IBHM 673 720 .pdf

Answers
Chapter 1

Exercise 1

1 a 135° b 20° c 72°
p
7p
3p
2 a
b
c
6
6
4
4 a 30.7cm2

b 4.71m2

˛

6 a 66.8cm
11 9.30cm

d 150° e 105° f 22.5° g 110° h 114.6° i 85.9° j 229.2° k 206.3° l 22.9°
7p
4p
7p
2p
3p
d
e
f
g
h
3 a 0.611 b 1.75 c 5.24 d 1.40 e 2.30
4
3
18
5
10
c 489cm2

˛

b 293cm
12 30cm

c 177mm

1
2

b 0.174

2 a

1
2

b

1
2

c

c 1

p 5p
,
3 3

1
22
23
2

d

˛

5 311m

2

˛

e

1
2

22

e

p 2p
,
3 3

b u

1

f

c u

2

9 46.9m

˛

b 99.1cm

c 295cm
2

˛

d 113mm
10 200°

2

˛

1

f

22

1

g

22

g

h 0.737

i 1

h 0

c x° 41.8°, 138.2°

p 3p
,
4 4

23
2

j 0

d x° 80.4°, 279.6°
p 11p
,
6 6

e u

d No Solution

e x° 22.0°, 158.0°

f u 0.290, 2.85

f x° 55.2°, 304.8°

g u 1.20, 5.08

h u 0.775, 2.37

Exercise 3

b 20.7cm2
˛

c 160cm2
˛

6 a x 5.82cm

10 a x 6.07cm

b x 22.7m

14 A 39.4°, B 46.5°, C 94.1°
17 21.6m

18 57.9°

Chapter 1
b 120°

d 26.0m2

2 8220m2

˛

b x 6.37cm

b x° 127.7°

1 a 180°

23
2

d 0.996

7 a x° 32.4°

16 312km

b 8.55m

8 52.2cm

˛

15 a 17.9cm

b x° 70.5°, 289.5°

Chapter 1
1 a 24cm2

14 7:2

b 2670cm

2

Exercise 2

3 a x° 30°, 150°
4 a u

5 a 6.28cm

˛

7 a 126cm

13 99.5cm

Chapter 1
1 a

d 2430cm2

˛

f 4.85

c x 7.64m

8 44.7° or 135.3°
c p 381mm

3 a 84.9cm2

˛

b 120cm2

˛

d n 7.35m

4 63.7m3

˛

˛

e a 14.2mm

9 10.75° or 169.25°
d t 10.7cm

15 a x 7.64m

11 2.63m

b x° 50.9°

12 a 59.6°

c x° 83.8°

b 143.2°

d x 15.8cm

13 68.7°
e x 16.9m

19 204 cm2
˛

Exercise 4
c

p
2

d 90°

e 360°

2 ay

f 120°
b

g 180°

h

p
2

i 36°

j 3°
c

y

d

y

y
1

1
360
0

120

240

x

0

1

x

f

300 x

360

4
0
4

180

360 x

y
3
1

120

x
0
1

1

2 e y

0

1
360

360
x

673

Answers
3 a

b

y

c

y

1

y
2
0



0

0

2



2



2

1
6

3 d y

e

y
8

1

3
2

0
1

4 a



0
2

b

y

2

c

y

y

1
2

3

x
0

180
x

0

180

x
0

2

1

4 d

3

e

f
y

y

y

6

2
180
x

0

0
60

x

180

150 180 x

6

4

5 a

b
y

y
0

8

2

0
4

674

180





5
8

c

y



11

3
0





Answers
5 d y

e



0

6

y



0

7

y

8

y

y

6

11
8

1
0

720

3

9 a y 4 cos 2u

b y sin x° 1

9 g y –cot 115x° – 45°2

1 a

1
2

23

3 a sin 43°

c

22

b cos 50°

4 a x° 30°, 150°
5 a u

1
2

c

1

b

p 2p
,
3 3

d

1
2

23
2

f

d 23

p 7p
,
6 6

23
2

j

y 2 csc 3x°

g 23
1

e

c u

f

22

d sin 50°

c tan 20°

Chapter 1
p 11p
,
6 6

e 1

b x° 30°, 330°

b u

1 a 60°, 240°

d y 4 cos x° 2
i



7
7
f y cos 4x°
2
2

e y 4 sin 2u 4

3
y 3 sec u
2

1

e cos 23°

d u

23
2

g 0

23

c x° 60°, 240°
p 5p
,
3 3

h

23
2

i

h

j

23
2

f tan 34°

i 1

23
2
j 4 23

g cos 15°

h sin 20°

i tan 46°

d x° 135°, 315°

5p 7p
,
6 6

Exercise 6

b 60°, 300°
7p 11p
,
6
6

b

3 a 15°, 75°, 195°, 255°
3 d 60°, 120°, 240°, 300°

c 60°, 120°
c

5p 11p
,
6
6

d 210°, 330°
d

p 5p
,
4 4

e

e 150°, 210°
p 5p
,
6 6

b 10°, 110°, 130°, 230°, 250°, 350°

p 5p 7p 11p 13p 17p 19p 23p
,
,
,
,
,
,
,
12 12 12 12 12 12 12 12

4 d

5p 7p 17p 19p
,
,
,
12 12 12 12

5 15°, 105°

6

b

7p 11p 19p 23p
,
,
,
24 24 24 24

b 41.4°, 318.6°

c 58.0°, 238°

10 f 31.3°, 288.7°

g 75.5°, 284.5°

h 45°, 135°

10 k 20.9°, 69.1°, 200.9°, 249.1°

p 7p
,
6 6

g

c

p 5p
,
2 6

h 30°, 150°
h

i 45°, 225°

p 11p
,
6 6

p 5p 13p 17p 25p 29p 37p 41p 49p 53p
,
,
,
,
,
,
,
,
,
30 30 30 30 30 30 30 30 30 30

7 1°, 5°, 13°, 17°

d 48.2°, 311.8°
i 228.7°, 341.3°

8 120°, 60°

9

8p 4p 2p 2p 4p 8p
, , ,
,
,
9
9
9
9 9 9

e 48.6°, 131.4°
j 36.9°, 143.1°

l 25.2°, 94.8°, 145.2°, 214.8°, 265.2°, 334.8°

10 m 24.9°, 60.9°, 96.9°, 132.9°, 168.9°, 204.9°, 240.9°, 276.9°, 312.9°, 348.9°,
10 o 8.2°, 171.8°

p 48.2°, 311.8°

11 a 0.253, 2.89

b 2.03, 4.25

11 i 1.66, 3.06, 4.80, 6.20

g 90°

f 40°, 80°, 160°, 200°, 280°, 320°

p 2p 7p 5p
,
,
,
6 3 6 3

10 a 19.5°, 160.5°

f

f 180°

c 11.25°, 56.25°, 101.25°, 146.25°, 191.25°, 236.25°, 281.25°, 326.25°

e 25°, 45°, 145°, 165°, 265°, 285°

4 a

11 n 0.14, 3.28


2

0

Exercise 5

b 1

1

c y tan 2u

y 3 sin 24u 8

h

Chapter 1

2 a

5

x

6

7

2 a

12

q 33.7°, 213.7°

c 1.17, 4.31

n 70.5°, 289.5°

r 15°, 75°, 105°, 165°, 195°, 255°, 285°, 345°

d 1.11, 5.18

e 2.01, 0.08

j 0.47, 1.26, 2.04, 2.83, 3.61, 4.40, 5.18, 5.97

o 0.11, 0.67, 1.68.2.25, 3.25, 3.82, 4.82, 5.39

f 3.34, 6.08

s 1.4°, 4.6°, 13.4°, 16.6°, p

g 1.23, 4.37

h 0.41, 1.68, 2.50, 3.78, 4.60, 5.87

k 0.28, 0.76, 2.38, 2.86, 4.47, 4.95

12 70.5°, 289.5°, 430.5°, 649.5°

13

l 1.14, 5.15

m 3.45, 5.98

35p 19p 11p 5p 13p 29p
,
,
,
,
,
36
36
36
36 36 36

675

Answers
14 a 3 minutes

b H(t)
39

21

3
0

14 c i 2 mins 15 secs

t

ii 45 secs

Chapter 1
1 a 30°

b 75°

6 a 15.1cm
8 a

1
2

b

15 a 7500

b 7060

c 12 years, 4500 fish

Review Exercise
2p
13p
b
3 7.01cm2 4 14.0m 5 20.6cm2
3
12
c 48.9° d 8.63 mm 7 2.35 6 BC 6 5

2 a

˛

b 44.8°
1
22

c

1
23

d

1
2

e

˛

1

f

22

9 a

23
2

g 23 h

1
22

b y

i

1

j 2

23

k

2
23

c y

4

5

11
2

0

0

360 x
5

4

2

7

9 d

y

e y

2

5

2

0



1

5

b y

10 a y 3 cos 2u 1
11 d u

5
5
cos1x 202°
2
2

p p 7p 2p 13p 7p 19p 5p
, ,
,
,
,
,
,
12 6 12 3 12 6 12 3

12 d x° 60°, 240°

16 a 14.5cm

17 a

4
4
4, 35
, 1 5
12
5

676

c u

12 a x° 135°, 315°

p 5p
,
4 4
b x° 90°

c 20.7°, 80.7°, 140.7°, 200.7°, 260.7°, 320.7°

c x = 1.57

y
4, 3
5

4, 3
5
4x

5p 7p
,
6 6

b 0.169s
b

16, 0
5

b u

c x° 210°, 330°

f x° 10°, 50°, 130°, 170°, 250°, 290°

b x° 109.3°, 160.7°, 289.3°, 340.7°

15 20.9°, 69.1°

p 5p
,
6 6

5p 11p 17p 23p
,
,
,
12 12 12 12

e x° 15°, 165°, 195°, 345°

13 a x° 64.6°, 295.4°
p p
14 , 0
3 3

e u

11 a u

16
5
4

4x

0

4, 3
5
12
5 , 1

4
5

d 64.6°, 295.4°

Answers

Chapter 2

Exercise 1
1
c x , x 8
2

1 a x 4, x 1

b x 3, x –2

2 a x 2, x 5

b x 3, x 8

3 a x 4.73, x 1.27

c x 4, x

b x 0.854, x 5.85

4 a x 5.45, x 0.551

3
2

3
2
d x ,x
2
3

2
e x ,x 3
3

c x 0.314, x 3.19

b x 1.61, x 5.61

Chapter 2

d x 1, x 3

d x 0.260, x 1.54

c x 1.77, 0.566

d x 3.91, x 1.41

Exercise 2

Exercise 2 2:

1 a 1x 12 4

3 2 3
b ¢x ≤
2
4

2 a 1x 32 2 5

Minimum 1 3, 52

2 b 1x 2 2 2 1

Minimum 12, 1 2

2

3 2 11
c ¢x ≤
2
4

e 5¢x

x intercepts: 3,1

5 17
Minimum ¢ , ≤
2
4

2 d 1x 22 2 7

Maximum 1 2, 72

y intercept: 3

x intercepts: 4.65, 0.646

2 e 1x 4 2 2 19

Maximum 14, 19 2

y intercept: 3

x intercepts: 0.359, 8.36

5 2 47
2 f 2¢x ≤
2
2

5 47
Minimum ¢ , ≤
2
2

7
3 2
2 g 4¢x ≤
8
16

3 7
Minimum ¢ , ≤
8 16

5 2 1
2 h 3 ¢x ≤
6
2

Minimum ¢

3 2 23
2 i 2¢x ≤
4
8

5
,
6

y intercept: 2

3 23
Minimum ¢ , ≤
4 8

3 a Minimum 1 1.25, 10.1252
3 b Maximum 12.5, 0.75 2
3 d Maximum (0.833,11.1)

Chapter 2

x intercepts: 3.5, 1
c Minimum 1 0.6, 14.22 y intercept: 16 x intercepts: none

x intercepts: none

b

c

26 x 6 1

Chapter 2

3 q

0, 5
2

Line not
included

8, 0
3

1
5

d x 7 3, x 6 5

j 3 x 2

1
k 8 x
2

3 a x 6 0.764, x 7 5.24

;28

3x 4y 5 0

(5, 0)

0, 5
4

Line not
included
x 2y 5

Line not
included

5, 0
3

y 3x 8

b x 6 2

1 a Imaginary roots

d

(0, 8)

Line not
included

5, 5
2
(0, 5)

2 p

x intercepts: 0.851, 2.35

x intercepts: 1.09, 2.76

y 2x 5

1
4

2
3

Exercise 3

1 a

2 i

x intercepts: 1,

y intercept: 4

y intercept: 7

x intercepts: 0.928, 5.93

x intercepts: none

y intercept: 2

y intercept: 7

y intercept: 9

x intercepts: 4.56,0.438

y intercept: 11

y intercept: 1

1
≤
12

7 2 349
≤
10
100

x intercepts: 0.764, 5.24

y intercept: 4
y intercept: 3

d 31x 1 2 2 11

5
17
2 c ¢x ≤
2
4

2 a x 7

e x 2.14, x 0.468

e x 1.78, x 0.281

c x

b x 6 0.854, x 7 5.85

1
3
f x
2
4
4
1
l 6 x 6
3
2

e 5 6 x 6 6

c x 6 0.157.x 7 1.59

g 3 x

7
2

d 2.14 x 0.468

h x 6 4, x 7 3

e 0.207 x 1.21

Exercise 4
b Real distinct roots
9
8

c Imaginary roots

5 a b 8ab3 16a2b 0
˛

˛

d Real distinct roots

7 p

1
2

9 m

32
9

e Real equal roots
11 a (ii)

b (iii)

f Imaginary roots
c (i)

d (iii)

677

Answers

Chapter 2

Review Exer cise

1
3 k 4.08, k 2.58 4 i. x , 2, y
3
6 m ;2 8 x 16, x 1 9 k 0, k 8
10
9
7
a , b . Maximum point is
4
2
c2 3
11 6 years old 12 cos C
. True for all
4c
real values of c.
1 4 1x 22 2

2 1 6 x 6 3

9
9
ii x , 6, y , 3
2
2

7
,0
3

5 2.46 6 k 6 0.458

7 9
7
¢ , ≤. Line of symmetry is x
2 4
2

˛

Chapter 3
1 a 11

b 1

Exercise 1

c 130

2 a 17

d 6

b 12

c 27

d

19
4

1
4 b 2, , 2, 4r
2

3 再 7, 1, 17冎

5 5y : 5 y 26

g(x)

f(x)
1

7

1

1

5

17

3
1

2
4

1
6

2

6 5y : 5 y 156

y

y

2

15
5

0

1
2

7 x

3

0

2

x

5

7 by :

25
y 24r
4

8 5y : 0.959 y 2216

y
24

y
96

9 5y : 5 y 6 q 6
10 a Domain 5x : 3 x 26, Range 5y : 9 y 116
10 b Domain 5x : 1 x 76, Range 5y : 2 y 186

0
6

6x

12

1 0

5x

10 c Domain 5x : 2 x 16, Range 5y : 0 y 46
12 c 36x2 18x

d 4x2 10x 4

Chapter 3
1 a 14

b 24

d 4x 24

ii 12x 12 2

˛

7 c x p
8 d i

678

2x
3 x

ii

ii 2x3 2x 17
˛

b 6x 8 p

3x 1
2

b 5

˛

˛

d cos1cos1x 2 2

2 a 61

2x
1 2x

b

1
2x 1

c

8 a i
e i

c p 2

˛

ii 3x2 7

3
2
x

c

x 2
x

d

f i 1x2

˛

6 a 12x2 3

42

3

ii

˛

ii

12 a 4x2 6x
˛

b x2 5x 4
˛

14 x

c

23
2
˛

˛

c 216x4
˛

c i

d sin ¢u

d i cos13x2 2

ii 1x6 2x 3 2 2 4

3
x2 3x
˛

23
2

˛

b 612x 3 2 2

˛

b

ii x6 6

˛

4 2 6 – 21x2

˛

p 3
3

3 a

c i 1x2 6 2 3

˛

x 3
9 9x
b i
x 3
x2
2x 4
ii
9 x
x

2
3x 2

1
x 1

d 3x2 7

c 46

b i 9x2 30x 21

4 e i 8x3 36x2 52x 31
5 a 6x 3p 4

x
x 2

b 3x 2

Exercise 2

c 2

4 a i 2x2 1
˛

13 a

˛

11 a 6x 2

d 4x 9

2 3x
x 1

ii

p
≤
3

ii 3 cos2 x

g i sin12x2 14 2
˛

7 a cos ¢x

2 5x
3 5x

p
≤
2

ii sin2 2x 7
b cos x

p
2

Answers

Chapter 3
1 a yes

b no

2 f f 1 1x2
˛

Exercise 3

c yes

9 x
4

4 c i x


2
3

˛

g f 1 1x2
˛

3 d x –5, x H ⺢
˛

2x 5
3x
x
B6

4 g i x 7 0

ii f 1 1x2

4 k i x 0

ii f 1 1x2 x4

˛

b f 1 1x2 x 5

c f 1 1x2 x 6

˛

d f 1 1x2

˛

˛

h f 1 1x2 1x 62 3
1
f x ,x H ⺢
2

h i x 7 0

˛

˛

˛

g 0 x p

ii f 1 1x2

d i x
2

x3
2

i f 1 1x2

1

˛

2x 7
x

ii f 1 1x2 2x 4

3 a x
3, x H ⺢

l i ⺢

˛

Chapter 3
1 a f 1 1x2
˛

x
2

˛

e i x


ii f 1 1x2 ¢
˛

x 5 3
≤ 5 a
2

1
c x
,x H ⺢
2

6x 1
3 7x
b i x
7 ii f 1 1x2
x
x
9x 8
6
4 6x
1
1
ii f 1x2
f i x

ii f 1x2
4x
5
5x

9
4

i i x 0

˛

b x
4, x H ⺢

˛

˛

˛

˛

ii f 1 1x2
˛

B

x 3
2

b h 1 1x2

h 1x2 3 x 6

ii f 1 1x2

j i x 16, x H ⺢

˛

B

16 x
9

1x 62
3

Exercise 4
b f 1 1x2 x 2

c f 1 1x2 x 3

˛

y

e f 1 1x2 7 x

ii f 1 1x2

4 a i x
6

1

1

˛

3
x
2

1

x 9
2

e x 9, x H ⺢

ii f 1 1x2

x
4

2 a f 1 1x2

d no

f (x) 2x

y

f 1(x) 1 x
2

2

˛

f 1(x) x 3

y

f (x) x 2

f(x) x 3

3
f 1(x) x 2

0

0

x

x

0

x

2
3

1 d f 1 1x2
˛

x 1
3
y

e f 1 1x2
˛

f (x) 3x 1

f 1(x) 12 x 2
2

0
13

f (x) 2x 4

y

f 1(x) 1 x 1
3
3
1

x 4
2

x

0

x

4
2 a f 1 1x2 1x

b f 1 1x2

˛

y

f (x) x2

˛

f 1(x) 冑x

x
B3

c f 1 1x2 2x 4

f (x) 3x2

y

d f 1 1x2 25 x

˛

y

˛

f (x) x2 4

y
5

f 1(x) 冑(x 4)

f 1(x) 13 x

0
3 a f 1 1x2
˛

x
1 2x
x

x

b f 1 1x2
˛

1 5x
x

f(x)

1
x 2
x

f 1(x) 1x 2

f 1(x) 冑5 x
0
c f 1 1x2
˛

x

0

x

2 x
x

y

y

y

0

0

f(x) 5 x2

f(x)

1
x 5
x

1
f 1(x) x 5

2
f(x) x
1
x
2
f 1(x) x 1

679

Answers
4 a

b

y

c

y

d

y

y
(3, 3)

No inverse
x

x

x

x

2

4 e y

f

y

x

1

x

Chapter 3

Exercise 5
12 x x2, 3 6 x 6 4

1
2x 1, x 6
2 x, x 6 2
2
1 f 1x2 b
2 f 1x2 μ
x 2, x 2
1
2x 1, x
2
˛

3 f 1x2 dx x 12, x 4

˛

5

6

y

0

4

9

˛

2

x

10

˛

˛

7

0

y

˛

1
2x2 5x 3, x
2

y

8

y

0

x

11

y

1
3 5x 2x2, 6 x 6 3
4 f 1x2 c
2
2x2 5x 3, x 3
˛

˛

˛

y

x

5
3

6

3

4

3 2 0

x

18 7 6 x 6 3

15 x 2, x 5

19 4 x 5

Chapter 3
1 a

13 x 4, x 6

x
2

14 x 4, x 1

x

y
12 x 5, x 1

0

2

01
3

x

3
17 x 2.91, x 2, x , x 2.41
2
1
21 6.80 x 5, 1 x 2.80 22 4.72 6 x 6 3, 6 x 6 2.22
2

