and containing C.
interpretation SL:
Practice Paper 1 c (2, 3) [6]

12 a a = −0.25, b = 2.25, c = −3.5

b £1560 [7]

13 a dl = 0.28
dt t
1 a 13 cm b 38
b 4 m
c 85 [6]
c Rate of growth slows but sharks continue

2 a f(x) > −1.10 b −0.41 to grow throughout their lifetime. [8]

c x = 21.1 [5] Applications and
interpretation SL:
3 a a + 9d = 285 Practice Paper 2

25a + 300d = 9000

b a = 60, d = 25 [5]

4 a 80 b 1 c 53 [5] 3
8 2
5 a AUS 444.89 1 a i − ii (0, 6) [3]

b AUS 6,693.40 [6] b 2x − 3y = 21 [3]

6 a H0: gender and favourite type of wine are c (6, −3) [1]
independent
d i 117 ≈ 10.8 ii 13 ≈ 3.61 [2]
H1: gender and favourite type of wine are not
independent e 19.5 [2]

b p = 0.0935 2 a p + q = 0.27 [1]

c 0.0935  0.05 so do not reject H0. Insufficient b 3p + 4q = 1 [3]
evidence at the 5% level that gender and wine
c p = 0.08, q = 0.19 [1]

preference are not independent. [6] d i 0.161 ii 0.467 [3]

7 a k = 6 e i 9.3 ii 1.88 [2]

b 806  V  941 [5] f 0.149 [3]

8 a i 500 ii 4220

Answers 491

3 a k = 6, c = 2 [3] f i 2.65 m
3 [4]
ii 12.2 m2 [4]
b y
5 a 0.978 [2]

y = S(x) b Yes, since value of r close to 1 so points lie

near to straight line. [2]

c a = 0.0444, b = 2.56

y = S− 1(x) c = 0.0152, d = 8.40 [5]

d i 6560 euros

ii 11 400 euros

ii 13 400 euros [3]

e Part i and ii are reliable as within the range
of the given data. Part iii is potentially

unreliable as had to extrapolate. [3]

x 6 a i − 1 [5]
p2
c i 0  S−1  125
( ) b i R 2
ii 0  S  150 [2] Q(2 p, 0), 0, p

d i 27 ii 2 [4]

ii The volume of the cuboid when the d 2 2 ≈ 2.83 [2]

surface area is 54 [2]

4 a p = 2.3, q = 22.5 [3]

b 1.63 m [1]

c 34.9% [2]

d a = 0.0323, b = −0.531, c = 2.18 [4]

e y [2]

(2, 2.5) (4, 2.3)

x
8

Glossary

χ 2 statistic In a χ2 test the value that measures how Edges (on a Voronoi diagram) The boundaries of a cell
far the observed data values are from what would be
expected if the null hypothesis were true Event A combination of outcomes

Alternative hypothesis In a hypothesis test, the stated Expected frequencies The frequencies of each
difference from the null hypothesis that is to be group in a frequency table or for each variable in a
investigated contingency table assuming the null hypothesis is true

Amortization The process of paying off debt through Exponent The number x in the expression b x
regular payments
Exponential decay The process of a quantity
Amplitude Half the distance between the maximum decreasing at a rate proportional to its current value
and minimum values of a sinusoidal function
Exponential equation An equation with the variable in
Annuity A fixed sum of money paid at regular intervals, the power (or exponent)
typically for the rest of the recipient’s life
Exponential growth The process of a quantity
Arithmetic sequence A sequence with a common increasing at a rate proportional to its current value
difference between each term
Exponential model A function with the variable in
Arithmetic series The sum of the terms of an the exponent
arithmetic sequence
Geometric sequence A sequence with a common
Asymptotes Lines to which a graph tends but that it ratio between each term
never reaches
Geometric series The sum of the terms of a geometric
Axis of symmetry The vertical line through the vertex sequence
of a parabola
Gradient function See derivative
Base The number b in the expression b x
Hypothesis testing A statistical test that determines
Biased A description of a sample that is not a good whether or not sample data provide sufficient evidence
representation of a population to reject the default assumption (the null hypothesis)

Cells (on a Voronoi diagram) The regions containing Incremental algorithm An algorithm for building a
the points which are closer to a given site than any Voronoi diagram one site at a time
other site
Indefinite integration Integration without limits – this
Contingency table A table showing the observed results in an expression in the variable of integration
frequencies of two variables (often x) and a constant of integration

Continuous Data that can take any value in a given range Inflation rate The rate at which prices increase over time

Cubic model A function of the form y = ax3 + bx2 + cx + d Initial value The value of a quantity when t = 0

