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756 Maths In Focus Mathematics Extension 1 Preliminary Course

Answers Chapter 1: Basic arithmetic

Problem

5

Exercises 1.1

1. (a) Rational (b) Rational (c) Rational (d) Irrational (e) Rational (f) Irrational (g) Irrational (h) Rational (i) Rational (j) Irrational

2. (a) 18 (b) 11 (c) 6 (d) 11 (e) .4 3- (f) −1 (g) 2157

(h) 12019

(i) 2 (j) 331

3. (a) 16.36 (b) 21.87 (c) 8.80 (d) 22.71 (e) .13 20-

(f) 0.17 (g) 0.36 (h) 1.20 (i) .4 27- (j) 8.16

4. 1300 5. 950 6. 3000 7. 11 000

8. 600 9. $8 000 000 10. $34 600 000

11. 844 km 12. 0.73 13. 33 14. 3.248 15. 4.21

16. 1.7 17. 79 cents 18. 2.73 19. 1.1 20. 3.6 m

21. $281.93 22. 1.8 g 23. $3.20

24. (a) 7.95 (b) 30.03 (c) 0.37 (d) 5.74 (e) 0.52 25. 0.2

Exercises 1.2

1. 1 2. 11- 3. 56- 4. 10 5. 4-

6. .1 2- 7. .7 51- 8. .35 52- 9. 6.57

10. 2154

- 11. 7- 12. −23 13. 10 14. 1

15. 5 16. 3 17. 1 18. 60 19. −20 20. 9

Exercises 1.3

1. (a) 2516

(b) 100051

(c) 5201

(d) 1154

2. (a) 0.4 (b) 1.875 (c) .0 416o (d) .0 63oo

3. (a) 501

(b) 83

(c) 1000

1 (d) 1

100097

4. (a) 0.27 (b) 1.09 (c) 0.003 (d) 0.0623

5. (a) 35% (b) %3331

(c) %22632

(d) 0.1%

6. (a) 124% (b) 70% (c) 40.5% (d) 127.94%

7. (a) . ;0 522513

(b) . ;0 071007

(c) . ;0 16812521

(d) . ;1 09 11009

(e) . ;0 434500217

(f) . ;0 122540049

8. (a) .0 83o (b) .0 07oo (c) .0 13oo (d) .0 16o (e) .0 6o (f) .0 15oo

(g) .0 142857o o or 0.142857 (h) .1 18oo

9. (a) 98

(b) 92

(c) 195

(d) 397

(e) 9967

(f) 116

(g) 457

(h) 6013

(i) 990217

(j) 149537

10. (a) .0 5o (b) 7.4 (c) 0.73o (d) .0 68oo (e) .1 72oo

11. (a) 85

(b) 281

(c) 118

(d) 2187

(e) 454

12. 74% 13. 77.5% 14. 17.5% 15. 41.7%

Exercises 1.4

1. 203

2. 207

3. (a) 2017

(b) 107

(c) 1201

(d) 283

(e) 53

4. $547.56 5. 714.3 g 6. 247

7. $65

8. 179 cm 9. (a) 11.9 (b) 5.3 (c) 19 (d) 3.2 (e) 3.5 (f) 0.24 (g) 0.000 18 (h) 5720 (i) 0.0874 (j) 0.376

10. $52.50 11. 54.925 mL 12. 1152.125 g 13. $10.71

14. 5.9% 15. 402.5 g 16. 41.175 m 17. $30.92

18. 3.2 m 19. 573 20. $2898

Problem

5115 minutes after 1 o’clock.

Exercises 1.5

1. (a) 500 (b) 145 (c) 641

(d) 3 (e) 2

2. (a) 13.7 (b) 1.1 (c) 0.8 (d) 2.7 (e) .2 6- (f) 0.5

3. (a) a17 (b) y 10 = (c) a 4- (d) w (e) x5 (f) p10

(g) y6 (h) x21 (i) x4 10 (j) y81 8- (k) a (l) y

x45

10

(m) w10 (n) p5 (o) x 3- (p) a ba

bor2 3

2

3-

(q) x yx

yor5 2

5

2

-

4. (a) x 14 (b) a 7- (c) m 4 (d) k 10 (e) a 8- (f) x (g) mn 2

(h) p 1- (i) 9 x 22 (j) x 21

5. (a) p 5 q 15 (b) b

a8

8

(c) b

a6412

3

(d) 49 a 10 b 2 (e) 8 m 17

(f) x 4 y 10 (g) k27

2 23

(h) 16 y 47 (i) a 3 (j) x y125 21 18-

Answer S1-S5.indd 756 7/11/09 1:26:16 AM

757ANSWERS

6. 421

7. 324 8. 22710

9. (a) a 3 b (b) 251

10. (a) pq 2 r 2 (b) 327

11. 94

12. 181

13. 274

14. 811

15. 108

1 16.

121

17. 2

558

22

18. 388849

Exercises 1.6

1. (a) 271

(b) 41

(c) 3431

(d) 10 000

1 (e)

2561

(f) 1

(g) 321

(h) 811

(i) 71

(j) 811

(k) 641

(l) 91

(m) 1

(n) 361

(o) 125

1 (p)

100 0001

(q) 1281

(r) 1

(s) 641

(t) 641

2. (a) 1 (b) 16 (c) 121

(d) 12511

(e) 1 (f) 125 (g) 131

(h) 49 (i) 383

(j) 32 (k) 231

(l) 1 (m) 13613

(n) 18119

(o) 1 (p) 16 (q) 1585

- (r) 237

- (s) 1 (t) 2516

3. (a) m 3- (b) x 1- (c) p 7- (d) d −9 (e) d −5 (f) x 2-

(g) 2x 4- (h) 3 y −2 (i) 21

z 6- or 2

z 6-

(j) 5

3t 8-

(k) 7

2x 1-

(l) 2

5m 6-

(m) 3

2y 7-

(n) 3 4x 2+ -] g (o) a b 8+ -] g

(p) 2x 1- -] g (q) 5 1p 3+ -^ h (r) 2 4 9t 5- -] g

(s) 41x 11+ -] g

(t) 9

5 3a b 7+ -] g

4. (a) 1

t5 (b)

1

x6 (c)

1

y3 (d)

1

n8 (e)

1

w10 (f)

2x (g)

3

m4

(h) 5

x7 (i)

8

1

x3 (j)

41n

(k) 1

1

x 6+] g (l) 8

1y z+

(m) 3

1

k 2-] g ( n) 3 2

1

x y 9+^ h (o) x 5 (p) y 10 (q) 2

p

(r) a b 2+] g (s) x y

x y

+

- (t)

2

3

w z

x y 7

-

+e o

Exercises 1.7

1. (a) 9 (b) 3 (c) 4 (d) 2 (e) 7 (f) 10 (g) 2 (h) 8 (i) 4 (j) 1 (k) 3 (l) 2 (m) 0 (n) 5 (o) 7 (p) 2

(q) 4 (r) 27 (s) 21

(t) 161

2. (a) 2.19 (b) 2.60 (c) 1.53 (d) 0.60 (e) 0.90 (f) 0.29

3. (a) y3 (b) y yor23 32_ i (c)

1

x (d) 2 5x +

(e) 3 1

1

x - (f) 6q r3 + (g)

x x7

1

7

1or

25 5 2+ +] ^g h

4. (a) 2t1

(b) 5y1

(c) 2x3

(d) x9 31

-] g (e) 2s4 1+1] g

(f) 2t2 3+-

1] g (g) -

2x y5 -

3^ h (h) 2x3 1+5] g

(i) 3x 2--

2] g (j) 2y21

7+-

1^ h (k) -

3x5 4+1] g

(l) y32 1 2

1

--

2a k (m) x53 2 4

3

+-

2_ i

5. (a) x 23

(b) x 21

-

(c) x 32

(d) x 35

(e) x 45

6. (a) x x x2 23

+ +2 (b) a b32

32

- (c) p p p2 21

+ +1-2

(d) 2x x 1+ +- (e) x x x321

23

25

- +- - -

7. (a) 2

1

a b3 - (b)

3

1

y 23 -^ h (c) 6 1

4

a 47 +] g

(d) 3

1

x y 54 +^ h (e) 7 3 8

6

x 29 +] g

Exercises 1.8

1. (a) .3 8 103# (b) .1 23 106

# (c) .6 19 104#

(d) 1.2 107# (e) .8 67 109

# (f) .4 16 105#

(g) 9 102# (h) .1 376 104

# (i) 2 107# (j) 8 104

#

2. (a) .5 7 10 2#

- (b) .5 5 10 5#

- (c) 4 10 3#

-

(d) 6.2 10 4#

- (e) 2 10 6#

- (f) 8 10 8#

-

(g) 7.6 10 6#

- (h) 2.3 10 1#

- (i) 8.5 10 3#

- (j) 7 10 11#

-

3. (a) 36 000 (b) 27 800 000 (c) 9 250 (d) 6 330 000 (e) 400 000 (f) 0.072 3 (g) 0.000 097 (h) 0.000 000 038 (i) 0.000 007 (j) 0.000 5

4. (a) 240 000 (b) 9 200 000 (c) 11 000 (d) 0.36 (e) 1.3 (f) 9.0 (g) 16 (h) 320 (i) 2900 (j) 9.1

5. (a) 6.61 (b) 0.686 (c) 8.25 (d) 1.30

6. 1.305 1010# 7. 6.51 10 10

#-

Exercises 1.9

1. (a) 7 (b) 5 (c) 6 (d) 0 (e) 2 (f) 11 (g) 6 (h) 24 (i) 25 (j) 125 2. (a) 5 (b) −1 (c) 2 (d) 14 (e) 4 (f) −67 (g) 7 (h) 12 (i) −6 (j) 10 3. (a) 3 (b) 3 (c) 1 (d) 3 (e) 1 4. (a) a (b) a- (c) 0 (d) 3 a (e) −3 a (f) 0 (g) 1a + (h) a 1- - (i) 2x - (j) 2 x-

5. (a) | | 6a b+ = | | | |a b 6+ = | | | | | |a b a b` #+ + (b) | | 3a b+ = | | | |a b 3+ = | | | | | |a b a b` #+ + (c) | | 1a b+ = | | | |a b 5+ = | | | | | |a b a b` #+ + (d) | | 1a b+ = | | | |a b 9+ = | | | | | |a b a b` #+ + (e) | |a b 10+ = | | | |a b 10+ = | | | | | |a b a b` #+ +

6. (a) | | 5x x2 = = (b) | | 2x x2 = = (c) | | 3x x2 = =

(d) | | 4x x2 = = (e) | | 9x x2 = =

7. (a) x x x x5 5 5 5for and for2 1+ - - - - (b) b b b x3 3 3 3for and for2 1- - (c) a a a a4 4 4 4for and for2 1+ - - - - (d) y y y y2 6 3 6 2 3for and for2 1- - (e) 3 9 3 3 9 3x x x xfor and for2 1+ - - - - (f) 4 4 4 4x x x xfor and for1 2- -

(g) 2 1 2 1k k k k21

21

for and for2 1+ - - - -

(h) 5 2 5 2x x x x52

52

for and for2 1- - +

(i) a b a b a b a bfor and for2 1+ - - - - (j) p q p q q p p qfor and for2 1- -

8. x 3!= 9. 1! 10. , x1 2! !

Answer S1-S5.indd 757 7/11/09 1:26:17 AM


Test yourself 1

1. (a) 209

(b) 0.14 (c) 0.625 (d) 200157

(e) 1.2%

(f) 73.3% 2. (a) 491

(b) 51

(c) 31

3. (a) 8.83 (b) 1.55 (c) 1.12 (d) 342 (e) 0.303 4. (a) 1

(b) 1 (c) 39 (d) 2 (e) 10- (f) 1- (g) 4 5. (a) x9

(b) 25y6 (c) a b11 6 (d) 27

8x18

(e) 1 6. (a) 4029

(b) 371

(c) 12 (d) 221

(e) 1221

7. (a) 4 (b) 6 (c) 19

(d) 641

(e) 4 (f) 3 (g) 71

(h) 2 (i) 1 (j) 4

8. (a) a 5 (b) x 30 y 18 (c) p 9 (d) 16 b 36 (e) 8 x 11 y 9. (a) 2n1

(b) x 5- (c) x y 1+ -^ h (d) 4x 1+1] g (e) 7a b+

1] g

(f) 2x 1- (g) 21

x 3- (h) 3x4

(i) 7x5 3+9] g (j) 4m

-3

10. (a) 1

a5 (b) n4 (c) 1x + (d)

1x y-

(e) 4 7

1

t 4-] g

(f) a b5 + (g) 1

x3 (h) b34 (i) 2 3x 43 +] g (j)

1

x3

11. | | 2a b+ = | | | | 8a b+ = | | | | | |a b a b` #+ +

12. 1 13. 192

1 14. 689 mL 15. (a) 6 h (b)

127

(c) 81

(d) 33.3% 16. $38 640 17. 70% 18. 6.3 1023#

19. (a) 2x1

(b) y 1- (c) 6x 3+1] g (d) 2 3x 11- -] g (e) 3y

7

20. (a) 1.3 10 5#

- (b) 1.23 1011# 21. (a)

97

(b) 33041

22. (a) 1

x3 (b)

2 51

a + (c) a

b 5c m 23. 14 500

24. | | ,2 5 7LHS = - + - = | | | | .2 5 7RHS = - + - = So | | | | | |a b a b#+ + since .7 7#

Challenge exercise 1

1. 4303278

2. 11811

3. . , %, , .0 502 519951

0 5o

4. 5331

% 5. 161

6. 3.04 1014# 7. 83% 8. 1

99903271

9. 18 h 10. 1.98

11.

2 2 1 22 2 22 2 22 2 1

2 2 1 2 2 2 1

LHS

RHS

k k

k k

k

k

k k k

1

1 1

1

1

1 1

:

`

= - +

= - +

= -

= -

=

- + = -

+

+ +

+

+

+ +

^

^^ ^

h

hh h

12. −2 4 .3 5 13. . , , . , ,0 34 2 1 5 073

- o 14. 632

%

15. ,x

xx

x1

11

11

1when when2 1-

--

- 16. 0.73

17. 0.6% 18 4.54 19. 4.14 10 20#

-

20.

| | | | | | , , ;| | | | | | , , ;a b a b a b a ba b a b b ba

0 0 0 00 0 0 0

when orwhen ora

2 2 1 11 2 1 1 2

+ = +

+ +

| | | |a b a b` #+ + for all a , b

Chapter 2: Algebra and surds

Exercises 2.1

1. 7 x 2. 3 a 3. z 4. 6 a 5. 3 b 6. −3 r

7. y- 8. −5 x 9. 0 10. 3 k 11. 9 t 12. 10 w

13. m- 14. x- 15. 0 16. 5 b 17. 11 b 18. 10x-

19. 6 6x y- 20. 3a b- 21. 4 2xy y+ 22. 6ab2-

23. m m6 122 - + 24. 2 6p p2 - - 25. 8 3x y+

26. 2 10ab b- + 27. 2bc ac- 28. 2 9 1a x5 3- +

29. 2 3 2x xy x y y3 2 2 3- + + 30. 3 7 6x x x3 2+ - -

Exercises 2.2

1. b10 2. xy8 3. p10 2 4. wz6-

5. ab15 6. xyz14 7. abc48 8. d12 2

9. a12 3 10. y27 3- 11. x32 10

12. a b6 2 3 13. a b10 3 2- 14. p q21 3 4

15. a b5 3 3 16. n8 10- 17. k p33 18. t81 12

19. 14m11- 20. x y24 6 3

Exercises 2.3

1. 6 x 2. 2 3. 4a2 4. 8 a 5. 4 a 6. 2

y 7. 3 p

8. 2ab

9. 34y

10. 3x3- 11. 3 a 12. 3

1

ab2 13.

2qs-

14. 3

2

c d2 15.

x

z

2 2

2

16. 6p q4 17. 4c

a b4 7

18. 2ab6

19. 3y

x z3 3

- 20. 2b

a6

13

Exercises 2.4

1. x2 8- 2. h6 9+ 3. a5 10- + 4. xy x2 3+

5. x x22 - 6. a ab6 162 - 7. a b ab2 2 2+ 8. n n5 202 -

9. 3x y x y63 2 2 3+ 10. k4 7+ 11. t2 17-

12. y y4 112 + 13. b5 6- - 14. x8 2-

15. m3 1- + 16. h8 19- 17. d 6- 18. a a2 42 - +

19. x x3 9 52 - - 20. ab a b b2 2 2- + 21. x4 1-

22. y7 4- + 23. b2 24. t5 6- 25. a2 26+

Exercises 2.5

1. 7 10a a2 + + 2. 2 3x x2 + - 3. 2 7 15y y2 + -

4. 6 8m m2 - + 5. 7 12x x2 + + 6. 3 10y y2 - -

7. 2 6x x2 + - 8. 10 21h h2 - + 9. 25x2 -

10. 15 17 4a a2 - + 11. 8 6 9y y2 + - 12. 7 4 28xy x y+ - -

13. 2 3 6x x x3 2- + - 14. 4n2 - 15. 4 9x2 -

16. 16 49y2- 17. 4a b2 2- 18. 9 16x y2 2- 19. 9x2 -

20. 36y2 - 21. 9 1a2 - 22. 4 49z2 -

Answer S1-S5.indd 758 7/11/09 1:26:18 AM

759ANSWERS

23. 2 11 18 18x xy x y2 - + - + 24. 2 2 7 6 3ab b b a2+ - - +

25. x 83 + 26. 27a3 - 27. 18 81a a2 + +

28. 8 16k k2 - + 29. 4 4x x2 + + 30. 14 49y y2 - +

31. 4 12 9x x2 + + 32. 4 4 1t t2 - +

33. 9 24 16a ab b2 2+ + 34. 10 25x xy y2 2- +

35. 4 4a ab b2 2+ + 36. a b2 2- 37. 2a ab b2 2+ +

38. 2a ab b2 2- + 39. a b3 3+ 40. a b3 3-

Exercises 2.6

1. 8 16t t2 + + 2. 12 36z z2 - + 3. 2 1x x2 - +

4. 16 64y y2 + + 5. 6 9q q2 + + 6. 14 49k k2 - +

7. 2 1n n2 + + 8. 4 20 25b b2 + + 9. 9 6x x2- +

10. 9 6 1y y2 - + 11. 2x xy y2 2+ + 12. 9 6a ab b2 2- +

13. 16 40 25d de e2 2+ + 14. 16t2 - 15. x 92 -

16. 1p2 - 17. 36r2 - 18. 100x2 - 19. 4 9a2 -

20. 25x y2 2- 21. 16 1a2 - 22. 49 9x2- 23. 4x4 -

24. 10 25x x4 2+ + 25. 9 16a b c2 2 2- 26. 44

xx

2

2+ +

27. 1

aa

2

2- 28. 2 4 4x y x y y2 2 2 2- - = - + -^ h

29. 2 2 2 2a b a b c c a ab b ac bc c2 2 2 2 2+ + + + = + + + + +] ]g g

30. 1 2 1 2 1 2 2x x y y x x xy y y2 2 2 2+ - + + = + + - - +] ]g g

31. 12a 32. 32 z2- 33. 9 8 3x x2 + -

34. 3 2x xy y x2 2+ + - 35. 14 4n2 -

36. 12 48 64x x x3 2- + - 37. x2 38. 2x x y y4 2 2 4- +

39. 8 60 150 125a a a3 2+ + +

40. 4 16 15 4 4x x x x4 3 2+ + - -

Problem

2,a = 7,b = 9,c = 4,d = 3,e = 8,f = 0,g = 6,h = 1i =

Exercises 2.7

1. y2 3+^ h 2. x5 2-] g 3. m3 3-] g 4. x2 4 1+] g 5. y6 4 3-^ h 6. xx 2+] g 7. m m 3-] g 8. y y2 2+^ h 9. a a3 5 -] g 10. ab b 1+] g 11. xy x2 2 1-] g 12. mn n3 32 +^ h 13. xy x z2 4 -] g 14. a b a6 3 2+ -] g 15. x x y5 2- +^ h 16. q q3 22 3 -_ i 17. b b5 32 +] g 18. a b ab3 22 2 -] g 19. 5)( 7)(m x+ + 20. 1 2y y- -^ ^h h 21. 7 )(4 3 )( y x+ - 22. 2 6 5a x- +] ]g g 23. 2 1t x y+ -] ^g h 24. 3 2 2 3x a b c- + -] ]g g 25. 3 2 3x x2 +] g 26. 3 2q pq3 2 -_ i 27. ab a b3 5 13 2 +^ h 28. 4 6x x2 -] g 29. 5 7 5m n mn2 3 -^ h 30. 4 6 4ab ab2 3 +^ h 31. r r h2r +] g 32. 3 2x x- +] ]g g 33. ( ) ( )x y4 22+ +

34. 1a- +] g 35. ( ) ( )a ab1 4 32 + -

Exercises 2.8

1. 4 2x b+ +] ]g g 2. 3y a b- +^ ]h g 3. 5 2x x+ +] ]g g 4. 2 3m m- +] ]g g 5. d c a b- +] ]g g 6. 1 3x x2+ +] ^g h 7. 5 3 2a b- +] ]g g 8. 2y x x y- +^ ^h h 9. 1 1y a+ +^ ]h g 10. 5 1x x+ -] ]g g 11. 3)(1 )(y a+ + 12. 2)(1 2 )(m y- -

13. 5 2 3x y x y+ -^ ^h h 14. 4a b ab2+ -^ ]h g 15. 5 3x x- +] ]g g 16. 7)( 4)(x x3+ - 17. 3 7x y- -] ^g h 18. 3 4d e+ -] ]g g 19. 4 3x y- +] ^g h 20. 3 2a b+ -] ]g g 21. 3)( 6)(x x2- + 22. 3q p q- +^ ^h h 23. 2 3 5x x2- -] ^g h 24. 3 4a b c- +] ]g g 25. 7 4y x+ -^ ]h g 26. 4)( 5)(x x3- -

27. (2 3)(2 4) (2 3)( )x x x x2 22 2- + = - +

28. ( ) ( )a b a3 2 3+ + 29. 5( 3)(1 2 )y x- +

30. r r2 3r+ -] ]g g

Exercises 2.9

1. 3 1x x+ +] ]g g 2. 4 3y y+ +^ ^h h 3. 1m 2+] g 4. 4t 2+] g 5. 3 2z z+ -] ]g g 6. 1 6x x+ -] ]g g 7. 3 5v v- -] ]g g 8. 3t 2-] g 9. 10 1x x+ -] ]g g 10. 7 3y y- -^ ^h h 11. 6 3m m- -] ]g g 12. 12 3y y+ -^ ^h h 13. 8 3x x- +] ]g g 14. a 2 2-] g 15. 2 16x x- +] ]g g 16. 4 9y y+ -^ ^h h 17. 6 4n n- -] ]g g 18. x 5 2-] g

19. 9 1p p+ -^ ^h h 20. 2 5k k- -] ]g g 21. 4 3x x+ -] ]g g 22. 7 1m m- +] ]g g 23. 10 2q q+ +^ ^h h 24. 5 1d d- +] ]g g 25. 9 2l l- -] ]g g

Exercises 2.10

1. 2 1)( 5)( a a+ + 2. 5 2 1y y+ +^ ^h h 3. 3 7)( 1)( x x+ + 4. 3 2)( 2)( x x+ + 5. 2 3)( 1)( b b- -

6. 7 2)( 1)( x x- - 7. 3 1 2y y- +^ ^h h 8. 2 3 4x x+ +] ]g g 9. 5 2 3p p- +^ ^h h 10. 3 5 2 1x x+ +] ]g g 11. 2 1)( 6)( y y+ - 12. 5 1 2 1x x- +] ]g g 13. 4 1)(2 3)( t t- - 14. 3 4)(2 3)( x x+ -

15. 6 1 8y y- +^ ^h h 16. 4 3 2n n- -] ]g g 17. 4 1 2 5t t- +] ]g g 18. 3 2 4 5q q+ +^ ^h h 19. r r r r4 1 2 6 4 12 3- + - +=] ] ] ]g g g g 20. 2 5 2 3x x- +] ]g g 21. 6 1 2y y- -^ ^h h 22. 2 3 3 2p p- +^ ^h h 23. 8 7)( 3)( x x+ +

24. 3 4 4 9b b- -] ]g g 25. 6 1)( 9)( x x+ -

26. 3 5x 2+] g 27. 4 3y 2+^ h 28. 5 2k 2-] g 29. 6 1a 2-] g 30. 7 6m 2+] g

Answer S1-S5.indd 759 7/25/09 1:31:06 PM


Exercises 2.11

1. 1y 2-^ h 2. ( 3)x 2+ 3. ( 5)m 2+ 4. ( 2)t 2-

5. ( 6)x 2- 6. 2 3x 2+] g 7. 4 1b 2-] g 8. 3 2a 2+] g 9. 5 4x 2-] g 10. 7 1y 2+^ h

11. 3 5y 2-^ h 12. 4 3k 2-] g 13. 5 1x 2+] g 14. 9 2a 2-] g 15. 7 6m 2+] g 16.

21

t2

+d n 17. 32

x2

-d n 18. 351

y2

+d n

19. 1

x x

2

+c m 20. 52

kk

2

-d n

Exercises 2.12

1. 2)( 2)(a a+ - 2. 3)( 3)(x x+ - 3. 1)( 1)(y y+ -

4. 5 5x x+ -] ]g g 5. 2 7)(2 7)( x x+ - 6. 4 3)(4 3)( y y+ -

7. 1 2 )(1 2 )( z z+ - 8. 5 1 5 1t t+ -] ]g g 9. 3 2 3 2t t+ -] ]g g 10. 3 4 3 4x x+ -] ]g g 11. 2 )( 2 )(x y x y+ -

12. 6 6x y x y+ -^ ^h h 13. 2 3 2 3a b a b+ -] ]g g 14. 10 10x y x y+ -^ ^h h 15. 2 9 2 9a b a b+ -] ]g g 16. 2 2x y x y+ + + -^ ^h h 17. 3)( 1)(a b a b+ - - +

18. 1 1z w z w+ + - -] ]g g 19. 21

21

x x+ -d dn n

20. 3

13

1y y+ -e eo o 21. 2 3 2 1x y x y+ + - +^ ^h h

22. ( )( ) ( )( )( )x x x x x1 1 1 1 12 2 2+ - = + + -

23. 3 2 3 2x y x y3 3+ -_ _i i 24. 4 2 2x y x y x y2 2+ + -_ ^ ^i h h 25. 1)( 1)( 1)( 1)(a a a a4 2+ + + -

Exercises 2.13

1. 2)( 2 4)(b b b2- + + 2. 3 3 9x x x2+ - +] ^g h 3. 1 1t t t2+ - +] ^g h 4. 4)( 4 16)(a a a2- + +

5. 1 )(1 )( x x x2- + + 6. 2 3 4 6 9y y y2+ - +^ _h i 7. ( ) ( )y z y yz z2 2 42 2+ - + 8. 5 )( 5 25 )(x y x xy y2 2- + +

9. 2 3 4 6 9x y x xy y2 2+ - +^ _h i 10. 1 1ab a b ab2 2- + +] ^g h 11. 10 2 )(100 20 4 )( t t t+ - + 2 12.

23

4 23

9x x x2

- + +d en o 13.

10 1 100 10 1a b a ab b2 2

+ - +d en o 14. 1 2 1x y x x xy y y2 2+ - + + + + +^ _h i 15. xy z x y xyz z5 25 306 362 2 2+ - +^ _h i 16. a a 19 2 - +- ^ h 17. 1

31

3 9x x x2

- + +d en o 18. 3 3 9 6x y y y xy x x2 2+ + - - + + +^ _h i 19. 1 4 5 7x y x x xy y y2 2+ - + - + - +^ _h i 20. 2 6 )(4 24 2 6 36)( a b a a ab b b2 2+ - + + + + +

Exercises 2.14

1. x x2 3 3+ -] ]g g 2. p p3 3 4+ -^ ^h h 3. y y y5 1 12- + +^ _h i 4. ) (ab a b a2 2 2 1+ -^ h 5. 5 1a 2-] g 6. x x2 3 4- -- ] ]g g 7. z z z3 5 4+ +] ]g g 8. ab ab ab3 2 3 2+ -] ]g g 9. x xx 1 1+ -] ]g g 10. x x2 3 2 2- +] ]g g 11. 5 3m n- +] ]g g 12. x7 2 1- +] g

13. 5 4 4y y y+ + -^ ^ ^h h h 14. 1 2 2 4x x x x2- + - +] ] ^g g h 15. x x x x x x1 1 1 12 2+ - + - + +] ^ ] ^g h g h 16. x x x2 5+ -] ]g g 17. ( )x x3 3 2+ -] g

18. ( ) ( )xy xyy 2 1 2 1+ - 19. b b b3 2 4 2 2- + +] ^g h 20. x x3 3 2 2 5- +] ]g g 21. x3 1 2-] g 22. 2)( 5)( 5)(x x x+ + - 23. 3z z 2+] g 24. 1 1 2 3 2 3x x x x+ - + -] ] ] ]g g g g 25. x x x y x xy y2 2 2 2 2+ - + - +] ] ^ _g g h i 26. ( ) ( )a a a4 3 3+ - 27. x x xx 2 4 25 2- + +] ^g h 28. 2)( 2)( 3)( 3)(a a a a+ - + - 29. 4 ( 5)k k 2+

30. 3( 1) 1) 3)( (x x x+ - +

Exercises 2.15

1. 4 4 2x x x2 2+ + = +] g 2. 6 9 3b b b2 2- + = -] g 3. 10 25 5x x x2 2- + = -] g 4. 8 16 4y y y2 2+ + = +^ h

5. 14 49 7m m m2 2- + = -] g 6. 18 81 9q x q2 2+ + = +^ h

7. 2 1 1x x x2 2+ + = +] g 8. 16 64 8t t t2 2- + = -] g 9. 20 100 10x x x2 2- + = -] g 10. 44 484 22w w w2 2+ + = +] g 11. 32 256 16x x x2 2- + = -] g 12. 3

49

23

y y y22

+ + = +d n

13. 74

4927

x x x22

- + = -d n 14. 41

21

a a a22

+ + = +d n

15. 94

8129

x x x22

+ + = +d n 16. 5

yy

y1625

45

22

2

- + = -d n

17. 11

kk

k16

1214

112

22

- + = -d n

18. 6 9 3x xy y x y2 2 2+ + = +^ h 19. 4 4 2a ab b a b2 2 2- + = -] g 20. 8 16 4p pq q p q2 2 2- + = -^ h Exercises 2.16

1. 2a + 2. 2 1t - 3. 3

4 1y + 4.

2 14

d - 5.

5 2xx-

6. 4

1y -

7. ab a

322

-

-] g 8.

31

ss+

- 9.

11

bb b2

+

+ +

10. 3

5p + 11.

31

aa+

+ 12.

2 4

3

x x

y2 + +

+ 13. 3x -

14. 4 2 1

2

p p

p2 - +

- 15.

2a ba b

-

+

Answer S1-S5.indd 760 7/11/09 1:26:21 AM

761ANSWERS

Exercises 2.17

1. (a) 45x

(b) 15

13 3y + (c)

128a +

(d) 6

4 3p + (e)

613x -

2. (a) 2 1a

b-

(b) 1

2 1

q

p q q2

+

- - +^ _h i (c)

b

x yb

2 1

2

10

2

-

+

]^

gh

(d) ab

x xy y2 2- + (e)

5 2

3 1

x x

x x

- -

- -

] ]] ]

g gg g

3. (a) 5x (b)

xx

x 12

-

- +

] g (c) 3

a ba b

+

+ + (d)

22

xx+

(e) p q

p q p q

p q

p q 11 2 2

+

+ -=

+

- ++^ ^h h (f)

x x

x

1 3

12

+ -

-

] ]]g gg

(g) 2 23 8

x xx

+ -

- +

] ]g g (h) 1

2

a

a2+

+

] g

(i) y y y

y y

2 3 1

3 14 132 2

+ + -

+ +

^ ^ ^_h h h

i (j)

x x x

x

4 4 3

5 22

+ - +

+-

] ] ]]g g g

g

4. (a) y y

xx

3 9

2

8

2

2 - +

+

_]

ig

(b) 15

2 1

y

y y+ +^ ^h h

(c) x x

x x2 3 4

10 242

- -

+ -

] ]g g (d) b

b bb 1

3 5 102

2

+

- -

] g (e) x

5. (a) 5 2 3

3 13x x x

x- - +

-

] ] ]g g g (b) 2 2

3 5x x

x+ -

-

] ]g g

(c) p q p q

p pq q

pq

3 5 22 2

+ -

+ -

^ ^h h (d) 2 1

a b a ba ab b2 2

+ -

- - +

] ]g g

(e) x y x y

x yy 1

+ -

+ +

^ ^^h h

h

Exercises 2.18

1. (a) 7.1- (b) 6.9- (c) 48.1 (d) 37.7- (e) 0.6

(f) 2.3 (g) 5.3- 2. 47 3. 7- 4. 375 5. 196-

6. 5.5 7. 377 8. 284 9. 40- 10. 51.935 11. 143

-

12. 22.4 13. 1838.8 14. 43

15. 15 16. 10

17. 2 3 18. 23.987 19. 352.47 20. 93 21. 4

Exercises 2.19

1. (a) 2 3 (b) 3 7 (c) 2 6 (d) 5 2 (e) 6 2

(f) 10 2 (g) 4 3 (h) 5 3 (i) 4 2 (j) 3 6

(k) 4 7 (l) 10 3 (m) 8 2 (n) 9 3 (o) 7 5

(p) 6 3 (q) 3 11 (r) 5 5

2. (a) 6 3 (b) 20 5 (c) 28 2 (d) 4 7 (e) 16 5

(f) 8 14 (g) 72 5 (h) 30 2 (i) 14 10 (j) 24 5

3. (a) 18 (b) 20 (c) 176 (d) 128 (e) 75

(f) 160 (g) 117 (h) 98 (i) 363 (j) 1008

4. (a) 45x = (b) 12x = (c) 63x = (d) 50x =

(e) 44x = (f) 147x = (g) 304x = (h) 828x =

(i) 775x = (j) 960x =

Exercises 2.20

1. 3 5 2. 2 3. 6 3 4. 3 3 5. 3 5- 6. 3 6

7. 7 2- 8. 8 5 9. 4 2- 10. 4 5 11. 2 12. 5 3

13. 3- 14. 2 15. 5 7 16. 2 17. 13 6

18. 9 10- 19. 47 3 20. 2 2 35 - 21. 7 5 2-

22. 2 3 4 5- - 23. 7 6 3 5+ 24. 2 2 3- -

25. 17 5 10 2- +

Exercise 2.21

1. 21 2. 15 3. 3 6 4. 10 14 5. 6 6- 6. 30

7. 12 55- 8. 14 9. 60 10. 12 2 3=

11. 2 48 8 3= 12. 15 28 30 7=

13. 2 20 4 5= 14. 84- 15. 2

16. 28 17. 30 18. 2 105- 19. 18

20 . 30 50 150 2= 21. 2 6 22. 4 3 23. 1 24. 6

8

25. 2 3 26. 3 10

1 27.

2 5

1 28.

3 5

1 29.

21

30. 2 2

3 31.

2

3 32.

2 5

9 33.

2 2

5 34.

