IBHM_673-720

673

Answers

1 a 135 b 20 c 72 d 150 e 105 f 22.5 g 110 h 114.6 i 85.9 j 229.2 k 206.3 l 22.9

2 a b c d e f g h 3 a 0.611 b 1.75 c 5.24 d 1.40 e 2.30 f 4.85

4 a b c d 5 a 6.28cm b 8.55m c 295cm d 113mm

6 a 66.8cm b 293cm c 177mm 7 a 126cm b 8 9 10 200

11 9.30cm 12 30cm 13 99.5cm 14 7:2 15 a 17.9cm b 99.1cm246.9m252.2cm22670cm2

2430cm2489cm24.71m230.7cm2

3p10

2p5

7p18

4p3

7p4

3p4

7p6

p

6

1 a b 0.174 c d e f g h

2 a b c 1 d e f g h 0 i j 0

3 a b c d e f

4 a b c d No Solution e f g h u 0.775, 2.37u 1.20, 5.08u 0.290, 2.85u p

6,

11p6

u p

4,

3p4

u p

3,

2p3

u p

3,

5p3

x 55.2, 304.8x 22.0, 158.0x 80.4, 279.6x 41.8, 138.2x 70.5, 289.5x 30, 150

11

221

2212

232

12

12

0.737232

1

22232

0.9961

2212

1 a b c d 2 3 a b 4

5 6 a b c d e

7 a b 8 44.7 or 135.3 9 38.8 or 141.2

10 a b c d 11 2.63m 12 a 59.6 b 143.2 13 68.7

14 15 a b c d e

16 312km 17 21.6m 18 57.9 19 56.25cm2

x 16.9mx 15.8cmx 83.8x 50.9x 7.64mA 39.4, B 46.5, C 94.1

t 10.7cmp 381mmx 22.7mx 6.07cm

x 127.7x 32.4

a 14.2mmn 7.35mx 7.64mx 6.37cmx 5.82cm311m2

63.7m3120cm284.9cm28220m226.0m2160cm220.7cm224cm2

1 a 180 b 120 c d 90 e 360 f 120 g 180 h i 36 j 3

2 a b c d

p

2p

2

Exercise 1Chapter 1

Exercise 2Chapter 1

Exercise 3Chapter 1

Exercise 4Chapter 1

y

x

1

1

0 120 240360

y

x

1

0360

1

y

x

1

0 360

1

y

x

4

0 3601804

2 e fy

x3001200

y

x

36013

Answers

675

Answers

674

3 d e

3 a b cy

1

0 21

y

2

0

y

2

2

0

6

y

2

10

1

y

2

3

8

20

4 a b c

4 d e f

y

180

1

0

1

x

y

1802

0

2

x

y

180

3

0

3

x

5 a b c

6 7

5 d ey

0

y

0

y

1720

3

7

y

6

0 x12

6

8 y

8

5

0

11

2

y

180

6

0

6

x

y

18060 x

y

180 x

4

2

0

y

4

8

0

y

2

0

5

8

y

3

0

11

9 a b c d e f

9 g h i j y 3 sec 32

uy 2 csc 3xy 3 sin 24u 8y tan 15x

y 72

cos 4x 72

y 4 sin 2u 4y 4 cos x 2y tan 2uy sin x 1y 4 cos 2u

Exercise 5Chapter 1

1 a b c d e 1 f g h i j

2 a b c d e f g 0 h i 1 j

3 a sin 43 b cos 50 c tan 20 d e f g h i

4 a b c d

5 a b c d u 5p6

, 7p6

u p

3,

5p3

u p

6,

7p6

u p

3,

2p3

x 135, 315x 60, 240x 30, 330x 30, 150

tan 46sin 20cos 15tan 34cos 23sin 50

423232

1

231

2223232

1

221

23

232

232

232

23232

12

12

112

Exercise 6Chapter 1

1 a 60, 240 b 60, 300 c 60, 120 d 210, 330 e 150, 210 f 180 g 90 h 30, 150 i 45, 225

2 a b c d e f g h

3 a 15, 75, 195, 255 b 10, 110, 130, 230, 250, 350 c 11.25, 56.25, 101.25, 146.25, 191.25, 236.25, 281.25, 326.25

3 d 60, 120, 240, 300 e 25, 45, 145, 165, 265, 285 f 40, 80, 160, 200, 280, 320

4 a b c

4 d 5 15, 105 6 7 1, 5, 13, 17 8 9

10 a 19.5, 160.5 b 41.4, 318.6 c 58.0, 238 d 48.2, 311.8 e 48.6, 131.4

10 f 31.3, 288.7 g 75.5, 284.5 h 45, 135 i 228.7, 341.3 j 36.9, 143.1

10 k 20.9, 159.1, 200.9, 339.1 l 25.2, 94.8, 145.2, 214.8, 265.2, 334.8

10 m 47.1, 83.1, 119.1, 155.1, 191.1, 227.1, 263.1, 299.1, 335.1 n 70.5, 289.5

10 o 8.2, 171.8 p 48.2, 311.8 q 33.7, 213.7 r 15, 75, 105, 165, 195, 255, 285, 345 s 1.4, 4.6, 13.4, 16.6,

11 a 0.253, 2.89 b 2.03, 4.25 c 1.17, 4.31 d 1.11, 5.18 e 2.01, 0.08 f 3.34, 6.08 g 1.23, 4.37 h 0.41, 1.68, 2.50, 3.78, 4.60, 5.87

11 i 0.08, 1.49, 3.22, 4.63 j 0.47, 1.26, 2.04, 2.83, 3.61, 4.40, 5.18, 5.97 k 0.28, 0.76, 2.38, 2.86, 4.47, 4.95 l 1.14, 5.15 m 3.45, 5.98

11 n 0.14, 3.28 o 0.11, 0.67, 1.68.2.25, 3.25, 3.82, 4.82, 5.39 12 70.5, 289.5, 430.5, 649.5 135p36

, 13p36

, 29p36

35p36

, 19p36

, 11p36

,

p

11.12,

2p9

, 4p9

, 8p9

8p9

, 4p9

, 2p9

, 120, 607p24

, 11p24

, 19p24

, 23p24

5p12

, 7p12

, 17p12

, 19p12

p

30,

5p30

, 13p30

, 17p30

, 25p30

, 29p30

, 37p30

, 41p30

, 49p30

, 53p30

p

6,

2p3

, 7p6

, 5p3

p

12,

5p12

, 7p12

, 11p12

, 13p12

, 17p12

, 19p12

, 23p12

p

6,

11p6

p

2,

5p6

p

6,

7p6

p

6,

5p6

p

4,

5p4

5p6

, 11p

67p6

, 11p

6p

6,

11p6

Answers

677

14 a 3 minutes b

Answers

676

14 c i 2 mins 15 secs ii 45 secs 15 a 7500 b 7060 c 12 years, 4500 fish

1 a 30 b 75 2 a b 3 4 14.0m 5

6 a 15.1cm b 44.8 c 48.9 d 8.63mm 7

8 a b c d e f g h i j 2 k

9 a b c

2

231

231

2223232

1

2212

1

231

2212

2.35 6 BC 6 5

20.6cm27.01cm213p12

2p3

Chapter 1 Review Exercise

1 a b c d

2 a b c d e

3 a b c d e

4 a b c d e

Exercise 2:x 1.78, x 0.281x 3.91, x 1.41x 1.77, 0.566x 1.61, x 5.61x 5.45, x 0.551

x 2.14, x 0.468x 0.260, x 1.54x 0.314, x 3.19x 0.854, x 5.85x 4.73, x 1.27

x 23

, x 3x 32

, x 23

x 4, x 32

x 3, x 8x 2, x 5

x 1, x 3x 12

, x 8x 3, x 2x 4, x 1

5

0

7

2

y

x

4

0

4

360

y

5

11

2

H(t)

21

t30

39

9 d ey

5

5

0 2

y

1

2

10 a b 11 a b c

11 d e 12 a b c

12 d e f

13 a b c 20.7, 80.7, 140.7, 200.7, 260.7, 320.7 d 64.6, 295.4

14 15 20.9, 69.1 16 a 14.5cm b 0.169s

17 a b c x 23p

3, 0

p

3

x 109.3, 160.7, 289.3, 340.7x 64.6, 295.4

x 10, 50, 130, 170, 250, 290x 15, 165, 195, 345x 60, 240

x 210, 330x 90x 135, 315u 5p12

, 11p12

, 17p12

, 23p12

u p

12, p

6,

7p12

, 2p3

, 13p12

, 7p6

, 19p12

, 5p3

u p

4,

5p4

u 5p6

, 7p6

u p

6,

5p6

y 52

cos1x 20 2 52

y 3 cos 2u 1

x4 445

354, 12

5 1,

165 0

,

354,

Exercise 1Chapter 2

Exercise 2Chapter 2

1 a b c d e

2 a Minimum y intercept: 4 x intercepts:

2 b Minimum y intercept: 3 x intercepts: 3,1

2 c Minimum y intercept: 2 x intercepts: 4.56,0.438

2 d Maximum y intercept: 3 x intercepts:

2 e Maximum y intercept: 3 x intercepts:

2 f Minimum y intercept: x intercepts:

2 g Minimum y intercept: 1 x intercepts: none

2 h Minimum y intercept: 2 x intercepts:

2 i Minimum y intercept: 4 x intercepts:

3 a Minimum y intercept: x intercepts:

3 b Maximum y intercept: x intercepts: none c Minimum y intercept: 16 x intercepts: none

3 d Maximum (0.833,11.1) y intercept: 9 x intercepts: 1.09, 2.76

10.6, 14.2 2712.5, 0.75 23.5, 1711.25, 10.125 2

0.851, 2.3534

, 2382x 3

42 23

8

1, 23

56

, 1123x 5

62 1

12

38

, 7164x 3

82 7

16

0.928, 5.931152

, 4722x 5

22 47

2

0.359, 8.3614, 19 21x 4 22 194.65, 0.64612, 7 21x 2 22 7

52

, 174x 5

22 17

4

12, 1 21x 2 22 10.764, 5.2413, 5 21x 3 22 5

5x 7102 109

2031x 1 22 11x 3

22 11

4x 3

22 3

41x 1 22 4

Exercise 3Chapter 21 a b c d

(0, 5)

y 2x 5

52 5

,

Line notincluded

y 3x 8

83 0,

(0, 8)

Line notincluded

x 2y 5

520,

(5, 0)Line notincluded

3x 4y 5 0

Line notincluded

53 0

,

540,

2 a b c d e f g h

2 i j k l

3 a b c d e 0.207 x 1.212.14 x 0.468x 6 0.157.x 7 1.59x 6 0.854, x 7 5.85x 6 0.764, x 7 5.24

43

6 x 6 12

8 x 12

3 x 2 2 6 x 6 1

x 6 4, x 7 33 x 72

12

x 34

5 6 x 6 6x 7 3, x 6 5x 15

x 6 2x 7 14

1 a Imaginary roots b Real distinct roots c Imaginary roots d Real distinct roots e Real equal roots f Imaginary roots

2 3 5 7 9 11 a (ii) b (iii) c (i) d (iii)m ;B329p 12a b 8ab3 16a2b 0q 98p ;14

Exercise 4Chapter 2

x

y

4

4,

4

35

4, 35 , 1125

165

45

0

1 a yes b no c yes d no 2 a b c d e

2 f g h i 3 a b c

3 d e f g 4 a i ii b i ii

4 c i ii d i ii e i ii f i ii

4 g i ii h i ii i i ii j i ii

4 k i ii l i ii 5 a b h11x 2 1x 6 23

h 1x 2 3 x 6f11x 2 x 5213f11x 2 x14x 0 f11x 2 B

16 x3

x 0f11x 2 Bx 32x 0f11x 2 2x 4x 7 0f11x 2 B x6x 7 0f11x 2 4 6x

5xx

65

f11x 2 9x 84x

x

94

f11x 2 2x 7x

x 2f11x 2 2x 53x

x

23

f11x 2 3 7xx

x 7f11x 2 6x 1x

x 60 x px 12

, x H x 0, x H x 0, x H

x

12

, x H x 4, x H x 3, x H f11x 2 x13

2f11x 2 1x 6 2 13f11x 2 x 9

2f11x 2 9 x

4

f11x 2 7 xf11x 2 32

xf11x 2 x 6f11x 2 x 5f11x 2 x4

Answers

679

Answers

678

1 2 3 4 i. ii 5

6 8 9 10 Maximum point is Line of symmetry is

11 6 years old 12 True for all real values of c.cos C c2 3

4c.

x 72

72

, 94.a 9

4, b

72

.k 0, k 8x 16, x 1m ;2

2.46 6 k 6 0.458x 92

, 6, y 92

, 3x 13

, 2, y 73

, 0k 4.08, k 2.581 6 x 6 34 1x 2 22Chapter 2 Review Exercise

1 a 11 b 1 c 130 d 6 2 a b 12 c d

3 4 5 6 5y : 5 y 1565y : 5 y 26b2, 12

, 2, 4r7, 1, 17 1942717Exercise 1Chapter 3

f(x)

