Math AA SL

574 Answers

5 a f ( )−1x = 1 (2 x1−) b f ( )−1x = 1 (3x1+) b f −(1)x2ln=( 1), x +x Ͼ −1
3 4
6 a f ( )−1x b f ( )−1x
7 a f ( )−1x = +x2 2 b f ( )−1x = x−2 3 22 a 4x1+
8 a f ( )−1x x−1
= 3x−2 = 3 x+3
b f ( )−12x >
= 3 x+2 b f ( )−1x = 3 x − 5
1
9 a f ( )−12x 3 = x+ b f ( )−13x 1 = +xx− 1 23 7
x −1 24 a
x+ x− y
10 a f ( )−12x 1 = 3x2− b f ( )−13x 1 = 2x3−

11 a x 2 b x −5

12 a x −1 bx 3
13 a x 2
bx 0

14 a −1 x 1 b −2 xϽ 2 −4 x

15 a x −1 bx 1
16 a x −2
17 a x Ͻ −2 bx 3

b x Ͻ −1
3x4−domain −4
18 a 5 3 b x x≠0

( )191ln 3x domain x > 0 b 2ln5
5 25 a 0

20 a y b − −3x

26 a x 5

b 5 −1 3 x
27 4

x Chapter 14 Mixed Practice

1 a f ( ) −1x = x+1
3
2a 5
b f ( )−1x = 3 5x15+ 3 0.462 b −4
21 a y 4a 3 b1

−1 5 x = −3
6a x>2

b 2 +e 3 x , g−(1x) Ͼ2

7 3 x − 1
3

x 8 a 2x33 +

b −1.15

9 a i −1
ii 0

b −3 x 3

Answers 575
c
y d i 2.5 ii =x0
5
e1

21 29
4

y = f (x) 5 x 22 a 0.577
−5 y = f − 1(x)
b Reflection in the line = y x
c 32

23 a i 1 , ≠x − 3
2x3+ 2
y=x
2 3,
−5 ii +x≠ x 0

1 b − ( 1, 1)
−
10 a x 1 − 2, x ≠ 1

11 f −(1)x4 1= x+ domain x ≠ 3 Chapter 15 Prior
3− x Knowledge

12 b 9.39

13 a 3 b − 14 1 3x120+8− x
14 a ex2+ , >y 2 b x− 2 2 (x1−0)( 1) x +
323
15 a a = − 1 b − −1 2 x 4 x > 73
16 a 28 2 b 9 −2 x
Exercise 15A
c i Reflection in the line = y x

ii x − 3 iii y 1 1ai2 ii 1 iii 3
bi3 ii 3 iii 1
17 a i 22 ii 4x1+1 ii 1 iii 2
iii 9 +x 4 x−1 2ai1 ii 2 iii 3
bi1 ii 3 iii 1
b f( )x can be 1, which is not in the domain of g ii 1 iii 3
3ai2 y
c ii y ≠ 1 bi2

18 a 9 b5 4a

19 a x+ 2
3

c i 2.5 y
ii

6

2x

23 x

576 y Answers
b by

12

8

x −3 −2 x
7a y
24

5a y

−3 2 x x
−3 5

b −6 −45 x
−3 y by 4

4x −5

−12 8a −60
6a y y
x
8 −3

−4 −1 x

Answers 577
b
y by
−3 −2 x 12

−18 −3 4 x

9a y 11 a y =x (−3+)( 1) x b y =x x(−4+)( 2)
18 12 a b y =x x3−(−1)( 4)
13 a y = 2−(x−1)x( 3) b y =x x2−(+1)( 4)
−2 3 x 14 a y =x 3−(+2)( 3) x
y = −3−(x1−)x( 2)
by b
x 15 a y = −2−(x1−)x( 3)
y = −4−(x1+)x( 3)
−6 b
16 a y = −3−(x3+)( 1) x
10 a y y = −(−x3)+( 2) x
6 b y = −(−x2)( 3) x +

−3 2 x 17 a (x3)−6+; (23, 6)
b (x4)−2+; (24, 2)

18 a (x2)+3+; (2−2, 3)
b (x3)+6+; (2−3, 6)

19 a ( x2)+3−; (2−2,−3)
b (x5)+2−0;2−( −5, 20)

20 a ( x2)−3−; (22, 3) −
b (x4)−1−3;2(4, 13) −

21 a ( x2)+9−; (2−2,−9)

b (x1)+9−; 2(−1−, 9)

x + 23+ 2 141 ; 3 , 11
2 4
−
( ) ( )22 a
x + 25+ 2 7 ; 5 2 , 47
( ) ( )b 4
−

3 2 5 3 5
2− 4 2 4
x − ; , −
( ) ( )23 a
x − 72− 2 37 ; 72 , −374
( ) ( )b 4

1 2 1265 ; 1 25
4− 4 16
x − , −
( ) ( )24 a
( ) ( )bx − 16− 2 13 ; 1 , − 13
36 6 36

578 Answers

25 a 2(x + 2)27+; x = −2 Exercise 15B

b 2(x + 3)25+; x = −3

26 a 3(x1−) 7; 2 + x = 1 1 a 2, 3 b 3, 5

b 3(x − 2)27−; x = 2 2 a −3, 2 b −3, 4
3 a3
27 a 2(x1)+3; 2 − x = −1 b −5
4 a − −1, 3 b − −2, 1
b 2(x + 2)21−1; x = −2
5 a −2, 3 b −4, 5
b2
( )28 a 1 2 1 1 6 a −3 b 0, 3−
2 2 2 7 a 0, 5
2 x − + ; x =

( )b 3 2 1 3 8 a 0, 4− b 0, 3
2 2 2 9 a 4, 4− b 7, 7−
2 x − + ; x = 10 a 2, 2− b 4, 4−
11 a 3±6 b 2 ±6
( )29 a 1 2 23 1
6 12 6
3 x + + ; x = −

1 2 11 1 12 a − ±5 2 7 b − ±4 2 3
6 12 6
( )3 x − + ; x = 3 ±5 5 ±33
2 2
b 13 a b

30 a − 2(x1)−3; 2 + x = 1 ±1 21 ±3 13
2 2
b − 2(x − 2)27+; x = 2 14 a − b −

31 a − +( x 2)22+; x = −2 15 a 6 ±33 b 2 ±5
2 2
b − +( x 3)21+2; x =− 3

32 a y = (−x3) 2 2 + b y = (−x 2)23+ 1 13 35
16 a − ± 4 b−± 4

33 a y = (+x1) 2 2 + b y = (+x2) 3 2 + 17 a − ±7 37 b − ±5 13
34 a y = 2(x − 2)23− b y = 3(x3−) 1 2 − 2 2

35 a y = −(x+2) 5 2 − b y = −(+x1) 2 2 − 5 33 3 29
18 a − ± 2 b−± 2

36 a y = −2(x1)−5 2 − b y = −3(x1−) 2 2 + 19 a 5 ±17 b 9 ±65
2 2
37 a y = −(x+3) 4 2 − b y = −2(x2+) 5 2 −
20 a − ±4 19 b − ±3 13

38 a −24 b −3, 8 5 17 7 33
b −3 21 a − ± 4 b−± 4
39 a 2, 3
bx= 52
c y = −3x2 + 3+x18 b 2(x + 3)25+ 22 a 7 ±89 b 3 ±89
10 10
5 2 21 b −5
2 4
( )x − − b −5 5 ±22 4 ±10
3 3
40 a 23 a b

41 a (0, 23) 27 12
24 a − ± 2 b−± 2
c (−3, 5) 25 a 3 Ͻ Ͻx 7
26 a −3 x 5 b 2 Ͻ Ͻx 6
42 a h = 2, k = 7 27 a x Ͻ −3 or x Ͼ −2 b −2 x 5
28 a x 0 or 6x b x Ͻ −4 or 3x Ͼ −
c y = −5x2 + 20x13− b x 0 or 7x
29 a ( x3)+( 4) x −
43 a 3(x1+) 5 2 −

44 a 5(x1)−2 2 − b f( x) −2 b −3, 4

Answers 579

30 a (x − 3)21−1 b 3 ±11

31 a (x + 5)22−5 b − ±5 4 2 12 k Ͻ 98
32 1 Ͻ Ͻx 5
33 x Ϫbor x b 13 k Ͼ 2254

34 −3, 7 14 b 4
35 x Ͼ 4 15 b − −1, 9
2± 7 16 k Ͻ −4 or 4k Ͼ
36 3 17 a 0 or 1a2

37 x Ͼq p+or Ͻ − x p q 18 ±6

38 b 5, 8 or −8, −5 19 c Ͼ 98
20 0 Ͻ Ͻb 8
39 b 22 cm 21 a Ͻ −1−4 3 or
22 k Ͻ −3 ork1Ͼ
40 (−2, 0) and (3, 5)

41 x = 3 b5m c455 s a Ͼ −1+4 3
42 a 2 s

) )3 − 2 23 , −3− 2 23 3 + 23 , −3+ 23 23 a 49
2 2 24 −4 Ͻ Ͻa
( (43 and 4

44 x = 5 or 2− 25 a Ͻ −1
3

45 −3 Ͻ Ͻx −2 or Ͻ2 Ͻ x 3 26 k = 274
46 1 Ͻ x 3 27 −5 Ͻ Ͻk 5
47 y x= or y = 2x
Chapter 15 Mixed Practice
48 x k= or x k = +1

49 x = 0, y = 1 1 a ( 9)(x2+) x −
b ( 9−, 0), (2, 0)
Exercise 15C

1 a −24, zero b 17, two ( ) ( )−3, 0 , 3 , 0
2 2
2 a 0, one b 0, one
b 49, two 2 −2−(x2+) 5 2
3 a 9, two
3a

4 a 56, two b −60, zero b (2, 5)

5 a k Ͻ 16 b kϽ9 4a 3 b1 c2
5 a a = 3=, 2b
6a k Ͼ 25 b kϾ 1
12 4
b ( 3 −2, 0 , 3 )2,( 0 + )
7 a k = 12 b k = 20
6 a − ( 3, 0), (5, 0)
8 a k = ±12 bx= 1
b k = ±20
8 c −16 b two
9 a −8Ͻ Ͻk 6 7 a 41
b −6Ͻ Ͻk 8 k Ͼ 45
16 b − 43