16 x 3.37, x 2.56, x 1.56, x 2.37

20 2 6 x 6

5
2

Exercise 6
b

y

c

y

d

y

y
7

3
4
4 0

680

x

0 1
1 2

x

0

3

x

0

7
2

x

Answers
1 e

f

y

g

y

h

y

y

12
12

4

3 0
72, 14

1

i

x

3

0 2

x

6

24

(4, 4)

j

y

y

8 x

0

2

5 , 121
2
4

k

9
17,
20 2740

0

4

4
5

15

1 , 15 1
12
2

1 m

2 a

y

x

y

1
4

x

3
2

l

4
3
8

, 25
1
3 3

y

20
53

0

2

x

x

5
2

x

3
2

b

y

c

y

y

5

3

( 1, 4)
0

5

x

1
2

4.30

6 x

( 2.5, 3.25)

4

30

2 d

e

y

(0.5, 30.25)
f

y

g

y

y
1

7
12

2
32

2

x

3

3

x

4

56
1 , 4
2 5

2 h

x

x

0.697

i

y

1
5

5

x

x

( 7 , 28)
2

j

y

y

(5.16, 0.969)

2
( 0.172, 0.299)

2 k

0.3

5

1
4

x
14
( 0.686, 0.275)
( 4.76, 22.7)

m

y

y

n

1.6

x

5

y
1

( 9.20, 2690)
7

(10.7, 21.5)
( 0.745, 1.49)

x

1
( 1.16, 0.430) 3

l

y

6

x

1.48
2

x

0

x

1

x

681

Answers

Chapter 3
1 a

Exercise 7
b y
4

y

c

y
5

2

2

2



2

x

360

4

2 a

b

y



1

c

y

y
8

0

3 a i

x

ii

y

3 b i

iii

y

0

x

0

ii

y

x

x

2

iii

y

3 c i

ii

y

0

3 d i

iii

y

0

x

ii

y

x

2

0

iii

y

iv

y

x

1

0

iv

y

x

2

x

0

y

0
0

x

0

y

x

0
1

x

2

iv

y

y

0

x

1

x

iv

y

x

y

4
3
0
0

4

x

6

x
0

(4, 1)

x

0
8

682

4

x

Answers
3 e i

ii

y

0

x

0

iii

y

2

iv

y

y

x

x

x

0

1

3 f i

ii

y

iii

y

iv

y

y
6

0

0

x

(2, 3)

3

3 g i

0

x

ii

y

x

x

0

4

iii

y

iv

y

y

3
2
0

4

34

x

0

0

x

6

x

4

1

x
4

0

7
4

3 h i

ii

y

iii

y

iv

y

y

1
2

1

0

x

x

6

x

1

0

0

3

2

1

x

2

4 a 1 2x x2
5 a i

b 1 3x 3x2 x3
ii

y

4

0

( 1, 6)

c

4 x

3x 1

d x 1

0

( 2, 6)

2x 1
x 1
iii

y

7

e

1

x

f 4 2x x2

y

( 1, 5)

g

2x 9
x 3

h

x 3
x

(2, 23)

(7, 5)

0

x

683

Answers
5 b i

( 4, 7)

ii

y

5

x

2

(3, 17)

y

6

( 3, 6)

6 1 0

iii

(1, 7)

y

2

(5, 5)

x

3

0

x

0

( 3, 4)

4

(2, 5)

( 3, 5)

13
( 1, 16)

5 c i

ii

( 1, 9) y

iii

y

( 6, 4)

( 2, 9)

( 3, 5)

(3, 4)

5

( 4, 2)

y
( 5, 5)

(7, 5)

(1, 2)

8

0

x

4

7

( 1, 1) 0

x

5

x
10

( 3, 7)

(2, 22)

6 a

b

y

c

y

d

y

y

1
2
x

0

1
3

6 e

f

y

0

7

6 i

1
2

j

y

2

y

3
8
0

0
45

684

0

h

x
8
5

3
2

0

4

x

34

y

x

0

x
5

x

0

1
x

0

g

y

5
7

2

x

3

x

y

1
7
0

x
7

Answers
y

7 a

b

1

y
2

1

0

3

x

1

0

6

7 d

c

y

e

y

x

43

f

y

y
5
2

4
x

2 0

0

x

3

7
2

5

2

1
2

b x

10
3

c x

9 a x 7 0

b x

13
4

c x 7 0

8 a x

y

h

y

x

1
2

1

7 g

x

6

16

1
5

2
3

d x

d x 7

e x

37
8

9
2

0
32

10 a

23

x

13

b

y

9 e x 7

c

y

y f(x)

x

3
2

4
3

f x 4

7
4

h x 7

11 a

y

y g(x)

g x 7

3
2

y f(x)

y

y h(x)

3

3

( 2, 3)

( 2, 3)

(6.16, 12.3)

0
x

2

x

x
( 0.8, 1.89)

(0.8, 1.11)

3

x

( 0.8, 1.89)
( 0.16, 0.324)

11 b

c

y

12 a

y

b

y

y

(8.16, 12.3)
(8.16, 12.3)

(1.83, 0.324)
x

5

x

5

(1.83, 0.324)

x

y h(x)

y p(x)
19

y g(x)

Chapter 3
1

7
2

0
1
5

1 , 19.75
2

19
12 , 19.75

1 , 19.75
2

Review Exer cise

7 1
11
2 b , , 3, r
6 2
3
4

x

0

3 a 21x 4
7
6
1
2
3
11
3

b 14x 11

5 a x 7 0, f 1 1x2 2x 6
˛

c

7 4x
x

4 a

5x 8
x 1

b x
5, f 1 1x2
˛

1 5x
x

b

8 3x
7 3x

c 9x 16

d x

3
7 3x
c x
, f 1 1x2
2
2x
˛

685

Answers
y

6 a

f(x)

b

2
x 1

7 a

y

b

y

y | f(x)|

y

y | f(x)|

2
1

x

x

0

x

0

6

1

2

x
12

8 x 8, x 1

9

4
6 x 6 2
5

10

11

12

y

2
3

y

(3, 1)

3

0

5.61

x

13 b

13 a
y

y

2

29

c

y

1.61

y

x

0

14 a

y

x

0

x

b

y

7
2
1
x
x

0

1

x

0

x

12 0

23

0
2

14 c

15 x

y

19
15

16 x

5
12

4

x

12
3

17

y

y

y f(x)

3, 12

1
2
2

y

x
(1.15, 1.02)

5

x

5

( 1.85, 1.02)
y g(x)

18 a

1
6 y 6 2
2

21 a x 6

686

1
22

b f 1 1x2
˛

b 0 y 6 q

2x 1
x 2

19 3 x

22 x 6 1 and x 7 4

1
3

x
( 1.85, 1.02)

20 f 1 1x2
˛

y h(x)

x 1
B x 1

Answers

Chapter 4
2 47

1 14

Exercise 1

3 21

4 a 51

b 1224

6 c Q 1x2 2x 13x 70, R 427
6 f Q 1x2
˛

3

2

˛

˛

˛

7 a f 1x2 13x 12 1x 22

˛

g Q 1x2
˛

˛

˛

˛

Chapter 4

g f 1x2

x
2

¢x3

˛

˛

˛

3x3
7x2
59x
177
2797



,R
4
16
64
256
256
˛

˛

˛

˛

7x
4

˛

˛

h Q 1x2

˛

7
≤ 12x
8

˛

˛

81
8

12

Exercise 2

8 a 1x 12 2 1x 12

7 (c) (d) and (f)

˛

c f 1x2 14x2 5x 6 2 1x 3 2 35

˛

2

˛

˛

e f 1x2 12x5 2x4 3x3 3x2 3x 3 2 1x 1 2 6

˛

x
13x
3
13

≤ 12x 12
2
4
8
8

˛

˛

˛

2

7 f f 1x2 ¢

˛

2

t
t
27
99

,R
2
4
8
8

7 d f 1x2 15x4 20x3 76x2 304x 1219 2 1x 4 2 4878
˛

˛

e Q 1x2 x x 4x 7, R 0
3

2

b f 1x2 1x2 11x 47 2 1x 5 2 242

˛

b Q 1x2 x2 3x 2, R 1

˛

˛

x
x
x
7x
87
439




,R
2
4
8
16
32
32
˛

6 a Q 1x2 x 8, R 13

2

˛

4

9
4

5

d Q 1x 2 x x 5, R 11

2

˛

d 70

c 674

8 e 12x 12 1x 32 1x 42

b 1x 12 1x 22 1x 32

f 12x 12 1x 4 2 1x 6 2

c 1x 2 2 1x 1 2 1x 5 2

g 12x 32 12x 1 2 13x 4 2

d 1x 1 2 1x 1 2 1x2 1 2
˛

h 1x 3 2 1x 3 2 1x 2 2
2

˛

i 1x 32 1x2 1 2 12x2 5 2
˛

˛

8 j 16x 1 2 13x 4 2 12x 5 2 1x2 4 2
˛

Chapter 4

Exercise 3

133
3 p 24 4 k 2 5 k 10, 1x 3 2 1x 1 2 12x 1 2
16
8 k 46, 12x 12 1x 4 2 1x 6 2 9 p 11, q 21 10 k 5

1 17

6 a 2, 1x 2 2 1x 3 2 1x 3 2

2

Chapter 4

Exercise 4
1
2 x ,x 6
2

1 x 1, x 3
4 c x 11, x 3

1
3 x 2, x 5, x , x 4
2

d x 3, x 2, x 4, x 5

5 x 2, x 1, x 9
10 a 1 x 7

7 x 3

b 2 x 5

Chapter 4

8 a p 30

c 0 x 1

˛

˛

˛

4

3

2

˛

˛

˛

Chapter 4

˛

6 y x3 2x2 11x 12
˛

˛

9 y x4 2x3 13x2 14x 24

˛

˛

˛

˛

11 y 2x 6x 18x 46x 24
4

3

˛

˛

2

˛

5 3x 11, 15

7 x 5

14x 21
x2 5

8 x

˛

18x 13
10 2x3 6x 2
x 3

47
9 3x3 6x2 10x 20
x 2
˛

13 x 4, x 3, x 1, x 2

Chapter 4

˛

˛

30x 85
9

2
21x2 7 2
˛

x2
27x
67
211
x3
11



2
4
8
16
1612x 1 2
˛

˛

12 x4 x2 5x 1
˛

˛

12 5x
x2 1
˛

14 x 2, x 2, x 6

Exercise 7

1
1 x 1, x , x 3
2

1
2 x 7, x 1, x , x 4
2

7 x 0.388, x 1

10 x 0.953

5 y 3x2 12x 12

˛

Exercise 6

4 2x 1

6 x 5.50

11 6 6 t 6 10

4 y 2x2 4x 6

˛

10 y 20x 180x 420x 20x 600

˛

9 a k 83

1
x 35
2

e

2
1
3
g x ,x ,x
3
2
2
1
b x –6, x , x 7
2

1
f x 2, x , x 8
2

b x 2, x 3, x 5

d 3 x 7

b x 7, x 2, x 2

Exercise 5

˛

3 x 3

4 a x 1, x 3, x 4

e x 2, x 4

1 y x2 3x 4 2 y x2 6x 3 y 10 3x x2
1
3
7 y x3 x2 5x 12 8 y 2x3 2x2 28x 48
2
2
˛

7 p 0, q 1

1
3 x 2, x , x 3
2

8 x 4.92, x 1.02, x 2.19, x 3.74

11 f 1x2 4x 10x 28x 16, x 2, x 0.388, x 3.86
˛

3

2

˛

˛

13 13x 42 13x 4 2 12x 1 2 12x 12

14 a x 6 6.964 and –0.239 6 x 6 1.203

15 a 1.575 6 x 6 2.130 and x 6 6.556

4 x 3

5 x 4.29, x 0.428, x 2.72

9 x 10.2, x 0.203, x 1.44
12 1x 5 2 1x 1 2 1x 4 2
b 3.877 6 a 6 59.877

b 7.89 6 a 6 29.1

687

Answers

Chapter 4
1 35

Review Exercise

˛

˛

5 f 1x2 12x 32 1x 4 2 1x 12
1
9 x 4, x , x 1
2

3 a x6

b

˛

˛

d 21y5

˛

4 4
y
3

e 16x12

˛

c 10p 6

˛

4 a

˛

˛

11 x 1.47

8 x 7, x 3

˛

12 x 6.59, x 2.38, x 1.97
16 a 6

17 a 4

18 k 6

Exercise 1
c x15

˛

7 k 1x2 1x 42 1x 1 2 1x2 5 2

4
6
15 a , b
5
5

3x
x2 1

14 x2 5

4 1x 3 2, 1x 9 2

˛

˛

Chapter 5
b p5

˛

˛

10 f 1x2 21x 32 12x 1 2 1x 42 2

˛

˛

˛

˛

11x
35
67
x2



2
4
8
812x 12

1 a p9

x3
x2
17x
17
95



≤12x 12
2
4
8
16
16

3 f 1x2 ¢x4

6 g 1x2 12x 12 1x 52 1x 22 13x 2 2

˛

13

45
x 2

2 3x3 4x2 8x 22

2

˛

h 6p7

˛

˛

f 12x5 15x

˛

b

y

g 2p2

˛

e 6m 3

d t2

˛

f t2

˛

2 a 4

g 2x 1

˛

b 3

1
10

c

d 1

1
3

f

1
4

g

1
8

h

i 9

h x 2 x 1
˛

c

y

e 125

d

y

y
(1, 5)

(1, 10)

(1, 6)

(1, 3)
1

1

1

0

x

0

x

0

1

x

x

0
e

f

y

1

1, 1
4
x

Chapter 5

0

x

0

x

c y

y

0

x

1

b

y 3x

y

0

x

1

y

x

0

c 2

d 3

3 g 3

h 6

i 2

j 2

x

1
b
2

2
3

6 a 1

3
4

h

5 d 3

1
c
2

c

e 3
k 3
1
d
2

f 5
l 1

b 3

1
3

g

f
1
2

d 1

m 1

1
e
2

5 a 4

e 4

b 2

x

1

b 2

4 g
1

0

3 a 2

1
4 a
3

1
1

x

1

(5, 1)

(3, 1)

(10, 1)

(5, 1)

y log 3x

(1, 3)
1

d y

(4, 1)

(1, 5)

688

1, 2
3

Exercise 2
b

0

1

1
x

(2, 1)

2 a

y

1, 3
2

0

y

0

h

y

1, 1
6

1

0

1 a

g

y

1
2

e 1

n 0

1
f
6
c 2

Answers

Chapter 5

Exercise 3

1 a loga 18

b loga 15

2 a log3 15

5
b log2 ¢ ≤
2

8
c log2 ¢ ≤
9

2
b loga ¢ ≤
x

c loga 1x 12

4 a loga 3x3
˛

7 y 9x4

8 y px5

11 a x 2

b x 4

˛

c loga 5

c x 3

Chapter 5
b 0.903

3 a 2

b 3.17

3 n 0.209

e loga 24

d loga ¢

xy2
˛

t3

≤

f loga 4

3 a 2.699

g loga 32

b 2

c 1

d

1
2

c

h loga 6
1
2

˛

10 a x 7

˛

1 a 3

d loga 64

d loga ¢

b x 57

d x 63

1x 2 2 2
3

≤

5 a 2

c x 6

e x

26
3

b

d x 9

f x 10

e 2

3
2

e x 8

f 2

2
3

d

i loga

1
6

c 1.415
c 2.18

o 1.07

e 0.301

d 0.477

d 2.04
p 0.824

e 2.72

2 a 2.30

f 1.39

q 1.49

g 1.13

j

k

4

1
1
logx y
2
log y x

g x 25

d 0.693

i

1
2

j 1.11

e 2.01

7
2

6 y 4x3
˛

h x 9

c

Chapter 5

0

m 0.226

y
5 a x 4, x 1024

(7, 1)
x

2
3

r 0.513

(5, 1)
1

f 1.11
l

k 1.07

(9, 1)

5 c x
0

x

1

b x

1
,x
4

1
8

1
, x 117649
49

x

1

Exercise 5

b x 3.36

c x 1.86

e x 0.981

4 a x 9

b x 6

8 a 2100g

b 1650g

11 a k 0.0133

c x 4

d x 6

e x 0.862

b x 7.39

5 a 40

9 a 20100 km

f x 0.416

c x 22000

b 132

c 3.84 days

b 22.7 years

3
19 x log4 5
2

2 a x 2.48

d x 8890000
6 37.8 months

10 a 220

12 a k 0.0114

c 9.6 hours longer

18 x 1 log4 3

Chapter 5

d x 3.53

3 a x 8100

c 57.8 years

b 67.2 hours

17 x 2 log2 5

1 a

i 3

h 0

˛

c 3.43

h 1.85

y

2 d x 1.50

k loga 12

12 S1 400

b 2.20

b
y

1 a x 8

f

j loga 72

Exercise 4

4 a

0

g 1

e 2

f x

1
8

b 57

b 60.6 years

20 x log6 4

c x 5.60

e x 1.22

7 a 80° C

c 2027
4
ln
3
13
ln 6

21 x 0

b x 3.40

b 10.8 mins

ln 2
14
ln 6

2
5
15
ln 5
ln

64
3
9
ln
2

ln
16

22 x 0.631

Exercise 6
b

y

c

y

(1, 2)
1

1
x

0

x

0

d

y

1, 1
2

( 3, 1)
x

0

x

0

y

2
3
2 a

b

y

c

y

d

y

y

(1, e)
4

1
0

x

0

(2, 1)
x

0

x

1

0

x

(1, e)

689

Answers
3 a

b

y

c

y

d

y

y
(1, 8)

4

1

( 2, 3)

x

0

3

x

4 a

b

y

c

y

x

0

1

0

5 a

b

1 0

c

y

0

x

x

1

(4, 1)

d

y

y
(1, 3)

(3, 2)

(3, 1)
x

1

2

x

0

y

0

(2, 1)

(1, 2)

x

1

y

d

y

(4, 3)
(4, 1)

x

0

0

x

1

0

2

x

1

1 0 1

x

b logp 2

c loga 25a

(3, 1)

6 k 5

7 k 2, p 3

8 a 3

Chapter 5
1

a x6

14

˛

2 a

10 p 3, a 6

Review Exercise

b 15p

˛

9 a 4, p 3

˛

c 2 4x 2
˛

b

y

y

3 a 5

(1, 6)

0

0

11 a

x

9 a x 5.25

c

y

b x 0.356

c x 2.08

d x 1.34

d

y

7 c 0.289

b

x

c

y

y

(1, 5)

690

(5, 1)
(1, 2)

x

0

d

0

2
0

x

6

y

0

d 2.32

c x 3

c 6

2

12 a

b x 81

(2, 6)
(1, 3)
0

x

8 a 8100

11
8
b 24200000

y

( 2, 6)
( 3, 1)

(1, 3)
0

8 c 44.7

b 5

4 a loga 48

6 a x 9

b 1

b 0.631

10 a 5

b

y

d 2

7 a 1.40
x

1

1
3

c

5 a log3 8x

(6, 1)
1

b 3

x

1

1
(7, 1)

0

x

x
5
(1, 8)

Answers
13 k 7

14 p 3, q 5

16 x ;6

Chapter 6

2 un 5n 4

˛

˛

13 un 4n 3

Chapter 6

˛

˛

2 u6
˛

x 11 xn 2

˛

9 Sq
˛

243
2

˛

36
7

10 1 6 x 6 1

1 x

8 a

1 3x

255
16

8 1050

9 5586

b 107

4 u6 160, un 51 2 2 n 1
˛

10

n 1n 12
2

18 k 8

d 425

20 n 5

f 3060

25
1 n 1
, un 100¢ ≤
8
2
˛

g 585936

14 un 41 32 n 1

˛

19 n 9

˛

e 66.4

1 n 1
13 un 270¢ ≤
3

˛

5 u6

˛

c 6560

12 un 512 2 n 1

17 k 1 or k 3

4 Sq
˛

4 22 years

˛

4
22 a , r 4
3

21 n 8

12

512
13
58
9

6 S q 12

5 Does not converge
13

214
99

14

727
99

7 S q 300

˛

15 a 1950

8 S q 50

˛

b

58025
32

c

˛

32
9

5 4.3%

6 £ 8820.36

7 a 17623 rats

b 14.2 months

8 a 187 leopards

b 2012

Exercise 6
5

c 896

2 a

a 4r

3 a 3n2 n
˛

2 3 3 2 17
n n
n
b
3
2
6
˛

˛

Chapter 6
b 336
b 120

q

n 2

b

r 1

8 a 216

7 354

b 10

b 252

1 a 30

6 203

˛

11 1 3x 2 n 2

˛

11 x 6 1 or x 7 1

3 98691 dkk

Chapter 6
1 a 35

20 k 9, k 6

Exercise 5

2 $2720.98

9 a 3.07m

11 Sn

3 Does not converge

Chapter 6
1 28 days

19 k 3

12 n 27

Exercise 4

2 Sq

˛

˛

˛

16 k 2 or k 10

Chapter 6
1 S q 40

18 k 8

11 n 23

˛

˛

3 u6 486, un 213 2 n 1

˛

1 1 x2 n

˛

1
1 4 2 n 1
8

5
1 n 1
, un 80 ¢ ≤
64
4

˛

10 Sn

˛

1 x

15 un

17 k 2

n
n2

3
6

5

˛

7 u6 18750, un 615 2 n 1

˛

˛

4 2014n 6n2

˛

6 u6 384, un 12122 n 1

˛

10 n 143

˛

˛

˛

Exercise 3

˛

9 Sn

˛

b 348

1
1 u6 , un 24 n
4

˛

˛

1
16 un n 4
2

˛

3
163n
3 n2
2
2

2 n2 7n

Chapter 6
˛

3
2
22 a x 6 , x 7 , x H ⺢
3
2

Exercise 2

12 a d 2

11 n2

15
10
,y
7
7

6 un 11n 4, u20 216

˛

1
1
9 un n , u15 8
2
2

˛

15 un 6n 50

˛

5 un 4n 8

˛

˛

14 un 9n 7

˛

4 un 9n 69

˛

8 un 7n 24, u19 109

˛

˛

21 x

b x 28

3 un 6n 2

˛

7 un 110n 90, u13 1520

n
3
n2
2
2

20 x = –5, y = 6; x 1, y 0

Exercise 1

1 un 2n 3

1

b Yes as it will be 34 units at the end of the trip. The pressure will be ok for a total of 8.07 hours.