32

35. 75

Exercises 2.22

1. (a) 10 6+ (b) 2 6 15- (c) 12 8 15+

(d) 5 14 2 21- (e) 6 4 18 6 12 2- + = - +

(f) 5 33 3 21+ (g) 6 12 6- - (h) 5 5 15-

(i) 6 30+ (j) 2 54 6 6 6 6+ = +

(k) 8 12 12 8 24 3- + = - + (l) 210 14 15-

(m) 10 6 120- (n) 10 2 2- - (o) 4 3 12-

2. (a) 10 3 6 3 5 9 3+ + +

(b) 10 35 2 14- - +

(c) 2 10 6 10 15 15 6- + -

(d) 12 18 8 1224 36 8 12

20 60 10 305 15 10 30

+ - - =

+ - -

(e) 52 13 10- (f) 15 15 18 10 6 6- + -

(g) 4 (h) 1- (i) 12- (j) 43 (k) 3 (l) 241-

(m) 6- (n) 7 2 10+ (o) 11 4 6- (p) 25 6 14+

(q) 57 12 15+ (r) 27 4 35-

(s) 77 12 40 77 24 10- = - (t) 53 12 10+

3. (a) 18 (b) 108 2 (c) 432 2 (d) 19 6 2+ (e) 9

4. (a) 21, 80a b= = (b) 19, 7a b= = -

5. (a) 1a - (b) p pp2 1 2 1- - -^ h 6. 25k = 7. 2 3 5x y xy- - 8. 17, 240a b= =

9. 107, 42a b= = - 10. 9 5 units2+

Answer S1-S5.indd 761 8/2/09 2:47:53 AM


Exercises 2.23

1. (a) 77

(b) 46

(c) 5

2 15 (d)

106 14

53 14

=

(e) 3

3 6+ (f)

212 5 2-

(g) 5

5 2 10+

(h) 14

3 14 4 7- (i)

208 5 3 10+

(j) 35

4 15 2 10-

2. (a) 4 4 3 243 2- -= ^ h (b) 47

6 7 3+- ^ h

(c) 19

2 15 4 1819

15 6 22-=

-- -^ ^h h

(d) 13

19 8 313

8 3 19-=

-- ^ h (e) 6 2 5 3 5 2+ + +

(f) 2

6 15 9 6 2 10 6- + -

3. (a) 2 2

(b) 2 3 32 6 3 2 3 3 6 2 3- + - + = - - + -^ h

(c) 39

22 5 14 2+

(d)

106 6 16 3 84 8 14

6 21 145

3 8 3 4

- - - +

=- + + -

^ h

(e) 4- (f) 4 2

(g) 15

20 12 19 6 25 3 615

19 6 65 3 6+ + -=

+ -

(h) 6

6 9 2 2 3+ + (i)

214 6 9 3+

(j) 415

30330 30 5- -

(k) 13

28 2 6 7 3- -

(l) 2

2 15 2 10 2 6 3 5+ - - -

4. (a) 45, 10a b= = (b) 1, 8a b= = (c) 21

,21

a b= - =

(d) 195

,98

a b= - = - (e) 5, 32a b= =

5.

2

2

3 2 23

2 1

2 1

2

4

2 1

2 1

2 1

2 1

2

4

2

2

2 1

2 1 2 12

4 2

2 12 2 2 1

2

13 2 2

2

2 2

2 2

# #

+

-+

=+

-

-

-+

=-

- -+

=-

- - ++

=-

+

= - +

=

^^ ^

hh h

So rational

6. (a) 4 (b) 14 (c) 16

7. 3

3 5 2 15 3- - -

8. 3 2 2

2

2

8

3 2 2

2

3 2 2

3 2 2

2

8

2

2

3 2 2

2 3 2 22

8 2

9 4 26 4 2

4 2

16 4 2

4 2

6 4 2 4 26

2 2

# #

#

++

=+ -

-+

=-

-+

=-

-+

=-

+

= - +

=

^^

hh

So rational

9. x 3 2= - +^ h 10. 4

4 4b

b b-

+ +

Test yourself 2

1. (a) 2y- (b) 4a + (c) 6k5- (d) 15

5 3x y+ (e) 3 8a b-

(f) 6 2 (g) 4 5

2. (a) 6 6x x+ -] ]g g (b) 3 1a a+ -] ]g g (c) ab b4 2-] g (d) 3)(5 )(y x- + (e) n p2 32 - +^ h (f) 2 )(4 2 )( x x x2- + +

3. (a) 4 6b - (b) 2 5 3x x2 + - (c) 4 17m +

(d) 16 24 9x x2 - + (e) 25p2 - (f) 1 7a- -

(g) 2 6 5 3- (h) 3 3 6 21 2 7- + -

4. (a) a ab 3 9

822 + +^ h (b)

2

15

m 2-] g

5. 157.464V = 6. (a) 17 (b) 17

6 15 9-

7. 3 2

4 5x x

x+ -

+

] ]g g 8. (a) 36 (b) 2- (c) 2 (d) 216 (e) 2

9. (a) 5

1 (b) 8 10. 11.25d =

11. (a) 15

2 3 (b)

22 6+

12. (a) 3 6 6 4 3 4 2- - + (b) 11 4 7+

13. (a) 3( 3)( 3)x x- + (b) x x6 3 1- +] ]g g (c) y y y5 2 2 42+ - +^ _h i

14. (a) 3y

x4

3

(b) 3 1

1x -

15. (a) 99 (b) 24 3

16. (a) a b2 2- (b) 2a ab b2 2+ + (c) 2a ab b2 2- +

17. (a) a b 2-] g (b) a b a ab b2 2- + +] ^g h 18.

23 3 1+

19. (a) 4 3

abb a+

(b) 10

3 11x -

20. 7

21 5 46 2- -

Answer S1-S5.indd 762 7/12/09 1:31:58 AM

763ANSWERS

21. (a) 6 2 (b) 8 6- (c) 2 3 (d) 3

4 (e) 30a b2

(f) 3n

m4 (g) 2 3x y-

22. (a) 2 6 4+ (b) 10 14 5 21 6 10 3 15- - +

(c) 7 (d) 43 (e) 65 6 14-

23. (a) 7

3 7 (b)

156

(c) 5 1

2+

(d) 15

12 2 6- (e)

5320 3 15 4 10 3 6+ + +

24. (a) 10

10x + (b)

2117 15a -

(c) ( 1)( 1)x x

x3 2+

-

-

(d) 1

1k -

(e) 3

15 6 15 3 15 2- - -

25. (a) 48n = (b) 175n = (c) 392n =

(d) n 5547= (e) 1445n =

26. 312171

27. (b), (c) 28. (d) 29. (a), (d) 30. (c)

31. (c) 32. (b) 33. (a) 34. (d) 35. (b)


1. (a) 2 8 6a b ab a2 2 3- + (b) 4y4 -

(c) 8 60 150 125x x x3 2- + -

2. 17

17 3 2 5 20+ + 3.

2 2

142

or

4. ab

xa

bx

ab

x4 2

2

2

2 2

+ + += d n

5. (a) x x4 9+ +] ]g g (b) ( ) ( )x y x y x y x y x y3 2 3 3 22 2 2- + = + - +_ _ _i i i (c) 5 7 25 35 49x x x2+ - +] ^g h (d) 2 2 2b a a- + -] ] ]g g g

6. 4 12 9 2 3x x x2 2+ + = +] g 7. x

y

1

1

2 -

+

] g 8. 2 5

9. 1

1

a a

a2

2

- +

+] g 10.

2 2x b

ax b

a+ -d dn n

11. x x x

x x x xy y

3 3 2

3 6 3 643 2

- + -

- + + -

] ] ]g g g

12. (a) 8 12 6 1x x x3 2- + - (b) 2 1

3 4

x

x2-

+

] g

13. x x x 97 153 2 + --

14. 13

66 6 4 2 15 4 5 65 3- + - + -

15. xx

x32

91

312

2

+ + = +d n 16. 2x =

17. 10

400 59 5- 18. (a) 3

12171

(b) ,a b2317

2314

= = -

19. 121

i = 20. 4

r4

3 3

r r

r= =

21. 2 6 3s = +

Chapter 3: Equations

Exercises 3.1

1. 5t = - 2. 5.6z = - 3. 1y = 4. 6.7w = 5. 12x =

6. 4x = 7. 151

y = 8. 35b = 9. 16n = - 10. 4r =

11. 9y = 12. 6k = 13. 2d = 14. 5x = 15. 15y =

16. 20x = 17. 20m = 18. 4x = 19. 7a = - 20. 3y =

21. 4b = - 22. 3x = 23. 132

a = - 24. 4t = -

25. 1.2x = 26. 1.6a = 27. 81

b = 28. 39t =

29. 5p = 30. 4.41x Z

Exercises 3.2

1. 331

b = 2. 35x = 3. 494

y = 4. 1359

x = 5. 585

k =

6. 36x = 7. 0.6t = 8. 3x = - 9. 1.2y = - 10. 69x =

11. 13w = 12. 30t = 13. 14x = 14. 1x = -

15. 0.4x = - 16. 3p = 17. 8.2t = 18. 9.5x = -

19. 22q = 20. 3x = - 21. 0.8b = 22. 0.375a = -

23. 3x = 24. 1y = 25. 132

t = -

Exercises 3.3

1. 8.5t = 2. 122l = 3. 8b = 4. 41a = 5. 4y =

6. 6.68r = 7. 6.44x = 8. 15n = 9. 332

y1 =

10. 3.7h = 11. (a) 25.39BMI = (b) 69.66w =

(c) 1.94h = 12. 0.072r = 13. 9x1 = - 14. 2.14t =

15. x 2!= 16. 2.12r = 17. 10.46r = 18. 1.19x =

19. 5.5x = 20. 3.3r =

Exercises 3.4

1. (a) x 32

-4 -3 -2 -1 0 1 2 3 4

(b) y 4#

-4 -3 -2 -1 0 1 2 3 4

2. (a) 7t 2 (b) x 3$ (c) 1p 2 - (d) x 2$ - (e) 9y 2 -

(f) a 1$ - (g) y 221

$ - (h) x 21 - (i) a 6# -

(j) y 121 (k) b 181 - (l) 30x 2 (m) x 343

#

(n) m 1432

2 (o) b 1641

$ (p) r 9# - (q) 8z 2

Answer S1-S5.indd 763 8/2/09 2:47:54 AM


(r) w 254

1 (s) x 35$ (t) t 9$ - (u) 6q52

2 -

(v) 1x32

2 - (w) b 1141

# -

3. (a) x1 71 1

0 1 2 3 4 5 6 7 8

(b) p2 51#-

-3 -2 -1 0 1 2 3 4 5

(c) x1 41 1

-3 -2 -1 0 1 2 3 4 5

(d) y3 5# #-

-3 -2 -1 0 1 2 3 4 5

(e) y61

132

1 1

-3 -2 -1 0 1 2 3 4 5

Exercises 3.5

1. (a) x 5!= (b) y 8!= (c) 4 4a1 1-

(d) ,k k1 1$ # - (e) 6, 6x x 12 -

(f) p10 10# #- (g) 0x = (h) ,a a14 142 1 -

(i) 12 12y1 1- (j) ,b b20 20$ # -

2. (a) ,x 5 9= - (b) ,n 4 2= - (c) ,a a2 212 -

(d) x4 6# # (e) ,x 3 6= - (f) ,x 5 475

= -

(g) 3 2y211 1- (h) ,x x9 6$ # - (i) x 12!=

(j) a2 10# #

3. (a) 141

x = (b) ,a 331

= - (c) 231

b =

(d) No solutions (e) 272

y = - (f) 7x = (g) ,m 5 132

=

(h) ,d 221

143

= - (i) ,y54

2= - (j) No solutions

4. (a) ,x 221

= - (b) 3, 231

y = (c) 10, 153

a = -

(d) ,x 4 731

= - (e) ,d 4 5= -

5. (a) ,t 3 152

= - (b) 1 3t521 1-

-3 -2 -1 0 1 2 3 4 5

Exercises 3.6

1. (a) 3x = (b) y 8!= (c) 2n != (d) 2x 5!=

(e) 10p = (f) x 5!= (g) y 3!= (h) 2w =

(i) n 4!= (j) 2q = -

2. (a) 6.71p != (b) 4.64x = (c) 2.99n = (d) 5.92x !=

(e) 1.89y = (f) .d 2 55!= (g) 4.47k != (h) 2.22x =

(i) .y 3 81!= (j) 3.01y =

3. (a) 27n = (b) 16t = (c) 32x = (d) 8t =

(e) 243p = (f) 625m = (g) 216b = (h) 27y =

(i) 128a = (j) 81t =

4. (a) 51

x = (b) 21

a = (c) 21

y = (d) x71

!=

(e) 32

n = (f) 2a = (g) 2x != (h) 9b =

(i) x32

!= (j) b 121

!=

5. (a) x512

1= (b) 6

41

x = (c) 811

a = (d) 625

1k =

(e) x81

= (f) 4x = (g) 8y = (h) n 73219

=

(i) 8b = (j) 1216127

m =

6. (a) 4n = (b) 5y = (c) 9m = (d) 5x = (e) 0m =

(f) 3x = (g) 2x = (h) 2x = (i) 1x = (j) k 2=

7. (a) 2x = (b) 1x = (c) 2x = - (d) 2n = (e) 0x =

(f) 6x = (g) y31

= (h) 2x = (i) 2x = (j) a 0=

8. (a) 21

m = (b) 31

x = (c) 31

x = (d) 21

k = -

(e) 32

k = - (f) 43

n = (g) 121

x = (h) 32

n =

(i) 61

k = - (j) 132

x =

9. (a) x 1= - (b) x 131

= - (c) k 4= - (d) n 3=

(e) x 212

= - (f) x32

= - (g) x 421

= - (h) x 1117

= -

(i) x 154

= (j) x 18=

10. (a) 41

m = (b) 243

k = - (c) 283

x = (d) 121

k =

(e) 181

n = (f) 21

n = - (g) 54

x = (h) 361

b = -

(i) 171

x = - (j) 5m =

Puzzle

1. All months have 28 days. Some months have more days as well. 2. 10 3. Bottle $1.05; cork 5 cents

4. 16 each time 5. Friday

Exercises 3.7

1. ,y 0 1= - 2. ,b 2 1= - 3. ,p 3 5= - 4. ,t 0 5=

5. ,x 2 7= - - 6. q 3!= 7. x 1!= 8. ,a 0 3= -

Answer S1-S5.indd 764 8/2/09 2:47:55 AM

765ANSWERS

9. ,x 0 4= - 10. x21

!= 11. ,x 1 131

= - -

12. ,y 1 121

= - 13. ,b43

21

= 14. ,x 5 2= - 15. ,x 032

=

16. ,x 1 221

= 17. 0, 5x = 18. 1, 2y = - 19. ,n 53=

20. 3, 4x = 21. 6, 1m = - 22. , ,x 0 1 2= - -

23. , ,y 1 5 2= - - 24. ,x 5 7= - 25. ,m 8 1= -

Exercises 3.8

1. (a) x 5 2!= - (b) 3a 7!= + (c) 4y 23!= +

(d) 1x 13!= - (e) p 44 7 2 11 7! != - = -

(f) x 28 5 2 7 5! != + = +

(g) 510y 88 2 22 2210 2! ! ! -= - = - = ^ h (h) 1x 2!= + (i) 12n 137!= -

(j) 3

y25!

=+

2. (a) . , .x 3 45 1 45= - (b) . , .x 4 59 7 41= - -

(c) . , .q 0 0554 18 1= - (d) . , .x 4 45 0 449= -

(e) . , .b 4 26 11 7= - - (f) . , .x 17 7 6 34=

(g) . , .r 22 3 0 314= - (h) . , .x 0 683 7 32= - -

(i) . , .a 0 162 6 16= - (j) . , .y 40 1 0 0749= -

Exercises 3.9

1. (a) . , .y 0 354 5 65= - - (b) , .x 1 1 5=

(c) . , .b 3 54 2 54= - (d) , .x 1 0 5= -

(e) . , .x 0 553 0 678= - (f) . , .n 0 243 8 24= -

(g) ,m 2 5= - - (h) ,x 0 7= (i) ,x 1 6= -

(j) . , .y 2 62 0 382=

2. (a) x2

1 17!=

- (b) x

65 13!

=

(c) q2

4 282 7

!!= =

(d) h8

12 1282

3 2 2! !=

-=

-

(e) s6

8 403

4 10! != =

(f) x2

11 133!=

- (g) d

125 73!

=-

(h) x2

2 321 2 2

!!= = (i) t

21 5!

=

(j) x4

7 41!=

Exercises 3.10

1. ,y y1 02 1 2. x021

1 1 3. 0, 1x x21

21

4. m072

1 # 5. ,x x53

0>1 - 6. b2 01#-

7. x1 141

1 1 8. z351

31 1- - 9. x2 243

1 #

10. ,x x2 261

1 2 11. x495

41#- -

12. x131

1157

1 1 13. ,a a341

221

1 2- -

14. x21

95

1 1 15. ,y y2 11 2- -

16. ,x x87

42# - 17. 4 26p21

11

18. ,x x151

2# - - 19. ,t t52

232

2#

20. 0m981 1- 21. 5, 0 1x x1 1 1-

22. ,n n0 2 41 # $ 23. ,x x5 3 01 12 -

24. 2, 1 6m m1# #- - 25. ,x x1 3 41# #-

26. ,x x221

32

1# #- 27. ,x x3 1 21$ #-

28. ,n n1 3 51 1 1- 29. ,x x432

74

1 1 1- - -

30. ,x x21

1 71# #

Exercises 3.11

1. 3 0x 11- 2. 0 4y1 1 3. ,n n0 1# $

4. ,x x2 2# $- 5. ,n n1 11 2- 6. n5 3# #-

7. ,c c1 21 2- 8. x4 2# #- - 9. 4 5x1 1

10. ,b b221

# $- - 11. ,a a131

1 2-

12. ,y y121

21 2- 13. ,x x32

1# $

14. ,b b352

1 2- 15. x121

31

# #- -

16. y4 3# #- 17. ,x x4 41 2- 18. a1 1# #-

19. 2 3x 11- 20. ,x x1 3# $- 21. 0 2x1 1

22. a1 121

# # 23. ,y y254

# $-

24. ,m m132

121

1 2- 25. x1 131

# #

26. 0 x21

11 27. ,x x021

1 $

28. ,y y154

1 2- - 29. 3 3n21

1 #

30. ,x x8 52# - - 31. ,x x52

73

1 2

32. ,x x451

52# 33. ,x x141

12# - -

34. ,x x3 21 2- 35. x43

53

1#- -

Answer S1-S5.indd 765 7/11/09 1:26:27 AM


Exercises 3.12

1. ,a b1 3= = 2. ,x y2 1= = 3. ,p q2 1= = -

4. ,x y6 17= = 5. ,x y10 2= - = 6. ,t v3 1= =

7. ,x y3 2= - = 8. ,x y64 39= - = - 9. ,x y3 4= = -

10. ,m n2 3= = 11. ,w w1 51 2= - = 12. ,a b0 4= =

13. ,p q4 1= - = 14. ,x x1 11 2= = -

15. ,x y1 4= - = - 16. ,s t2 1= = -

17. ,a b2 0= - = 18. ,k h4 1= - =

19. ,v v2 41 2= - = 20. , .x y2 1 41Z=

Problem

23 adults and 16 children.

Exercises 3.13

1. ,x y0 0= = and ,x y1 1= =

2. ,x y0 0= = and ,x y2 4= - =

3. ,x y0 3= = and ,x y3 0= =

4. ,x y4 3= = - and ,x y3 4= = - 5. ,x y1 3= - = -

6. ,x y3 9= = 7. ,t x2 4= - = and ,t x1 1= =

8. ,m n4 0= - = and ,m n0 4= = -

9. ,x y1 2= = and ,x y1 2= - = -

10. ,x y0 0= = and ,x y1 1= =

11. ,x y2 1= = and ,x y1 2= - = - 12. ,x y0 1= =

13. ,x y1 5= = and ,x y4 11= =

14. ,x y41

4= = and ,x y1 1= - = - 15. ,t h21

41

= - =

16. ,x y2 0= =

17. ,x y0 0= = and ,x y2 8= - = - and ,x y3 27= =

18. ,x y0 0= = and ,x y1 1= = and ,x y1 1= - =

19. ,x y21

243

= = 20. 135

,1312

x y= - = -

Exercises 3.14

1. , ,x y z2 8 1= - = - = - 2. , ,a b c2 1 2= - = - =

3. , ,a b c4 2 7= - = = 4. , ,a b c1 2 3= = = -

5. , ,x y z5 0 2= = = - 6. , ,x y z0 5 4= = - =

7. , ,p q r3 7 4= - = = 8. , ,x y z1 1 2= = - =

9. , ,h j k3 2 4= - = = - 10. , ,a b c3 1 2= = - = -

Test yourself 3

1. (a) 10b = (b) 116a = - (c) 7x = -

(d) ,x x431

32# - - (e) p 4#

2. (a) 1262.48A = (b) 8558.59P =

3. (a) x x x8 16 42 2- + = -] g (b) k k k4 4 22 2+ + = +] g 4. (a) ,x y2 5= - = (b) ,x y4 1= = and ,x y

21

8= - = -

5. (a) 2x = (b) 41

y =

6. (a) ,b 2 131

= - (b) ,g 241

= (c) ,x x4 3$ #

7. (a) 36A = (b) 12b = 8. ,x21

1=

9. 1 3y1 #-

10. (a) . , .x 0 298 6 70= - - (b) . , .y 4 16 2 16= -

(c) . , .n 0 869 1 54= -

11. (a) 764.5V = (b) 2.9r = 12. x 7141

2

13. ,x x2 91 2 14. . , .x y2 4 3 2= = 15. (a) 2100V =

(b) 3.9r = 16. (a) ii (b) i (c) ii (d) iii (e) iii

17. , ,a b c3 2 4= = = -

18. ,n n0 331

2 1 -

19. 4x = - 20. 2x = - 21. (a) 3y 2 (b) n3 0# #-

(c) 2x = (d) 2x = (e) ,x 3 152

= - (f) 1, 2t t$ # -

(g) 4 2x# #- (h) 3x = - (i) ,y y2 22 1 -

(j) 1, 1x x# $- (k) 65

x = (l) 21

2b# #-

(m) No solutions (n) 231

,53

t = (o) 1 3x1 1-

(p) ,m m3 2# $- (q) ,t t1 01 2- (r) 1 3y1 1

(s) 2 252

n1 # (t) x21

51

1 #-


1. 1y = 2. ,x a x a1 2- 3. ,a b3 2!= =

4. . , .x 2 56 1 56= - 5. ,y y2 0 31# #-

6. ; ,x x x x x x3 3 2 2 4 3 22!+ - - + + =] ] ] ^g g g h

7. ,x y1 2= = and ,x y1 0= - =

8. ; . , .b x4 17 4 8 12 0 123! Z= = + - 9. x 1!=

10. 1 1t1 1- 11. x3 8# #- 12. 41

x =

13. 2.31r = 14. No solutions 15. x b a a2!= + +

16. ,y y221

32

1# #- 17. 2247.36P =

18. x3

4 102 !=^ h

19. , . .x x4 2 2 0 71 1 1- -

20. ,y y153

1 2-

Answer S1-S5.indd 766 7/25/09 1:31:09 PM

767ANSWERS

Chapter 4 : Geometry 1

Exercises 4.1

1. (a) 47y c= (b) 39x c= (c) 145m c= (d) 60y c= (e) 101b c= (f) 36x c= (g) 60a c= (h) 45x c= (i) 40y c= (j) 80x c= 2. (a) 121c (b) 72 29c l (c) 134 48c l 3. (a) 42c (b) 55 37c l (c) 73 3c l

4. (a) (i) 47c (ii) 137c (b) (i) 9c (ii) 99c (c) (i) 63c (ii) 153c (d) (i) 35c (ii) 125c (e) (i) 52c (ii) 142c (f) (i) 15c7l (ii) 105c7l (g) (i) 47c36l (ii) 137c36l (h) (i) 72c21l (ii) 162c21l (i) (i) 26c 11 l (ii) 116c 11 l (j) (i) 38c 15 l (ii) 128c 15 l 5. (a) 49x c= (b) 41c (c) 131c 6. (a) ,y x z15 165c c= = =

(b) , ,x y z142 48 28c c c= = =

(c) , ,a b c43 137 101c c c= = =

(d) ,,a b d c97 41 42c c c= = = =

(e) , ,a b c68 152 28c c c= = = (f) ,a b10 150c c= =

7. 0x x x x

xx

8 10 2 10 10 7 10 36

18 36020

- + - + + + + =

=

=

(angleof revolution)

( )

( )

ABE x

EBC x

ABE EBC

8 108 20 101502 102 20 1030150 30180

c

cc cc

+

+

+ +

= -= -

== -= -

=+ = +

=

ABC`+ is a straight angle

( )

DBC x

DBC EBC

7 107 20 10150150 30180

cc cc

+

+ +

= +

= +

=

+ = +

=

DBE`+ is a straight angle AC and DE are straight lines

8.

AFC x

CD bisects

`

`

+ =

AFE+

( )

( )

( )

( )

DFB x

x

CFE x x

xAFC CFE

AFB

AFB

180 180

180 180 2

is a straight angle

(vertically opposite angles)

is a straight angle

`

c c

c c

+

+

+ +

+

+

= - -

=

= - + -

=

=

9. ABD DBC+ ++

110 3 3 70180

x xc

= - + +

=

So ABC+ is a straight angle. AC is a straight line.

10. AEB BEC CED+ + ++ +

y y y50 8 5 20 3 60

90c= - + - + +

=

So AED+ is a right angle.

Exercises 4.2

1. (a) ,a b e f c d g148 32c c= = = = = = =

(b) ,x z y70 110c c= = =

(c) , ,x y z55 36 89c c c= = = (d) ,y x z125 55c c= = =

(e) ,n e g a c z x 98c= = = = = = = 82o m h f b d y w c= = = = = = = =

(f) , ,a b c95 85 32c c c= = =

(g) , ,a b c27 72 81c c c= = =

(h) , , ,x y z a b56 124 116 64c c c c= = = = =

(i) 61x c= (j) 37y c=

2. (a) CGF

BFG CGF

180 121

5959

( is a straight angle)FGH

`

c c

cc

+

+ +

= -

=

= =

These are equal alternate angles. AB CD` < (b) BAC 360 292 68

(angle of revolution)c c c+ = - =

BAC DCA 68 112180

` c cc

+ ++ = +

=

These are supplementary cointerior angles.

AB CD` <

(c) 180 76104

104

BCD

ABC BCDc

c

+

+ +

= -

=

= =

( BCE+ is a straight angle)

These are equal alternate angles.

AB CD` ;

(d) 180 12852

52

CEF

CEF ABEc

c

+

+ +

= -

=

= =

( CED+ is a straight angle)

These are equal corresponding angles.

AB CD` ;

(e) 180 23 115CFH+ = - +] g ( EFG+ is a straight angle)

42c=

42BFD` c+ = (vertically opposite angles)

ABF BFD 138 42

180c cc

+ ++ = +

=

These are supplementary cointerior angles. AB CD` ;

Answer S1-S5.indd 767 7/12/09 1:32:20 AM


Exercises 4.3

1. (a) 60x c= (b) 36y c= (c) 71m c= (d) 37x c=

(e) 30x c= (f) 20x c= (g) 67x c= (h) 73a c=

(i) , ,a b c75 27 46c c c= = =

(j) , ,a b c36 126 23c c c= = =

(k) , ,x y z w67 59 121c c c= = = =

2. All angles are equal. Let them be x . x x x 180Then (angle sum of )D+ + =

xx

3 18060

=

=

So all angles in an equilateral triangle are 60 .c

3. x90 c-] g

4. 50

180 (50 45 )ACBABC

DEC ABC85

85

(vertically opposite angles)(angle sum of )

`

cc c cc

c

++

+ +

D

=

= - +

=

= =


AB DE` <

5.

124 68

ACB

CBACBA

CBA ACBABC

180 12456

68 124

5656

is isosceles

( is a straight angle)

(exterior angle of )

DCB

`

`

c cc

c cc cc

c

+

++

+ +D

D

= -

=

+ =

= -

=

= =

6. 38y c=

7. (a) x 64c= (b) ,x y64 57c c= = (c) 63x c=

(d) ,a b29 70c c= =

8. 180 (35 25 )120180 12060180 (90 30 )60180 (60 60 )60

HJI

IJL

JIL

ILJ

(angle sum of )

( is a straight angle)

(angle sum of )

(angle sum of )

HJI

HJL

IKL

JIL

c c ccc ccc c cc

c cc

+

+

+

+

D

D

D

= - +

=

= -

=

= - +

=

= - +

=

Since 60 ,IJL JIL ILJ c+ + += = = IJLD is equilateral

( )

( )

( )

KJL

JLK

KJI

JKL

180 60120180 30 12030

is a straight angle

angle sum of

c cc

c c cc

+

+ D

= -

=

= - +

=

°JLK JKL 30`+ += =

JKL` D is isosceles

9. BC BD=

BDC 46` c+ = (base angles of isosceles triangle)

CBD 180 2 4688

#c

+ = -=

CBD BDE 88` c+ += = These are equal alternate angles.

AB ED` ;

10. 18032

OQP 75 73c

+ = - +

=

] g (angle sum of triangle)

MNO OQP 32` c+ += =


MN QP` ;

Exercises 4.4

1. (a) Yes

5AB EF cm= = (given)

6BC DF cm= = (given)

8AC DE cm= = (given)

ABC DEF` /D D ( SSS )

(b)Yes

4.7XY BC m= = (given)

XYZ BCA 110c+ += = (given)

2.3YZ AC m= = (given)

XYZ ABC` /D D ( SAS )

(c) No

(d) Yes

PQR SUT 49c+ += = (given)

PRQ STU 52c+ += = (given)

8QR TU cm= = (given)

PQR STU` /D D ( AAS )

(e) No

2. (a)

,

AB KLB L

BC JLABC JKL

438

5by SAS

(given)(given)(given)

`

c+ +

/D D

= =

= =

= =

(b)

,

Z BXY ACYZ BC

RHS XYZ ABC

9072

by


`

c+ +

/D D

= =

= =

= =

(c)

,

MN QRNO PRMO PQ

MNO PQR

885

by SSS


` /D D

= =

= =

= =

(d)

.

Y TZ S

XY TRXYZ STR

90351 3

by AAS,


`

cc

+ ++ +

/D D

= =

= =

= =

(e)

,

BC DEC E

AC EFABC DEF

490

7by SAS


`

c+ +

/D D

= =

= =

= =

3. (a) B CBDA CDA

ADABD ACD

90is common

by AAS,

(base angles of isosceles )(given)

`

c+ +

+ +

/D D

D=

= =

Answer S1-S5.indd 768 8/2/09 2:47:56 AM

769ANSWERS

(b) BD DCAD BCbisects

(corresponding sides in congruent s)`

`

D=

4. , )AB CDABD BDC ernate angles+ + <= (alt

( , )ADB DBCBD

ABD CDBAD BC

AD BCis common

by AAS,

alternate angles

(corresponding sides in congruent s)

`

`

+ +

/

<

D D

D

=

=

5. (a) OA OC= (equal radii)

OB OD= (similarly)

AOB COD+ += (vertically opposite angles)

AOB COD` /D D ( SAS )

(b) AB CD= (corresponding sides in congruent

triangles)

6. (a) AB AD= (given)

BC DC= (given)

AC is common

ABC ADC` /D D ( SSS )

(b) ABC ADC+ += (corresponding angles in congruent

triangles)


OB is common

AOB COB 90c+ += = (given)

OAB OBC` /D D ( SAS )

(b) OCB OBC+ += (base angles of OBC, an isosceles

right angled triangle)

But OCB OBC 90c+ ++ = (angle sum of triangle)

So OCB OBC 45c+ += =

Similarly 45OBA c+ =

45 45 90OBA OBC` c c c+ ++ = + =

So ABC+ is right angled

8. (a) 90AEF BDC c+ += = (given)

AF BC= (given)

FE CD= (given)

AFE BCD` /D D ( RHS )

(b) AFE BCD+ += (corresponding angles in

congruent triangles)


OB is common

AB BC= (given)

OAB OBC` /D D ( SSS )

(b) OBA OBC+ += (corresponding angles in

congruent triangles)

But 180OBA OBC c+ ++ = ( ABC is a straight angle)

So 90OBA OBC c+ += =

OB is perpendicular to AC.

10. (a) AD BC= (given)

ADC BCD 90c+ += = (given) DC is common ADC BCD` /D D ( SAS )

(b) AC BD= (corresponding sides in congruent

triangles)

Exercises 4.5

1. (a) .x 15 1= (b) 4.4x = (c) 6.6m =

(d) , ,76 23 81c c ca i b= = = (e) 4.5b =

(f) , , .x y115 19 3 2c ca = = = (g) 9.7p =

2. . , .a b1 81 5 83= =

3. ( , )

( )

BAC EDCABC DECACB ECD

AB EDalternate angles(similarly)vertically opposite angles

+ ++ ++ +

<=

=

=

since 3 pairs of angles are equal, | CDED||ABCD

4.

.

..

..

.

GFE EFD

EFGF

DFEF

EFGF

DFEF

2 71 5

0 5

4 862 7

0 5

(given)

`

+ +=

= =

= =

=

o

o

Since two pairs of sides are in proportion and their included angles are equal, then | FGED||DEFD

5. ..

.

..

.

..

.

DEAB

DFAC

EFBC

DEAB

DFAC

EFBC

1 821 3

0 714

5 884 2

0 714

6 864 9

0 714

`

= =

= =

= =

= =

Since three pairs of sides are in proportion, | DEFD||ABCD

y 41c=

6. (a) OA OBOC OD

ODOA

OCOB

AOB COD

(equal radii)(similarly)


`

+ +

=

=

=

=

Since two pairs of sides are in proportion and their included angles are equal, | OCD3||OAB3

(b) 5.21AB cm=

7. (a) A+ is common

( , )ABC ADE

ACB AEDBC DEcorresponding angles

(similarly)+ ++ +

<=

=

Answer S1-S5.indd 769 8/2/09 2:47:58 AM


since 3 pairs of angles are equal, | ADED||ABCD

(b) . , .x y2 17 2 25= =

8. ( , )( , )( )

ABF BECCBE BFA

C A

s AB CDBC AD

s

alternate anglesimilarlyangle sum of`

+ ++ ++ +

z

z

D

=

=

=

since 3 pairs of angles are equal, | CEBD||ABFD

9. A+ is common

..

..

ABAD

ACAE

ABAD

ACAE

31 2

0 4

20 8

0 4

`

= =

= =

=

Since two pairs of sides are in proportion and their included angles are equal, | , .ABC m 4 25D =||AEDD

10. .

.

..

..

.