Decimal places The number of digits after the decimal Integration The process of reversing differentiation
point
Intercept A point at which a curve crosses one of the
Definite integration Integration with limits – this coordinate axes
results in a numerical answer (or an answer dependent
on the given limits) and no constant of integration Interest The amount added to a loan or investment,
calculated in each period either as a percentage of
Degrees of freedom (v) The number of independent the initial sum or as a percentage of the total value
values in the hypothesis test at the end of the previous period

Depreciate A decrease in value of an asset Interquartile range The difference between the upper
and lower quartiles
Derivative A function that gives the gradient at any
point of the original function (also called the slope Inverse proportion Two quantities are inversely
function or gradient function) proportional when one is a constant multiple of the
reciprocal of the other
Differentiation The process of finding the derivative
of a function Limit of a function The value that f(x) approaches as x
tends to the given value
Direct proportion Two quantities are directly proportional
when one is a constant multiple of the other Linear model A function of the form y = mx + c

Discrete Data that can only take distinct values Local maximum point A point where a function has a
larger value than at any other points nearby
Discrete random variable A variable with discrete
output that depends on chance Local minimum point A point where a function has a
smaller value than at any other points nearby

Glossary 493

Long term behaviour The value of a function when x Significant figures The number of digits in a number
gets very large that are needed to express the number to a stated
degree of accuracy
Lower bound The smallest possible value of a quantity
that has been measured to a stated degree of accuracy Simple interest The amount added to an investment
or loan, calculated in each period as a percentage of
Normal to a curve A straight line perpendicular to the the initial sum
tangent at the point of contact with the curve
Sinusoidal model A function of the form
Null hypothesis The default position in a hypothesis test y = a sin(bx) + d or y = a cos(bx) + d
Observed frequencies In a χ2 test the sample values
Sites (on a Voronoi diagram) The given points around
on the variable which cells are formed

One-tailed test A hypothesis test with a critical region Slope function See derivative
on only one end of the distribution
Spearman’s rank correlation coefficient A measure
Outcomes The possible results of a trial of the agreement between the rank order of two
variables
p-value Assuming the null hypothesis is correct, the
probability of the observed sample value, or more Standard deviation A measure of dispersion, which
extreme can be thought of as the mean distance of each point
from the mean
Parabola The shape of the graph of a quadratic function
Standard index form A number in the form a × 10k
Period The smallest value of x after which a sinusoidal where 1 ഛ a < 10 and k ∈
function repeats
Subtended The angle at the centre of a circle
Perpendicular bisector The perpendicular bisector of subtended by an arc is the angle between the two
the line segment connecting points A and B is the line radii extending from each end of the arc to the centre
which is perpendicular to AB and passes through its
midpoint t-statistic In a t-test the value that measures how far
the sample mean is from what would be expected if
Piecewise linear model A model consisting of different the null hypothesis were true
linear functions on different parts of the domain
Tangent to a curve A straight line that touches the
Polynomial A function that is the sum of terms curve at the given point but does not intersect the
involving non-negative integer powers of x curve again (near that point)

Pooled sample t-test A two-sample t-test conducted Toxic waste dump problem A problem in which the
when the two populations are assumed to have object is to find the point which is as far as possible
equal variance from any of the sites

Pooled variance In a two-sample t-test, an estimate Trapezoidal rule A rule for approximating the value of
of each of the common variance of each of the two a definite integral using trapezoids of equal width
populations formed by combining both samples
Trial A repeatable process that produces results
Population The complete set of individuals or items of
interest in a particular investigation Turning point A local maximum or minimum point

Principal The initial value of a loan or investment Two-tailed test A hypothesis test with a critical region
on either end of the distribution
Principal axis The horizontal line halfway between
the maximum and minimum values of a sinusoidal Upper bound The largest possible value of a quantity
function that has been measured to a stated degree of accuracy

Quadratic model A function of the form y = ax2 + bx + c Value in real terms The value of an asset taking into
account the impact of inflation
Quartiles The points one quarter and three quarters of
the way through an ordered data set Variance The square of the standard deviation

Range The difference between the largest and smallest Vertex (or vertices) of the graph The point(s) where
value in a data set the graph reaches a maximum or minimum point and
changes direction
Relative frequency The ratio of the frequency of
a particular outcome to the total frequency of all Vertices (on a Voronoi diagram) The points at which
outcomes the edges of the cells intersect

Roots of an equation The solutions of an equation Voronoi diagram A diagram that separates an area into
regions based on proximity to given initial points (sites)
Sample A subset of a population
Zeros of a function/zeros of a polynomial The values
Sample space The set of all possible outcomes of x for which f(x) = 0