7

1

17

1

1

3

g(x)

24212

3116

7 8 9 5y : 5 y 6 q65y : 0.959 y 2216by : 254

y 24rx

y

0

5

2

7 x

y

0

5

15

23

10 c Domain Range 11 a b c 12 a b

12 c d 13 a b c d 14 xx 2

x1

2x 12x

1 2xx

x 24x2 10x 436x2 18x

x2 5x 44x2 6x3x

23x 26x 25y : 0 y 465x : 2 x 16,

x

y

0 6

24

6

x

y

0

96

5

12

1

Exercise 2Chapter 3

1 a 14 b 24 c 2 d 2 a 61 b 5 c 46 d 3 a b c d

4 a i ii b i ii c i ii d i ii

4 e i ii f i ii g i ii

5 a b c 6 a b c d 7 a b

7 c d 8 a i ii b i ii c i ii

8 d i ii e i ii 9 x2x 4

x1

x 13x 1

22x

3 x

2 5x3 5x

2 3xx 1

3

x2 3x

9 9x

x2x 3x 3

23x 2

cos1cos1x 2 2x pcos x

p

2cos x p

24x 9216x4612x 3 2212x2 3p 26x 8 p6x 3p 4 sin2 2x 7sin12x2 14 21x6 2x 3 22 41x2 4 26 21x2 4 2 32x3 2x 178x3 36x2 52x 31

3 cos2 xcos13x2 2x6 61x2 6 233x2 79x2 30x 2112x 1 222x2 1sin u p

323

2232

p 33

3x2 74x 24

Exercise 3Chapter 3

Exercise 4Chapter 3

1 a b c f11x 2 x 3f11x 2 x 2f11x 2 x2

x

y

0

f (x) 2x

f1(x) 12 x

x

y

0

4

2

f (x) 2x 4

f1(x) 12 x 2

2 a b c d f11x 2 25 xf11x 2 2x 4f11x 2 B x3f11x 2 1x

3 a b c f11x 2 2 xx

f11x 2 1 5xx

f11x 2 1 2xx

x

y

0

f (x) x2f1(x) x

x

y

0

f (x) 3x2

13xf1(x)

x

y

0

f (x) x2 4

f1(x) (x 4)

x

y

0

f (x) 5 x2

f1(x) 5 x

5

x

y

0

2

2

f (x) x 2

f1(x) x 2x

y

0

3

3

f (x) x 3

f1(x) x 3

x

y

01

f (x) 3x 1

13

f1(x) 1313x

1 d e f11x 2 x 42

f11x 2 x 13

x

y

0

f1(x) 1x 2

f(x) 1x 2

x

y

f1(x) 1x 5

f(x) 1x 5

x

y

f1(x) 2x 1

f(x) 2x 1

10 a Domain Range

10 b Domain Range 5y : 2 y 1865x : 1 x 76,5y : 9 y 1165x : 3 x 26,

2 d e f g

Answers

681

1 2 3 4

5 6 7 8

f1x 2 c3 5x 2x2, 12 6 x 6 32x2 5x 3, x 3

f1x 2 d12 x x2, 3 6 x 6 4x2 x 12, x 42x2 5x 3, x

12

f1x 2 b2x 1, x 6 12 OR 2x 1, x x2 x 12, x 3

f1x 2 b2 x, x 6 2x 2, x 2

Answers

680

4 a b c d

Exercise 5Chapter 3

x

y

04 x

y

0 x

y

0 53 x

y

6 2

x

y

0 3 4x

y

03 2

9 10 11

14 15 16 17

18 19 20 21 22 4.72 6 x 6 3, 12

6 x 6 2.226.80 x 5, 1 x 2.802 6 x 652

4 x 57 6 x 6 3

x 2.91, x 2, x 32

, x 2.41x 3.37, x 2.56, x 1.56, x 2.37x 2, x 5x 4, x 1

Exercise 6Chapter 3

x

y

2 0 13

1 a b c d

x

y

04

4

x

y

04 3

12

72

14

,

x

y

0 62

12

(4, 4)

x

y

0

24

3 8

52

1214

,

x

y

0

8

2 43

13

253

,

x

y

01 12

x

y

0

3

3 x

y

0

7

72

1 e f g h

x

y

15

32

53

112

, 0.0833

0x

y

20

4552

172

108940

,

x

y

14

4 x

y

32

2

x

y

12

4

0

1 i j k l

1 m 2 a b c

x

y

30

5 6

(0.5, 30.25)

x

y

0.6974.30

3

(2.5, 3.25)

x

y

5(1, 4)

x

y

2

32 x

y

32

65

12

45

,

x

y

43

(35, 28)

x

y

1

551

x

y

2 5

(0.172, 0.299)0.3 x

y

(1.16, 0.430)

(5.16, 0.969)

31 x

y

5(0.745, 1.49)(10.7, 21.5)

1.6

x

y

(4.76, 22.7)(0.686, 0.275)

6 4

1

41 x

y

(9.20, 2690)

7 21.48

x

y

0 x

y

1

1

2 h i j

2 k l m n

x

y

x

y

2

x

y

No inverse

x

y

(3, 3)

4 e f

x

y

1x

y

12 13 x 4, x 6x 5, x 1

12

3 e i ii iii iv

Answers

683

1 a b c

Answers

682

Exercise 7Chapter 3

x360

y

2

2

y

4

4

2

y

1

2

5

2 a b cy

x0

y

x2

y

x

8

3 a i ii iii ivy

x0

y

x20

y

x

10

y

x0

3 b i ii iii ivy

x0

y

x20

y

x10

y

x0

3 c i ii iii ivy

x0

y

x20

y

x1

0

y

x0

3 d i ii iii ivy

x

4

40

y

x60

y

x

3

(4, 1)0

y

x4

8

0

y

x0

y

x20

y

x

1

y

x

0

3 f i ii iii ivy

x

3

0

y

x

(2, 3)0

y

x

4

0

y

x

6

0

3 g i ii iii ivy

x3440

y

x3

1

0

y

x41

74

0

y

x

4

32

0

3 h i ii iii ivy

x

1

22 1

y

x3

1

0

y

x13

0

y

x1

2

0

4 a b c d e f g h

5 a i ii iii

x 3x

2x 9x 3

4 2x x22x 1x 1

x 1 3x 11 3x 3x2 x31 2x x2

y

x

(1, 6)

4 40

y

x

(2, 6)

7 10

y

x

(1, 5) (7, 5)

(2, 23)

0

7 a b c

Answers

685

Answers

684

x

y

4

6 1

(3, 6)(4, 7)

20 x

y (1, 7)6

(3, 4)

2 35 0

x

y

(3, 5)

(1, 16)

13

(3, 17)

(5, 5)(2, 5)

0

x

y

(4, 2)4

(3, 5)(6, 4)

(1, 9)

8 0x

y

(2, 9)

(1, 2)5

(3, 4)5

7x

y

(5, 5)

(3, 7)

(2, 22)

(7, 5)

10

(1, 1) 0

x

y

0 x

y

313

0 x

y

2

12

0 x

y

434

0

x

y

7

57

0 x

y

1

12

0 x

y

2

32

0x

y

7

170

x

y

545

0x

y

38

85

0

5 b i ii iii

5 c i ii iii

6 a b c d

6 e f g h

6 i j

x

y

6

1

1

0 x

y

3

1

43

0 x

y

2

616

x

y

4

21

0 x

y

30 x

y

52

212

x

y

5

13

32

x

y

23

32

72

0

7 d e f

7 g h 8 a b c d e

9 a b c d

9 e f g h x 732

x 774

x 4x 743

x 792

x 7 0x 134

x 7 0

x 378

x 23

x 15

x 103

x 12

10 a b c 11 a

x

yy f(x)

(0.8, 1.11)

20

x

yy g(x)

(0.8, 1.89)

(2, 3)3

x

yy h(x)

(0.8, 1.89)

(2, 3)3

x

y y f(x)

(0.16, 0.324)

(6.16, 12.3)

3

x

y

y g(x)

(1.83, 0.324)(8.16, 12.3)

5 x

y

y h(x)(1.83, 0.324)

(8.16, 12.3)5 x

y

y p(x)

, 19.7519

12

x

y

, 19.7512 , 19.75

1912

0

11 b c 12 a b

1 2 3 a b c 4 a b c d x

4

301

5

12

76

113

9x 168 3x7 3x

5x 8x 1

7 4xx

14x 1121x 4b76

, 12

, 3, 113r7

2


5 a b c x 32

, f11x 2 7 3x2x

x 5, f11x 2 1 5xx

x 7 0, f11x 2 2x 6

1 14 2 3 21 4 a 51 b 1224 c 674 d 5 6 a b

6 c d e

6 f g h

7 a b c

7 d e

7 f g f1x 2 x3 x22

7x4

78 12x 1 2 81

8f1x 2 x2

2

13x4

38 12x 1 2 13

8

f1x 2 12x5 2x4 3x3 3x2 3x 3 2 1x 1 2 6f1x 2 15x4 20x3 76x2 304x 1219 2 1x 4 2 4878f1x 2 14x2 5x 6 2 1x 3 2 35f1x 2 1x2 11x 47 2 1x 5 2 242f1x 2 13x 1 2 1x 2 2

Q1x 2 3x34

7x2

16

59x64

177256

, R 2797256

Q1x 2 x22

x4

278

, R 998

Q1x 2 x42

x3

4

x2

8

7x16

8732

, R 43932

Q1x 2 x3 x2 4x 7, R 0Q1x 2 x2 x 5, R 11Q1x 2 2x2 13x 70, R 427Q1x 2 x2 3x 2, R 1Q1x 2 x 8, R 139

47047

Answers

687

Answers

686

x

y

1

2

f(x) 2x 1

x

y

6 2

y | f(x)|

x

y

23

3 x

y

29

25.61 1.610

x

y

0x

y

(3, 1)

0

1

x

y

0 x

y

10

6 a b 7 a b

10 11 12 13 a

x

y

12

72

0x

y

23

2

0

13 b c 14 a b

x

y

(1.15, 1.02)

12

2

y f(x)

x

y

(1.85, 1.02)5

y g(x)

3, 12

x

y

(1.85, 1.02)5

y h(x)

15 16 x 7 0x 1915

Exercise 1Chapter 4

Exercise 2Chapter 4

7 (c) (d) and (f) 8 a b c d

8 e f g h i

8 j 16x 1 2 13x 4 2 12x 5 2 1x2 4 21x 3 2 1x2 1 2 12x2 5 21x 3 2 1x 3 2 1x2 2 212x 3 2 12x 1 2 13x 4 212x 1 2 1x 4 2 1x 6 212x 1 2 1x 3 2 1x 4 2

1x 1 2 1x 1 2 1x2 1 21x 2 2 1x 1 2 1x 5 21x 1 2 1x 2 2 1x 3 21x 1 221x 1 2

Exercise 3Chapter 4

1 17 2 3 4 5 6 7

8 9 10 k 5p 11, q 21k 46, 12x 1 2 1x 4 2 1x 6 2p 0, q 1a 2, 1x 2 2 1x 3 2 1x 3 2k 10, 1x 3 2 1x 1 2 12x 1 2k 2p 24133