CDAB

ACBC

ADAC

CDAB

ACBC

ADAC

2 62

0 769

3 93

0 769

5 073 9

0 769

`

= =

= =

= =

= =

Since three pairs of sides are in proportion,

,c| ,ACD x y109 47cD = =||ABCD

11. (a) 7.8x = (b) . , .m p4 0 7 2= = (c) 6.5x =

(d) . , .x y6 2 4 4= = (e) . , .x y1 4 9 2= =

12. (a) BCAB

DEAD

DEAD

FGAF

BCAB

FGAF

Also

`

=

=

=

(b)ACAB

AEAD

AEAD

AGAF

ACAB

AGAF

Also

`

=

=

=

(c) CEBD

AEAD

AEAD

EGDF

CEBD

EGDF

Also

`

=

=

=

13. . , .a b4 8 6 9= = 14. 0.98y = 15. . , .x y3 19 1 64= =

Exercises 4.6

1. (a) 6.4x = (b) 6.6y = (c) 5.7b = (d) 6.6m =

2. (a) 61p = (b) 58t = (c) 65x = (d) 33y =

3. .s 6 2 m= 4. .CE 15 3 cm=

5. 81, 144, 225AB CB CA2 2 2= = =

AB CB

CA

81 144225

2 2

2

+ = +

=

=

ABC` D is right angled

6. 1XY YZ= = XYZ` D is isosceles

,YZ XY XZYZ XY

XZ

1 21 12

2 2 2

2 2

2

= = =

+ = +

=

=

XYZ` D is right angled

7. AC AB BC

BCBC

BCBC

AC

BC

2 34 311

22 12

2 2 2

2 2 2

2

2

`

#

= +

= +

= +

=

=

=

=

=

^ h

8. (a) 5AC =

(b) , ,AC CDAD

25 144169

2 2

2

= =

=

25 144169

AC CD

AD

2 2

2

+ = +

=

=

ACD` D has a right angle at ACD+ AC` is perpendicular to DC

9. AB b3= 10. xx y2 2+

11. d t tt t t t

t t

20 3 15 2400 120 9 225 60 413 180 625

2 2 2

2 2

2

= - + -

= - + + - +

= - +

] ]g g

12. 1471 mm 13. 683 m 14. 12.6 m 15. 134.6 cm

16. 4.3 m 17. 42.7 cm

18. 1.3 1.1 2.9 1.5 2.25and2 2 2+ = =

. . .1 3 1 1 1 52 2 2!+ so the triangle is not right angled the property is not a rectangle

19. No. The diagonal of the boot is the longest available space and it is only 1.4 m.

20. (a) 6 4BC2 2 2= - 20= 20BC = 6AO cm= (equal radii) So 6 4AC2 2 2= - 20= 20AC = Since ,BC AC= OC bisects AB

(b) OCA OCB 90c+ += = (given) OA OB= (equal radii) OC is common OAC OBC` /D D ( RHS ) So AC BC= (corresponding sides in congruent triangles) OC bisects AB

Exercises 4.7

1. (a) x 94c= (b) y 104c= (c) x 111c= (d) x 60c= (e) y 72c= (f) °, °x y102 51= = (g) °, °x y43 47= =

Answer S1-S5.indd 770 7/11/09 1:26:31 AM

771ANSWERS

2. ABED is isosceles.

( s )

( )

B ECBE DEB

76180 76104

base equal

straight s

` cc cc

+ ++ +

+

+

= =

= = -

=

D

DD

62 104 104 360270 360

90

(angle sum of quadrilateral)c c c cc c

c

++

+

+ + + =

+ =

=

CD is perpendicular to AD

3. (a)

( )( , )

( , )

( , )

D x

C x

xx

A C xB xB D x

A D AB DC

C D AD BC

B C AB DC

180

180 180

180 180

180180

and cointerior angles

and cointerior angles

and cointerior angles`

`

c

c c

c c

cc

+

+

+ +++ +

+ +

+ +

+ +

<

<

<

= -

= - -

= - +

=

= =

= -

= = -

(b) x x x x180 180360

Angle sum c cc

= + + - + -

=

4. ,a b150 74c c= =

5. (a) 5 , 3 , 108 , 72a b x z ym m c c= = = = = (b) , ,x y z53 56 71c c c= = = (c) 5 , 68x y cm ca b= = = = (d) , ,121 52 77c c ca b i= = = (e) 60x c= (f) ,x y3 7= =

6. ( ), ),

ADB CDBCDB ABDADB DBCABD DBC

BD ABC

BD ADCAB DCAD BC

bisects

bisects(alternate angles(alternate angles )

`

`

+ ++ ++ ++ +

+

+

<

<

=

=

=

=

7. (a) ..

AD BCAB DC

3 85 3

cmcm

(given)(given)

= =

= =

Since two pairs of opposite sides are equal, ABCD is a parallelogram.

(b) AB DCAB DC

7cm (given)

(given)<

= =

Since one pair of opposite sides is both equal and parallel, ABCD is a parallelogram.

(c) 54 126180

X M c cc

+ ++ = +

=

These are supplementary cointerior angles. XY MN` <

XM YNAlso, (given)<

XMNY is a parallelogram

(d) AE ECDE EB

56

cmcm

(given)(given)

= =

= =

Since the diagonals bisect each other, ABCD is a parallelogram.

8. (a) ,x 5 66cm ci= = (b) , ,90 25 65c c ca b c= = = (c) ,x y3 5cm cm= = (d) ,x y58 39c c= = (e) x 12 cm=

9. 6.4 cm 10. 59 , 31 , 59ECB EDC ADEc c c+ + += = =

11. 4 2 cm 12. 57x y c= =

Exercises 4.8

1. (a) 540c (b) 720c (c) 1080c (d) 1440c (e) 1800c (f) 2880c 2. (a) 108c (b) 135c (c) 150c (d) 162c (e) 156c 3. (a) 60c (b) 36c (c) 45c (d) 24c

4. 128 34c l 5. (a) 13 (b) 152 18c l 6. 16 7. 3240c

8. 2340c 9. 168 23c l

10. ( )

.

n nn n

nn

145145 180 360

3510 3

2 180Sum # c= = -

= -

=

=

But n must be a positive integer. no polygon has interior angles of 145 .c

11. (a) 9 (b) 12 (c) 8 (d) 10 (e) 30

12. (a) ABCDEF is a regular hexagon. AF BC= (equal sides) FE CD= (equal sides) AFE BCD+ += (equal interior angles) AFE BCD` /D D ( SAS )

(b) ( )

S n

AFE

6720

6720

120

2 1802 180#

#

cc

cc

c

+

= -

= -

=

=

=

] g

Since ,AF FE= triangle AFE is isosceles. So FEA FAE+ += (base angles in isosceles triangle)

FEA2

180 120

30

`c

c

+ =-

=

(angle sum of triangle)

EDA 120 3090

cc

+ = -

=

Similarly, DEB 90c+ =

So ED DEA B 180c+ ++ = These are supplementary cointerior angles AE BD` <

13. A regular octagon has equal sides and angles. AH AB= (equal sides)

GH BC= (equal sides) AHG ABC+ += (equal interior angles)

AHG ABC` /D D ( SAS )

So AG AC= (corresponding sides in congruent triangles)

( )S n

81080

2 1802 180#

#

cc

c

= -

= -

=

] g

AHG

81080

135

`c

c

+ =

=

HGA HAG+ += (base angles in isosceles triangle)

Answer S1-S5.indd 771 8/2/09 2:47:59 AM


HAG2

180 135

22 30

`c

c

+ =-

= l


GAC 135 2 22 30

90# c

c+ = -

=

l

We can similarly prove all interior angles are 90c and adjacent sides equal . So ACEG is a square .

14. EDC5

5

108

2 180# c

c

+ =-

=

] g

ED CD= (equal sides in regular pentagon)

So EDC is an isosceles triangle. DEC ECD`+ += (base angles in isosceles triangle)

36

DEC2

180 108c

c

+ =-

=


108 3672

AEC cc

+ = -

=

Similarly, using triangle ABC , we can prove that 72EAC c+ = So EAC is an isosceles triangle. (Alternatively you could prove EDC and ABC congruent triangles and then AC EC= are corresponding sides in congruent triangles.)

15. (a) p

360

(b) Each interior angle:

180360

180 360

180 360

180 2

p

p

p

p

p

p

p

p

-

= -

=-

=-^ h

Exercises 4.9

1. (a) .26 35 m2 (b) .21 855 cm2 (c) .18 75 mm2 (d) 45 m2 (e) 57 cm2 (f) 81 m2 (g) .28 27 cm2 2. .4 83 m2

3. (a) .42 88 cm2 (b) .29 5 m2 (c) .32 5 cm2 (d) .14 32 m2 (e) .100 53 cm2 4. (a) 25 m2 (b) .101 85 cm2 (c) .29 4 m2 (d) .10 39 cm2 (e) 45 cm2

5. 7 51 98 7 51 14 cm2+ = +^ h 6. .22 97 cm2

7. $621.08 8. (a) .161 665 m2 (b) 89 m2 (c) 10.5 m

9. (a) 48 cm (b) 27 cm 10. w12 units2

Test yourself 4

1. (a) , ,x y z43 137 147c c c= = = (b) 36x c= (c) , ,a b c79 101 48c c c= = = (d) 120x c= (e) 7.2r cm= (f) 5.6 , 8.5x ycm cm= = (g) 45ci =

2. )AGF HGB(vertically opposite+ +i=

AGF CFESo+ + i= =

These are equal corresponding .s+ AB CD` <

3. 118.28 cm 2

4. (a)

( )

DAE BACADE ABCAED ACB

ABC ADE AAAand are similar

(common)(corresponding angles, DE BC)(similarly)

`

+ ++ ++ +

<

D D

=

=

=

(b) 3.1 , 5.2x ycm cm= =

5. 162c 6. 1020.7 cm 3 7. 36 m

8. (a) AB ADBC DC

(adjacent sides in kite)(similarly)

=

=

AC is common Δ ABC and Δ ADC are congruent (SSS)

(b) AO COBO DO

AOB COD

(equal radii)(similarly)(vertically opposite angles)+ +

=

=

=

Δ AOB and Δ COD are congruent (SAS)

9. 73.5 cm 2

10. 6 2 7 36 28 64 82 2 2+ = + = =^ h ` Δ ABC is right angled (Pythagoras)

11. AGAF

AEAD

AEAD

ACAB

AGAF

ACAB

(equal ratios on intercepts)

(similarly)

`

=

=

=

12. (a) (base s of isosceles+ D)( , )

AB ACB C

BD DC AD BC

(given)

bisects given+ +

=

=

=

ABD ACD SAS` /D D ] g (b)

180ADB ADC

ADB ADCBut(corresponding s in congruent s)

(straight )c+ ++ +

+

+

D=

+ =

So 90ADB ADC c+ += =

So AD and BC are perpendicular.

13.

34˚ 34

( )( )

ACBCAD

CAD ADC

6868 34

base s of isoscelesexterior of

`

cc cc

c

++

+ +

+

+

D

D

=

= -

=

= =

So Δ ACD is isosceles base s equal+^ h 14.

Answer S1-S5.indd 772 7/11/09 1:26:32 AM

773ANSWERS

( , )

, )DAC ACBBAC ACD

AD BCs AB DC

alternate s(alternate

+ ++ +

+

+

<

<

=

=

AC is common

ABC ADC

AB DC(AAS)


`

/D D

D=

Similarly, AD BC= opposite sides are equal

15. (a) 24 cm 2 (b) 5 cm 16. 9

17. BFG FGD x x109 3 3 71180c cc

+ ++ = - + +

=

These are supplementary cointerior .s+ AB CD` <

18. 57 cm 2

19. (

(( )

)

)

ACB A Bx y

ACD ACBz x y

x yx y

180180180180 180180 180

sum of

straight

cccc cc c

+ + +

+ +

+

+

D= - +

= - -

= -

= - - -

= - + +

= +

] g

20. (a)

..

.

..

.

A E

EFAC

DEAB

EFAC

DEAB

2 72 97

1 1

3 63 96

1 1

given

`

+ +=

= =

= =

=

^ h

So Δ ABC and Δ DEF are similar (two sides in proportion, included s+ equal). (b) 4.3x cm=


1. 94c 2. , ,x y z75 46 29c c c= = = 3. ,1620 32 44c c l

4. , )

( )

BAD DBCABD BDCADB DCB

AB DC(given)(alternate anglesangle sum of`

+ ++ ++ +

<

D

=

=

=

since 3 pairs of angles are equal, BCDD;ABD <D

6.74d cm=

5. AB DCA D 131 49

180

(given)c cc

+ +=

+ = +

=

A+ and D+ are supplementary cointerior angles AB DC` <

Since one pair of opposite sides are both parallel and equal, ABCD is a parallelogram.

6. .27 36 m2

7.

Let ABCD be a square with diagonals AC and BD and

D

AD DC90

(adjacent sides of square)c+ =

=

°

°°

ADCDAC DCA

DAC DCADAC DCABAC BCA

904545

is isosceles

Similarly,

(base angles of isosceles )(angle sum of )

(other angles can be proved similarly)

`

`

`

+ ++ +

+ ++ +

D

D

D

=

+ =

= =

= =

8.

Let ABCD be a kite

AD ABDC BC

(given)(given)

=

=

AC is common

, ADC ABCDAC BAC

AD ABDAE BAE

by SSS

(corresponding angles in congruent s)(given)(found)

`

` + +

+ +

/D D

D

=

=

=

AE is common

,(

( )

ADE ABEDEA BEADEA BEADEA BEA

DEB18090

by SAS

But

the diagonals are perpendicular

corresponding angles in congruent s)is a straight angle

`

`

`

`

cc

+ ++ ++ +

/D D

D=

+ =

= =

9. 84 (15 112 ) )

( )

MNYMNY

XYZXYZMNY XYZ

MNZ

XYZ43

69 11243

43

(exterior angle of

exterior angle of`

`

`

c c cc

c cc

c

++

+++ +

D

D

+ = +

=

+ =

=

= =

These are equal corresponding angles. MN XY` <

10. .x 2 12 m= 11. (a) 6 m2 (b) 10 2 5 2 5 5 m+ = +^ h

12. . , .x y28 7 3 8cm cm== 13. 7.40 , 4.19x ym m= =

Answer S1-S5.indd 773 8/2/09 2:48:00 AM


14. (a) AB BCABE CBE 45

(adjacent sides in square)

(diagonals in square make 45 with sides)c

+ +=

= =

EB is common.

, ABE CBE

AE CEby SAS


`

/D D

D=

Since AB BC= and ,AE CE= ABCE is a kite.

(b) BD x x

xx

DE BD

x

22

21

22

units

2 2

2

= +

=

=

=

=

Practice assessment task set 1

1. 9p = 2. 2 5 y x y+ -^ ^h h 3. (a) x 1- (b) 3x4

4. 6 10y - 5. 23

25 5 2+ 6. 2 16 3x x x3 2+ - +

7. 72

x = 8. 3

2x -

9. °ABC EDCACB ECD

AB EDABC EDC

AC ECACE

90

by AAS

is isosceles

(given)(vertically opposite angles)(given)

(corresponding sides in congruent triangles)`

`

`

+ ++ +

/D D

D

= =

=

=

=

10. 231.3 11. 3- 12. 135c 13. 7.33 10 2#

-

14. 3 10 4- 15. 3.04 16. 3x + 17. . , .x 1 78 0 281= -

18. 1.55r = 19. x 12

20. 157

21. x2

42 3

12!!= = 22.

491

23. 4, 11 1, 4x y x yor= = = - = - 24. ,x y2 1= = -

25. 7 26. 7.02 cm 27. 2 1 4 2 1x x x2- + +] ^g h

28. 43

6 15 2 6+ 29. 7 30. $643.08 31. 1.1

32. 2 10 3 5 2 2 3- + - + 33. $83.57

34. , ,x y w z22 29 90c c c= = = = 35. 56.7 cm2

36. a ba

b21 10

21

10

=- 37. ,x x6 252

2 1 - 38. 81

39. x 7- - 40. 41

x = 41. ,x x3 3# $- 42. 61

43. Given diagonal AC in rhombus ABCD :

)

)

AB BCDAC ACBBAC ACBDAC BAC

AD BCABC

(adjacent sides in rhombus)(alternate s,(base s of isosceles

`

+ ++ ++ +

+

+

<

D

=

=

=

=

` diagonal AC bisects the angle it meets. Similarly, diagonal BD bisects the angle it meets.

44. x 3 1+ -] g 45. 6 12 8x x x3 2+ + + 46. 2

517

4

47. 53x c=

48. ,x y98 41c c= = 49. x0 51 1 50. 3 2

1

x +

51. (a) 12 8x y- (b) 2 31 (c) 3 9

3

x x

x2 - +

- (d) 3 2 1+

(e) 1 1

5

x x

x

+ -

- +

] ]]g g

g (f)

611 3

(g) x y zx z

y14 7 11

14 11

7

=- -

(h) 5 1 2

3a a b b+ +] ]g g (i) 8 5 (j) 13

21

52. . , .x y2 7 3 1= = 53. 25x = 54. r2

cm3 r

=

55. 17.3 cm

56. DEA xEAD xCD x x

xABC xABC DEA

A222

LetThen (base s of isosceles )

(exterior of )

(opposite s of gram are equal)

EAD

`

`

+++

++ +

+

+

+ <

D

D

=

=

= +

=

=

=

57. 52

58. 5% 59. 2.2 10 kmh8 1#

- 60. 20k =

61. 9xy y 62. 147 16c l 63. 5.57 m2

64. (a) a b a a ab b b5 2 2 4 2 4 4 42 2+ - - - + + +] ^g h (b) 3 4 6 2a b a b c+ - +] ]g g

65. x181

543

1#-

66. (BCEF is a gram)<

(BC AD ABCDBC FEAD FE

is a gram)

`

< <

<

<

BC ADBC FEAD FE

Also opposite sides of gram

similarly`

<=

=

=

^^

hh

Since AD and FE are both parallel and equal, AFED is a parallelogram.

67. 11.95b m= 68. (a) 34 cm (b) 30 cm 2

Answer S1-S5.indd 774 7/11/09 1:26:34 AM

775ANSWERS

69. 75

18 3 31 2 25 5+ - 70. 20 71. 32 m

72. BD bisects AC So AD DC= 90BDC BDA c+ += = (given) BD is common BAD BCD` /D D ( SAS ) AB CB` = (corresponding sides in congruent

triangles) So triangle ABC is isosceles

73. 2

x y2 2+ 74. (b) 75. (c) 76. (a) 77. (b) 78. (b)

79. (d) 80. (d)

Chapter 5 : Functions and graphs

Exercises 5.1

1. Yes 2. No 3. No 4. Yes 5. Yes 6. Yes 7. No

8. Yes 9. Yes 10. No 11. Yes 12. No 13. Yes

14. No 15. Yes

Exercises 5.2

1. 4, 0f f1 3= - =] ]g g 2. , ,h h h0 2 2 2 4 14= - = - =] ] ]g g g

3. 25, 1, 9, 4f f f f5 1 3 2= - - = - = - - = -] ] ] ]g g g g 4. 14

5. 35- 6. 9x = 7. x 5!= 8. x 3= - 9. ,z 1 4= -

10. 2 9, 2 2 9f p p f x h x h= - + = + -^ ]h g

11. 1 2g x x2- = +] g 12. f k k k k1 12= - + +] ] ^g g h 13. ; ,t t1 2 4= - = - 14. 0

15. 125, 1, 1f f f5 1 1= = - = -] ] ]g g g

16. 0 4 1 3f f f2 2 1- - + - = - + = -] ] ]g g g

17. 10 18. 7 19. 28-

20. (a) 3 (b) 3 3 3 0x - = - = Denominator cannot be 0 so the function doesn’t exist for .x 3= (c) 4

21. 2 5f x h f x xh h h2+ - = + -] ]g g 22. 4 2 1x h+ +

23. x c5 -] g 24. 3 5k2 + 25. (a) 2 (b) 0 (c) 2n n4 2+ +

Exercises 5.3

1. (a) x -intercept 32

, y -intercept -2

(b) x -intercept -10, y -intercept 4 (c) x -intercept 12, y -intercept 4 (d) x -intercepts 0, -3, y -intercept 0 (e) x -intercepts 2! , y -intercept -4 (f) x -intercepts -2, -3, y -intercept 6

(g) x -intercepts 3, 5, y -intercept 15

(h) x -intercept 53- , y -intercept 5 (i) x -intercept -3, no y -intercept (j) x -intercept ,3! y -intercept 9

2. 2

( )

f x xxf x

2

even function

2

`

- = - -

= -

=

2] ]g g

3. (a) 1f x x2 6= +^ h (b) f x x x2 12 6 3= + +] g7 A

(c) 1f x x3- = - +] g (d) Neither odd nor even

4.

( )

g x x x xx x xg x

3 23 2

even function

8 4 2

8 4 2

`

- = - + - - -

= + -

=

] ] ] ]g g g g

5. f x x f x- = - = -] ]g g odd function

6. 1

( )

f x xxf x

1

even function

2

2

`

- = - -

= -

=

] ]g g

7. f x x xx xx x

f x

444

odd function

3

3

3

`

- = - - -

= - +

= - -

= -

] ] ]^]

g g gh

g

8. f x x xx xf x

even function

4 2

4 2

`

- = - + -

= +

=

] ] ]]

g g gg

0f x f x- - =] ]g g

9. (a) Odd (b) Neither (c) Even (d) Neither (e) Neither

10. (a) Even values i.e. , , ,n 2 4 6 f=

(b) Odd values i.e. , , ,n 1 3 5 f=

11. (a) No value of n (b) Yes, when n is odd (1, 3, 5, …)

12. (a) (i) x 02 (ii) x 01 (iii) Even

(b) (i) x 21 (ii) x 22 (iii) Neither

(c) (i) x2 21 1- (ii) ,x x2 21 2- (iii) Neither

(d) (i) All real x 0! (ii) None (iii) Odd

(e) (i) None (ii) All real x (iii) Neither

Exercises 5.4

1. (a) x -intercept 2, y -intercept -2

(b) x -intercept 121

- , y -intercept 3

(c) x -intercept 21

, y -intercept 1

(d) x -intercept -3, y -intercept 3

(e) x -intercept 32

, y -intercept 31

-

Answer S1-S5.indd 775 7/11/09 1:26:34 AM


2. (a)

-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

(b) y

x-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

(c) y

x-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

(d) y

x-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2-1

1

(e) y

x-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2-1

112

(f) y

x-4

-5

-3 -2-1 2 3 4

2

1

3

4

5

-3

-4

-2-1

1

(g) y

x-4

-5

-3-2 -1 2 3 4

2

1

3

4

5

-3

-4

-2-1

1

23

-

(h) y

x-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

Answer S1-S5.indd 776 7/11/09 1:26:35 AM

777ANSWERS

(i) y

x-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

(j) y

x-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-1111

2

3. (a) ,x yall real all real" ", , (b) :,x y y 2all real =" ", , (c) : ,x x y4 all real= -! "+ , (d) : ,x x y2 all real=! "+ , (e) , :x y y 3all real =! "+ ,

4. (a) Neither (b) Even (c) Neither (d) Odd (e) Odd

5. y

x-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-1111

2

(3, -1)

Exercises 5.5

1. (a) x -intercepts 0, -2, y -intercept 0 (b) x -intercepts 0, 3, y -intercept 0 (c) x -intercepts ! 1, y -intercept -1 (d) x -intercepts -1, 2, y -intercept -2 (e) x -intercepts 1, 8, y -intercept 8

2. (a) y

x-4

-5

-3 -2-1 2 3 4 5

2

1

3

4

5

6

-3-4

-2-1

1

(b) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

6

-3

-4

-2

-11

(c) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

6

-3

-4

-2-1

1

Answer S1-S5.indd 777 7/11/09 1:26:36 AM


(d) y

x-4

-5

-3 -2-1 2 3 4 5

2

1

3

4

5

6

-3

-4

-2

-1 1

(e) y

x-4

-5

-3 -2-1 2 3 4 5

21

3

4

5

6

-3

-4

-2

-11

(f) y

x-4

-10

-3 -2 -1 2 3 4 5

4

6

8

2

10

12

-6

-8

-4-2

1

(g) y

x-4

-5

-3 -2 -1 2 3 4 5

21

3

4

5

-3

-4

-6

-2

-11

(h) y

x

-5

-3-4 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-6

-2

-1 1

(i) y

x-4

-5

-3 -2 -1 3 4 5

2

1

3

4

5

-3

-4

-6

-2-1 2111

2

(j) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3-4

-6

-2-1

1

3. (a) (i) x -intercepts 3, 4, y -intercept 12 (ii) {all real x },

:y y41

$ -( 2 (b) (i) x -intercepts 0, -4, y -intercept 0 (ii) {all real x }, :y y 4$ -" , (c) (i) x -intercepts -2, 4, y -intercept -8 (ii) {all real x }, : 9y y $ -" , (d) (i) x -intercept 3, y -intercept 9 (ii) {all real x }, :y y 0$" , (e) (i) x -intercepts ,2! y -intercept 4 (ii) {all real x }, :y y 4#" ,

4. (a) {all real x }, :y y 5$ -" , (b) {all real x }, :y y 9$ -" ,

Answer S1-S5.indd 778 7/11/09 1:26:37 AM

779ANSWERS

(c) {all real x }, :y y 241

$ -( 2 (d) {all real x }, :y y 0#" , (e) {all real x }, : 0y y $" ,

5. (a) y0 9# # (b) y0 4# # (c) y1 24# #-

(d) y4 21# #- (e) y18 241

# #-

6. (a) (i) x 02 (ii) x 01 (b) (i) x 01 (ii) x 02

(c) (i) x 02 (ii) x 01 (d) (i) x 21 (ii) x 22 (e) (i) x 52 - (ii) x 51 -

7.

( )

f x xx

f xeven

2

2

`

- = - -

= -

=

] ]g g

8. (a) Even (b) Even (c) Even (d) Neither (e) Neither (f) Even (g) Neither (h) Neither (i) Neither (j) Neither

Exercises 5.6

1. (a) x -intercept 0, y -intercept 0 (b) No x -intercepts, y -intercept 7 (c) x -intercepts ,2! y -intercept -2 (d) x -intercept 0, y -intercept 0 (e) x -intercepts ,3! y -intercept 3 (f) x -intercept -6, y -intercept 6

(g) x -intercept 32

, y -intercept 2

(h) x -intercept 54

- , y -intercept 4

(i) x -intercept 71

, y -intercept 1

(j) No x -intercepts, y -intercept 9

2. (a) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(b) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(c) y

-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(d) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(e) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(f) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

Answer S1-S5.indd 779 7/13/09 10:16:03 AM


(g) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(h) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(i) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(j) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

3. (a) {all real x }, :y y 0$" , (b) {all real x }, :y y 8$ -" , (c) {all real x }, :y y 0$" , (d) {all real x }, :y y 3$ -" , (e) {all real x }, :y y 0#" ,

4. (a) (i) x 22 (ii) x 21 (b) (i) x 02 (ii) x 01

(c) (i) x 121

2 (ii) x 121

1 (d) (i) x 02 (ii) x 01

(e) (i) x 01 (ii) x 02

5. (a) 0 2y# # (b) y8 4# #- - (c) 0 6y# #

(d) 0 11y# # (e) y1 0# #-

6. (a) x 32 - (b) x 01 (c) x 92 (d) x 22 (e) x 21 -

7. (a) x 3!= (b) ,x x1 12 1 - (c) x2 2# #-

(d) ,x 1 3= - - (e) 3x = (f) ,x 1 2= (g) x3 51 1-

(h) x4 2# #- (i) ,x x4 02 1 (j) ,x x2 4# $

(k) x4 1# #- (l) ,x x0 1# $ (m) ,x 221

= -

(n) No solutions (o) 0x = (p) 1x = (q) ,x 0 2= -

(r) No solutions (s) 31

x = ( t) 0, 6x =

Exercises 5.7

1. (a) (i) {all real x : x ! 0}, {all real y : y ! 0} (ii) no y -intercept

(iii) y

x-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

(b) (i) {all real : },x x 0! {all real :y y 0! } (ii) no y -intercept

(iii) y

x-2 -1 2

2

1

-2

-1

1

Answer S1-S5.indd 780 7/12/09 2:04:41 AM

781ANSWERS

(c) (i) {all real :x x 1! - }, {all real : 0y y ! } (ii) 1

(iii) y

x-2 -1 2

2

1

-2

-1

1

(d) (i) {all real :x x 2! }, {all real : 0y y ! } (ii) 121

-

(iii) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(e) (i) {all real :x x 2! - }, {all real : 0y y ! } (ii) 61

(iii) y

x-2 -1 2

2

1

-2

-1

1

(f) (i) {all real :x x 3! }, {all real :y y 0! } (ii) 32

(iii) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(g) (i) {all real : 1x x ! }, {all real : 0y y ! } (ii) -4

(iii) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

(h) (i) {all real : 1x x ! - }, {all real : 0y y ! } (ii) -2

(iii) y

x-4

-5

-3 -2-1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

Answer S1-S5.indd 781 7/11/09 1:26:40 AM


(i) (i) :21

x xall real !' 1 , {all real : 0y y ! } (ii) 32

-

(iii) y

x-2 -1 2

2

1

-2

-1

1

23

-

12

(j) (i) {all real :x x 2! - }, {all real :y y 0! } (ii) -3

(iii) y

x-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-2

-11

2.

( )

f x x

xf x

2

2

odd function`

- =-

= -

= -

] g

3. (a) 1y91

## (b) 1y31# # (c) y2

21

21

# #- -

(d) 3y73

## (e) 2 y81

# #- -

4. (a) 1 3x# # (b) 1 4x# # (c) 6 0x# #-

(d) 1 4x# # (e) 1 2x# #

Exercises 5.8

1. (a) (i) y

x-3

3

3

-3

(ii) : , :x x y y3 3 3 3# # # #- -! "+ , (b) (i) y

x-4

4

4

-4

(ii) : , :x x y y4 4 4 4# # # #- -! "+ , (c) (i)

(2, 1)

-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

y

x

Answer S1-S5.indd 782 7/11/09 1:26:41 AM

783ANSWERS

(ii) : 0 4 , : 1 3x x y y# # ## -! "+ , (d) (i)

-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

y

x

(ii) : , :x x y y4 2 3 3# # # #- -! "+ , (e) (i)

-4 -3 -2 -1 2 3 4

2

1

3

4

5

-2

-1

(-2, 1)

1

y

x

(ii) : , :x x y y3 1 0 2# # # #- -! "+ , 2. (a) (i) Below x -axis

(ii) y

x-5 5

-5

(iii) : , :x x y y5 5 5 0# # # #- -! "+ , (b) (i) Above x -axis

(ii) y

x-1

1

1

(iii) : , :x x y y1 1 0 1# # # #-! "+ , (c) (i) Above x -axis

(ii) y

x-6

6

6

(iii) : , :x x y y6 6 0 6# # # #-! "+ , (d) (i) Below x -axis

(ii) y

x-8 8

-8

(iii) : , :x x y y8 8 8 0# # # #- -! "+ ,

Answer S1-S5.indd 783 7/11/09 1:26:42 AM


(e) (i) Below x -axis

(ii) y

x- 7

- 7

7

(iii) : , :x x y y7 7 7 0# # # #- -" #, - 3. (a) Radius 10, centre (0, 0) (b) Radius 5 , centre (0, 0)

(c) Radius 4, centre (4, 5) (d) Radius 7, centre (5, -6) (e) Radius 9, centre (0, 3)

4. (a) 16x y2 2+ =

(b) 6 4 12 0x x y y2 2- + - - =

(c) 2 10 17 0x x y y2 2+ + - + =

(d) 4 6 23 0x x y y2 2- + - - =

(e) 8 4 5 0x x y y2 2+ + - - =

(f) 4 3 0x y y2 2+ + + =

(g) 8 4 29 0x x y y2 2- + - - =

(h) 6 8 56 0x x y y2 2+ + + - =

(i) 4 1 0x x y2 2+ + - =

(j) 8 14 62 0x x y y2 2+ + + + =

Exercises 5.9

1. (a) {all real x }, {all real y } (b) {all real x }, {y: y = -4} (c) {x: x = 3}, {all real y } (d) {all real x }, { y : y $ -1 }

(e) {all real x }, {all real y } (f) {all real x }, : 1241

y y #' 1 (g) { : 8 8}, { : 8 8}x x y y# # # #- -

(h) {all real :t t 4! }, {all real ( ): ( )f t f t 0! }

(i) {all real : 0!z z }, {all real :g g 5!zz^ ^h h }

(j) {all real x }, { :y y 0$ }

2. (a) { x : 0x $ }, { y : y 0$ } (b) { x : x 2$ }, { y : y 0$ } (c) {all real x }, { y : y 0$ } (d) {all real x }, { y : y 2$ - }

(e) : 221

, { : }x x y y 0$ #-' 1

(f) {all real x }, { :y y 5# } (g) {all real x }, { : }y y 02

(h) {all real x }, { : }y y 01

(i) {all real :x x 0! }, {all real :y y 1! } (j) {all real :x x 0! }, {all real :y y 2! }

3. (a) ,x 0 5= (b) , ,x 3 1 2= - (c) , ,x 0 2 4=

(d) ,x 0 4!= (e) x 7!= 4. (a) x1 1# #-

(b) { : }x x1 1# #-

5. (a) { : , }x x x1 2# $- (b) { : , }t t t6 0# $-

6. (a) { y : y9 3# #- }

(b) { y : y0 9# # } (c) { y : y8 1# #- }

(d) :51

1y y# #' 1 (e) { y : 0 4y# # }

(f) { y : y1 15# #- } (g) { y : y1 0# #- }

(h) :y y1 8# #-" , (i) { y : 4 21y# #- }

(j) :y y61

64

# #-' 1 7. (a) {all real :x x 1! - }

(b) x -intercept: 0y =

01

3x

=+

0 3= This is impossible so there is no x -intercept (c) {all real :y y 0! }

8. (a) {all real :x x 0! } (b) {all real :y y 1!! }

9. (a) y

x-4 -3 -2 -1 2 3 4 5

10

5

15

20

25

-15

-10

-51

(b) y

x-4 -3 -2 -1 2 3 4

4

2

6

8

-6

-8

-4

-21

(c) y

x-4 -3 -2 -1 2 3 4 5

10

5

15

20

25

-15

-10

-51

Answer S1-S5.indd 784 7/11/09 1:26:43 AM

785ANSWERS

(d) y

x-4 -3 -2 -1 2 3 4

4

2

6

8

-6

-8

-4

-21

(e) y

x-4 -3 -2 -1 2 3 4

4

2

6

8

-6

-8

-4

-21

(f) y

x-10 10

10

-10

(g) y

x-1

1

2

3

-1

1

10. (a) : : 0x x y y1$ $" ", , (b) y

x2 3

2

1

-11

11. y

x-1

4

3

2

1

5

6

-1 1

12. (a) (i) {all real x }, {all real y } (ii) All x (iii) None (b) (i) {all real x }, :y y 22 -" , (ii) x 02 (iii) x 01 (c) (i) {all real :x x 0! }, {all real : 0y y ! } (ii) None (iii) All 0x ! (d) (i) {all real x }, {all real y } (ii) All x (iii) None (e) (i) {all real x }, :y y 02" , (ii) All x (iii) None

13. (a) 2 2x ##- (b) (i) { x : 2 2x# #- }, { y: 0 2y# # } (ii) { x : 2 2x# #- }, { y: 2 0y# #- }

Exercises 5.10

1. (a) 21 (b) 10- (c) 8 (d) 3 (e) 3 (f) 75 (g) 0

(h) 6- (i) 41

(j) 1 (k) 7- (l) 3x x2 -

(m) 2 3 5x x3 + - (n) 3c2

2. (a) Continuous (b) Discontinuous at 1x = - (c) Continuous (d) Continuous (e) Discontinuous at x 2!=

Answer S1-S5.indd 785 8/2/09 2:48:01 AM


3. (a)

(b)

(c)

Exercises 5.11

1. (a) 0 (b) 0 (c) 0 (d) 2 (e) 1 (f) 6 (g) 32

(h) 0 (i) 5 x (j) 3

2. (a) x x

x

x x

11 3

3

RHS

LHS

2

2

2

= + +

=+ +

=

(b) 1 from above (c) 1 from below

3. (a) 2 from below (b) 2 from above

4. (a) 3x

(b) 4

5x2

5. (a)

(b)

(c)

(d)

(e)

Answer S1-S5.indd 786 7/12/09 1:34:32 AM

787ANSWERS

(f)

(g)

(h)

(i)

(j)