Significance level In a hypothesis test the value
specifying the probability that is sufficiently small to
provide evidence against the null hypothesis

Index

3D shapes see three-dimensional shapes calculus  282, 404 data points, plotting  332
abstract reasoning  xv trapezoidal rule  412–14 decimal places  280
accumulations  256–7 see also differentiation; integration definite integrals  262–4, 412
accuracy depreciation  40
Cartesian coordinates  78 derivative function  225, 406
levels of  278–80, 283–4 cell sequences  22–3 differential equations  320
of predictions  309, 334 central tendency  140–2 differentiation  222–3, 406
see also approximation; estimation
algebra chi-squared  χ 2 tests  372, 375–9 anti-differentiation  258–60
basic skills  xxiv–xxv derivatives  225, 230–3, 240–3, 245–50
computer algebra systems (CAS)  xx circles, arcs and sectors  344–6 gradient of a curve  225–7, 231–3,
definition  2 coefficients  325
exponential equations  7 common difference, arithmetic 245–50, 406–8
algorithms  279, 297, 353, 414 limits  224–5
alternative hypothesis  372–3 sequences  25–7 maximum and minimum points  407
amortization  287–90 common ratio, geometric sequences  34–5 rate of change  227, 257
amplitude  328–9 compound interest  39, 288–90 see also integration
angles computer algebra systems (CAS)  xx direct proportion  323–4
of depression  119 computer algorithms  279 discrete data  132, 142–3, 202–3
of elevation  119–20 cone discrete random variable  202–3
see also triangles; trigonometry dispersion  143–4
annuities  289–90 surface area  94 distribution
approximation  40, 136, 278, 285 volume  95 binomial  207–10
decimal places  280 conjecture  x–xi normal  212–15, 372–3, 377, 385–7
percentage errors  282–3 contingency tables  378–9 dynamic geometry packages  xxi
rounded numbers  280–3 continuous data  132
significant figures  280 contrapositive statements  xvi e (number)  15
trapezoidal rule  414 convenience sampling  134–5 elevation  119–20
see also estimation converse  xvi encryption algorithms  297
arcs  344 coordinate geometry enlargement symmetry  63
area Cartesian coordinates  78 equations
of a sector  345–6 midpoint  87
spatial relationships  343 Pythagoras’ theorem  86–7 defining quantities  299
surface  94–5 three-dimensional  86–7 differential  320
of a triangle  107–8, 345 two-dimensional  74–82 linear  298–9
upper and lower bounds  281 correlation  159–61 normal to a curve  248–9
arithmetic sequences  24–9 cosine  101, 328 polynomial  297, 301–3
applications of  28–9 rule  105–6, 345 quadratic  316
common difference  25–7 cubic functions  325–6 roots of  67–8, 301–3
sigma notation  27 cumulative frequency graphs  151–2 simultaneous  298–9
sum of integers  26–8 curves see parabola solving using graphs  67–8
arithmetic series  25 cycle of mathematical inquiry  x solving using technology  297
assumptions  xvii, 23, 29, 189, 209 of straight lines  76–82, 310
see also hypothesis testing data see also exponential equations
asymptotes  62, 318–19 bivariate  160 error analysis  282–3
average see central tendency contingency tables  378–9 estimation  283–5
axiom  xii continuous  132 exponential
axis correlation  159–61 decay  318–19
principal  328–9 degrees of freedom  375 equations  7, 17
of symmetry  315–16 discrete  132, 142–3, 202–3 growth  318–19
effect of constant changes  145 models  318–21
base  4 frequency distributions  139, 141, 375–9 exponents
bearings  118–19 frequency tables  375–7 applied to fractions  5–6
bias, in sampling  132–3 goodness of fit  375 laws of  4–8, 20, 320–1
binomial distribution  207–10 grouped  141–2 negative  5
bivariate data  160 measure of dispersion  143–4 simplifying expressions  4–7
box-and-whisker diagrams  153 measures of central tendency  140–2
modal class  142 Fermi, Enrico  284
calculations outliers  133–4, 144 financial mathematics
levels of accuracy  278–80 patterns in  164
number line  281 presenting  150–3, 159–66 amortization  287–90
rounded numbers  280–3 quartiles  142–3 annuities  289–90
significant figures  12 regression  162–4 compound interest  39, 288–90
reliability of  132–3 depreciation  40
summarizing  139–40 inflation  40
validity  133, 144 interest  28–9, 39, 287–90
see also statistics

