Mathematics - 4 & 5 Extended - Harrison, Huizink, Sproat-Clements, Torres-Skoumal - Oxford 2017

6 a h (t) > 0 b 3.56 seconds d y

6

7 x > 2.73 cm

5

8 4.47 cm × 4.47 cm × 2 cm

4 x=2
3
Practice 2

1 a x > 1 or 1<x<0 b 1
0 <x≤

4 2
1
y > ln(x − 2) + 3
0
c 1<x<4 d x > 5.09

(2.05, 0)

e 1.5 < x < −1 or x > 1.57 f 1.5 < x < −1 or x < −5

x

g 3<x<0 h 4 < x < −2 or 1 < x < 6 2 3 5 6

2a R= 2R ⇒ 2R > 1, R >2
1 1
1 y
2+ R 2+ R 3
1 1 e

b R > 7.48

2 2
1
x+1

y≤e −3

Practice 3

(0.099, 0)

1a y x
8
4 3 2 1 1 2 3 4 5 6
1

6 2
3
y = −3

4 2

y>x + 3x − 1

( 3.3, 0) 2 (0.303, 0)

2 x fy
2 8

6 4 2 4 6

(0, 1)

6 x+3
4
y<

( 1.5, 3.25)

x−3

6

2 x=3
( 3, 0)

8

x

8 6 4 2 4 6 10 12
2
b y
1.0 (0, 1)

4

0.5 (1.5, 0.25)

6

(1, 0) (2, 0)

0 x

1.0 1.0 2 2

2a y < −x − 4x + 5 b y ≥ 0.5 x + 2x − 1

0.5

x

2x + 1

c y ≤e −4 d y>

1.0 2 x 2
1.5
y ≤ −x + 3x − 2

2

3a i f (x ) = x −4 ii y > f ( x )

2.0 (0, 2) 1 2
(x − 1)
bi f (x ) = +3 ii y > f ( x )
ii y < f ( x )
3

2.5

ci 2 6x + 5
f (x) = x

3

2

4a f (x ) = − ( x − 45) +3

2025

b Domain {x 0 ≤ x ≤ 90}, range {y 0 ≤ y ≤ 3}

c 0 ≤ y ≤ f (x )

c y
6
( 0.75, 5.13) Practice 4
5

1a 1 ≤ x≤ 3 b x < 2 or x > 8

4 (0, 4) c 3 ≤ x≤ 4 d 10 ≤ x ≤ 4
3
2
y ≥ 4 − 3x − 2x

e 1 f x< 1
3 < x< or x > 1

2 3

2

1 2

g < x<

1 2 3

( 2.35, 0) 0 (0.851, 0)
1 x
2 0 < x ≤ 1.70 cm

2.0 1.0

3 Student’s own answers

2

3

34 4 Answers

Practice 5 c y
8 2
1a x < 0.209; x > 4.79 ( 0.345, 0) (4.35, 0)
x
b 6.32 < x < 0.317 0

6 4 2

c 2.19 ≤ x ≤ 0.687 2

d x < 0.333 or x > 2

e 2.79 ≤ x ≤ 1.79 4
6
2

y < 2x − 8x − 3

f x < −0.414 or x > 2.41

g 0.290 < x < 0.690

8

h 9.12 ≤ x or x ≥ −0.877

10

i x < −1.18 or x > 0.425

j 0.209 < x < 4.79 12 (2, 11)

2 a ∆ = 0, hence only one root, x = −1. Since the graph lies

entirely above the x-axis except for this one value, there is d y
2.0 1.5

only solution, f ( x ) = 0 for x = −1.

b ∆ = 0, hence only one root, x = 2. Since the graph lies 1.0

0.5 y < log (x + 1)
(0, 0)
10

entirely above or on the x-axis, the solution is x ∈ 

c Since ∆ < 0, the quadratic has no roots. Since a > 0, its graph

is always above the x-axis for all values of x. Hence x∈ 

x

1.0 1.0 2.0 3.0

d Since ∆ < 0, the quadratic has no roots. Since a < 0, its graph 0.5

is always below the x-axis for all values of x. Hence x ∈ 

1.0

Practice 6 1.5

1a x < −1 or x > 1 b 1<x≤4

e y
10
c x < −1 or x > 4 d x ≥ 3 or x < 2

e 1 2
0 <x < f x > 5 or x ≤ −

3 8

4

2 3 < x < −1

6

3 About 17

(0, 5.14)