16

Exercise 4Chapter 4

1 2 3 4 a b x 7, x 2, x 2x 1, x 3, x 4x 2, x 5, x 12

, x 4x 12

, x 6x 1, x 3

Exercise 5Chapter 4

1 2 3 4 5 6

7 8 9

10 11 y 2x4 6x3 18x2 46x 24y 20x4 180x3 420x2 20x 600

y x4 2x3 13x2 14x 24y 2x3 2x2 28x 48y 12

x3 32

x2 5x 12

y x3 2x2 11x 12y 3x2 12x 12y 2x2 4x 6y 10 3x x2y x2 6xy x2 3x 4

Exercise 6Chapter 4

3 4 5 7 8

9 10 11 12

13 14 x 2, x 2, x 6x 4, x 3, x 1, x 2

x4 x2 5x 1 12 5x

x2 1

x3

2

x2

4

27x8

6716

211

1612x 1 22x3 6x 18x 13

x2 33x3 6x2 10x 20

47x 2

x 92

30x 85

21x2 7 2x 5 14x 21

x2 53x 11, 152x 1x 3

Exercise 7Chapter 4

1 2 3 4 5

6 7 8 9

10 11 12

13 14 a and b

15 a and b 7.89 6 a 6 29.1x 6 3.830.961 6 x 6 1.63

3.42 6 a 6 1.271 6 x 6 3.30x 6 0.30313x 4 2 13x 4 2 12x 1 2 12x 1 21x 5 2 1x 1 2 1x 4 2f1x 2 4x3 10x2 28x 16, x 2, x 0.388, x 3.86x 0.953

x 10.2, x 0.203, x 1.44x 4.92, x 1.02, x 2.19, x 3.74x 0.388, x 1x 5.50

x 4.29, x 0.428, x 2.72x 3x 2, x 12

, x 3x 7, x 1, x 12

, x 4x 1, x 12

, x 3

x

y

10

8 945

6 x 6 2x 8, x 1

x

y

12

y | f(x)|

0

x

y

12

3

4

14 c

17

18 a b 19 20

21 a b 22 and x 7 4x 6 10 y 6 q14

x 14

f11x 2 Bx 1x 13 x 13f11x 2 2x 1x 20 6 y 6 2

4 c d e f g x 23

, x 12

, x 32

x 2, x 12

, x 8x 2, x 4x 3, x 2, x 4, x 5x 11, x 3

5 7 8 a b 9 a b x 12

, x 7k 83x 3, x 5p 30x 3x 2, x 1, x 9

10 a b c d e 11 6 6 t 6 1012

x 103

3 x 70 x 12 x 51 x 7

1 a b c d e f g h i j k

2 a b c d 3 a 2.699 b 2 c 1 d e 2 f 2 g 1 h 0 i j 4 k

4 a b c d 5 a 2 b c d e 2 f 6

7 8 10 a b c d e f g h

11 a b c d e f 12

1 a 3 b 0.903 c 1.41 d 0.477 e 2 a 2.30 b 2.20 c 3.43 d e 2.01 f

3 a 2 b 3.17 c 2.18 d 2.04 e 2.72 f 1.39 g 1.13 h 2.24 i j k l m

3 n o 1.07 p 0.824 q r

4 a b c

0.5131.490.209

0.22623

1.071.1112

1.110.6930.301

S1 400x 10x 263

x 15x 3x 4x 2

x 9x 25x 16

x 8x 9x 6x 57x 7y px5y 9x4

y 4x3logx y 1

logy x12

23

32

12

loga 1x 2 223 loga1x 1 2loga 2xloga 3x3

52

312

loga xy2t3log2 89log2 52log3 15 loga 12loga 72loga

18

loga 6loga 32loga 4loga 24loga 64loga 5loga 15loga 18

Answers

689

Answers

688

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16 17 18 k 6a 4a 6a 45

, b 65

x2 5 3x

x2 1

x2

2

11x4

358

67

812x 1 2

x 6.59, x 2.38, x 1.97x 1.47f1x 2 21x 3 2 12x 1 2 1x 4 22x 4, x 12

, x 1

x 7, x 3k1x 2 1x 4 2 1x 1 2 1x2 5 2g1x 2 12x 1 2 1x 5 2 1x 2 2 13x 2 2f1x 2 12x 3 2 1x 4 2 1x 1 21x 3 2 , 1x 9 2f1x 2 x4 x3

2

x2

4

17x8

171612x 1 2 95

163x3 4x2 8x 22

45x 2

35


Exercise 1Chapter 5

1 a b c d e f g h 2 a 4 b 3 c d 1 e 125 f g h i 9

3 a b c d e f g h

4 a b c d

x 2 x12x 112x5 15x6m23t210p6

43

y4x6

18

14

13

110

6p72p2t 216x1221y5x15p5p9

x

y

1

(1, 10)

0x

y

1

(1, 6)

0x

y

1

(1, 5)

0x

y

1

(1, 3)

0

e f g h

x

y

1

0

231,

x

y

1

0

321,

x

y

1

0

161,

x

y

1

0

141,

1 a b c d

Exercise 2Chapter 5

y

x

(10, 1)

10

y

x

(5, 1)

10

y

x

(4, 1)

10

y

x

(2, 1)

10

y

x

(5, 1)

(1, 5)

1

1

0

y

x

(3, 1)(1, 3)

y 3x

y log 3x

1

1

0

2 a b 3 a 2 b 2 c 2 d 3 e 3 f 5

3 g 3 h 6 i 2 j 2 k 3 l 1 m 1 n 0

4 a b c d e f

4 g h 5 a b c

5 d e f g

6 a 1 b 2 c d e 1112

12

13

43

23434

23

16

12

12

12

12

13

Exercise 3Chapter 5

Exercise 4Chapter 5

x0

y

1

(5, 1)

x0

y

1

(7, 1)

x0

y

1

(9, 1)

Exercise 5Chapter 5

1 a b c d e f 2 a b c

2 d e 3 a b c d e

4 a b c d 5 a 40 b 132 c 3.84 days 6 37.8 months 7 a 80 C b 10.8 mins

8 a 2100g b 1650g c 57.8 years 9 a 20100 km b 22.7 years 10 a 220 b 57 c 2027

11 a b 67.2 hours c 9.6 hours longer 12 a b 60.8 years 13 14 15 16

17 18 19 20 21 22

1 a b c d

2 a b c d

x

y

(2, 1)0x

y

(1, e)1

0x

y

1

(1, 4e)

0x

y

(1, e)

10

x

y

(3, 1)0x

y

10

121,

x

y

2

3

0x

y

1(1, 2)

0

x 0.631x 0x log6 4x 32

log4 5x 1 log4 3x 2 log2 5

ln 643

ln 92

ln 25

ln 80ln 2ln 6

ln 12ln 5

k 0.0114k 0.0133

x 6x 4x 6x 9

x 1.22x 8890000x 22000x 7.39x 8100x 0.981x 1.50

x 5.60x 3.40x 2.48x 0.416x 0.862x 3.53x 1.86x 3.36x 8

5 a b

5 c x 117649

x 8x 0, x 1024

Exercise 6Chapter 5

13 14 15 a b Yes as it will be ok for 8.07 hours 16 17 18

19 20 21 22 a b 23

24 a b x 28k 32

, m 1

x ;8y 0, y H x 6 23

, x 732

, x H x 157

, y 107

x 1, y 0x 4, y 4

1 log5 3ln 9ln 8

x ;6k 0.0633p 3, q 5k 7

Answers

691

3 a b c d

4 a b c d

5 a b c d

6 7 8 9 10

1 a b c

2 a b

x

y

(6, 1)

0 1

x

y(1, 6)

0

1

2 4x215p14x6

p 3, a 6a 4, p 3a 3k 2, p 3k 5

x

y

(1, 3)

0 112x

y

(3, 1)0 1

y

x

(3, 2)

10

y

x

(3, 1)

10

y

x

(4, 1)10

y

x

(2, 1)

2 1 0

y

x

(1, 2)

(4, 3)

0

y

x1

(4, 1)

0

y

x

3

(1, 8)

0

y

x

(2, 3)

0x

y

1

0x

y

4

0

Answers

690

Review ExerciseChapter 5

3 a 5 b 3 c d 4 a b c

5 a b 1 6 a b c d

7 a 1.40 b 0.631 7 c 0.289 d 2.32 8 a 8100 b 24200000

8 c 44.7 9 a b c d

10 a 5 b 5 c 6

x 1.34x 2.08x 0.356x 5.25

118

x 3x 81x 9log3 8x

loga 25alogp 2loga 48213

11 a b c d

12 a b c

x

y

(1, 5)

01x

y

(7, 1)

0 1x

y

(1, 2)(5, 1)0

x

y

(1, 8)

0

65x

y

(2, 6)(1, 3)

02

x

y

(2, 6)(3, 1)

0x

y

(1, 3)0

2

Exercise 1Chapter 6

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18 19 20 k 9, k 6k 3k 8k 2un 12

n 4un 6n 50un 9n 7un 4n 3

n 27n 23n 143un 12

n 12

, u15 8un 7n 24, u19 109un 110n 90, u13 1520

un 11n 4, u20 216un 4n 8un 9n 69un 6n 2un 5n 4un 2n 3

1 2 3 4 5 6 203 7 354 8 1050 9 5586 10

11 12 4, 10, 16 13 14 15 1344 16 a b 308d 2n 6n 16n2

n1n 1 22

n3

n2

62014n 6n2

32

n2 163n

2n2 7n

n2

32

n2

Exercise 2Chapter 6

Exercise 3Chapter 6

1 2 3 4 5

6 7 8 a b 107 c 6560 d e 66.4 f 3060 g 585936

9 10 11 12 13 14

15 16 or 17 or 18 19 20 21 22 a 43

, r 4n 8n 5n 9k 8k 3k 1k 10k 2un 18

14 2n1

un 413 2n1un 27013n1un 512 2n1Sn 11 13x 2n 21 3xSn 1 1x 2n1 xSn x11 xn 21 x425

25516

u6 18750, un 615 2n1u6 384, un 1212 2n1

u6 25

8, un 10012n1u6 160, un 512 2n1u6 486, un 213 2n1u6 564, un 8014n1u6 14, un 24n

1 2 3 Does not converge 4 5 Does not converge 6 7 8

9 10 11 or 12 13 14 15 a 1950 b c329

5802532

72799

21499

589

x 7 1x 6 11 6 x 6 1Sq 367

Sq 50Sq 300Sq 12Sq 51213

Sq 2432

Sq 40

Exercise 4Chapter 6

Exercise 5Chapter 6

1 28 days 2 $2720.98 3 98691 dkk 4 20 years 5 4.3% 6 8820.36 7 a 17623 rats b 14.2 months 8 a 187 leopards b 2012

9 a 3.07m b 10

Exercise 6Chapter 6

1 a 35 b 252 c 896 2 a b c

3 a b c d12

1k 1 2 17k 8 283

n3 2n2 533

n23

n3 32

n2 176

n3n2 n

aq

r14r 5a

n2

r15r 7a

5

r14r

Exercise 7Chapter 6

1 a 30 b 336 c 56 d 126 e 70 2 360 3 2002 4 a 39916800 b 990 c 330 5 1712304 6 19068840 7 12870

8 a 216 b 120 9 a b c d 10 a b c 11 a b c n 3n 3n 7n 9n 10n 4n 6n 4n 5n 6

Answers

693

Answers

692

Exercise 8Chapter 6

1 a b c

1 d 2 a b

2 c d 3

4 a 40 b 7000 c 3840 d 6480 e 150994944 f 21 g h 90720 5 a 58 b c d 5

6 a b

6 c 7 a 22 b 7688 c 8 1360 9 a 1.04 b 0.210 c 15800000

10 11 Proof: p2 15

22688x2 12480x 3200

5888x7 2x5 6x3 8x 17x

6

x3

20

x5

8

x7

x6 6x5 15x4 26x3 39x2 42x 37 30x

12

x2

8

x3x7 x6 69x5 109x4 1616x3 3360x2 12800x 32000

24313928

x6 9x5 30x4 45x3 30x2 9x 116t4 8t2 32

1

8t2

1

256t4x6 6x4 15x2 20

15

x2

6

x4

1

x6

x5 10x3 40x 80x

80

x3

32

x5x3 3x 3x1 x332p5 240p4q 720p3q2 1080p2q3 810pq4 243q5

1 4x 6x2 4x3 x4729x6 2916x5 4860x4 4320x3 2160x2 576x 64a4 4a3b 6a2b2 4ab3 b4

1 2 a b 3 a b 4 a b

5 a b 6 a b 5 7 a b 50 8 9 10 a b

11 12 a b 32.8080401 13 a b 14 a 59 b

14 c i 99 ii 100 15 280 and 84 16 17 34642080 18 4455 19 20 n 106n 56

n1n 1 2 12n 1 2 , 972k 36p3

n 12, d 0.25d 920

r 12

x5 10x4 40x3 80x2 80x 32a 2, b 3

un 1514 2n1n 6a 9a ;38n 3x 6 32un 4n 31,9a 9r

23

n 30Sn 3n2

2

n2

Sn 1614n 1 2r 47


1 a b c d No possible value 2 a b c No possible value d 4

5 a b c d 7 a 0.464, 2.68, 3.61, 5.82 b c 0.785, 2.90, 3.93, 6.04

7 d e 3.53, 5.90 f 1.05, 5.24p, p

4,

5p4

p

2,

3p2

7 cot2 usin utan usin u

;13

;226

5;

35

;12

;226

7;

1

22;232

Exercise 1Chapter 7

Exercise 2Chapter 7

1 a b c 2 a and b 3 4 8 a 0 b

9 a b c 10 11 0.951 12 13 0.2829, 0.3877 14 52.5, 232.5 16 17 a b

17 c d 18 a 40.9, 220.9 b 0, 180 c 7.6, 187.6 19 a 2.88, 6.02 b 2.36, 5.50 c 1.70, 4.84527625

336625

725

2425

610

72

1565

44

117117125

45

232

11 23 2

2221 23

22223 1

2221 2323 1

23 1222

1 23222

Exercise 3Chapter 7

1 a b c 2 3 4 a b 5

6 7 119169

2 cos u16 sin5 u 16 sin3 u 3 sin u

3 cos5 u 4 cos4 u 4 cos3 u 3 cos2 u2

251

25161289

120169

2312

232

Exercise 4Chapter 7

3 4 5 a b 6 a 30, 90, 150, 270 b 0, 180

6 c 60, 90, 270, 300 d 90 e 60, 300 f 60, 300 g 120, 240 h 210, 330

7 a b c d e f g

8 a b c No solution d 9 a 0.841, 1.82, 4.46, 5.44 b 0, 1.82, 4.46p, 2p3

, 2p3

p

3, p

3p, 0

0, p, 7p6

, 11p

6p0,

2p3

, 4p3

p

6, p

2,

5p6

, 3p2

p

3,

2p3

, 4p3

5p3

0, 2p3

, p, 4p3

p

3, p,

5p3

p

6,

5p6

, 7p6

, 11p

6p

6, p

3,

7p6

, 4p3

22 122

23 1222

Exercise 5Chapter 7

1 a b c d 2 a b

2 c d 3 a b 4 a b

5 a b 6 a b c d

7 a Minimum Maximum b Minimum (150, 3) Maximum (330, 7)