Index 495

levels of accuracy  278–80 parallel lines  80 lines of symmetry  63
modelling  278–85, 290 period of  328–9 local maximum points  407–8
simple interest  28–9 perpendicular lines  80–2 local minimum points  407–8
forces  257 plotting data points  332 logarithms  14, 20, 52
fractal  63 principal axis  328–9
fractions, with exponents  5–6 proportion  323 evaluating  16
frequency distributions  139, 141, 375–9 quadratic models  314–16 natural logarithm  15–16
frequency tables  375–7 scatter diagrams  159–61, 391 logic  xv–xvi
function notation  50 simultaneous equations  82 lower bounds, of rounded numbers  281–2
functions  48 sine  328
derivative  406 sinusoidal models  328–9, 332 mathematical
domain  52 sketching  65–7 induction  24
graphical representation  50–1, 53, 55–6 solving equations using  67–8 inquiry, cycle of  x
inputs  49–52 symmetries  63
inverse  54–5 tangents  225, 246–7 mean  140–2
and mathematical modelling  53–4 turning points  314 median  140
outputs  49, 52–3 vertices  62, 315 modal class  142
range of  52 x-intercept  77–9, 315–16, 319, 328 mode  140, 142
zeros of  67 y-intercept  62, 77–9, 315–16, 319, 325, 328 modelling  xvii–xix, 22

Gauss, Carl Friedrich  26 histograms  150–1 accuracy of predictions  309, 334
generalizations  22, 24 horizontal asymptote  318–19 arithmetic sequences  28–9
geometric sequences  33–6 hypotenuse  101 assumptions  xvii, 23, 29, 189, 209
hypothesis testing  370–9, 385–8, 391–4 cubic models  325–6
applications of  36, 39–40 differential equations  320
common ratio  34–5 alternative hypothesis  372–3 exponential models  318–21
financial applications of  39–40 Fermi estimate  284
geometric series  34–5 chi-squared  χ 2 tests  372, 375–9 geometric sequences  36
geometry  92–3 linear  310
arcs and sectors  344–6 null hypothesis  372–3, 375, 378, 385 mathematical model of a process  53–4
origins of  118 one-tailed test  373 normal distribution as  214
perpendicular bisectors  349–54 Pearson’s correlation coefficient  patterns  23–4
spatial relationships  343, 349–58 piecewise linear models  310–11, 332,
Voronoi diagram  342–3, 349–58 161–2, 391–3
see also coordinate geometry; p-value  374–5, 385 334
significance level  373–7, 385–8 quadratic models  314–16
geometric sequences; Spearman’s rank correlation  391–4 qualitative and quantitative
trigonometry t-tests  385–8
Godfrey, Charles  67 two-tailed test  373 results  230
gradient of a curve  225–7, 231–3, 245–50, representations  308–10
406–8 implication  xvi sinusoidal models  328–9, 332
gradient of a straight line  76–80, 310 incremental algorithm  353 skills  332–4
Graham’s Number  11 indices see exponents straight-line models  75
graphical interpretation, of infinity, reasoning with  xiv using equations  297
derivatives  230–3 inflation  40 variables in  309, 314
graphical representation integration see also financial mathematics
and data  150–4 modulus sign  282–3
functions  50–1, 53, 55–6 accumulations  256–7
solving equations  68 anti-differentiation  258–60 Napier, John  14
graphs  61–8 definite integrals  262–4, 412 natural logarithm  15–16
amplitude  328–9 trapezoidal rule  412–14 nearest neighbour interpolation  355
asymptotes  62 see also differentiation negation  xvi
axis of symmetry  315–16 interest negative exponents  5
box-and-whisker diagrams  153 amortization  287–90 negative gradient  76
cubic functions  325–6 compound  39, 288–90 negative numbers, and the domain of a
cumulative frequency  151–2 simple  28–9
direct proportion  323–4 see also financial mathematics function  52
equations of straight lines  76–82 interquartile range  143 neural networks  297
exponential models  318–19 inverse normal distribution  212–15, 372–3, 377, 385–7
finding points of intersection  66–7 functions  54–5 normals  248–50
gradient of a curve  225–7, 231–3, proportion  324–5 notation  xv
245–50, 406–8 irrational numbers  15 n terms
gradient of a straight line  76–80, 310
histograms  150–1 Lagrange’s notation  226 arithmetic sequences  24–5
intercepts  62, 77–80 large numbers  11 geometric sequences  33–5, 321
intersection of two lines  82 laws of exponents  4 null hypothesis  372–3, 375, 378, 385
inverse proportion  324–5 Leibniz, Gottfried  225, 259 number line  281
parabola  315–16 level of significance  373–7, 385–8
limits  224–5 Occam’s razor  xviii, 326
linear equations  298–9 one-tailed test  373
linear models  310–11 operations, large numbers  11
oscillating patterns  332
outliers  133–4, 144

