Answers for maths in focus 3u

8/20/2019 Answers for maths in focus 3u

1/95

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) 215

7

(h) 1

20

19 (i) 2 (j) 3

3

1

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. 215

4- 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) 5

201 (d) 11

54

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

3. (a)50

1 (b)

8

3 (c)

1000

1 (d) 1

1000

97

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

5. (a) 35% (b) %333

1 (c) %226

3

2 (d) 0.1%

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

7. (a) . ;0 5225

13 (b) . ;0 07

100

7 (c) . ;0 168

125

21

(d) . ;1 09 1 100

9

(e) . ;0 434 500

217

(f) . ;0 1225 400

49

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)9

8 (b)

9

2 (c) 1

9

5 (d) 3

9

7 (e)

99

67 (f)

11

6 (g)

45

7

(h)60

13 (i)

990

217 (j) 1

495

37

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

11. (a)8

5 (b) 2

8

1 (c)

11

8 (d) 2

18

7 (e)

45

4

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

Exercises 1.4

1.20

3 2.

20

7 3. (a)

20

17 (b)

10

7 (c) 1

20

1

(d)28

3 (e)

5

3 4. $547.56 5. 714.3 g 6.

24

7 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

511

5 minutes after 1 o’clock.

Exercises 1.5

1. (a) 500 (b) 145 (c)64

1 (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) y 6 (h) x21 (i) x4 10 (j) y 81 8- (k) a (l) y

x45

10

(m) w 10 (n) p5 (o) x 3- (p) a ba

bor2 3

2

3-

(q) x y x

y or5 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) 9x 22 (j) x 21

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

a8

8

(c)b

a6412

3

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

(f)x 4 y

10 (g)k

27

2 23 (h) 16

y 47 (i)

a 3 (j)

x y 125 21 18-


2/95

757 ANSWERS

6. 42

1 7. 324 8. 2

27

10 9. (a) a 3 b (b)

25

1

10. (a) pq 2 r 2 (b)32

7 11.

9

4 12.

18

1 13.

27

4

14.81

1 15.

108

1 16.

12

1 17.

2

558

22

18. 3888

49

Exercises 1.6

1. (a)27

1 (b)

4

1 (c)

343

1 (d)

10 000

1 (e)

256

1 (f) 1

(g)32

1 (h)

81

1 (i)

7

1 (j)

81

1 (k)

64

1 (l)

9

1 (m) 1

(n)36

1 (o)

125

1 (p)

100 000

1 (q)

128

1 (r) 1

(s)64

1 (t)

64

1

2. (a) 1 (b) 16 (c) 12

1 (d) 1

25

11 (e) 1 (f) 125 (g) 1

3

1

(h) 49 (i) 383 (j) 32 (k) 2

31 (l) 1 (m) 1

3613 (n) 1

8119

(o) 1 (p) 16 (q) 158

5- (r)

23

7- (s) 1 (t)

25

16

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)2

1z 6- or

2

z 6- (j)

5

3t 8- (k)

7

2x 1-

(l)2

5m 6- (m)

3

2 y 7-

(n) 3 4x 2+ -] g (o) a b 8+ -] g (p) 2x 1- -] g (q) 5 1 p 3+ -^ h (r) 2 4 9t 5- -] g (s)

4

1x 11+ -] g (t)

9

5 3a b 7+ -] g

4. (a)1

t 5 (b)1

x6 (c)1

y 3 (d)1

n8 (e)1

w 10 (f)2x (g)

3

m4

(h)5

x7 (i)

8

1

x3 (j)

4

1

n (k)

1

1

x 6+] g (l) 81

y z+

(m)3

1

k 2-] g (n) 3 21

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)

2

1 (t)

16

1

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

3. (a) y 3 (b) y y or23 32_ i (c) 1

x (d) 2 5x +

(e)3 1

1

x - (f) 6q r 3 + (g)

x x7

1

7

1or

25 5 2+ +] ^g h

4. (a) 2t 1

(b) 5 y 1

(c) 2x3

(d) x9 31

-] g (e) 2s4 1+ 1] g (f) 2t 2 3+

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

(i) 3x 2--

2] g (j) 2 y 2

17+

-1^ h (k) - 3x5 4+ 1] g

(l) y 3

21

21

-

-2

a k (m) x

5

32

43

+

-2

_ i

5. (a) x23

(b) x 21

-

(c) x32

(d) x35

(e) x45

6. (a) x x x2 23

+ +2 (b) a b3

2

3

2

- (c) p p p2 21

+ +1-2

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

2

3

25

- +

- - -

7. (a)