10 a 3

11 ±20

580 Answers

9ax= 2 y 2 x = 1, y = 1 b x4
b 3 ax2 y

4 y=x

A(0, 6)

y = f − 1(x)

y = f (x) x

B(1, 0) D x

C(2, −2)
c3

10 a 3( 1)x7− 2 + b f( ) 7x Exercise 16A
b f( ) 7x
11 a −(+x2) 7 2 + 1a y
12 − ±2 2 6 8
6
13 b 9cm 2 4
14 −6 2 р рk 6 2 2

15 a i −5 ii 78 −8−6−4−2
by −2
(−1,11) −4
−6
(3,7) −8 x
8642
4 (5 ), 87 by
4 8 x
6 8642
x 4
2
16 a − 10( x4)+( 6) x − b −10(x1)−2520+
17 b ±5 b12 −8−6−4−2
18 a ( 3)x6− 2 + −2
−4
19 ± 15 −6
20 a Ͼ 1 −8

Chapter 16 Prior
Knowledge

1 a x −1 b f( ) x0

Answers 581

2a y by
8
8 6
6 4
4 2
2
x −10 −8 −6 −4 −2 −2 x
−8−6−4−2 2468 −4 642
−2 4a y −6
−4 8 −8 x
−6 6 10 8 6 4
−8 4
2
by
8

6

4

2

−8−6−4−2 x −6−4−2 2
−2 2468 −2
−4
−4 −6
−8
−6

−8

3a y by x
−10 −8 −6 −4 −2 10 8 6 4 2
8 8
6 6 b y x = 2+ 53
4 4 b y x =x 8− 7−2 1
2 2 b y x= 3−( 6)3

x −6−4−2
246 −2
−2 −4
−4 −6
−6 −8
−8
5 a y x = 3+ 32
6 a y x x= 8− 7−2 4
7 a y = 4−(x3)2

582 Answers

8a y = (+x3) 6( 23)+2 x + + by
b y = (+x2) 5( 22)+4 x + + 6

9a y 4

8

6 2

4 −6−4−2 2 4 6x

2 −2

−8−6−4−2 x −4
−2 2468

−4 −6

−6 11 a y
6
−8

by 4
10

8 2

6 −6−4−2 2 4 6x

4 −2

2

2 4 6 8x −4

−8−6−4−2 −6
−2

−4 by
6
−6

10 a y 4

8 2
6
4 −6−4−2 2 4 6x
2
−2
−8−6−4−2
−2 2468 x −4
−4
−6 −6
−8

Answers 583

12 a y y
21 a
12
10 8
8 6
6 4
4 2
2
−8−6−4−2
−12−10 −8 −6 −4 −2 2 4 6 8 10 12 x −2 8 6 4 x2
−2 −4
−4 −6
−6 −8
−8
−10
−12

by by
6
16
12 4
8 2
4
4 8 12 16 x −6−4−2 x
−20 −16 −12 − 8 −4 246
−4
−8 −2
−4
−12
−16 −6

13 a vertical stretch, sf 3 22 a y
b vertical stretch, sf 2
12
14 a vertical stretch, sf 1 2 10
8
b vertical stretch, sf 1 3 6
15 a horizontal stretch, sf 1 2 4
2
b horizontal stretch, sf 1 3 10 8 6x4 2
16 a horizontal stretch, sf 2 −10 − 8 − 6 − 4 − 2
−2
b horizontal stretch, sf 3 b (5, 1)− −4
17 a (2, 3)− b ( 3−, 1) −6
18 a ( 2−, 4) b ( 1−, 5) −8
19 a ( 2−, 3) b (5, 1)−
20 a (2, 3)−

584 Answers
10 8 6x4 2
by 24 a y
8
12
6 10
4 8
6
2 4
2
−6−4−2 2 4 6x
−10 −8 −6 − 4 − 2
−2 −2
−4
−4 −6
−8
23 a y
b
16 8
14
12 6
10
8 4
6
4 2
2
−6−4−2 246
−12−10−8 −6 −4 −−22 −2
−4 2 4 6 8 10 12 x
−6
−8 −4

−6

by 25 a vertical stretch with scale factor 3, vertical
8
translation −2 units
6 b vertical stretch scale factor 5, vertical
4
translation 1 unit

26 a horizontal stretch scale factor 2, vertical

2 translation −1 unit
b horizontal stretch scale factor 2, vertical

translation 3 units

−6−4−2 2 4 6x 27 a horizontal translation 2 units, vertical
translation 5 units

−2 b horizontal translation −3 units, vertical
−4
translation −4 units , reflection in
3
28 a horizontal stretch scale factor 1
y-axis

b horizontal stretch scale factor 1 2 , reflection in
y-axis

Answers 585

29 a y 32 a (x5)−1−42
6 b horizontal translation 5 units, vertical
translation −14 units
4
33 a 5( 3)x2+
2
b horizontal translation −3 units,
vertical stretch scale factor 5

2 4 6 8x 34 a y x = 9 2 b3

−4−2 35 a y = 16x3 b 12
−2
36 y x=x2+ 53 2
−4 37 a y = 5f(x) 3+

−6 b translation 3 5 units, stretch sf 5
38 y = 2f( x) 1−
by

8 Exercise 16B

6 1a y
x = −1
4

2

2 4 6x 3

−6−4−2 y=2
−2

−4 3 x
2
−

cy

8 by
6
4 2 4 6 8 10 x y=3
2 x
− 1 1
−8 − 6 − 4 − 2 3 2
−2
−4 x = −2
−6
−8

30 y = 12x2 + 56x6+0
31 y = 3ex2−

586 Answers
2a y
4a y

x=5

y = 3
2

1x y=1 x
−13
by
x = − 21 x = −3
by
y=1
y = 4 x
3

3x
4

x = − 1 −3
3

3a y 5 a x = 0, 0y =
x = −21 3
2 y

x

x

by

b vertical translation 2 units up

x
y= 0

−2

x = − 1
3

Answers 587

c x = 0, y = 2 9 a x = −1=, 2y
y
b − 32
y
10

y=2 y = 32

x x
− 21

− 31 1
3

6 a x = 0, y = 0 x = −23
b horizontal translation 3 units to the right
c x = 3, y = 0 11 y

7 a horizontal translation by −2 units
by

x = −2

y = 1x 3 y = 43 x
4
x −

y =x1+ 2 −3 x = 31

( )8 a 3 12 a 2 +5 x
(−3, 0), 0, −2 b vertical stretch with scale factor 5, vertical
translation by 2 units
b x = 2, y = 1 c x = 0, 2y =

cy 13 b horizontal translation by 3 units, vertical
translation by 2 units

cy

−3 − 23 y=1 x y=2
x
5
x=2 3

5
2

x=3

588 y Answers
b x
14 a 5 −1 x x
b reflection in the x-axis, vertical translation by a x
5 units b x
cy y=1

a

y=5

1 x x=b
5
y Exercise 16C

1a y

15 a x
b x−2

1

y=1 x
16 a y
x=2 by
1

a y=1
b x

a

2a y
1
x=b

Answers 589
b
y by
1
1
x x

3a y 5a y
b y=x

4a 1 y = 2x 1 x
x 1

y = log 2 x

y b y
y=x
1
x y = 4x 1 x
1

y 6a y = log 4 x

1 y
x y=x

y = 10x 1 x
1

y = log x

590 y by Answers
b y=x y = 4−x y=x

y = ex 1 1 x
1 1
x

y = log 0.25 x

7a y = lnx 9 a k = 0.742 b k = 0.531
10 a k = 1.44 b k = 1.63
y = 0.5x y b k = −0.693
y=x b k = −1.39

11 a k = −0.511 y = 2x
12 a k = −1.10

1x 13 A = ii, B = i, C = iii
1
y = log 0.5x ( )14 x y

y= 1
2

by y=x
y = 0.2x
1
y=0 y x

1 x 15
1

y = log 0.2x

8a y y=e x+2 3
y = 5−x y=x 1
y=2
y = ex x
y=0

1
x

1
y = log 0.2x

Answers 591

16 y b x = −5 y

ln 5 y = ln(x + 5)
−4 x

3 y = ex
y = 3e x

1

y=0 x

24 C = 3, 4a =

17 y 25 a −2 b 0.85
y=x
26 a 1.65

b horizontal stretch with scale factor 0.607

27 a f( x) 2Ͼ −

y = 6x 1 b f −(1)xln = ( )x+ 2 ,x Ͼ −2
c 3 y
x
y=0 1 y=x
y = log 6x

x=0 ( )2 1 y = f −1(x) x

18 A = iii, B = i, C = ii y = f(x) ln 3 1

19 A = ii, B = i, C = iii ln (32 )
20 y
x=2
x=0

y = lnx 28 a y = ln 3x
b vertical translation by ln 3 units
13 x
29 a vertical stretch scale factor 5
b horizontal translation − ln5 units

Chapter 16 Mixed Practice

y = ln(x −2) 1a y
x=1
b3
21 a 2 by=6 (8, 4)
22 a 8
23 a x = −5 3x
7

592 Answers
b
y 6a y
x = −1

2x 1 x
(2.5, −2) x

c y x=0
x = −6 by

15 x x=2
(12,−2)
−2

3

2 2x3 − 18x2 + 50x4−2 cy

3 a ( 2) x5+ 2 + 1x
b horizontal translation by −2 units, vertical 3 e5
translation by 5 units

4a yx = +32
by
x = −2

y=0 3 x=0
2x

7 a = 3, 4b =

8 vertical stretch scale factor 13, translation 5 units
to the left

9 y = ln(e6( 2x)−) 2

5 b translation 5 units to the right and 2 units up. 10 a x = −72 , 2y = ( ) ( )3,0 , 0, 3
c x = 5, y = 2 4 −7
b