18 1 log5 3 19 x 4, y 4

3
23 x ;8 24 a k , m 1
2

y H

b

ln 9
ln 8

17

15 a k 0.0633

a 5r 7

c

r 1

a 4r 5

r 1

53
8
n
c n3 2n2
3
3
˛

˛

d

1
1k 1 2 17k 8 2
2

Exercise 7
c 56

d 126

9 a n 6

e 70

2 360

b n 5

3 2002

c n 4

4 a 39916800

d n 6

b 990

10 a n 4

c 330
b n 10

5 1712304

6 76275360

c n 9

11 a n 7

7 12870
b n 3

c n 3

691

Answers

Chapter 6

Exercise 8

1 a a4 4a3b 6a2b2 4ab3 b4
˛

˛

˛

˛

˛

b 729x6 2916x5 4860x4 4320x3 2160x2 576x 64

˛

˛

˛

˛

˛

˛

˛

˛

˛

˛

˛

c 1 4x 6x2 4x3 x4

˛

2 a x3 3x 3x 1 x 3

1 d 32p5 240p4q 720p3q2 1080p2q3 810pq4 243q5
˛

˛

˛

˛

˛

b x5 10x3 40x

˛

˛

˛

˛

˛

80
80
32
3 5
x
x
x
˛

˛

1
3
1
d 16t4 8t2 2
2
256t4
8t

15
6
1
2 c x6 6x4 15x2 20 2 4 6
x
x
x
˛

˛

˛

˛

˛

4 a 40

b 7000

c 3840

˛

˛

d 6480

e –150994944

g 8

f 21

˛

˛

˛

˛

˛

˛

˛

˛

˛

˛

˛

6 a x7 x6 69x5 109x4 1616x3 3360x2 12800x 32000
˛

3 x6 9x5 30x4 45x3 30x2 9x 1

˛

h 90720

b –54 c 243

5 a 242

d –54

b x6 6x5 15x4 26x3 39x2 42x 37

˛

˛

˛

˛

˛

˛

30
12
8
2 3
x
x
x
˛

17
6
20
8
6 c x7 2x5 6x3 8x
3 5 7
x
x
x
x
˛

˛

˛

˛

10 22688x2 12480x 3200

Chapter 6

8 1360

˛

b Sn 1614n 1 2
a 冟x冟 6

6

˛

˛

3
2

b 5

3n2
n

2
2

a 8n 3

7

˛

˛

˛

15 280 and 84

˛

54
5p3

16 k

8 a ;3

b 50

Chapter 7

5 a 3 sin u
7 d

1

c ;

22

226
7

c sin u

e 3.53, 5.90

1 23

17 c

23 1

117
b
125

336
625

d

23
2

b

1 23

222

23 1

44
c
117

16
65

527
625

1
2

10

13
2

c 23

4

p
5p
, p,
3
3

8 a p, 0

692

b n 12, d 0.25

20 n 10

˛

3
5

b ;

c No possible value
b

p 3p
,
2 2

d ;

2 26
5

4 ;

1
3

c 0.785, 2.90, 3.93, 6.04

23 1

2 a and b

2 22
2
7

12

b 0°, 180°

3

1 23
2 22

4

13 0.2829, – 0.3877

c 7.6°, 187.6°

11 232
2 22

8 a 0

14 52.5°, 232.5°

19 a 2.88, 6.02

b 2.36, 5.50

16

b
6
10

23
2
17 a

c 1.70, 4.84

2

120
169

3

161
289

7

1

4 a

25

b

2
25

5 cos u(1 – 8 sin2 u cos2 u) – 4 sin u cosu(1 – 2sin2 u)

119
169

Exercise 4
22 12
2

6 c 60°, 90°, 270°, 300°
7 a

14 a 59

Exercise 3

Chapter 7
22

1
2

7 a 0.464, 2.68, 3.61, 5.82

11 0.951

18 a 40.9°, 220.9°

6 2 cos u冤16 sin5 u 16 sin3 u 3 sin u冥

3

2 a ;

d No possible value
d 7 cot2 u

c

Chapter 7
1 a

9
20

b d

Exercise 2

b

2 22

4
9 a
5

1
2

b A = 15, r = 4

f 1.05, p, 5.24

Chapter 7
1 a

10 a n 6

Exercise 1

b tan u

p 5p
,
4 4

9 a 9

5
n 1n 1 2 12n 1 2 , 972
6

19 6n

18 4455

˛

b ;

c 15800000

b a 9

13 a r

b 32.8080401

17 34642080

2
3

4 a r

b n 30

˛

12 a x5 10x4 40x3 80x2 80x 32

14 c i 99 ii 100

23
2

b 0.210

1
5

3 a Sn

˛

b un 4n 3

11 a 2, b 3

1 a ;

9 a 1.04

Review Exercise

2 a r 4

5 a 1,5,9

c 5888

b 952

˛

11 Proof: p 2

˛

1 7

7 a 22

˛

˛

b 0,

d 90°

2p
4p
, p,
3
3

p p
b ,
3 3

5 a

p p 7p 4p
, ,
,
6 3 6 3

e 60°, 300°
c

p 5p 7p 11p
,
,
,
6 6 6
6

f 60°, 300°

p 2p 4p 5p
,
,
3 3 3 3

c No solution

b

d p,

d

g 41.41°, 180°, 318.59°

p p 5p 3p
, ,
,
6 2 6 2

2p 2p
,
3 3

6 a 30°, 90°, 150°, 270°

e 0,

b 0°, 180°

h 210°, 330°

2p 4p
,
3 3

9 a 0.841, 1.82, 4.46, 5.44

f p

g 0, p,

b 0.605, 2.54

7p 11p
,
6
6

24
25

b

7
25

Answers

Chapter 7

Exercise 5

1 a 10 cos1x 53.1 2°

b 13 cos1x 67.4 2°

2 c 25 cos1u 4.25 2

d 2 cos ¢u

5 a 210 sin1x 108.42 °

2p
≤
3

c 210 cos1x 288.4 2 °
3 a 17 cos1x 28.1 2 °

b 2 sin1x 240 2 °

d 25 cos1x 153.42 °
37
cos1x
B2

b

6 a 22 cos12x 45 2 °

7 a Minimum 1135°, 5222 Maximum 1315°, 5 222

4 a 2 sin ¢u

54.52 °

b 2 sin(3x +60)°

2 a 2 cos ¢u

c 6 cos ¢u

11p
≤
6

2p
≤
3

11p
≤
6

7p
≤
4

b 22 cos ¢u

b 22 sin ¢u

d 22 cos ¢30u

3p
≤
4

3p
≤
4

b Minimum (150°, 3) Maximum (330°, 7)

7 c Minimum 1157.5°, 8.90 2, 1337.5°, 8.90 2 Maximum (67.5°, 10.9), (247.5°, 10.9)
8 a Minimum 15.33, 2122 Maximum 12.19, 2122

8 c Minimum ¢

p
,p
2

¢

35p
41p
47p
29p
, 5.83≤, ¢
, 5.83≤, ¢
, 5.83≤, ¢
, 5.83≤
24
24
24
24

p
p
7p
5p
, 0.172≤, ¢ , 0.172≤, ¢ , 0.172≤, ¢ , 0.172≤
12
3
12
6

¢

13p
4p
19p
11p
, 0.172≤, ¢ , 0.172≤, ¢
, 0.172≤, ¢
, 0.172≤
12
3
12
6

5p
3

b 0,

c 0,

3p
2

Chapter 7

12

35
12

d 0,

ii –

119
169

6

13 2112 sin1u 3.86 2

Chapter 8
1 f¿ 1x2 5

10 a 102.4°, 195.7°

p p p 2p 5p
, , ,
,
6 3 2 3 6

7 a 5 cos1u 0.644 2

14 11.25°, 22.5°, 56.25°, 67.5°

3 f¿1x 2 2

3
11 f¿ 1x 2 2
x

b u 0.644

8 0.905

1 23

9

2 22

17 b a 0.464, k 0.559, p 0.5

4 f¿ 1x 2 2x

12 f¿ 1x 2 2x

5 f¿ 1x 2 3x2
˛

13 f¿ 1x2 3

6 f¿ 1x2 4x3

7 f¿1x 2 4x

˛

10

13
2
11
85
22 12

25 +2
,
4

c A

u

1.34

8 f¿ 1x 2 10x

9 f¿1x2 12x2
˛

1
15 f¿ 1x2 2 2
x

14 f¿ 1x 2 2x 4

˛

˛

Chapter 8
1 f¿1x2 18x

b 7.3°, 34.0°, 127.3°, 154.0°, 247.3°, 274.0°

Exercise 1

2 f¿ 1x 2 8

10 f¿1x2 0

p p 5p
4p 3p 11p
, ,
, p,
,
,
3 2 6
3 2
6

Review Exercise

23
120
2 b i
2
169

1 ;

7p
p
13p
25p
19p
31p
, 2≤, ¢
, 2≤, ¢
, 2≤ Maximum ¢ , 2≤, ¢
, 2≤, ¢
, 2≤
18
18
18
18
18
18

5p
11p
17p
23p
, 5.83≤, ¢
, 5.83≤, ¢
, 5.83≤, ¢
, 5.83≤
24
24
24
24

8 c Maximum ¢
9 a

b Minimum ¢

Exercise 2

2 f¿ 1x2 30x2
˛

3 f¿ 1x2 24x3
˛

4 f¿ 1x2 15x4
˛

5 f¿ 1x 2 0

6 f¿ 1x 2 7

7 f¿ 1x2 11

8 f¿ 1x2 8

9 f¿ 1x2

8
x3
˛

10 f¿ 1x2
17

dy
dx



5
2 1x

4
4
2
3
x
˛

11 f¿ 1x 2 2x 5

12 f¿ 1x2

25
2 2x7

13

dy
dx

3x2 10x 7

14

˛

dy
dx

12x

dy
dx



27
1 2x7 1x2
10
˛

19 f¿ 13 2 2

20 g¿ 16 2

10
9

21

dy
dx

6

15



3081
128

dx

˛

˛

18

2
x2

dy

22

dy
dx



1
4 3
4 2x

16

dy
dx

5 2x
3
3



2

˛

˛

23 (5, 7)

1 199
217
≤ and ¢5,
≤
24 ¢ ,
2 24
6

Chapter 8

Exercise 3

1
19
5
1
1 a Tangent y 6x 3, Normal y x
b Tangent y x 4, Normal y x c Tangent y 4x 3, Normal y x
6
6
4
4
1
3
1
31
43
1
1 d Tangent y x , Normal y 6x 57 e Tangent y 8x 12, Normal y x
f Tangent y 18x 47, Normal y x
6
2
8
8
18
6
2 At A, y 4x 4, At B, y 3x 3
8 y 3x

38
9

Chapter 8

3 y

1
4
x
3
3

4 Q1 3, 20 2

5 y 2x 11

9 y 5x 15 and y 5x 10, y 5 x 15 and y 5x 10

1 9
6 ¢ , ≤
4 4

7 y 7x

10 y

8, y

2x

28
and y 19x 57
3

x + 4 , 17 1
7
7
7

Exercise 4

1 a 14, 13 2 Minimum Turning Point

b 1 2, 232 Maximum Turning Point and 12, 92 Minimum Turning Point

1 25
1 d ¢ , ≤ Minimum Turning Point
3
3

c (0, 0) Minimum Turning Point

1
1
e ¢ , 4≤ Maximum Turning Point and ¢ , 4≤ Minimum Turning Point
2
2

693

Answers
2 a 1 2, 132 Minimum Turning Point

4 256
c ¢ ,
≤ Maximum Turning Point and (4, 0) Minimum Turning Point
3 27
e (0, 0) Rising Point of Inflexion

b 1 1, 252 Maximum Turning Point

2 d (1, 10) Maximum Turning Point and (2, 9) Minimum Turning Point

8
3 a ¢1, ≤ Maximum Turning Point and 13, 4 2 Minimum Turning Point
3
3 d (0, 0) Minimum Turning Point

Chapter 8
1 a ¢ 2,

5
b ¢ , 0≤ Minimum Turning Point
2

1
c ¢ , 12≤ Maximum Turning Point
2

e 1 1.57, 12.5 2 Maximum Turning Point and (0, 2) Minimum Turning Point

4 2

Exercise 5

224
224
≤ Non-Stationary, (0,0) Stationary, ¢2,
≤ Non-Stationary
3
3

iii None Quadratic functions have no points of inflexion

b 1 1, 152 Non-Stationary
b 11, 3 2 Non-Stationary

2 a (0,0) Stationary

c i None

ii None

c 1 2, 35 2 Non-Stationary

3

2 d x

b
8b
bc
,y

d There is always one point of inflexion for cubic functions
3a
3a
27a2
˛

˛

1
b ¢ , 4.11≤ Non-Stationary and 1 1, 9 2 Non-Stationary
6

3 a 1 1, 32 Non-Stationary and (1, 3) Non-Stationary

d No points of inflexion, 10, 32 is a minimum turning point.

3 c (0,0) Stationary

Chapter 8

4 y 21x 36

5 21312 ⯝ 36.2

Exercise 6

1 Vertical: x 0, Horizontal y 0

2 Vertical: x 3, Horizontal y 1

4 Vertical: x 2, Horizontal y 1

5 Vertical: x 2, Oblique y x 2

7 Vertical: None, Oblique y x 2

8 Vertical: None, Oblique y x

10 Vertical: x 1 and x 1, Horizontal y 1

3 Vertical: x 0, Oblique y x
6 Vertical: x 3, Oblique y 2x 9

9 Vertical: x 1 and x 4, Horizontal y 0

11 Vertical: x 3 and x 3, Horizontal y 3

12 Vertical: x 2 and x 3, Oblique y 4x 4
Graphs of the following functions, including asymptotes, stationary points and intercepts:
13

14

y

1
0

1
1

y

(2.41, 0.172)
1

1 0

x

x 1
x 1

y

1

4

x

x 1
x 1x 12

y
y

18

1 0

–1

x

(2, 4)

1
0

x

y

2x2
x 1
˛

y

20

10
3

x
52

2

2
0

3

4x

0

3

1
12
1 , 0.0816
2

y

x
x 1
2

˛

694

y

x2
1 x
˛

y

12x 5 2 1x 4 2
1x 2 2 1x 3 2

x

0

( 2, 8)

x
x 4
(0.124, 3.33)
y

19

0

16

1

˛

y

17

y

y

15

y
( 0.414, 5.83)

y

1
x2 x 12
˛

4 x

Answers
21

22

y

0

冑2

y

冑2

2

20

x

20

0

x

y2
x2

25
16
9

4
x2 2

˛

˛

˛

Chapter 8
1 a

y

Exercise 7
b

y

c

y

d

y

y

4

x

0

1 e

f

y

x

0

x

0
1

g

y

h

y

0

x

0

x

y
3

1
x

0

1 i

2 a

y

b

y

1
3
( 2, 1)

x

0

x

0

c

y

y

3

x

0

y

2 d

e

1 0

2 h

2

x

y

3

6 x

0

0

j

2

x

x

4

y

5x

1

y

2

0

f

0

i

y

1 0

x

0

0

g

4 x

1

x

3

y

0 a

c

x

y

1 0

1

3

6

x

695

Answers
3 a

b

y

1 0

4 a

2

x

b

y

c

y

1 0

x

0

d

y

c

y

5

x

7

1
2

1

4 e

x

2

x

f

g

y

5 a

0

c

0, 1
16
1, 1
27
4

x

y

3
2


2

1

y

b

y

x

1

0

x

6

0

3 x

h

1
x

1

0

y

1
0

y

12
0

y

0 1

d

y

1
4
0

y

0
2

x

y

d

y

x

1
2

1
2
3 0

e

2

x

0

f

y
3, 1 1, 1
4
7

4

x

0

2

x

3

0

3

x

y
1, 1
5

(4, 1)

3, 1
8
x

1 0

4

Chapter 8
1 f¿1x2 3x2 4
˛

2

3
16

y

696

7

Review Exercise
3

dy
dx



3 12
17 152
x
x
4
4
˛

˛

4 x

0, x 7 4

1 9
2 50
8 ¢ , ≤ max TP, ¢ , ≤ min TP 9 y
2 2
3 27

7

1 0

2

x

7

3

2

6x

5 h 1x2 45x2 60x 19, h¿1x2 90x 60
˛

5x + 1, y

˛

1
15 13 76
– x+
,¢ , ≤
2
2 11 11

6 x = 2, y

6x

16

Answers
11 y 1, x 1, x 4

10

12 a 4, b 18

y
13 a

(5.36, 8.87

2

y

2, 1
4

x

3

b

y

1, 1
3

1
3

3, 1
7

4

4

x
1, 1
7

x

( 3.36, 4.39)