Exercises 5.12

1. 0x21

11- 2. 0 x31

1 1 3. x0 11 #

4. x21

01#- 5. 1 1x31

11 6. ,x x1 21$ - -

7. x2 252

1 # 8. ,x x6 31 2- -

9. ,x x32

12# 10. x232

21#- -

Exercises 5.13

1. (a) y

x-4 -3 -2 -1 2 3 4

2

1

3

4

5

6

-3

-4

-2

-11

(b) y

x-4 -3 -2 -1 2 3 4

2

1

3

4

5

6

-3

-4

-2

-11

(c) y

x-4 -3 -2 -1 2 3 4

2

1

3

4

5

6

-3

-4

-2

-11

Answer S1-S5.indd 787 7/11/09 1:26:51 AM


(d) y

x-4 -3 -2 -1 2 3 4

2

1

3

4

5

6

-3

-4

-2

-11

(e)

y = x +1

y

x-4 -3 -2 -1 2 3 4

2

1

3

4

5

6

-3

-4

-2

-11

(f)

y = 2x-3

y

x-4 -3 -2 -1 2 3 4

2

1

3

4

5

6

-3

-2

-11

(g)

x + y = 1

y

x-4 -3 -2 -1 2 3 4

2

1

3

4

5

6

-3

-2

-11

-4

(h)

3x - y - 6 = 0

y

x-4 -3 -2 -1 2 3 4

2

1

3

4

5

6

-3

-2

-11

-4

-5

-6

(i)

x + 2y - 2 = 0

y

x-4 -3 -2 -1 2 3 4

2

1

3

4

5

6

-3

-2

-11

-4

-5

-6

Answer S1-S5.indd 788 7/11/09 1:26:53 AM

789ANSWERS

(j)

x-4 -3 -2 -1 2 3 41

y

2

1

3

4

5

6

-3

-2

-1

-4

-5

-6

x =12

2. (a) x 32 - (b) y 2$ - (c) y x 1$ + (d) y x 422 -

(e) y 2x$

3. (a) y

x-4 -3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-2

-11

-4

-5

y = x2 - 1

(b)

-3 3

3

-3

y

x

(c) y

x-1 1

1

-1

(d)

x-3-4 -2 -1 2 3 4 51

y = x 2

y

1

2

3

4

5

-3

-2

-1

-4

-5

(e) y

x-4 -3 -2 -1 2 3 4

4

2

6

8

-6

-8

-4

-21

y = x3

4. (a) y x3 21 - (b) y x 222 +

(c) x y 492 21+

(d) x y 812 22+

(e) ,x y5 21 2

Answer S1-S5.indd 789 7/11/09 1:26:55 AM


5. (a) y

x-4 -3 -2 -1 2 3 4

3

1

2

4

5

-2

-11

(b) y

x-4 -3 -2 -1 2 3 4

3

1

2

4

5

-2

-11

(c) y

x-4-5 -3 -2 -1 2 3 4

3

1

2

4

5

-2

-11

6. (a) y

x-4 -3 -1-2 2 3 4

3

1

2

4

5

6

-2

-3

-4

-11

(b) y

x-4 -3 -1-2 2 3 4

3

1

2

4

5

6

-2

-3

-4

-5

-6

-11

y = x - 3

(c) y

x-4 -3 -1-2 2 3 4

3

1

2

4

5

6

-2

-3

-4

-5

-11

y = 3x – 5

-6

(d) y

x-4 -3 -1-2 2 3 4

3

1

2

4

5

6

-2

-3

-4

-5

-6

-11

y = x + 1

y = 3 – x

Answer S1-S5.indd 790 8/2/09 4:42:53 AM

791ANSWERS

(e) y

x-3 3

3

-3

y = 1

(f) y

x-1-2 2

1

2

-2

x = – 1

(g) y

x-4 -3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-2

-11

-4

-5

y = x2

y = 4

(h) y

x-4 -3 -2 -1 2 3 4

4

2

6

8

-6

-4

-21

-8

y = x3

y = 3

x = -2

(i) y

x-1 1

1

1

-1

(j) y

x-4 -3 -1-2 2 3 4

3

1

2

4

5

6

-2

-3

-4

-5

-6

-11

x - y = 2

x - y = -1

Answer S1-S5.indd 791 7/11/09 1:26:59 AM


7. (a) y

x-4 -3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-2

-11

-4

-5

y = x2

(b) y

x-4 -3 -2 -1 2 3 4

4

2

6

8

-6

-4

-21

-8

y = x3

y = 1

(c) y

x1-2 2

2

-2

x = 1

(d)

1-1 2 3 4

1

2

-2

y

x

y =2x

(e)

-1 2 3-2-3-4 1 4

1

2

-1

-2y =

1x + 2

x

y

8. (a)

x2 3 4 51-1-3-4 -2

y

y = x2

y = 5

x = 2

3

2

1

4

5

-2

-1

-3

-4

-5

Answer S1-S5.indd 792 7/11/09 1:27:01 AM

793ANSWERS

(b)

x2 3 41-1-3-4 -2

y

x=3

y=-1

y= x-2

3

2

1

4

5

6

-2

-1

-3

-4

-5

-6

(c)

x2 3 41-1-3-4 -2

y

y= 2x+ 1

2x- 3y= 6

3

2

1

4

5

6

-2

-1

-3

-4

-5

-6

(d)

-3 3

3

-3

x

x=-3

y= 2

y

(e)

x2 3 41-1-3-4 -2

y

y= 3

y= |x |

x= 2

3

2

1

4

5

6

-2

-1

-3

Test yourself 5

1. (a) f 2 6- =] g (b) f a a a3 42= - -] g (c) ,x 4 1= -

2. (a)

(b)

(c)

(d)

(e)

Answer S1-S5.indd 793Answer S1-S5.indd 793 8/11/09 11:31:52 AM8/11/09 11:31:52 AM


(f)

(g)

(h)

3. (a) Domain: all real x ; range: y 641

$ -

(b) Domain: all real x ; range: all real y (c) Domain: 1 1;x# #- range: 1 1y# #- (d) Domain: 1 1;x# #- range: 0 1y# # (e) Domain: 1 1;x# #- range: 1 0y ##- (f) Domain: all real ;x 0! range: all real y 0! (g) Domain: all real x ; range: all real y (h) Domain: all real x ; range: y 0$

4. 15 5. (a) 4 (b) 5 (c) 9 (d) 3 (e) 2

6.

7.

8.

9.

10.

11. (a) y 3# (b) y x 22 + (c) ,y x y 02$ #-

12. (a) Domain: all real ,x 3! range: all real y 0!

(b)

13. (a)

(b) (i) ,x 2 4= - (ii) 4 2x 11- (iii) ,x x2 42 1 -

Answer S1-S5.indd 794 7/11/09 1:27:04 AM

795ANSWERS

14. (a) 2 (b) 332

x = (c) 131

15. (a) x -intercept ,10- y -intercept 4 y(b) x-intercepts , ,2 7,- y-intercept y 14-

16. (a) i (b) iii (c) ii (d) i (e) iii

17. (a) 4 (b) 52

(c) 121

- (d) 3

18.

19. (a) Domain: 2,x $ range: 0y $

(b)

20. (a) ( ) 1( )

( )

f x( x x3f x(

xf x(

13 1x

4 2334 2

4 23x

= x4

=)= x4

=

] x-x g gxxx]34 34

So f x] g is even.

(b) ( )( ) ( )

( )( )

f x( x xf x( (

x x

f x(

3

3

3

3

= x=)= - +

= -

= -

] gx-x

So f x] g is odd.

2211.. (aaa)) y

x

1

(b)

x

y

-1 1

(c)

x

y

-4 4

2

(d) y

x

1

(e) y

x

-4 -3 -1

-2

-2 3 4 5

1

2

2114-

Answer S1-S5.indd 795 7/12/09 1:35:04 AM



1. ,b32

3= -

2.

3.

x2

y

-2

4.

5.

6. , ,f f f3 9 4 16 0 1= - = =] ] ]g g g

7. Domain: all real ;x 1!! range: ,y y1 02# -

8.

9. Domain: ;x 0$ range: y 0$ 10. , ,x 0 3 2= -

11.

12. h h h2 1 0 3 0 1 2+ - - = - + - - = -] ] ] ]g g g g

Answer S1-S5.indd 796 7/11/09 1:38:34 AM

797ANSWERS

13.

14.

15. ( ) ( )

( )

f a aa

f a

2 12 1

2 2

2

2

- = - -

= -

=

^ h

16. x4

1 41!=

17. (a) 2

2

x

xx

x

xx

xx

xx

x

31

32 3

31

32 6 1

32 7

32 7

31

RHS

LHS

`

= ++

=+

++

+

=+

+ +

=+

+

=

+

+= +

+

] g

(b) Domain: all real ;x 3! - range: all real y 2!

(c)

18.

19.

20. Domain: ;x 3$ range: 0y $ 21. Domain: x2 2# #-

22.

23. (a) 0 (b)

24.

Answer S1-S5.indd 797 7/11/09 1:38:38 AM


Chapter 6: Trigonometry

Exercises 6.1

1. , ,cos sin tan135

1312

512

i i i= = =

2. , ,sin cot sec54

43

35

b b b= = =

3. , ,sin tan cos74

757

74

5b b b= = =

4. , ,cos tan cosecx x x95

556

56

9= = =

5. ,cos sin53

54

i i= =

6. , ,tan sec sin25

23

35

i i i= = =

7. ,cos tan635

35

1i i= =

8. ,tan sin751

1051

i i= =

9. (a) 2 (b) 45c

(c) , ,sin cos tan452

145

2

145 1c c c= = =

10. (a) 3 (b) , ,sin cos tan3021

3023

303

1c c c= = =

(c) , ,sin cos tan6023

6021

60 3c c c= = =

11. .sin cos67 23 0 92c c= = 12. .sec cosec82 8 7 19c c= =

13. .tan cot48 42 1 11c c= = 14. (a) 2 61 2 29cos sinorc c

(b) 0 (c) 0 (d) 1 (e) 2

15. 80x c= 16. 22y c= 17. 31p c= 18. 25b c=

19. 20t c= 20. 15k c=

Exercises 6.2

1. (a) 47c (b) 82c (c) 19c (d) 77c (e) 52c

2. (a) 47 13c l (b) 81 46c l (c) 19 26c l

(d) 76 37c l (e) 52 30c l

3. (a) 77.75c (b) 65.5c (c) 24.85c

(d) 68.35c (e) 82.517c

4. (a) 59 32c l (b) 72 14c l (c) 85 53c l

(d) 46 54c l (e) 73 13c l

5. (a) 0.635 (b) 0.697 (c) 0.339 (d) 0.928 (e) 1.393

6. (a) 17 20c l (b) 34 20c l (c) 34 12c l

(d) 46 34c l (e) 79 10c l

Exercises 6.3

1. (a) 6.3x = (b) 5.6y = (c) 3.9b = (d) 5.6x = (e) 2.9m = (f) 13.5x = (g) 10.0y = (h) 3.3p = (i) 5.1x = (j) 28.3t = (k) 3.3x cm= (l) 2.9x cm= (m) 20.7x cm= (n) 20.5x mm= (o) 4.4y m= (p) 20.6k cm= (q) 17.3h m= (r) 1.2d m= (s) 17.4x cm= (t) 163.2b m=

2. 1.6 m 3. 20.3 cm 4. 13.9 m

5. (a) 18.4 cm (b) 13.8 cm 6. 10 cm and 10.5 cm

7. 47.4 mm 8. 20.3 m 9. (a) 7.4 cm (b) 6.6 cm (c) 9.0 cm

10. (a) 6.8 cm (b) 6.5 cm 11. 38 cm

Exercises 6.4

1. (a) x 39 48c= l (b) 35 06ca = l (c) 37 59ci = l (d) 50 37ca = l (e) 38 54ca = l (f) 50 42cb = l (g) x 44 50c= l (h) 3 10 5ci = l (i) 29 43ca = l (j) 45 37ci = l (k) 57 43ca = l (l) 43 22ci = l (m) 37 38ci = l (n) 64 37ci = l (o) 66 16cb = l (p) 29 56ca = l (q) 54 37ci = l (r) 35 58ca = l (s) °59 2i = l (t) 56 59cc = l

2. 37 57c l 3. 22 14c l 4. 36 52c l 5. 50c

6. (a) 11.4 cm (b) 37 52c l 7. ,31 58 45 44c ca b= =l l

8. (a) 13 m (b) 65 17c l 9. (a) 11 19c l (b) 26 cm

10. 4.96 cm and 17.3 cm 11. (a) 12.9 m (b) 56 34c l

Exercises 6.5

1. (a)

100c

Boat

Beachhouse

North

S6.indd 798 8/2/09 1:33:24 AM

799ANSWERS

(b)

320c

Campsite

Jamie

North

(c)

200c

Seagull

Jetty

North

(d)

50c

Alistair

Bus stop

North

(e)

B Hill285c

Plane

North

(f)

12c

Dam

FarmhouseNorth

(g)

160cHouse

Mohammed

North

(h)

80c

Town

Mine shaft

North

(i)

349cSchool

YvonneNorth

S6.indd 799 7/11/09 1:23:44 PM


(j)

Island

Boat ramp

280c

North

2. (a) 248c (b) 145c (c) 080c (d) 337c (e) 180c

3. 080c 4. 210c 5. 160c 6. 10.4 m

7. 21 m 8. 126.9 m 9. 72 48c l

10. (a) 1056.5 km (b) 2265.8 km (c) 245c

11. 83.1 m 12. 1.8 km 13. 12 m 14. 242c 15. 035c

16. 9.2 m 17. 171 m 18. 9.8 km 19. 51 41c l 20. 2.6 m

21. 9 21c l 22. 1931.9 km 23. 34.6 m 24. 149c

25. 198 m 26. 4.8 km 27. 9.2 m 28. 217c

29. (a) 1.2 km (b) 7.2 km 30. (a) 13.1 m (b) 50 26c l

Exercises 6.6

1. (a) 2

3 1+ (b) 1 (c) 2 (d) 4 (e)

34 3

(f) 3

2 3

(g) 141

(h) 4

6 24

2 3 1+=

+^ h (i) 3

(j) 2 3- +^ h (k) 0 (l) 1 (m) 2 2 1-^ h (n) 6

(o) 131

(p) 3 2 2- (q) 2 3 (r) 21

- (s) 632

(t) 2 3

2-

2. (a) 2

3 2x = (b)

29 3

y = (c) 2 3p =

3. 60c 4. 2 m 5. 3 m 6. 3

10 3m

7. (a) 6 2 m (b) 4 m 8. 0.9 m 9. 3

5 3 3m

+^ h

10. 100 3 m

Exercises 6.7

1. (a) 1 st , 4 th (b) 1 st , 3 rd (c) 1 st , 2 nd (d) 2 nd , 4 th (e) 3 rd , 4 th (f) 2 nd , 3 rd (g) 3 rd (h) 3 rd (i) 2 nd (j) 4 th

2. (a) 3 rd (b) 21

- 3. (a) 4 th (b) 2

1-

4. (a) 2 nd (b) 3- 5. (a) 2 nd (b) 2

1

6. (a) 1 st (b) 23

7. (a) 1 (b) 2

1 (c) 3- (d)

21

(e) 21

- (f) 21

- (g) 23

(h) 3

1- (i)

23

- (j) 2

1-

8. (a) 2

1- (b)

23

- (c) 3 (d) 23

- (e) 23

-

(f) 3- (g) 21

(h) 3

1- (i)

2

1 (j)

2

1-

9. (a) 23

- (b) 3 (c) 23

(d) 21

(e) 21

- (f) 3

(g) 2

1 (h)

2

1 (i) −1 (j)

21

10. ,sin cos53

54

i i= - = -

11. ,cos tan733

33

4i i= - = -

12. ,cos cosecx x89

8589

= = -

13. , ,cosec cot tanx x x21

5

21

2221

= - = - = -

14. ,cos sinx x74

7 7474

5 74= - = -

15. ,tan sec65

4

65

9i i= - =

16. , ,tan sec cosecx x x355

38

55

8= = - = -

17. (a) 103

sinx = (b) 1091

,91

3cos tanx x= - = -

18. , ,cot sec cosec65

561

661

a a a= - = = -

19. ,sin cot1051

51

7i i= = -

20. (a) sin i (b) cos x (c) tan b (d) sin a- (e) tan i-

(f) sin i- (g) cos a (h) tan x-

Exercises 6.8

1. (a) ,20 29 159 31c ci = l l (b) ,120 240c ci = (c) ,135 315c ci = (d) ,60 120c ci = (e) ,150 330c ci = (f) ,30 330c ci =

(g) , , ,30 120 210 300 0 2 720c c c c c c# #i i= ] g (h) 70 , 110 , 190 , 230 , 310 , 350

0 3 1080c c c c c c

c c# #

i

i

=

] g

S6.indd 800 7/12/09 1:45:15 AM

801ANSWERS

(i) , , ,30 150 210 330c c c ci = (j) , , , , , , , ,

, , ,15 45 75 105 135 165 195 225255 285 315 345c c c c c c c c

c c c c

i =

2. (a) 79 13! ci = l (b) ,30 150c ci = (c) ,45 135c ci = -

(d) ,60 120c ci = - - (e) ,150 30c ci = -

(f) ,30 150! !c ci =

(g) , , ,22 30 112 30 67 30 157 30c c c ci = - -l l l l

(h) , , , , ,15 45 75 105 135 165! ! ! ! ! !c c c c c ci =

(i) ,135 45c ci = - (j) , , ,30 60 120 150! ! ! !c c c ci =

3.

4. 1-

5.

6. , ,x 0 180 360c c c= 7. 1- 8. 1

9. ,x 0 360c c=

10.

11. 0 12. 270x c= 13. , ,x 0 180 360c c c=

14. , ,x 0 180 360c c c= 15. ,x 270 90c c= -

16.

17.

Exercises 6.9

1. (a) cos i (b) tan i- (c) cos i (d) tan i (e) sec a-

2. (a) sin i (b) sec i (c) cosec x (d) cos 2 x (e) sin a

(f) cosec 2 x (g) sec 2 x (h) tan2 i (i) cosec5 2 i

(j) sin 2 x (k) 1 (l) sin cosi i

3. (a) 1cos xLHS 2= -

sin

sinx

x1 1

RHS

2

2

= - -

= -

=

So cos sinx x12 2- = -

(b) sec tanLHS i i= +

cos cossin

cossin

1

1

RHS

i i

i

i

i

= +

=+

=

So sec tancos

sin1i i

i

i+ =

+

(c) 3 3 tanLHS 2 a= +

( )tansec

cos

sin

3 13

3

1

3

RHS

2

2

2

2

a

a

a

a

= +

=

=

=-

=

So tansin

3 31

32

2a

a+ =

-

S6.indd 801 7/11/09 1:24:10 PM


(d) sec tantan tan

cosec cot

x xx x

x x

11

LHS

RHS

2 2

2 2

2 2

= -

= + -

=

= -

=

So sec tan cosec cotx x x x2 2 2 2- = -

(e) sin cossin cos sin cossin cos sin sin cos cossin cos sin cos

sin sin cos cos sin cos

x xx x x xx x x x x xx x x xx x x x x x

21 2

2 2

LHS

RHS

2 2

2 2

= -

= - -

= - - +

= - -

= - - +

=

3

2

]] ]] ^] ]

gg gg hg g

So sin cos sin sin cos cossin cos

x x x x x xx x2

2

2

2

- = - -

+

3] g

(f) sin cossin sin

sin coscos sin

sin coscos

sin cossin

sincos

coscot sec

1 2

2

2

2

2

RHS

LHS

2

2

2

i i

i i

i i

i i

i i

i

i i

i

i

i

ii i

=- +

=+

= +

= +

= +

=

So cot secsin cossin sin

21 22

i ii i

i i+ =

- +

(g) cos cotsin cot

sinsincos

sin cos

90LHS

RHS

2

2

2#

c i i

i i

ii

i

i i

= -

=

=

=

=

] g

So 90cos cot sin cos2 c i i i i- =] g

(h) cosec cot cosec cotcosec cot

cot cot

x x x xx xx x1

1

LHS

RHS

2 2

2 2

= + -

= -

= + -

=

=

] ]g g

So cosec cot cosec cotx x x x 1+ - =] ]g g

(i)

( )

cos

sin cos

cos cos

sin cos

sec sintan costan costan cos

1

1

1 11 1

LHS

RHS

2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

i

i i

i i

i i

i i

i i

i i

i i

=-

= -

= -

= + - -

= + - +

= +

=

So cos

sin costan cos

12

2 22 2

i

i ii i

-= +

(j) cosec

cotcos

cosec

cot cos cosec

cosec

cot cossin

cosec

cot cot

cosecsin

1

1

11

1

1

LHS

#

b

bb

b

b b b

b

b bb

b

b b

b

b

=+

-

=+ -

=

+ -

=+ -

=

=

tan cot

sec

cos

sin

sin

cos

sec

sin cos

sin cos

sec

sin cos

sec

seccos sin

cos

cos sin

sin

1

1

11

RHS

2 2

#

#

b b

b

b

b

b

b

b

b b

b b

b

b b

b

bb b

b

b b

b

=+

=

+

=+

=

=

=

=

LHS RHS=

So cosec

cotcos sin

1

b

bb b

+- =

4.

( )

cos sincos sincos sin

x y2 2

4 444 14

LHS

RHS

2 2

2 2

2 2

2 2

i i

i i

i i

= +

= +

= +

= +

=

=

=

] ]

]

g g

g

So 4x y2 2+ =

5.

( )

cos sincos sincos sin

x y9 9

81 818181 181

LHS

RHS

2 2

2 2

2 2

i i

i i

i i

= +

= +

= +

= +

=

=

=

2 2] ]

]

g g

g

So 81x y2 2+ =

S6.indd 802 7/11/09 1:24:20 PM

803ANSWERS

Exercises 6.10

1. (a) 8.9x = (b) 9.4y cm= (c) 10.0a =

(d) 10.7b m= (e) 8.0d =

2. (a) 54 57ci = l (b) c61 23a = l (c) x 43 03c= l

(d) 87 04ca = l (e) 150 56ci = l

3. 126 56c l 4. (a) 13.5 mm (b) 25 mm

5. (a) 1.8 m (b) 2.7 m 6. 5.7 cm

7. (a) 10.3 m (b) 9.4 m 8. (a) 60 22c l (b) 57 9c l

9. (a) 14.1 cm (b) 15.6 cm

10. (a) 54.7 mm (b) 35.1 mm

Exercises 6.11

1. (a) 5.8m = (b) 10.4b m= (c) 7.4h cm=

(d) 16.4n = (e) 9.3y =

2. (a) 51 50ci = l (b) 60 27ci = l (c) x 57 42c= l

(d) 131 31cb = l (e) 73 49ci = l

3. 32.94 mm 4. 11.2 cm and 12.9 cm

5. (a) 11.9 cm (b) 44 11c l (c) 82 13c l

6. ,XYZ XZY YXZ66 10 47 40c c+ + += = =l l

7. (a) 18.1 mm (b) 80 49c l 8. (a) 6.2 cm (b) 12.7 cm

9. 12.9 cm 10. (a) 11 cm (b) 30c

Exercises 6.12

1. 12.5 cm and 4.7 cm 2. (a) 040c (b) 305c 3. 16.4 m

4. 103c 5. 1.97 m 6. 11c

7. (a) 1.21 km (b) 1 minute 8. 32 m 9. 107 m

10. (a) .sin

sinAC

101 365 8 42 29

c

c=

l

l (b) 74 50ci = l

11. 8.5h = 12. 7.7 km 13. 5.7 km and 5.4 km

14. 1841 km 15. 35.8 m 16. 89 52c l 17. 9.9 km

18. 163.5 km 19. 64.1 m 20. 3269 km

21. (a) 11.3 cm (b) 4044c l 22. 141c

23. (a) 11.6 cm (b) 73 14c l

24. (a) 265.5 km (b) 346 33c l

25. (a) 35 5c l (b) (i) 4.5 m (ii) 0.55 m

Exercises 6.13

1. (a) 7.5 cm2 (b) 32.3 units2 (c) 9.9 mm2 (d) 30.2 units2 (e) 6.3 cm2

2. 2

15 3m2 3. 7.5 cm2 4. 15.5 cm2 5. 34.8 cm2

6. 1.2 m2 7. 42 cm2 8. 247.7 mm2

9. (a) 7.8 cm (b) 180.8 cm2

10. (a) 5.6 cm (b) 18.5 cm2 (c) 19.1 cm2

Exercises 6.14

1. (a) 2 m (b) 2.2 m (c) 65 21c l 2. (a) 1.9 m (b) 49 46c l

3. (a) 109 cm2 (b) 16 20c l 4. 965c l

5. (a) 9 m (b) 25 7c l 6. (a) 56 m (b) 89.7 m

7. (a) 48 m (b) 128.6 m (c) 97.7 m 8. 84 m

9. 16 50c l 10. 11 10c l

Exercises 6.15

1. (a) sin cos cos sina b a b- (b) cos cos sin sinp q p q-

(c) tan tan

tan tan

1 a b

a b

-

+ (d) sin cos cos sinx x20 20c c+

(e) tan tan

tan tanx

x1 48

48c-

+ (f) cos cos sin sin2 2i a i a+

(g) cos cos sin sinx x75 75c c- (h) tan tan

tan tan

x y

x y

1 5 7

5 7

+

-

(i) sin cos cos sin4 4a b a b- (j) tan tan

tan tan

1 3

3

a b

a b

+

-

2. (a) sin a b+] g (b) tan 65c (c) cos 55c (d) 2 3sin x y+^ h

(e) tan 2i (f) sin 32c (g) sin cosa b2 (h) cos sinx y2

(i) sin sinx y2 (j) cos cosm n2

3. (a) 2 2

1 34

2 6+=

+ (b)

2 2

1 34

2 6+=

+

(c) 3 1

1 32

2 3 43 2

-

+=

+= +

(d) 1 3

1 32

2 3 43 2

-

+=

- += - +

^^

hh

(e) 2 2

1 34

2 6-=

- (f)

2 2

3 14

6 2-=

-

(g) 2 2

1 34

2 6+=

+

S6.indd 803S6.indd 803 8/11/09 11:38:03 AM8/11/09 11:38:03 AM


(h) 1 3

1 32

4 2 32 3

-

+=

- += - +

^ ^h h

(i) sin cosx x2

3 12

1 3-+

+e eo o

(j) cos cosy y2

22=

4. tan x2 5. (a) 12

6 35+ (b)

123 5 2 7+

(c) 3 5 2 7

6 3517

32 5 27 7

-

+=

+

6. (a) sin cos2 i i (b) cos sin2 2i i- (c) tan

tan

1

22 i

i

-

7. (a) sin cos sin3 2 3i i i-

(b) cos sin cos33 2i i i- (c) tan

tan tan

1 3

32

3

i

i i

-

-

8. (a) tan 4i (b) sin cos cos sin7 3 7 3i i i i-

9. cos cos sin sinx x x x2 7 2 7- 10. (a) 2

1 (b) 3

(c) 23

- (d) 23

- (e) 3

1

11. (a) 54

(b) 1312

(c) 6533

- (d) 5

12 (e) 3

1615

-

12. (a) cos cosx y2 (b) [ 115 ]cos cos21

15c c+ -] g

13. (a) sin cosx y2 (b) sin sinx y2- (c) cos sinx y2

(d) cos cos sin sin sin cos cos sinx y x y x y x y- + -

(e) tan tan

tan tan

x y

x y

1

2 12 2

2

-

+_ i (f)

tan tan

tan tan

x y

y x

1

2 12 2

2

-

+^ h

14. (a) sin cosb b2 (b) tan

tan

1

22 i

i

- (c) cos sin2 2i i-

(d) sin cos cos sinsin cos sin cos sin cosx y x yx y y x y y

2 222 2

+

= - +_ i

(e) cos cos sin sin

cos sin cos sin cos sin

2 2

22 2

a b a b

a a b a a b

-

= - -^ h

(f) tan tan

tan tan

tan tan tan

tan tan tan tanx y

x y

y x y

x x y y1 2

2

1 2

22

2

-

+

=- -

- +

(g) sin cos cos sinsin cos cos cos sin sin sin2 2

2 2 2

i d i d

i i d i d i d

-

= - +

(h) cos cos sin sin

cos cos sin sin sin cos

2 2

22 2

i c i c

i c c i c c

+

= - +_ i

(i) tan tan

tan tan

tan tan tan

tan tan tan tanx z

x z

z x z

x x z z1 2

2

1 2

22

2

+

-=

- +

- -

(j) sin cos cos sinsin cos cos sin

sin cos cos sin

x y x yx x y y

y y x x

2 2 2 22

2

2 2

2 2

-

= -

- -

_^

ih

15. (a) sin x6 (b) cos y14 (c) tan 10i (d) cos y2

(e) sin21

12i (f) sin x1 2+ (g) cos 6a (h) cos 80c

(i) tan 2b (j) sin x1 6-

16. (a) 2 2

142

= (b) 21

(c) 3

1 (d)

21

(e) 3 (f) 23

(g) 2

1 (h) 1 (i)

2 2

1 (j)

21

-

17. ,cos sinx x2327

232

5 39= - =

18. (a) 6563

(b) 257

(c) 169120

(d) 5633

-

19. 4 sin cos cos sinsin cos sin cos4 4

2 2

3 3

i i i i

i i i i

- =

-

^ h

20. (a) tan x (b) 2 3

12 3

+= -

21. 2 1-

22. (a) 2

( )

sin tan

sin cos tan

sin coscossin

sin

sin sin tan

21

21

2

21

2

RHS

LHS

2

2`

i i

i i i

i ii

i

i

i i i

=

=

=

=

=

=

(b)

2

2

2

sincos

sin cos

cos sin

sin cos

cos sin

sin cos

sin sin

sin cos

sin

cos

sin

tan

tansin

cos

1

22 2

12 2

2 2

12 2

2 2

2 2

2 2

22

2

2

2

21

RHS

LHS

2 2

2 2

2 2

2

`

i

i

i i

i i

i i

i i

i i

i i

i i

i

i

i

i

i

i

i

=-

=

- -

=

- +

=

+

=

=

=

=

=-

d n

S6.indd 804 7/11/09 1:24:22 PM

805ANSWERS

23. 11 3( ) ( )

( )( )

( ) ( )

sin sinsin sinsin cos cos sinsin cos cos sin

sin cos cos sinsin sin sin sinsin sin sin sin sin sinsin sin

7 4 7 47 4 7 47 4 7 47 4 7 47 1 4 1 7 47 7 4 4 7 47 4

RHS

LHS

2 2 2 2

2 2 2 2

2 2 2 2 2 2

2 2

i ii i i ii i i ii i i i

i i i i

i i i i

i i i i i i

i i

=

= + -

= +

-

= -

= - - -

= - - +

= -

=

sin sin sin sin7 4 11 32 2` i i i i- =

24. ( )

( )

( )

coscoscos cos sin sincos sin cos sin cos

cos sin cos sin coscos sin coscos cos coscos cos cos

cos cos

32

2 22

233 13 3

4 3

LHS

RHS

2 2 2

3 2 2

3 2

3 2

3 3

3

ii ii i i i

i i i i i

i i i i i

i i i

i i i

i i i

i i

=

= +

= -

= - -

= - -

= -

= - -

= - +

= -

=

cos cos cos3 4 33` i i i= -

25. sin sinx x3 4 3-

Exercises 6.16

1. (a) tan i (b) cos i (c) tan 20c (d) cos 50c

(e) sin 2i (f) cos i

2. (a) 23

(b) 2

1 (c)

21

(d) 0

3. (a) tt

21 2+

(b) 1

1

t

t2

2

-

+ (c)

21

tt2-

(d) 1

2 1

t

t t2

2

+

+ -

(e) 1

1 2

t

t t2

2

-

- + (f)

1

1

t

t2

2

-

+ (g)

1

3 3 8

t

t t2

2

+

- + (h)

1t

(i) 11

tt

-

+ (j)

1

4 1

t

t t2 2

2

+

-

^^

hh

4. sin cossin cos

t

t

t

t

t

t

t

t

t

t t t

t

t t t

tt t

t

t t

t

11

11

2

1

1

11

2

1

1

1

1 2 1

1

1 2 1

2 22 2

2 1

2 1

LHS

RHS

2 2

2

2 2

2

2

2 2

2

2 2

2

i i

i i=

+ +

+ -

=

++

++

-

++

-+

-

=

+

+ + + -

+

+ + - +

=+

+

=+

+

=

=

]]

gg

sin cossin cos

t11

`i i

i i

+ +

+ -=

5. t

t t t t

1

4 4 1 62 2

3 2 4

+

- - + -

^ h

6. (a) sin5 26 34ci + l] g (b) sin2 60ci +] g (c) sin2 45ci +] g (d) sin29 21 48ci + l] g (e) sin17 14 2ci + l] g (f) sin10 18 26ci + l] g (g) sin13 56 19ci + l] g (h) sin65 60 15ci + l] g (i) sin41 38 40ci + l] g (j) sin34 59 2ci + l] g

7. (a) sin2 45ci -] g (b) sin5 63 26ci - l] g (c) sin2 60ci -] g (d) sin2 30ci -] g (e) sin29 21 48ci - l] g

8. cos10 18 26ci - l] g 9. cos2 60ci +] g 10. (a) sin85 12 32ci + l] g (b) cos85 77 28ci - l] g Exercises 6.17

1. (a) ,x 45 225c c= (b) ,x 30 210c c=

(c) , , , ,x 0 60 180 300 360c c c c c=

(d) , , , ,x 0 45 180 225 360c c c c c=

(e) , ,x 90 210 330c c c= (f) 0 , 60 , 300 , 360x c c c c=

( g) 0 , 45 , 180 , 225 , 360x c c c c c= (h) 0 , 180 , 360x c c c=

(i) 30 , 135 , 150 , 315x c c c c= (j) 0 , 360x c c=

2. (a) ,126 52 306 52c ci = l l (b) ,35 58 189 16c ci = l l

(c) 60 , 240c ci = (d) 180 , 270c ci =

(e) ,240 43 327 21c ci = l l (f) 90 , 180c ci =

(g) ,90 340 32c ci = l (h) ,56 34 176 34c ci = l l

(i) ,51 2 190 54c ci = l l (j) ,160 32 270c ci = l

3. (a) 180 30n 1 n# ci = + -] g (b) 180 60n ca = +

(c) n360 30! ci = (d) x n180 1 30n# c= - -] g

(e) 180 45n ci = - (f) n 45360 ! cb =

(g) n180 60! cc = (h) 180 30n ci = +

(i) n360 75 49! ci = l (j) n180 1 23 31n# ca = + - l] g

4. , , ,x 52 30 82 30 97 30 127 30c c c c= - -l l l l

5. ,x n n180 1 30 0 9036# !c c= + - n] g

6. , , ,x 180 0 90 180c c c c= -

7. (a) n180i = (b) 360x n= (c) 180x n=

(d) ( )n 1 270180 n ci = + - (e) n360 90! c

8. (a) (i) ,x 30 150c c= (ii) x n180 1 30n! c= + -] g

(b) (i) ,x 41 25 318 35c c= l l (ii) x n360 41 25! c= l

(c) (i) ’,x 71 34 251 34c c= l (ii) x n180 71 34c= + l

(d) (i) ,x 161 34 341 34c c= l l (ii) x n180 18 26c= - l

(e) (i) 45x c= (ii) 180 ( 1)x n 4590n c c= + - -

S6.indd 805 7/25/09 1:47:50 PM


9. 180 30 , 180 ( 1) 270x n n1 n# c c= + - + -n] g

10. (a) , , ,x 0 120 240 360c c c c= (b) , nn 360360 120! c

Test yourself 6

1. ,cos sin34

5

34

3i i= =

2. (a) cos x (b) 2 (c) cosec A (d) cos i (e) cos 20i

3. (a) 0.64 (b) 1.84 (c) 0.95

4. (a) 46 3ci = l (b) 73 23ci = l (c) 35 32ci = l

5.