Answers 593
c
y 21 y

y=2 3 x
−1 1

4x
3
− 7

x = − 27 22 y = 2− f( ) x

11 a x = −5, y = 3 b x ≠ −5, f( x) 3≠ Chapter 17 Prior
Knowledge
12 a translation 3 units to the left
y
b x = −3 x = ln (x2 + 6x + 9)

1 x = 0, 2
3
2ln3
ln 3 y = ln(x + 3) 2 x = −4, 7
3 x = l−n ≈7 2− 0.05410
−2 x
4 x= 12

5 £113.14

Exercise 17A

13 a p = −4, q = 6 b y = −x+2 12x31− 1 a 0,ln 4 b 0,ln5
c y=7 2 a 0, e2
14 a 3 b7 3 a 1,ln 2 b 0, e3
4 a 0, 1±
15 a 840 b2 c 40 5 a 0, 8 b 1,ln 3

ln( )16 e2 6 a 0, 1,ln5 b 0, 3±
x−3
7 a ± ±1, 2 b 0, 23
17 a x = 5, y = 2 b α = 2, β = 1 8 a 3 32, 3
( )b 0, 2−, ln 20
c horizontal translation 5 to the right, vertical 9 a ±2 3
translation 2 up 10 a 4, 25
11 a 16 b ± ±2, 3
d x 5 9− e reflection in y = x 12 a 0,ln 6 b 1, 73
x−2 13 a ln 3,ln5
18 horizontal stretch factor 2 14 a ln 2
15 a ln 3 b ±3
19 vertical stretch factor 1ln10 16 a no solutions b 9, 16
20 y 17 a 2 b 25
18 a 3 b 0,ln 3
x=4 19 0, 5± b ln 3,ln 4
b ln 2
log16 b ln 4
b no solutions
35 x b1
b5

594 Answers

20 1, 8 13 a y y = x1 − 3
5 y = ln(x − 2)
y=0 x
21 ± 2±, 6
22 e4, e −2

23 ln 3,ln 7
24 ln 3
25 ln 2
26 2

27 −1, 27

28 3

29 10 x=2 x=3
b zero
30 2, 3 b two

31 8, 1 14 0.0922, 0.643, 13.8
2 15 −3.08 < k< 3.08
16 0.667
32 −1 log 23
17 k > 2.15
33 ±4, 8, 6 18 a three

Exercise 17B

1 a −1.52 b 1.24 Exercise 17C
2 a −1.09, 2.32 b −3.04, 0.650
3 a −0.836, 0.910, 1.84 1 1.54 seconds
b 0.300, 2.36 2 2.6 months
b − 1.71, −0.731, 0.698 b 2.42 3 2.20 seconds
4 a 0.144, 3.26 b 0.191, 4.12 4 17.2 hours
b −1.32 5 72 cm2
5 a 1.17 b 4.81, 10.2 6 12
6 a − 3.02, 0−.175
7 −0.257, 0.361, 0.896
7 a −1.75 8 a 600 1.03 n
8 a 1.59, 4.41 9 58 years
10 a 12 m
9 −2.02, 0.379, 1.12 y b 60 km per hour
10 4.42, 11.0 c 50 km per hour

11 a 11 a ln(2 3± )

2 12 b 29.4
y = 2 − x2
y = e− x 13 b − ln 2
x
Chapter 17 Mixed Practice

b two 1 0, e51−
12 −1.84, 1.15
2 −1.81, 2.12
3 a 42 units

b 2.47 and 6.93 hours
c 30 hours

Answers 595

4 ±3 Exercise 18A
5 b 0.17 s
6a y 1 a3π b 4π

y = 4 −x2 2 a5 π b 2 π
6 3

y = ln (x) 3 a2π b 3 π
x 2

4 a 0.489 b 0.628
5 a 1.17
b 1.36
6 a 3.42
x=0 7 a 35.5° b 4.12
8 a 72.2° b 47.6°
b one 9 a 264.1° b 77.3°
10 a 36° b 300.2°
7 a 10 500 b 13 400 c 33 11 a 105° b 22.5°
8 1.51, 2.10 b two 12 a 420° b 48°
9 1, 2 c zero b 330°
10 ln 2,ln5 c 10 13
11 3 y
12 1
13 0 < k< 10.1 a
b
14 a two
x

15 x = −1.89 b 78732
16 x = ln 3

17 a 1 3

18 3 14 y

19 2 or 3 a x

20 ln (k ± k2 − 1) b

21 2
3

22 a −0.796 b −0.898
b one
23 a none
25 2 −1

Chapter 18 Prior
Knowledge

1 θ = ° 21 x − = − +x2 4 x3

2 x = − −2, 3
3 y = (−x3) 2( 23)+

4 2x
x2− 1

596 Answers
15 y
19 a 3.5cm, 8.75cm 2 b 7.2 cm, 28.8cm 2
b
20 a 7.2 cm, 14.4 cm 2 b 14.7 cm, 51.5cm 2
x
21 a 33.6 cm, 134 cm 2 b 25.5cm, 63.8cm 2
a
22 a 19.3cm , 228.0 cm b 4.73cm , 215.9 cm
16 y
23 a 41.2 cm , 233.7 cm b 15.5cm , 219.2 cm
x
24 a 177 cm ,257.3cm b 8.56 cm ,213.2 cm
b
a 25 per 2=0.8cm, area 19.2=cm 2

17 y 26 per 2=7.9 cm, area 48.=1cm 2

a 27 10.3cm b 17.5cm 2
b 28 a θ = 1.4

x 29 0.699

30 15.5cm

31 a 11.6 cm b 38.2 cm

32 area 5=7.1cm , p2er 34.3=cm

33 6.84 cm

34 1.32 cm, 13.7 cm
35 12πcm

36 0.241cm c 13.1cm

37 per 1=4.3cm, area 2.55=cm 2

38 b 1.74
39 3
40 r = 23.4 cm, θ2.1=5

41 per 3=3.5cm, area 78.=6 cm 2

Exercise 18B

1 a −0.6 b −0.6
2 a 0.6 b −0.6
b 0.6
3 a −0.6
4 a 0.6 b −0.6
b −0.940
18 y 5 a −0.940 b −0.940
b 0.940
ab 6 a −0.940 b 0.940
7 a 0.940
b −0.73
x 8 a −0.940 b 0.73

12 a −0.73 b −0.73
13 a 0.73
b − 12
14 a −0.73
b−22
15 a 1 2
b−3
16 a 2
2

17 a − 3

Answers 597

18 a 3 b−32 9a 15 b 4 29
2 8
b−22
19 a − 2 2 10 a − 5 2 b−32
b1 3
20 a − 1 3 b 49 , °131 ° 11 a 2 2− b 2 2−
b 57° 2 2
b 66 ,°114 °
21 a 64 , °116 ° b 50° 12 a 2 3− b 2 3+
22 a 55° 2 2
23 a 59 , °121 °
24 a 57° 13 a 2 3− b 2 3+
2 2

25 0 14 a 2 2− b 2 2+
2 2

27 2 −2 3 15 a − 21 5 b − 4 2215

28 5 3
6
29 cos x b − 3 1200
16 a − 2 107
30 61 cm

31 52.7, 127 17 a 4981 b ± 659

32 4.42 cm, 6.37 cm 18 7 −4sin2 x
19 9cos 2 x − 5
33 18.2 cm
34 39 2
35 y = 3x 23 a 2

36 c −1 24 33
6

Exercise 18C 25 3
3

1 a ± 15 4 b±53 26 − 7891

2 a ± 2 23 b±74 27 a 1 −2sin2 x

32 14
4
3 a − 12 b − 79
33 a 2 2−
4
4 a19 b 78
Exercise 18D
5 a cos x=± 21 , tan x = ± 2
5 21
1a y
b cos x=± 7 , tan x 3
4 = ± 7

6 a sin x=± 15 , sin 2x = 15 x = −π3 x = 2π
4 8 π 3
6
b sin x = ± 26 , sin 2x = 46 7π x
5 25 6

7 a 4 59 b 3 78 x = 5π
8 a − 2 21 b − 12 2 3

598 Answers
by
by

x = −6π x = 56π x = 6π x = π2 x = 5π
π 4π x π π 6
3 3 3 3
πx
x = 161π

2a y

x=0 x=π 4a y
π
2 3π x (π,√ 3 )
2
x = 2π

x

by

x = −34π x = 4π by
x = −π
− 54π − 4π x
5a y
3π 4 x
4 x=π
x = 54π

3a y

x = π x = 34π x = 54π x = 74π
4 3π

π π 2 2π x
2

1 x
2π

π

−2

Answers by 599
by x
1 2π x 4 x
3 3 2π
π x 2 x
−1 3 x 2π
π
−3 4

6a y 8a y
4 5

2 2

π 2π
23

by π −1
1 3π by
4
−2 5

−5 x 3
7a y
√2 1
π 2
2 π
2
−2 9 a a = 3, b2,=1 d =
−3 b a = 2, 3b,=d1 =
−4 10 a a = 2, c = π2 , 2d =
b a = 3=, πc 3 , 1d =
(2π, −3 − ) 11 a a = 4, 2b,= d = −3
b a = 2, 3b,=d2 =
12 a a = −2=, cπ 2 , 2d =
b a = −1=, cπ =4 , d3

600 Answers
13 y
Exercise 18E
π
4 x = 34π 1 a 0.386, 1.71, 2.48 b 0.912, 2.23
x 2 a no solutions b no solutions
3 a −0.795, 0.205 b 5.20, 11.5

4 a 6 π ,π56 b 3 π,π23

5 a − 7π4 −, 5π 4π ,π4 π, 34π , 94 , 141

b − 3π2 π, 2π, 52

14 π2 6 a 4 3π,π53 b 5 4π,π74
7 a − 5π2 −, 2π
b 7 π ,π161
8 a 60°, 300 ° 6
15 −17, 3
9 a − ° 3−30° °, 30 , 30 b 45°,°315
16 1.4 m 10 a − ° 1°35 , 135
11 a − ° 1°80 , 180 b − °3−00° ,° 60 , 60
17 a 6.6 m b 6 a.m. b − °6°0 , 60
b − °3°6°0 , 0 , 360
18 a 8 minutes b 11m c 8.6 m
19 a = 5, b = 2