2

1

a b3

-

(b)

3

1

y 2

3 -^ h (c)

6 1

4

a47

+] g

(d)3

1

x y 54 +^ h (e) 7 3 86

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) 3a

(e) −3a (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 y 2 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 k2

1

2

1for and for2 1+ - - - -

(h) 5 2 5 2x x x x5

2

5

2for 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! !


3/95


Test yourself 1

1. (a)20

9 (b) 0.14 (c) 0.625 (d)

200

157 (e) 1.2%

(f) 73.3% 2. (a)49

1 (b)

5

1 (c)

3

1

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) 25 y 6 (c) a b11 6 (d)27

8x18 (e) 1 6. (a)

40

29

(b) 37

1 (c) 12 (d) 2

2

1 (e) 12

2

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

(d)64

1 (e) 4 (f) 3 (g)

7

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

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

2

1x 3- (h) 3x

4

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

10. (a) 1a5

(b) n4 (c) 1x + (d) 1x y -

(e)4 7

1t 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)

12

7 (c)

8

1

(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) 3 y 7

20. (a) 1.3 10 5# - (b) 1.23 1011# 21. (a) 9

7 (b)

330

41

22. (a)

1

x3 (b) 2 5

1

a + (c) a

b 5

c m 23. 14 500 24. | | ,2 5 7LHS = - + - = | | | | .2 5 7RHS = - + - =

So | | | | | |a b a b#+ + since .7 7#

Challenge exercise 1

1. 4303

278 2. 1

18

11 3. . , %, , .0 502 51

99

510 5o

4. 533

1% 5.

16

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

9990

3271

9. 18 h 10. 1.98

11.

2 2 1 2

2 2 22 2 2

2 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. −24.35 13. . , , . , ,0 34 2 1 5 07

3-

o 14. 63

2%

15. ,x

xx

x1

11

1

11when when2 1

--

-- 16. 0.73

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

20. | | | | | | , , ;

| | | | | | , , ;

a b a b a b a b

a b a b b ba

0 0 0 0

0 0 0 0

when or

when ora

2 2 1 1

1 2 1 1 2

+ = +

+ +

| | | |a b a b`

#+ +

for all a , b

Chapter 2: Algebra and surds

Exercises 2.1

1. 7x 2. 3a 3. z 4. 6a 5. 3b 6. −3r

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

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

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

23. m m6 122 - + 24. 2 6 p 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 y 3 2 2 3- + + 30. 3 7 6x x x3 2+ - -

Exercises 2.2

1. b10 2. xy 8 3. p10 2 4. wz6-

5. ab15 6. xyz14 7. abc 48 8. d 12 2

9. a12 3 10. y 27 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. t 81 12

19. 14m11- 20. x y 24 6 3

Exercises 2.3

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

y 7. 3 p

8.2

ab 9.

3

4

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

3

1

ab2 13.

2qs

-

14.3

2

c d 2 15.

x

z

2 2

2

16. 6 p q4 17. 4c

a b4 7 18.

2a

b6

19.3 y

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 y 63 2 2 3+ 10. k4 7+ 11. t 2 17-

12. y y 4 112 + 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. y 7 4- + 23. b2 24. t 5 6- 25. a2 26+

Exercises 2.5

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

4. 6 8m m2 - + 5. 7 12x x2 + + 6. 3 10 y y 2 - -

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

10. 15 17 4a a2 - + 11. 8 6 9 y y 2 + - 12. 7 4 28xy x y + - -

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

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

20. 36 y 2 - 21. 9 1a2 - 22. 4 49z2 -


4/95

759 ANSWERS

23. 2 11 18 18x xy x y 2 - + - + 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 49 y y 2 - +

31. 4 12 9x x2 + + 32. 4 4 1t t 2 - +

33. 9 24 16a ab b2 2+ + 34. 10 25x xy y 2 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 t 2 + + 2. 12 36z z2 - + 3. 2 1x x2 - +