Chapter 9
1
7

dy
dx
dy
dx

sec2 x

Exercise 1
dy

2

cos x

dx

18x 4 sin x

dy

8

dx

cosec x cot x

12
17
23
27

dx
dy
dx
dy
dx
dy
dx
dy
dx

2 f¿ 1x 2 412x 32
dy

3
3
13x 22 2
2

13 f¿ 1x2 2015x 4 2 2

18

dy
dx

4812x 32 5

dx

3 sin 3x

12x3 3 cos2 x sin x

9 f¿ 1x2

22

dy
dy
dx

1
x

6
10
14
18
21
23

dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx

10 f¿ 1x 2
17

cosec x sec x

23

2
x

3
1
13x 8 2 2
2

1
1
cos x
2
2

dx
24

dy
dx

10

dy



dx

dy

20

dx

dx

7 cosec2 x

6

dy

3 sec x tan x

dx

7

dy
dx

dy

29

dy

21

25

dx

6 f¿ 1x2 817 2x2 3

216x 5 2 3
2

75
dN
5

18 5p2 2
dp
2

16

9 sec 9x tan 9x

dx

3 sin2 x cos x

26

p
≤
4

30

3 sin¢3x

dx

dy

11

dP
1814 3k 2 3
dk

15

5 cos 5x 3013x 4 2 2

5 f¿ 1x 2 315 x2 2

10
2
12x 9 2 3
3

6 sec2 6x

3013x 4 2 6 2 sec2 2x tan 2x

11 f¿ 1x 2

dx



1
x ln 2



1 2
sec 1ln x 2
x

dy
dx

dy

22
dy
dx

dx

6 3 cosec2 3x

8 tan 4x sec2 4x

dy

1
1

2 1x

dx

12 2

sec2 1 2x

12

18

2x
x2 2
dy
dx



5 f¿ 1x 2 54e 9x

˛

dy

12

dx

1
x ln 8

ln 4 # 4x

19

dy
dx

6 f¿ 1x 2 2xex

13

dy
dx

ln 4 # 4x

˛

ln 10 # 10x

1
x ln 5

20

2

˛

˛

dy

14

dy
dx

dx

8 f¿ 1x2

7 f¿1x 2 2e2x 3
˛

6 ln 5 # 5x

4e4x 2 cos 2x
˛

dy

15
1
x

dx

3e3x ln 3 # 3x
˛

dy

21

1
x

dx

tan x

Exercise 4
dy

2

˛

cos2 x sin2 x

4 f¿ 1x 2 10e 5x

˛

dy

x 12 sin x x cos x 2

dy

7

dx

6x2 13x 22 15x 22
˛

dx

x2 13 cos x x sin x2

˛

11

dy
dx

12x 1 2 2 18x 11 2

x 1x 12
˛

x2 2x 5
˛

3x 5 ¢4 tan ¢3x

dy
dx

3 cos 3x cos 2x 2 sin 3x sin 2x

15

312x 1 2 2 cosec 3x 3 2 12x 12 cot 3x4
8x Bln1x2 2x 5 2

3

˛

13x 42 2 3 9 sin x 13x 42 cos x 4

˛

dy

5

4 f¿ 1x 2 2015x 4 2 3

14 f¿ 1x 2 5613 8x2 2

˛

Chapter 9
1

dx

dy



dx

3 f¿ 1x2 4e4x

˛

1
ln 2 # 2x
x



5 sin x

dx

Exercise 3
2 f¿ 1x 2 7e7x

˛

dy

28

˛

9

19

2 cosec 2x cot 2x 1213x 22 3

1 f¿1x2 3e3x

dx

dy

8

Chapter 9

16

3 f¿ 1x 2 613x 4 2

2019 4x 2 4

4 cos 4x

dy

4

Exercise 2

1 f¿ 1x2 21x 42
dy

cos x 12x

dx

7 5 cos sec x tan x

Chapter 9

7

dy

3

R

22

dy
dx

12

3xex 12 x 2
˛

dy

8

dx

dy
dx

dy
dx

dy
dx
˛

dx

e3x 13 sin x cos x 2
˛

˛

dy
9

16

dy
dx

1
≤
x ln 8

x2 a 3 log6 x
˛

20

dy
dx

x2 1x
˛

dx

21x 5 2 13x 2 2 3 19x 28 2

4x ¢ln 4 log8 x

e3x sec ¢2x

dy

2x 1x 1 2 12x 1 2

5x 1ln 5 # cos x sin x2

19

4

dy

13

dx

1
≤
ln 6

ln12x 3 2

5

dy
dx

2 2 3 1 7x



sin x
ln x cos x
x
62

615 2x2 2 13x 4 2 11 5x2
17

dy
dx

e4x sec 3x 14 3 tan 3x2
˛

˛

2x
2x 3

p
p
≤ B3 2 tan ¢2x ≤R
4
4

dy
dy
p
p
≤ 3x sec2 ¢3x ≤≤ 24
x 1 12 ln x 1 2 sin x x ln x cos x2 25
e3x 1x 2 2 冤13x 8 2 tan x 1x 22 sec2 x冥
2
2
dx
dx
˛

˛

697

Answers

Chapter 9

Exercise 5

ex 1cos x sin x2

1 f¿1x2

cos2 x
6
1x 3 2 2

6 f¿1x2

dx



1x 32 2

8 f¿ 1x2

2x 2x 9
3

7 tan x 7x sec2 x
tan2 x

4x 12x ln 4 1 2
2x

1
1
ln1x 4 2
ln x
x
x 4

12
dx
1ln1x 4 2 2 2

16

12x 32 4

11

dy

x 2 sin x

x cos x

dx

x 2
1x 1 2

2
1e e x 2 2



˛

dy
dx

˛

˛

5 f¿ 1x 2

1x

dy



dx

ex 1x 52
˛

1x 4 2 2

e3x 13x 2 2
˛

9x3
˛

21cos 2x 3 sin 2x 2



e6x
˛

p
≤ 2
4

18
dx
p
e2x cos¢x ≤
4

10 2

15x

˛

dx

˛

10

3
2

14

x

xe3x 13x2

dy

17

ex

dx

dy

13

1 ln x
4x2
˛

9 f¿ 1x 2

3
2

dy

1213x 22 4 14x 19 2

4 f¿ 1x 2

˛

˛

1
1x 62 log6 x
x ln 6

11
dx
1x 6 2 2

15

3 f¿ 1x 2

˛

31x 122

7 f¿ 1x 2

dy

dy

6x 1x 62

2 f¿ 1x 2

˛

tan¢x

dy

523

˛

1) cosec2 ¢2x

2(3 x

dy

p
≤ ln13x
3
e2x

19 dx

3

3 cot¢2x

˛

Chapter 9
1

12

p
≤
3

dy

3x2

dx

x

y

˛

dy

1

dx

6 2xy2

Exercise 6
x

2a

2

˛

3y

3y 2 2

12

,

12

3 x2

cos x

˛

3y 2

b

3

4y

4x

1

1 2 sin x

14x

,

8 cos x

14x

32y

1
1
x , x2

c

122

˛

˛

Chapter 9
dy

1

dx



1
225 x

2

Exercise 7
dy

2

dx



˛

dy

7

dx



1
264 x

1

dx

dy

8

22 1x 4 2 1x 3 2

1 f¿1x2 2x 5
dy

dy

2

dx

x 12 ln x 12

dx



10
100 x2

1
2x 2x 1

14 f¿ 1x2

3 cos 3x sin 3x
ex

dy

11



dx

x
2 cos 1 x≤
21 x2
3x3
˛

p 2
64

19

dy
dx



Chapter 9
2 1

3



dx

2
29 4x

dy

5

2

dx



3
21 9x

6

2

dy
dx



2ex
4 e2x
˛

˛

˛

1



x21 1ln 5x2 2

3 f¿ 1x2 8 sin 8x
dy

8

dx



˛

dx

12

x 2e y
x

˛

˛

20

dy
dx

˛

˛

˛

9 f¿ 1x 2 3x 1ln 3 # sin x cos x2

2 cos 2x cos 4x 2

dy

13

dx



12
21 4x2
˛

˛

˛

2

5 f¿ 1x2 x2e 4x 13 4x 2

sec x tan x 5e5x

61sin 2x sin 4x

dx

sin x 1ln x 1 2 x ln x cos x

y 12x exy 2
˛

dx

dy

˛

dy

dy

4

˛

˛

1x 9 2 2

15

3 12
x
2

sin x 4x sin x x cos x
e4x

x ln x 1x 18 2 x 1x 9 2

˛

18 f¿ 142

dx

˛

dy

˛

dy

9

˛

¢

4

˛

2

612x 7 2 2

7 f¿ 1x 2

˛

x 4
3 log2 x
x ln 2

10
dx
1x 42 4

3
4



Exercise 8

dy

1

dx

˛

Chapter 9

6

dy

3

2



16

dy
dx



1sec x cosec x ln1cot x 2 2

17 f¿ 122

ex
˛

y 14x3y2 cos x 2
˛

˛

˛

1sin x 3x4y2 2
˛

˛

Exercise 9

4 ln 4 3
e2

4
x
5

4 y

˛

9 (0, 0) rising point of inflexion and ¢3,

9
6
x –
5
5

14
,y
5

5

4
4 4p
6
3
ep

7 (0,0) and ¢2,

2
l
≤
8 ¢e- 2 ,
e

4
≤
e2

˛

˛

˛

27
≤ max TP
e3
˛

Chapter 9
1 a 6013x 2 2 3
2 c

e5x
2x 4

698

¢5

Review Exercise
b 14x 13 2x2 2 2
3

˛

1
≤
21x 42

˛

d

c 6 3 tan 3t sec 3t

3
2

3x 4
2x 1

e

d 48e8x

e

1
3x ln 3
x

1
2
¢2 3 tan 3x
≤
ln 10
x 4

2 a 4e4x sin 3x 3e4x cos 3x
3 a

6xy
8y 3x

2

˛

b

y
3x2

ln x
x ln x
˛

b

1
cot x
x

20
3

Answers

4

6

4x 1cos x

(2 sin x

2x 2cos x–7 2

sin x 2

16 1x 4p 1x
p2

b i 3 5p

7

3
4

8

2 All three sides are 250cm

˛

9 x y 75cm. It is a maximum value.
11 t 6.29 seconds, t 12.6 seconds

Chapter 10
˛

10 a 1r u

15
32p

r sin u, r

3

2
h

˛

˛

t2 cos t t cos t sin t
1t 12 2
e2t 1t2 2t 12
1t2 1 2 2

e4
9

b v
b a

4e2t
t2

2
1

t 2
t
t 1

31s2 2st 2t2 2
˛

˛

1t s 2 3

Chapter 10

6 a A 2prh pr2
˛

9

b 4.30 cm
p 2 1
cm s
3

˛

Chapter 11

8
7
5

b A

168x 24x2
7
˛

c 42cm2
˛

9

8 0.011cms–1

5
3

˛

˛

˛

3 a 411 2t 2

˛

b t

1
second
2

c 0ms 2
˛

˛

˛

1t2

1

c 20ms 2

t3 sin t 2t2 sin t 2t cos t 2 cos t t sin t 2 sin t
7 Period p
1t 1 2 3

(8t

2) e2t

4t2 e2t
1t2

122

1
ln 300
100

c a

ln 300 2

b. v e 100 t ¢1
˛

˛

2
1

t 2
t2
1t 1 2 2

2
27e4

d a

123

˛

ln 300 2
t≤
50

d v

˛

˛

10 a v 3k cos1kt c2

ln 300 ln100300 t2
e
¢3t
50

c a

˛

˛

ln 300 3
t≤
50
˛

d 1.927s

4
31
ms 1 a
ms 2
45
4050
˛

˛

c s t 1 1 < 232
˛

c 1120 cm2

10

56 2
ms
27

c 2h (x

h ¢2 –

3

1
cms 1
2p

7 a 6t sin 5t 15t2 cos 5t
˛

11 a 6.28 hours.
1
23

≤

˛

b 0.457 hours

b 0.524 hours.

14 a t 2 minutes

4 a = b = 24.42 cm

b

5 1.6

c 0.304 hours

12 256 cm3 s–1
112t3
cm min 1
18 t4 2 2
˛

8 a 1p 2x 2 sin x

13 a BC 21x h csc u2
c 19.55 cm2 min–1

d 1.48 minutes

˛

Exercise 1

b 2 3 c 3 3

The operation is addition.
Magazines
Alan
Bill
Colin

4r
3

b t 4.70 seconds and t 1.05 seconds

13 b 2xh h2 1cot u 2 csc u 2

1 a 1 3

˛

b 16 12t

2 c Maximum velocity is 0.541 ms 1 Minimum velocity is 1.35 ms 1

8 b 1.12 units2

8 b

Review Exercise

2 a 13 ms 2

1 120 kmh–1

7 78.125

4p 2 1
cm s
3

b

˛

c a

11 a k

˛

2t s
t s

2
cms 1
9

˛

c a

˛

c a 0.0213ms 2

12 a Min is (2,1.39)
13 a v

y
7 x

7
24

15 a

7 2.988cm2s–1

˛

2 a 8ms 1

˛

b v

˛

c

14 20

6 0.954ms 1

˛

13 a

˛

b 1.74ms 1

˛

k

6 3.414 cm2

5 7cm, 3.5cm

10 c x 1.58cm. It is a maximum value.

˛

5 216cm3s 1

˛

p

450x 60x3
≤
13

13 50 Yen, 14.14 Kmh–1

11 100cm3s 1

˛

10 b t

4 5.62cm, 4.22cm, 16.9cm

Exercise 3

˛

˛

3 14.65cm

12 11100

1
2 23
1 v 3 ms 1 a 3ms 2 t
seconds
4
3

8 a s

˛

10 b V 60 ¢

4 611N0

r cos u 2

Chapter 10

4 a v

10 (0, 0)

Exercise 2

2

˛

5
3

Exercise 1

1 v 2500ms 1

1 128pcm2s 1

4
x
3

9 y

4
5

b

Chapter 10

1
3

211 csc x cot x cot2 x2
, which reduces to –1
1 cos x 2
1 ¢
≤
sin x
–4
11 2.533 12 y = x – 4 13 .
14 a 6 sec 2t tan 2t 5
(2x 1) 2

5

y

e

15 a 1

ii 5

˛

2x

e


x2 cos x

2x sin x

x

d 4 1

3
2 Week 1 £2
4

1
2≥
4

1
Week 2 £4
0

2
1≥
1

4
Week 3 £1
1

2
0≥
1

Newspapers
5
3
6

699

Answers
8
3 a £20
28

4
8
16
1
2

4 e k

18
6
18

12
12≥
4

f k 5

18
5 g £ 8
4

5
15≥
37

24
12≥
24

b £

3k
c £ 4k
12k

g x 0, y 11

15
h ¢
1

7 3k2 4k
5
¥
6 f §
4 k
2
1 2k 2k

31k 1 2
1 k

21k 1 2
≤
0

d ¢

11
≤
29

2
≤
4

10
0

b ¢

5 a not possible

24
6 a ¢
6

1
≤
5

6k
k≥
4k

29
22 ≥
7

b k 0, 1

11
d £ 6
8

c not possible

11
c £6
12

b 1 26 2

4 a k 6

29
d £11
2

5
0≥
1

e ¢

11
9 ≥
2

39
41
29

c k 1

7
1

1
≤
3

d k 0, 3

2
≤
3

9
f ¢
0

6 k2
e ¢ 2
k 3

2 2k
≤
2k 1

˛

˛

˛

45k 4
≤
39k 18

45
39

45
39

4
≤
18

6
6

84 9k
≤
52 5k

7 a ¢

b ¢

45k 4
≤
39k 18

45
39

c ¢

45 39k
d ¢
39

4 18k
≤
18

e ¢

9k 12
≤
27k 6

9
27

˛

0
11

0
≤
7 11k

7 f ¢

13
12

9 A2 ¢
˛

g ¢

4
40
≤ A3 ¢
3
39

13
≤
12

˛

264
99

168
≤
63

h ¢

3
5

2
7

10 a A ¢

8 a x 2, y 1
1900
b B £1300≥
1100

4
≤
2

Minimum total number of calories consumed on day 1 and day 2.

b x 0, y 1

c x 10, y 1

d x 4, y 3

12700
≤
20800

c AB ¢

d i (33500). Minimum total calories consumed.

9
ii ¢ ≤ Total number of people dieting on each day.
14
3
6
11 a P •10
3
8

iii (4300) Total number of calories consumed by one man, one women and one child.

1
4

7
6

6
≤
1

a ¢

12

6
c ¢
0

9
≤
4

b ¢

1 3c
≤ QP £
c

4
5c 4

18 PQ ¢

0
≤
6

5 c
2
3 c2

1 a

1 5
¢
11 3

2
≤
1

5
≤
k

b sin 2u

15
,y
7

2 c x

5
,y
7

3 c x

c 20 units2

13
20
c •31μ
18
27

3
b Q £1≥
0

17 c 2

b x 5, y 0

3
14
¥
5
14

5
Y £2
7

2
37
3
e ¶
74
5
74

b Not collinear

1 1
B 12A B 2
3

c x

25
333
1
37
5
111

13
,y 0
3

a
0

20 ¢

37
304
1
f ¶
38
17

152

8
333
11
∂
74
19

222


31
18
¥
11

18

3 a 30

f y2 y 2

b
≤
a

14 b

1 x 3
¢
8 x 1



b 44

33
152
5
19
9
76

c 78

˛

˛

˛

22
7

12
,y
11k 5

b p
d x
13k
11k
d k

21, q
16
,y
5
7
,k
5

14
69
5

5
11

d x

5
,1 5 a c
3

5
,y
6

c x
3 a x

2
9

d x

1
3k

1

(5k
3k

,y

4k 14
,y
k2
k 3
˛

7, y

17

2 a x

10 c 5, 4.5

x 1
≤
3 x

2)
,k
1

10k 12
, k H
k2 k
3

1, 0. Lines are parallel.

1
3

17
,y
4
b x

k2

313
3

˛

k2

2. Lines are parallel.

61
,b
11

b a
,y

4 a Unique solution

˛

b c

11
20

d 190

˛

Exercise 3
17
7

9
76
1
∂
19
1
38

g a3 a2b ab2 b3

˛

8 k 1.22, 1.08

c Collinear

b X I

˛



43
18
Z §
19
18

1
2≥
5

e cos3 u sin3 u 3 sin2 u tan3 u

6 a Collinear
13 a

29
274
11

∂
274
1

274



2 X §
1

d 2abc

12 x y 1 or x = y = –1

4 c No unique solution

700

c 2

6
137
7
137
38
137



1

k
≤
3k

˛

Chapter 11
1 a x

16 c 5

8
19 a x , y 10
5

11
137
10
d ¶
137
24

137

8
≤
4

1 1
¢
76 9

c

1
k 2
h 2
¢
k 7k 1 2k

5 b 4 units2

11 y 3.36, 3.86

7
≤
2

1 10
¢
41 3

4 a cos 2 u sin 2 u 1
5 a 4 units2

14 k 10

7
6
3μ
2
3

Exercise 2

b

1
1
1 g
¢
14k 3k

15
6 ≥
9

22
8
12 2c

˛

Chapter 11

13 m 1, n 6

4
2
1
9
3

12

31k
3

˛

k2

,k

9
11
;23

b Unique solution

Answers
6 a Consistent. Lines intersect giving unique solution.
7 p 3

b Consistent. Same line giving infinite solutions.

Chapter 11

Exercise 4

1 a x 1, y 9, z 13

b x 10, y 10, z 36

c x 2, y 4, z 3

d x 2, y 1, z 2

2 b x 2, y 1, z 4

c x 1, y 1, z 2

d x 4, y 5, z 2

1
3 c x , y 1, z 2
2

d x 2, y 3, z 2

4 a Determinant 6. Unique solution.

4 c Determinant 0. No unique solution.

6 c x

35
43
13
,y ,z
66
66
33

7 c x

19 5l
7l 11
, y l, z
13
13

7 h x 4, y 4, z 6

e x

8 d No solution.
b c 3
4b
b

12 a

38
20

3l
44 4l
, y l, z
11
11

13 a 2k2 11k 37
˛

5l 10
2

b x

5
x
0
k 1≥ £y≥ £ k 2≥
k
z
2k 1
˛

50
20≥
22

13 a a 7, b 2

2 l 1 or 6

˛

1
k
¢
k2 1 1

3 a

a 1

˛

b
3
≤
2

3 a AB ¢

c z

k2 5k 4
8 3k

1
≤
k

0

2
15
1
∂
10
1
18


b x

˛

5l 11
17 3l
, y l, z
51
17

˛

1
c=4
2

15 a 1

8 b k 3

16

b x 1, y k

d If k

˛

˛

b x 1, y 2, z 1

˛

8
3

c Since M is singular, A must be singular.

Chapter 12
1 a

b k

˛

y2 18z1 11z2 3z3

17

3
5
1
5

c No solution.

2
1
7
,y ,z
15
5
9

14 a 36

k3 2k2 29k 22
k2 6k 14
k2 5k 4
,y
z
8 3k
8 3k
8 3k

7 b Solution is not unique. x
39
16
18

3
59
28
,y ,z
19
19
19

Review Exercise

5 d Otherwise x

36
10 b £14
15

b x

b x 2, y 1, z 1

1
5
1
9 a ¶
10
1

6

˛

k 3
3
1

b Determinant 0. No unique solution.