4

x−2

y≥e +5

Mixed practice 2

1 a x < −3 or x > −1 b x ≤ −8.58 or x ≥ 0.583

0 x

10 8 6 4 2 2 4 6 8 10

c x < −3.59 or x > 2.09 d x > 0 or x < −3

2

e 0 < x < 0.286 f 0.333 < x < 5

2a y fy
8

6

6 2
4
y>x − 4x + 2 4
2
x−2

y>

x+4

2 x = −4
(0.586, 0)
(3.41, 0)

x

12 10 8 6 4 2 2 4 6 8
2
0 x

6 4 2 2 4 6 (0, 1)

2

(2, 2)

4

4

6

6

3a 6<x<1 b x < −2 and x > 12

y c 7 < x< 5 d 8 < x< 2

b

6

(1.5, 4.5) 3

e 0 < x < 5.5 f x≤ or x ≥ 2

4 5

2

y ≥ −2x + 6x

g x > 3 or x < −5 h 1<x<1

2 (3, 0)
(0, 0)

4 36.2 meters

0 x

4 2 2 4 6

5 Between 11 and 30 tourists

2

6 4.6 cm < radius < 9.7 cm; 6.8 cm < height < 29.8 cm

4

7 1.66 in < x < 6.16 in

6

Review in context

1, 2 Student’s own answers.

Answers 34 5

d

E15.1

You should already know how to: A B
B
1a Song 2
1980
9
17

10 Song 1
18 1980

8 e
17
1990

A

10 1980
17

8 1990
18

7 1990
17

5 Practice 1
1
b
1
17 2 1

1 1 P (B |A) =
2 36
B
35
36 36

32 3 4 1
36
2a b c

1

52 169 221 P (B |A ) =

A 36
A
B 1 1 1

15

P (B ) = + =

3a P ( A) = b P(A ∪ B ) = 1

72 72 36

27

B

4

A U
B

26 7 21 35 B
36

P (B A) = P (B A ) = P(B )

∴ independent events

2 5 5
9 A) =
5a P (B
4
9 B 9
5 B
3 A 9 P (B 5
5 A A)=

2 9
5
P (B ) = P (A ∩ B ) + P (A′ ∩ B )

3 2 5

= + =

B 9 9 9

b

4 B
9
A B
B
P (B A) = P (B A ) = P (B )

∴ independent events

3 4
9
B P (B 4
5 B A) =
9 B
c 5 9
A 9
5 A P (B 5
10 A A)=

5 9
10
2 5 1

P (B ) = + =

9 18 2

4 B
9

P (B A) ≠ P (B A ) ≠ P (B )

The events are not independent.

34 6 Answers

4 6
10
17 19 17
ii
B 5a i iii
35
35 35
13
1 A 9 ii 7
7 A iii
6 bi 17
6 18
7 P( B A) =

19

4 B 10
10
7
1
2 c

28

B 1 Practice 3
B A)=
P( B

2

1a

1 Male (M ) Female (F ) Total
2
The events are not independent. Professional (P ) 5 12 17

5 Amateur(A) 5 6 11
A) =
5 P (B

9

Total 10 18 28

6

P (B A )= , therefore not independent events.

9 5 5 10
A) = iii P (M ) =
b i P (M P) = ii P (M
11 28
5 17
A) =
6 P (B

11

c No they are not independent events

P (B A )= 6
, therefore not independent events.

2

11

soccer No soccer Total

7 P (B A ) = 90%

Left 10 30 40

P (B A ) = 65%, therefore not independent events.

Right 15 45 60

8 P (Y X ) = 0.8

P (Y X ) = 0.8 Total 25 75 100

∴ P (Y X ) = P (Y X ) = P (Y )

P (L 10
S) =

∴ independent events

25

Practice 2 30
NS ) =
P( L

75

11 11

1a b

26 32 2
P( L ) =
2a
5

They are independent events.

K C

3 Yes, because P (F and S ) = P (F ) × P (S )

20 12 14

Practice 4

1 a P( LS operation) = P ( LS 1
No operation) = P ( LS ) =
4
9

The results are independent events the claim is not valid.

bi 32 ii P (C 12
P( K ) = K) =
b 1
50 32 9

LS

iii 26 iv P (K 12
P (C ) = C) =

50 26 1 O
4

3a 8
9
LS

A D

7 5 10 1
9
LS

3 O
4

8

8 LS
9

12 5
b i P ( A) =
ii P (D A) =
30
12 It is evident from the tree diagram ( LS operation) =

P ( LS No operation)

22 5
P (D ∪ A ) =
iii iv P (D ∩ A D ∪ A) = c Student’s own answer with justication.