4. 16 64 y y 2 + + 5. 6 9q q2 + + 6. 14 49k k2 - +

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

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

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

16. 1 p2 - 17. 36r 2 - 18. 100x2 - 19. 4 9a2 -

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

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

xx

2

2+ +

27.1

aa

2

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

29. 2 2 2 2a b a b c c a ab b ac bc c 2 2 2 2 2+ + + + = + + + + +] ]g g 30. 1 2 1 2 1 2 2x x y y x x xy y y 2 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 y 4 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. y 2 3+^ h 2. x5 2-] g 3. m3 3-] g 4. x2 4 1+] g 5. y 6 4 3-^ h 6. xx 2+] g 7. m m 3-] g 8. y y 2 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 y 5 2- +^ h 16. q q3 2

2 3

-_ i 17. b b5 32

+] g 18. a b ab3 22 2 -] g 19. 5)( 7)(m x+ + 20. 1 2 y 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 y 4 22+ + 34. 1a- +] g 35. ( ) ( )a ab1 4 32 + -

Exercises 2.8

1. 4 2x b+ +] ]g g 2. 3 y 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. 2 y x x y - +^ ^h h 9. 1 1 y 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 4 y 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 r 2 3r+ -] ]g g

Exercises 2.91. 3 1x x+ +] ]g g 2. 4 3 y 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 3 y y - -^ ^h h 11. 6 3m m- -] ]g g 12. 12 3 y y + -^ ^h h 13. 8 3x x- +] ]g g 14. a 2 2-] g 15. 2 16x x- +] ]g g 16. 4 9 y y + -^ ^h h 17. 6 4n n- -] ]g g 18. x 5 2-] g 19. 9 1 p 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 1 y 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 2 y y - +^ ^h h 8. 2 3 4x x+ +] ]g g 9. 5 2 3 p 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 8 y 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 r 4 1 2 6 4 12 3- + - +=] ] ] ]g g g g 20. 2 5 2 3x x- +] ]g g 21. 6 1 2 y y - -^ ^h h 22. 2 3 3 2 p 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 3 y 2+^ h 28. 5 2k 2-] g 29. 6 1a 2-] g 30. 7 6m 2+] g


5/95


Exercises 2.11

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

2

1t

2

+d n 17. 3

2x

2

-d n 18. 35

1 y

2

+d n 19.

1x

x

2

+c m 20. 5 2kk

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.

2

1

2

1x x+ -d dn n

20.3

13

1 y 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 y 3 3+ -_ _i i 24. 4 2 2x y x y x y 2 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 t 2+ - +] ^g h 4. 4)( 4 16)(a a a2- + + 5. 1 )(1 )( x x x2- + + 6. 2 3 4 6 9 y y y 2+ - +^ _h i 7. ( )( ) y z y yz z2 2 42 2+ - + 8. 5 )( 5 25 )(x y x xy y 2 2- + +

9. 2 3 4 6 9x y x xy y 2 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 2

39

x 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 y 2 2+ - + + + + +^ _h i 15. xy z x y xyz z5 25 306 362 2 2+ - +

^ _

h i 16. a a 19 2 - +- ^ h

17. 13

13 9

x 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 y 2 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 y 5 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 4 y 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 xy y 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 y 2 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 4 y y y 2 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 t 2 2- + = -] g 9. 20 100 10x x x2 2- + = -] g 10. 44 484 22w w w 2 2+ + = +] g 11. 32 256 16x x x2 2- + = -] g 12. 3

4

9

2

3 y y y 2

2

+ + = +d n

13. 74

49

2

7x x x2

2

- + = -d n 14. 4

1

2

1a a a2

2

+ + = +d n

15. 94

81

2

9x x x2

2

+ + = +d n 16. 5 y y y 16

25

4

5

22

2

- + = -d n

17.11

k k

k16

121

4

11

22

2

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

Exercises 2.16

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

4 1 y + 4.

2 1

4

d - 5.

5 2x

x

-

6.4

1

y - 7.

a

b a

3

22

-

-] g 8.

3

1

s

s

+

- 9.

1

1

b

b b2

+

+ +

10.3

5 p + 11.

3

1

a

a

+

+ 12.