5
15
7
,y
,z
44
44
22

f x 1, y 2, z 3

b k 7.86, 2.36

1 R is an n p matrix S is an m p matrix
1
5 a £1
1

55
40
64
,y
,z
47
47
47

5

7 a x

c x l, y l, z 2l

Chapter 11

b x

4l
2l 3
2
8
b x
, y l, z
+ l, y l , z
–l
2
4
5
5
4 2l m
2
7
d x l, y m, z
e x , y 0, z
f No solution. g No solution.
3
5
5

d x 0, y 2, z 3

a x 2l 3, y l, z

8

2 a x 3, y 2, z 4

5 a x 1, y 1, z 1

74
3
9
,y ,z
19
19
19

6 a x

1
2
,z
3
3

3 a x 4, y

d Determinant 0. No unique solution.

1
1
d x ,y ,z 2
4
2

5 c x 1, y 2, z 4

10 a 0

c Consistent. Same line giving infinite solutions.

8 l 6, x 1, y 1

11 k 5

4 x 1, y 8

8
there is no solution.
3

6 a 1, b 3

7 a 3p 3q r 0

9 p 3, q 5

10 a c 2.5, 0.5, 2

12 b c 3

a 4, b 1

c x

1 7l
11l 7
, y l, z
2
2

y1 16z1 36z2 58z3

17

˛

˛

˛

˛

y3 13z1 12z2 17z3
˛

˛

˛

˛

Exercise 1
b

a

b 冟AB冟 213

1
3

b 0,

1
4 ¢ ≤
3

1
2

c 8

5 a 234

c a

8
7

b 3

b 253

c 290

c c 6

13
2
8 §
¥
13 23
2

2 a PQ i 4j

d 229

e 221

b 冟PQ冟 217

f 257

6 a Parallel

b Parallel

5
6 c Not parallel

d Not parallel

7 a c 6

b c 7

9 ¶

262
6
262
1

∂

10 a Not parallel

b Not parallel

c Parallel

262

701

Answers
4
QR ¢ ≤
6

2
11 PQ ¢ ≤
3

1
QR £ 4 ≥
2

4
£ 2≥
1

12 b PQ

2
PR ¢ ≤
3

冟PQ冟 213

221
14 SU a b

221

冟RP 冟 214

4 c CB

b 6i 2j 9k

a2

c 2i 11j 2k

9 a BE

ii CA b a
7 b EF

b FH b a

12 c FA b c d

b 1

3 c 24.8°
5

d 2

e 54.0°

5 a

c 1a b2

iv AX

c CF

c AH b c

i

235

e 7
f 50.0°

c

1
b
3

v XD

3b
m n 3

1 a

14i 5j 8k

2 a

2123
2
3

219

i

3

15 2234 units2

j

d AG a b c

f 54

g 40

h 25

2 a 5

2
¢ ≤
8

b AC

6
¢ ≤
10

6 a

1
1b a2
2

b 2136

–25

4
c ¢ ≤
8

d §

b AC ¢

6k
≤
4c

1m n 2b

e ED a

m n 3

12m 3n 92b
m n 3

12 a DG d a b

b AH = d

b They are perpendicular.

4 p . q 14 , cos u

7
B 19

c 30

d 9

e 3

f 55

5 a and d, a and f, b and c, b and e.

g 26
6 a

3
2

h 1 3 a 58.7°
b –11

c

15
2

b 86.6°
d 3, 2

12 It is not a rhombus

b 14i 5j 8k
c 6
1
219

k

7 a
10

c 14i 5j 8k

18
2817

i

274
units2
2

3
2817
11

j

d 28i 10j 16k
22

2817

k

2341
units2
2

b

817
B 986

e 14i 5j 8k
8 a

12 26 units2

2
26

i

1
26

f 42i 15j 24k
j

1
26

k

b

13 4 2189 or 23024 units2

g 0

54
B 55
14 2850 units2

Area 10 21734 units2

Review Exercise

1
3
137
5 cos u
21314 units2 c 10.8i 9.6j 1.2k d
2146 or
2 a 6i 12j 12p 12 k b p 4 4 22 2 cos u 5 ¢
2
2
5 sin u
224817
p
6 0 8 0.70210 a 2u 11 a A has coordinates (2, 4, 6) B has coordinates 16, 3, 02 C has coordinates 14, 7, 6 2
2
3
3
10
2
11 b 50.2 units 11 c § ¥ d 96.3° e 22.6 units2 12 m
2
3
0

1 b

702

4
¥
–47
4

e BH b c a

b 6

16 PQ 25i 5j 10k and PS 6i 14j 2k

Chapter 12

4 a AB b a

Exercise 4

9239
2

219

d

2
b
3

d AF a

j

Chapter 12
b

3
£ 2 ≥
6

f 20i 25j 84k

ka b
1 k

d

iii BD b a
nb
m n 3

b 2425

1
k 9 x 17.9 or x 6.5 10 70.5°
235
16 It is a rectangle since ABˆ C is 90° but we do not know if AB BC
235

3

5
¢ ≤
20

e 3i 6j 49k

Exercise 3

c 29

d 129°

69
75
,y
53
53

3 q 1 Ratio is 1:2

c 74.7°

14 a AB c, BC a, AC c a, OB a c

Chapter 12
1 a 11

d 17i – 21j – 28k

b 2421

b2

k
1a b2
1 k

b

mb
m n 3

10 a BC b

1
1a
2

d OC

1
1b a 2
1 k

8 a i CD a

17 x

Exercise 2

1 g 13mi + 117mj – 91mk 2 a 17i 16j
1
1b
2

214

4
4
6
16 p , q , r
7
7
7

15 a 1, b 1

1 a 3i 4j k

9

冟SR冟 221

3

4

22

Chapter 12

8

冟QR冟 221

221
221
221
214
2
2
4
2
PQ ¶
∂ QR ¶
∂ SR ¶
∂ RP ¶
∂
221
221
221
214
1
2
1
1

12 d

7 a

3
12 a R £3≥
5

冟PR冟 213

3
RP £ 2≥ c 冟PQ冟 221
1

4
SR £ 2≥
1

1

4

冟QR 冟 252

3
≤
2

Answers

Chapter 13

Exercise 1

0
1
£ 2 ≥ + l £ 2≥
3
1

1 a r

b r i 2j l1i 4j 2k 2

3
7
b r £ 1 ≥ l £ 2≥
0
2

2
4
2 a r £1≥ l £ 3≥
2
1

5
4
≤ l¢ ≤
1
7

1 f r ¢

4
3
≤
2 e r ¢ ≤ l¢
3
0

3 a

4
0
c r £4≥ l £ 5≥
3
12

r i 2j 4k l13i j 5k2

3 b

3
4
r £ 2≥ l £ 7≥
3
3

x 3 4l, y 2 7l, z 3 3l

3 c

r j k l1i 3k 2

x l, y 1, z 1 3l

3 d

4
1
r £1≥ l £ 2 ≥
0
2

4 a

x 1 2l, y 1 3l, z 2 2l

4 c

x 2 4m, y 8 7m, z 1 6m

x 4 l, y 1 2l, z 2l 4 x

7
3
5 a r £ 6 ≥ l £1≥
4
2

6 b

r

7 a No

b Yes

c No

d Yes

e No

f Yes

3
1
d r £ 4 ≥ l £ 9≥
2
1
z 4
x 1
y 2
3
5

1 z
3
y 1
2



4
5
c r £ 0 ≥ m £3≥
3
4

z
2
y 5
z 1
x 2


3
1
4

x 2 3m, y 5 m, z 1 4m

b

x 1 2n, y 1 3n, z 7 n

d

f x 1 2t, y 6 5t

2
4
5
5
5
c r ¶ ∂ l • 2μ
3
3
7
2

2

2
4
l§ ¥
3
2

e r 3i j l12i j2

y 2
x 3
z 3


4
7
3

8 y
x 2
z 1


4
7
6

4
1
b r £ 5≥ m £ 3 ≥
1
5

2
2
c r £ 2≥ l £ 9 ≥
3
6

x 1 3l, y 2 l, z 4 5l

y 1
1 x
2 z


2
3
2

6 y
x 4

3
5

4 e x 4 3s, y 6 5s

5
2
§ ¥
1
1

y 1, x

d r 5i 2j k l13i 6j k 2

y 6
x 1

2
5
4
0
d r £ 1≥ n £ 1 ≥
5
1

1
6
1
r • 4 μ
d
l£ 1 ≥
3
9
4

3
4
6 a r £ 5≥ l £ 3 ≥
1
3

5
3
1
2
e r ¶ ∂ l £2≥
3
3
1
3



y 1
x 1

7 z
2
3

5
49
3
f r § ¥ l £ 4≥
7
0
2

6
3
6
5
8 r £ 7 ≥ l £ 9≥ Position vector is § ¥
2
1
2
0

9
11
7
9 Crosses the xy plane at 1 9, 7, 02 Crosses the yz plane at ¢0, 11, ≤ Crosses the xz plane at ¢ , 0, ≤
2
2
4
1 13
10 a r 2i j 5k l13i j k 2 Crosses the xy plane at 1 13, 4, 02 Crosses the yz plane at ¢0, , ≤ Crosses the xz plane at 1 1, 0, 42
3 3
2
1
10 b r £6≥ l £ 1≥ Crosses the xy plane at (9, –1, 0) Crosses the yz plane at (0, 8, 9) Crosses the xz plane at (8, 0, 1)
7
1
11 Crosses the xy plane at ¢

17
29 14
17 29
14
, , 0≤ Crosses the yz plane at ¢0, , ≤ Crosses the xz plane at ¢ , 0, ≤
15 5
4
12
3
3

Chapter 13
1 a Skew
2210
2 a
6

Exercise 2

b Intersect at the point (2, 1, 6)
b 217

c 1

0
3
3 AB: r £ 1 ≥ l £4≥
2
7
4 a 60.5°

b 36.3°

5
d
22

c Parallel

e Lies on line.

y 1
x
z 2


3
4
7

c 71.2°

d 88.4°

d Skew
f

e Skew

g Skew

27

0
1
AD: r £ 1 ≥ l £ 2 ≥
2
1

e 62.8°

f Parallel

223

x

y 1
2

z 2 Coordinates of C are (4, 7, 4)

5 a 2 Point of intersection is 11, 5, 3 2 6 r ¢

3
3
≤ t¢ ≤
8
2

7 a r = 2i – 3j + k + l(i + 8j – 3k) b p 34

703

Answers

Chapter 13

Exercise 3

1
1 a r. ¶

218
4
218
1

2
∂

1

b r. ¢

3

235

3

i

235

5

j

235

25
c r. • 0 μ
1

8

k≤

235

1
274

8

i

274

3

j

19

k≤

274

2354
17
2354
4

7
265

32

∂

i r. ¶

2354

1
230

4

j

265

2

i

2557
6
2557
11

∂

k≤

230

0

3

b 4x y 6

j

5
230

9 b r. 1i 5j k 2 1

86

11 r. 1i k2 1
i

2
r. £ 3≥
2

13

k

b

4
242

i

1
242

8
219

5

13

k≤

17

5

j

242

8

4

g r. ¢

219

226

1

j

10

i

21133

226

k r. ¢

k≤ 0

3
21133

3
213

i

15
c r. £ 3 ≥ 3
11

j

2
213

32
21133

63

k≤

21133

j≤

15
213

k

3
6
2
i j k
7
7
7

c

d

5
245

d r. 1i 11j 9k2 16

1
4 r. £ 3≥
7

d 4x 3y 8z 21

41

2

i

1

245

j

4
245

4

7 r. 113i 4j 11k2 31

k

units Distance between p1 and p2 is

227
14

units Distance of P2 to origin is
˛

265

13
217

17
289

units Distance between P1 and P2 is
˛

265

7
227

units

55

˛

265

units

units.

units.

14 r 14i 3j 7k 2 l12i 2j 5k 2

17
8
9
units Distance of P2 to origin is
units Distance between P1 and P2 is
units
23
23
23
b The line and the plane are parallel. c The line and the plane intersect. d The line and the plane intersect.
˛

˛

˛

˛

Exercise 4
1
b ¢4, , 2≤
2
c 11.0°

b 74.2°

c 13, 2, 92

d 64.5°

12
2178

e 90°

d a

f 55.9°

e ¢

3 a 11.7°

b 57.8°

2
7
b r £ 2≥ l £ 1≥
0
4

f r ¢

20 13 4
,
, ≤
3
9 3

51 1
, , 1b
7 7

f 116, 10, 72

c 17.6°

d 32.5°

c r 17i 3j 2 l12i k2

56
29
i
j≤ l136i 26j 25k2
25
25

6 48.5°

e 34.1°

f 5.51°

d r 1i 6j2 l1 3i 2j k2

1
3
7 a r £ 2 ≥ l£ 1 ≥
3
3

b 1 2, 1, 02

c 13.9 units.

Chapter 13
i

226

i

c Distance of P1 to origin is

4 e r = (–4i – 22j) + l(7i + 31j + 2k)

704

3

units Distance of p2 to origin is

227

Distance of plane from the origin is

b 23.2°

8 a a 3

3

Distance of plane from the origin is

289

4 a r 14i j2 l114i 17j 13k 2

2178

k≤

˛

289

1
2
, b
2
3

3

218

units.

˛

Chapter 13

1 i

218

15

k≤

r1 is not contained in the plane. r2 is contained in the plane.

16 a The line and the plane intersect.

2 a 45.0°

219

c 15x 13y 8z 38

c Distance of p1 to origin is

15 b r. 1i j k 2 9

1 a a 9,

1

j

b r. 15i 4j 15k 2 42

2

˛

8

219

j r. ¢

2557

10 P1 = r.(2i + 5j – 6k) = –41. Distance of P1 to origin is

289

3

i

219

r. 15j + 2k 2

2 a

8 The direction normals are equal. Distance

12 r. ¢

218

1

j

2557

3 a x 2y 7z 9

5 a

218

4

i

20

2354
1 l r. ¢

f r. ¢

274

7
1 h r. ¶

1

25

218
1 e r. ¢

d r. ¢

6

Review Exercise
j

5
2178

k

ii r.¢

3
2178

i

12
2178

j

5
2178

k≤

7
2178

iii 73.5°

2 b

230
units2
2

c

1
230

i

2
230

j

5
230

k

Answers
1 y

2 d AD has equation x 1
1
l £ 2≥
1

2
£1≥
1

3 c r

x

5 d 62.7° 6 a

2

3

18
4 b ¢ ,
5

36
, 4≤
5

y

1

5

1

8 b i j 2k

z

1

2
c r. £ 4 ≥
9

1
d r. £ 1 ≥ 3
2

9 a A lies in the plane. B does not lie in the plane
10 a 2i 2j k

b 1 4, 1, 4 2

12 a ii r 1 j k 2 l13i 11j k 2

4

c r. 1i 4j 2k2 9

221

13

d

5 13
16 a ¢4, , ≤
2 2

221

2
£3 ≥
0

f 326

1, 42

1
l£ 1 ≥
1
i

g

j
26



25
l£ 8 ≥
8

1
£ 3≥ BC
1

8 a AB



26

6
£ 1≥
4

c r

2k

1
£1≥
0

h 1 4, 5, 62

26

e r. 1i 3j 10k 2 87

d 3.16 units.

e ¢

722
6

c ii

c 2x 3y 4z 4 0

b (8, 5, 13)

˛

b 16,

d i 18, 20, 12 2

d 10, 13, 182

c 0.716

b c 2. Line of intersection.

14 a r 12i 3j 7k 2 t 13i j 3k 2

c 43.6°

1
2
£ 1 ≥ – l£ 2 ≥ b 3x 2y z 5
1
1

3 a ii r

13 12
, ≤
5 5

7 r

z 6 2l

ii 79.0°

˛

e 16.4°

5 a ¢1,

d 8.52

y 3
2 x

z 8
2
4

b

˛

13
≤
3

y 1 l

b i n1 6i 3j 2k and n2 2i 2j k

11 a r 12i j 4k 2 l13i j2

15 b

4 5
, ,
3 3

c ¢

e x 2 l

51
, 4≤
2

d ¢ 33,

0

1 10 8
b ¢ , ,
≤
3 3 3

1

26
2

c

4 2y
1 z
6 2z

BD has equation 2x 4
4
9
7



13

3
0
f r £ 13≥ l• 13 μ
18
26

3

13
28
, 0, ≤
3
3

7
£ 0 ≥
4

13 r

e i i j k

b x 2y 9

c ¢

7
l£ 1 ≥
5

ii PO i 2j 4k

37 4 9
, , ≤
5 5 5

iii

12
214

d (6. 63, 3.50, 4.65)

16 (6.87, 3.99, 4.15) or (4.53, –0.694, 4.15)

Chapter 14
1 5x

2 10x

Exercise 1

3 2x

4 2x2

Chapter 14
1 x2 x c
9

2 32
x c
3
˛

22 y

˛

1 4
x c
4

17 3x 2 c

˛

˛

28 y

˛

32 94
k c
9
˛

7 y 16x 46
˛

1 4
x 2 ln冟x冟 c
4
˛

1

12 2x2 c

13

˛

˛

5

˛

˛

˛

30 y t 1 3t 2 3t 3 c
˛

˛

25 y

˛

˛

˛

8 x 1 c

˛

20 y 8x

˛

˛

5 3
x 4x c
3

˛

15 x4 2x2 9x c

˛

2 32
1
x 2x2 c
3

19 y

7

14 2x2 2x 2 c

˛

1
5
24 y x 2 x 4 c
2
4

1 6 1 2
z z c
6
2
˛

˛

˛

˛

˛

29 y

6 2x4 2x2 3x c

2 7
x x5 c
7

1
18 y x 4 c
2

˛

1
3
4
818
8 y t7 t5 t3
7
5
3
105
˛

˛

9 23
x c
2
˛

21 y 4x4 12x2 c
˛

8 72
14 32
x
x c
7
3
˛

26 y

˛

31 y

˛

14 194
3
x 2x2 c
19
˛

˛

2 12
4
t t3 c
3
9
˛

˛

˛

˛

4 y x2 5x

9 Q

˛

2 72
p
21
˛

8 112
p
33

5 y x4 2x3 7x 3
˛

˛

6 y

509
4 3
x 6x 1
3
6
˛

˛

2

˛

Exercise 4
2 4ex cos x c
˛

6 5ex 2 cos x 3 ln冟x冟 c
˛

˛

˛

3 y 4x2 3x 18

˛

Chapter 14
1

˛

Exercise 3

2 y 2x2 3

1
2

10 4x 1

9 3x3

˛

5 2x3 5x c

˛

˛

Chapter 14
1 y 6x 4

8 x5

23 y 3x3 12x2 16x c

˛

27 y 6p 2 c

˛

1 5
x c
5

˛

2 3 21 2
x
x 27x c
3
2
˛

4

˛

11 7x 2x 2 c

1

˛

16 x x2 2x3

7 x4

˛

1 4
x c
4

3

˛

10 3x3 c

˛

6 x3

˛

Exercise 2
1 3
x c
3

2

˛

5 6x2

˛

7

3 5 ln冟x冟 sin x c

1 x 5
e ln冟x冟 7 cos x c
3
2
˛

8

4 6 cos x

6 5
x c
5
˛

ex
3
10x2 sin x c
15

5 8 cos x 7ex c
˛

˛

705

Answers

Chapter 14

Exercise 5

1
1
1
1
5
1 6x
1 cos 5x c 2
sin 6x c 3 cos 2x c 4 2 cos x c 5 2 sin 4x c 6 2 cos 3x c 7 sin 2x c 8
e c
5
6
2
2
2
6
1 5x
4 6t
5
1
1
9
e c 10 e4x c 11
e c 12 e6p c 13 4x2 e2x c 14 2e 2x c 15 y ln冟2x 3冟 c
5
3
6
2
2
1
1
1
1
16 y ln冟8x 7冟 c 17 y 2 ln冟2x 5冟 c 18 y
13x 1 2 6 c 19 y
14x 72 7 c 20 y 14x 32 2 c
8
18
28
8
˛

˛

˛

21 y

˛

1
13 2x2 5 c
10

Chapter 14
2 38

16 0.490

4 0

17 0.0429

5

2
3

17

4
3

18 9

64
3

2 61

4 70.4

4
3

1
6

10 1.60
17

256
3

3

˛

2
ln冟3x 2冟 c
3

7 201

8 216468

20 3.47

19 1.85

9 1490

10

2
3

11 0

12

1 3p
3

14 200

13 21

21 8.56

22 0, because both functions are odd 23 2 ln 冟 2k 1冟

8 0.619

9 22

15 312.6

6 6.39

7 5.55

2 3p 5
ln 2
2
3
8

21

1
(1 – e
2

22

2p
˛

)