7 c Minimum Maximum (67.5, 10.9), (247.5, 10.9)

8 a Minimum Maximum b Minimum Maximum

8 c Minimum

8 c Maximum

9 a b c d 10 a 102.4, 195.7 b 7.3, 34.0, 127.3, 154.0, 247.3, 274.0p

3, p

2,

5p6

, p, 4p3

, 3p2

, 11p

60,

3p2

0, 5p3

p

2, p

13p12

, 0.172, 4p3

, 0.172, 19p12

, 0.172, 11p6

, 0.172 p12

, 0.172, p3

, 0.172, 7p12

, 0.172, 5p6

, 0.172 29p24

, 5.83, 35p24

, 5.83, 41p24

, 5.83, 47p24

, 5.835p24

, 5.83, 11p24

, 5.83, 17p24

, 5.83, 23p24

, 5.83 7p18

, 2, 19p18

, 2, 31p18

, 2 p18

, 2, 13p18

, 2, 25p18

, 212.19, 212 215.33, 212 21157.5, 8.90 2 , 1337.5, 8.90 21315, 522 21135, 522 2

22 cos 30u 3p46 cos u 11p

62 cos13x 300 222 cos12x 45 22 sin1x 240 2210 sin1x 108.4 2 22 sin u

3p42 sin u 2p

35

2 cos1x 36.9 217 cos1x 28.1 22 cos u 2p

325 cos1u 4.25 2 22 cos u

7p42 cos u 11p

625 cos1x 153.4 2210 cos1x 288.4 213 cos1x 67.4 210 cos1x 53.1 2

1 2 b i ii 6 7 a b 8 0.905 9 10 11

12 13 14 11.25, 22.5, 56.25, 67.5 17 b c A 1.059, u 1.80a 0.464, k 0.559, p 0.52112 sin1u 3.86 23512

2

22 121385

1 23222u 0.6445 cos1u 0.644 2

p

6, p

3, p

2,

2p3

, 5p6

119169

120169

;232


1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 f 1x 2 2 1x2

f 1x 2 2x 4f 1x 2 3f 1x 2 2xf 1x 2 3x2

f 1x 2 0f 1x 2 12x2f 1x 2 10xf 1x 2 4xf 1x 2 4x3f 1x 2 3x2f 1x 2 2xf 1x 2 2f 1x 2 8f 1x 2 5

Exercise 1Chapter 8

Exercise 2Chapter 8

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

17 18 19 20 21 22 23 (5, 7)

24 and 5, 21761

2,

19924

dy

dx

3081128

dy

dx 6g 16 2 10

9f 13 2 2dy

dx

2710

12x7 1x 2dydx

43

4

x2

dy

dx

5 32x23

dy

dx

1

4 42x3dy

dx 12x

2

x2dy

dx 3x2 10x 7f 1x 2 25

22x7f 1x 2 2x 5f 1x 2 5

21x

f 1x 2 8x3

f 1x 2 8f 1x 2 11f 1x 2 7f 1x 2 0f 1x 2 15x4f 1x 2 24x3f 1x 2 30x2f 1x 2 18x

1 a Tangent Normal b Tangent Normal c Tangent Normal

1 d Tangent Normal e Tangent Normal f Tangent Normal

2 At A, At B, 3 4 5 6 7 and

8 9 and and 10 1207

y 5x 10y 5x 10, y 5 x 15y 5x 15y 3x 389

y 19x 57y 7x 283

14

, 94y 2x 11Q13, 20 2y 1

3 x

43

y 3x 3y 4x 4,

y 1

18 x

436

y 18x 47,y 18

x 318

y 8x 12,y 6x 57y 16

x 32

,

y 14

x 54

y 4x 3,y xy x 4,y 16

x 196

y 6x 3,

Exercise 3Chapter 8

Exercise 4Chapter 8

1 a Minimum Turning Point b Maximum Turning Point and Minimum Turning Point c (0, 0) Minimum Turning Point

1 d Minimum Turning Point e Maximum Turning Point and Minimum Turning Point12

, 412

, 413

, 253 12, 9 212, 23 214, 13 2

21 22

x2

16

y2

9 25y

4

x2 2

x

y

0 2020x

y

0

222

Answers

695

2 a Minimum Turning Point b Maximum Turning Point c Maximum Turning Point and (4, 0) Minimum Turning Point

2 d (1, 10) Maximum Turning Point and (2, 9) Minimum Turning Point e (0, 0) Rising Point of Inflexion

3 a Maximum Turning Point and Minimum Turning Point b Minimum Turning Point c Maximum Turning Point

3 d (0, 0) Minimum Turning Point e Maximum Turning Point and (0, 2) Minimum Turning Point 4 211.57, 12.5 21

2, 125

2, 013, 4 21, 8

3

43

, 2562711, 25 212, 13 2Answers

694

Exercise 5Chapter 8

1 Non-Stationary, (0,0) Stationary, Non-Stationary 2 Non-Stationary 3 a None b None

3 c None Quadratic functions have no points of inflexion 4 a (0,0) Stationary b Non-Stationary c Non-Stationary

4 d There is always one point of inflexion for cubic functions

5 Non-Stationary and (1, 3) Non-Stationary 6 Non-Stationary and Non-Stationary

7 (0,0) Stationary 8 No points of inflexion, is a minimum turning point. 9 10 21312 36.2y 21x 3610, 3 211, 9 21

6, 4.1111, 3 2x b3a, y 8b

3

27a2

bc3a

d

12, 35 211, 3 211, 15 22, 224

32, 224

3

Exercise 6Chapter 8

1 Vertical: Horizontal 2 Vertical: Horizontal 3 Vertical: Oblique

4 Vertical: Horizontal 5 Vertical: Oblique 6 Vertical: Oblique

7 Vertical: None, Oblique 8 Vertical: None, Oblique 9 Vertical: and Horizontal

10 Vertical: and Horizontal 11 Vertical: and Horizontal

12 Vertical: and Oblique

Graphs of the following functions, including asymptotes, stationary points and intercepts:

13 14 15 16

17 18 19 20

y 1

x2 x 12y

12x 5 2 1x 4 21x 2 2 1x 3 2y

x2

1 xy

x

x2 1

x

y

043 1

12

, 0.081612

x

y(0.124, 3.33)

0 3 422

52

103

x

y

(2, 4)0x

y

01

y 2x2

x 1y

xx 4

y x 1

x1x 1 2y x 1x 1

x

y

(2, 8)

01

x

y

04

1

x

y(0.414, 5.83)

(2.41, 0.172)01 1x

y

01 1

1

1

y 4x 4x 3,x 2

y 3x 3,x 3y 1x 1,x 1

y 0x 4,x 1y xy x 2

y 2x 9x 3,y x 2x 2,y 1x 2,

y xx 0,y 1x 3,y 0x 0,

Exercise 7Chapter 8

1 a b c d

1 e f g h

1 i 2 a b c

2 d e f g

2 h i j

x

y

0 11 63x

y

0 22x

y

30 61

x

y

a0 cx

y

40 1x

y

50 1x

y

201

x

y

0 3x

y

0 4x

y

0

3

x

y

0

x

y

(2, 1)0

13

3

x

y

0x

y

0

1

x

y

0

x

y

0x

y

0x

y

01

x

y

0

4

10 11 12

13 a b

y

x2 3

(3.36, 4.39)

(5.36, 8.87)

a 4, b 18y 1, x 1, x 4

Answers

697

3 a b c d

4 a b c d

4 e f g h

5 a b c d

e f y

x

2 7

151, 1

83,

y

x3 1 4

(4, 1)143,

171,

0

y

x3 30

y

x20

y

x2 40

y

x

12

3 0

x

y

012

x

y

01

1

2

32

2x

y

0 1x

y

01

x

y

0 4

12

x

y

0 12 3x

y

0

14

x

y

0

1

12

x

y

05 1 6x

y

01 7x

y

02x

y

01

Answers

696


1 2 3 4 5 6

7 8 9y

x1 20 6

y 5x 1, y 12

x 172

, 53

, 2831

2,

92 max TP, 2

3,

5027 min TP y 6 x 16h1x 2 45x

2 60x 19, h 1x 2 90x 60x 7 4dydx

34

x12

174

x152

116

f 1x 2 3x2 4

y

x4

1131, 173,

y

x3 4

142,

171,

1 2 3 4 5 6

7 8dy

dx 7 5 cos sec x tan x

dy

dx 18x 4 sin x

dy

dx 3 sec x tan x

dy

dx 7 cosec2 x

dy

dx 5 sin x

dy

dx cos x 12x

dy

dx cos x cosec x cot x

dy

dx sec2 x

Exercise 1Chapter 9

1 2 3 4 5 6

7 8 9 10 11

12 13 14 15 16

17 18 19 20 21 22

23 24 25 26

27 28 29 30dy

dx 1x 1 212 sec212x 1 2dy

dx 3 sin3x p

4dy

dx 3013x 4 26 2 sec2 2x tan 2xdy

dx 12x3 3 cos2 x sin x

dy

dx 8 tan 4x sec2 4x

dy

dx 3 sin2 x cos x

dy

dx 5 cos 5x 3013x 4 272dy

dx 2 cosec 2x cot 2x 1213x 2 23

dy

dx 6 3 cosec2 3x

dy

dx 9 sec 9x tan 9x

dy

dx 6 sec2 6x

dy

dx

12

cos 12

xdy

dx 3 sin 3x

dy

dx 4 cos 4x

dNdp

752

18 5p 252dPdk

1814 3k 23f 1x 2 5613 8x 22f 1x 2 2015x 4 22dydx

32

13x 2 232

dy

dx 216x 5 223dy

dx

103

12x 9 2 23dydx

32

13x 8 212dydx

4812x 3 25dydx

2019 4x 24f 1x 2 817 2x 23f 1x 2 315 x 22f 1x 2 2015x 4 23f 1x 2 613x 4 2f 1x 2 412x 3 2f 1x 2 21x 4 2

Exercise 2Chapter 9

Exercise 3Chapter 9

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15

16 17 18 19 20 21

22 23dy

dx

1x

sec21ln x 2dydx

cosec x sec x

dy

dx tan x

dy

dx 4e4x 2 cos 2x

1x

dy

dx ln 4 # 4x 1

x ln 5

dy

dx

1x ln 8

dy

dx

1x ln 2

dy

dx

1x

ln 2 # 2x

dy

dx 3e3x ln 3 # 3xdy

dx 6 ln 5 # 5xdy

dx ln 10 # 10xdy

dx ln 4 # 4xf 1x 2 2x

x2 2f 1x 2 2

xf 1x 2 1

x

f 1x 2 1x

f 1x 2 2e2x3f 1x 2 2xex2f 1x 2 54e9xf 1x 2 10e5xf 1x 2 4e4xf 1x 2 7e7xf 1x 2 3e3x

1 2 3 4 5

6 7 8 9

10 11 12 13

14 15 16 17

18 19 20

21 22

23 24 25dy

dx e3x1x 2 2 13x 8 2 tan x 1x 2 2 sec2 xdy

dx x1 12 ln x 1 2sin x x ln x cos x 2dy

dx 3x5 4 tan 3x p

2 3x sec2 3x p

2

dy

dx e3x sec 2x p

4 B3 2 tan 2x p

4Rdy

dx 8x Bln1x2 2x 5 2 x1x 1 2

x2 2x 5R

dy

dx ln12x 3 2 2x

2x 3

dy

dx 4x ln 4 log8 x 1x ln 8dydx 312x 1 22 cosec 3x 32 12x 1 2 cot 3x 4

dy

dx e4x sec 3x14 3 tan 3x 2dy

dx x2 a3 log6 x 1 ln 6dydx 5x1ln 5 # cos x sin x 2dydx 13x 4 22 39 sin x 13x 4 2 cos x 4

dy

dx 615 2x 2213x 4 2 11 5x 2dy

dx 21x 5 2 13x 2 2319x 28 2dy

dx 12x 1 2218x 11 2dy

dx 6x213x 2 2 15x 2 2

dy

dx x21x 2 2313 4x 2dy

dx 2x1x 1 2 12x 1 2dy

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

dy

dx cos2 x sin2 x

dy

dx

sin xx

ln x cos xdy

dx e3x13 sin x cos x 2dy

dx 3xex12 x 2dy

dx x213 cos x x sin x 2dy

dx x12 sin x x cos x 2

Exercise 4Chapter 9

Answers

699

Answers

698

Exercise 5Chapter 9

1 2 3 4 5

6 7 8 9 10

11 12 13 14

15 16 17 18

19dy

dx

2 cosec2 x2x p3 ln13x 1 2 3

3x 1 cot2x p

3

1ln13x 1 2 22

dy

dx

tanx p4 2

e2x cosx p4dydx xe3x13x2 15x 10 21x 5 22dydx 1x 1 2 sin x x cos xexdydx 1213x 2 2414x 19 212x 3 24