( )

sincos

sinsin

sinsin sin

sinsin

12

12 1

12 1 1

2 12 2

LHS

RHS

2

2

i

i

i

i

i

i i

ii

=-

=-

-

=-

+ -

= +

= +

=

^

] ]

h

g g

2 2sin

cossin

12

So2

i

ii

-= +

6. b 40c= 7. (a) 2

1 (b)

23

- (c) 3-

(d) 21

- (e) 221140

-

8. ,x 120 240c c=

9.

,x 90 270c c=

10. 122 km 11. 5 3 12. (a) 6.3 cm (b) 8.7 m

13. (a) 65 5ci = l (b) 84 16ci = l (c) 39 47ci = l

14. 65.3 cm2 15. (a) ,x 60 120!! c c=

(b) , , ,x 15 105 75 165c c c c= - -

(c) , , ,x 0 180 30 150!c c c c= -

16. ,sin cot53

34

i i= - = 17. (a) 209c (b) 029c

18. n180 1 30 53 8n c ci = + - + l] g

19. (a) sin

sinAD

9920 39

c

c= (b) 8.5 m

20. 2951 km 21. (a) 2 2

3 14

2 3 1+=

+^ h

(b) 2 2

1 34

2 1 3-=

-^ h (c)

2 2

142

=

22. (a) x n360 60! c= (b) 180 45x n c= +

(c) 180 60x n 1 n# c= + -] g

23. , ,0 120 360c c ci = 24. 51 40ca = l

25. (a) cos x y+^ h (b) cos cos cos sin sincos sin

sin sinsin

x x x x x xx x

x xx

11 2

2 2

2 2

2

+ = -

= -

= - -

= -

]

^

g

h


1. 92 58c l 2. 50.2 km 3. 12.7x cm=

4. (a) .sin

sinAC

41 2125 3 39 53

c

c=

l

l (b) 25.2h cm= 5. 4.1 km

6. cos x- 7. 16 3 cm2 8. 2

1

9. , , ,x 22 30 112 30 202 30 292 30c c c c= l l l l 10. 75 45ci = l

11. 5.4 m 12. ,110 230c ci = 13. 6.43 km

14. 956

- 15. 31 m 16. sin

cos sin cos

coscos

sin cos

cossin cos

tan

1

1

LHS

RHS

2

2

i

i i i

ii

i i

i

i i

i

=-

+

=+

=+

= +

=

]

]

g

g

17. 4 5 0x y y2 2+ + - = 18. (a) 65 m (b) 27 42c l

19. (a) 52 37c l (b) 9 m 20. 30 8c l

21. 6 4 6 4( )

( )

cos cos sin sincoscoscos sincos cos

cos

6 4105 55 1 5

2 5 1

LHS

RHS

2 2

2 2

2

i i i ii ii

i i

i i

i

= -

= +

=

= -

= - -

= -

=

6 4 6 4 2 5 1cos cos sin sin cos2` i i i i i- = -

22. 30.1 , 0.5m ms 1- 23. °, °, °30 150 270i =

24. 180 ( 1) 270n n ci = + - 25. t-

S6.indd 806 7/11/09 1:24:26 PM

807ANSWERS

Chapter 7: Linear functions

Exercises 7.1

1. (a) 5 (b) 10 (c) 13 2. (a) 13 (b) 65

(c) 85 (d) 52 2 13=

3. (a) 9.85 (b) 6.71 (c) 16.55 4. 12 units

5. , 134 128Two sides side= =

6. Show 85AB BC= =

7. Show points are 17 units from ,7 3-^ h 8. 3 , 9x yRadius units equation 2 2= + =

9. Distance of all points from ,0 0^ h is 11, equation

11x y2 2+ = 10. 3a = 11. a 6 2!= -

12. All 3 sides are 2 units. 13. ,a 10 2= -

14. , ,MQ NP QP MN37 20= = = = so parallelogram

15. 98BD AC= = 16. (a) ,AB AC BC40 4= = =

(b) OC OB 2= = 17. 2 101 18. 61 units

19. 29, 116, 145AB BC AC= = =

AB BC

AC

29 116145

2 2

2

+ = +

=

=

So triangle ABC is right angled (Pythagoras’ theorem)

20. , ,XY YZ XZ65 130 65= = =

Since XY YZ= , triangle XYZ is isosceles.

XY XZ

YZ

65 65130

2 2

2

+ = +=

=

So triangle XYZ is right angled. (Pythagoras’ theorem)

Problem

30.2

Exercises 7.2

1. (a) ,2 4^ h (b) ,1 1-^ h (c) ,2 1-^ h (d) ,3 2-^ h (e) ,1 1-^ h (f) ,3 2-^ h (g) ,3

21d n (h) ,1

21

1d n (i) ,

21

221d n (j) ,0 5

21d n

2. (a) ,a b9 3= = - (b) ,a b5 6= - =

(c) ,a b1 2= - = - (d) ,a b1 2= - = -

(e) ,a b6 1= =

3. ,2

3 30

24 4

0+ -

=- +

=] g

4. ,P Q 2 1= = -^ h 5. ,4 3^ h 6. 3x = is the vertical line through

midpoint ,3 2^ h .

7. Midpoint of , .AC BD 221

321

midpoint of= = d n

Diagonals bisect each other

8. 125,AC BD= = midpoint AC midpoint=

,BD 421

= - ;d n rectangle 9. ,8 13-^ h

10. (a) , , ,X Y Z21

321

21

21

1 1= - = =, ,d d ^n n h

(b) , ; ,XY BC XZ10 40 2 10234

= = = =

; ,AC YZ AB3422

2= = =

11. 4x y2 2+ = 12. 1x y2 2+ =

Exercises 7.3

1. (a) 2 (b) 131

(c) 131

- (d) 252

- (e) 32

(f) 81

-

(g) 421

- (h) 32

- (i) 241

(j) 2- 2. 21y1 =

3. 1.8x = 4. 9x = 5. (a) Show 53

m m1 2= =

(b) Lines are parallel .

y

-3 -2 -1 3 4 5 6 7

1

3

4

-2

-1 2(2, -1)

(-2, 1)

(7, 2)

(3, 4)

1

2

6. Gradient of 1AB CD21

gradient of= =

Gradient of 0BC ADgradient of= =

7. Gradient of 1AB CD31

gradient of= = -

Gradient of BC AD43

gradient of= =

Gradient of ,AC 521

= -

gradient of 21

BD = -

8. Gradient of 1,AC = gradient of BD 1= -

9. (a) Show AB BC AC2 2 2+ =

(b) Gradient of 45

,AB =

gradient of 54

BC = -

Answer S7-S8.indd 807 7/12/09 3:00:43 AM


10. (a) , , ,F G1 2 421

= - =^ dh n (b) Gradient of FG BC

65

gradient of= =

11. 4 3 11 0x y- - = 12. Gradient of ,2 4-^ h and , ,3 1 3 1gradient of- = -^ ^h h and ,5 5 3=^ h

13. 1 14. 0.93 15. 21 16. 50 12c l 17. 108 26c l

18. (a) 3 (b) 3

1 (c) 3-

19.

tantan

m

m

7 45 2

33

1

1180 45 2135

nd quadrant` c c

c

ii

i

=-

- - -

=-

= -

=

- =

= -

=

]

^

g

h

20. 3

2 3 3x =

+^ h

Exercises 7.4

1. (a) (i) 3 (ii) 5 (b) (i) 2 (ii) 1 (c) (i) 6 (ii) 7-

(d) (i) 1- (ii) 0 (e) (i) 4- (ii) 3 (f) (i) 1 (ii) 2-

(g) (i) 2- (ii) 6 (h) (i) 1- (ii) 1 (i) (i) 9 (ii) 0

(j) (i) 5 (ii) 2- 2. (a) (i) 2- (ii) 3 (b) (i) 5- (ii) 6-

(c) (i) 6 (ii) 1- (d) (i) 1 (ii) 4 (e) (i) 2- (ii) 21

(f) (i) 3 (ii) 121

(g) (i) 31

- (ii) 2- (h) (i) 54

- (ii) 2

(i) (i) 321

(ii) 21

- (j) (i) 132

(ii) 32

3. (a) 4 (b) 2-

(c) 0 (d) 2- (e) 1- (f) 3- (g) 2 (h) 41

- (i) 121

(j) 141

(k) 32

(l) 21

(m) 51

(n) 72

(o) 53

-

(p) 141

- (q) 15 (r) 121

- (s) 61

(t) 83

-

Exercises 7.5

1. (a) 4 1y x= - (b) y x3 4= - + (c) 5y x=

(d) 4 20y x= + (e) 3 3 0x y+ - = (f) x y4 3 12 0- - =

(g) 1y x= - (h) 5y x= + 2. 8 0x y+ - =

3. (a) 4 3 7 0x y- + = (b) 3 4 4 0x y- + =

(c) 4 5 13 0x y- + = (d) 3 4 25 0x y+ - =

(e) 2 2 0x y- + = 4. 4 8 0x y+ - = 5. (a) 3y =

(b) x 1= - 6. y x2= - 7. 3 4 12 0x y- - =

8. 2 3 0x y+ - = 9. 4x = - 10. 3 8 15 0x y+ - =

Exercises 7.6

1. (a) 3- (b) 31

(c) 43

(d) 121

(e) 1 (f) 65

- (g) 3

1

(h) 31

(i) 3

1 (j)

51

2. (a) 1 0x y- + = (b) 3 16 0x y- + = (c) 5 0x y+ - =

(d) 2 5 0x y+ + = (e) 2 4 0x y- + =

(f) 3 1 0x y+ - = (g) 3 4 13 0x y+ + =

3. 3m m1 2= = so parallel

4. m m51

5 11 2 #= - = - so perpendicular

5. 151

m m1 2= =

6. m m37

73

11 2# #= - = - 7. 32

k = - 8. 4m m1 2= =

9. AB CD m m 31 2< = =_ i and BC AD m m85

1 2< = = -d n 10. Gradient of : ,AC m

21

1 = gradient of BD : 2,m2 = -

m m21

2 11 2# #= - = -

11. (a) y x= - (b) 5 8 0x y- - = (c) 2 2 0x y+ + =

(d) 2 3 16 0x y- + = 12. 7 6 24 0x y+ - =

13. 3 0x y+ - = 14. 2 5 0x y- - =

15. 2 3 18 0x y- + =

Exercises 7.7

1. (a) ,2 4-^ h (b) ,1 3- -^ h (c) ,4 4^ h (d) ,0 2-^ h (e) ,5 1-^ h (f) ,1 1-^ h (g) ,3 7^ h (h) ,4 0^ h (i) ,41 26^ h (j) ,

191

197

-d n 2. Substitute ,3 4-^ h into both lines

3. , , ,2 5 4 1^ ^h h and ,1 1- -^ h 4. All lines intersect

at ,2 3-^ h 5. All lines meet at ,5 0-^ h 6. 11 6 0x y+ =

7. 5 6 27 0x y+ - = 8. x y4 7 23 0+ =+

9. 1 0x y+ - = 10. 2 2 0x y+ - =

11. 3 0x y+ - = 12. 2 3 0x y- - =

13. x y 1 0- + = 14. 3 2 0x y- + =

15. 3 7 0x y+ - = 16. 5 13 0x y+ + =

17. 27 5 76 0x y- - = 18. 3 14 0x y- - =

19. 2 1 0x y- - = 20. 3 11 0x y- - =

21. 5 17 0x y- + =

Answer S7-S8.indd 808 8/2/09 1:50:05 AM

809ANSWERS

Exercises 7.8

1. (a) 2.6 (b) 1133

(c) 2.5 (d) 2.4 (e) 138

2. (a) 3.48 (b) 1.30 (c) 0.384 (d) 5.09 (e) 1.66

3. (a) 13

7 13 (b) 5 (c)

2054 205

(d) 13

5 26 (e)

1314 13

4. d d d 11 2 3= = =

5. : , :A d B d5

14

5

3= =

-

Opposite signs so points lie on opposite sides of the line

6. , : , , :d d2 310

139 2

10

5- = =^ ^h h

Same signs so points lie on the same side of the line

7. , : , , :d d3 2 4 4 1 251

- = - =^ ^h h


8. 2d d1 2= = so the point is equidistant from both lines

9. , : , , :d d8 337

551 1

37

9- = =^ ^h h

Same signs so points lie on same side of the line

10. , : , , :d d3 25

64 1

5

7- =

-=^ ^h h


11. 4d d1 2= = so same distance 12. 5

8 5 units

13. 1 14. 4.2 15. 9 17x32

or= - 16. 3 1b41

121

or= -

17. m 1 1832

31

or= - -

18. Show distance between ,0 0^ h and the line is 5

19. Show distance between ,0 0^ h and the line is greater than 1

20. (a) , , , , ,3 1 374

71

2 2- -^ d ^h n h (b) , ,5

2 105

13 5119

26 34

Exercises 7.9

1. (a) 18 26c l (b) 29 45c l (c) 82 52c l (d) 26 34c l

(e) 10 29c l (f) 41 49c l (g) 72 15c l (h) 18 26c l

(i) 74 56c l (j) 36 52c l

2. (a) 149 2c l (b) 119 45c l (c) 143 58c l (d) 172 14c l

(e) 135c 3. 12 20c l 4. 53 58c l

5. , ,21 2 120 58 38c c cl l 6. ,m 331

= -

7. . , .m 5 4 1 53= - 8. . , .k 1 64 0 095Z -

9. (a) ,A C B D63 26 116 34c c+ + + += = = =l l

(b) 124 31c l 10. ,A B C61 56 59 2c c+ + += = =l l

Exercises 7.10

1. (a) ,53

152

-d n (b) ,251

353d n (c) ,2

94

198

-d n

(d) ,471

172

-d n (e) ,2109

221

-d n (f) ,5 241

-d n

(g) ,276

776d n (h) ,3

114

1111

- -d n (i) ,76

174

-d n

(j) ,132

132

-d n

2. (a) ,4 321

-d n (b) ,654

2d n (c) ,19 25^ h (d) ,12 521d n

(e) ,40 12^ h (f) ,9 174

-d n (g) ,621

- -d n (h) ,9 132d n

(i) ,58 30-^ h (j) ,10 13^ h

3. (a) ,E32

2= d n (b) ,F 132

2= d n (c) ,EF AC AC EF1 3 3`= = =

4. A B(3, 2) (-1, 6)(1 , 3 )2

313 ( , 4 )1

323

5. , , ,P Q PQ153

51

16 19 24 units= = - =,d ^n h

6. ,B 954

1252

= -d n 7. ,p q453

20= =

8. (a) ,32

132d n (b) Each ratio gives , .

32

132d n This means

that the intersection of the medians divides each median in the ratio : .2 1

9. ,a b8 18= = 10. 92

, 398

P = d n

Test yourself 7

1. 6.4 units 2. ,221

2-d n

3. (a) 151

- (b) 2 (c) 3

1 (d)

53

4. (a) 7 11 0x y- - = (b) 5 6 0x y+ - = (c) 3 2 0x y+ =

(d) 3 5 14 0x y+ - = (e) 3 3 0x y- - =

5. 5

6 5units

6. ,m m41

41 2= - = so m m 11 2 = -

` lines are perpendicular.

7. x -intercept 5, y -intercept 2-

8. (a) 2 1 0x y+ - = (b) 21

(c) 25

units

9. 5,m m1 2= = so lines are parallel 10. 3 4 0x y- =

Answer S7-S8.indd 809 8/2/09 1:50:06 AM


11. ,1 1-^ h 12. ,a b6 1= = 13. 66 48c l

14. Solving simultaneously, 4 0x y- - = and

2 1 0x y+ + = have point of intersection , .1 3-^ h

Substitute ,1 3-^ h in 5 3 14 0:x y- - =

5 1 3 3 14 0LHS RHS# #= - - - = =

point lies on 5 3 14 0:x y- - =

Substitute ,1 3-^ h in 3 2 9 0:x y- - =

3 1 2 3 9 0LHS RHS# #= - - - = =

point lies on 3 2 9 0:x y- - =

lines are concurrent

15. ,295

131

-d n 16. 0.499- 17. ,c 13 65= - -

18. 3y = 19. ,4 7^ h 20. 154

x = 21. 93 22c l

22. , : , , :d d2 113

86 3

13

2- =

-=^ ^h h


23. 63 26c l 24. 4 0x y- - = 25. 3 7 14 0x y- - =


1. 2k = - 2. 3 3 3 0x y- - = 3. 10 10 81x y2 2+ =

4. Show AC and BD have the same midpoint ,1 2^ h and m m 1AC BD# = -

5. Show distance of all points from ,0 0^ h is 3; radius 3; equation 9x y2 2+ =

6. 13

4 13 7. 45 ; ( )OBA a b sides of isoscelesc+ D= =

8. 13

12 13 9. 113 12c l 10. 2 3 13 0x y+ + =

11. .angled

, , ;,

BC AC ABm m

18 61

so is isoscelesso is rightBC AC#

D

D

= = =

= -

12. ,3 5-^ h

13. ,a b2 3= = 14. 2 5 14 0x y+ + = 15. 45c

16. 3 3 2 3 0x y+ + - = 17. 6 0x y- + =

18. ,b 231

21= - 19. , , ,231

231

132

332

- -d dn n 20.

m m

m m

m m

m m

m m m m

m m m m

m m

m m

m m m m

m m m m

11

11

11

11

11

or

1 2

1 2

1 2

1 2

1 2 1 2

1 2 1 2

1 2

1 2

1 2 1 2

1 2 2 1

`

+

-=

+

-=

+ = -

= - -

+

-= -

- = - -

= - -

21. ,Pp

p

p

p

1

4 1

1

7 3=

-

- -

-

-f p

22. (a) AB : 7 5 14 0x y+ + =

,7 7-^ h lies on the line (show by substitution)

(b) :1 2- or :1 2-

23. ,x y1632

17= = - 24. . , .m 0 059 9 2= - -

25. (a) ,P 132

331

= d n (b) ,Q 431

331

= d n (c) PQ has gradient 0m1 =

AC has gradient 0m2 =

Since ,m m PQ AC1 2 <=

(d) ,R 631

0= d n

(e) PR has gradient 75

m1 = -

BC has gradient 75

m2 = -

Since ,m m PR BC1 2 <=

Chapter 8: Introduction to calculus

Exercises 8.1

1.

2.

3.

Answer S7-S8.indd 810 7/25/09 2:05:37 PM

811ANSWERS

4.

5.

6.

7.

8.

9.

10.

Exercises 8.2

1. Yes, 0x = 2. Yes, x x1= 3. No 4. Yes, 0x =

5. Yes, ,x x x x1 2= = 6. Yes, 0x = 7. Yes, x 3= -

8. Yes, 2x = 9. Yes, ,x 2 3= - 10. Yes, x1 01#-

11. Yes, ,x 90 270c c= 12. Yes, 0x = 13. No 14. No

15. Yes, x 3!=

Exercises 8.3

1. (a) 3 (b) 7- (c) 3 (d) 8 (e) 2 (f) 3- (g) 2 (h) 1- (i) 10 (j) 1-

2. (a) 2 4x x2 - - (b) 2 1x x3 + - (c) 7 1x- - (d) 4x x4 2- (e) 4 3x- + (f) 2 6x2 + (g) 2x- (h) 4x2 (i) 3 1x - (j) 2 9x x2 - +

Exercises 8.4

1. (a) 4.06 (b) 3.994 (c) 4

2. (a) 13.61 (b) 13.0601 (c) 12.9401 (d) 13 3. 6

4. (a) 2f x h x xh h2 2+ = + +] g

(b) ( ) ( )f x h f x x xh h xxh h

22

2 2 2

2

+ - = + + -

= +

(c) h

f x h f x

hxh h

hh x h

x h

2

2

2

2+ -=

+

=+

= +

] ]

]

g g

g

(d) ( )

( )

lim

lim

f xh

f x h f x

x h

x

2

2

h

h

0

0

=+ -

= +

=

"

"

l] ]g g

5. (a) ( ) ( )( )

f x h x h x hx xh h x hx xh h x h

2 7 32 2 7 7 32 4 2 7 7 3

2

2 2

2 2

+ = + - + +

= + + - - +

= + + - - +

] g

(b) ( ) ( ) ( )( )

f x h f x x xh h x hx x

x xh h x hx x

xh h h

2 4 2 7 7 32 7 3

2 4 2 7 7 32 7 3

4 2 7

2 2

2

2 2

2

2

+ - = + + - - +

- - +

= + + - - +

- + -= + -

Answer S7-S8.indd 811 7/11/09 1:21:39 PM


(c)

h

f x h f x

hxh h h

hh x h

x h

4 2 7

4 2 7

4 2 7

2+ -=

+ -

=+ -

= + -

] ]

]

g g

g

(d) f x x4 7= -l] g

6. (a) f 2 11=] g (b) 2 5 11f h h h2+ = + +] g

(c) f h f h h2 2 52+ - = +] ]g g

(d)

h

f h f

hh h

hh h

h

2 2 5

5

5

2+ -=

+

=+

= +

] ]

]

g g

g

(e) f 2 5=l] g

7. (a) f 1 7- = -] g

(b) f h f h h h1 1 4 12 123 2- + - - = - +] ]g g (c) 12

8. (a) f 3 8=] g (b) f h f h h3 3 6 2+ - = +] ]g g (c) f 3 6=l] g

9. (a) f 1 13= -l] g (b) 17

10. (a) 2y x x2= +

Substitute ,x x y yd d+ +_ i :

( )

2

y y x x x xx x x x x x

y x xy x x x x

22 2 2

2 2Since

2

2 2

2

2

d d d

d d d

d d d d

+ = + + +

= + + + += +

= + +

] g

(b) x

y

xx x x x

x

x x x

x x

2 2

2 2

2 2

2

d

d

d

d d d

d

d d

d

=+ +

=+ +

= + +

] g

(c) 2 2dx

dyx= +

11. (a) 2 (b) 5 (c) 12- (d) 15 (e) 9-

12. (a) f x x2=l] g (b) 2 5dx

dyx= +

(c) f x x8 4= -l] g (d) 10 1dx

dyx= -

(e) 3dx

dyx2= (f) f x x6 52= +l] g

(g) 3 4 3dx

dyx x2= - + (h) x xf 6 2= -l] g

13. (a) 0.252 (b) 0.25 (c) 0.2498

14. (a) 0.04008- (b) 0.03992- (c) 0.04- 15. 1-

Exercises 8.5

1. (a) 1 (b) 5 (c) 2 3x + (d) 10 1x - (e) 3 4 7x x2 + - (f) 6 14 7x x2 - + (g) 12 4 5x x3 - + (h) 6 25 8x x x5 4 3- - (i) 10 12 2 2x x x4 2- + - (j) 40 63x x9 8-

2. (a) 4 1x + (b) 8 12x - (c) 2 x (d) 16 24x x3 - (e) 6 6 3x x2 + -

3. (a) x3

1- (b) x x2 3 2- (c) 3

86

xx

75- (d) 4 x (e)

41

(f) 2 2 2x x2 - +

4. f x x16 7= -l] g 5. 56-

6. 60 40 35 3dx

dyx x x9 7 4= - + - 7. 10 20

dtds

t= -

8. g x x20 5= - -l] g 9. 30dtdv

t= 10. 40 4dtdh

t= -

11. drd

rV

4 2r= 12. 3 13. (a) 5 (b) 5- (c) 4x =

14. (a) 12 (b) x 2!= 15. 18

Exercises 8.6

1. (a) 72 (b) 13- (c) 11 (d) 18- (e) 18 (f) 27

(g) 11 (h) 136 (i) 4- (j) 149

2. (a) 261

- (b) 251

(c) 201

(d) 431

- (e) 101

(f) 71

(g) 711

- (h) 201

(i) 81

- (j) 51

-

3. (a) (i) 6 (ii) 61

- (b) (i) 8 (ii) 81

-

(c) (i) 24 (ii) 241

- (d) (i) 8- (ii) 81

(e) (i) 11 (ii) 111

-

4. (a) 27 47 0x y- - = (b) 7 1 0x y- - = (c) 4 17 0x y+ + = (d) 36 47 0x y- - = (e) 44 82 0t v- - =

5. (a) x y24 555 0+ - = (b) 8 58 0x y- + = (c) 17 516 0x y- - = (d) 45 3153 0x y- + = (e) 2 9 0x y+ - =

6. (a) (i) 7 4 0x y- + = (ii) 7 78 0x y+ - = (b) (i) 10 36 0x y- + = (ii) 10 57 0x y+ - = (c) (i) 10 6 0x y+ - = (ii) 10 41 0x y- - = (d) (i) 2 2 0x y+ + = (ii) 2 19 0x y- - = (e) (i) 2 2 0x y- + = (ii) 2 9 0x y+ - =

7. x 3!= 8. (1, 2) and ( 1- , 0) 9. ( 5- , 7- )

10. (0, 1) 11. (1, 2) 12. ,143

41615

- -d n 13. (a) (1, 1- ) (b) 6 7 0x y- - =

14. 10 7 0t h- - = 15. x y4 2 19 0- - =

Answer S7-S8.indd 812 8/2/09 1:50:06 AM

813ANSWERS

Exercises 8.7

1. (a) 3x 4- - (b) 1.4x0.4 (c) 1.2x 0.8- (d) 2x21 -

1

(e) 2x x3 2+-

-

1

(f) 3x-

2

(g) 4x6-

1

(h) 2x-

3

2. (a) x

12

- (b) 2

5

x (c)

6

1

x56 (d)

10

x6- (e)

15

x4

(f) 2

1

x3- (g)

3

x7- (h)

23 x

(i) 3

2

x2-

(j) 2

1 12

x x3 5- -

3. 271

4. −3 5. 321

6. −3 7. 2 3 1x x+ +

8. 81

9. 3 16 8 0x y+ - = 10. 9 0x y- + =

11. (a) 2

1

x3- (b)

161

- 12. x y16 016+ - = 13. (9, 3)

14. 4x = 15. , , ,552

552

- -d dn n

Exercises 8.8

1. (a) 4 3x 3+] g (b) 6 2 1x 2-] g (c) 70 5 4x x2 6-^ h

(d) 48 8 3x 5+] g (e) 5 1 x 4- -] g (f) 135 5 9x 8+] g (g) x4 4-] g (h) 4 6 3 2 3x x x2 3 3

+ +^ ^h h (i) 8 2 5 5 1x x x2 7

+ + -] ^g h (j) 6 6 4 2 3x x x x5 6 2 5

- - +^ ^h h (k) 2x23

3 1--

1] g

(l) 2 4 x 3- -] g (m) 6 9x x2 4- -

-^ h (n) -

3x35

5 4+2] g

(o) -

4x x x x x43

3 14 1 72 3 2- + - +

1^ ^h h (p) 2 3 4

3

x +

(q) 5 2

5

x 2-

-] g (r) 1

8

x

x2 5

-+^ h (s)

7 3

2

x3-

-

(t) 2 4

5

x 3-

+] g (u) 4 3 1

3

x 3-

-] g (v) 2 2 7

27

x 10-

+] g

(w) 3 3

4 9 3

x x x

x x4 3 2

3 2

-- +

- +

^^

hh (x)

316 4 1x3 +

(y) 4 7

5

x 94 -] g

2. 9 3. 40 4. (4, 1) 5. ,x 2 121

= - 6. 8 7 0x y+ + =

Exercises 8.9

1. (a) 8 9x x3 2+ (b) 12 1x - (c) 30 21x +

(d) 72 16x x5 3- (e) 30 4x x4 -

(f) 5 2 1x x x 2+ +] ]g g (g) 8 9 1 3 2x x 4- -] ]g g (h) x x x3 16 7 43 2- -] ]g g (i) 10 13 2 5x x 3+ +] ]g g (j) x x x x x x x

x x x x x

10 5 3 1 3 10 1

13 60 3 20 1

3 2 2 4 2 2 5

3 2 2 4

+ - + + + +

= + + - +

^ ^ ^ ^^ ^

h h h hh h

(k) x

xx

x

x

2 22

2 2

4 3-

-+ - =

-

-

(l) x

xx x2 1

2 5 32 1

5

2 1

112 2-

- ++

-= -

-]]

]gg

g

2. 26 3. 1264 4. 77

1

7

8+ = 5. 176

6. 10 9 0x y- - = 7. 69 129 0x y- - =

8. x3

6 30!=

- 9. 34 29 0x y- + =

Exercises 8.10

1. (a) x2 1

22-

-

] g (b) 5

15

x 2+] g (c) x

x x

x

x x

4

12

4

122 2

4 2

2 2

2 2

-

-=

-

-

^ ^^

h hh

(d) 5 1

16

x 2+] g (e) 14 14

x

x x

x

x4

2

3

- +=

- + (f)

3

11

x 2+] g

(g) 2

2

x x

x2 2

2

-

-

^ h (h) 2

6

x 2-

-

] g (i) x4 3

342-

-

] g (j) x3 1

142+

-

] g

(k) 3 7

3 6 7

x

x x2 2

2

-

- - -

^ h (l) x

x x

x

xx

2 3

4 12

2 3

342

2

2-

-=

-

-

] ]]

g gg

(m) x

x

5

182 2-

-

^ h (n) x

x x

x

x x

4

2 12

4

2 62

3 2

2

2

+

+=

+

+

] ]]

g gg

(o) x

x x

3

2 9 72

3 2

+

+ +

] g (p) 3 4

3 8 5

x

x x2

2

+

+ -

] g

(q) x x

x x x

1

2 4 12 2

4 3 2

- -

- - -

^ h (r)

-2 2

xx x x

52 5 5

+

+ - +

1 1

] ]g g

(s)

(t) 28

x

x x x

x

x

7 2

7 1 7

7 2

21 302 28 5

4 3

+

-=

+

- ++ - +

]] ] ]

]gg g g

g

(u) x

x x x x

x

x x2 5

15 2 5 3 4 6 3 4 2 5

2 5

3 3 4 4 33

6

3 4 5 2

4

4

-

- + - + -

=-

+ -

]] ] ] ]

]] ]

gg g g g

gg g

(v) x

x x

x

x

x1

1 2 1

3 1

2 1

3 53+

+ +

+

=+

+-3

] g

(w) x

x

x

x

x x

x

2 3

2 1

2 3

1

2 1 2 3

2 12 2-

-

-

-=

- -

- +2-

] ]g g

(x) x

x

x x

x x

x x

x x

9

1

9

9 1

1 9

9 24

2

2

2

2 3

2

-

+

-

- +=

+ -

- - -2-

]

]]

]g

gg

g

x

x x x

x

x x

5 1

6 5 1 2 9 5 2 9

5 1

2 9 20 512

2 3

2

2

+

+ - - -=

+

- +

]] ] ]

]] ]

gg g g

gg g

Answer S7-S8.indd 813 8/2/09 1:50:07 AM


2. 81

3. 195

- 4. 0, 1x = 5. 9, 3x = -

6. 18 8 0x y- + = 7. 17 25 19 0x y- - =

Exercises 8.11

1. (a)

(b) Substitute Q into both equations .

(c) 4y x2= - has 4m1 =

8 12y x x2= - + has 4m2 = -

(d) 28 4c l

2. (a)

(b) ,P 3 9= ^ h (c) 6m = (d) 0c 3. 8 8c l

4. 71 34c l 5. 162 54c l 6. (a) , , ,X Y4 16 1 6= = -^ ^h h (b) : ,

: ,X m mY m m

12 78 3

AtAt

1 2

1 2

= =

= - = -

(c) : :X Y3 22 11 19At Atc cl l

7. ,71 34 8 58c cl l 8. (a) (0, 0), (2, 8), ( 1- , 1- )

(b) 63 26c l at (0, 0), 4 42c l at (2, 8), 71 34c l at ( 1- , 1- )

9. At (0, 0), 0 4m mand1 2= = At (2, 4), 4 0m mand1 2= = Angle at both is 75 58c l

10. 164 45c l at (0, 0), 178 37c l at ( 3- , 33- ), 146 19c l at (1, 3)

Test yourself 8

1. (a)

(b)

2. 10 3dx

dyx= - 3. (a) 42 9 2 8

dx

dyx x x5 2= - + -

(b) 2 1

11dx

dy

x 2=

+] g (c) 8( ) ( )dx

dyx x x9 2 4 4 22= + + -

(d) 40 5 5 (10 1)dx

dyx x x x x2 1 2 1 2 13 4 3= - + - = - -] ] ]g g g

(e) 2

5dx

dy x3

= (f) 10

dx

dy

x3= -

4. dtdv

t4 3= - 5. (a) 1 (b) 20 6. 10 7. 42

8. (a) 2x = - (b) 1x = (c) 2x =

9. (a) 32 4 9f x x 3= +l] ]g g (b) 3

5dx

dy

x 2= -

-] g

(c) dx

dyx x9 1 3 1= - -] ]g g (d)

4dx

dy

x2= -

(e) f xx5

145

=l] g

10. y

11. 9 7 0x y- - = 12. (2, 3) 13. drdS

r8r=

14. ( 2- , 71), (5, 272- ) 15. 4 6 0x y- - = 16. 3525

17. 9 18. x y12 4 0+ - = 19. ,51

dtds

u at t= + =

20. 107

21. 17 6c l at (3, 9), 853c l at ( 1- , 1)

22. 175 26c l at (2, 4), 177 40c l at (4, 16)

Answer S7-S8.indd 814 7/11/09 1:21:45 PM

815ANSWERS


1. ,f f1 3 1 36= - = -l] ]g g 2. 1813

-

3. ; , .dtdx

t t t8 300 0 37 53 2= + = -

4. , ,x y x y x y2 0 3 3 0 6 12 0+ = - - = - + =

5. , , , , 12 26 0, 12 170 0x y x y2 2 2 14- - + - = + + =^ ^h h 6.

43

7. 5 5 1 9 15 9 5 110 5 1 9 (4 13)x x x x

x x x

3 4 5 2

2 4+ - + - +

= + - -

] ] ] ]] ]g g g gg g

8. x

x x x

x

x4 9

2 4 9 16 2 1 4 9

4 9

2 12 17

8

4 3

5

-

- - + -

=-

- +

]] ] ]

]]

gg g g

gg

9. x12

6 2046

3 51! !=

-=

- 10. 2 25 0x y+ - =

11. 271

a = - 12. ,P 241

6161

= -d n 13. ,x31

31 13!

=

14. 21

15. , , ,x y Q PQ3 5 0 0 5 10- + = = =^ h

16. (a) Substitute (1, 1) into both curves:

3 2 :y x 5= -] g

13 1 2

11

LHSRHS

LHS

5

5

#

=

= -

=

=

=

] g

So (1, 1) lies on the curve 3 2y x 5= -] g

15 3

yxx

=+

- :

1

1 15 1 3

22

1

LHS

RHS

LHS

#

=

=+

-

=

=

=

So (1, 1) lies on the curve 1

5 3y

xx

=+

-

(1, 1) is a point of intersection

(b) 22 45c l

17. 8n = 18. , , x y11211

23 3

12 3 012 31- + =e o

19. , ,x21

121

153

= - 20. (a) ,x 90 270c c=

(b) y

x1

90c 180c 270c 360c

21. ,4 73- -^ h 22. 3 9 14 0x y- - = 23. x x

x

4 3 2

4 534 -

-] g

24. (a) ,x y x y16 32 1 0 4 2 1 0+ + = - - =

(b) 2m m21

1

1 2$ #= -

= -

So perpendicular

25. 0, 2, 6x = 26. ,a b14 7= - = 27. 22

5 22

28. 121

p = 29. drdV

38 3r

= 30. 4k = 31. 4 0x y- - =

32. 4 13 0x y- - = 33. 481

- 34. , ,a b c1 2 4= - = =

35. 8 8 2S r rhr r r= - +

36. (a) 6 5 3 1 3 5x x x2 3- - -] ]g g (b) x x

x

3 2 1

5 64- +

+-

]]g

g

37. x6

4 13!=

38. (a) 7 80 0x y+ - =

(b) ,Q 471

12491

= -d n


1. 0.77- 2. 1 3. 5 2 1 0x y+ - = 4. ,2 2-^ h 5. 0.309- 6. (a) 3 cm2 (b) , 1AC BD13 cm cm= =

7. 1; ,m m A43

68

1 121

1 2 #= - = - = -d n 8. x 15c=

9. 127

Answer S7-S8.indd 815 8/2/09 3:28:23 AM


10.