20 3 12 a 3 π ,π4 b 6 π ,π7
21 a y 3 6

y =3 sin(x) 13 a − 7π4 −, 3π4π, π4 , 54

b 5− π3 −, 2π 3π ,π3 , 43

14 a 5 π ,π161π , 167 b 3 4π,π74π, 141
6

x 15 a − 8π3 −, 5π3−, π3 π, 23 b 2−, ,π0 −π

y = cos(2x) 16 a 230°, 350 ° b 265°,3° 55
17 a 65°,°155 b 90°, °150

18 a −π, π0, b 5− π6 π, 6

19 a −1π138−, π111−8π, 1π8π, π18, 1118 , 1138

b two c six b − 2π3 −, 6π π, 6π, 23
b7
22 a 1.6 m 5 , 34 7π ,π1214π ,π1294 , 23
c 0.279 seconds 20 a π 1π2 b 24 24

23 π 21 a 22.5°, 112.5 ° b 20°,°8°0 , 140

24 a no 3π 22 a 6 π ,π3 ,π76π, 43 b 8 π,π38π, 9π8 , 181
b 34 4
, for x =

Answers 601

23 a 105°, 165 °, 285 ,° 345 ° c h(t)
26
b 120°, 150 °, 300 ,° 330 ° (20, 26)

24 a π ,π43 b π6 ,π76
3

25 a − °135 , 45 ° b − °45 , 135 °
26 a 6 π ,π2 ,π56 ,π32π, 5
b 6 π 6 (10, 14) (14, 30)
14

27 a 45°, 135 ° b ° 0 , 135 °, 180 °
b 3 π,π53
28 a 2π ,π43 b 6 π ,π32π, 56 (40, 2)
3 b 3 π,π, 5π3 2 10 20 30 40 t

29 a 6 π,π2 ,π56 bπ d a = 12, b π , 1c4=
20
30 a 0, 2 π ,π43π,2 =
3
ei
31 a 2 π
C
32 a 0, , 2π π b 2 π ,π3 12 m A6 m
2 BO
33 0.702, 4.41
34 30°, 150 ° θ
35 ± °45± ,° 135
36 130°, 250 ° 20 m
37 b 60°, 90 °, 270 ,°300 °
38 45°, 225 ° 2m
39 0, 3π,π
ii 120° iii 13.3 seconds
b − 18
40 30°, 150°, 270° 9a−7 4
b4
41 −3+ 57 10 ± 5π , ±6 π b 6.9.1 cm2
8 6

42 − 2π3 , 0, 23π 12 (5 6 1 − )

43 k р 0 13 17.8 cm 2

Chapter 18 Mixed Practice 14 b 0, 3 π,π23π,

16 a = 3, c = π4

1 a 1.3m b 2.51m 17 a 5

3 60°, 120° c p = − π , 5q =
4 55.2°, 125 ° 4

18 π3, 5π
3
5 0.695, 3.54, 2.45, 5.88
2π 5π 19 3.7 cm and 3.6 cm
6 0, 3 , ,π 3 , 2π 20 a 68.9 (1°.20 rad)
23 10p
7 a − 1213 b − 512 c − 121069 24 0 f( ) 25x
ii 26 m 25 0.207 amps
8 a i 14 m
b 10 seconds, 30 seconds 26 6p

602 Answers
27 a y
8 76
(−1,2)
9 a It predicts chemistry mark for a given maths
mark

b 72 c No, extrapolation

10 a $1573 b $4150, $280
11 a −1 b x =y −2+8

1 x 12 a 6 cm/minute. The fact that the regression
(21 , −41) line has a non-zero estimate suggests that
k2 the model is not perfect, either due to
measurement error or because the snail is not
b 111 cm moving at a constant speed.

b 0.−25 b Y = 3.6T0.+05
c 0.6
28 a 82.3°

29 7 Exercise 19B
9

Chapter 19 Prior 1 a 0.5 b 0.25
2 a 0.8 b 0.25
Knowledge 3 a 0.4 b 0.63
4 a 0.21 b 0.06
1 a y = −0.393x1+1.1 b 5.99 5 a 0.25 b 0.5
2 37 b x = 8.32 6 a 0.625 b 0.5
3 0.125 7 a no b yes
4 a 0.106 8 a yes b no
9 a yes b yes
Exercise 19A 10 a no b no
11 a no b yes
1 a x = 1.66 y − 0.0984 b x = 0.952 +4.02 12 a yes b no
13 a no b yes
2 a x = −1.31 +9.81 b x = −1.22 +30.7 14 a yes b no
3 a i x = 21.9 15 a yes b no
b 0.3
ii reliable 17 a 0.15 c 0.25
18 a 0.1 b 0.15
b i x = −4.26 19 a P(A| B) (P)=A

ii not reliable (extrapolation) c 0.65

4 a i x = 73.5

ii not reliable (no correlation)

b i x = 87.9 20 yes
21 x = 0.14, 0y.3=9 or 0.39, 0x.1=4
ii not reliable (no correlation) y=

5 a x = −0.0620 y15+.5 22 2
5
b 2.8 km
8
c Reliable; correlation is high and the values are 23 71
within the range or data

6 363 24 0.75
7 a 65.5 cm
b 149 cm 25 5
9

Answers 3 a 1 10 b 12 603
c no
Exercise 19C

1 a 1.33 b 0.25 4 a 0.2 b no

2 a −1.5 b −2.38 5 b µ = 24.7, 1σ.3=1

3 a −1 b −3.25 6 a 1 10 b 1720
4 a 0.833
b −0.476 7 a y = 10.7 x12+1
5 a 71 b 143
b i additional cost per box
6 a 3.59 b 0.11
ii fixed costs
7a5 b9
c $760 d 13
8 a 133 b 12
e i extrapolation
9 a P(Z Ͻ 1.5) b P(Z 1.Ͻ4)
10 a P(−1.52 Ͻ ϽZ 0.2) ii the line is y on x

b P(−1.67 Ͻ ϽZ −1.13) b P(Z Ͼ 1.42) 9 4.34 cm
10 µ = 19.2, σ = 5.61
11 a P(Z 0.6Ͼ25)
12 a P(−0.286 Ͻ ϽZ 0.357) 11 19.6%

b P(1Ͻ ϽZ 1.28) 12 159 b 0.217
13 a 0.309 b 10.1%
13 a 12 b 22
b 0.761
14 a 16 b 8.6 14 a 2.28%

15 a 68 b 18 15 51.3
16 µ = 766, σ = 10
16 a 6.2 b 7.5
17 a µ = 21, σ = 2.5 b µ = 13, σ = 1.6 17 a 0.785
18 a µ = 5.5, σ = 0.82 b µ = 6.2, σ = 0.95 1
19 a µ = 35, σ = 12 b µ = 40, σ = 15 18 2
20 a µ = 28, σ = 8.5 b µ = 24, σ = 7.3
19 a 1 2 b no

21 0.0918 20 0.284

22 2.58cm 21 a −0.0209 b 0.983
23 0.927
c y = 2.07 x12.5−2
24 5.79
25 2.43 22 a 0.819 ii 0.924
c 48.7
26 7.80
27 µ = 51.6, σ 2 = 126 d i 0.360

28 4.62% Chapter 20 Prior
29 99.7% Knowledge
30 90.0

Chapter 19 Mixed Practice 1 f ( ʹ) =x2 x
2 y − 1=3−( 1)x
b 55 3 x = l≈n5 1.61

1 a 0.949

c reliable; correlation is high and the value is

within the range of the data

2 a low correlation
b the line is y on x

c extrapolation

604 Answers

Exercise 20A 25 x Ͼ 0.5
26 0.5
1 a f ( )' x = 2 x− 1 1 2
2 a f ( )' x 3 3 2
3 a f ( ) '9x b f'(x) 3= 4 x−4 bπ 4
27 a π 4
3 7
= − 1 b f (')x = − 4 28 (4, 2)
2 x− 2 3 x− 3

1 3 ( )2913π , 0
6
= x2 b f (')x2 = x− 4

5 7 30 0.5

4 a f ( ) ' x = −6 x−3 b f (')x = −4x− 5

1 b f (')x = 3 x− 3 Exercise 20B
b f (')x 2 4
5 a f ( ) ' x = x− 2
6 1a dy = 12(3x2)+3
4 dx
= x− 5
6 a f ( ) '4x = x− 3

3x− 43 4 x−73 x− 3 3 x− 5 b dy = 10(2 x7)−4
3 2 2 2 dx
7a− − b− +

8a 1 1 3 b 4 1 2 2a dy = 3 x(x32) + 1
20 21 dx 2 −4
3x4 x− 4 2x3 x− 3
− +

9a− 2 x− 4 − 1 b− 5 5 1 b dy = − 4 x(x4 )− 1
3 dx 3 2 −3
2x−3 x−4 + 6x−4
3 4

10 a f ( ')x3co=s x b f (')x = −4sin x 3a dy = 1 (2 x3x+x 1 1) 2 +
dx 2
)−(26

11 a f ( ')x = − 1 sin x − 5cos x dy 1 (5x −x 2 x2
2 dx 3
b = 3)−(35 3 −)

b f (')x = 3 cos x + 2sin x 4 a f ( )'2xcos=2 x
4

12 a dy x= 3 b dy = − 4 1 1
dx dx x 3 3
b f (')x = cos x
dy dy
13 a dx x= 2 b dx = − 1
x
5 a f ( )' x = −πsπin x

14 a ʹ =y 5e x b ʹ =y −6e x b f (')x5si=n5− x

15 a ʹ =y − e x b ʹ =y 3e x 6 a f ( )'3xcos=(3 1) x +

2 4

16 0.5 b f (')x4co=s(−1 4 ) − x

17 −5 m s−1 . Falcon is descending. 7 a f ( )'3xsin=(2 3 ) − x

18 (0,1) ( )b 1 1
2 2
19 5 f' (x) = − sin x + 4

20 y = − 3+2x4 π + 3 3 x
2
8 a 3e3x b 12e 2
b 8 ex4 x2
21 y =x + −3 3ln 3 9 a − 3xe2x − 3