10 1.69

11 0.825

1 3
p
3

23 k 4

9 23

10 408

˛

12 2.19

13 27

14 10.4

15 2

16 2.27

24 a 1

4

85
4

253
12

5

6

2401
16

863
6

7

8 22.1

11

21
4

12 2

13 6

14 15.3

Exercise 9

1
2

11 1.85
18 18

˛

Exercise 8

3 16

1
8

2

8
3

5 2

20 1.48

Chapter 14
1

˛

Exercise 7

Chapter 14
1

˛

18 0.549

3 50
19

32 2x3

6

Chapter 14
1 32

˛

Exercise 6

3 20

343
2
6

˛

3
4
23 y 12t 12 1 c 24 y ln冟3x 1冟 c 25 y 2 ln冟3x 5冟 c
2
3
3 4x
1
1
28
e c 29 cos 3x 2x2 c 30 e 8x 2 sin 2x c
2
3
2

27 y 3 ln冟6 t冟 c

1
1
ln冟2x 1冟
13x 42 6 c
2
18

1 3

˛

2
22 y 13x 22 2 c
3

26 y 8 ln冟4 p冟 c
31

˛

4

1
3

12 3.08
19

125
6

5

13

1025
3

Chapter 14

5
2

6

407
4

14 3.92

20 3.62

7 36

15

21 4.53

8 6.43

8
3

16

9

y

160
3

32
3

0

x

22 3.21

Review Exercise

4 3
1
1
3
1
1 a
x 7x c
b 3x3 2x2 5x c c 4x 2 c
d 13 2x2 3 c
2 a y x 4 c
b y p3 p8 c
3
6
x
2
8
1
3 2 1 12
3 2
2 c y t t c
3 y x 8x 2
4 a 4ex cosx c
b 7 sinx 4 ln冟x冟 c c
e6x 5 ln冟x冟 4 cosx c
8
2
2
3
5 a 3 sin2x c
7 a
10

382
25

407
4

b

b 2e2x c

1
1
p
2
4

11 a 0.753

c 3 sin2k
b 2.45

Chapter 15
1 cos¢u

3p
≤ c
4

1
3 12p4 12 2 14
16
1
13
3 16e2a 7 2 2 14
24
7

˛

˛

706

1
ln冟4x 3冟 c
2

c

8 a e5 e2

c 1.78

12 a

9
2

1
7 3x 1
e
13x 22 7 c
e 13x 42 4 c
21
3
3
3
2
b 36 c
9
ln12p 5 2
3
2
d

b 24.3

13 a 13.3

b 1.93

5

6

4x2
3
c
5
x

14 30.2

Exercise 1
1
p
sin¢3x ≤ c
3
4

2

3

1 32x 7
e
c
32
˛

4 2esin 2 2
˛

5

1
ln冟8x 9冟 c
4

6

1 6
1x 9 2 9 c
54
˛

1
1
1
1
3
3
1
11 x2 2 2 c 9
14x2 3 2 2 c 10
13 tan x 4 2 4 c 11 1x2 1 2 2 c 12
31 12 cos 0.5 12 5 4
3
12
12
10
3
1
1 2
3
14 1cos 2x 12 2 c 15
1x 2x 42 6 c 16 ln冟x2 3x 5冟 c 17
c
3
12
21cos x 82 2

8.

˛

˛

˛

˛

˛

Answers
18

1
ln冟2ex 4冟 c
2

23

15
128

1
ln冟3 sin x 12冟 c
3

19

˛

1
ln冟3x2 3x 4冟 c
3

24

Chapter 15
1
8
14

25

˛

x
1
tan 1 c
3
3

2
5
13x3 6x 19 2 2 c
15

20

1
c
913x2 3x 42 3

2

tan 1¢

1
ln冟2p2 2p 5冟
6

28

˛

˛

Exercise 2
2 sin 1

x
c
5

2x

3

27

≤

3 cos 1

15 4 sin 1¢

c

Chapter 15
1 sin x

27 ln冟e2x 1冟 c

26 1ln冟p冟 2 2

1
ln冟3 tan 2x 7冟 c
6

22

˛

x
c
6

4 3 tan 1

x
c
3

2x 3
≤ c
7

x

5 2 sin 1

222

23 1
1
x 1
sin x23 c 9 sin 1 1x 12 c 1 0 tan 1 a
b c
3
8
2
27

1
7
11 3 cos 2x2 2 c
21

21

˛

p23
p23
23
tan 1

3
3
18

6

18 sin 1¢

17 0.0623

p 3
23

1
1
2 cos 2x cos3 2x c
2
6

3 cos x

2
1
cos3 x cos5 x c
3
5

1
tan 2p p
2

4

sin 4x
x

c
2
8

5

x
sin 4x
1
4
6
4
1

c 7
112x 8 sin 2x sin 4x2 c 8 cos x cos3 x cos5 x cos7 x cos9 x c
2
8
32
3
5
7
9
1
1 3
1
1
1
1
1
14x sin 4x2
c
10
11
tan4 x tan6 x c 12 cos3 2x
cos5 2x c 13
sin p sin5 p
32
4
6
6
10
3
5

Chapter 15

6

c

5

12 7x2 4

2

28

3 21 2x c

1
ln冟x4 3冟 c
4

23

1
3
11 x2 2 8 c
12

29

1
ln冟x2 2x 3冟 c
2

Chapter 15
12

x 1
22

3

2

15

22

213x

7 tan 1 x2 c

8

˛

1

41p 22 2

˛

16

2627
3

1x 52 2

13

2

1
tan 1 12 tan x2 c
2

3

4 2 2 19x

˛

˛

1
x
1sin 2x 32 5 c 22 2e1 cot 2 c
10
1
1
124x 8 sin 4x sin 8x2
27 21ex 2 2 2 c 28
64
21

˛

˛

1

32 21 x2 4x 5 2 2 c

33

˛

17

c

1 x sin x cos x c

2
3
1x 22 2 13x 4 2 c
15

4

382

5

c

1
1sin2 x 3 2 5 c
5

1p 22 18p 12 7

12p 12 2 1p 12 2
3

4

c

9

1 6x
c
2412x 12 3

10

20

3

151x 52 75 ln冟x 5冟
3
(sin 1 x
2

p
1
1
1
p cos ¢sin 1 ≤ 2 sin 1 cos ¢sin 1 ≤
2
2
4
2
4

20

x2 )

4x21

1
25

˛

c

125
c
x 5
18

215p 22 2 115p 4 2

14

375



11

1 5x
c
101x 3 2 5

416 23
125

1
x
tan 1¢2 tan ≤ c
2
2

tan 1 1 25 tan x2 c

21 3tan

1

1
5 tan p
£
2

3

≥ –3tan

1

3
4

4

8
tan 1 2xn 1 c
3n

23

˛

Chapter 15

Exercise 6
2

e2x
12x 12 c
4
˛

6 x2 cos x 2x sin x 2 cos x c
˛

p
1
10 tan¢ 2x≤ c
2
3

1
ln冟x2 2x 3冟 c
2

16

˛

26 ln冟ex 2冟 c
x 2
≤ c
3

c

30

15 ln 冟x2 4冟 c

˛

31 2 sin 1¢

13 5x2 2

5

1
ln冟3 4 cos x冟 c
4

9

1 2
1x 6x 8 2 7 c
14

20

135

p

1
tan 2x
32

1
ln冟3x 1冟 c
3

3 21 sin 2x c

˛

c

x2 4x 8
c
x 2

19 2 sin 1

≤ c

1
ln冟6x2 4x 13冟 c
4

3

3

c

1

c

12x 12 1x 8 2
21p 22 2

1
ln cos 3x
3

Exercise 5

1
2

12

8 tan 1 2x c

1
c
21cot x 32 2

25

30 22 tan 1¢

˛

14

3
tan 1 2x c
4

19

1
24 cos4 x c
4

˛

2x
c
ln 2

13

˛

18 sin 1 3x c

˛

˛

1
tan2 3x
6

3

11 2x2 2
3

c
2
3
412x 1 2

4

3
p
sin¢4x ≤ c
4
2

7

1 4x 1
e
c
4

12

17

6

3

c

2
1
3
1
11 x2 2 211 x2 2
c
3
1 x

1x2 32 6

9

Exercise 4

2
11 cos13x a 2 c
3

1

23
≤
3

Exercise 3

1 3
sin x c
3

1x 22 5

1
3x 1
tan 1¢
≤ c
6
2

13

≤ sin 1¢

6

1

7 0.0203

1
x 3
x 2
tan 1¢
≤ c 12 5 cos 1¢
≤ c
3
3
3

11

1 1 x 1
sin ¢
≤ c
3
2 23

16

c

7

x5
15 ln x 12 c
25
˛

3

e2x
12x2 2x 1 2 c
4
˛

˛

8

4

x cos 2x
sin 2x

c
2
4

x3
13 ln 3x 1 2 c
9
˛

5

1p

12 10
110

9 x3 ln 8x
˛

1 10p

x3
c
3
˛

10

4608
55

12

e 3x
19x2 6x 2 2 c
27
˛

˛

707

Answers
11

ex
1cos x sin x2 c
2

16

e2x
12 sin 3x 3 cos 3x2 c
13

12 x sin 1 x 11 x2 2 2 c
˛

˛

1 2p

20

12

1 np
12 1n

n

1

21n

1

3
x
ln冟x2 4冟 2 tan 1 c
2
2

2

˛

˛

Chapter 15

11

x2 8
36

17

7 2ex

c

2

x

˛

5

c

˛

12 x

c

1

17

˛

8

24

e 2x
1sin 2x cos 2x2 c
4

28

eax
1a sin 2x 2 cos 2x2 c
a 4

21

˛

˛

38

1
tan3 x
3

41 ln2

1 3
sin 2x
6

3x2
2
˛

35

c

1
tan5 x
5
2x

1x 32

7

4

c

2

˛

11 5 x2

c

x
4

1 3
¢ x
4 2

2
3
13x 52 2 c
9

˛

2512 5x2 2

3

24 3.16

25

708

2 4.48

19x

p
1
tan¢2x ≤ c
2
3

1

c

42

40
x3
3

211
x2
2

˛

˛

x

1

x
2
1

sin

2 ln x

10 2 sin 1¢

c
1
80

325

15 tan

1

4

x
5

ex

≤

1 4x 5
e
c
4

30 1.44

4

˛

7

23

3

˛

˛

˛

12x3
192x12
32x2


c
7
23
3
10

˛

x

c

˛

1
ln冟2x 1冟 c
2

27 x tan 1

32

u
sin 2u

c
2
4

33

x

c

1
1
ln冟x2 1冟 c
x
2
˛

sin 6x
x

c
2
12

sin 4ax
≤
4a

1
¢x
4

37

x2
4

x
4

13 ln 2x
8

1
cosec 4x
4

c

6 0.0791

c

1

p
p
≤ cos¢x ≤ c
6
6

c

c

19 sin 1¢

31

x 2
233

20

˛

11 3a 2
3



1
3

11

43

˛

˛

1

c

27

32 x cos 1 2x

1 a b cos p
2
12 ln2
b
a b

1
12ex 1 2 2 c
4
˛

16 0.142

x
1ln x 1 2 c
ln 4

1x
12x2 70 2 c
25

26

31a 12

≤ c

5

1 1 2x
sin
c
2
5

1
112 8 sin 2x sin 4x2 c
32

15

17 16.5

21 sin 1 1x 12 c

22

x5
15 ln 2x x2 c
25
˛

1
1
tan5 x tan7 x c
5
7

28

1 1
sin 2x 221 4x2 c
2

21 4x2
c
2

34

pa4
16

˛

33 1

˛

˛

35 0.0280

36 1.50

38 0

Exercise 10
3 0.148

5 9.42

6 214

1

x2 2 2
˛

7 0.227a2
˛

10

4 e3
2
˛

11

c

˛

2 tan

˛

12

5p

101p

23 x sin¢x

x2 2 2

2 x1

1

14

5

˛

3
tan 1 x2 c
2

1
tan4 x c
4

2
x
tan3
3
2

36

c

c

26 x ln冟2x 1冟 x
31

x2
3x 10 ln冟x 3冟 c
2

4 2 tan 1 2x c

19

˛

1
x 2
tan 1¢
≤ c
2
2

8

82

e3x
13x 12 c
9

22

c

˛

Chapter 15
1 0.215

135

1
x
18 cot¢sin 1 ≤ c
4
2

3 2x
sin ≤
3
4

e 3x
1sin x 3 cos x 2 c
10

x
tan 1¢25 tan ≤ c
2
25

1
37 ln冟cos 2x冟 c
4

5

4
x
cos5
5
4

22 ≤

3
x 3
tan 1¢
≤ c
4
4

14

2

29

x
3

˛

3

422

413x

13

9 ln冟x2 1冟 3 tan 1 x c

8 0.169

˛

23 2.44

c

1
30 cos3 x c
3

8
x
cos3
3
4
6 sin

3x2
3x
3
cos 2x sin 2x cos 2x c
4
4
8

18

x

4 2 ln冟x2 4x 8冟

Exercise 9

e 2x
12x2 2x 12 c
4

13

˛

7 x 7 ln冟x 4冟 c

˛

x
x
x
8x sin 16 cos c
2
2
2

2214
214
tan 1¢
13x
7
14

32

4

4 cos

39

c

Chapter 15
1

˛

2

˛

c n =/ 1, ln cos x + c for n = 1 9 tan

25

5

1 3
sin x c
3

29

˛

25

≤ c

1
ln冟3x2 1冟 c
3

3

x

1
1ln冟x冟 1 2 c
x

25 2x2 cos

2

sin 2x
2

1

x
223
tan
tan 1£
2≥ c
3
23

22
x 2
tan 1
c
2
B 2

˛

1
12 cos n

1n

10 ln x

5

x 3

Exercise 8.

˛

16 x 11 x2 2 2 sin 1 x c

34

1

˛

1
3
3
18x2 81x 12 2 11 3x2 4 2 c
3

2

8x2

20 2x 2

˛

eax
1a sin bx b cos bx 2 c
a b2

19

˛

6 21 x2 6x 4 2 2 sin 1¢

Exercise 8

sin 3x
cos 2x

c
3
2

11

˛

1
5 23
x23
ln冟x2 3冟
tan 1
c
2
3
3

3

˛

22
x 2
tan 1¢
≤ c
2
22

5 ln冟x2 4x 6冟

sin x

xn 1
3 1n 1 2 ln x 1 4 c
1n 12 2

18

e3x
1sin x 3 cos x2 c
10

15

˛

Exercise 7

1 11 x2 2 2 sin 1 x c

6 221

˛

14 e2x 1x 1 2 c

˛

1

12
22

Chapter 15

1

ex
1sin 2x 2 cos 2x 2 c
5

17

1
ln冟1 x2冟 c
2

13 x tan 1 x

1

˛

212a

3

1 2 2 13a
15

12

12 5.66

13 11.7

14 3.03

Answers

Chapter 15

Review Exercise
3

1k2 4 2 2

3

1 e ek 1

4 x 2 2
¢
≤ (3x
15
2

2

˛

7 a 1.07

8

e 2x
12x2 2x 12 k
4
˛

b y 0

13 a (0,1) is a maximum

Chapter 16
1 y

x3
cos x k
3

120

15

6 y

y2

2x

2x2 ln x

˛

˛

239
240

16

y

˛

1

13

˛

x2 5

˛

12

1
5

1

2x tan

np
1n 1 2
9

c

8 y

˛

1
14x sin 4x2 k 9 y
8

4
13x
135

5

22 2

˛

1
1
ln冟4x2 3冟 ln 19
2
2

14

y=

p
¢ ≤
4

3x

5

˛

x

ln 1

k

3

1
ln 3
2

5
≤
c – 6ln x

1

x2 – 2x tan
˛

3 23
y
2

2x

e
2

1
ln 131
2

3
ln 3
8

4 sin

t

k

3
2

˛

˛

1

x
2

3x

4x 2 + 3
1
ln ¢
≤
2
19

e
2

15

t

k≤

12 ln y

1

ln 4

ln x

sin x
1

11

t2 2 2

k

k2

˛

2x

˛

cot y

ln cos x

1

31t sin

˛

ln ¢

4 y

k

8 s3

k

11 y

1
2e2

3
ln x2
2

2x

˛

k

˛

u3
3

17

1
ln 4
3

2y

7 ln y

sin 2at
2a

˛

14

1

˛

Chapter 16

e2t
12 sin t
5
˛

p3
24

cos t2

1
5

1

18 2 sin

3s

3( t2 – 1) + p
˛

Exercise 3

1 y ln冟cos x冟 k

2 x

2t3
kt c
6

3 x

˛

dy
dz
1
tan 1 z x k
dx
dx

6 a

1
1
7 y 11 15x2 2 2 k
3

13 y

e

10 v2

c

ln 4

3
ln 4x2
8

3

6p
9

iii

12 0.690

4 y x cos x sin x k

˛

˛

1
ln x
2

y

6 y = ln ¢

k

˛

5

1
ln 2y2
2

16

2 tan

110

13 ln y

4p
9

ii

2
3
3 y 11 x2 2 2 k
3

k

˛

dx

12 10 130x

13x

y

9

k
x2

˛

2p
9

a
11 ln13 b sin x2 k
b

Exercise 2

ln cos x

5 y4

1
x 3
arctan¢
≤ k
2
2

10

x 2
b c
5

13x4
kx2
x4
ln x

cx d
24
288
2

cx2
2

˛

2

1
10

b

15 b i

˛

kx3
6

Chapter 16
1

14 1

3 2kx3
3
e B1 R c
2k
2k

11 y
˛

x cos x – 4 sin x
12x 12 6

13x 7 2 5

2 y

˛

10 y ln冟cos x冟 kx c

15 y

ex
2

6 a 25, b 2, sin 1 a

5 0.307

˛

Exercise 1

5 y ln冟1 sin x冟 k

12 y

p
1
2

d

3 25

9 a y

˛

1
ln冟1 x2冟 k
2

4 x arctan x

˛

3

c

4)

sin nt
kt c
n2

y3
˛

4

3

˛

c y tan1x k 2 x

7 y

ln冟1 x3冟 9
˛

Bx4
Bx3
Bx


24A
12A
24A
˛

5 tan y tan x k

8 V 4 cos ¢t

˛

p
≤ 6
4

9 r 4t ln t 4t 24 20 ln 5

Chapter 16
1

v

4t3
3
˛

t4
3

;

2se2s
˛

D

11 a 2.60

t2
2

˛

s

sin 3t
,s
3

4 a v

7 v

t

Exercise 5
˛

cos 3t
9
e2s
2

11 b v

˛

9

220e

2 a v

3t4
2

10

2 b

11
18

p
seconds
3

5 s

8 v
2.60t
˛

˛

4 b

;

12s
C

c 108m

12 5

6134 ms

223

˛

t

2

12 b

1

2p

b

3t5
10

10t

2 d

14

6 v

B

˛

2 c s

2 ln l

3
9 a 7 ms
4

5
12 a v

t2
2

1

1
1
t

13 v

7

1
1
S 7 ms
2
2

g

g

k

kekt
˛

1

29900 m
2 cos ¢s

4

as t S ∞

, Yes v S

3 26.6m
p
≤
4

10 b

e8

137

˛

4

g
k

709

Answers

Chapter 16
1 a 176

1 b 107

2 d 0.622

9

49p
16

12 a

Exercise 6
1 c 57.4

2 e 113

1 d 8.90

2 f 0.965

1 e 104

2 g 12.8

3

1 f 2060

pa
5

1 g 327

1 h 1.23

1 i 1.08

2 a 8p

1 j 274

2 b 5.98

˛

4 145

5 1940

6 84.2

7 5cm, 196 cm2

8

y
y | x2 1 |

10 p2b2a 11 p3(n2 + (n – 1)2)

1
12 sin 1 x 2x21 x2 2 k
4

p
1p 2 sin 1 a 2a 21 a2 2
4

b.