30 22

15 9 4
4a
b c
25
25 9

Answers 347

2a Mixed practice
1a
T T Total
45
G 10 15 25 100 44
99
40 O
100 40

G 30 10 40 15
100
99

O A
A
Total 40 25 65 B

15 B
99

b The events are not independent. 45 O
c Student’s own answer. 99 39

3a i P (E ) = 0.4 ii P (L E ) = 0.8 99

A

iii P (L E ) = 0.4 iv P (E ∩ L ) = 0.32

v P (E ∩ L ) = 0.08 vi P (E′ ∩ L ) = 0.24 15
99
B

vii P (E′ ∩ L ) = 0.36 viii P (L) = 0.56

b No, because the conditional probabilities are dierent. 45
99
O
40

Practice 5 99
1a
A A

14 B
99
W

23 10 8

3750 25 40 55
99
b = c d

9900 66 99

6 12 2 1
5 d
10
3
2a b c

13 13

33 18 5 1 3
P (A ) = P (W ) = 13 f
bi ii e g
51 2
51 4

iii P (A 10 iv P (W 10 3a 2
W)= A) = 3
I
18 33

8

v P (W A)= 4 S
5
18

1 I
3
c A A

W 10 8 18
33
51 1 I
3

W 23 10 1
5
S

33 18

2 I
3

12

12 4 2 8

2a b b × =

70

30 5 3 15

c P (M ) does not equal P (M FT ) does not equal P (M FT ), c P( I S ) ≠ P (I S)

therefore not independent. ∴ not independent

3 a 88% 4 4a S
b
0.2
12

c No, from the tree diagram the second branches are not R
the same.
0.5

0.8

S

14 14
4a
b
30
70

S

c Yes they are independent as 0.8

0.5

16 64 R

P (M L) = = P (M L )=

30 120

0.2

S

b Th 2: P (R ∩ S ) = 0.1

P (R ) = 0.5 P (R ∩ S ) ≠ P (R ) × P (S )
∴ Not independent
P (S ) = 0.1 + 0.4 ∴ Not independent

= 0.5

Th 3: P (S R) = 0.2

P (S R) = 0.8

34 8 Answers

5a Review in context
B
T T 45 9 1
B 19 29 ii P (N ) =
11 11 48 1a i P (D ) = =
30 40 22 2
70 200 40

1 35 7

iii P (D N) = iv P (D N )= =
10
100 20

b They are not independent from Theorem 3.

30 29

b c d Not independent, by Th 3

2a T

70 48

0.98

6a B
H
0.01

B 0.02

T

4 4 2

T

0.05

0.99

4 B

8 6 10
i iii
b ii 0.95
14 14
T

14

4 6 8 b 0.0098 + 0.9405 = 0.9503
3 The events are independent.
iv v vi

14 14 14

4 4 4 Smoker ( A )
ix
12 Non-smoker ( A )
vii 14
viii
14
14

Cancer (B ) 1763 981 2744
197 256
c P (H 4 P (H 4 200 000
B) = B )=

6 8

(B ) 73 237 124 019

P (B 4 P (B H )= 75 000 125 000
H) =

8 6

7a 1763

Should have Should not have Total a P (B A) =
dress code dress code
75000

b P (B 981
A)=
Middle school 15 30 45
124 019

High school 40 80 120 c They are not independent from Theorem 2

d Students’ own answer

Total 55 110 165

10

5 a P(B A) = 0.69 b P(A) = 0.01 c P(B ) =

b Theorem 2 330
P (MS and Dress code ) = P (MS ) × P (dress code)
b The events are not independent.

15 45 55

= ×

165 165 165

Theorem 3

P (MS DC ) = P (MS DC )

15 30
=

55 110

Therefore they both work.

Answers 34 9

Index

A cosine r ule 183–4, 189–94
application 195–9
A4 paper 203–4 formula 192, 199
additive systems 3
ambiguous case for the sine r ule 243–5 CPCTC 273
amplitude of a sinusoidal function 135–7 cuboids 168–176
Angle-Side-Angle Congr uence (ASA/AAS) 269, 277
angular displacement 127 D
areas 183–4
data 76–7
area of a parallelogram 188 dispersion 77–87
area of a segment 188 mean 82
area of a triangle 184–9 normal distribution 86–7, 94
area of a triangle formula 185, 199
area of an equilateral triangle 188 decay problems, growth and 114–116
sine and cosine r ules 189–94, 199 decimal exponents 98, 105–6
sine and cosine r ules, applying 195–9 decimal number system 5, 9–10
argument of a logarithm 111 decomposing functions 32
arithmetic series 146, 151, 164 diagonals 170, 171, 180
asymptotes 218, 226, 229, 231 dilation 211, 213, 221–2, 229