496 roots  62 Index
of an equation  67–8, 301–3
parabola  315–16 Graphical Display Calculator
paradox  xv rotational symmetry  63 (GDC)  xix–xx
parallel lines  80 rounded numbers  280–3
patterns  23–4, 30 programming  xxi
Pearson’s correlation coefficient  161–2, upper and lower bounds of  281–2 terminology  xvi
Russell, Bertrand  xii–xiii tetration  11
391–3 three-dimensional shapes
percentage errors  282–3 sample space diagrams  185
perpendicular sampling  132–3 finding angles  113–18
surface area  94–5
bisectors  349–54 bias in  132–3 volume  94–5
lines  80–2 pooled variance  387 toxic waste dump problem  356
perspective  119 random  375 trajectory  75
piecewise linear models  310–11, 332, 334 representative  375 translational symmetry  63
point-gradient form  79 techniques  134 trapezium  413
Polya, George  ix–x see also normal distribution trapezoidal rule  412–14
polynomial equations  297, 301–3 scatter diagrams  159–61, 391 tree diagrams  184–5, 202
pooled variance  387 sector, area of  345–6 triangles
positive gradient  76 sequences  22–3 area of  107–8, 345
powers see exponents standard notation  24 in the real world  92–3
predictions see also arithmetic sequences; right-angled  101, 113
accuracy of  309, 334 see also coordinate geometry
see also modelling geometric sequences trigonometry  92, 101–8, 308
principal axis  328–9 sets  xv applications of  113–20
probability shapes, three-dimensional solids  94–5 area of a triangle  107–8
binomial distribution  207–10 Shapiro-Wilk test  214 bearings  118–19
combined events  186–7 sigma notation cosine  101
complementary events  181 cosine rule  105–6
concepts in  179–82 arithmetic sequences  27 finding angles  101–2, 113–18
conditional  187–8 geometric sequences  35–6 measuring angles  118–19
discrete random variable  202–3 significance level  373–7, 385–8 sine  101
distributions  201–15 significant figures  12, 280 sine rule  103–4
events  180 simple interest  28–9 tan rule  106
independent events  188–9 simplifying exponent expressions  4–7 three-dimensional shapes  113–14
mutually exclusive events  187 simultaneous equations  82, 298–9 true value  282–3
normal distribution  212–15 sine  101, 328 t-tests  385–8
outcomes  180 rule  103–4, 345 two-tailed test  373
relative frequency  180 sinusoidal models  328–9, 332
sample space  180–1 spatial relationships  343, 349–58 upper bounds, of rounded numbers  281–2
techniques  184–5 Spearman’s rank correlation  391–4
trials  180 spreadsheets  xx validity  133, 144
problem solving  ix–xii square root  14 value in real terms  40
programming  xxi see also logarithms variables
proportion  323–5 standard deviation  143
p-value  374–5, 385 standard index form  11–12 correlation between  391–4
pyramid standard notation, sequences  24 modelling  xvii–xviii, 309
finding angles  115–16 statistics  130 variance  143
surface area  96 approximation  136 velocity  257
volume  96–7 misleading  130, 133 Venn diagrams  184, 186, 188
Pythagoras’ theorem  xiii, 86–7, 96, 116 outliers  133–4, 144 and sets  xv
sampling  132–4, 375 vertices  62, 315
Q-Q plots  214 validity  133, 144 volume, three-dimensional solids  94–5
quadratic see also data; hypothesis testing Voronoi diagrams  342–3, 350–8
straight-line models  75, 310
equations  316 stratified sampling  135–6 Watson selection test  xv
models  314–16 surface area
quadrilateral  413 three-dimensional solids  94–5 x-axis  62
quantities, defining  299 see also area x-intercept  77–9, 315–16, 319, 328
quota sampling  135 syllogism  xvi
symmetries  63 y-axis  62
Ramanujan  xiv systematic measurement error  282 y-intercept  62, 77–9, 315–16, 319, 325, 328
random sampling  134, 375 systematic sampling  135
rate of change  227, 257 Zeno’s paradoxes  227
reasoning  xv tangents  101, 225, 246–7, 249–50, 406 zero  xv, 4, 52, 224
representations  12, 203, 308–10 tan rule  106, 345
representative sampling  375 technology  xix–xxi of a function  67
right-angled triangles  101, 113 gradient of a curve as  406
computer algebra systems (CAS)  xx of a polynomial  301
see also trigonometry dynamic geometry packages  xxi Zipf’s Law  53