˛

(0, 1)

˛

( 1, 0)
3.35 ,

Chapter 16
p
1 12
6

v

B

3ln ¢

s2

p 1e8a – 12
˛

3 4m

4

˛

˛

4a

4e

5x 2

1

≤

y = e2

˛

1

•

2T – 1
T

–

p

–

4

1

9 Correct

2

x

10 169.8 years

12 a Correct b 0, 3, 6 seconds c i distance =

t sin
0

p
(p + 2)
14
4

15 a

y
6a

b Vx = 72pa3

0

d 60i

2

d 16i

Chapter 17
3 a 4 19i

Vy
Vx

6

t sin
3

≤, s

3

3
2p

¢t –

pt
36
dt ii
metres
3
p

3
2p

2pt

sin

3

≤

13 a v = v0e–kt b t

32
45

a 12i

b 15i

5 a 9

b 1

c 24i
6 a 9i

d 52i 3 a 240i b 32 c 45i d 96i
3
5
7i
1
b
c i d 1 e 1 i
3
3
2
4

e 72i

f 35

g 90i

Exercise 2

b 18 28i
b 47 35i

c 7 i

d 14 21i

c 7 2i

e 7 11i

d 115 111i

e 306

2 a 3 5i
f a2 b2
˛

˛

b 13 + 22i
g 1 70i

c 20 10i
h 117 44i

d 7 13i
i x2 y2 2ixy
˛

˛

3 i
15 10i
7 26i
3 j m(m2 – 3n2) + in(3m2 – n2) k (m3 – 18m2 – 12m + 24) + i(3m3 + 6m2 – 36m – 8) 4 a
b
c
5
13
29
12x 5y2 i 15x 2y2
x 1y2 3x2 2ixy2
2x2 y2 3ixy
10 33i
3 4i
3
i
4 d
e 6 2i f
g
h
i
j
29
5
2
2
4x2 y2
x2 y2
x2 y2
˛

b 11 2i

6 d x 5, y 12

710

e

˛

˛

˛

5 a 4

1
ln 2
k

Exercise 1

1 a 18i b 38i c 112i
15
5
4 a
b 2 c
i
2
2

1 a 8 16i

c

pt
dt –
3

2pt

¢1 – cos

2p

x

4a

Chapter 17

3

11a v

3

b t = 3 seconds

p
2

5x 2

˛

1
8
ln ¢ ≤ ii 45.3 mins
7 a Correct b i A = 78, k =
15
13

2T

x

y

2
5 y = Ce

1
≤ + 4. At t = 5, v = 23 ln 13 + 4

2

(1, 0)

Review Exercise

x
–1
B2

2 y

8 cos–1 ¢

2 c 0.592

5

c 237 3116i
x 6, y 3

˛

˛

d 8432 5376i
f x

5
14
,y
17
17

6 a x 15, y 7
g x 21, y 20

˛

˛

b x 8, y 0
15
h x y ;
B 2

˛

˛

˛

˛

c x 0, y 3
i x

29
72
,y
25
25

Answers
x 3, y 1

6 j

7 d Re1z2

x

k

33
2591
,y
169
169

61
72
, Im1z2
65
65

e Re1z2

x

c 2.12 0.707i, 2.12 0.707i

1; i23
2

c x

8 d 3.85 1.69i, 3.85 1.69i

3 ; i 251
2

d x

3 ; i 26
3

d x3 3x2 7x 5 0

11 a x2 4x 53 0

b x2 8x 25 0

˛

˛

13 3, 1 i
18 z1
˛

14 a 3, 1 2i, 1 2i
19

Chapter 17
1 a

e x

3 ; i295
4

˛

2xy
˛

3 i23
2

˛

f x4 10x3 20x2 90x 261 0
˛

˛

˛

˛

214 735 i
15
53

3
9
20 p , q
5
5

9 a x 3 ; i

b x2 6x 10 0

˛

˛

d x2 2ax a2 b2 0

˛

b 2, 1 3i, 1 3i

f 0.734 0.454i, 0.734 0.454i

10 a x2 4x 13 0

˛

c x2 14x 85 0

˛

˛

41
117
, Im1z2
145
145

x2 y2
˛

i 0.541 0.0416i, 0.541 0.0416i

e x3 8x2 25x 26 0

˛

26 2i
21 i
, z2
17
17

f Re1z2 0, Im1z2

˛

e 1.92 1.30i, 1.92 1.30i

h 1.59 1.42i, 1.59 1.42i

10 c x2 8x 25 0
˛

˛

c Re1z2

2p
2p
, Im1z2 sin
i Re1z2 128, Im1z2 12823
3
3
xy
9x
8 a 1 4i, 1 4i b 1.10 0.455i, 1.10 0.455i
1 y2 16 + 9x 2

8 g 0.704 0.369i, 0.704 0.369i
9 b x

˛

53
89
, Im1z2
185
185

h Re1z2 cos

12
, Im1z2
16 + 9x 2

y2

1

b Re1z2

2a
12
ab
3a

, Im1z2

4 b2
16 a2
4 b2
16 a2
˛

7 g Re1z2 597, Im1z 2 122
7 j Re1z2

7 a Re1z2 27, Im1z2 8

12 x4 10x3 42x2 82x 65 0

˛

˛

˛

˛

348 115i
17
13

16 8 2i, 8 i

88 966i
25

21

Exercise 3
b

iy

c

iy

d

iy

iy

z2 z3 2 7i
z1 z3 3 i

z3
z1 z3

z1 z4 4 3i

z1 z4 6 i
z2 z3

x

z1

z3

x

z1

z4
z1 z4

z2

z1 z4

x

e

2 a 2¢cos

iy
z3 z4 1 4i

p
p
i sin ≤
3
3

z3 z4

x

z4

4 a 1 i23

b

4 h 1.67 4.03i

215
25
i
2
2

p

b 5e 2.21i

3 a 422ei 4
˛

c 522 5i 22

5 a r 2313, u 0.825

7 vii r = 25, u = –1.33

˛

d 3.74 1.00i

b r 2701, u 1.38

6 c Root 1 r 27, u 0.714 Root 2 r 27, u 0.714
ii r 325, u 1.46

c 253e1.85i

˛

6 a Root 1 r 2.65, u 2.17 Root 2 r 2.65, u 2.17

7 b i r 65, u 1.04

b 28 Bcos¢

2 d 2413cos1 0.8962 i sin1 0.8962 4

z3

x

z1

z4

e

3p
3p
≤ i sin¢ ≤R
4
4

e 101cos 0 i sin 02
d 282e 1.46i

22370, u

f 6¢cos

e 8e0i

˛

322
322

i
2
2

c r

c 2263cos12.942 i sin12.942 4

˛

p
p
i sin ≤
2
2
p

f 2ei 2
˛

f

25
215
i
2
2

0.487

d r 4

g

1523
15

i
2
2

2370
, u 1.41
5

b Root 1 r 25, u 1.11 Root 2 r 25, u 1.11
7 a i z1 13e 1.18i
˛

iii r 26, u 0.129

˛

ii z2 5e2.21i
˛

iv r 2.5, u 1.17

˛

iii z3 25e 0.284i
˛

˛

v r 5, u 2.50

p

iv z4 2ei 3
˛

˛

vi r = 6.5, u = –2.22

viii r 150, u 0.763

711

Answers
8 a

and

c Rotation of

b

iy

iy

27 3cos1 0.7142

9 a z1
˛

x
z2 2 3i

p
radians clockwise.
2
i sin1 0.7142 4 z2

x

z3

9 b 冟z1z2冟 214, arg1z1z2 2 3.07, 2

z4

˛

˛

˛

22Bcos ¢

˛

˛

3p
≤
4

i sin ¢

3p
≤R
4

z1
z1
7
2
, arg¢ ≤ 1.64
z2
B2
z2
˛

˛

˛

˛

z1 3 5i

10 z1
˛

23
7
1
4

i, z

i
53
53 2 13
13
iy
10 67i
˛

14 25i

1 i23 1 i23
,
2
2

11 b z 1 0i,

106z1 39z2

ii 10

14 a 13 cos10.8412 i sin10.8412 4, 13 cos1 0.8412 i sin1 0.8412 4
3
15 a r , u 1.23
2

106z1 39z2

12 b i 16

15 b

b 1.68 radians

c No.

x
3

1 2
4

Chapter 17

Exercise 4

1 a 10241cos 10u i sin 10u2
1 f

b cos 25u i sin 25u

1
1
Bcos¢ u≤ i sin¢ u≤R
3
3
24
1

3

c

1
3cos1 5u 2 i sin1 5u 2 4
243

p
p
h cos¢ ≤ i sin¢ ≤
2
2

g cos 0 i sin 0

c 61cos u i sin u 2 3

1

1
1
e cos u i sin u
2
2

d cos1 9u 2 i sin1 9u 2

i cos¢

3p
3p
≤ i sin¢ ≤
4
4

d 1cos u i sin u 2 4
1

j cos¢

e 1cos u i sin u 2 2

p
p
≤ i sin¢ ≤
10
10

f 1cos u i sin u2 8
1

2 a 1cos u i sin u2 7

b 41cos u i sin u 2 2

3 a cos 8u i sin 8u

5
5
b cos u i sin u c cos 3u i sin 3u d cos u i sin u e cos 11u i sin 11u f cos 3u i sin 3u
2
2
3
3
3
3
p
p
h cos p i sin p i cos p i sin p j cos
4 a 2 3i, 2 3i b 0.644 1.55i, 0.644 1.55i
i sin
4
4
4
4
12
12

1
1
3 g cos u – i sin u
6
6
4 c

1

26 Bcos¢

3p
3p
p
p
1
≤ i sin ¢ ≤R, 26 Bcos ¢ ≤ i sin ¢ ≤R,
4
4
12
12

4 e 1.456 + 0.347i,

0.347 + 1.456i, 0.347

4 f 0.132 1.67i, 1.62 0.389i
1

4 g 26 Bcos¢

p
p
≤ i sin¢ ≤R,
36
36

5 a

1

26 Bcos¢

1

g 26 Bcos¢

1

712

2
3
2 1
3

x

0.347i

7p
7p
≤ i sin ¢ ≤R
12
12

d

1.69 0.606i, 0.322 1.77i, 1.37 1.16i

f 1.54 0.640i, 1.09 1.27i, 0.872 1.42i,

35p
23p
35p
23p
1
≤ i sin¢
≤R, 26 Bcos¢
≤ i sin¢
≤R,
36
36
36
36

1

26 Bcos¢

11p
11p
≤ i sin¢
≤R,
36
36

13p
13p
25p
25p
1
≤ i sin¢
≤R, 26 Bcos¢
≤ i sin¢
≤R
36
36
36
36
c
1

2
3

1.456

b

iy

1

1.456i,

1

26 Bcos ¢


3

3
1


3

1

3

3
3

1

d
1


1
1
4 4
4
4


1
1
4 4
1
1
4 4
1

1

1

2
2 9 2
1
9
9
2
1 2
9 1
9
2
2
9
1 9 2 2 1
9
9

1

1

Answers

p
p
6 a 16Bcos¢ ≤ i sin¢ ≤R
2
2

7 b Real part is –215 Imaginary part is 215 23
11 a z2 53 cos1 0.9272 i sin1 0.9272 4
2p
2p
i sin ≤
7
7

˛

z41 16Bcos¢
˛

15 c

9 b tan¢
4
25

b

˛

15 a z1 2¢cos

b z21 4¢cos
˛

15p
7p
p
9p
≤, tan¢ ≤, tan¢ ≤, tan¢ ≤
16
16
16
16

4p
4p
i sin ≤,
7
7

z31 8¢cos
˛

15 d Rotate

8 Real part

˛

–2.06
˛

˛

x y
2

2

˛

˛

2 k 2

˛

˛

3 a 2, b 5

, Imaginary part

˛

15
6
1
4 6
z
z2
z
˛

5 b 416

9 23

˛

x2 y2

ii 223

2 A B
2
4 3
2
3

4 4

x2y y3 y
˛

iy

C

2p
2p
≤ i sin¢ ≤R,
7
7

z71 1281cos 0 i sin 02
˛

c cos¢

d a

˛

2p
2p
≤ i sin¢ ≤
3
3

1
3
15
5
,b
,c
,d
32
16
32
16

Review Exercise

x3 xy2 x

10 c i

˛

p
p
p
p
16 b cos¢ ≤ i sin¢ ≤, cos¢ ≤ i sin¢ ≤
3
3
3
3
17 c z6 6z4 15z2 20

Chapter 17
214.8, u

z61 64Bcos¢

2p
anticlockwise. Enlargement scale factor 2.
7

˛

1 r

6p
6p
i sin ≤
7
7

˛

128

1
23
10 i
2
2

12 Product 4, Sum 2

6p
4p
6p
4p
≤ i sin¢ ≤R, z51 32Bcos¢ ≤ i sin¢ ≤R
7
7
7
7

2
2
7
7
4
2
2 8
2
7
7
16
2
2
7
7 2
7
32
64

p
p
7 a 2Bcos¢ ≤ i sin¢ ≤R
3
3

b 1.85 0.765i, 0.765 1.85i, 0.765 1.85i, 1.85 0.765i

6 x

47
1
,y
65
65
p
3

10 a 冟z冟 2, arg1z2

˛

iii 223

3
2

11 c cos 0 i sin 0, cos

11 b 1,

7 x3 5x2 10x 12 0
˛

˛

b 冟z2冟 4, arg1z2 2
˛

˛

1 i23 1 i 23
,
2
2

2p
2p
2p
2p
i sin
, cos¢ ≤ i sin¢ ≤
3
3
3
3
1
2
3

x

2
3 1
2
3

1

11 d Each side has length 23. Area of triangle
13 d i a 32, b 32, c 6
16 a i 1

19 a 1,

ii

2p
3

c

3
3 23

i
2
2

1 i23 1 i 23
,
2
2

21 c 1 23, 2232, 1 23, 2232

ii 6

5
16
12 a a , b
3
9

14 z 5 i, 6 i

17 z
3
c £0
0

323
4

5,
0
3
0

p
3

0
0≥
3

arg1z2

2p
3

p, 2

b a

17
31i
17
31i

,b

4
4
4
4

15 a i cos3 u 3 cos u sin2 u i 13 cos2 u sin u sin3 u2
˛

Re1z2

d x 1, y z 2

5
2

10k i 1k2 21 2
˛

18 a

˛

k2 49

21 a 2 i, 2 i

c

23 22
20

b k ;221

˛

b

iy
B
2 i
0

x
2 i
A

713

Answers

Chapter 18

Exercise 3

Answers are given when asked to form a conjecture
2n 1
≤
2n

n

2

a 3r

r

a 4r 7 n 12n 52

3

72

˛

1

Chapter 18

n

1
n 13n
2

2

˛

Chapter 19

6 n2 2n 2

7 n2 4

˛

˛

Exercise 1

n
≤
1 n 1 a Continuous
mode 4., median 4, mean 4.15 3 mode Blue

n 1
n

16 Mn ¢

5 16.7

6 a 1.58m.

25000 32000 40000 45000


x 18

x 26

x 34

x 42

x 50

x 58

x 66

x 74

x 82
10
18
26
34
42
50
58
66
74
Age

Weight
5

65000

Heights of boys

6 a

Weights of eggs

44 49.5

8

57.5

69

74

Heights of girls
Grade 7

160

168

176

192

133

145

151 157

133

145 148 151

169
Grade 8

714

16
14
12
10
8
6
4
2
30.5 x
31.5

Frequency

Frequency

20

Swedish
British
American
Norwerian
Danish
Chinese
Polon
Others

Frequency

30

10

Salaries of teachers

144

7 0.927

Weight of bags of nuts

3

40

Nationality of student

7

2

b Mean 1.79m

Age of members of a golf club

2

100
90
80
70
60
50
40
30
20
10

4

d Continuous

Exercise 2

Nationality of students

1

c Continuous

b There is no information about the ages or gender of the students.

8 a People below this height are not allowed on the ride.

Chapter 19

b Discrete

29.5 x
30.5

c 78.2

˛

28.5 x
29.5

4 b 81-90

18p

Review Exercise

5 sum of the first n odd numbers n2

8 1n 12! 1

4 Any value

˛

r 1

27.5 x
28.5

˛

26.5 x
27.5

1
1 Dn ¢
0

163

b 21.0

Spanish test marks

8

16

21

27

30

Answers
9 median = 13,

10 estimate median = 25
Age of children
at a drama workshop

Age of mothers giving birth
160
140

50
Cumulative frequency

Cumulative frequency

60

40
30
20
10

120
100
80
60
40

11 12 13 14 15 16 17
Age

20
18 22 26 30 34 38 42
Age

11 estimate Q1 12, Q2 21, Q3 24
˛

˛

˛

12 estimate Q1 20, Q2 20.5, Q3 21.25, 10th percentile 19
˛

˛

˛

13 estimate Q1 35m, Q2 42m, Q3 48m, 35th percentile 38m, 95th percentile 54m
˛

˛

˛

Chapter 19

Exercise 3

1 a Q2 10, IQ 8

b Q2 59.5, IQ 6.5

˛

c Q2 67, IQ 39

˛

˛

d Q2 176, IQ 89
˛

e Q2 40, IQ 3
˛

2 The two sets have a similar spread as IQ 3 for both sets. The average age for set B is less as the median is 18 and the median for set A is 19.
3 Box and whisker plots. Medicine IQ 1, median 2. Law IQ 2, median 3. Law students rated the lecturing higher but there was a greater
spread of opinion among this group.
Rating
Medicine
1

2

3

5
Law

1

2

4 a 2.28

3

4

b 3.74

5

c 33.4

d 5.50

e 9.17

5 Mean = 19.01, standard deviation = 6.28, estimated standard deviation = 6.31
6 x 501, variance 2.61
7 a Q2 = 105, IQ = 16

b x 110, s 16.2

Chapter 19

Exercise 4

1 a i Q1 8.3, Q2 9.9, Q3 11.9
˛

˛

˛

ii x 9.8, s 1.79

1 c i Q1 34000, Q2 45500, Q3 57250
˛

2 Graph IQ 3

˛

˛

b i Q1 183, Q2 263, Q3 298
˛

ii x 44500, s 14300

˛

˛

ii x 238, s 60.2

d i Q1 0.62, Q2 0.755, Q3 0.845
˛

˛

˛

ii x 0.724, s 0.189

3 Daniel Graph, median 185.5, range 167 Paul Graph, median 198.5, range 71

4 x 40.2, s 29.1, New mean 46.2, s 33.5

5 x 16.6, s 1.44

715

Answers

Chapter 19
1 a Continuous

Review Exercise

b Discrete c Discrete
Height of students

5

d Continuous

2 Red

3 x 76.0, s 16.4

4 1.49

6 a

b 17.5

c 40.9

100
12

33.5 39.5

51

60

Frequency

80
60

9 x

1.70 x
1.80

1.60 x
1.70

1.50 x
1.60

1.40 x
1.50

1.30 x
1.40

1.20 x
1.30

20

£264, s

Chapter 20
1
4

1
2

b

c

b 0.183

13 a 156

b x 44 minutes

13
15

11 a 0.1

d

13
15

b 1

£264, s

x

£46.70

b IQ 11

12 a 31.3

£41.70. After bonus, x

£296, s

b 9.84

£46.70

Exercise 1
1
2

2a

1
10

b

1
2

c

3
10

d

2
5

1
4

3a

b

13
20

e
c 0.5

2
3

1
4

9 a
12

1
10

5
8

b
13

1
4

c

1
8

d 1

14 0.72

3
4

e

1
3

15 a

c

9
10

d

13
20

e 0

4 a 0.48

5 No. P 1A´B2 1

4 b Because the probability of either a novel or a mathematics book is 1.
8 c

£296, s

11 a median 135

10 a 29.9

b x 1.63, s 1.51

˛

£41.70. After bonus, x

Height

1a

8 a Q2 1, IQ 1

7 estimate Q1 = 18, Q2 = 24, Q3 = 28

40

6 a

˛

f

1
2

g 0

b

13
18

c

10 a

1
9

5
7
d
18
9

e

15 f Because it is not possible to have one die showing a 5 and for the sum to be less than 4.