2 4

3

x x

y

2+ +

+

13. 3x -

14.4 2 1

2

p p

p

2- +

-

15.2a b

a b

-

+


6/95

761 ANSWERS

Exercises 2.17

1. (a)4

5x (b)

15

13 3 y + (c)

12

8a + (d)

6

4 3 p + (e)

6

13x -

2. (a)2 1a

b

- (b)

1

2 1

q

p q q2

+

- - +^ _h i (c)

b

x y b

2 1

2

10

2

-

+

]^

gh

(d)ab

x xy y 2 2

- + (e)

5 2

3 1x x

x x

- -

- -

] ]] ]g gg g

3. (a)5x

(b)x

x

x 1

2

-

- +

] g (c)3

a b

a b

+

+ + (d)

2

2

x

x

+

(e) p q

p q p q

p q

p q 11 2 2

+

+ -

=+

- ++^ ^h h (f)

x x

x

1 3

12

+ -

-

] ]]g g

g

(g)2 2

3 8

x x

x

+ -

- +

] ]g g (h) 12

a

a2

+

+

] g

(i) y y y

y y

2 3 1

3 14 132 2

+ + -

+ +

^ ^ ^_

h h hi

(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) 152 1

y

y y + +

^ ^h h (c)

x x

x x

2 3 4

10 242

- -

+ -

] ]g g (d) bb b

b 1

3 5 10

2

2

+

- -

] g (e) x 5. (a)

5 2 3

3 13

x x x

x

- - +

-

] ] ]g g g (b) 2 23 5

x 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 b

a ab b2 2

+ -

- - +

] ]g g

(e)x y x y

x y y 1

+ -

+ +

^ ^^

h hh

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. 14

3-

12. 22.4 13. 1838.8 14. 4

3 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.

2

1

30.2 2

3 31.

2

3 32.

2 5

9 33.

2 2

5 34.

3

2 35.

7

5

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 12

24 36 8 12

20 60 10 30

5 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 p p2 1 2 1- - -^ h 6. 25k = 7. 2 3 5x y xy - - 8. 17, 240a b= =

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


7/95


Exercises 2.23

1. (a)7

7 (b)

4

6 (c)

5

2 15 (d)

10

6 14

5

3 14=

(e)3

3 6+ (f)

2

12 5 2- (g)

5

5 2 10+

(h) 14

3 14 4 7- (i) 20

8 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 18

19

15 6 22-=

-- -^ ^h h

(d)13

19 8 3

13

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)

10

6 6 16 3 84 8 14

6 21 14

5

3 8 3 4

- - - +

=- + + -

^ h

(e) 4- (f) 4 2

(g)15

20 12 19 6 25 3 6

15

19 6 65 3 6+ + -=

+ -

(h)6

6 9 2 2 3+ + (i)

21

4 6 9 3+

(j)415

30

330 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)2

1,

2

1a b= - =

(d) 19

5,

9

8a b= - = - (e) 5, 32a b= =

5.

2

2

3 2 2

3

2 1

2 1

2

4

2 1

2 1

2 1

2 1

2

4

2

2

2 1

2 1 2 1

2

4 2

2 1

2 2 2 12

1

3 2 22

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 2

2

8 2

9 4 2

6 4 24 2

1

6 4 24 2

6 4 2 4 26

2 2

# #

#

+

+

=

+ -

-+

=

-

-+

=-

-+

=-

+

= - +

=

^

^

h

h

So rational

9. x 3 2= - +^ h 10.4

4 4

b

b b

-

+ +

Test yourself 2

1. (a) 2 y - (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) 25 p2 - (f) 1 7a- -

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

4. (a)a ab 3 9

822

+ +^ h (b) 215

m 2-] g

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

6 15 9-

7.3 2

4 5

x x

x

+ -

+

] ]g g 8. (a) 36 (b) 2- (c) 2 (d) 216 (e) 29. (a)

5

1 (b) 8 10. 11.25d =

11. (a)15

2 3 (b)

2

2 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 y 5 2 2 42+ - +^ _h i

14. (a)3 y x

4

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.