11.

12. ’45 49c 13. Domain: all real ;x21

! range: all

real y 0!

14.

15.

16. sin4 i 17. 2 units 18. 8 15 0x y- + =

19. ,120 240c ci = 20. 132

- 21. 2 22. 11 565ca = l

23. .y 16 5= 24. 3 5 0x y+ - = 25. x132

31 1

26. 7 27. 3x = 28. 3-

29. Show perpendicular distance from ,0 0^ h to the line is 2 units, or solving simultaneous equations gives only one solution.

30. (a) ,g g2 1 3 6= - = -] ]g g

(b)

31. 3 4x x2 - 32. 2

1- 33. 17.5 m

34. ,x y2 17= - = - 35. (a) 7.0AB m= (b) 27.8 m 2

36. cos3 i 37. (a) 2 4 0x y- + = (b) , ,,P Q2 0 0 4-^ ^h h (c) 4 units 2

38. 127 m 39. 15 units 2 40. ( )

( )

f x x xx xf x

33

6 2

6 2

- = - - - -

= - -=

] ]g g

41. 16x x x x x1 1 18 1 2 12 22 2 2 2 2 33 4+ + + ++ =^ ^ ^ ^h h h h

42. y431

9# #- 43. 3

x2-

44. (a) 3 4 0x y- - = (b) 2 0x y- - =

(c) 3 10 0x y+ + = (d) ,R 10 0= -^ h 45.

138

units 46. Domain: all ;x 4!- range: all y 0!

47. 2 7

1

x - 48. 4.9 km 49. 8 7 10x x 3- - -

50. 1

5

x 2+] g 51. 2 3x - 52. x x

x

x x

x

5

17 2

5

17 22 2+

- -=

+

+- ] g

53. 6 56 0x y+ - = 54. ,f f2 45 2 48- = - - =l] ]g g

55. ,a b2 9= = - 56. 7 5 9 0x y- + =

57. 47 109 0x y- + = 58. 0.25x = - 59. ,33 17-^ h

60. 2 2

3 14

6 2+=

+ 61. 67 37c l

62. ,x 63 26 243 26c c= l l

Answer S7-S8.indd 816 7/11/09 1:22:03 PM

817ANSWERS

63.

64. (a) cos i (b) cos i b+^ h (c) tan 14a 65. 3

66. , .x x4 4 61 2 67. 12 32c l at both points

68. (a) domain: x21

$ range: y 0$

(b) domain: all real x 7!- range: all real y 0!

(c) domain: x2 2# #- range: y2 0# #-

69. ,a b15 1= - = - 70. cos 2i

71. (a) (0, 0), (1, 3), ( 1- , 1- ), (2, 20)

(b) 6 263c l at (0, 0), 2 20c l at (1, 3), 40 36c l at ( 1- , 1- ), 20 2c l at (2, 20)

72. (a) x n360 45! c= (b) x n180 30c= +

(c) x n180 1 60n# c= + -] g

73. (a) (1, 1) (b) 2 13 units (c) 121

-

(d) 3 2 5 0x y+ - =

74. (a) 75. (b), (d) 76. (a) 77. (c) 78. (c)

79. (b), (d) 80. (c)

Chapter 9: Properties of the circle

The proofs given as answers to this chapter are informal. Also, they may not be the only way to answer the question.

Exercises 9.1

1. (a) 32ci = (b) 8x cm= (c) 68 30ci a= = l (d) 31ci =

(e) 9x mm= (f) 3022ci = l

2. 9

16cm

r 3. (a) 29ci = (b) 18x c=

(c) ,83 42c ca b= = (d) 68x c= (e) 10x cm=

(f) 97y c= (g) , ,x y z15 150 75c c c= = =

(h) , ,x y z47 43 94c c c= = = (i) 40cb =

(j) 39x y c= =

4. (a) , ,x y z112 56 34c c c= = = (b) 49x c=

(c) ,x y55 43c c= = (d) ,x y166 7c c= =

(e) ,x 62 31c cb= =

(f) , , ,x y z v w32 58 32 17c c c c= = = = = (g) 5x c=

(h) 102y c= (i) 57 30 , 32 30x yc c= =l l

(j) , ,x y z75 77 13c c c= = =

5. (a) (vertically opposite )( s in the same segment)(similarly)

DCE ACBEDC BACDEC ABC

s+ + ++ + ++ +

=

=

=

Since all pairs of s+ are equal,

DECD;<ABCD

(b) 5.5x cm=

6.

° (angle at centre is double the at the circumference)( ° °) ( sum of isosceles )

xy

30180 30 2

75'

c

++ D

=

= -

=

7. ° ° ( at the centre is double theat the circumference)

°° (similarly)

x

xy

360 2 110

14070

#

`

++

- =

==

8. ( in semicircle)( sum of )

( in same segment)

ABCBAC

x

9090 296161

`

`

cc ccc

+ ++ +

+

D

=

= -

=

=

9. ( in same segment)(similarly)(vertically opposite )

STV WUVTSV UWVTVS UVW s

+ + ++ ++ + +

=

=

=

Since all pairs of angles are equal,

| WUVD||

.STV

x 2 4 cmD

=

10. ( )BAC AB BC

AC

AC

90

6 336 945

453 5

21

23 5

in semicircle

Radius

cm

2 2 2

2 2

c+ +=

= +

= +

= +

=

=

=

=

=

11. (Base s of isosceles )(similarly)

( at the centre is double theat the circumference)

OACBAOCAB

x CAB

302530 25552

2 55110

`

#

ccc cc

cc

+ +++

+++

D=

=

= +

=

=

=

=

Answer S7-S8.indd 817 7/25/09 2:05:40 PM


12. (a) ,x y2 765 c c= =

(b) AC BD= (equal diameters) Diagonals are equal so ABCD is a rectangle. AD BC` = (opposite sides of a rectangle)

13. 33ECB c+ = (angles in same segment)

EBC 180 114 33+ = - +] g (angle sum of triangle)

°33= ECB ADE`+ += These are equal alternate angles. AD BC` <

14. (a) 90AOB c+ = (given)

ABC 90c+ = (angle in semi-circle)

AOB ABC+ += A+ is common | ABCD||AOB`D ( AAA )

(Note 2 pairs of angles equal means 3 pairs will be equal by angle sum of triangle.)

(b) AO BO= (equal radii)

AB r r

r

r

r

2

22

2 2

2

2#

= +

=

=

=

By similar triangles

ABAO

BCBO

=

But soAO BO AB BC= =

So BC r2=

15. Obtuse 2BOD+ i= (angle at centre double

angle at circumference)

Refl ex 360 2BOD+ i= - (angle of revolution)

21

BCD BOD+ += (angle at centre double angle at circumference)

(360 2 )

18021

i

i

= -

= -

So BCD+ and DAB+ are supplementary (add to180c)

Exercises 9.2

1. (a) 5x cm= (b) 15y cm= (c) 2.4x m= (d) 42x c= (e) 90z c= (f) 10.3x mZ (g) 6 , 3x ym m= = (h) .m 13 4 cmZ (i) 5y cmZ (j) 5x mm=

2. 41 cm 3. 144 mm 4. 25.6 cm

5. . ..

..

(perpendicular from bisects chord)O

CE

CD

AB

11 5 6 99 22 9 218 4

2 2

#

= -

=

=

=

=

6. 8.3OB cm= 7. . , .x y4 7 1 8m m= =

8. 4.4 , 78 , 38 , 64x m c c cZ a b i= = =

9. OA r=

2

ACx

= (perpendicular from the

centre bisects a chord)

2

OC rx2

2

= - d n (Pythagoras’ theorem)

r

x

r x

r x

r x

4

44

4

44

24

22

2 2

2 2

2 2

= -

= -

=-

=-

CD r

r x

r r x

24

22 4

2 2

2 2

= +-

=+ -

10. (a) ECD ACB+ += (vertically opposite angles)

A E+ += (angles in same segment)

CDED|||ABC`D ( AAA )

(b) By similar triangles

CEAC

CDBC

=

. .AC CD BC CE=

Exercises 9.3

1. (a) ,x y107 94c c= = (b) ,134 90c ci c= = (c) , ,x y z112 112 68c c c= = = (d) ,x y92 114c c= = (e) , ,73 107 107c c cb a c= = = (f) ,x y141 63c c= = (g) ,x y65 43c c= = (h) , , ,w x y z89 86 54 35c c c c= = = = (i) , , ,w x y z69 111 82 98c c c c= = = = (j) 118x c=

2. (a) ,x y62 31c c= = (b) ,x y75 105c c= = (c) ,x y88 65c c= = (d) , ,x y z62 82 36c c c= = = (e) ,x y90 113c c= = (f) ,x y38 71c c= = (g) ,x y85 95c c= = (h) ,x y48 78c c= = (i) ,x y107 73c c= = (j) , , , ,a b c d e81 55 83 16 28c c c c c= = = = =

3. (a) 180 58A c c+ = - ( A+ and B+ cointerior angles,

AD BC; )

180 58D c c+ = - ( C+ and D+ cointerior angles, AD BC; )

So A C180c+ += - and D B180c+ += -

Since opposite angles are supplementary, ABCD is a cyclic quadrilateral.

(b) 90B D c+ += = (given)

180B D` c+ += -

Let A x+ =

360 90 90C x+ = - + +] g (angle sum of quadrilateral)

Answer S9-S10.indd 818 7/12/09 4:40:20 AM

819ANSWERS

xx

A

360 180180180 +

= - -

= -

= -

Since opposite angles are supplementary, ABCD is a cyclic quadrilateral.

(c) CDA 180+ i= - (straight angle)

B CDA180` c+ += -

Let A x+ =

360 90 90C x+ = - + +] g (angle sum of quadrilateral)

xx

A

360 180180180 +

= - -

= -

= -

Since opposite angles are supplementary, ABCD is a cyclic quadrilateral .

Exercises 9.4

1. (a) 47ci = (b) 5x m= (c) 11.3y cm= (d) 26x y c= = (e) ,a b64 32c c= = (f) 57ci = (g) 12p 145 cmZ= (h) 10y mm= (i) 5.79x cmZ (j) ,x y33 33c c= =

2. (a) 10x cm= (b) ,x y64 26c c= = (c) 13x cm= (d) ,x y27 54c c= = (e) 5y cm= (f) ,x y32 7c c= = (g) ,x y72 42c c= = (h) ,x y35 90c c= = (i) , , ,m n p q23 67 67 23c c c c= = = = (j) ,x y71 62c c= =

3.

( )

(tangent o radius)( sum of )

( radii)

(base s of isosceles )

( m of )

(opposite s of cyclic quad.)

( at centre twice at circumference)

OABz

OA OCOAC OCA y

y

ACD AED

y uuu

BAC OAB OACx

v AOC

AOB

OAC

9090 4842

180 48 2

66180

180 6266 118

52

90 6624

21

21

48

24

t

equal

su

`

`

'

`

`

#

cc cc

c c

cc

c cc c

c

c cc

c

c

=+

+ +

+ +

+ + +

+

+

+

+

+

+ +

D

D

D

=

= -

=

=

= =

= -

=

= -

+ = -

+ =

=

= -

= -

=

=

=

=

4. 21 cm

5. . ..

..

( ’ )

AC BC

AB

AB AC BCACB

3 9 5 242 256 542 25

90 by Pythagoras theorem

2 2 2 2

2 2

2 2 2`

` c+

+ = +

=

=

=

= +

=

A lies on a diameter of the circle (tangent ⊥ radius)

6. (a) x 67c= (b) 7.5y cmZ (c) ,x y72 121c c= = (d) ,x y63 126c c= = (e) 8.9 , 5.1x ym mZ= (f) ,x y63 63c c= = (g) , ,x y z98 65 17c c c= = = (h) ,x y57 57c c= = (i) ,x y72 15c c= = (j) , ,x y z61 70 52c c c= = =

7. (a) , ,x y z26 74 48c c c= = = (b) 68 , 44 , 68x y zc c c= = = (c) 45x y z c= = = (d) ,x y70 31c c= = (e) , ,x y z20 57 103c c c= = = (f) 5.4x cmZ (g) .x 7 7 cmZ (h) ,x y77 13c c= = (i) 1.2 , 2.1x ycm cmZ Z (j) , ,x y z55 112 57c c c= = =

8. 13AB mZ

Test yourself 9

1. 56ci = 2. 2.3y mm= 3. 7.2x m=

4. 12x y cm= =

5. c

cc ccc

( )( )

( )

( )

zy

x

19180 131 193030

s in same segmentsum of

s in same segment

+

+

+

D

=

= - +

=

=

6. 10x cm=

7. , ,3 44 136c c ca b c= = =

8.

90

(

a

OCAb

OC OEOCE

21

100

50

90 837

is isosceles

at centre twice at circumference)

(tangent perpendicular to radius)

(equal radii)

#

`

`

c

ccc cc

+

+ +

D

=

=

=

= -

=

=

( )OCE OEC c

ccc

2 100 1802 80

40

sum of`

c ccc

+ ++ D

= =

+ =

=

=

360 100

( )

(

(

COE

d260360 260 50 7

43

Reflex of revolution)

sum of quadrilateral)

c ccc c c c

c

+ +

+

= -

=

= - + +

=

9. 17 cm 10. 5.3 m 11. ,a b101 98c c= =

12. ,61 29c ca b= = 13. 14.9 cm 14. 4.9x m=

15. 18 cm 16. 127 , 53c ca b= =

17. ( )

47

( )

( )

D

y

180 80 5347

sum of

s in same segment`

c c ccc

+ + T

+

= - +

=

=

x 47c= ( s+ in alternate segment)

18. , ,x y z55 56 54c c c= = =

Answer S9-S10.indd 819 7/12/09 4:40:57 AM


19. C+ is common

A CBD+ += ( s+ in alternate segment)

| ABCD||BCD AAA` D ] g 20. (a)

( )OCB OCA

OA OB90 (given)

equal radiic+ += =

=

OC is common OAC OBC RHS` /D D ] g (b) AC BC= (corresponding sides in )s/ D

∴ OC bisects AB


1. 6 cm

2. Then

(base s of isosceles )( )( radii)

( of s of isosceles )

DOB DCB xEDO x

EO DOOED EDO x

ODC

EOD

2

2

Letext. ofequal

base`

+ ++

+ +

+

+

+

D

D

D

= =

=

=

= =

180 ( )

( )

( )

( sum of )

( straight )

EOD OED EDOx

AOE EOD DOB

x xx

AOE DCB

EOD

AOC

180 4180

180 180 433`

ccc

c c

+ + +

+ + +

+ +

+

+ +

D= - +

= -

= - +

= - - +

=

=

3. andThen

180 ( )180 ( )180

( in alternate segment)(similarly)

( sum of )( sum of )

( is straight )

DAB x CAB yDAC x yACB DAB xADB CAB yDBA x yCBA x y

DBA CBA

ADBACB

DBC

Let

s

ccc

+ +++ ++ +++

+ +

+

+

+

+

D

D

= =

= +

= =

= =

= - +

= - +

+ =

( ) ( )x y x y180 180 180` c c c- + + - + =

( )x yx y

DAC

180 290

90

`

`

`

cc

c+

= +

= +

=

4. (a) AD DB BE EC CF FA (equal radii)= = = = =

AB BC CA` = = ABC is equilateral`D

(b) r unitsr

(c) r r r321

22 3

units2 2 2 2rr

- =-e o

5. ( )BDE ABD BADABDABD

BAD2

ext. of`

+ + +++

+

a aa

D= +

= +

=

BAD AD BDis isosceles with` D =

( )CDE ACD CADACDACD

CAD

2ext. of

`

+ + +++

+

b b

b

D= +

= +

=

CAD AD CDAD BD CD

is isosceles with`

`

D =

= =

So a circle can be drawn through A , B and C with centre D .

6. Let ODC x+ = and .OAB y+ = Then you can fi nd all these angles (giving reasons).

AOC COB BOD AOD 360c+ + + ++ + + = ( + of revolution)

y x COB y x

AOD90 90

360c c

c+

+- + + + + - +

=

180 360

180COB AOD

COB AOD`

c c

c

+ ++ ++ + =

+ =

7. B

A

D

C

Let ABCD be a kite with AB AD= and ,BC DC= and °.ADC ABC 90+ += = AC is common.

∴ by SSS (or RHS) ABC ADC/D D

BAC DAC BCA DCAand+ + + += =

(corresponding s1 in congruent sD )

Then

90 ( sum of )

BAC DACBADBCA DCABCD

2

180 2

Let

`

c

c

+ +++ ++

+

aa

aa

D

= =

=

= = -

= -

Opposite angles are supplementary.

∴ ABCD is a cyclic quadrilateral, and A , B , C and D are concyclic points

Since ,ABC 90c+ = AC is a diameter. ( + in semicircle)

8. r

2825

units2

2r

Answer S9-S10.indd 820 7/12/09 4:41:15 AM

821ANSWERS

9. Let interval AB subtend angles of x at ADB+ and .ACB+

Assume A , B , C and D are not concyclic. Draw a circle through A , B and C that cuts AD at E .

Then AEB BCA x+ += = ( s+ in same segment)

But AEB+ and EDB+ are equal corresponding angles.

| DB|EB` (this is impossible!)

∴ A , B , C , D must be concyclic

10. Let ABCD be a quadrilateral with opposite angles supplementary. i.e. A C 180c+ ++ = and B D 180c+ ++ =

Assume the points are not concyclic. Draw a circle through A , B and C , cutting CD at E .

Now ABCE is a cyclic quadrilateral, so

180AEC B c+ ++ = (opposite s+ supplementary)

Also, 180D B c+ ++ = (given)

D AEC+ +=

These are equal corresponding angles, so DA EA< (this is impossible!)

∴ A , B , C and D must be concyclic

∴ ABCD is a cyclic quadrilateral.

Chapter 10: The quadratic function

Exercises 10.1

1. Axis of symmetry 1,x = - minimum value 1-

2. Axis of symmetry 1.5,x = - minimum value 7.5-

3. Axis of symmetry 1.5,x = - minimum value 0.25-

4. Axis of symmetry 0,x = minimum value 4-

5. Axis of symmetry 83

,x = minimum point ,83

167d n

6. Axis of symmetry 1,x = maximum value 6-

7. Axis of symmetry 1,x = - maximum point ,1 7-^ h 8. Minimum value ,1- 2 solutions

9. Minimum value 3.75, no solutions

10. Minimum value 0, 1 solution

11. (a) ;x 3= - (-3, -12) (b) ;x 4= - (-4, 17)

(c) ; ,x 141

141

381

= d n (d) ; ,x 141

141

1341

= - - -d n (e) ; ,x 3 3 23= - - -^ h

12. (a) (i) x 1= - (ii) -3 (iii) (-1, -3)

(b) (i) 1x = (ii) 1 (iii) (1, 1)

Answer S9-S10.indd 821 7/12/09 4:41:43 AM


13. (a) Minimum (-1, 0) (b) Minimum (4, -23) (c) Minimum (-2, -7) (d) Minimum (1, -1) (e) Minimum (2, -11)

(f) Minimum ,41

381

- -d n (g) Maximum (-1, 6)

(h) Maximum (2, 11)

(i) Maximum , 721

43d n

(j) Maximum (1, -3)

14. (a) (i) -2 (ii) Minimum 0 (iii) y

x-4 -3 -2 -1 2

3

2

1

4

5

-2

-3

-11

(b) (i) -1, 3 (ii) Minimum -4

(iii) y

x-4 -3 -2 -1 2 3 4

2

1

3

4

5

-3

-2

-11

-4

-5

(c) (i) 5.83, 0.17 (ii) Minimum -8

(iii) y

x-4 -3 -2 -1 2 3 4 5 6

4

2

6

8

10

-6

-4

-21

-8

-10

(d) (i) -2, 0 (ii) Minimum -1

(iii) y

x-4 -3 -2 -1 2

2

1

3

4

5

-3

-2

-11

(e) (i) 3! (ii) Minimum -18

(iii) y

x-2-3-4 -1 1 2 5

1

2

-6

-8

-10

-12

-14

-16

-18

-4

-243

(f) (i) -1, 32

(ii) Minimum 21

12-

Answer S9-S10.indd 822 7/12/09 5:04:29 AM

823ANSWERS

(iii) y

x-4 -3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-5

-6

-2

-1

-2

1

112

23

(g) (i) 1.65, -3.65 (ii) Maximum 7

(iii) y

x-4 -3 -2 -1 2 3 4 5

2

1

3

4

5

7

6

-3

-2

-11

(h) (i) 1.3, -2.3 (ii) Maximum 341

(iii) y

x-4 -3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-2

-11

3 14

(i) (i) 0.56, -3.56 (ii) Minimum 441

(iii) y

x-4 -3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-2

-11

4 14

(j) (i) 2.87, -0.87 (ii) Maximum 7

(iii) y

x-4 -3 -2 -1 2 3 4 5

2

1

3

4

5

6

7

-3

-2

-11

15. (a) 4 (b) None

(c) y

x-4 -3 -2 -1 2 3 4 5

2

1

3

4

5

6

7

-3

-2

-11

Answer S9-S10.indd 823 8/2/09 2:58:15 AM


16. (a) None (b) 643

(c) y

x-4 -3 -2 -1 2 3 4 5

4

2

6

8

10

12

14

-3

-2

-11

17. (a) 387

- (b) None

(c)

18. (a) y

x-4 -3 -2 -1 2 3 4 5

4

2

6

8

-3

-2

-11

(b) ,x x2 31 2 (c) x2 3# #

19. y

x-4 -3 -2 -1 2 3 4 5

4

2

6

8

-6

-4

-21

Graph is always above the x -axis so y 02 for all x x x3 2 4 02` 2- + for all x

20. y

x-4 -3 -2 -1 2 3 4 5

4

2

6

8

-6

-4

-21

Graph is always above the x -axis so y 02 for all x x x 2 02` 2+ + for all x

21. y

x-4 -3 -2 -1 2 3 4 5

2

4

-18

-10

-12

-14

-16

-8

-6

-4

-21

Graph is always below the x -axis so y 01 for all x x x2 7 02` 1- + - for all x

y

x-4 -3 -2 -1 2 3 4 5

1

2

-18

-16

-14

1

-12

-10

-8

-6

-4

-2

Answer S9-S10.indd 824 7/12/09 5:10:54 AM

825ANSWERS

22.

Graph is always below the x -axis so y 01 for all x x x5 4 1 02` 1- + - for all x

Exercises 10.2

1. ,x x3 31 2- 2. 1 0n ##- 3. 0, 2a a# $

4. ,x x2 21 2- 5. y0 6# # 6. 0 2t 11

7. 4, 2x x1 2- 8. 3, 1p p# $- - 9. ,m m2 41 2

10. 3, 2x x# $- 11. h121

21 1 12. 4 5x ##-

13. 2 7k21# #- 14. ,q q 631 2 15. All real x

16. ,n n4 3# $- 17. x3 51 1- 18. t6 2# #-

19. ,y y31

51 2- 20. ,x x2 4# $- 21. x21

01 1-

22. x031

1 1 23. x0 11 # 24. 0x21

1#-

25. x1 131

1 1 26. ,x x1 21$ - - 27. 2 2x52

1 #

28. ,x x6 31 2- - 29. ,x x32

12#

30. x222

21#- -

Exercises 10.3

1. (a) 20 (b) -47 (c) -12 (d) 49 (e) 9 (f) -16 (g) 0 (h) 64 (i) 17 (j) 0

2. (a) 17 unequal real irrational roots (b) -39 no real roots (c) 1 unequal real rational roots (d) 0 equal real rational roots (e) 33 unequal real irrational roots (f) -16 no real roots (g) 49 unequal real rational roots (h) -116 no real roots (i) 1 unequal real rational roots (j) 48 unequal real irrational roots

3. 1p = 4. k 2!= 5. b87

# - 6. p 22 7. k 2121

2 -

8. a 3 02=

b ac4 1 4 3 7

830

2 2

1

- = - -

= -

] ] ]g g g

So x x3 7 02 2- + for all x

9. ,k k5 3$# - 10. k0 41 1 11. ,m m3 31 2-

12. ,k k1 1# $- 13. 3

p1

1 - 14. b0 221

# #

15. ,p p2 6# $-

16. Solving simultaneously: 2 6y x= + (1)

3y x2= + (2)

Substitute (2) in (1):

x xx xb ac

3 2 62 3 04 2 4 1 3

160

2

2

2 2

2

+ = +

- - =

- = - - -

=

] ] ]g g g

So there are 2 points of intersection

17. 3 4 0x y+ - = (1) 5 3y x x2= + + (2) From (1): 3 4y x= - + (3) Substitute (2) in (3):

5 3 3 48 1 0

4 8 4680

x x xx x

b ac 1 1

2

2

2 2

2

+ + = - +

+ - =

- = - -

=

] ]g g

So there are 2 points of intersection

18. 4y x= - - (1) y x2= (2) Substitute (2) in (1):

44 0

4 1 415

0

x xx x

b ac 1 4

2

2

2 2

1

= - -

+ + =

- = -

= -

] ]g g

So there are no points of intersection

19. 5 2y x= - (1) 3 1y x x2= + - (2) Substitute (2) in (1):

x x xx x

b ac

3 1 5 22 1 0

4 2 4 1 10

2

2

2 2

+ - = -

- + =

- = - -

=

] ] ]g g g

So there is 1 point of intersection the line is a tangent to the parabola

20. 341

p =

21. (c) and (d)

y

x-4 -3 -2 -1 2 3 4 5

1

2

-5

-6

-7

-4

-3

-2

-11

Answer S9-S10.indd 825 8/2/09 2:58:20 AM


Exercises 10.4

1. (a) , ,a b c1 2 6= = = - (b) , ,a b c2 11 15= = - = (c) , ,a b c1 1 2= = = - (d) , ,a b c1 7 18= = = (e) , ,a b c3 11 16= = - = - (f) , ,a b c4 17 11= = = (g) , ,a b c2 12 9= = - = - (h) , ,a b c3 8 2= = - = (i) , ,a b c1 10 24= - = = - (j) , ,a b c2 0 1= - = = -

2. , ,m p q2 5 2= = - =

3. 4 5 2 2 1 3 4x x x x x2 - + = - - + + +] ]g g

4. a x x b x cx x x

x x x xx x

2 3 21 2 3 1 2 17

3 2 6 2 172 9

RHS

RHS

2

2

= - + + - +

= - + + - +

= + - - + - +

= + +

=

] ] ]] ] ]

g g gg g g

true

5. , ,A B C1 5 6= = = - 6. , ,a b c2 1 1= = = -

7. , , .K L M1 6 7 5= = = 8. 12 5 2 3 65 2x x 2+ + - - -] ]g g

9. , ,a b c0 4 21= = - = -

10. (a) 5y x x2= - - (b) 3y x x2= -

(c) 2 3 7y x x2= - + (d) 4 9y x x2= + -

(e) 2 1y x x2= - - +

Exercises 10.5

1. (a) ,2 1a b ab+ = - = (b) . ,1 5 3a b ab+ = = - (c) . , .0 2 1 8a b ab+ = = - (d) ,7 1a b ab+ = - =

(e) ,232

1a b ab+ = =

2. (a) 3 (b) 6- (c) 0.5- (d) 21

3. (a) 3 10 0x x2 + - = (b) 4 21 0x x2 - - = (c) 5 4 0x x2 + + = (d) x x 08 112 - + = (e) 2 27 0x x2 - - =

4. 0.5m = 5. 32k = - 6. 4b = 7. 1k = 8. 13p =

9. 5k = - 10. m 3!= 11. 1k = - 12. ,n 1 3= -

13. ,p r2 7= = - 14. ,b c6 8= - = 15. ,a b0 1= = -

16. 11

`ab ba

= =

17. (a) 1k = - (b) 1, 0k = - (c) 1.8k = - (d) 3k =

(e) ,k k1 0# $-

18. (a) p 2 3!= (b) ,p p2 3 2 3# $-

(c) p2

3 3!=

19. (a) k 2= (b) 3k = - (c) 2k =

20. (a) 1m = (b) ,m m2

3 102

3 101 2

- +

(c) 3m = -

Exercises 10.6

1. (a) ,x 1 4= - - (b) 2, 5y = (c) 4, 2x = - (d) 1, 4n = - (e) 3, 5a = - (f) 3, 4p = (g) ,x 2 4= - (h) 5, 12k = (i) ,t 6 4= - (j) ,b 12 4= - -

2. (a) 2, 3x = - (b) 2, 3x = (c) 4, 5x = (d) 3, 5x =

(e) 121

x = , 4

3. (a) x 3!= (b) ,y 2 2! != (c) x2

1 5!=

(d) . , . , . , .x 1 37 4 37 0 79 3 79= - - (e) ,a 2 2 6!= - -

4. (a) 0, 3x = (b) 1p = (c) 1x = (d) 1x = (e) 1, 3x =

5. 2,x 1! != 6. 1x = -

7. . , . , . , .x 2 19 0 46 1 93 0 52! ! ! !=

8. (a) , , ,x 0 90 180 360c c c c= (b) , ,x 90 180 270c c c= (c) , ,x 90 210 330c c c= (d) , , ,x 60 90 270 300c c c c= (e) , , ,x 0 180 270 360c c c c=

9. (a) , , , ,x 0 45 180 225 360c c c c c= (b) , ,x 0 180 360c c c= (c) , , , ,x 0 30 150 180 360c c c c c= (d) 45 , 60 ,135 , 120 , 225 , 240 , 315 , 300x c c c cc c c c= (e) 30 , 60 , 120 , 150 , 210 , 240 , 300 , 330x c c c c c c c c=

10.

( ) ( ) ( ) ( )

xx

x xx

x x

x xx x

33

25

3 33

23 5 3

3 2 5 33 5 3 2 0

2

2

# # #

+ ++

=

+ + ++

+ = +

+ + = +

+ - + + =

]] ]

] ]

gg g

g g

Let 3u x= +

u ub ac

5 2 04 5 4 1 2

170

2

2 2

2

- + =

- = - -

=

] ] ]g g g

So u has 2 real irrational roots. x 3` + and so x has 2 real irrational roots

Test yourself 10

1. (a) x0 3# # (b) ,n n3 31 2- (c) 2 2y ##-

2. , ,a b c1 9 14= = - = 3. (a) 2x = (b) 3-

4. ab ac1 0

42 4 1 724

0positive definite

2

# #

`

2

1

D

=

= -

= - -

= -

2] g

Answer S9-S10.indd 826 7/12/09 4:43:24 AM

827ANSWERS

5. (a) 6 (b) 3 (c) 2 (d) 18 (e) 30 6. ,x 132

31

=

7. (a) iv (b) ii (c) iii (d) ii (e) i

8.

( ) ( )

ab ac

1 04

3 4 1 47

0

2

2# #

1

1

D

= -

= -

= - - -

= -

x x4 3 02` 1- + - for all x

9. (a) 41

x = - (b) 681

10. 3 2 12 3 41x x2- + + -] ]g g 11. , ,x 30 150 270c c c=

12. (a) 341

k = (b) 1k = (c) 3k = (d) 3k = (e) 2k =

13. ,x21

3= - 14. m169

1 - 15. ,x 0 2=

16. (a) i (b) i (c) iii (d) i (e) ii

17. (a) iii (b) i (c) i (d) ii

18.

ac

kk

1

1

1

For reciprocal roots

LHS RHS

ba

ab

aa

=

=

=

= =

∴ roots are reciprocals for all x .

19. (a) 3 28 0x x2 + - = (b) 10 18 0x x2 - + =

20. 1, 3x =

21. (a) ,x x174

1 2- - (b) ,n n3 32# -

(c) y51

31

1 1 (d) ,x x10 221

2# - - (e) x4 71 #


1. k 4 02$D= -] g and a perfect square ∴ real, rational roots

2. y x x5 42= - + 3. , ,a b c4 3 7= = - = 4. x 2!=

5. 11 6. 2.3375n = - 7. .p 0 752 8. Show 0D =

9. x 1!=

10. 2, 19, 67 2, 13, 61A B C A B Cor= = - = = - = = -

11. 2

4 12

31

1

x x

xx x2 - -

+=

-+

+

12. ,k k2

1 212

1 21# $

- +

13. , ,x 30 90 150c c c= 14. ,x 12

3 5!=

15. , , ,x 60 90 270 300c c c c= 16. 23-

Chapter 11: Locus and the parabola

Exercises 11.1

1. A circle 2. A straight line parallel to the ladder.

3. An arc 4. A (parabolic) arc 5. A spiral

6. The straight line 2 2 | | 2x xor1 1 1-

7. A circle, centre the origin, radius 2 (equation 4x y2 2+ = i

8. lines y 1!= 9. lines x 5!= 10. line 2y =

11. Circle 1x y2 2+ = (centre origin, radius 1)

12. Circle, centre , ,1 2-^ h radius 4 13. 5y = -

14. Circle, centre (1, 1), radius 3 15. x 7= - 16. 3x =

17. y 8!= 18. x 4!=

19. Circle, centre , ,2 4-^ h radius 6

20. Circle, centre , ,4 5-^ h radius 1

Exercises 11.2

1. x y 12 2+ = 2. 2 2 79 0x x y y2 2+ + + - =

3. 10 4 25 0x x y y2 2- + + + = 4. 8 6 13 0x y- + =

5. 12 26 1 0x y- - = 6. y x!=

7. 3 32 3 50 251 0x x y y2 2- + - + =

8. 5 102 5 58 154 0x x y y2 2- + + - =

9. 4 20 36 0x x y2 - + - = 10. 20 0x y2 - =

11. 8 32 0y x2 + - = 12. 2 8 7 0x x y2 - + - =

13. 12 0x y2 + = 14. 5 2 11 0x x y y2 2- + - - =

15. 3 4 0x x y y2 2+ + - - =

16. 2 17 0x x y y2 2+ + - - =

17. 2 4 2 6 47 0x x y y2 2+ + - + =

18. 2 2 2 4 27 0x x y y2 2+ + + + =

19. 3 4 25 0, 3 4 15 0x y x y+ + = + - =

20. ,x y x y12 5 14 0 12 5 12 0- - = - + =

21. x y2 3 5 5 0!- - =

22. 7 9 0, 7 5 0x y x y- + = + - =

23. 7 4 30 0, 32 56 35 0x y x y- - = + - =

24. 16 7 40 0xy x y- - + =

25. 6 3 12 9 0x x y y2 2- - - + =

Answer S9-S10.indd 827 7/12/09 4:43:36 AM


Problem

,x y x y12 5 40 0 12 5 38 0+ - = + + =

Exercises 11.3

1. (a) Radius 10, centre (0, 0) (b) Radius 5 , centre (0, 0) (c) Radius 4, centre (4, 5) (d) Radius 7, centre (5, −6) (e) Radius 9, centre (0, 3)

2. (a) 16x y2 2+ = (b) 6 4 12 0x x y y2 2- + - - = (c) 2 10 17 0x x y y2 2+ + - + = (d) 4 6 23 0x x y y2 2- + - - = (e) 8 4 5 0x x y y2 2+ + - - = (f) 4 3 0x y y2 2+ + + = (g) 8 4 29 0x x y y2 2- + - - = (h) 6 8 56 0x x y y2 2+ + + - = (i) 4 1 0x x y2 2+ + - = (j) 8 14 62 0x x y y2 2+ + + + =

3. 18 8 96 0x x y y2 2- + + + =

4. 4 4 8 0x x y y2 2+ + + - = 5. 2 48 0x x y2 2- + - =

6. 6 16 69 0x x y y2 2+ + - + =

7. 10 4 27 0x x y y2 2- + + + = 8. 9 0x y2 2+ - =

9. 2 10 25 0x x y y2 2- + - + =

10. 12 2 1 0x x y y2 2+ + - + =

11. 8 6 22 0x x y y2 2- + - + = 12. 6 1 0x y y2 2+ + + =

13. (a) Radius 3, centre (2, 1) (b) Radius 5, centre (−4, 2) (c) Radius 1, centre (0, 1) (d) Radius 6, centre (5, −3) (e) Radius 1, centre (−1, 1) (f) Radius 6, centre (6, 0) (g) Radius 5, centre (−3, 4) (h) Radius 8, centre (−10, 2) (i) Radius 5, centre (7, −1) (j) Radius 10 , centre (−1, −2)

14. Centre ,3 1-^ h , radius 4 15. Centre ,2 5^ h , radius 5

16. Centre ,1 6- -^ h , radius 7 17. Centre (4, 7), radius 8

18. Centre ,121

1-d n , radius 221

19.