22 (9,3) b 4 hours 10 a 1 x b 1x
23 a y = 2x5/−2 8x
11 a 2x
24 a 1 million x2 + 1 b − −3 4 2x
c 1.5 million/hour

d no upper limit to the number of bacteria

Answers 605

12 a ʹ =y 6sin 3 coxs3 x 7 a f ( )'4xsin=(4−1)xcos(4x1−) + x−
b ʹ =y −8cos 4 sxin 4 x
b f'(x) 5=si−n5x2−2cos5 x −x −2 x
8a
13 a ʹ =y −2sin 2 ecxos2 x b ʹ =y 5cos5 esixn5 x f'(x) 4=e 1 1 1 5x+
2
1 − 1 − 2 x2 4 5x+ + x− 2e4
2x 2 3
14 a yʹ = (ln3 x) b y' =1 3x (ln 2x)

e b f (')x = − x−2−e1 2xe− x−3−1 x

15 − − x

16 23 9 a f ( )' x = 2x−1 − x−2 ln(2 x3)−
2x3−
17 y = 4x + 9
5 5 x3
b f (')x = − 5− x + 3xl2n(5 ) − x
18 (3, 0)
π2x
19 y = 4 − π 10 a yʹx=x( 2+ 4 b ʹ =y − 12
2 x2+) ( x4)−2
2

20 y =x −3 + 5 ln 3 1
3 2
x + 2
yʹ yʹ x−5
21 a 2sin coxs x 11 a = 3 b = ( x3)−3

( ) ( )π , 1 , (π, 0), 3π , 1 ( x1+) 2
2 2
b ex(3 5x)− ex(2 7x)−
25 x = 3 12 a yʹ = (3x1+)3 b yʹ = (2 x1)−4

26 b no, e.g. = y3e2 x π4 4 πx2
−
Exercise 20C 13 y =

1a dy = xcos x + sin x 14 2 x1 − 2
dx π

b dy = 1 cos x + 1 1 sin x 15 7e6
dx 2 16 ex(( 1x)+ln 1) x +
x2 x− 2 17 ex(sxsinin x +x x cos )

2a dy = − x2 sin x + 2xcos x 18 (ln 2, 8 −11ln 2)
dx

b dy = −x−1 sin x +x −2 cos x ( )1, e1
3a dx
19
dy = 1 ex − 1 3 ex
dx 2 20 (e, e)
x−2 x− 2 k

b dy = x3ex3+e x2 x 21 (k +x 2 )1.5
dx
22 a = 3, b4 =
4a dy = 1 + 2 1 x 23 a = 0, b =− 2
dx 3
x−3 x−3 ln 12e−n
24 a (1 −3 )e2−n
b dy = +x3 4xln3x
dx b i 1000 rabbits
ii 750 rabbits per year
1 1
2 c4
5 a f ( ) ' x = x(x2 1)+ −2 + (2x1)+

b f (')x = 9 x2x(3 4−) 1 3
2 2
− 2 + 2x(3x4)−

6 a f'(x) 3=cosx32 2 sinx3+ x x

3 x + 3 x− 1 sin 2x
4 4
b f'(x) 2=cosx24

606 Answers
dP 6a y

4 x

1

n

25 b y =x by
x
Exercise 20D

1 a 10 b 12
2 a 62 b8
3 a 4e−4 b 18e6

4a−1 4 b1
5a
y

7 a i increasing, concave-down

x ii concave-up
iii  concave-down

b i decreasing, concave-up

ii increasing

iii  decreasing, concave-down

8a xϽ 4 b xϾ −3
3

by 9 a x < −1 or 3x > b 0 Ͻ Ͻx 3
x
10 2
11 a = 1, b =− 1

12 (x2)+e x

Answers 607

13 y 2 a x = 91 local max

x b x = 41 local min
3a
x = π local max
b 3

4a x = 5π local min
3

x = π local max
6

x = 5π local min
6

x = ln5 local min

14 y b x = ln 21 local max
x
5 a x = 23 local max
b x = 4 local min b 100 cm2

6 b = −2=, 3c
7 b = −4=, 1c

8 $2.5 million

9 a 40 cm

10 (0, 0) and (−2, −4)
11 (1, 1), local min

15 k = 3 )π3 , 32 6− π
(12
16 4, −3
13 108 cm3
19 − (+a 1 )2 1 ln x
x
14 a − x2 ( )e, e1
( )1, 1− e2
e d Local max b

20 5 ( )π, e , local max
3 2
21 xϽ

15

22 a −6 ( )3π , e1 , local min
b b Ͼ 12 2

23 a 6 16 a 6 million b 5.61 million
24 5x2 +10x2+
17 a $7
b It does not predict negative sales if Ͼ x10 .

Exercise 20E c $9

1 a x = −1 local max 18 a 800 −20m0 2 b 0.5 kg
x = 3 local min m

b x = 1 local max 19 f(x) 10 3ln−3
x = 2 local min
20 f(x) 30 −

608 Answers

21 (0, 3) local max 13 a x = −2, 1
(ln 2,ln16) local min b x = −2 local minimum

22 1.5 32 m x = 1 point of inflection

Exercise 20F 14 a x = 0, 4
b local minimum at x = 0

local maximum at x = 4

1 a x = 2, non-stationary 2 , e 1 − 2 , 1
b x = 4, non-stationary − 2
)2 e−2
( ) (15 2 ,
2 a x = 0, stationary )e−,233 − 2 e−3
x = 3, non-stationary (16

b x = 0, non-stationary 17 (0, 2)
x = 2, stationary

3 a x = 0, stationary Chapter 20 Mixed Practice

x = π, non-stationary

x = π2, stationary 1 a 3ax2 − 3 b −3 c14

b x = π , non-stationary 2 (−2, 25)
2 3 b = −4=, 7c
4 b = −6=, 4c
x = 3π , stationary
2 5 0.25
6 y =x −1
4 a x = ln 21, non-stationary 7 y = 1−2 x

b x = 0, non-stationary

5 (−3, 50)

6 (2, −288) 8 a A(e−,2−42e ), (1,B0)
7 b = −3, c = 5 b (e−,1−e1 )
8 a = 6, b =− 3
9 a = −3, b = 3, c = 2 3
3
10 a (2, 4 + 8ln2) b non-stationary 9 mg l −1
11 b (1, 1)
y 10 ds = 25
12 a dt

R 11 a 0
P b xϾ0
ci2

d f (x)

Q P x
R

d No. xf ʹ( ≠) 0at these points.

1.3 x

Answers 609

( )−1, 1 c 1 Ͻ Ͻk 1 5a 3 4 b 5 6
9 9 8 18
x3c+ x5c+

1132 by =x 3 1

14 2e2(x2 x2 + 4x1)+ 6 a 8x4c+ b 14x2c+

15 a 4.5cm 2 b 6 +3 2 7 a y = 3sin x +c b y = −s+in x c

16 6e5 8 a y = 2cos x +c b y = − 1 cos x +c
2
17 k = 12
18 ex(( 1x)+sin x +x cos x) 9 a y = 5+ex c b y = − 34+ex c
19 a = 1, b = 8, c = 1.5
20 a x Ͻ 2 b (2,2e −)2 10 a y = 2 ln x +c b y = 1 ln x +c
2

21 ⎛⎜⎝ π4 , e −24π⎞⎠⎟ 11 a 1 (2 x1)+ 3 c b − −83+(1 2 )x 4
3
+2 c3

22 a 5 million m3 12 a − 2 (3 5 )x 1 + b 2 (2 x7)− + 3 c
−5 3 4
c2

b +5 2 million m3 13 a − 13sin(2 3− )+x c 1
e 4
b sin(4 x3+)c +
c 2 hours after the storm

b 34.3 ( )14 a 1 5+ c 1 cos(5 −2 c
24 a 10051 e5 −2cos 2 x − b 2 )+x

c 0 f(Ͻ) 100x Ͻ b±2 15 a 1 e5x+2 c + b− 1 e1− 3xc+
25 b Not if q < b < p 5 3
26 a −2 р рx 2
27 a 0 16 a 1 4 ln 4 x5− + c b − 12−ln+ 3 2 x c

Chapter 21 Prior 17 a 1 ( x2 4+ 4+) c b 2 (x24) − + 5 c
Knowledge 4 5 2

18 a 1 sin4 x+c b 1 cos3 x+c
4 3

1 y = 1 x3 + 2+x4 19 a ln x3x+c2+ b ln 5 x4x−c+
3
3 4
2 48.4 20 a 2 ( x3 + c b − −83+(1 )
9 4+)2 x2c 3
dy
3 dx = 2e2x + cos x 21 a − 2e−x2 + c b 3ex3 + c

1 22 a 1 3 ln x3 + 5+ c b 1 4 ln 3x4 − + c

4 ʹ =f (x) x(x 2 + 3)− 2 3

Exercise 21A 23 y =x 2 2 − 42

1a 3 5 b 4 7 + c 24 2sin 3xc+os 2 x +
5 7 25 y = 2−ex5ln 2e x −
x3c+ x4

1 1

2 a 2x2 + c b − 3x−3 + c

5 5 26 x2 + 3 ln x c+
2
3 a 4x2 + c b 4x4 + c

1 3 27 V = 1 t2 + 1 cos t3+
2 2
4 a 12x3c+ b 5x5c+

2

610 Answers

28 x = 5−t2e 7 t + 9a4 3 b 43
10 a 272
29 − 1 (5 −2 )7x +c 11 a 1.63 b 52
14 12 a 2.83 b 9.77
b 1.83
30 − 3 cos(2 x) 2− 3sin(3x) c+ 13 a 36
2 b 92
14 a 1 3 b9
31 1 ( x2 + 1)6 + c
6