dy

dx

21cos 2x 3 sin 2x 2e6x

dy

dx

2

1ex ex 22dy

dx

1x

ln1x 4 2 1x 4

ln x

1ln1x 4 2 22dy

dx

1x ln 6

1x 6 2 log6 x1x 6 22

dy

dx

e3x13x 2 29x3

f 1x 2 x 21x 1 2 32f 1x 2 4x12x ln 4 1 2

2x32

f 1x 2 31x 12 22x32x 9f 1x 2

6

1x 3 22

f 1x 2 ex1x 5 21x 4 22f 1x 2

1 ln x

4x2f 1x 2 7 tan x 7x sec2 x

tan2 xf 1x 2 6x1x 6 21x 3 22f 1x 2

ex1cos x sin x 2cos2 x

1 2 3 4 5 6

7 8 9 10 11

12 13

14 15 16 Answers given19.y 32

x 52

, y 12

x 52

d2y

dx2

1

x2 dy

dx

1x

d2y

dx2

14x 1 2sin x 4 dydx 41cos x 4y 2

14x 1 22 dy

dx

cos x 4y

4x 1 d2y

dx212 3y 22 3x12 3y 2 x2

12 3y 23 dy

dx

x2 3y

dy

dx

y218 ex 2 4y1x y 231x y 2313y x 2

dy

dx

31x y 2231x y 22 ey

dy

dx

4x3

1 ln y

dy

dx

sin1x y 21 sin1x y 2

dy

dx

13 cos 3x 2e2xy3 23e2xy2

dy

dx

21x y 2ey 2x 2y

dy

dx

y

2y x

dy

dx

1 xy 2x

x1x 3 2dy

dx

1

62xy2dy

dx

4xy

dy

dx

3x2 y

x

1 2 3 4 5 6

7 8 9dy

dx

1

x21 1ln 5x 22dy

dx

1

2x2 2x 1

dy

dx

1

221x 4 2 1x 3 2

dy

dx

2ex

4 e2xdy

dx

3

21 9x2dy

dx

2

29 4x2dy

dx

10

100 x2dy

dx

1

264 x2dy

dx

1

225 x2

Exercise 6Chapter 9

Exercise 7Chapter 9

Exercise 8Chapter 9

1 2 3 4 5

6 7 8 9

10 11 12 13

14 15 16 17

18 19 20dy

dx

y14x3y2 cos x 21sin x 3x4y2 2

dy

dx

y12x exy 2x2 2exy

f 14 2 p 264

f 12 2 203

dy

dx

1sec x cosec x ln1cot x 2 2ex

dy

dx sin x1ln x 1 2 x ln x cos xf 1x 2

x21 x2 2 cos1 x3x3

dy

dx

12

21 4x2dy

dx 61sin 2x sin 4x 2 cos 2x cos 4x 2dy

dx

x ln x1x 18 2 x1x 9 21x 9 22

dy

dx

x 4x ln 2

3 log2 x

1x 4 24

f 1x 2 3x1ln 3 # sin x cos x 2dydx

sin x 4x sin x x cos x

e4xf 1x 2 3 cos 3x sin 3x

exdy

dx x12 ln x 1 2

f 1x 2 x2e4x13 4x 2dydx

sec x tan x 5e5xf 1x 2 8 sin 8x 32

x12

dy

dx 612x 7 22f 1x 2 2x 5

1 2 0 3 4 6 7 (0,0) and 8 (0, 0) and

9 (0, 0) rising point of inflexion and max TP3, 27e3 2,

4

e22 2p

epy

75

x 175

, y 35

x 135

4 ln 4 3

e234

1 2 All three sides are 3 6.83cm 4 5.62cm, 4.22cm, 16.9cm 5 4.16cm, 6.56cm 6 7 78.5 8 b

9 It is a maximum value. 10 b 10 c It is a maximum value.

11 12 11100 13 12.5 Yen, 14 20 15 a b c 42cm2A 168x 24x2

77 x

7

y

2424.1Kmh1t 6.29 seconds, t 12.6 seconds

x 1.58cm.V 60 450x 60x313

x y 75cm. 4r33.66cm2250cmv 2500ms1Exercise 1Chapter 10

Exercise 9Chapter 9

1 2 3 4 5 6 7 8 9

10 a 11 13 a b4p3

cm2s129

cms1100cm3s11au a sin u, a a cos u 2

53

0.283cms11.49cm2s10.954ms1216cm3s11640N02h

1532p

128pcm2s1

Exercise 2Chapter 10

1 2 a b c 3 a b c

4 a b c 7

8 a b c d 10 a

10 b c 11 a b. c d 0.629

12 a Min is (2,1.39) b c d

13 a b c s t11 < 23 2a 31s2 2st 2t2 21t s 23v 2t st s

v 445

ms1 a 31

4050 ms2a

2

t2

1

1t 1 22 t 2v 2t

1

t 1 t 2

a ln 300

50 e

ln 300100 t

23t ln 30050

t2v eln 300100 t21 ln 30050

t2k 1100

ln 300a 0.0213ms2t p 2c

2k

v 3k cos1kt c 2a 10e49

a 2e2t1t4 6t3 4t2 6t 1 2

1t2 1 23v e4

9s

e2t1t2 2t 1 21t2 1 22

Period pa t3 sin t 2t2 sin t 2t cos t 2 cos t t sin t 2 sin t

1t 1 231.74ms1v t2 cos t t cos t sin t

1t 1 22

0ms2t 12

second411 2t 220ms216 12t8ms1v 3 14

ms1 a 3ms2 t 223

3 seconds


Review Exercise Chapter 10


1 2 a b

2 c Maximum velocity is Minimum velocity is 3 4 5

6 a b 4.30 cm c 7 a b 0.457 hours c 0.304 hours 8 a

8 b 9 10 11 a 6.28 hours. b 6.28 hours. 12 13 a BC 21x h csc u 27043

cm3 s15627

ms2p

3 cm2 s11.12 units2

1p 2x 2 sin x6t sin 5t 15t2 cos 5t1120 cm2A 2prh pr21.6b 22.7 cma 26.2 cm

12p

cms11.35 ms10.541 ms1

t 4.70 seconds and t 1.05 seconds13 ms2240 kmh1


1 a b c d 2 Week 1 Week 2 Week 3

The operation is addition.

4 21 01 1

1 24 10 1

3 12 24 4

4 13 32 31 3Magazines Newspapers

Alan 8 5Bill 7 3Colin 5 6

e , 2e l2-

1 a b c d e 2 a b

2 c d e 3 a b

4 5 6 7 34

161x 4p1x

p2211 csc x cot x cot2 x 2

1 1 cos xsin x

22 sin x 4x cos x 2x sin x x2 1cos x sin x 2ex 2x sin x x2 cos xe2x y3x2

ln x

y

x ln x

6xy

8y 3x21

ln 10 2 3 tan 3x 2

x 43

3x 4

22x 1

e5x

2x 4 5 121x 4 2 1x

cot x4e4x sin 3x 3e4x cos 3x1x

3x ln 348e8x6 3 tan 3t sec 3t14x13 2x2 2326013x 2 23

13 b c 14 a b c d 1.48 minutes27.7 cm2 min1112t3

18 t4 22 cm min1t 2 minutesh12x h23 22xh h21cot u 2 csc u 2

8 9 10 11 2.533 12 13 . 14 a b i ii 5 15 a b 45

13 5p6 sec 2t tan 2t 5212x 1 22y x4

52

x 043

13

6 a Consistent. Lines intersect giving unique solution. b Consistent. Same line giving infinite solutions. c Consistent. Same line giving infinite solutions.

7 8 l 6, x 1, y 1p 3

Answers

701

3 a b c d 4 a b c d

4 e f g 5 a not possible b c not possible d e f

5 g h 6 a b c d e

6 f 7 a b c d e

7 f g h 8 a b c d

9 10 a b c

Total number of calories consumed on day 1 and day 2. d i (33500). Total calories consumed. ii Total number of people dieting on each day.

iii (4300) Total number of calories consumed by one man, one women and one child. 11 a b c

12 a b c 13 14 16 17

18 19 a b c 20 a b0 a

x 133

, y 0x 5, y 0x 85

, y 10QP 5 c 22 152 8 63 c2 12 2c 9

PQ 4 1 3c5c 4 c

c 2c 5k 10m 1, n 66 00 6

7 96 4

1 64 1

132031

1827

Q 310P 3 4 76 2 610 1 3

3 9 28 3 3

914AB

1270020800

B 190013001100

A 3 2 45 7 2

A3 40 1339 12

A2 13 412 3

x 4, y 3x 10, y 1x 0, y 1x 2, y 1264 16899 63

6 84 9k6 52 5k

0 011 7 11k

9 9k 1227 27k 6

45 39k 4 18k39 18

45 45k 439 39k 18

45 45k 439 39k 18

45 439 18

7 3k2 4k54 k

1 2k 2k2

6 k2 2 2kk2 3 2k 1

29 39 1111 41 92 29 2

11 296 2212 7

126 224 116 29

15 11 5

18 58 154 37

9 20 3

7 11 3

11 56 08 1

10 20 4

x 0, y 11k 5k 12

k 0, 3k 1k 0, 1k 631k 1 2 21k 1 21 k 0

3k 6k4k k12k 4k

18 246 1218 24

8 4 1220 8 1228 16 4

Answers

700


1 a b c d e f

1 g h 2 3 a b 44 c 78 d 190

4 a b c d 2abc e f g

5 a 5 b c 6 a Collinear b Not collinear c Collinear 8 10

11 12 13 a b 14 b18

x 3 x 1x 1 3 x

X I13

B112A B 2x y 1y 3.36, 3.86c 5, 4.5k 1.22, 1.0820 units24 units24 units2

a3 a2b ab2 b3y2 y 2cos3 u sin3 u 3 sin2 u tan3 u2sin 2ucos 2 u sin 2 u 1

30 Z 4318 31181918

1118

Y 52 12

7 5

X 1 3141

514

1k2 7k

k 2 k1 2k 3k

114k

1 53k k

37304 33152 976 1

38519

119

17152

976

138

237 25333 8333374

137

1174

574

5111

19222

11137 6137 2927410137

7137

11274

24137

38137

1

274

176

1 89 4

141

10 73 2

111

5 23 1

1 a b c d 2 a b

2 c d 3 a b

3 c d 4 a Unique solution b Unique solution

4 c No unique solution d No unique solution 5 a Lines are parallel. b Lines are parallel.c 2.c 1, 0.

x 31 33k

3k2 2k 5, y

5k 6

3k2 2k 5, k

53

, 1x 12

11k 5, y

13k 711k 5

, k

511

x 313 k 23 k2

, y 31k 1 23 k2

, k ;23x 13k 1

, y 5k 23k 1

, k

13

x 165

, y 695

x 57

, y 227

a 1911

, b 9

11x

174

, y 1120

x 7, y 17x 56

, y 29

p 21, q 14x 157

, y 177


Exercise 4Chapter 111 a b c d 2 a

2 b c d 3 a b

3 c d 4 a Unique solution. b No unique solution.

4 c No unique solution. d No unique solution. 5 a b

5 c d 6 a b

6 c d 7 a b

7 c d e f No solution. g No solution.

7 h 8 a b c No solution.

8 d No solution. e f 9 a b

10 a 0 b c

12 13 a b 14 a 36k 7.86, 2.362k2 11k 37a 2 4bb 20

x l, y l, z 2lc 3

x 7

15, y

25

, z 19

15 35 215 110

15

110

16

0118

x 1, y 2, z 3x 44 4l11

, y l, z 3l11

x 544

, y 1544

, z 722

x 2l 3, y l, z 5l 10

2x 4, y 4, z 6

x 75

, y 0, z 25

x l, y m, z 4 2l m

3x

19 5l13

, y l, z 7l 11

13

x 215

, y 415

, z 43

x 5 24l

2, y l, z

2l 34

x 0, y 2, z 3x 3566

, y 4366

, z 1333

x 2, y 1, z 1x 7419

, y 3

19, z

919

x 14

, y 12

, z 2x 1, y 2, z 4

x 3

19, y

5919

, z 2819

x 1, y 1, z 1Determinant 0.Determinant 0.

Determinant 0.Determinant 6.x 2, y 3, z 2x 12

, y 1, z 2

x 5547

, y 4047

, z 6447

x 4, y 13

, z 23

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

x 3, y 2, z 4x 2, y 1, z 2x 2, y 4, z 3x 10, y 10, z 36x 1, y 9, z 13


1 R is an matrix R is an matrix 2 3 a b 4

5 a b c d If there is no solution.