20. Show perpendicular distance from the line to ,4 2-^ h is 5 units, or solve simultaneous equations.

21. (a) Both circles have centre ,1 2-^ h (b) 1 unit

22. 2 2 23 0x x y y2 2+ + + - =

23. 56 units 24. 34 units

25. (a) 5 units (b) 3 units and 2 units (c) XY is the sum of the radii. The circles touch each other at a single point, ,0 1^ h .

26. Perpendicular distance from centre ,0 0^ h to the line is equal to the radius 2 units; perpendicular distance from centre ,1 2-^ h to the line is equal to the radius 3 units.

27. (a) 2 6 15 0x x y y2 2+ + - - = (b) , ,,2 7 1 2- -^ ^h h (c) ,Z 1 8= -^ h (d) m m

31

3

1

zx yx# #= -

= -

ZXY 90` c+ =

28. (a) 4 units (b) 4 10 13 0x x y y2 2- + + + =

Exercises 11.4

1. (a) 20x y2 = (b) 36x y2 = (c) 4x y2 = (d) 16x y2 =

(e) 40x y2 = (f) 12x y2 = (g) 24x y2 = (h) 44x y2 =

(i) 8x y2 = (j) 48x y2 =

2. (a) x y42 = - (b) 12x y2 = - (c) 16x y2 = -

(d) 28x y2 = - (e) 24x y2 = - (f) 36x y2 = -

(g) 32x y2 = - (h) 8x y2 = - (i) 60x y2 = -

(j) 52x y2 = -

3. (a) (i) (0, 1) (ii) y 1= - (b) (i) (0, 7) (ii) 7y = - (c) (i) (0, 4) (ii) 4y = - (d) (i) (0, 9) (ii) 9y = - (e) (i) (0, 10) (ii) 10y = - (f) (i) (0, 11) (ii) 11y = -

(g) (i) (0, 3) (ii) 3y = - (h) (i) (0, 121c m (ii) 1y

21

= -

(i) (i) 0, 221c m (ii) 2y

21

= - (j) (i) 0, 343c m

(ii) 3y43

= -

4. (a) (i) (0, −1) (ii) 1y = (b) (i) (0, −6) (ii) 6y = (c) (i) (0, −2) (ii) 2y = (d) (i) (0, −12) (ii) 12y = (e) (i) (0, −5) (ii) 5y = (f) (i) (0, −4) (ii) 4y = (g) (i) (0, −8) (ii) 8y = (h) (i) (0, −10) (ii) 10y =

(i) (i) 0,21

-c m (ii) 21

y = (j) (i) 0, 521

-c m (ii) 521

y =

5. (a) 28x y2 = (b) 44x y2 = (c) 24x y2 = - (d) 8x y2 = (e) 12x y2

!= (f) 32x y2!=

(g) 32x y2 = (h) 71

x y2 =

6. (a) Focus , ,0 2^ h directrix 2,y = - focal length 2 (b) Focus , ,0 6^ h directrix 6,y = - focal length 6

(c) Focus , ,0 3-^ h directrix 3,y = focal length 3

(d) Focus , ,021d n directrix ,y

21

= - focal length 21

(e) Focus , ,0 143

-d n directrix 143

,y = focal length 143

(f) Focus , ,081d n directrix ,y

81

= - focal length 81

Answer S11-S12.indd 828 7/12/09 3:09:35 AM

829ANSWERS

7. 2y = 8. ,4 4^ h 9. ,X 121

83

= - -d n 10. 4, 2-^ h and 4, 2- -^ h ; 8 units

11. (a) 12x y2 = - (b) 3y = (c) 3331

units

12. (a) Substitute the point into the equation.

(b) 3 4 3 0x y+ - = (c) ,243

-d n 13. (a) 4 2 0x y- + = (b) 0, 1^ h does not lie on the line

(c) 4 2 1 0x x y y2 2- + - + = (d) Substitute ,0 1^ h into the equation of the circle.

14. (a) Substitute Q into the equation of the parabola. (b) 1 2 2 0q x qy aq2 - - + =_ i (c) Equation of latus rectum is .y a= Solving with 4x ay2 = gives two endpoints , , ,A a a B a a2 2-^ ^h h . Length of 4AB a= .

Exercises 11.5

1. (a) 8y x2 = (b) 20y x2 = (c) 56y x2 = (d) 36y x2 = (e) 32y x2 = (f) 24y x2 = (g) 28y x2 = (h) 12y x2 = (i) 16y x2 = (j) 4y x2 =

2. (a) y x362 = - (b) 16y x2 = - (c) 40y x2 = - (d) y x242 = - (e) 8y x2 = - (f) 48y x2 = - (g) 44y x2 = - (h) y x202 = - (i) 12y x2 = - (j) 28y x2 = -

3. (a) (i) (2, 0) (ii) x 2= - (b) (i) (3, 0) (ii) 3x = - (c) (i) (4, 0) (ii) 4x = - (d) (i) (1, 0) (ii) 1x = - (e) (i) (7, 0) (ii) 7x = - (f) (i) (8, 0) (ii) 8x = - (g) (i) (6, 0) (ii) 6x = - (h) (i) (9, 0) (ii) x 9= -

(i) (i) 41

, 0c m (ii) 41

x = - (j) (i) 421

, 0c m (ii) 4x21

= -

4. (a) (i) (−2, 0) (ii) 2x = (b) (i) (−3, 0) (ii) 3x = (c) (i) (−7, 0) (ii) 7x = (d) (i) (−1, 0) (ii) 1x = (e) (i) (−6, 0) (ii) 6x = (f) (i) (−13, 0) (ii) 13x =

(g) (i) (−15, 0) (ii) 15x = (h) (i) 21

, 0-c m (ii) 21

x =

(i) (i) 621

, 0-c m (ii) 621

x = (j) (i) 141

, 0-c m (ii) 141

x =

5. (a) 20y x2 = (b) 4y x2 = (c) 16y x2 = - (d) 12y x2 =

(e) 36y x2!= (f) 8y x2

!= (g) 12y x2 = (h) 21

y x2 =

6. (a) Focus , ,2 0^ h directrix 2,x = - focal length 2

(b) Focus , ,1 0^ h directrix 1,x = - focal length 1

(c) Focus , ,3 0-^ h directrix 3,x = focal length 3

(d) Focus , ,121

0d n directrix 121

,x = - focal length 121

(e) Focus , ,141

0-d n directrix 141

,x = focal length 141

(f) Focus , ,121

0d n directrix ,x121

= - focal length 121

7. 4x = (latus rectum) 8. , , ,,12 3 6 3 6-^ ^h h 9. , ,,9 6 81 18-^ ^h h 10. (a) 5 12 25 0x y- - = (b) ,5 4

61

- -d n (c) 10125

units 2

(d) 4132

units (e) 11.7 units 2

Exercises 11.6

1. (a) yx 3 8 32- = +] ^g h (b) 5 4 6x y2- = +] ^g h (c) x y1 4 32 = +-] ^g h (d) 12x y4 32- = - -] ^g h (e) 6 8 7x y2- = +] ^g h (f) 16x y7 32+ = - -] ^g h (g) 4x y2 52- = - -] ^g h (h) 9 12 6x y2+ = +] ^g h (i) x y1 4 22+ = - -] ^g h (j) 3 8 1x y2- = +] ^g h

2. (a) 4 4 4y x2- = +^ ]h g (b) 1 8 2y x2- = +^ ]h g (c) y x2 12 12+ = +^ ]h g (d) 10 4 29y x2- = - -^ ]h g (e) 3 16 1y x2+ = - -^ ]h g (f) 6 8 4y x2- = +^ ]h g (g) 5 24 2y x2+ = - -^ ]h g (h) 12 4 36y x2+ = +^ ]h g (i) y x2 20 12- = - -^ ]h g (j) 4 8 2y x2+ = - -^ ]h g

3. (a) 2 8 9 0x x y2 + - + = (b) x x y8 4 16 02 + - + =

(c) 4 8 12 0x x y2 - - - = (d) 6 8 41 0x x y2 - - + =

(e) 4 16 20 0x x y2 + - + = (f) 2 16 1 0x x y2 + + + =

(g) x x y8 20 4 022 - + - = (h) 10 8 1 0x x y2 + + + =

(i) 6 12 45 0x x y2 + + + = (j) x y4 24 02 + + =

(k) 6 12 3 0y y x2 - - - = (l) 8 4 8 0y y x2 - - + =

(m) 8 32 0y x2 - + = (n) y y x4 16 2 012 + - =-

(o) 2 8 7 0y y x2 + - - = (p) y y x8 12 042 + + + =

(q) 2 4 11 0y y x2 - + - = (r) 6 16 25 0y y x2 - + + =

(s) 4 2 5 0y y x2 - + + = (t) y y x2 2 062 - + =-

4. (a) (i) (3, −2) (ii) 4y = - (b) (i) (1, 1) (ii) y 3= -

(c) (i) (−2, 0) (ii) 2y = - (d) (i) (4, 2) (ii) 4y = -

(e) (i) (−5, −1) (ii) 5y = - (f) (i) (3, 1) (ii) 3y =

(g) (i) (−1, 0) (ii) 4y = (h) (i) (2, 0) (ii) 2y =

(i) (i) (4, −2) (ii) 4y = (j) (i) (−2, −3) (ii) 5y =

5. (a) (i) (0, −1) (ii) 2x = - (b) (i) (2, 4) (ii) 4x = - (c) (i) (0, 3) (ii) 4x = - (d) (i) (3, −2) (ii) x 5= - (e) (i) (7, 1) (ii) 5x = - (f) (i) (1, −5) (ii) 5x = (g) (i) (11, −7) (ii) 13x = (h) (i) (−3, 6) (ii) 7x =

(i) (i) (−7, 2) (ii) 9x = (j) (i) 1021

, 3- -c m (ii) 921

x =

6. 12 36 0x y2 - + =

7. ,x x y x x y4 8 4 0 4 8 12 02 2+ - - = + + + =

8. 2 4 19 0x x y2 - - - = 9. 12 12 12 0y y x2 - + + =

10. x x y2 1 1 022 - - + = 11. 2 28 29 0x x y2 - - + =

12. 4 24 44 0y y x2 + + - = 13. 6 32 9 0y y x2 - - + =

14. 6 8 15 0x x y2 - + - = 15. 2 16 49 0y y x2 + - + =

16. 6 4 7 0x x y2 + + - = 17. 4 12 8 0x x y2 - - - =

Answer S11-S12.indd 829 7/25/09 2:27:10 PM


18. 2 16 95 0y y x2 + + - =

19. (a) Vertex ,2 1-^ h , focus ,2 3-^ h , directrix 1y = -

(b) Vertex ,3 2^ h , focus ,3 5^ h , directrix 1y = -

(c) Vertex ,1 1-^ h , focus ,1 2-^ h , directrix 0y =

(d) Vertex ,3 4^ h , focus ,7 4^ h , directrix x 1= -

(e) Vertex ,0 2-^ h , focus ,6 2-^ h , directrix 6x = -

(f) Vertex ,5 0-^ h , focus ,7 0-^ h , directrix x 3= -

20. Vertex ,1 4-^ h , focus 1, 3- -^ h , directrix 11,y = axis 1,x = - maximum value 4

21. 4 8 12 0x x y2 - - + = or 4 8 36 0x x y2 - + - =

22. (a) 8 9 72 0x y2 + - = (b) , , y0 73223

8329

=d n

23. (a)

(b) 1, 8 , y43

49

1- - = -d n

24. 4 8 20 0x x y2 + + - = 25. 0.3 m

Exercises 11.7

1. 31

m = 2. m 4= - 3. m 1= - 4. 21

m =

5. dx

dyx= 6. 2 0x y- - = 7. 2 12 0x y- + =

8. 6 0, 18 0x y x y+ - = - - =

9. 2 2 0, 2 9 0x y x y- - = + - =

10. ,,x y M 187

21

4 8 0+ - = = d n 11. , ,x y P9 0 18 27+ - = = -^ h 12. 33, 60.5Q = ^ h 13. , ,x y x y4 144 0 4 2 9 0+ + = + + = , .18 40 5-^ h ; show

the point lies on the parabola by substituting it into the equation of the parabola

14. , ,x y R4 0 4 0- - = = ^ h 15. (a) Substitute P into the equation of the parabola

(b) 2 0x py p p3+ - - = (c) Substitute 0, 1^ h into the equation of the normal.

( )Since 0, 1 0

p p pp pp p

p p

0 2 00

1

3

3

2

2!

+ - - =

= +

= +

+ =

Exercises 11.8

1. (a)

(b)

(c)

(d)

(e)

Answer S11-S12.indd 830 7/12/09 3:09:45 AM

831ANSWERS

(f)

2. (a) 2 2 0x y- - = (b) 2 11 0x y- - = (c) 3 2y x x2= + + (d) 16 1y x2= - (e) 2xy =

3. (a) ,x t y t2 2= = (b) ,x t y t6 3 2= =

(c) ,x t y t4 2 2= - = - (d) ,x t y t8 4 2= =

(e) ,x t y t18 9 2= - = - (f) ,x t y t10 5 2= =

(g) ,x t yt

32

3 2

= - = - (h) ,xt

yt

2 4

2

= =

(i) ,xt

yt

4 8

2

= = (j) ,x t yt

52

5 2

= - = -

4. (a) 16x y2 = (b) 20x y2 = (c) 4x y2 = (d) 28x y2 = - (e) 8x y2 = - (f) 4x ay2 = (g) x y42 = - (h) 24x y2 = (i) x y22 = - (j) 4x ay2 =

5. (a) Substitute ,t t6 3 2-_ i into the equation (b) ,P 12 12= - -^ h (c) 2 12 0x y- + =

6. (a) ,Q 8 4-= ^ h (b) 12 0x y- + =

7. , , x4 0 4= -^ h 8. , ; x yP 4 4 4 3 4 0= - + - =^ h

9. (a) 24x y2 = (b) 41

10. 3 18 0x y- - =

Exercises 11.9

1. (a) (i) 2

t n+ (ii)

21

4 0y t n x tn- + + =] g

(b) (i) 2

p q+ (ii) 0y p q x pq

21

2- + + =^ h

(c) (i) 2

m n+ (ii) 0y m n x mn

21

3- + + =] g

(d) (i) 2

p q+ (ii) 0y p q x pq

21

5- + + =^ h

(e) (i) 2

a b+ (ii) 0y a b x ab

21

- + + =] g

(f) (i) 2

p q-

+ (ii) y p q x pq

21

2 0+ - =+ ^ h

(g) (i) 2

a b-

+ (ii) y a b x ab

21

6 0+ - =+ ] g

(h) (i) p q

2

+ (ii) y p q x pq

21

4 0+ =- -^ h

(i) (i) 2

s t-

+ (ii) y s t x st

21

0+ - =+ ] g

(j) (i) p q

2

+ (ii) y p q x pq

21

7 0+ =- -^ h

2. (a) (i) p (ii) 1p

- (iii) 0y px p2- + =

(iv) 2x py p p3+ = +

(b) (i) q (ii) 1q- (iii) 3 0y qx p2- + =

(iv) 3 6x qy q q3+ = +

(c) (i) t (ii) 1t

- (iii) 2 0y tx t2- + =

(iv) 2 4x ty t t3+ = +

(d) (i) n (ii) 1n- (iii) 5 0y nx n2- + =

(iv) 5 10x ny n n3+ = +

(e) (i) p (ii) 1p

- (iii) 6 0y px p2- + =

(iv) 6 12x py p p3+ = +

(f) (i) − k (ii) 1k

(iii) 4 0y kx k2+ - =

(iv) 4 8x ky k k3- = +

(g) (i) q (ii) 1q- (iii) 0y qx q2- - =

(iv) 2x qy q q3+ = - -

(h) (i) − t (ii) 1t

(iii) 2 0y tx t2+ - =

(iv) 2 4x ty t t3- = +

(i) (i) m (ii) 1m- (iii) 3 0y mx m2- - =

(iv) x my m m3 63+ = - -

(j) (i) − a (ii) 1a (iii) 8 0y ax a2+ - =

(iv) 8 16x ay a a3- = +

3. (a) (i) ,p q pq+^ h (ii) ,pq p q p pq q 22 2- + + + +^ h7 A (b) (i) 4 , 4p q pq+^ h7 A (ii) ,pq p q p pq q4 4 22 2- + + + +^ _h i8 B

(c) (i) 2 , 2a b ab+] g6 @ (ii) ,ab a b a ab b2 2 22 2- + + + +] ^g h7 A (d) (i) 3 , 3s t st+] g6 @ (ii) 3 , 3st s t s st t 22 2- + + + +] ^g h7 A (e) (i) 5 , 5t w tw+] g6 @ (ii) 5 , 5tw t w t tw w 22 2- + + + +] ^g h7 A (f) (i) ,p q pq6 6+ -^ h7 A (ii) ,pq p q p pq q6 6 22 2- + - + + +^ _h i8 B

(g) (i) ,m n mn4 4+ -] g6 @ (ii) 4 , 4mn m n m mn n 22 2- + - + + +] ^g h7 A (h) (i) ,p q pq10 10+ -^ h7 A (ii) 10 , 10pq p q p pq q 22 2- + - + + +^ _h i8 B

(i) (i) ,h k hk5 5+ -] g6 @ (ii) 5 , 5hk h k h hk k 22 2- + - + + +] ^g h7 A (j) (i) 3 , 3p q pq- + -^ h7 A (ii) ,pq p q p pq q3 3 22 2+ - + + +^ _h i8 B

4. (a) (i) 4xx y y1 1= +_ i (ii) 4

y y x x x11

1- = - -_ i (b) (i) 6xx y y1 1= +_ i (ii)

6y y x x x1

11- = - -_ i

Answer S11-S12.indd 831 7/12/09 3:09:46 AM


(c) (i) 8xx y y1 1= +_ i (ii) y y x x x8

11

1- = - -_ i (d) (i) 2xx y y1 1= +_ i (ii) y y x x x

21

11- = - -_ i

(e) (i) 10xx y y1 1= +_ i (ii) y y x x x10

11

1- = - -_ i (f) (i) 2xx y y1 1= - +_ i (ii)

2y y x x x1

11- = -_ i

(g) (i) xx y y41 1= - +_ i (ii) 4

y y x x x11

1- = -_ i (h) (i) 12xx y y1 1= - +_ i (ii)

12y y x x x1

11- = -_ i

(i) (i) 22xx y y1 1= - +_ i (ii) 22

y y x x x11

1- = -_ i (j) (i) 14xx y y1 1= - +_ i (ii)

14y y x x x1

11- = -_ i

5. (a) 8xx y y1 1= +_ i (b) 2xx y y1 1= +_ i (c) 4xx y y1 1= +_ i (d) 6xx y y1 1= +_ i (e) 10xx y y1 1= +_ i (f) xx y y21 1= - +_ i (g) 12xx y y1 1= - +_ i (h) xx y y41 1= - +_ i (i) 8xx y y1 1= - +_ i (j) xx y y181 1= - +_ i

6. (a) 0y px ap2- + = (b) 2xx a y y0 0= +_ i 7.

21

2 0y t r x tr- + + =] g 8. 2 36 0x y+ - =

9.

,

yx

dx

dy x

tt

dx

dy t

t

18

9

92

9

99

At

2

2

= -

= -

- -

= --

=

e

do

n

For normal, m m 11 2 = -

mt1

2` = -

The equation is given by

( )

( )

( )

y y m x x

yt

tx t

ty t x tx t

x ty t t

29 1

9

2 9 2 92 18

2 2 9 18 0

1 12

3

3

`

- = -

+ = - +

+ = - +

= - -

+ + + =

10. 2x ty at at3+ = + 11. 3 4 4 0x y- + =

12. Substitute focus ,0 1-^ h into equation 3 4 4 0x y+ + = .

13. Equation of chord

y p q x apq21

0+ + =- ^ h

Substitute , a0^ h into equation

a apq

pq

0

1

+ =

= -

( )a p q apq

apq a

21

0 0- + + =

= -

14. ,2 1- -^ h 15. Equation of tangent at P :

0y px ap2- + = (1) Equation of tangent at Q : 0y qx aq2- + = (2)

:1 2-] ]g g

0( ) ( ) 0

( ) ( )( ) 0( ) 0

( )

px qx ap aqx q p a q p

x q p a q p q px a q p

x a q p

2 2

2 2

- + + - =

- - - =

- - + - =

- + =

= +

Substitute in (1):

( ) 000

y pa q p apy apq ap ap

y apqy apq

2

2 2

- + + =

- - + =

- =

=

16. (a) 3 4 8 0x y+ - = (b) Substitute ,0 2^ h into equation.

17. (a) For proof, see no. 9 above (b) ,N ap a0 22= +_ i 18. (a) 15 8 4 0x y+ + = (b) ,N

41

321

= - -c m 19. (a) 3 3 0x y+ - = (b) , ,6 3 2

31

-^ ch m

20. (a) ,F 0 6= ^ h

(b) 3 4 24 0x y+ - =

(c) ,Q 24 24-= ^ h (d) : 2 3 0; : 2 24 0P x y Q x y- - = + + =

(e) ,m m21

2 11 2 #= - = - ` tangents at P , Q are

perpendicular (f) 9, 6R = - -^ h (g) directrix: 6,y a= - = - R lies on directrix

21. , .P 2 1 5= - -^ h 22. 9 0x y- + =

23.

1(since 1 for focal chord)m m pq

pq1 2 =

= - = -

tangents are perpendicular

24. Tangents intersect at ,a p q apq+^ h6 @

(since for focal chord)Directrix:

y apqa pq

y a1

i.e. =

= - = -

= -

tangents meet on the directrix

25. 4

2

ya

x

dx

dy

ax

2

=

=

At , ,x yP 0 0_ i

2dx

dy

a

x0=

Answer S11-S12.indd 832 7/12/09 3:09:47 AM

833ANSWERS

The equation is given by

( )

( )

( )

( )

( )

y y m x x

y ya

xx x

ay ay x x x

xx x

xx ay x ay

ay ay xx

a y y xx

22 2

4 42 22

since

1 1

0

0

0

0 0 0

0 02

0 0 02

0

0 0

0 0

`

- = -

- = -

- = -

= -

= - =

+ =

+ =

Exercises 11.10

1. (a) 3 4 8 0x y+ - =

(b) ,Q 221

= d n

(c) (−3, −2)

(d) 4dx

dy x=

At P (−8, 8):

dx

dy

m48

21`

=-

= -

At ,q 221c m :

42

21

dx

dy

m2`

=

=

m m 2

21

1

1 2 #= -

= -

So the tangents are perpendicular.

2. (a) 0y px p2- + = (b) 1p2 + (c) ,R p0 2= -_ i and 0, 1F = ^ h 1FR p

PF

2= +

=

3. (a) 3 0y tx t2- + = (b) ,Y t0 3 2= -_ i (c) ,F 0 3= ^ h 3 1TF FY t2= = +^ h

4. (a) 5 0y qx q2+ - = (b) 0, 5R q2= _ i (c) ,F 0 5= -^ h 5 1FR FQ q2= = +_ i So triangle FQR is isosceles. FQR FRQ`+ += (base angles of isosceles triangle)

5. (a) 4 3 9 0x y+ - = (b) Focus (0, 3) Substitute into equation:

4 3 90 30

LHS

RHS

= + -

=

=

] ]g g

So it is a focal chord.

(c) Directrix y 3= - Point of intersection 8, 3= - -^ h So the point lies on the directrix.

6. 2 2x a y a2 = -^ h 7. ; ;y px p y qx q y2 0 2 0 22 2- + = - + = = -

8. x y16 62 = -^ h 9. x a y a22 = -^ h 10. (a) y a= - (b) 2x a y a2 = -^ h 11. 4 4x y2 = - +^ h 12. (a) PO has gradient

2;

p QO has gradient

2

q

m mp q

pq2 2

1

4

1 2 #

`

= = -

= -

(b) 2 4x a y a2 = -^ h (c) 2 4x a y a2 = -^ h is a parabola in the form ( ) 4x h a y k2

0- = -^ h where ,h k^ h is the vertex and a0 is the focal length

vertex is , a0 4^ h and focal length is 2a

13. 2x ay a2 2= - or 22

x a ya2 = -d n

14. (a) ,a p q apqT += ^ h6 @ (b) 6y a= -

15. (a) 5a y ax 92 = -^ h Test yourself 11

1. 8 6 29 0x y+ - = 2. 4 8 4 0x x y2 - - - =

3. Centre , ,3 1^ h radius 4 4. (a) ,1 3-^ h (b) 4, 3-^ h 5. (a) ,8 8^ h (b) 2 8 0x y- - =

6. 25x y2 2+ = 7. (a) 2y = (b) ,0 2-^ h 8. 3 10 0x x y y2 2+ + - - = 9. 8 16 16 0x x y2 - + - =

10. (a) (i) ,1 1^ h (ii) ,1 2^ h (b) 0y =

11. 2 3 6 0x y+ + = 12. 14 units

13. 24y x2 = - 14. 8 16 0x y2 - + =

15. ,x y x y4 3 16 0 4 3 14 0- - = - + =

16. ,y x y x= = - 17. 20y x2 = 18. (a) 21

- (b) 2

19. (a) 12x y2 = (b) 32y x2 = -

20. (a) 4 72 0x y- + = (b) ,9 2041d n

21. Sub ,0 4^ h: 7 0 3 4 12 0LHS RHS# #= - + = =

22. ,92

7-d n 23. 3 2 40 0x y- + =

24. 10 100 0x y2 - + = 25. 3 9 0y x a- + =

26. (a) 3 0x y- - = (b) ,R 0 3= -^ h (c) ,F FP FR0 3 6= = =^ h

Answer S11-S12.indd 833 8/2/09 3:03:59 AM


27. (a) 21

0y p q x apq- + + =^ h (b) Sub , a0^ h :

a apq

pq

0

1

+ =

= -

( )a p q apq

apq a

21

0 0#- + + =

= -

28. 2 48 0x y- + =

29. (a) 3 0x y- + = (b) ,6 9^ h and ,2 1-^ h 30. y a= -


1. (a) 8 6 29 0x y+ - = (b) Midpoint of AB lies on line; m m 11 2 = -

2. (a) 2 6 15 0x x y y2 2- + - - = (b) Put 0y = into equation

3. 1 2y x2= - 4. ,221

3-d n 5. (a) ;x y x y4 2 9 0 2 24 0- + = + - =

(b) 1m m1 2 = - (c) , .X 3 10 5= ^ h (d) 3 4 8 0;x y- + = focus ,0 2^ h lies on the line

6. ,0 0^ h 7. (a) ;x y x y2 4 1 0 2 4 0- - = + + =

(b) Point lies on line 1y = -

8. 2 4 2y x x2= - + - 9. 3 2 0x y+ + =

10.

11. (a) 4 10 21 0x x y y2 2+ + - + =

(b) 2 5 8;x y2 2+ + - =] ^g h centre , ;2 5-^ h 28 2radius = =

12. 3

2 3-

13. (a) 4 16 52 0y y x2 + - + = (b) 2 6 0x y- - =

14. 4 2 units 15. 2 2 0x y y2 2+ - - =

16. 696 mm from the vertex

17. ;x y x y141 127 32 0 219 23 58 0+ + = + + =

18. (a) ,Nap aq ap aq

5

6 4

5

3 22 2

=+ +f p

(b) 2 2x a y a2 = +^ h 19. 0y =

20. (a) ,T 6 20= -^ h (b) ,P t s atsa= +] g6 @

(c)

1

1

1

( )

( )

tan

tan

m t m s

m m

m m

tst s

tst s

tst s

ts t ss t ts

t s

ss

t

tst s

ts t ss t ts

t s

ss

t

45

1

11

11

1

11

11

11

1

11

and

or

1 2

1 2

x y

c

i

= =

=+

-

=+

-

=+

-

=+

-

+ = -

+ = -

= -

-

+=

- =+

-

- - = -

- = +

= +

+

-=


1. ≤ , ≥m m2 3 2. 4 3 16 0x y+ - = 3. 8x y2 =

4. 24 cm 5. Centre , ,3 5-^ h radius 7

6. (a) 32

(b) 31

- (c) 191

7. Focus , ,0 2-^ h directrix 2y =

8. 5x = - or 6- 9. 1k = -

10.

180

( equal to opp. interiorin cyclic quadrilateral)

( s supplementary in cyclicquadrilateral)

AFE CBE

CBE EDC

AFE EDC180

ext.

opp.

`

c

c

+ +

+ +

+ +

+

+

+

=

= -

= -

These are supplementary cointerior angles. | CD|AF`

11. ,x y x y3 4 14 0 3 4 16 0- - = - + =

12. Vertex ,4 17- -^ h , focus , .4 16 75- -^ h 13. ,x 0 3= 14. 7.2k cm= 15. 2 2 0x y+ + =

16. b 2$ - 17. 16ci = 18. 16,x y2 2+ = circle centre ,0 0^ h and radius 4

19. 4 6 12 0x x y y2 2+ + + - =

Answer S11-S12.indd 834 7/12/09 3:09:49 AM

835ANSWERS

20. x x y y3 6 17 02 2- + - - =

21. ( in semicircle)(similarly)

( s in same segment)( sum of )

BCDDABDBC DAEBDC DBC BDCBDC DAEDAE BDC

9090

909090

`

`

cc

ccc

+ +++ + ++ + ++ ++ +

D

=

=

=

= -

= -

= -

22. 0.75- 23. 5 54 5 20 79 0x x y y2 2- + + - =

24. , ,a b c2 1 0= = = 25. °, °x y33 57= =

26. 9 x

x2

--

27. (a) 0y px ap2- + = (b) ,Rp

pa

a 12

=-

-_f i p

(c) 2 1 0px p y a ap2 2+ - + - =_ i

28. 4 16 20 0x x y2 - - + =

29. and (given)


AC BC CD CE

CDAC

CEBC

ACB ECD

`

+ +

= =

=

=

since two sides are in proportion and their included angles are equal, Δ ABC is similar to Δ CDE 5.3 cmy =

30. 4 0x y- - =

31. 2 16 15 0x x y2 + - - = 32. ,x 0 2=

33. 04

1 4( 1)( 9)35

0

ab ac2

2

1

1

D = -

= - - -

= -

Since a 01 and 0, 9 0x x21 1D - + - for all x

34. ( )( ) ( )x x x8 3 2 5 3 2 51 3 4+ + +- ( )x x30 7 2 5 3= + +] g

35. sec cosecx x

36.

Let ABCD be a cyclic quadrilateral of circle, centre O . Join AO and CO .

Obtuse 2Reflex

( at centre double at circumference)AOC ADCAOC ABC2 (similarly)

+ ++ +

+ +=

=

Obtuse reflex

It can be proved similarly that

by drawing and .

( of revolution)AOC AOCADC ABC

ADC ABC

BAD BCDBO DO

3602 2 360

180

180

`

ccc

c

+ ++ ++ +

+ +

++ =

+ =

+ =

+ =

opposite angles in a cyclic quadrilateral are supplementary

37. Centre , ,5 3-^ h radius 2

38. ( , )

( )( )

( )( )

DBA x EBC yEDB x DEB yFDE xGED yFGB DBA xGFB EBC yFDE FGBGED GFB

DE ACFDBGEB

180180

180180

Let andThen and

and

alternate sstraightstraight

s in alternate segmentsimilarly

`

cc

cc

+ ++ ++++ ++ ++ ++ +

+

+ +

+ +

+

<

= =

= =

= -

= -

= =

= =

= -

= -

Since opposite angles are supplementary, FGED is a cyclic quadrilateral.

39.

( )( )

ab ac0

41 4 1 3

110

2

2

2

1

D = -

= - -

= -

] g

Since 0a 2 and ,01D x x 3 02 2- + for all x

40. 1k = 41. 3 2 9 0x y+ - =

42. (a) 217 km (b) 153c

43. , ,a b c3 18 34= = - = - 44. ,x x4 32 1

45. (a) 1y x2= - (b) ,4 15-^ h (c) 8 124 0x y- + =

46. ’95 44ci = 47. x 11c=

48. 361 0 and a perfect squareT 2= ^ h 49. 2 9 0x y+ + = 50. k 3#

51.