5
6

b

c

7
11

b

922
1155

7

1
2

1
6

d

1
6

1
9

f

e

1
4

8 a

1
5

g

2
3

b

9
15

7
18
16

23
32

9
1
b
c 1
16
2
18 a Events X and Y are not mutually exclusive because 2 fish of type A and 2 fish of type B fit both.
17 a

18 b Events X and Z are not mutually exclusive because 2 fish of type A, 1 fish of type B and 1 fish of type C fit both.
exclusive because the event Y does not allow a fish of type C and event Z does.
1
83
19 0.15 20 a
b
500
25

Chapter 20

Exercise 2

1 a

5
42

b

3
7

2 a

4 a

1
14

b

1
6

c

9
15

1
6

2
2
5 a
7
5

7 b No, because P 1A´B2 1
˛

b

c
b

8

1
2

14
17

1
2

3 a
c

9 a

1
2

7
16

b

6 0.0768

1
4

c
7 a

b

2
3

2
7

d

1
2

181
208
c

1
2

D
C

CD

B
A
A

716

AB
B

BC

BD

AC
C

AD
D

c Events Y and Z are mutually

Answers

Chapter 20
1 a 0.3

b 0.5

Exercise 3
1
6

2 a
1
6

5
6

3
1
6

5
6

3

b 0.12
0.6 W
0.2

R

0.7
0.8

c 0.32

R
W

d 0.3

H

(R W) 0.24

1
2

1
2

e 0.176

2
3 H
1
2

1
2

2
3

H

1
2

T

H
T

(T T) 1
4

5 d

5
12

e

2
3

H

1
3

T

1 a 0.974

b 6

3
4

1
2

6 a

6 c

56
121

7 a

8 a

108
455

b

1
(T T T) 12

13 a 0.2

T

1
5

c

b 0.44

9

5
29

11

41
55

c

2
3

1
6

b

15 a

c 0.25

d 0.12

b 0.34

2
15

12 a

14 0.521

(C C) 49
121

C

d 0

a

49
121

28
(V C) 121
(C V) 28
121

V

7
11

c 0.652
2
5

(V V) 16
121

C

3
11

b

128
455

2
5

b

1
22

b

4
11 V
7
11
C

V

7
11

10 a

3
4

7
8

d

4
11

16 a 0.03

Chapter 20

1
4

c

4
11

1
(T H T) 12
(T T H) 2
12

f

(H T) 1
4
(T H) 1
4

1
2

1
(H T T) 12
(T H H) 2
12

1
3

H

T
1
2

3
8

b

T
1
T
3 2
3 H

1
2

1
8

4 a

1
2

e

1
4 ( 4 4) 4

1
2

1
(H H T) 12
(H T H) 2
12

T
H

4

(H H) 1
4

T

(H H H) 2
12

1
3

H


4

b

1
2 H
1
2
T

5 c

1
2

1
2

1
2

5 a

(R W) 0.08
(R W) 0.56

1
2
4 (
4
4) 1
4
1
2
4 (
4 4) 1
4

4 ( 4
4) 1
4

d

1
2

1
2
0.3

25
36

(R W) 0.12

0.4
W
W

c

(3 3) 25
36

3

3 a

1
36

5
(3 3) 36
(3 3) 5
36

3

5
6

3

b

(3 3) 1
36

3

e
b
b

21
55

7
15

f

8
15
2
9

c

3
4

1
2

17 a 0.815

b 0.149

Exercise 4
c 6

9 0.84

10 a 0.222

1
13 a
16

1
b
64

2 a 0.832
b 0.074

c 19

b 24

c 0.144

14 a 0.0191

3 0.985
d 6

4

3
4

5

11 a 0.00877

b i 0.00459

ii 0.0255

2
3

6

45
53

b 0.322

7

32
45

c 0.632

1
15 a
27000

8 a i
d 0.561

b 90

c

11
609

ii 0.0212

12 a 0.3

iii 0.0153

b 0.35

b 0.0839

c 0.075

d 0.3

1
900

717

Answers

Chapter 20
1 720

Exercise 5

2 16065

b 5040

3 24

11 c 2520

16 a 9000000
21 a 60

4 60

12 a 40320

b 30

9 c

1
30

b 5040

22 a 28

b 28

c 35

7 241920

c 20160, 2520

13 a 2494800

17 a 831600

23 a 120

8 2177280

b 176400

b 90

9 a 36

b 635040

c 151200

24 756756

25 70

b 6

c 12

14 a 5040

18 119
26 210

d 24

10 70

11 a 90720

b 4320

c 720

15 a 36

19 38760

20 a 4

27 3185325

b 48

c 24

b 10

d 256

28 27

Exercise 6

1
1
1
1
2
7
2
2
c
d
2 a 362880 b i
ii
c
3 a 15 b
4 a 360 b
2
5
5
9
9
189
3
3
5
12
1
1
3
3
1
c
6 a 10440 b
c
7 a 16 b i
ii
iii
8 a 3150 b
39
29
15
2
8
16
30

1 a 100
3
20

6 604800

16 b Increases by 16000000

Chapter 20

5 b

5 4838400

b

10 a 3628800

1
30

b

Chapter 20

7
15

c

8
15
1
c
45
c

5 a 3838380
d

1
6

9 a 720

b

1
15

3
28

d

Review Exercise

1
1
1
b Events are not independent since P 1Black 2 P 1Brown2 and P 1Black ¨ Brown2
6
3
6
Events are not mutually exclusive because P 1Black ¨ Brown2 0 Events are exhaustive since P 1Black ´ Brown2 1

1 a

˛

˛

˛

˛

2 a 0.984

b 7

7 b 90720

3 a 151200

c 362880

13 b 216

14 0.048

5
16 b ii
6

7
iii
25

3n 1
2

18 a

21 a 0.581

b

8

b 10080

19
30

n 1
3n 1

b 0.0918

8
15
1
c
56

9 a 15

15 a 20160
28
iv
51

˛

1
4 a
6
b

1
4

b

17 a 252
20 a 0.549

b 0.0670
4
15

c

16 a

b 196

1
5 2n 2
¢ ≤
c
6
6

10 0.888

11 a 252

1
2

c 0.439

3 a

X

c 0.0663

X

0

P 1X x2

0

1
1
55

2 3
4 9
55 55

X

12
9
42

13
10
42

14
11
42

˛

P 1X x2
˛

5 c k

1
225

Y
P 1Y
˛

718

e 2

b

4
3
36

˛

2
3
8

d 0.8

3
2
36

P 1X x2

4 d

c 0.65

2
1
36

3 d X

4 a

b 0.6
1
3
8

˛

d 32

Y

1
3

Y

13 a 360
1
2

b i

Y

1
2
2
3

4
5

12

Y

A

3
10

Y

1
2
2
3

Y

1
3

Y

Y

B

6
10
1
10

N

N

Exercise 1

0
1
8

P 1X x2

c 126

B

2
7

1 a b 0.2

7 a 453600

A

3
7

2
7

Chapter 21

6 62

b 250

17 9
,
42 17

c 186

b 0.368

10
5
13

3
1
8

2 a a 0.33

b 0.87

c 0.75

d 0.73

c

X

0
1
2
3
27 108 144 64
P 1X x2
343 343 343 343

15
12
42

6
5
36

7
6
36

4
16
55

5
25
55

8
5
36

5 a k

1
2
3
4
1
8
3
64
y2
225 225 225 225

9
4
36
b

10
3
36

X
P 1X x2
˛

1
9

X
P 1X x2
˛

5
5
9

11
2
36

0
1
2 3
125 75 15 1
P 1X x2
216 216 216 216
˛

e

12
1
36

X
0
1
2
3
P 1X x2 0.512 0.384 0.096 0.008
˛

1
1
21

2
2
21

3
3
21

3
2
9

4
3
9

5
4
9

d k 35

f 7

X

˛

5
4
36

e 0.25

B
P 1B b2
˛

4
4
21

5
5
21

6
6
21
1
b k
74 X

c

X
P 1X x2
˛

P 1X x2
˛

1
5
35

2
6
35

3
7
35

4
8
35

5
9
35

4
15
74

7
6
30
5
24
74

8
7
30
6
35
74

9
8
30

10
9
30

(A Y) 3
20
(A Y) 3
20
(B Y) 2
5
(B Y) 1
5

Answers
6 a X
P 1X x2
˛

249
252

b

0
1
2
3
6 108 270 120
504 504 504 504

7 aX

0
27
64

P 1X x 2
˛

1
27
64

2
9
64

5
32

b

3
1
64

125
0
1
2
3
4 b 126
1
20 60 40
5
P 1Y y2
126 126 126 126 126

8 a Y
˛

Chapter 21
1 a b 0.3
2 E 1X2
˛

5

1
2

E 1X2 4.7

1

P 1X x 2

˛

441
91

4 a E 1X 2
˛

9 E 1X2 1

10 a 2.05

˛

c 6

d 0.986

b 5.45

13 a 7

Y

0.782

˛

7

18 a can be a probability density function Mean
19 a 0, 1, 2, 4

b

X

0
15
24

P 1X x2
˛

20 a 10, 11, 12, 13, 14

b

1
4
24

X
˛

4
, Var 1X2
7

b E 1X2
˛

c

d 8.15
35
6

110
35

4
1
39

2
1
10

4
1
10

b 14
15
7

b
1
2

17 a

c Mean

4
1
24

0
12
30
90

13
24
90

c

b

6
4
10

c 7

d 8

24
49

15 a

26
7

1
14

b 2.53

c 1.65

c 1.44

2
3

Variance 1.06

c E 1X 2 12, Var 1X 2

14
6
90

˛

˛

16
15

d

83
225

8
, Var 1Y2
7

e E 1Y2

0
1
2
3
4
5
6
81 54 27 24
7
2
1
x2
196 196 196 196 196 196 196

X
˛

Chapter 21

˛

Standard deviation 1.05

3

11
24
90

9
7

P 1X

1 a 0.27

0
4
10

11 a 3

14 a

c 0, 1, 2, 3, 4, 5, 6 d

0.816

˛

2
4
24

10
6
90

P 1X x2

329
6

b

E 1X 2 2.6

b x 3

P 1Y y 2

c 3.10

d b 0.1

˛

Y

0
1
2
3
2
56 84
8
39 195 195 39

˛

1
14

x 3
4

E 1X 2 3.06

˛

P 1Y y2

21 a

c b 0.28

˛

1 1
X
3
P 1X x2 0.15 0.25 0.6

6

28
, Var 1 Y 2
15

˛

˛

0.7

b 5.8

16 E 1Y 2

b E 1X2

1

0.3

25
9

691
400

12
91

E 1X 2 3.35

˛

˛

8 E 1X2

b b 0.15

˛

3 a c

X

12 a

Exercise 2

˛

˛

1.63

Exercise 3

b 0.532

b 0.833 c 0.834 3 a 0.208 b 0.0273 c 0.973 d 0.367 4 0.751 5 a 2.4 b 1.44 c 2
3
1
6 a 2.4 b 1.68 c 2 7 a n 7, p , q
b 0.445 c 1 or 2 8 a 0.00345 b 0.982 c 0.939 9 a 0.0872 b 0.684
4
4
5
9 c 0.684 d 0.847 10 a 3.52 10
b 0.0284 c 0.683 d 0.163 11 a 0.238 b 0.0158 12 a 0.245 b 0.861 c 0.997
4
1
13 a 0.060 b 0.00257 c 0.00191 d 0.230 e 0.129 14 a 10 b 2.74 c 0.416 15 a
b 17 16 a 0.0258 b 0 c
d 4
7
3
16 e 0.00258
8
18 c
3

c 0.0556

17 a Mean 2.1 Variance 1.81

f 0.00858

d 0.961

2 a 0.201

19 a 8

b 0.822

Chapter 21

c 8

b 0.423

c 0.353

d 3

4 a 1.68

b 0.0618

c 0.910

5 a 2.48

7 a 2.10

b 0.0991

c 0.0204

11 a 0.874

c 0.204

d 0.148

e 0.473

1
18 a X ~ Bin ¢8, ≤
3

a 0.125

b 0.332

c 0.933

b 0.156

d 0 0.0108

Exercise 4

1 a 0.224

14 a 0.0183

b 2

b 0.191 c 0.223
b 0.215

c 0.927

2 a 0.134

d 0.350
d 0.0426

b 0.151

b 0.213
8 a 7.62

c 0.554

c 0.763

12 a 0.0804

16 a i 0.839

ii 21.4, 21.4 iii 21 b 0.000241

18 a 0.0823

b 0.0499

c 0.00363

9 a 1

b 0.751

17 a 0.195

d 0.0000314

d 0.0404

b 0.996

d Mean 80 Variance 80

d 6

c 2

e 0.849
b 0.785

3

6 a 2.69

b 0.0799

b 0.981

10 a 0.905

d 0.182

e 0.454

f 0.0262

c 0.152

c 0.136

d 0.505

b 0.00468

13 a 0.0149

15 a 0.0324

d 0.166

d 10
e 0.944

c 0.000151
b 0.223

b 0.992

c 0.112

d 0.868

e 0.654

e 8

719

Answers

Chapter 21
1
10

1 a

b 3

Review Exercise
2 a l 2.99

c 1

6 a 0.176

b 0.905

10 a 0.0729

b 22

c 7

d 6.5

11 a

1
9

i

b 0.424

e 0.0158
ii

1
81

b i

3 a 0.225

b 3 or 4

7 a

12
25

b

73
648

ii

575
1296

48
25

c 17.5

8 a 0.175
c ii

˛

b 0.0226
P 1A¨B2

15 a P 1A B 2

P 1B2

˛

c 1

d 10

e 0.0232

13 30

b P 1A1 ´A2 2 P 1A1 2 P 1A2 2

˛

˛

˛

˛

˛

˛

˛

14 a 0.222

b 0.141

b 12.4

c 42.2

9 a 0.160

5 a 0.191

b 4

c 0.271

b 0.246

d 0.0808
iii

6
1
2
3
4
5
15
65
175
369 671
1
x2
1296 1296 1296 1296 1296 1296

X
P 1X

12 a 0.0105

4 a 20

b 0.939

c 0.104

d 0.00370

e 0.332

f 0.0145

6797
1296

g 0.995

c P 1E1 ¨E2 2 P 1E1 2 P 1E2 2

˛

˛

˛

˛

˛

˛

˛

˛

˛

15 d This is a distribution that deals with events that either occur or do not occur, i.e. there are two complementary outcomes. We are usually told the
number of times an event occurs and we are given the probability of the event happening or not happening.
Cj u j 11 u2 4 j

4

15 e i

Ckuk 11 u 2 4 k

ii E 1X2 4u, Var 1X2 4u11 u 2

4

˛

15 e vi 0.545, 0.0488

˛

˛

vii 0.969

Chapter 22
1 a k 1

1

2

3

4

5

6

Amount
received
in Euros
X

30

25

15

12

18

25

40

0.219

0.329

0.263

0.0878

x2

0.00137 0.0165 0.0823

˛

18

b i 0.0977 ii 0.885

Exercise 1
c

3
4
1
4

1
2

d

27
32

2 a c=4

y 1 x
4

1

3

y

c

2

d 0.634 4 a k 1.08

3
16

d 0

y x 1
2

x

c 0.707

冑2

b

1

y

b

6u 11 u2 2 4u4 11 u2 u4
2

0

1
123
d 305 17 a , E 1X 2
7
49

b y

3 a k 22

v

No heads

˛

c 59.5

iv 0.688, 0.996

16 a

P 1X
16 b Gain of 20.3 Euros

˛

iii 6u2 11 u 2 2 4u4 11 u 2 u4

b

x

4

y

c 0.306

(1.15, 1.54)

y 冑2 cos x

y 1 x(4 x2)
2

1

4

5 a

3
269

b

x

c 0.494

y

0.279

y 3 t2
269
y 6 (2 t)
1345

0.0758
0.0312
5

720

1.08

15 t

d 0.375

2

x

d 0.621

Answers

Chapter 22

Exercise 2

1 a k

1
2

b

5 a k

1
9

b 2.25

8 a k 1.56
10 c 0.273

4
3

2
9

c

27
80

c

1
ln 3

2 a

2
ln 3

b

c

1
(e – 1)

6 a c

d 2.38

4
4

ln 3
1ln 32 2

3 a c 0.755

1
(e – 1)

b

7 a k 1

9 a 0

b 0.616

c 0.422

d 0.546

e 0.616

12 a i 7.71

ii 0.947

b 0.318

13 a k

1
2

x

4 a k

b 0.530

b 0.571

c 0.141

b 0 6 x 6 0.421

1
16

b

p
6

e 0.169

d

c 0.115

23
6

c 1.97

d 0.0209

d 4

10 a 1

e

15
32

b 0

y

e2
2
31e 1 2
˛

b 0.148

c 0.00560

14 a

˛

14 b 2135

3
15 a c 0, k
8

c 12.4

b 0.15

c 1.587

d 0.305
0.05
y

1 (x 60)2
72000
x

60

Chapter 22
1 a 0.775

Exercise 3

b 0.589

2 c 0.396

c 0.633

d 0.678

d 0.0392

e 0.9234

f 0.485

e 1.69

f 0.0973

g 0.999

g 0.203

h 1.56

h 0.562

i 0.509

i 0.841

j 0.813

j 0.5392

3 a 0.00332

2 a 0.121

b 0.901

b 1.53

c 0.00332

d 0.968

4 a 0.0912 b 0.997
6 c 0.683 d 0.0279

c 0.952 d 0.122 e 0 5 a 0.106 b 0.809 c 0.998 d 0.101 e 0 6 a 0.275 b 0.00139
7 a 0.840 b 0.0678 c 0.683 d 0.997 8 a 40.6 b 38.9 c 41.9 d 39.2 9 a 93.9 b 84.6

9 d 82.2

b 13.7

15 0.939

10 a 5.89

11 Upper quartile Z 0.674 Lower quartile Z 0.674

d 4.18

12 0.935

c 86.6

13 0.912

14 0.999

16 0.134

Chapter 22
1 15.1

2 75.6

Exercise 4

3 30.5

11 m 290, s 11.1

4 39.0

b 2.29 kg

4 e 0.606

f 0.292

7 14.1

8 11.2

b 0.432

13 11.7

5 a 440

b 82.3 kg

9 m 11.6, s 4.53

b 0.227

1
1
≤, ¢m s,
≤
s22ep
s22ep
b 564g c 114 12 a 19.0 b 117

11 a 0.235

16 a m 28.5, s 21.49

21 0.00123

22 m 59.3, s 18.6

Chapter 22

3 a 0.309

6 ¢m s,

b 0.423

15 a 194 X 303
b 0.161

b m 589g, n 600g

2 a 10.6%

10 m = 64.56, s = 11.36

10 m 46.3, s 4.26

b 0.587
23 a 4.82

17 a 126

c 0.440

4 a 0.106

7 0.886

8 a 5

b 0.734

b 57.4

c 98.2

13 a 90.9%

b 280g

18 a 1.43

c 0.599

9 a 7.93

d 0.159

b 48.9

b 94.7

14 0.338

b 0.0146

19 4.14

b 0.0173

Review Exercise

1 a m 34.5, s 3.93
3 c 0.000109

6 7.81

Exercise 5

1 a 0.453

20 a 32.8

5 6.81

12 a m 23.6, s 6.13

Chapter 22

9 c 7

c 18.1

b 0.996

d 0.999851

4 2

2 a 0.0668

1
2

5 a 99.9%

b 11.4

d 0.350

e 0.348

7 c 0.268
0.637

9 a 1.63

冑3

1
冑3

10 d ii

b 142 cm

c 0.434

36
125

iii

c q 140 cm, r 180 cm
c 3.96%
8 a

1
4

d 0.00110

6 a 0.0327

˛

6 0.43

e 0.332

b 8.00

b E 1X2 p, Var 1X2 2.93
˛

d $6610 10 a m 28.6, s 14.3

98
10 e
125

d 0.121

b 12.6%

3 a 0.946

b 0.798

c Day 1: 2620. Day 2: 2610.

7 b 0

c 0.323
c x 0 Model is not perfect.

d i

8
125

f Either the events are not independent or the distribution is not continuous

2

11 a i E 1X2
˛

冮

1
x 18x x3 2 dx
12
˛

˛

ii 1.24

b ii 1.29

c 1.63

12 a i 1.355

ii 110.37

b A 108.63, B 112.11

13 a

4
81

b 0.6

0

13 c 0.24

d 9000 cents.

14 0.783

1

1

15 b e4 e2
˛

˛

1
e
4

c E 1X 2
˛

e
e
e2
1, Var 1X 2 1
2
3
4
˛

˛

d 0.290

e 0.0243

f 0.179

721