B logarithmic functions 211, 213
rational functions 221–2, 229
base of a logarithm 111 dimensions 167–8
bases, number 3–17 describing space 168–75
bell cur ve 86 distance formula 172, 180
binar y numbers 16–18 dividing space 175–9
face diagonals 170
C nding the midpoint 176, 180
higher dimensional space 179–80
change of base formula 111 planes 169, 180
circles 124–5 section formula 178, 180
space diagonals 170–1, 180
area of a segment 188 dispersion 77–87
radian measure and the unit distance formula 172, 180
dot product 68, 73
circle 125–34, 142
column vectors 56–7, 73 E
common ratio, r 153–4
components, of vectors 59 Euclid 273
composition of functions 29–32 exponential equations 109, 121
compounded interest 117 exponential functions 202, 203
conditional probabilities 299–300
from exponential to logarithmic functions 203–5
conditional probability axiom 302, 312 inverses 205
making decisions with probability 309–12 exponents
mathematical notation 301, 312 decimal exponents 98, 105–6
representing conditional probabilities 305–9 fractional exponents 97–8, 98–102, 106
Theorem 1 302
Theorem 2 302, 312 F
Theorem 3 302, 312
what is conditional probability? 301–4 face diagonals 170
congr uence 264–5, 277 four dimensions 179–80
conditions for congr uence 269, 277 fractals 163
CPCTC 273, 277 fractional exponents 97–102, 106
establishing congr uent triangles 265–73 frequency of a sinusoidal
shapes 266
sign 270 function 135–7
using congr uence to prove other function operations 22
functions 21–2, 35–6
results 273–6
using similarity to prove other results 276–7 decomposing functions 32–3
doing and undoing 50–1

35 0 Index

exponential functions 202–5 logarithmic functions 202–3, 213
function operations 22–9 dilation 211, 213
inverse functions 42, 44–50, 51 from exponential to logarithmic functions 203–5
logarithmic functions 202–12, 213 graphs of logarithmic functions 206–7, 209, 213
modelling using compositions of functions 29–32 proper ties of logarithmic functions 205–10
one-to-one functions 38–44, 51 reection 211, 213
onto functions 37–8, 51 transformations of logarithmic
rational functions 216–31 functions 210–12, 213
reversible and irreversible changes 44–50 translation 211, 213
sinusoidal functions 134–41, 142
types of 37–43 logarithms 108–9, 109–14
argument 111, 121
G base 111, 121
change of base formula 111, 121, 257, 261
Gallivan, Britney 204 creating generalizations 250–4
Gauss’s method 149 laws of logarithms 249–54, 261
generalizations 250–4 natural logarithms 117–21, 121
power r ule 251, 261
proving generalizations 255–8 product r ule 251, 261
geometric series 146, 154, 164 proving generalizations 255–8
quotient r ule 251, 261
innite geometric unitar y r ule 251, 261
series 162, 164 using logarithms to solve equations 114–17, 258–61
zero r ule 251, 261
grad/gradian 125
graphs of logarithmic M

functions 206–7, 209, 213 magnitude, vectors 60–61
graphs of sinusoidal
mapping diagram 49
functions 134–41
growth and decay problems 114–116 Mayan number system 11–12

H mean 82

hexadecimal 17 midpoint, nding 176, 180
hieroglyphic number
modelling
system, 3–14
horizontal asymptote 218, 231 compositions of

equation 226, 229 functions 29–32
horizontal dilations 254
horizontal line test 37, 51 multiple transformations,
horizontal shift 136, 142
horizontal translation of a sinusoidal function 135–7 order of 229
horizontal translations 254
hyperbolas 219 multiplication tables, dierent bases 16

I N

indices 97–8, 102–5, 106 Napier, John 118, 251, 256
innite geometric series 162, 164 non-linear inequalities 282–3, 297
inter net search ranking 255
inverse functions 42, 44–50, 51 algebraic solutions to non-linear inequalities 291–6
non-linear inequalities in one variable 283–8
exponential functions 205 non-linear inequalities in two variables 288–91
inverse of a rational function 230–1 solving a quadratic inequality algebraically 292, 297
isosceles triangle theorem 274 solving a quadratic inequality graphically 284, 297
solving non-linear equalities in two variables
K
graphically 289, 297
Koch snowake 163 normal distribution 86–7, 94
number systems 2–3
L
binar y numbers 16–18
Leibniz, Gottfried 6 numbers in dierent bases 3–12
line of symmetr y 42 performing operations 12–16