32 − 1 cos (3x)2 +c
6

33 3 x2 + 1+ c 15 a 8 3 b 13

34 − 1 3) + 3 16 112 b 36
2(2x + 2 9 b 3712

35 (31)ln+x 2 1 17 3

36 2ln3 5+ 18 3.29
37 − cos4 x +kx 3x3c+ +
19 25
20 a − ( 3, 0), (3, 0)

38 ex1+ 3 e−3 xc + 21 a 8 ,−512
3

1 3 22 11.0
3 52
39 ( x2 + 1)2 + c 23 81

40 b x − 1 sin2 x +c 24 −5ln3
2 4
( ) ( )4π, 22 b 2 −2
41 a − 1 2sin2 x b 1 x − 1 sin 2x +c b 0.115
2 4
26 a 1
42 − ln cos x c+ 27 a A(0,1), 1B, − e
43 − cos x + 13cos3x +c
28 a y

44 ln ln x c+ 2
y = 2 − 2sin (x2 )

Exercise 21B 1

1 a 14 3 b 12 π x
2 a 33 5 2
3 a 26 3 b − 16 √
4a1 2 b −60
5a2 b1 y = cos(x2 )
b5
6 a ln 5 b ln 32 b 0.0701
b −0.152
7 a 0.774 b 3.02 29 5.33
8 a 1.12
30 2
7

31 3

32 3 (1 −e )− 2
2

Answers 611

( )331ln11 20 a 0 s, 6 s b4s
2 5
c4s b 0 m s−1
d 2 m s−2
( )34ln2 21 a sintc+tots
3
c 4+ πm− s 1
42
35 27
36 17.5 e 0.556 s, 1.57 m, 5.10 m
37 a ln x1+
b1 f 13.3 m
38 −2
24 a −10 m s−2 b 0.5 s, 61.25 m
c4s d 62.5 m

Exercise 21C e e.g. no obstructions to path of ball

1 a v = 3t2 + 6t b v = 4t3 − 5 25 a 18 m s−1 b3s
a = 6+t6 a = 12t 2 c −8 m s−2 d 108 m
f 36 m
2 a v = −2sin 2t b v = 21 cos t2 e −36 m
a = −41sin t4
a = −4cos 2t 26 10 m s−1
b v = −4−e 4 t
3 a v = 3e3t a = 16e−4 t 27 8 m s−1
a = 9e3t 28 0.5 m
b v t= 1
4 a v =t 1 29 a 0 m b 50 m
a = −t12 b0
a = −t12 b 124 30 a 6.06 m s−1
5 a 45
b 45 c 71 ≈ 2.96 m s−1
6a4 b3 24

7a33 b ln 163 31 a i = v At u 2 + ii =s +At ut3
b 0.935 2 6
8 a ln 5 3 b 0.984
9 a 1.21 b 2.53 32 7 seconds
10 a 1.27 b 0.495
11 a 0.672 b 1.435 33 0.9375 m
12 a 1.54 b 4 m s−1
13 a 0.657 Chapter 21 Mixed Practice
14 a 16 m
1 51 1
c −2 m s−2 2
2 y = x+2
15 2 −2 e≈ 1.55m
3 3 −1
16 a −48m 2

17 23 m s−1 4 3 ln x + 2+1x c
18 a 256 m s−1 4

c4s 5 v = −10.3m s−,1 a = −3.10 m s−2
19 2.39
( )6 6ln 2 5x3−

b 176 ≈ 58.7 m 7 a −31.7 cm s −1
3 b 7.41 cm

b −32 m s−2 8 3 + 4x−c21 +
d 819.2 m
2x2

9 y = sin3 x + 2

612 Answers
y
Analysis and approaches
10 a SL: Practice Paper 1

1 y = cos x 1 a 13 cm b 38 c5 8 [6]
2 a 15 b (5, 0) c3 5 [5]
[5]
c (2.5, 7.5) b 18 [5]
3 a 80
[6]
πx 4 a −2 b 8, 14
2 5a1 b decreasing
c y = − 2πx + 1
( )b A(0,1), 2B π,0 c maximum (1, 4), minimum (4, 3−)
d (2, 1)

d −1π4 6a1 2 [6]
74 [7]
11 4 8 a (3, 0) [1]
3
b i (5, 4) [3]
12 a A(0,1), (B2, 5), (7C, 0) ii x2 − 1+0x29 [3]

b 17.2 c ( 5, 4)

13 a − 9m s−2 d ii 134 [6]
b 18.5 m [2]
[2]
14 a y
[3]
y = 2sin x + 1 9a 1 (1−) p
2

b a = 2, 2b =

1 ( )c i 1 3
x 2
(1 −) p 3
2π
( )ii 1k
(1 −) p k 2

b 2.20 m s− 1 c 6.28 m d 9.02 m d 23 [6]
[1]
15 1 log17 10 a f ʹ(x) 0 for all x Ͼ −1
2 c f ʺ(0=) 0 and ʹ Ͼf (x) 0 on either side [2] [3]
[4]
16 5π d ( 3 2,ln 3)
3
e f −(1) xe 1=, −3 ∈x x , y Ͼ −1

17 a displacement = A, acceleration = B

b =t3

18 6 Ͼ

19 a 0 Ͻx dϽ
b a x Ͻb Ͻ and x c
c 15

21 a x − 1 sin6 x b x + 1 sin6 x
2 12 2 12

Answers 613

Analysis and Approaches 5 a x ≠ −1 b f(x) ≠ 2 [6]
SL: Practice Paper 2
c f −(1)x = 3− x t [6]
x− 2 b 26.8 cm [7]
[3]
1 a 9.75 cm b 16.4 cm2 [5] 6 a a = etcos 2t2−e sint 2 [3]

2 [6] b 6.38 m [5]
x [5] [7]
2a f'(x) = ln x 7 a 51.8 cm [2]
[5]
c x = 0.340, 1.86 c 17 900 cm3 [5]

3 44.8°or 135° 8ai−3 2 ii (0, 6) [4]
b 2x3−y21− 0= ii 13 3≈.61 [2]
4a y
c (6, −3) ii 0.135
5 d i 117 1≈0.8
C e 19.5
9 a ii µ − 1.6449σ3 =

iii µ = 12.0, 5σ .4=6

b 7.40 minutes

c i 0.291

10 a i − 1 p2

( )b i Q(p2R, 0), 0, 2
p

b 0.977 5x ii 2
[6] d 2 2 2.≈83

Glossary

Amplitude Half the distance between the maximum Exponential equation An equation with the variable in
and minimum values of a periodic function the power (or exponent)

Arithmetic sequence A sequence with a common Factorial The product of all integers from 1 to n,
difference between each term denoted by n!

Arithmetic series The sum of the terms of an Geometric sequence A sequence with a common
arithmetic sequence
ratio between each term

Asymptotes Lines to which a graph tends but that it Geometric series The sum of the terms of a geometric

never reaches sequence

Axis intercepts The point(s) where a graph crosses the Gradient function See derivative

axes Gradient of a curve The gradient of the tangent to a

Base The number b in the expression bx curve at the given value

Biased A description of a sample that is not a good Hyperbola The shape of the graph ofy=x 1

representation of a population Identity A statement that two expressions are equal for

Binomial coefficients The constants of each term in all values of the variable(s)

the expansion of (a + b)n given by n C r Identity function A function that has no effect on any

Chain rule A rule for differentiating composite value in its domain

functions Indefinite integration Integration without limits – this

Completing the square The process of writing a results in an expression in the variable of integration

quadratic in the form a(x - h) 2 + k (often x) and a constant of integration

Compound interest The amount added to an Inflation rate The rate at which prices increase over

investment or loan, calculated in each period as time

a percentage of the total value at the end of the Integration The process of reversing differentiation
previous period
Intercept A point at which a curve crosses one of the
Concave-down The part(s) of a curve where the second coordinate axes
derivative is negative
Interquartile range The difference between the upper
Concave-up The part(s) of a curve where the second and lower quartiles
derivative is positive
Limit of a function The value that f(x) approaches as x
Continuous Data that can take any value in a given
range tends to the given value

Definite integration Integration with limits – Limits of integration The lower and upper values used
this results in a numerical answer (or an answer for a definite integral

dependent on the given limits) and no constant of Normal to a curve A straight line perpendicular to the
tangent at the point of contact with the curve
integration

Depreciate A decrease in value of an asset Outcomes The possible results of a trial

Derivative A function that gives the gradient at any Outlier An extreme value compared to the rest of the
point of the original function (also called the slope data set: one that is more than 1.5 standard deviations
function or gradient function) above the upper quartile or below the lower quartile

Differentiation The process of finding the derivative of Parabola The shape of the graph of a quadratic
function
a function

Discrete Data that can only take distinct values Period The smallest value of x after which a function

Discrete random variable A variable with discrete repeats
output that depends on chance
Point of inflection A point on a curve where the
Discriminant The expression b2 - 4ac for a quadratic concavity changes

Equation A statement that two expressions are equal Population The complete set of individuals or items of
for certain values of the variable(s) interest in a particular investigation

Event A combination of outcomes Product rule A rule for differentiating a product of two
functions
Exponent The number x in the expression bx

Glossary 615

Quartiles The points one quarter and three quarters of Standard index form A number in the form a × 10k
where 1 a < 10 and ∈k
the way through an ordered data set

Quotient rule A rule for differentiating a quotient of Standard normal distribution The normal distribution

two functions with mean 0 and variance 1

Radians An alternative measure of angle to degrees: Stationary point A point on a curve where the

2π radians = 360° gradient is zero

Range The difference between the largest and smallest Stretch Multiplication of the x (or y) values of all points

value in a data set on a curve by a given scale factor

Reflection The mirror image of a curve in a given line Subtended The angle at the centre of a circle
subtended by an arc is the angle between the two
Relative frequency The ratio of the frequency of radii extending from each end of the arc to the centre
a particular outcome to the total frequency of all
outcomes Sum to infinity The sum of infinitely many terms of a
geometric sequence
Roots of an equation The solutions of an equation

Sample A subset of a population Tangent to a curve A straight line that touches the

Sample space The set of all possible outcomes curve at the given point but does not intersect the
curve again (near that point)
Self-inverse function A function that has an inverse
the same as itself Translation The addition of a constant to the x (or y)

values of all points on a curve
Simple interest The amount added to an investment othr eTrial
loan, calculated in each period as a percentage of A repeatable process that produces results

initial sum Value in real terms The value of an asset taking into

Slope function See derivative account the impact of inflation

Standard deviation A measure of dispersion, which Variance The square of the standard deviation

can be thought of as the mean distance of each pointVertices of a graph Points where the graph reaches a
maximum or minimum point and changes direction
from the mean

Zeros of a function Values of x for which f(x) = 0

Acknowledgements

The Publishers wish to thank Pedro Monsalve Correa for his valuable comments and positive review of the
manuscript.