5 d Otherwise 6 7 a

7 b Solution is not unique. 8 b 9 10 a

10 b c Since M is singular, A must be singular. 11 12 b c

13 a b 15 16

17 y3 13z1 12z2 17z3 y2 18z1 11z2 3z3

y1 16z1 36z2 58z3 a 4, b 1a 1x 1, y 2, z 1a 7, b 2

x 1 7l

2, y l, z

11l 72

c 3k 536 39 5014 16 2015 18 22

c 2.5, 0.5, 2p 3, q 5k 3x 17 3l

51, y l, z

5l 1117

3p 3q r 0a 1, b 3x k3 2k2 29k 22

8 3k, y

k2 6k 148 3k

z k2 5k 4

8 3k

k 83

z k2 5k 4

8 3kk

83

1 k 3 51 3 k 11 1 k

xyz 0k 2

2k 1 x 1, y 8x 1, y k

1

k2 1 k 1

1 kl 1 or 6m pn p

1 a b c 2 a b

3 a b 4 5 a b c d e f 6 a Parallel b Parallel

6 c Not parallel d Not parallel 7 a b c 8 9 10 a Not parallel b Not parallel c Parallel 526262621

262

1321323

2

c 6c 7c 62572212292902532341

3AB 213AB 3

2 PQ 217PQ i 4j b 3 a 47 c 8 b 0, 12 a 13 c 6 b 12 a 1


Answers

703

11 12 a

12 b c RP 214 SR 221 QR 221PQ 221 RP 321 SR 42

1QR 14

2PQ 42

1

R 335 PR 213 QR 252 QR 221 PQ 213 PR 2

3 QR 4

6PQ 2

3

Answers

702

1 a b c d e f

1 g 2 a b c 43.0 3 Ratio is 1:2 4 a b AC 12

1b a 2AB b aq 1242117i 16j5mi 27mj 17mk20i 25j 84k3i 6j 49k17i 21j 16k2i 11j 2k6i 2j 9k3i 4j k


1 a 11 b c 29 d 2 e 7 f 54 g 40 h 25 2 a b 6 c 30 d e 3 f 55 g h 1 3 a 58.7 b 86.6

3 c 24.8 d 129 e 54.0 f 50.0 4 , 5 a and d, a and f, b and c, b and e. 6 a b 11 c d 2

8 9 or 10 70.5 16 It is a rectangle since is 90 but we do not know if AB BCABCx 6.5x 17.95

235 i 3

235 j 1

235 k3,

152

32

cos u B 719p . q 1426951


1 a b c d e f g 24

2 a b c 6 7 a b 8 a b

9 10 11 12 13 14 15

16 and Area 1021734 units2PS 6i 14j 2kPQ 25i 5j 10k2234 units22850 units22756 units226 units22341

2 units2

2742

units23

219 i 3

219 j 1

219 kB5455 226 i

1

26 j 1

26 kB817986

18

2817 i 3

2817 j 22

2817 k9239

22123

2

42i 15j 24k14i 5j 8k28i 10j 16k14i 5j 8k14i 5j 8k 14i 5j 8k



1 ii iii iv 29.6 2 a b 4 5

6 0 8 0.70210 11 a A has coordinates (2, 4, 6) B has coordinates C has coordinates

11 b 11 c d 96.3 e 12 m 103

22.6 units2 332

0

50.2 units214, 7, 6 216, 3, 0 2a p

2 2u

5 cos u 35 sin u 2

22 2 cos up 46i 12j 12p 1 2k10.8i 9.6j 1.2k2378 units2

1 a b c d e

1 f 2 a b c d

2 e 3 a

3 b

3 c

3 d

4 a b

4 c d

4 e f

5 a b c d 6 a

6 b c d e f

7 a No b Yes c No d Yes e No f Yes 8 Position vector is

9 Crosses the xy plane at Crosses the yz plane at Crosses the xz plane at

10 a Crosses the xy plane at Crosses the yz plane at Crosses the xz plane at

10 b Crosses the xy plane at (9, 1, 0) Crosses the yz plane at (0, 8, 9) Crosses the xz plane at (8, 0, 1)

11 Crosses the xy plane at Crosses the yz plane at Crosses the xz plane at 173

, 0, 1430, 17

4,

291229

15,

145

, 0r 267 l 11

1 11, 0, 4 20,

13

, 133113, 4, 0 2r 2i j 5k l13i j k 2

112

, 0, 740, 11, 9

219, 7, 0 2

6520

r 371 l 69

2

r 5372

l 4940r 532

313

l 123r 164

34

l 11

9r 255

3

72

l 45232

r 5211

l 24

32

r 35

1 l 43

3r 41

5 n 01

1r 40

3 m 53

4r 45

1 m 13

5r 76

4 l 31

2

x 12

y 6

5x 1 2t, y 6 5t

x 43

6 y

5x 4 3s, y 6 5s

x 1

2

y 1

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

x 24

8 y

7

z 16

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

x 2

3

y 5

1

z 14

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

2

y 1

3

2 z2

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

4 x y 1

2

z2

x 4 l, y 1 2l, z 2l r 410 l 12

2

y 1, x 1 z

3 x l, y 1, z 1 3l r j k l1i 3k 2

x 34

y 2

7

z 33

x 3 4l, y 2 7l, z 3 3l r i 2j 4k l13i j 5k 2

x 1

3 y 2

z 45

x 1 3l, y 2 l, z 4 5l r 323 l 47

3r 4

3 l 3

0

r 342 l 1 9

1r 22

3 l 29

6r 31

0 l 72

2r 21

2 l 43

1r 5

1 l 4

7

r 3i j l12i j 2r 5i 2j k l13i 6j k 2r 443 l 05

12r i 2j l1i 4j 2k 2r 02

3 l 12

1 Exercise 1Chapter 13


1 a Skew b Intersect at the point (2, 1, 6) c Parallel d Skew e Skew f Parallel g Skew

2 a b c 1 d e Lies on line. f

3 Coordinates of C are (4, 7, 4)

4 a 60.5 b 36.3 c 71.2 d 88.4 e 62.8 5 Point of intersection is 6 7 p 34r 38 t 3

211, 5, 3 2a 2

x y 1

2 z 2AD: r 01

2 l 12

1x

3

y 1

4

z 27

AB: r 012 l 34

7

22357

21542

23933

2262

12 f RP 3214 2214 1

214

SR 4221 22211

221

QR 122142212

22

PQ 4221 22211

221

14 15 16 17 x

6953

, y 7553

p 47

, q 47

, r 67

a 1, b 1SU a b

4 c d 5 a b c d 6 a b c d

7 a b c d

8 a i ii iii iv v b c

9 a 7 b c d e

10 a b c d e 12 a

12 c 14 a b They are perpendicular.AB c, BC a, AC c a, OB a cFA b c d

DG d a bBH b c aAG a b cAH b cFH b aBC b

ED a 12m 3n 9 2b

m n 3AF a

1m n 2bm n 3

CF 3b

m n 3EF

nbm n 3

BE mb

m n 3

BX 2k3cAC 6k

4cXD 2

3 bAX

13

bBD b aCA b aCD a

ka b1 k

1a b 2k1 k

1a b 211 k

1b a 2

3454

352136 6

10 2

8 52

10

2425 520OC 1

2 1a b 2CB 1

2 1b a 2

Answers

705

1 a b c d

1 e f g

1 h i j k r. 3213 i 2213 j 15213r. 3226 i 4226 j 1226 k 0r. 20

25576

255711

2557

862557r. 7

235417

23544

2354

322354r. 1021133 i 321133 j 3221133 k 6321133r. 3219 i 3219 j 1219 k 3219r. 1274 i 8274 j 3274 k 19274

r. 1218 i 4218 j 1218 k 15218r. 2250125 625r. 1235 i 3235 j 5235 k 8235r.

1

2184

2181

218

3218Answers

704



1 a b c d e f

2 a 45.0 b 23.2 c 11.0 d 64.5 e 90 f 55.9 3 a 11.7 b 57.8 c 17.6 d 32.5 e 34.1 f 5.51

4 a b c d r 1i 6j 2 l13i 2j k 2r 17i 3j 2 l12i k 2r 220 l 71

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

116, 10, 7 2203

, 13

9,

43a 51

7 ,

17

, 1 b13, 2, 9 24, 12

, 2a9, 12

,23

b

2 iv AD has equation BD has equation v 16.4 3 b

3 c 4 ii iii iv 5 i ii iii

5 iv 116 6 a b c d 8.52 7 8 a

8 b c d e f g h

9 i A lies in the plane. B does not lie in the plane ii iii 43.6 iv 3.16 units. v

10 a b i and ii 79.0 d i

11 i ii iii 0.716 iv v vi r 01318 l13313

263

133

, 0, 28310, 13, 18 214, 1, 4 2r 12i j 4k 2 l13i j 2

18, 20, 12 2n2 2i 2j kn1 6i 3j 2k2i 2j kr. 1i 3j 10k 2 872 x

2

y 3

4 z 8

14, 5, 6 2i26 j

26 2k

26326z 6 2ly 1 lx 2 lr. 112 3262i j 2kAB 13

1 BC 11

0r 13

0 l11

14

3,

53

, 13

31

3,

103

, 83x 2

1

y 5

1

z 11

r 614 l258

616, 1, 4 21, 13

5,

12533, 51

2, 4r. 48

5 0 2

21,

421

, 821r 21

1 l 12

1

3x 2y z 52x 4 4 2y

9

6 2z7

x 1 1 y

3

1 z4


1 5x 2 10x 3 4 5 6 7 8 9 10 4x13x3x5x4x36x22x22x


1 2 3 4 5 6 7 8

9 10 11 12 13 14 15

16 17 18 19 20 21

22 23 24 25 26

27 28 29 30 31 y 23

t12

49

t3 cy t1 3 t2 3t3 cy 16

z6 12

z2 cy 329

k94 cy 6p2 c

y 1419

x194 2x

32 cy

87

x72

143

x32 cy

12

x2 54

x4 cy 3x3 12x2 16x cy 23

x3 212

x2 27x c

y 4x4 12x2 cy 8x 92

x23 cy

23

x32 2x

12 cy

12

x4 c3x2 cx x2 2x3 14

x4 c

x4 2x2 9x c2x52 2x2 c

27

x7 x5 c2x12 c7x 2x2 c3x

13 c

23

x32 c

x1 c53

x3 4x c2x4 2x2 3x c2x3 5x c15

x5 c14

x4 c13

x3 cx2 x c


1 2 3 4 5 6

7 8 9 Q 2

21 p

72

833

p72 2y

17

t7 35

t5 43

t3 818105

y 16x12 46

y 43

x3 6x1 5096

y x4 2x3 7x 3y x2 5xy 4x2 3x 18y 2x2 3y 6x 4


12 a ii b Line of intersection. c ii 13

14 a b (8, 5, 13) c e i ii iii

15 b c d 16 a b c d (6. 63, 3.50, 4.65)375

, 45

, 95x 2y 94, 5

2,

13213221r. 1i 4j 2k 2 94221

12

214PO i 2j 4ki j k2x 3y 4z 4 0r 12i 3j 7k 2 t13i j 3k 2

r 704 l71

5722

6c 2.r 1j k 2 l13i 11j k 2

1 l 2 a b c d

3 a b c d 4

5 a b c d 7

8 The direction normals are equal. Distance units.

9 b c Distance of to origin is units Distance of to origin is units Distance between and is units

10 Distance of to origin is units Distance of to origin is units Distance between and is units

11 is not contained in the plane. is contained in the plane.

12 Distance of plane from the origin is units.

13 Distance of plane from the origin is units. 14

15 b c Distance of to origin is units Distance of to origin is units Distance between and is units

16 a The line and the plane intersect. b The line and the plane are parallel. c The line is contained in the plane. d The line and the plane intersect.