2

3 3 1+ 19. (a)

4 3

ab

b a+ (b)

10

3 11x -

20.7

21 5 46 2- -


8/95

763 ANSWERS

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)

15

6 (c)

5 1

2

+

(d)15

12 2 6- (e)

53

20 3 15 4 10 3 6+ + +

24. (a)10

10x + (b)

21

17 15a - (c)

( 1)( 1)x x

x3 2

+

-

-

(d)1

1

k - (e)

3

15 6 15 3 15 2- - -

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

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

26. 3121

71 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) 4 y 4 -

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

2.17

17 3 2 5 20+ + 3.

2 2

1

4

2or

4.ab

x

a

bx

a

bx

42

2

2

2 2

+ + += d n 5. (a) x x4 9+ +] ]g g

(b) ( )( )x y x y x y x y x y 3 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 13 4xx 2

-

+

] g 13. x x x 97 153 2 + --

14.13

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

15. x x

x3

2

9

1

3

122

+ + = +d n 16. 2x =

17.10

400 59 5- 18. (a) 3

121

71 (b) ,a b

23

17

23

14= = -

19. 12

1i = 20.

4r

4

3 3

r r

r= =

21. 2 6 3s = +

Chapter 3: Equations

Exercises 3.1

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

6. 4x = 7. 15

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

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

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

21. 4b = - 22. 3x = 23. 13

2a = - 24. 4t = -

25. 1.2x = 26. 1.6a = 27. 8

1b = 28. 39t =

29. 5 p = 30. 4.41x Z

Exercises 3.2

1. 33

1b = 2. 35x = 3. 4

9

4 y = 4. 1

35

9x = 5. 5

8

5k =

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

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

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

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

23. 3x = 24. 1 y = 25. 13

2t = -

Exercises 3.3

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

6. 6.68r = 7. 6.44x = 8. 15n = 9. 33

2 y

1=

10. 3.7h = 11. (a) 25.39 BMI = (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) 1 p 2 - (d) x 2$ - (e) 9 y 2 -

(f) a 1$ - (g) y 22

1$ - (h) x 21 - (i) a 6# -

(j) y 121 (k) b 181 - (l) 30x 2 (m) x 34

3#

(n) m 143

22 (o) b 16

4

1$ (p) r 9# - (q) 8z 2


9/95


(r) w 25

41 (s) x 35$ (t) t 9$ - (u) 6q

5

22 -

(v) 1x3

22 - (w) b 11

4

1# -

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) y 3 5# #-

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

(e) y 6

11

3

21 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 12 y 1 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 47

5= -

(g) 3 2 y 2

11 1-

(h) ,x x9 6$ # -

(i) x 12!=

(j) a2 10# #

3. (a) 14

1x = (b) ,a 3

3

1= - (c) 2

3

1b =

(d) No solutions (e) 27

2 y = - (f) 7x = (g) ,m 5 1

3

2=

(h) ,d 22

11

4

3= - (i) , y

5

42= - (j) No solutions

4. (a) ,x 22

1= - (b) 3, 2

3

1 y = (c) 10, 1

5

3a = -

(d) ,x 4 7 3

1= - (e) ,d 4 5= -

5. (a) ,t 3 15

2= - (b) 1 3t

5

21 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) 10 p = (f) x 5!= (g) y 3!= (h) 2w =

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

2. (a) 6.71 p !=

(b) 4.64x =

(c) 2.99n =

(d) 5.92x !=

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

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

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

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

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

4. (a)5

1x = (b)

2

1a = (c)

2

1 y = (d) x

7

1!=

(e)3

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

(i) x3

2!= (j) b 1

2

1!=

5. (a) x512

1= (b) 6

4

1x = (c)

81

1a = (d)

625

1k =

(e) x8

1= (f) 4x = (g) 8 y = (h) n 7

32

19=

(i) 8b = (j) 1216

127m =

6. (a) 4n = (b) 5 y = (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) y 3

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

8. (a)2

1m = (b)

3

1x = (c)

3

1x = (d)

2

1k = -

(e)3

2k = - (f)

4

3n = (g) 1

2

1x = (h)

3

2n =

(i)6

1k = - (j) 1

3

2x =

9. (a) x 1= - (b) x 13

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

(e) x 21

2= - (f) x

3

2= - (g) x 4

2

1= - (h) x 1

11

7= -

(i) x 15

4= (j) x 18=

10. (a)4

1m = (b) 2

4

3k = - (c) 2

8

3x = (d) 1

2

1k =

(e)18

1n = (f)