The Publishers would like to thank the following for permission to reproduce copyright material.

Photo credits

p.xiii © https://en.wikipedia.org/wiki/Principia_Mathematica#/media/File:Principia_Mathematica_54-43.png/
https://creativecommons.org/licenses/by-sa/3.0/; p.xxiv © R MACKAY/stock.adobe.com; pp.2-3 left to right ©
3dmentat - Fotolia; © GOLFCPHOTO/stock.adobe.com; © Janis Smits/stock.adobe.com; pp.22-23 © Reineg/
stock.adobe.com; p.26 © The Granger Collection/Alamy Stock Photo; p.30 left to right © glifeisgood - stock.
adobe.com; The Metropolitan Museum of Art, New York/H. O. Havemeyer Collection, Bequest of Mrs. H. O.
Havemeyer, 1929/https://creativecommons.org/publicdomain/zero/1.0/; © Iarygin Andrii - stock.adobe.com; ©
Nino Marcutti / Alamy Stock Photo; pp.48-9 left to right © Julija Sapic - Fotolia; © Getty Images/iStockphoto/
Thinkstock; © Marilyn barbone/stock.adobe.com; © SolisImages/stock.adobe.com; pp.74-5 left to right ©
Evenfh/stock.adobe.com; © Andrey Bandurenko/stock.adobe.com; © Arochau/stock.adobe.com; © Tony
Baggett/stock.adobe.com; © Peter Hermes Furian/stock.adobe.com; p.93 left to right © Shawn Hempel/stock.
adobe.com; © Innovated Captures/stock.adobe.com; p.133 © Bettmann/Getty Images; pp.178-9 left to right
© Lrafael/stock.adobe.com; © Comugnero Silvana/stock.adobe.com; © Talaj/stock.adobe.com © tawhy/123RF ;
pp.200-201 left to right © Daniel Deme/WENN.com/Alamy Stock Photo; © Maxisport/stock.adobe.com; ©
The Asahi Shimbun/Getty Images; pp.222-3 left to right © Shutterstock/Pete Niesen; © Luckybusiness - stock.
adobe.com; © Shutterstock/Muellek Josef; p.225 © Caifas/stock.adobe.com; p.230 © Dmitry Nikolaev/stock.
adobe.com; pp.256-7 left to right © Alena Ozerova/stock.adobe.com © Alena Ozerova/123RF; © Vesnafoto/
stock.adobe.com; © Drobot Dean/stock.adobe.com; pp.278-9 left to right © Africa Studio/stock.adobe.com; ©
Norman/stock.adobe.com; © Hoda Bogdan/stock.adobe.com; © Artem Merzlenko/stock.adobe.com; pp.286-7
left to right © Ollirg - Fotolia; © Cristian/stock.adobe.com; © Christian/stock.adobe.com; © Analysis121980/
stock.adobe.com; pp.314-15 left to right © AntonioDiaz/stock.adobe.com; © AntonioDiaz/stock.adobe.com; ©
AntonioDiaz/stock.adobe.com; pp.326-7 left to right © Pink Badger/stock.adobe.com; © Vchalup/stock.adobe.
com; © Louizaphoto/stock.adobe.com; © Preserved Light Photography/Image Source/Getty Images; © Wajan/
stock.adobe.com; pp.354-5 left to right © https://en.wikipedia.org/wiki/File:Wallpaper_group-p2-3.jpg/https://
creativecommons.org/publicdomain/mark/1.0/deed.en; https://en.wikipedia.org/wiki/File:Wallpaper_group-
pmm-4.jpg/https://creativecommons.org/publicdomain/mark/1.0/deed.en; https://en.wikipedia.org/wiki/
File:Wallpaper_group-p4m-1.jpg/https://creativecommons.org/licenses/by-sa/3.0/; https://en.wikipedia.org/wiki/
File:Alhambra-p3-closeup.jpg/https://creativecommons.org/licenses/by-sa/3.0/deed.en; pp.382-3 left to right ©
Granger Historical Picture Archive/Alamy Stock Photo; © Nahhan/stock.adobe.com; © Peter Hermes Furian/
stock.adobe.com; pp.394-5 left to right © Vitaliy Stepanenko - Thinkstock.com; © L.R. Greening/Shutterstock.
com; © Ingram Publishing Limited / Ultimate Lifestyle 06; © Howard Taylor/Alamy Stock Photo; pp.440-1 left
to right © Eyetronic/stock.adobe.com; © Seafarer81/stock.adobe.com; © Gorodenkoff/stock.adobe.com; pp.458-9
left to right © Ipopba/stock.adobe.com; © Alekss/stock.adobe.com; © Inti St. Clair/stock.adobe.com; © Vegefox.
com/stock.adobe.com; pp.490-1 left to right © Andrey ZH/stock.adobe.com; © Maximusdn/stock.adobe.com; ©
Phonlamaiphoto/stock.adobe.com; © Alexey Kuznetsov/stock.adobe.com

Text Credits

All IB Past Paper questions are used with permission of the IB © International Baccalaureate Organization

p.441 Data from https://commons.wikimedia.org/wiki/File:Population_curve.svg [accessed 16.7.2019]; p.442
Data from GISTEMP Team, 2019: GISS Surface Temperature Analysis (GISTEMP), version 4. NASA Goddard
Institute for Space Studies. Dataset accessed 2019-07-16 at https://data.giss.nasa.gov/gistemp/; p.442 Data from:
https://commons.wikimedia.org/wiki/File:FTSE_100_Index.png [accessed 16.7.2019]

Every effort has been made to trace all copyright holders, but if any have been inadvertently overlooked, the
Publishers will be pleased to make the necessary arrangements at the first opportunity.

Index

3D shapes see three-dimensional common difference 25–7 in kinematics 507–9
shapes common ratio 34–5 second derivative 470–4
completing the square 332, 614 see also integrals
abstract reasoning xv composite functions 316–17, 464–5 differential geometry 473
accumulations 256–7, 490–1 composite transformations 361–3 differentiation 222–3, 459, 614
algebra 2 compound interest 39, 390, 614 anti-differentiation 258–60
computer algebra systems (CAS) xx chain rule 464–5, 467
computer algebra systems (CAS) concave-down 471–4, 614 of composite functions 464–5
xx concave-up 471–3, 614 concave-down 471–4
conditional probability 187–8, 446–7 concave-up 471–3
and confidence xxiv cone, surface area and volume 94–5 derivatives 225, 230–3, 240–3,
exponential equations 7 confidence intervals 444
and generalization 283 conjecture x–xi 245–50, 460–79
in proofs 281–2 continuous data 132 gradient of a curve 225–7, 231–3,
amplitude 224, 419–22, 614 contrapositive statements xvi
angles convenience sampling 134–5 245–50, 470–4
of depression 119 coordinate geometry graphical representations 470–4
of elevation 119–20 limits 224–5
see also triangles; trigonometry Cartesian coordinates 78 local minimum and maximum
approximation 40, 131, 136, 164, 264 midpoint 87
arc 396, 398–400 Pythagoras’ theorem 86–7 points 476–9
arithmetic sequences 24–9, 614 three-dimensional 86–7 optimization 478–9
applications of 28–9 two-dimensional 74–82 points of inflection 481–3
common difference 25–7 correlation 159–61 product rule 466–7
sigma notation 27 cosine (cosθ) 101, 405–6, 413–22 quotient rule 467–8
sum of integers 26–8 cosine rule 105–6, 400 rate of change 227, 257
arithmetic series 25, 614 cumulative frequency graphs 151–2 second derivative 470–4
Aryabhatiya 406 cycle of mathematical inquiry x trigonomic functions 461
asymptotes 62, 367, 370–1, 614 see also integration
average see central tendency data discrete data 132, 142–3, 202–3, 614
axiom xii bivariate 160, 442 discrete random variable 202–3, 614
axis intercepts 62, 77–80, 614 continuous 132 discriminant 346–8, 614
correlation 159–61 disguised quadratics 385
base 4, 614 discrete 132, 142–3, 202–3, 614 dispersion 143–4
bearings 118–19 effect of constant changes 145 displacement 508–9
bias 132–3, 614 extrapolation 443 dynamic geometry packages xxi
binomial frequency distributions 139, 141
grouped 141–2 elevation 119–20
coefficients 306–10, 614 measure of dispersion 143–4 enlargement symmetry 63
distribution 207–10 measures of central tendency equations 280, 614
expansions 306–10 140–2
theorem 300, 306 modal class 142 applications of 389–90
bivariate data 160, 442 outliers 133–4, 144 exponential 7, 17, 287, 294–5, 390, 614
box-and-whisker diagrams 153 patterns in 164 factorizing 384–6
Brahmagupta 343 presenting 150–3, 159–66 leading to quadratics 386, 431–2
quartiles 142–3 normal to a curve 248–9
calculus 458 regression 162–4 roots of 67–8
see also differentiation; reliability of 132–3, 444 solving analytically 383–6
integration summarizing 139–40 solving using graphs 67–8, 383,
validity 133, 144
Cartesian coordinates 78 see also statistics 387–8
catenary 333 of straight lines 76–82
cell sequences 22–3 decimalization 396 of tangents 246–7, 249–50, 463
central tendency 140–2 definite integration 262–4, 614 trigonomic 426–30
chain rule 464–5, 467, 614 depreciation 40, 614 see also quadratics
coefficients derivatives 225, 460–79, 490, 614 equivalence 292, 386, 400
event 180, 614
binomial 306–10 exponential equations 7, 17, 287,
quadratics 332–3 294–5, 390, 614