52. 5 4 41 0x y- - = 53. ,352

252

-d n 54. 3 1

1 3

-

+

55. y141

11#- - 56. 22

3 6 10 3 3 5- + -

Answer S11-S12.indd 835 7/25/09 2:27:11 PM


57. 4.9 , 11.1x ycm cm= = 58. 1x = 59. 8.25 units

60. 4.5 m 61. 2187128

62. °, °, °, °x 60 120 240 300=

63. 2 3 3 0x y+ - = 64. ,y 131

21

= - 65. 162c

66. 9090

( semicircle)( straight )

ACBDCA DCB

in`

c

c

++

+

+ +

=

=

AD is a diameter of the circle

67. °, °, °, °x 45 135 225 315=

68. 1, 2 or , 4x y x y41

41

= - = = - =

69. a b a ab b2 2 42 2+- +] ^g h 70. 43x = 71. 311

-

72. 1.8 units 73. tan i

74. 8 2 5 ( 1) 2( 1)x x x x2 3 2 4+ - + -] g ( ) ( )x x x2 1 9 20 12 3 2= - + -

75. 41

76. 2 3 25 0x x y y2 2+ + - - =

77. Focus (2, 1), directrix 5y =

78. 9 0px y p2- - = 79. 2 36 0x y- - =

80. (a) 21

0y p q x apq- + + =^ h (b) 2 2x a y a2 = -^ h (c) Concave upward parabola, vertex (0, 2 a )

81. (c) 82. (d) 83. (b) 84. (a) 85. (c) 86. (a)

87. (c) 88. (a) 89. (a), (d) 90. (c)

Chapter 12: Polynomials 1

Exercises 12.1

1. (a) 7 (b) 4 (c) 1 (d) 11 (e) 3 (f) 0 (g) 4

2. (a) 19- (b) 10- (c) 1- 3. (a) 6- (b) 5 (c) 2 (d) 1 (e) 2 4. (a) 5 (b) 4 (c) 3- (d) 0

5. (a) 3! (b) 5- (c) ,2 1- (d) 4 (e) 0

6. (a) ;P x x x x12 6 2 4 33 2= - - +l ] g (b) ;P x x10 1=l] g (c) ;P x x x108 35 8 1111 4= - +l] g (d) ;P x x x x7 9 2 7 66 2= - + -l ] g (e) 8; 0P x =l] g

7. (a), (b), (g) 8. (a) 0a = (b) 10b = (c) c 6= -

(d) a 1= - (e) 4a = 9. (a) 221

- (b) ,x 2 1= -

(c) 3 (d) 3 (e) x5

10. (a) b ac48

8 0

2

1

D = -

= -

-

f x` ] g has no zeros

(b) 9x3 (c) 2- (d) 9 (e) ,x32

1= -

11. (a) 2 (b) 0 (c) 2 (d) 0 (e) 2 (f) 4 (g) 3

12. ,x 3 2= - 13. 0, 1x =

14. P x x x3 2 92= - +l] g b ac4 104 02 1- = -

So ( )P xl has no real roots

15. Q x x x3 6 32= - +l] g 4 0b ac2 - =

So Q xl] g has equal roots

Exercises 12.2

1. 3 2 5 4 3 10 45x x x x2 + + = + - +] ]g g

2. 7 4 1 6 2x x x x2 - + = - - -] ]g g

3. 2 1 3 4 14 41x x x x x x3 2 2+ + - = - + + +] ^g h

4. 4 2 3 2 3 2 2 3x x x x2 + - = + - +] ]g g

5. 5 2 3 8 25 2x x x x x x x3 2 2- + + = + - + +^ ] ]h g g 6. 3 2 3 5 7x x x x x x3 2 2+ - - = - + + +] ^g h

7. 5 2 3 1 5 7 10 1x x x x x x x3 2 2- + + = + - + +^ ] ]h g g 8. 2 3

281x x x x

x x x x4 5 18 71

4 3 2

3 2

- - + -

= + - + - +] ^g h

9. 2 5 2 2 5 2 2 2 5x x x x x x x x x4 3 2 2 2- + + - = - - + -^ ^ ]h h g 10. 4 2 6 1 2 1 2 2 4 5x x x x x x3 2 2- + - = + - + -] ^g h

11. 6 3 1 3 2 231

132

x x x x2 - + = - + +] dg n

12. x x x x x x x x2 2 2 2 24 3 2 2 2- - - = - - - + - -^ ^ ]h h g 13. 3 2 3 1x x x x x

x x x x x2 3 8 13 25 49 99

5 4 3 2

4 3 2

- - + - -

= + - + - + -] ^g h

14. 5 2 1 4 6x x x x2 + - = + + -] ]g g

15. 2 5 4 3 3 7 26 82x x x x x x x4 2 3 2- + + = - + + + +] ^g h

16. 2 5 2 2 3 6 12 5x x x x x x x4 3 2 2- + = - + + + +^ ^ ]h h g 17. 3 3 1 5 3 2 14x x x x x x3 2 2- + - = + - + - +^ ] ]h g g 18. 2 4 8 3 2 2 2 12x x x x x x x3 2 2+ - + = + + - + +^ ] ]h g g 19. 2 4 2 5x x x x

x x x x x2 1 4 13 28 18

4 3 2

2 2

- + + +

= + - - + + - +^ ^ ]h h g

20. 3 2 1 1 3 3 2 3x x x x x x x x5 3 4 3 2- + - = + - + - + -] ^g h

Exercises 12.3

1. (a) 41 (b) 3- (c) 43- (d) 9424 (e) 0 (f) 37 (g) 47 (h) 2321 (i) 31 174 (j) 3-

2. (a) 8k = (b) 72

k = (c) k 15 299= (d) k9

68

= (e) k 2!=

3. (a) 0 (b) Yes (c) 4 6 2 2 3x x x x x x3 2 2- + + = - - -] ^g h (d) 2 3 1f x x x x= - - +] ] ] ]g g g g

Answer S11-S12.indd 836 7/25/09 2:27:11 PM

837ANSWERS

4. (a) 3 81 81 81 81 0P - = - - + =] g x 3` + is a factor (b) P x x x x3 32= + -] ] ]g g g

5. ,a b1127

14817

= - = - 6. 6a = -

7. (a) P 3 140 0!=] g x 3` - is not a factor of P x] g (b) 39k = - 8. ,a b2 1= - = -

9. (a) ,a b3 11= = (b) 1 3 8 7 2f x x x x x3 2= + + + +] ] ^g g h (c) g 1 0- =] g (d) 3 2 1f x x x 3= + +] ] ]g g g

10. (a) 2 4x x+ -] ]g g (b) 2 1x x x+ -] ]g g (c) 1 4 2x x x- + -] ] ]g g g (d) 5 3 2x x x+ - +] ] ]g g g (e) 3 1 7x x x- - -] ] ]g g g (f) 2 9 5x x x+ - -] ] ]g g g (g) 3 2x x 2- -] ]g g (h) x x x4 1 2+ +] ]g g (i) 1 2x x 2- +] ]g g (j) 1 3 2x x x+ - +] ] ]g g g

11. (a) 1 3 2P x x x x= - + -] ] ] ]g g g g (b) , ,3 1 2- (c) Yes

12. (a) Dividing f x] g by 5 2x x+ -] ]g g gives 5 2 7 12f x x x x x2= + - + +] ] ] ^g g g h x x5 2` + -] ]g g is a factor of f x] g (b) 5 2 3 4f x x x x x= + - + +] ] ] ] ]g g g g g

13. 1 4 3P x x x x 2= + - +] ] ] ]g g g g 14. (a) P P6 5 0- = =] ]g g (b) 4 6 5P x x x x= - + -] ] ] ]g g g g 15. (a) P u u u2 1 2= - -] ] ]g g g (b) 2, 3x =

16. (a) 1 2 3f p p p p= - + -^ ^ ^ ^h h h h (b) , ,x 0 121

1= -

17. (a) 2 1 1P k k k 2= - +] ] ]g g g (b) , ,x 30 150 270c c c=

18. (a) 1 3 9f u u u u= - - -] ] ] ]g g g g (b) 0, 1, 2x =

19. , ,x 5 4 2= - - -

20. , , , , ,0 90 120 240 270 360c c c c c ci =

21. (a) , , ,a b c d1 3 4 2= = = = - (b) , , ,a b c d1 1 8 12= = - = = - (c) , , ,a b c d2 0 1 6= = = - = (d) , , ,a b c d1 1 11 12= = = = - (e) , , ,a b c d3 0 1 8= = = - = (f) , , ,a b c d1 1 4 7= = = - = - (g) , , ,a b c d5 2 19 43= = - = - = - (h) , , ,a b c d1 4 1 1= - = = - = - (i) , , ,a b c d1 3 6 4= - = = = - (j) , , ,a b c d1 10 27 20= - = - = - = -

22. 12P x x x x3 2= - -] g 23. , ,a b c1 3 6= = - = -

24. 2 4 10 12P x x x x x4 3 2= - - +] g

25. P ( x ) has degree 3. Suppose P ( x ) has 4 zeros, a 1 , a 2 , a 3 and a 4 . Then x a x a x a x a1 2 3 4- - - -_ _ _ _i i i i is a factor of P ( x ) . So P x x a x a x a x a Q x1 2 3 4= - - - -] _ _ _ _ ]g i i i i g . P ( x ) has at least degree 4 But P ( x ) only has degree 3. So it cannot have 4 zeros .

Exercises 12.4

1. (a) y

x- 21 3

6

(b)

y

x-4 2

(c) y

x1 3

(d) y

x-2

Answer S11-S12.indd 837 7/25/09 2:27:12 PM


(e)

2. (a) (i) 4 2P x x x x= - +] ] ]g g g

(ii)

(b) (i) 1 5f x x x x= - - +] ] ]g g g (ii)

y

x-5 1

(c) (i) 1 2P x x x x2= + +] ] ]g g g (ii) y

x-1-2

(d) (i) 2 5 3A x x x x= - +] ] ]g g g (ii)

(e) (i) 3 1P x x x x2= - - +] ] ]g g g (ii) y

x-1 3

3. (a) , ,x 0 1 2= -

(b) y

x-2 1

4. (a) P 2 8 12 8 120

= - - +

=

] g

(b) 2 3 2P x x x x= - - +] ] ] ]g g g g (c) y

x-2 2 3

12

y

x-2 5

50

-5

y

x21

2-3

y

x-2 4

Answer S11-S12.indd 838 7/12/09 3:18:58 AM

839ANSWERS

5. (a) y

x-4 -2

-24

3

(b) y

x-3 -1

-9

3

(c) y

x

12

1 3 4

(d) y

x

12

1 3-4

(e) y

x2 3-3

-18

(f) y

x2-2

-8

(g) y

x1 2

-4

(h) y

x1

3

-3

Answer S11-S12.indd 839 7/12/09 3:22:46 AM


(i) y

x4-2

(j) y

x

1

1-1

Exercises 12.5

1. (a) 3,x = double root (b) 0, 2, 7,x = single roots (c) 0,x = double root, 3,x = single root (d) 2,x = - single root, 2,x = double root (e) 2,x = - triple root (f) , ,x 0 2= single roots,

1,x = double root (g) 1, 3,x = - double roots (h) 0,x = triple root, 4,x = double root (i) 1,x = triple root, 5,x = -

single root (j) 121

,x = triple root

2. (a) (i) Positive (ii) Even (b) (i) Negative (ii) Odd (c) (i) Negative (ii) Even (d) (i) Negative (ii) Odd (e) (i) Positive (ii) Odd (f) (i) Positive (ii) Even (g) (i) Positive (ii) Odd (h) (i) Negative (ii) Even (i) (i) Positive (ii) Odd (j) (i) Positive (ii) Even

3. P x x 4= +2] ]g g Yes, unique

4. (a) P x k x 1= -3] ]g g Not unique (b) 5 1P x x 3= -] ]g g

5. (a) 4 8 16x x x2 2- = - +] g Dividing by 8 16x x2 - + gives 7 8 16 8 16 1x x x x x x3 2 2- + + = - + +^ ]h g so x 4-

2] g is a factor (b) 1 4P x x x 2= + -] ] ]g g g

(c) 4 4 1 4 4P 2= + -] ] ]g g g 0=

( )( ) ( )

P x x xP

3 14 84 3 4 14 4 8

0

2

2

= - +

= - +

=

l

l ] g

6. (a) x x x x3 9 27 273+ = + + +3 2] g

Dividing by 9 27 27x x x3 2+ + + gives 7 9 27 54 9 27 27 2x x x x x x x x4 3 2 3 2+ + - - = + + + -^ ]h g so x 3 3+] g is a factor (b) f x x x2 3= - + 3] ] ]g g g (c) f 3 3 2 3 3- = - - - + 3] ] ]g g g 0=

( )

( ) ( )f x x x x

f4 21 18 27

3 4 3 21 3 18 3 270

3 2

3 2

= + + -

- = - + - + - -

=

l

l ] ]g g

7. (a) P x x k Q x3= -] ] ]g g g where Q ( x ) has degree 3n - (b) P k k k Q k3= -] ] ]g g g 0=

( )P xl u v v u= +l l

( )P kl ( ) ( )( ) ( )

Q x x k x k Q xQ k k k k k Q k

33

0

3 2

3 2

= - + -

= - + -

=

l

l

] ]] ]

g gg g

8. (a) y

x

(b) y

x

Answer S11-S12.indd 840 7/12/09 3:22:47 AM

841ANSWERS

(c) y

x

(d) y

x

(e)

y

x

9. y

x2

10. y

x-1

11. y

x2

12. y

x-3

13. y

x1

Answer S11-S12.indd 841 7/12/09 3:22:48 AM


14. y

x

15. y

x-2

16. y

x4

17. Odd function with positive leading coeffi cient starts negative and turns around at the double root. It then becomes positive as x becomes very large so it must cross the x -axis again. So there is another root at k 12 -

k

y

x-1

18. Even function with negative leading coeffi cient is negative at both ends. The triple root has a point of infl exion so the curve must cross the x -axis to turn negative again. So there is another root at k 22 -

y

xk-2

19. Odd function with positive leading coeffi cient starts negative and turns around at both the double roots. It then becomes positive as x becomes very large so it must cross the x -axis again. So there is another root at k 22

y

xk-3 2

Answer S11-S12.indd 842 7/12/09 3:22:48 AM

843ANSWERS

20. Odd function with negative leading coeffi cient starts positive and turns around at the double root. It then becomes negative as x becomes very large so it must cross the x -axis again. So there is another root at k 12

y

xk1

Exercises 12.6

1. (a) (i) 2 (ii) 8 (b) (i) 2- (ii) 32

- (c) (i) 7- (ii) 1

(d) (i) 241

(ii) 3- (e) (i) 3- (ii) 0 2. (a) (i) 1- (ii) 2-

(iii) 8- (b ) (i) 3 (ii) 5 (iii) 2 (c) (i) 21

(ii) 3

(iii) 1- (d) (i) 3- (ii) 0 (iii) 11- (e) (i) 0 (ii) 7 (iii) 3

3. (a) (i) 2- (ii) 1- (iii) 1 (iv) 5 (b) (i) 1 (ii) 3-

(iii) 2- (iv) 7- (c) (i) 1 (ii) 3- (iii) 2- (iv) 4-

(d) (i) 1 (ii) 2- (iii) 121

- (iv) 1- (e) (i) 6 (ii) 0

(iii) 0 (iv) 321

4. (a) 5 (b) 5- (c) 1- (d) 35

(e) 200 5. (a) 23

(b) 21

- (c) 25

- (d) 31

-

(e) 21

-

6. (a) 3- (b) 5- (c) 132

7. 26k = -

8. ,2 7a b ab+ = = - 9. ,221

21

a b ab+ = = -

10. (a) 0k = (b) 4k = (c) ±1k = (d) ,k21

1= -

(e) 0k = 11. m 9= - 12. ,a b183

941

= - = -

13. (a) 1 0P =] g (b) ,1 6a b c abc+ + = = -

14. ;a 1 2a b= + = - 15. (a) 154

(b) ,p q8154

17152

= = - 16. 1 17. 5-

18. ,x21

121

= - 19. ,x31

21

!= 20. , ,x 3 121

32

!= -

Test yourself 12

1. 3 3 5 1p x x x x x= + - + -] ] ] ] ]g g g g g 2. (a) 3 (b) 9 (c) 1 (d)

91

3. ( ) ( ) ( )( )P x x x xx x x

6 1 25 8 123 2

= - - +

= - - +

4. (a) 3 2x x2 + + (b) 5 3 1 2p x x x x x= - + + +] ] ] ] ]g g g g g 5. (a) 3 (b) 3- (c) , ,3 0 1- (d) x3

6.

7. (a) 3a = (b) 5-

8. ( ) ( )p 7 7 7 7 5 7 4725 0

3 2

!

- = - - - + - -

= -

] ]g g

9. ,x 1 3!= - 10. , ,a b c2 18 40= = - =

11. x -intercepts , , ;3 2 4- y -intercept 24

12. ( ) ( )

x x xx x x x x

3 7 8 52 3 6 5 18 36 67

5 3 2

4 3 2

- + -

= - + + + + +

13. 60 , 90 , 180 , 270 , 300x c c c cc= 14. 7.4k =

15. 4, 5

16.

17. 14k = - 18. 4

Answer S11-S12.indd 843 7/12/09 3:22:49 AM


19. ( ) ( )

( ) ( ) ( )( ) ( ) ( )

P a A a a a

P x A x x a A x x aP a A a a a A a a a

033

0

3

2 3

2 3

= -

=

= - + -

= - + -

=

l l

l l

]] ]] ]

gg gg g

20. f 5 5 6 5 12 5 35 03 2= - + - =] ] ]g g g

21. (a) f 5 5 7 5 5 5 753= - - +2] ] ]g g g 0=

(b) f x x x3 14 52= - -l] g

( ) ( )f 5 3 5 14 5 50

2= - -

=

l ] g

(c) Double root at 5x = (d) f x x x3 5= + - 2] ] ]g g g

22.

3

y

x

23. (a) P x x Q x6= + 3] ] ]g g g (b) y

x-6

24. y

x

25. (a) , , ,a b c d2 3 4 5= = - = =


1. 1 1 1P x x x x x2 2= - + + +] ] ] ^g g g h 2. (a) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

P b b b Q b

P x x b Q x Q x x bP b b b Q b Q b b b

077

0

7

7 6

7 6

= -

=

= - + -

= - + -

=

l l

l l

]] ]] ]

gg gg g

(b) ,a b 17= - = -

3. , , , , , , , ,0 45 60 120 180 225 240 300 360c c c c c c c c ci =

4. (a) 3 2 0x y- + = (b) ,2 8^ h 5. (a) 4

33a -

(b) a 14= -

6. (a) 3- (b) 17 7. 90 , 210 , 330c c ci = 8. 5a = -

9. If x a- is a factor of P x] g

( ) ( ) ( )( ) ( ) ( )

P x x a Q xP a a a Q a

0

Then`

= -

= -

=

10. , , ,1 1 3 5- - -^ ^h h 11. P x x x1 2= - + -2 3] ] ]g g g

12. y

xa1 a2

Answer S11-S12.indd 844 7/12/09 3:22:50 AM

845ANSWERS

Chapter 13: Permutations and combinations

Exercises 13.1

1. 10 000

1 2.

3316

3. 92

4. 20 000

1

5. 98.5% 6. (a) 74

(b) 73

7. 203

8. 31

9. (a) 61

(b) 31

(c) 65

10. (a) 621

(b) 313

(c) 21

(d) 12499

11. (a) 151

(b) 158

(c) 53

12. 8

13. (a) 8629

(b) 4319

(c) 8667

14. 32

15. (a) 61

(b) 21

(c) 31

(d) 21

(e) 21

16. (a) 185

(b) 91

17. (a) 54

(b) 36 18. (a) 4423

(b) 4421

19. 196

20. 982329

21. (a) 3119

(b) 3120

(c) 314

(d) 1131

22. (a) 5914

(b) 5935

(c) 2459

(d) 5938

23. 245

24. 19% 25. 0.51

Exercises 13.2

1. 456 976 2. 67 600 3. 26 105 4# 4. 260

5. 26 1010 15# 6. 1 000 7. 1 000 000

8. 300 9. 64 10. 10 000

3

11. (a) 84 (b) 841

12. (a) 10 000 000 (b) 1000

13. Yes 14. 67 600 000

1 15. 7 16. Yes

17. 5184

1 18. 6 19. 6840 20. 360

21. 7 880 400 22. 210 23. 271 252 800

24. (a) 9900 (b) 9900

1 25.

7201

Exercises 13.3

1. (a) 720 (b) 3 628 800 (c) 1 (d) 35 280 ( e) 120 (f) 210 (g) 3 991 680 (h) 715 (i) 56 (j) 330

2. 362 880 3. 720 4. 479 001 600 5. 120

6. (a) 39 916 800 (b) 479 001 600 7. 40 320

8. 5040 9. 6 10. 720 11. 5040

12. 1.3 1012# 13. (a) 39 916 800 (b) 3 628 800

14. (a) 720 (b) 120 (c) 48 15. 5040

16. (a) 41

(b) 241

17. (a) 479 001 600 (b) 121

18. 120

1

19. 6 227 020 800

20. (a) !! 8 7 6 ... 2 1

48

4 3 2 18 7 6 5

# # #

# # # # #

# # #

=

=

(b) !! 11 10 9 ... 2 1

611

6 5 4 3 2 111 10 9 8 7# # # # #

# # # # #

# # # #

=

=

(c) .

.

.

.!!

...

... ...

...( 1)( 2) ... ( )

rn

r r r

n n n r r r

n n n rn n n r

1 2 3 2 1

1 2 1 1 3 2 1

1 2 11

# # # #

# #

=- -

- - + -

= - - +

= - - +

] ]] ] ] ]] ] ]

g gg g g gg g g

(d) . .

. .

!!

...

... ...

...n r

nn r n r n r

n n n n r n r

n n n n r1 2 3 2 1

1 2 1 3 2 1

1 2 1-

=- - - - -

- - - + -

= - - - +

] ] ] ]] ] ] ]] ] ]

g g g gg g g gg g g

Exercises 13.4

1. (a) 6 3 !

6!120

-=] g (b)

5 2 !5!

20-

=] g (c) 8 3 !

8!336

-=] g

(d) !

!10 7

10640 800

-=] g (e)

!!

9 69

60 480-

=] g

(f) 7 5 !

7!2520

-=] g (g)

!!

8 68

20 160-

=] g

(h) !

!11 8

116 652 800

-=] g (i)

9 1 !9!

9-

=] g

(j) 6 6 !

6!720

-=] g

2. (a) 650 (b) 15 600 (c) 358 800 (d) 7 893 600

3. (a) 648 (b) 432 (c) 144 4. (a) 20 (b) 4 (c) 12 (d) 8

5. (a) 24 (b) 24 6. (a) 4536 (b) 2016 (c) 3528

7. (a) 120 (b) 48 (c) 96 (d) 72 (e) 60

8. (a) 479 001 600 (b) 1320

9. (a) 56 (b) 336 (c) 1680

10. (a) 60 480 (b) 2520 (c) 907 200 (d) 151 200 (e) 60 (f) 453 600 (g) 360 (h) 2520 (i) 59 875 200 (j) 90 720

11. (a) 24 (b) 5040 (c) 40 320 (d) 3 628 800 (e) 39 916 800

12. (a) 6 (b) 720 (c) 5040 (d) 362 880 (e) 3 628 800

Answer S13.indd 845 7/25/09 2:38:37 PM


13. (a) 181 440 (b) 19 958 400 (c) 20 160 (d) 1 814 400 (e) 239 500 800

14. (a) 720 (b) 120 15. (a) 362 880 (b) 40 320

16. (a) 3 628 800 (b) 362 880 (c) 181 440

17. (a) 24 (b) 12 (c) 24

18. (a) 720 (b) 240 (c) 480 (d) 144

19. (a) 3 628 800 (b) 362 880 (c) 28 800

20. 92

21. (a) 20! (b) 5!8!7!3! (c) 207

22. (a) 60 (b) 48 (c) 36 (d) 51

23. 16

24. 1

336 25. (a) 40 320 (b) 30 240 (c) 21 600

26. (a) 20 (b) 60 27. (a) 720 (b) 120 (c) 192

28. (a) x ! (b) 1 !x -] g (c) 2! 2 !x -] g (d) ! !x3 2-] g (e) 3 2 !x x- -] ]g g

29. (a) ! !

!!

!!

!

P

3 38 3

8

58

3

8

3

'

=-

=

] g

8

8 8

!!

!

! !!

! !!

!

!!

!

!!

!

! !!

! !

P

P P

58

31

5 38

5 58 5

8

38

5

38

51

5 38

3 5

5

3 5

#

'

#

=

=

=-

=

=

=

` =

] g

(b)

n

! !!

!

!!

!

!!

!

! !!

! !

( ) !!

r

P

rn rn

n rn

r

n rn

r

n r rn

n r

P

n r

n n rn

1

n

r

n r

'

#

=-

=-

=-

=-

-=

-

- --

]

]

]

]

] ]

g

g

g

g

g g5 ?

`

n n

!!

( ) !

!!

!

! !!

! !

n n rn

n r

rn

n r

n r rn

r

P

n r

P

1

r n r

'

#

=- +

-

=-

=-

=-

-

]

]

]

]

g

g

g

g

30. n

n

n

1+

n

!

!

!!

!

!

!!

!

!

!!

!!

!

!

!!

!

!

!!

!! ! ! !

!! !

!

!

!

!

Pn r

n

P r Pn rn

rn r

n

n rn

rn r

n

n rn

n rrn

n r n r

n r n

n rrn

n r

n r n

n rrn

n rn n n rn rn

n rnn n

n r

n n

n r

n

P P r P

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

r

rn

r

n

r r r

1

1

1

$

`

=+ -

+

+ =-

+- -

=-

+- -

=-

+- +

=+ - -

+ -+

+ -

=+ -

+ -+

+ -

=+ -

+ - +

=+ -

+

=+ -

+

=+ -

+

= +

-

+

-

]]

] ^] ^] ]

] ]]

]

]]

]

]

]

]]

]]

gg

g hg hg g

g gg

g

gg

g

g

g

gg

gg

5

5

?

?

Exercises 13.5

1. (a) 9 5 !5!

9!126

-=] g (b)

12 7 !7!12!

792-

=] g

(c) ! !

!8 3 3

856

-=] g (d)

10 4 !4!10!

210-

=] g

(e) 11 5 !5!

11!462

-=] g

2. (a) (i) 1 (ii) 1 (iii) 1 (iv) 1 (v) 1 (b) (i)

nC 10 = (ii)

nC 1n =

3. (a) 28 (b) 84 (c) 462 (d) 5005 (e) 38 760

4. (a) Number of arrangements 15=

R1R2 R2R3 R3B1 B1B2 B2B3

R1R3 R2B1 R3B2 B1B3

R1B1 R2B2 R3B3

R1B2 R2B3

R1B3

(b) 77 520

Answer S13.indd 846 7/11/09 2:47:39 PM

847ANSWERS

5. 15 504 6. 210 7. 2 598 960

8. (a) 720 (b) 120 9. (a) 2184 (b) 364

10. 296 010 11. 4845 12. 2925

13. 23 535 820 14. (a) 792 (b) 125

(c) 335

15. (a) 100 947 (b) 462 (c) 924 (d) 36 300 (e) 26 334 (f) 74 613 (g) 27 225

16. $105

17. (a) 2 042 975 (b) 55 (c) 462 462 (d) 30 030

18. (a) 3003 (b) (i) 2450 (ii) 588 (iii) 56 (iv) 1176

19. (a) 1.58 1010# (b) 286 (c) 15 682 524 (d) 5 311 735

(e) 12 271 512

20. (a) 395 747 352 (b) 32 332 300 (c) 4 084 080 (d) 145 495 350 (e) 671 571 264

21. (a) 170 544 (b) 36 (c) 20 160 (d) 17 640 (e) 6300

22. (a) 7 (b) 27 132 (c) 13 860 (d) 20 790 (e) 27 720

23. (a) 5 (b) 360 (c) 126

24. (a) 792 (b) 792

(c) 12

12

12

! !!

! !!

! !!

! !!

C

C

C C

12 5 512

7 512

12 7 712

5 712

5

7

125 7`

=-

=

=-

=

=

]

]

g

g

25. 9

8

9

8

8 8

C

C C

C C C

84

28 5684

6

6 5

6 6 5`

=

+ = +

=

= +

26. ! !

!

! !!

! !!

! !!

137 13 7 7

13

6 713

136 13 6 6

13

7 613

137

136`

=-

=

=-

=

=

b ]

b ]

b b

l g

l g

l l

27. ! !

!

! !!

! !!

! !!

! !!

! !!

! !!

! !!

! !!

! !!

! !! !

! !!

! !!

104 10 4 4

10

6 410

94

93 9 4 4

99 3 3

9

5 49

6 39

6 5 46 9

4 6 34 9

6 46 9

6 44 9

6 46 9 4 9

6 410 9

6 410

104

94

93

#

#

#

#

# #

# #

#

`

=-

=

+ =-

+-

= +

= +

= +

=+

=

=

= +

b ]

b b ] ]

b b b

l g

l l g g

l l l

28. ! !

!

( ) ! !!

! !!

! !!

nr n r r

n

nn r n n r n r

n

n n r n rn

r n rn

nr

nn r`

=-

-=

- - -

=- + -

=-

=-

b ]b ]

] ]

]b b

l gl g

g g

gl l

5 ?

29.

!r

!r

n

!!

!! !

!

!!

Pn rn

C rn r r

n

n rn

P

r

nr

nr

nr

#

`

=-

=-

=-

= C

]

]

]

g

g

g

30. ! !

!nk n k k

n=

-b ]l g

! !

!

! !

!

! !

!

! !

!

! !

!

! !

!

! !

!

! !

!

! !

!

! !

!

! !!

nk

nk n k k

n

n k k

n

n k k

n

n k k

n

k n k k

k n

n k n k k

n k n

n k k

k n

n k k

n k n

n k k

k n k n

n k k

n n

n k kn

nk

11

11 1 1

1

1

1

1

1

1

1

1

1

1

1

1 1

1

1

--

+-

=- - - -

-+

- -

-

=- -

-+

- -

-

=- -

-+

- - -

- -

=-

-+

-

- -

=-

+ - -

=-

-

=-

=

b b ]] ]]

]]

] ]]

]]

] ]]

] ]] ]

]]

]] ]

]] ]

]]

]b

l l gg gg

gg

g gg

gg

g gg

g gg g

gg

gg g

gg g

gg

gl

Answer S13.indd 847 7/11/09 2:47:47 PM


Test yourself 13

1. (a) 5040 (b) 720 2. (a) 114

(b) 2213

(c) 2217

3. (a) 24 (b) 12 4. (a) 720 (b) 120

5. (a) 65 780 (b) 25 200 (c) 252 6. 29%

7. 120 8. 2.4 1018# 9.

91

10. 142 506

11. 990 12. (a) 40 320 (b) 362 880 (c) 80 640 (d) 168

13. (a) 19 958 400 (b) 4 989 600 (c) 181 440 (d) 9 979 200 (e) 181 440

14. ! !

!nn k k

nk =

-b ]l g 15. (a) 151 200 (b) 881 280

16. 1.08 1017#

17. (a) 1 709 316 (b) 203 490 (c) 167 580

18. (a) 15 (b) 181 440 19. 37 015 056

20. (a) 1

(b) n

! !!

! !!

! !!

! !!

nn

nn

nn n n n

n

nn

n nn

0 0

01

01

0`

=-

=

=

=-

=

=

=

0c ]

c ]

c c

m g

m g

m m


1. (a) 60 (b) 72 (c) 30 2. (a) 360 (b) 60

3. ! !

!nk n k k

n=

-b ]l g

( ) ! !

!

! !

!

! !

!

! !

!

! !

!

! !

!

! !

!

! !

!

! !

! !

! !

!

! !

!

! !!

nk

nk n k k

n

n k k

n

n k k

n

n k k

n

k n k k

k n

n k n k k

n k n

n k k

k n

n k k

n k n

n k k

k n n k n

n k k

n k n k

n k k

n n

n k kn

11

11 1 1

1

1

1

1

1

1

1

1

1

1

1

1 1

1 1

1

1

--

+-

=- - - -

-+

- -

-

=- -

-+

- -

-

=- -

-+

- - -

- -

=-

-+

-

- -

=-

- + - -

=-

- + -

=-

-

=-

b b ]]

]]

] ]]

]]

] ]]

] ]] ]

]]

]] ]

]] ] ]

]]

]]

]

l l gg

gg

g gg

gg

g gg

g gg g

gg

gg g

gg g g

gg

gg

g

5

5

?

?

nk

nk

nk

11

1` =

--

+-b b bl l l

4. (a) 1 !n -] g (b) !

!k

n k 1- +] g 5. (a) 90 720 (b) 246

6. (a) 792 (b) 445

7. n

!!

! !! !

!

!!

!

Pn r

n

r C rn r r

n

n rn

P r C

r

nr

nr

nr`

=-

=-

=-

=

]]]

ggg

8. (a) 1 860 480 (b) 41

(c) 403

(d) 4021

9. (a) 94 109 400 (b) 7920 (c) 5 527 200 (d) 93 024 (e) 37 643 760 (f) 23 289 700

10. (a) 354

(b) 3517


1. 21

2. 1 1 4P x x x x= - + +] ] ] ]g g g g 3. 3y x4=

4. (a) 362 880 (b) 4320 (c) 282 240 5. ,1 2-

6. ,19 10^ h 7. (a) 4- (b) −2 (c) 43

(d) 10

8. 2;x y2 2+ = circle centre ,0 0^ h radius 2

9. ;3060 161 3c c l

10. Distance from centre ,0 0^ h to line is

| |d

a b

ax by c

1040

4radius

line is tangent

2 2

1 1

`

=+

+ +

=

=

=

11. k 221

= -

12. x 74c= ( s+ in alternate segment)

( )

( )y 180 74 2

53 sum in isosceles'c c

c + D

= -

=

13. 120 14. ,x x2 21 2-

15. ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )

P x x Q xP x x Q x x Q xP Q

P Q Q

22 2 2

2 2 2 20

2 2 2 2 2 2 2 20

= -

= - + -

= -

=

= - + -

=

2

2

2

2

l l

l l

] ] ]]]]

g g gggg

16. (a) 1 (b) 3 (c) ,101

0ab a b= - + =

17. 126 18. 7 19. 7.1 m 20. 131 38c l

Answer S13.indd 848 7/25/09 2:38:41 PM

849ANSWERS

21. (a) 1 3P x x x 2= - -] ] ]g g g (b) y

x31

-9

22.

23. P

x x x xP

2 55

2 2 11 232 is the remainder

2

`

- = -

+ - +

-

P 55 on division= -

]] ] ^]

gg g hg

24. (a) 0ad

abc acd bcd abd+ + + = - = (b) 1 (c) 1-

25.

( ) ( ) ( ) ( )( ) ( ) ( )

P x x Q xP Q

Q

P x x Q x x Q xP Q Q

Q Q

33 3 3 3

0 302 3 3

3 2 3 3 3 3 3 32 0 3 0 30

2

= -

= -

=

=

= - + -

= - + -

= +

=

2

2

2l l

l l

l

] ] ]] ] ]

]] ] ] ] ]

]

g g gg g g

gg g g g g

g

26. 1 884 960 27. Radius 3; 9x y2 2+ =

28. , ,a b c3 14 9= = - =

29. (a) 8.1 m (b) 35 46c l

30. (a) 1!

(b) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

P

P x x x x x x

P

1 1 1 1 50

6 1 5 1 2

1 6 1 1 1 1 5 1 1 2 10

2 3 2

2 2 2 2 3

2 2 2 2 3

$

$

= - +

=

= - + + -

= - + + -

=

l

l

^

^ ^^ ^

h

h hh h

31. Domain: all real x ; range: y 3$ -

32. )|ED(

,

ACB ECDABC CEDAC CD

ABC CDEby AAS

vertically opposite anglesalternate angles

given

AB|

`

+ ++ +

/D D

=

=

=

^^

hh

33. 46 m 2 34. 3 0x y+ - =

35. x x x12 36 62 2- + = -] g 36. 41 38c l

37. . , .y y2 5 6 5$ # -

38.

39. ,174

771

-d n 40. (a) x y9 16 0- + = (b) x y9 20 0+ + =

(c) ,Q 20 0= -^ h (d) 27 21c l

41. Domain: all real ;x 2!! range: all real y

42. (a) sin a b-^ h (b) 45cos2

1c= (c)

83 1

2+^ h

43. (a) 149.1 m (b) 46 48c l 44. 7.5,17.5X = ^ h 45. 3 46. , .x x1 1 61 2 47. x 150c=

48. (a) 8 129 0x y- + = (b) ,R 781

17641

= d n

49. t

t t t t

1

2 6 2 12

4 3 2

+

+ - + +2^ h

50. f x x x xf

3 7 5 33 3 3 7 3 5 3 3

81 63 15 30

3 2

3

= - - -

= - - -

= - - -

=

2

]] ] ] ]gg g g g

So x 3- is a factor of f x x x x3 7 5 33 2= - - -] g

51. , ,a b c1 3 1= = - = -

52. ,x y x y3 1 0 3 7 0- - = + - =

53. 2175 cm 3 ; 1045 cm 2

54. ,y y3 2 11 1 1- - - 55. 6556

56. (a) ,, ,x 60 120 240 300c c c c= (b) , ,x 0 90 360c c c=

(c) x 270c=

Answer S13.indd 849 7/11/09 2:47:51 PM


57. (a) 0

(b)

58. n90 1 135# #c ci = + - n] g 59. 8 8c l

60. . , .x y6 5 2 8= - = 61. y x4= -

62. (a) 4 (b) −2 (c) −3 (d) 121

(e) 22

63. 2 1 5P x x x2= - + +] ] ^g g h 64. 15 504

65. P x x x5 1= - - + 2] ] ]g g g 66. 63 67. (a)

68. (b) 69. (c) 70. (a), (b), (d)

71. (b) 72. (b) 73. (a) 74. (d)

75. (d)

Answer S13.indd 850 7/11/09 2:47:55 PM

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