O

one-to-one functions 38–44, 51
onto functions 37–8, 51
ordered pairs 41

Index 3 51

P scalars 61, 73
scalar multiplication of a vector 65, 73
paper folding formula 204
section formula 177–8
parallelograms, area of 188 sequences 146
series 145–55, 164
per pendicular height 185
arithmetic series 146, 151, 164
per pendicular vectors 69, 73 geometric series 146, 154, 164
innite geometric series 162, 164
phase shift 135, 136, 142 series and fractions 159–63
series in real-life 155–9
place value system 4 Side-Angle-Side Congr uence (SAS) 269, 277
Side-Side-Side Congr uence (SSS) 269, 277
planes 169, 180 Sier pinski Triangle 163
sine r ule 183–4, 189–94
populations 79–80 application 195–9
formula 189, 199
population growth sinusoidal functions 134–41, 142
horizontal shift 136, 142
equation 117, 121 order of transformations 141, 142
phase shift 135, 136, 142
sample versus population 91–4 SOHCAHTOA 132
space 167–8
position vectors 57, 73 describing space 168–75
distance formula 172, 180
positive numbers 111, 121 face diagonals 170
nding the midpoint 176, 180
positive real numbers 99, 101, 106 higher dimensional space 179–80
planes 169, 180
power r ule of logarithms 250–1 section formula 178, 180
space diagonals 170, 171, 180
proving 255 special triangles 132
standard deviation 76–7, 94–5
product r ule of logarithms 250–1 calculating from summar y statistics 89–90
dierent formulae for dierent pur poses 87–91
proving 255 sample versus population 91–4
ultimate measure of dispersion 77–87
proper ties of graphs of logarithmic functions 206 sum of a sequence 147

pyramid scheme 157 T

Pythagoras’ theorem 170, 192 theodolite 198
transformation
Pythagorean identity 239
exponential functions 211–12
Pythagorean triples 171 logarithmic functions 210–13
rational functions 224–9
Pythagorean quadr uples 171 sinusoidal functions 141, 142
translation
Q logarithmic functions 211, 213
rational functions 229
quadrants 128–9 vectors 63, 73
quotient r ule of logarithms 250–1 tree diagrams 301–11
trial and improvement 110–111
proving 255 triangles 183–4
area 184–9
R area of a triangle formula 185, 199
area of an equilateral
radians 125–34, 142
angles 127 triangle 188
radian measures 126 CPCTC 273, 277
establishing congr uent triangles 265–73
radicals 97–8, 102–5, 106
radicands 104, 106
Ratio identity 239
rational functions 216–24, 231

dilation 221–2, 229
inverse of a rational

function 230–1
parent function, domain and range 226
transforming rational functions 224–9
translation 229
reciprocal functions 217, 219
rectangular hyperbola 219
reection
logarithmic functions 211, 213
reversible and irreversible changes 44
Right Angle-Hypotenuse-Side Congr uence

(RHS) 269, 277

S

samples 79
sample versus population 91–4

35 2 Index

triangular numbers 147 operations with vectors 62–70
trigonometr y 233–4 per pendicular vectors 69, 73
position vectors 57, 73
ambiguous case 242–6 resultants 63, 73
cosine r ule 183–4, 189–99 scalar multiplication of a vector 65, 73
Pythagorean identity 239, 246 translations 63, 73
ratio identity 239, 246 vector addition law 63–64
sine r ule 183–4, 189–99 vector geometr y 70–2
trigonometric equations 234–8 zero vectors 57, 73
trigonometric identities 235, 238–42, 246 Venn diagrams 305–310
ver tical asymptote 218, 231
U equation 226, 229
volume diagonal 171
unbiased estimate 92
unit circle 126–133 X
unitar y r ule of logarithms 250–1
x y plane 169
V
Z
vectors 54–60
column vectors 56–7, 73 zero r ule of logarithms 250–1
dot product 68, 73 zero vectors 57, 73
equal components 59, 73
magnitude 60–2, 73

Index 353

MYP Mathematics

A concept-based approach

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students to develop a deep and engaged understanding of mathematics.
An inquir y-based approach combined with links to Global contexts equips Extended
learners to acquire and practice the essential knowledge and skills while
exploring the wider applications of mathematics.

This text will

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● Explore the wider meanings of mathematical topics through

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