618 Index

exponential functions 370–3, 461–2 Godfrey, Charles 67 infinity, reasoning with xiv
exponents 4, 614 gradient function see derivatives inflation 40, 614
gradient of a curve 225–7, 231–3, integrals 490–502
applied to fractions 5–6
laws of 4–8, 20, 287–9, 294 245–50, 614 see also derivatives
negative 5 gradient of a straight line 76–80 integration 490–502, 614
rational 288 Graham’s Number 11
simplifying expressions 4–7 graphical interpretation, of accumulations 256–7
extrapolation 443 anti-differentiation 258–60, 491
derivatives 230–3 area between curves 501–2
factorial 308, 614 graphical representation definite integrals 262–4
factorial function 308–9 indefinite 492–4, 614
factorizing 331–3, 342, 384–6 and data 150–4 by inspection 494–5
financial calculations functions 50–1, 53, 55–6 limits of 614
solving equations 68 see also differentiation
compound interest 39 graphs 61–8, 355 intercepts 62, 614
depreciation 40 asymptotes 62 see also x-intercepts; y-intercepts
inflation 40 box-and-whisker diagrams 153 interest
simple interest 28–9 composite transformations 361–3 compound 39, 390, 614
forces 257 cumulative frequency 151–2 simple 28–9, 615
fractal 63 derivatives 470–4 interquartile range 143, 614
fractions, with exponents 5–6 equations of straight lines 76–82 inverse xvi
frequency distributions 139, 141 exponential 370–3 inverse functions 54–5, 319–20
function notation 50 finding points of intersection 66–7 irrational numbers 15
functions 48 gradient of a curve 225–7, 231–3,
commutative 316 kinematics 507–9
composite 316–17, 464–5 245–50 Koch snowflake 301
domain 52 gradient of a straight line 76–80
exponential 370–3, 461–2 histograms 150–1 Lagrange’s notation 226
graphical representation 50–1, hyperbola 366–8 large numbers 11
intercepts 62, 77–80 laws of exponents 4
53, 55–6 intersection of two lines 82 Leibniz, Gottfried 225, 497
identity 319 parabola 327–8, 333–5 limit of a function 497, 614
inputs 49–52 parallel lines 80 limits 224–5
inverse 54–5, 319–20 perpendicular lines 80–2 limits of integration 262, 614
and mathematical modelling 53–4 quadratic 328–34 linear regression 442–4
one-to-one 320 rational functions 366–8 lines of symmetry 63
outputs 49, 52–3 reciprocal functions 366–7 local maximum points 476–9
periodic 419 reflections 356, 360–1, 615 local minimum points 476–9
range of 52 scatter diagrams 159–61 logarithms 14, 20, 52, 290–1
rational 366–8 simultaneous equations 82
reciprocal 366–7 sketching 65–7 change of base 293–4
self-inverse 367 solving equations using 67–8, 383, evaluating 16
trigonomic 405–10, 418–22, 461 laws of 287, 291–5
zeros of 67 387–8 natural 15–16, 461–2
see also quadratics stretches 356, 358–9, 615 see also exponential equations
symmetries 63 logic xv–xvi
Gauss, Carl Friedrich 26 of a tangent 246–7
generalizations 22, 24, 283, 307 tangents to a curve 225 mathematical
geometric sequences 33–6, 302–4, 614 transforming 356–63 induction 24
translations 356 inquiry x
applications of 36, 39–40 vertices 62, 615
binomial expansions 306–10 mean 140–2
common ratio 34–5 histograms 150–1 median 140
financial applications of 39–40 horizontal translations 356–8 modal class 142
infinite convergent 302–4 hyperbola 366–8, 614 mode 140, 142
geometric series 34–5, 614 hypotenuse 101 modelling xvii–xix, 22
geometry 92–3
origins of 118 identities 280, 413–17, 614 arithmetic sequences 28–9
see also coordinate geometry; identity function 319, 614 assumptions xvii
implication xvi geometric sequences 36
geometric sequences; indefinite integration 492–4, 614 mathematical model of a process
trigonometry indices see exponents
53–4

Index 619

patterns 23–4 concepts in 179–82 representations 12, 203, 334
qualitative and quantitative conditional 187–8, 446–7 right-angled triangles 101, 113
discrete random variable 202–3
results 230 distributions 201–15 see also trigonometry
straight-line models 75 events 180 roots 62
independent events 188–9 roots of an equation 67–8, 342, 615
Napier, John 14 mutually exclusive events 187 rotational symmetry 63
natural logarithm 15–16, 461–2 normal distribution 212–15 Russell, Bertrand xii–xiii, 281
negation xvi outcomes 180
negative exponents 5 relative frequency 180 sample space diagrams 185
negative gradient 76 sample space 180–1 sampling 132–3, 615
negative numbers, and the domain techniques 184–5
trials 180 bias in 132–3
of a function 52 problem solving ix–xii techniques 134
normal distribution 212–15, 449–51 product rule 466–7, 614 scatter diagrams 159–61
normals 248–50, 463 programming xxi self-inverse functions 367, 615
normal to a curve 248, 614 proof xii–xvi, 279 sequences 22–3, 301
notation command terms 282 binomial expansions 306–10
deductive 281–3 standard notation 24
equations 280 structures of 280–3 see also arithmetic sequences;
function 50 using coordinate geometry 86–7
identities 280 pyramid geometric sequences
Lagrange’s 226 finding angles 115–16 sets xv
number sets xv surface area 96 shapes, three-dimensional solids
sigma 27, 35–6 volume 96–7
standard 24 Pythagoras’ theorem xiii, 86–7, 96, 116 94–5
n terms Pythagorean identity 413–14 sigma notation
arithmetic sequences 24–5
geometric sequences 33–5 quadratics 326 arithmetic sequences 27
completing the square 332–3, 342 geometric sequences 35–6
one-to-one functions 320 discriminant 346–8, 614 significant figures 12
operations, large numbers 11 disguised 385 simple interest 28–9, 615
optical illusions 382 equations leading to 386 simplifying exponent expressions 4–7
optimization 478–9 factorizing 331–3, 342 simultaneous equations 82
outcomes 180, 614 graphs of quadratic functions sine (sinθ) 101, 405–6, 413–22
outliers 133–4, 144, 614 328–34 sine rule 103–4, 410
inequalities 344 slope function see derivatives
parabola 327–8, 333–5, 614 parabola 327–8, 333–5 space 508
paradox xv quadratic equations 342–4, 431–2 spatial frame of reference 382, 389
parallel lines 80 quadratic formula 343, 346–7 spreadsheets xx
Pascal’s triangle 308 in the real world 326–7 square root 14, 346
patterns 23–4, 30, 291, 354–5 see also logarithms; quadratics
quartiles 615 standard deviation 143, 449–51, 615
see also symmetries quota sampling 135 standard index form 11–12, 615
Pearson’s product-moment quotient rule 467–8, 615 standard notation, sequences 24
stationary point 240, 477, 481–2, 615
correlation coefficient (r) 161–2 radians 396–400, 615 statistics 130
period 419, 614 Ramanujan xiv approximation 136
periodic functions 419 range 143, 615 confidence intervals 444
perpendicular lines 80–2 rate of change 227, 257 linear regression 442–4
perspective 119 rational exponents 288–9 misleading 130, 133
point-gradient form 79 rational functions 366–8 normal distribution 212–15, 449–51
points of inflection 481–3, 614 reasoning xv outliers 133–4, 144
Polya, George ix–x sampling 132–4
population 132, 614 see also proof standard deviation 143, 449–51
reciprocal functions 366–7 validity 133, 144
see also data; statistics reflection 356, 360–1, 615 see also data
positive gradient 76 relative frequency 180, 615 straight-line models 75
powers see exponents reliability, of data 132–3 stratified sampling 135–6
probability stretches 356, 358–9, 615
subtended 398, 615
binomial distribution 207–10 sum to infinity 302, 615
combined events 186–7
complementary events 181

620 Index

surface area, three-dimensional in the real world 92–3 validity 133, 144, 443
solids 94–5 right-angled 101, 113 value in real terms 40, 615
see also coordinate geometry variables, modelling xvii–xviii
syllogism xvi trigonometry 92, 101–8, 395, 405–10 variance 143, 615
symmetries 63, 354–5 applications of 113–20 velocity 257
area of a triangle 107–8 Venn diagrams 184, 186, 188
see also patterns bearings 118–19
systematic sampling 135 cosine (cosθ) 101, 405–6 and sets xv
cosine rule 105–6, 400 vertical translations 356–7
tangent degrees 409 vertices 62, 615
of an angle 101 finding angles 101–2, 113–18 volume, three-dimensional solids
to a curve 225, 615 identities 413–17
equation of 246–7, 249–50, 463 measuring angles 118–19 94–5
function 407–8 radians 396–400, 409
ratios 408, 415 Watson selection test xv
tangent (tanθ) 101, 407–8, 418–22 real-life contexts 422
tan rule 106 sine (sinθ) 101, 405–6 x-axis 62
tetration 11 sine rule 103–4, 410 x-intercepts 77–9, 331–2, 346–7
three-dimensional shapes tangent (tanθ) 407–8
tan rule 106 y-axis 62
finding angles 113–18 three-dimensional shapes 113–14 y-intercepts 62, 77–9, 328–30
surface area 94–5 trigonomic equations 426–30
volume 94–5 trigonomic functions 405–10, Zeno’s paradoxes 227
trajectory 75 zero xv, 4, 52, 224
translational symmetry 63 418–22, 461 zeros of a function 67, 615
translations 356, 615 Zipf’s Law 53
tree diagrams 184–5, 202 U-shaped curve see parabola z-value 449
trial 180, 615
triangles
area of 107–8

www.hoddereducation.com/dynamiclearning

[email protected]
hoddereducation.com