17

23P2P19

23P28

23P1r. 1i j k 2 9r 14i 3j 7k 2 l12i 2j 5k 213217r. 232 13217

17

289r. 8289 i 5289 k 17289 r2r1r. 1i k 2 155

265P2P114

265P241

265P1

7

227p2p11

227p28

227p1r. 1i 5j k 2 1

8

219

r. 113i 4j 11k 2 315245 i 2

245 j 4

245 k37

i 67

j 27

k4

242 i 1

242 j 5

242 k1

230 i 2

230 j 5

230 k

r. 137 104x 3y 8z 2115x 13y 8z 384x y 6x 2y 7z 9

r. 1i 11j 9k 2 16r. 15311 3r. 15i 4j 15k 2 42r. 15j 2k 2 2r. 7265 i 4265 k 0

4 e f 6 48.5 7 a b

8 a b 74.2 c 13.9 units.a 3

12, 1, 0 2r 123 l 31

3r 56

25 i

2925

j l136i 26j 25k 2r 18i 14j 2 l137i 61j 10k 21 i ii iii 73.5 2 ii iii

1

230 i 2

230 j 5

230 k230

2 units2r. 32178 i 122178 j 52178 k 7217832178 i 122178 j 52178 k

1 2 3 4 5

6 7 8 ex

15 10x

32 sin x c

13

ex 52

lnx 7 cos x c5ex 2 cos x 3 lnx c

8 cos x 7ex c6 cos x 65

x5 c5 lnx sin x c4ex cos x c14

x4 2 lnx c


18 19 20 21 22

23 24 25 26 27 2816

ln2p2 2p 5lne2x 1 c1lnp 22 1913x2 3x 4 23 c

13

ln3x2 3x 4 c15

128

16

ln3 tan 2x 7 c1

21 11 3 cos 2x 2 72 c2

15 13x3 6x 19 2 52 c1

3 ln3 sin x 12 c

12

ln2ex 4 c

Answers

707

Answers

706


1 2 3 4 5 6 7 8

9 10 11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

26 27 28 29 30

31 32 2x3 23

ln3x 2 c12

ln2x 1 1

18 13x 4 26 c

12

e8x 2 sin 2x c13

cos 3x 2x2 c32

e4x cy 3 ln6 t cy 8 ln4 p c

y 2 ln3x 5 cy 43

ln3x 1 cy 32

12t 1 21 cy 23

13x 2 22 cy 110

13 2x 25 cy

18

14x 3 22 cy 128

14x 7 27 cy 118

13x 1 26 cy 2 ln2x 5 cy 18

ln8x 7 c

y 12

ln2x 3 c2e2x c4x2 12

e2x c56

e6p c43

e6t ce4x c15

e5x c

16

e6x c52

sin 2x c2 cos 3x c2 sin 4x c2 cos 12

x c12

cos 2x c16

sin 6x c15

cos 5x c


1 3 2 38 3 20 4 0 5 6 7 201 8 216468 9 1490 10 11 0 12 13 21 14 15 312.6

16 0.490 17 0.0429 18 0.549 19 1.85 20 21 8.56 22 23 2 ln 2k 114

cos p 3p2 14

3.47

2001 3p

323

83

23


1 32 2 3 50 4 5 2 6 6.39 7 5.55 8 0.619 9 22 10 1.69 11 0.825 12 2.19 13 27 14 10.4 15 2 16 2.27

17 18 9 19 20 1.48 21 22 23 24 a 1k 412

e2p 13

p323

ln 23p 58

243

43

854

3436

1 2 61 3 16 4 5 6 7 8 22.1 9 10 408 11 12 2 13 6 14 15.3214

713

8636

240116

25312

854

643

1 2 3 4 5 6 7 36 8 6.43 91603

4074

1256

13

12

18

16



x

y

010 1.60 11 1.85 12 3.08 13 14 3.92 15 16

17 18 18 19 7.45 20 3.62 21 4.53 22 3.212563

323

83

52

1 2 3 4 5 6

7 8. 9 10 11 12

13 14 15 16 173

21cos x 8 22 clnx2 3x 5 c1

12 1x2 2x 4 26 c1

3 1cos 2x 1 2 32 c1

24 3 16e2a 7 22 1 4

110

31 12 cos 0.5 1 25 41x2 1 2 12 c112

13 tan x 4 24 c112

14x2 3 2 32 c13

11 x2 2 32 c116

3 12p4 1 22 1 4

154

1x6 9 29 c14

ln8x 9 c2esin 2 2132

e32x7 c13

sin3x p4 ccosu 3p

4 c Exercise 1Chapter 15


1 2 3 4 5 6 7 0.0203

8 9 1 0 11 12 13

14 15 16 17 0.0623 18 sin1p 323 sin1233 13 sin1x 1223 c4 sin12x 37 c2211 tan12x 327 c16

tan13x 12 c5 cos1x 2

3 c1

3 tan1x 3

3 c1

8 tan1 a x 1

2b csin11x 1 2 c23

3 sin1 x23 c

233

tan1 p23

3p23

182 sin1

x

222 c3 tan1 x3

ccos1 x6

csin1 x5

c13

tan1 x3

c


1 2 3 4 5

6 7 8 9

10 11 12 1313

sin3 p 15

sin5 p16

cos3 2x 110

cos5 2x c14

tan4 x 16

tan6 x c1

64 18x sin 4x 2 c

16

tan3 3x 13

lncos 3x ccos x 43

cos3 x 65

cos5 x 47

cos7 x 19

cos9 x c1

32 112x 8 sin 2x sin 4x 2 cx

2

sin 4x8

c

x2

sin 4x

8 c

12

tan 2p pcos x 23

cos3 x 15

cos5 x c12

cos 2x 16

cos3 2x csin x 13

sin3 x c

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15 16

17 18 19 20 21 22

23 24 25 26 27 28

29 30 31 32 3315

1sin2 x 3 25 c21x2 4x 5 2 12 c2 sin1x 23 c22 tan1x 122 c12 lnx2 2x 3 c

132

148x 4 sin 4x sin 8x 2 c21ex 2 2 12 clnex 2 c121cot x 3 22 c

14

cos4 x c112

11 x32 28 c2e1cot

x2 c

110

1sin 2x 3 25 c114

1x2 6x 8 27 c34

tan1 2x csin1 3x c14

lnx4 3 c

12

lnx2 2x 3 cln x2 4 c13

ln3x 1 c2x

ln 2 c

14

e4x1 c23

cos13x a 2 c

12

tanp3

2x c14

ln3 4 cos x ctan1 2x c34

sin4x p2 c2

3 11 x 2 32 211 x 2 12 1

1 x c

13 5x 2 3230

c3

412x 1 22 11 2x 2 32

3 c21 2x c12 7x 24

28 c

1x 2 255

c

1 2 3 4 5

6 7 8 9 10 11

12 13 14

15 16 17 18

19 20 21

22 2383n

tan12xn 1 c161 tan 2x

c

7532

Btan1 425

5 tan p2

3R 7532

tan1 1225

1

25 tan1125 tan x 2 c2 sin1 p

2 p cossin1 p

2 2 sin1 1

4

12

cossin1 14 12 tan12 tan x2 c34 3sin1 x 2x21 x2 4 c12 tan112 tan x 2 cx

2 4x 8x 2

c

215p 2 2 32 115p 4 2375

41623

125

1x 5 222

151x 5 2 75 lnx 5 125x 5 c2627

3 41p 2 2 12 21p 2 2

32

3

1 5x

101x 3 25 c1 6x

2412x 1 23 c1p 2 2418p 1 2 7

20

213x 4 2 32 19x 38 213

ctan1 x2 c12x 1 2 12 1x 8 2

3 c

12p 1 2 12 1p 1 2 23

215

1x 2 2 32 13x 4 2 c21 sin 2x c14

ln6x2 4x 13 c1x2 3 26

12 c



1 2 3 4 5

6 7 8 9 10e3x

27 19x2 6x 2 2 cx3 ln 8x x3

3 c

x3

9 13 ln 3x 1 2 ce2x

4 12x2 2x 1 2 cx2 cos x 2x sin x 2 cos x c

1p 1 210110

1p 10 2 111

x cos 2x2

sin 2x

4 c

x5

25 15 ln x 1 2 ce2x

4 12x 1 2 cx sin x cos x c



1 a b c d 2 a b

2 c 3 4 a b c13

e6x 5 lnx 4 cosx c7 sinx 4 lnx c4ex cosx cy 32

x2 8x 2y 38

t2 12

t12 c

y p3 18

p8 cy 1x

32

x4 c16

13 2x 23 c4x2 c3x3 2x2 5x c43

x3 7x c

5 a b c d e 6 c

7 a b c 3 sin2k 8 a b 36 c 932

ln12p 5 223

e5 e214

p12

38225

4x52

5

3x

73

e3x 13

13x 4 24 c121

13x 2 27 c ln4x 3 c12

2e2x c3 sin2x c

10 11 a 0.753 b 2.45 c 1.78 12 a b 24.3 13 a 13.3 b 1.93 14 30.292

4074

1 2 3 4 5 0.307 6

7 8 9 a b 10 11 12 0.690

13 a (0,1) is a maximum b d 14 1 15 b i ii iii cnp9

1n 1 26p9

4p9

2p9

p

2 1y 0

ab

ln13 b sin x 2 k12

arctanx 32 k1

10y

ex2

e2x

4 12x2 2x 1 2 ka 1.07

a 25, b 2, sin1 a x 25b cx arctan x 1

2 ln1 x2 k

1k2 4 2 32325

415

x 22323x 4

30 ke ek 1

Answers

709

11 12 13 14 15

16 17 18 19

201

41n 1 22 3 12p 1 2n112np 1 2 2n 1 4

eax

a2 b2 1a sin bx b cos bx 2 cxn11n 1 22 3 1n 1 2 ln x 1 4 c

ex

5 1sin 2x 2 cos 2x 2 ce2x

13 12 sin 3x 3 cos 3x 2 c

e3x

10 1sin x 3 cos x 2 ce2x1x 1 2 cx tan1 x 1

2 ln1 x2 cx sin1 x 11 x2 2 12 ce

x

2 1cos x sin x 2 c

Answers

708

1 2 3 4

5 6 7 8

Exercise 8.

1 2 3 4 5

6 7 8 9 10 11

12 13 14 15

16 17 18 19

20 21 22 23

24 25 26 27

28 29 30 31 32 33

34 35 36 37

38 39 40

41 42 43 x3

4

x4

13 ln2x 1

cx3

3

x2

2 x 2 lnx 1 cln23x22 2x 3 2 22147 tan121414 13x 2 2 c

211 x2 2 12 sin1 x c316

2x 4 sin x3

sin 2x3 c1

3 tan3 x

15

tan5 x c

14

x sin 4ax4a

c23

tan3 x2

2 tan x2

x c4 cos x4

83

cos3 x4

45

cos5 x4

csin 2x

2

16

sin3 2x c

x2

sin 6x

12 c

u

2

sin 2u4

c14

tan4 x c13

cos3 x c13

sin3 x ceax

a2 4 1a sin 2x 2 cos 2x 2 c

x tan1 1x

12

lnx2 1 cx ln2x 1 x 12

ln2x 1 c2x2 cos x2

8x sin x2

16 cos x2

ce2x

4 1sin 2x cos 2x 2 c

x sinx p6 cosx p

6 ce3x

9 13x 1 2 c1

x 1lnx 1 2 c2x 2 22

2 tan1 Bx 22 c

32

tan1 x2 c14

cotsin1 x2 c223

3 tan1tan x2

23 cx 11 x2 2 12 sin1 x c

tan1 ex c15p 1 2101p 3 25

180

413x 4 2 32135

19x 8 2 cx 25x 5

10 lnx 525

x 5 c

11 x 2836

17 8x 2 c2 sin1x 45 ctan1 21x 1 2 c11n 1 2 cosn1 x c2ex2x5 c221 sin x c

12

tan2x p3 c2 tan1 2x c1

3 ln3x2 1 c

13

18x32 81x 1 2 32 11 3x 24 2 csin 3x3

cos 2x

2 c

x2

2 3x 10 lnx 3 cx 7 lnx 4 c21x2 6x 4 2 12 sin1x 325 clnx2 4x 6 222 tan1x 222 c

2 lnx2 4x 8 12

tan1x 22 c1

2 lnx2 3

5233

tan1 x23

3 c

32

lnx2 4 2 tan1 x2

c11 x2 2 12 sin1 x cExercise 7Chapter 15



1 2 3 4 5 6 0.0791

7 8 0.169 9 10 11 12

13 14 15 16 17 16.5

18 19 20 21 22

23 2.44 24 2.10 25 26 27 28

29 30 1.44 31 32 33 1 34 35 36 1.50

37 38 014

lncos 2x c

0.0280pa4

16x cos1 2x

21 4x22

c11 3a 231a 1 23

13

2

25 tan125 tan x2 c12

sin1 2x 221 4x2 c15

tan5 x 17

tan7 x c1x25

12x2 70 2 ce3x10

1sin x 3 cos x 2 c

x5

25 15 ln 2x x 2 csin11x 1 2 cx

ln 4 1ln x 1 2 csin1x 2233 c3x24 cos 2x 3x4 sin 2x 38 cos 2x c

0.1421

32 112 8 sin 2x sin 4x 2 c3

4 tan1x 3

4 c11 5x 2

2512 5x 22 c

1b

ln2a b cos pa b

214

12ex 1 22 c12

sin1 2x5

clnx2 1 3 tan1 x ce2x

4 12x2 2x 1 2 c

cosec 4x c12x

73

7

192x2312

23

32x32

3 c

14

e4x5 c29

13x 5 2 32 c1x 3 24

4 c


1 0.215 2 4.48 3 0.148 5 9.42 6 7 10 11 12 5.66 13 11.7 14 3.0312a 1 2 32 13a 1 2 4

154 e3

20.227a2

14

14 x2 2 12



1 2 3 4

5 6 7 8

9

10 11

12 13 14

15 16 y 2x tan1 x ln1 x2 x 5y 12x 1 26

120 2x

239240

y 34

cos4x p4 2y 1

2 ln4x2 3

12

ln 19y x cos x_4 sin x kx3

6

cx2

2 dx e

y x4

24 ln x

13x4

288

kx2

2 cx dy lncos x kx c

y 4

45 13x 2 2 52 kx c

y 18

14x sin 4x 2 ky 13

11 15x2 2 12 ky 32k

e2kx3 B1 3

2kR cy ln1 sin x k y x cos x sin x ky

23

11 x2 2 32 ky 13x 7 25

15 ky

x3

3 cos x k

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15

16 17 1813

sin1 3s t2

2

12

u3

3

e2t

5 12 sin t cos t 2 p3

24

15

12

ln2y2 3 38

ln4x2 1 12

ln 131 38

ln 3

cot y ln x 1 ln 432

y23

e2x

2

32

1

2e2ln y

13 x 255

ln 4 15

lny 32

lnx2 2x ky lne2x2

kv2 sin 2at2a

t k3y3 13x 1 211

1113x 1 22

2 k

s3 31t sin1 t 11 t2 2 12 k 2ln y 4 sin1 x2

key 56

ln x ky4 2x2 ln x x2 k y lncos x sin x k12 ln3 2y 13 ln4 3x ktan1 y 12 lnx ky2

2 lncos x k



1 2 3 4 5

6 a 6 c 7 8

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

V 4 cos t p4 6y Bx4

24A

Bx3

12A

Bx24A

y tan1x k 2 xtan1 z x kdzdx

1 dy

dx

tan y tan x ky3

3 ln1 x3 9x

sin nt

n2 kt cx

2t3

6 kt cy lncos x k

1 2 a 2 b 2 c 2 d 3 26.6m

4 a 4 b 5 6

7 8 9 a 10 b 11 a 2.60 11 b c 108m

12 a 12 b 13 Yes v Sg

kv

g

k

g

kekt,

2p

v 2

v 220e2.60te8 137

47

34

ms1v ;C12s 1 25 223

5v ;D2se

2s e2s 92

v ; B2 2 cos s p4s t22 t 2 ln l 1 14p3 sscondsv sin 3t3 , s cos 3t9 111829900 ms

3t5

10 10t6134 ms1v

3t4

2 10 s

t4

3

t2

2 v

4t3

3 t


8

1 a b c d

e

x

iy

z3

z4

z3 z4 1 4i

z3 z4

x

iy

z4 z1

z1 z4 6 i

z1 z4

x

iy

z3

z2

z2 z3 2 7i

z2 z3x

iy

z1

z3

z1 z3

z1 z3 3 i

Answers

711

Answers

710

1 a 176 1 b 107 1 c 57.4 1 d 8.90 1 e 104 1 f 2060 1 g 327 1 h 1.23 1 i 1.08 1 j 274 2 a 2 b 5.98 2 c 0.592

2 d 0.622 2 e 113 2 f 0.965 2 g 12.8 3 4 145 5 1940 6 84.2 7 5cm, 8196 cm22pa5

5

8p


x

y

(0, 1)

(1, 0)(1, 0)

y | x2 1 |9 10 11

12 a b. p

4 1p 2 sin1 a 2a21 a2 21

4 12 sin1 x 2x21 x2 2 k

4n cos1n 1 2pp2b249p16

3.35 , p

2

1 a 18i b 38i c 112i d 60i 2 a 12i b 15i c d 3 a 240i b c 45i d 96i e f 35 g

4 a b 2 c d 5 a 9 b 6 a 9i b c d 1 e 1 34

i12

i53

7i3

116i52

i152

90i72i3252i24i



1 a b c d e 2 a b c d

3 a b c d e 306 f g h i

3 j k 4 a b c

4 d e f g h i j

5 a b c d 6 a b c

6 d e f g h i

6 j k 7 a b c

7 d e f

7 g h i

7 j 8 a b c

8 d e f

8 g h i 9 a

9 b c d e 10 a b

10 c d e f

11 a b c d 12

13 3, 14 a 3, b 2, 15 16 17

18 19 20 2188 966i

25p

35

, q 95

3 i232

z1 26 2i

17, z2

21 i17

348 115i13

8 2i, 8 i214 735 i

531 3i, 1 3i1 2i, 1 2i1 i

x4 10x3 42x2 82x 65 0x2 2ax a2 b2 0x2 14x 85 0x2 8x 25 0x2 4x 53 0

x4 10x3 20x2 90x 261 0x3 8x2 25x 26 0x3 3x2 7x 5 0x2 8x 25 0

x2 6x 10 0x2 4x 13 0x 3; i295

4x

3; i263

x 3; i251

2x

1; i232

x 3 ; i0.541 0.0416i, 0.541 0.0416i1.59 1.42i, 1.59 1.42i0.704 0.369i, 0.704 0.369i

0.734 0.454i, 0.734 0.454i1.92 1.30i, 1.92 1.30i3.85 1.69i, 3.85 1.69i

2.12 0.707i, 2.12 0.707i1.10 0.455i, 1.10 0.455i1 4i, 1 4iRe1z 2 x1 y2

1225

, Im1z 2 xy1 y2

9x25

Re1z 2 128, Im1z 2 12823Re1z 2 cos 2p3

, Im1z 2 sin 2p3

Re1z 2 597, Im1z 2 122

Re1z 2 0, Im1z 2 2xyx2 y2

Re1z 2 2a4 b2

12

16 a2, Im1z 2 ab

4 b2

3a

16 a2Re1z 2 72

65, Im1z 2 61

65

Re1z 2 117145

, Im1z 2 41145

Re1z 2 53185

, Im1z 2 89185

Re1z 2 27, Im1z 2 8x 33169

, y 2591169

x 3, y 1

x 7225

, y 2925

x y ;B152x 21, y 20x 517, y 1417x 6, y 3x 5, y 12x 0, y 3x 8, y 0x 15, y 78432 5376i237 3116i11 2i4

x1y2 3x2 2ixy 2x2 y2

12x 5y 2 i15x 2y 2x2 y2

32

i2

3 4i5

2x2 y2 3ixy

4x2 y26 2i

10 33i29

7 26i29

15 10i13

3 i5

1m3 18m2 12m 24 2 i13m3 6m2 36m 8 2m1m2 3n2 2 i13m n 2x2 y2 2ixy117 44i1 70ia2 b2115 111i7 2i47 35i4 19i

7 13i20 10i13 2i3 5i7 11i14 21i7 i18 28i8 16i


x

iy

z4 z1

z1 z4 4 3i

z1 z4

2 a b c

2 d e f

3 a b c d e f 2ei p28e0i282e1.46i253e1.85i5e2.21i422ei p4

6cos p2

i sin p

2101cos 0 i sin 0 2241 3cos10.896 2 i sin10.896 2 4 226 3cos12.94 2 i sin12.94 2 428 Bcos

3p4 i sin3p

4R2cos p

3 i sin

p

3

4 a b c d e f g

4 h 5 a b c d

6 a b

6 c 7 a i ii iii iv

7 b i ii iii iv v vi

7 vii viii8 a and b

10 11 b 12 b i 16 ii 10

x

iy

106z1 39z2

14 25i

10 67i

106z1 39z2

z 1 0i, 1 i23

2,

1 i232

z1 1

53

2353

i, z2 4

13

713

i

x

iy

z3z4

x

iy

z2 2 3i

z1 3 5i

r 150, u 0.763r 2.5, u 1.33

r 7.5, u 2.22r 5, u 2.50r 2.5, u 1.17r 26, u 0.129r 325, u 1.46r 65, u 1.04

z4 2ei p3z3 25e

0.284iz2 5e2.21iz1 13e

1.18iRoot 1 r 27, u 0.714 Root 2 r 27, u 0.714Root 1 r 25, u 1.11 Root 2 r 25, u 1.11Root 1 r 2.65, u 2.17 Root 2 r 2.65, u 2.17

r 4 2370

5, u 1.41r 2370, u 0.487r 2701, u 1.38r 2313, u 0.8251.67 4.03i

15232

152

i252

i 215

2

3222

322

2 i3.74 1.00i522 5i22215

2 i 252

1 i23

9 a

9 b z1z2 214, arg1z1z2 2 3.07, 2 z1z2 2 B72, argz1z2 1.64z1 27 3cos10.714 2 i sin10.714 2 4z2 22Bcos3p4 i sin3p4 R

14 a b 1.68 radians

15 a 15 b c No.r 32

, u 1.23

1 3cos10.841 2 i sin10.841 2 4 , 1 3cos10.841 2 i sin10.841 2 4

12

4

3

c Rotation of radians clockwise.p

2

10 c i ii iii 11 b

x

iy

B2

A4

C

23

23

1, 1 i23

2,

1 i232

223 32

223Answers

713

1 a b c d e

1 f g h i j

2 a b c d e f

3 a b c d e f

3 g h i j 4 a b

4 c d

4 e f

g

5 a b c d

6 a b 7 a

7 b Real part is Imaginary part is 9 b 10

11 a b 12

15 a b

15 c

248

16

32 64

128272

727

27

27

27

27

z17 1281cos 0 i sin 0 2 z16 64Bcos2p7 i sin2p7 R, z14 16Bcos6p7 i sin6p7 R, z15 32Bcos4p7 i sin4p7 R

z13 8cos 6p

7 i sin

6p7z12 4cos 4p7 i sin 4p7 , z1 2cos 2p7 i sin 2p7 Product 4, Sum 2

425

z2 5 3cos10.927 2 i sin10.927 2 4

12

i 232

tan15p16, tan7p

16, tan p

16, tan9p

1621523215. 2Bcos

p

3 i sinp

3R1.85 0.765i, 0.765 1.85i, 0.765 1.85i, 1.85 0.765i16Bcosp

2 i sinp

2R

29

29

29

29

29

29

29

292

911

1

11

1

1

1 1

4

44

4

4

4

4

4 1

111

1

1

1

1

3

3

33

3

3

1

1

1

1x

iy

23

23

23

1

1 1

216Bcos13p

36 i sin13p

36R, 216Bcos25p

36 i sin25p

36R216Bcos p

36 i sin p

36R, 2

16Bcos11p

36 i sin11p

36R, 216Bcos35p

36 i sin35p

36R, 216Bcos23p

36 i sin23p

36R,0.132 1.67i, 1.62 0.389i 1.54 0.640i, 1.09 1.27i, 0.872 1.42i, 1.45 0.354i, 0.354 1.45i, 0.354 1.45i, 1.45 0.354i

1.69 0.606i, 0.322 1.77i, 1.37 1.16i216 Bcos7p

12 i sin7p

12R216 Bcos3p

4 i sin 3p

4R, 216 Bcos p

12 i sin p

12R, 0.644 1.55i, 0.644 1.55i2 3i, 2 3icos p12 i sin p12cos 34 p i sin 34 pcos 34 p i sin 34 pcos 16 u i sin 16 u

cos 3u i sin 3ucos 11u i sin 11ucos u i sin ucos 3u i sin 3ucos 52

u i sin 52

ucos 8u i sin 8u

1cos u i sin u 2181cos u i sin u 221cos u i sin u 21461cos u i sin u 2341cos u i sin u 2 121cos u i sin u 27cos p

10 i sin p

10cos3p

4 i sin3p

4cosp

2 i sinp

2cos 0 i sin 01

324 Bcos13 u i sin13 uRcos

12

u i sin 12

ucos19u 2 i sin19u 21243

3cos15u 2 i sin15u 2 4cos 25u i sin 25u10241cos 10u i sin 10u 2

Answers

712


15 d Rotate anticlockwise. Enlargement scale factor 2.

16 b c

17 c d a 132

, b 316

, c 1532

, d 5

16z6 6z4 15z2 20

15

z2

6

z4

1

z6

cos2p3 i sin2p

3cosp

3 i sinp

3, cosp

3 i sinp

32p7

1 2 3 4 4 5 b 4294967296 6 7

8 Real part Imaginary part 9 10 a b z2 4, arg1z2 2 2p3

z 2, arg1z 2 p3

23 x2y y3 yx2 y2

x3 xy2 x

x2 y2,

x3 5x2 10x 12 0x 4765

, y 1

65a 2, b 5k 2r 214.8, u 2.06


11 c

23

23

23

1

1

1

cos 0 i sin 0, cos 2p3

i sin 2p3

, cos2p3 i sin2p

3

11 d Each side has length Area of triangle 12 a b

13 d i ii 6 14 15 a i c

16 a i 1 ii c 17 18 a b

19 a c d 21 a b

21 c 123, 223 2 , 123, 223 2

2 i, 2 ix 1, y z 23 0 00 3 00 0 3

1, 1 i232

, 1 i23

2

k ;22110k i1k2 21 2k2 49

z 5, p

3 arg1z 2 1.98, 2 Re1z 2 5

232

323

2 i

2p3

232220

cos3 u 3 cos u sin2 u i13 cos2 u sin u sin3 u 2z 5 i, 6 ia 32, b 32, c 6

a 174

31i4

, b 174

31i4

a 53

, b 169

323

423.

Answers are given when asked to form a conjecture


1 2 3 4 Any value

5 sum of the first n odd 6 7 n2 4n2 2n 2numbers n2

21pan

r14r 7 n12n 5 2a

n

r13r 2

12

n17n 3 2Dn 1 2n 10 2n


8 16 Mn n 1 nn 1 n

1n 1 2 ! 11 a Continuous b Discrete c Continuous d Continuous 2 3

4 b 81-90 c 78.2 5 16.7 6 a 1.58m. b There is no information about the ages or gender of the students. 7 0.927

8 a People below this height are not allowed on the ride. b Mean 1.79 m

mode Bluemode 4., median 4, mean 4.15


4 f

4 g

x

iy

0

2 i

2 i

A

B

1 i a b ii a b

Documents

IBHM_673-720