2

1n = - (g)

5

4x = (h) 3

6

1b = -

(i) 17

1x = - (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= -


10/95

765 ANSWERS

9. ,x 0 4= - 10. x2

1!= 11. ,x 1 1

3

1= - -

12. , y 1 12

1= - 13. ,b

4

3

2

1= 14. ,x 5 2= - 15. ,x 0

3

2=

16. ,x 1 22

1= 17. 0, 5x = 18. 1, 2 y = - 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) 4 y 23!= +

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

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

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

(j)3

y 2

5!=

+

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

6

5 13!=

(c) q2

4 282 7

!!= =

(d) h8

12 128

2

3 2 2! !=

-=

-

(e) s6

8 40

3

4 10! != =

(f) x2

11 133!=

- (g) d

12

5 73!=

-

(h) x2

2 321 2 2

!!= = (i) t

2

1 5!=

(j) x4

7 41!=

Exercises 3.10

1. , y y 1 02 1 2. x02

11 1 3. 0, 1x x

2

121

4. m07

21 # 5. ,x x

5

30>1 - 6. b2 01#-

7. x1 14

11 1 8. z3

5

131 1- - 9. x2 2

4

31 #

10. ,x x2 26

11 2 11. x4

9

541#- -

12. x13

1115

71 1 13. ,a a3

4

122

11 2- -

14. x2

1

9

51 1 15. , y y 2 11 2- -

16. ,x x87 42# - 17. 4 26 p

21 11

18. ,x x15

12# - - 19. ,t t

5

223

22#

20. 0m9

81 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 x22

1

3

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

28. ,n n1 3 51 1 1- 29. ,x x4 3

2

7

41 1 1- - -

30. ,x x2

11 71# #

Exercises 3.11

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

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

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

10. ,b b22

1# $- - 11. ,a a1

3

11 2-

12. , y y 1 2

121 2- 13. ,x x3

21# $

14. ,b b35

21 2- 15. x1

2

1

3

1# #- -

16. y 4 3# #- 17. ,x x4 41 2- 18. a1 1# #-

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

22. a1 12

1# # 23. , y y 2

5

4# $-

24. ,m m13

212

11 2- 25. x1 1

3

1# #

26. 0 x2

111 27. ,x x0

2

11 $

28. , y y 15

41 2- - 29. 3 3n

2

11 #

30. ,x x8 52# - - 31. ,x x5

2

7

31 2

32. ,x x45

152# 33. ,x x1

4

112# - -

34. ,x x3 21 2- 35. x4

3

5

31#- -


11/95


Exercises 3.12

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

4. ,x y 6 17= = 5. ,x y 10 2= - = 6. ,t v 3 1= =

7. ,x y 3 2= - = 8. ,x y 64 39= - = - 9. ,x y 3 4= = -

10. ,m n2 3= = 11. ,w w 1 51 2

= - = 12. ,a b0 4= =

13. , p q4 1= - = 14. ,x x1 11 2= = - 15. ,x y 1 4= - = - 16. ,s t 2 1= = -

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

19. ,v v 2 41 2

= - = 20. , .x y 2 1 41Z=

Problem

23 adults and 16 children.

Exercises 3.13

1. ,x y 0 0= = and ,x y 1 1= =

2. ,x y 0 0= = and ,x y 2 4= - =

3. ,x y 0 3= = and ,x y 3 0= =

4. ,x y 4 3= = - and ,x y 3 4= = - 5. ,x y 1 3= - = -

6. ,x y 3 9= = 7. ,t x2 4= - = and ,t x1 1= =

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

9. ,x y 1 2= = and ,x y 1 2= - = -

10. ,x y 0 0= = and ,x y 1 1= =

11. ,x y 2 1= = and ,x y 1 2= - = - 12. ,x y 0 1= =

13. ,x y 1 5= = and ,x y 4 11= =

14. ,x y 4

14= = and ,x y 1 1= - = - 15. ,t h

2

1

4

1= - =

16. ,x y 2 0= =

17. ,x y 0 0= = and ,x y 2 8= - = - and ,x y 3 27= =

18. ,x y 0 0= = and ,x y 1 1= = and ,x y 1 1= - =

19. ,x y 2

12

4

3= = 20.

13

5,

13

12x y = - = -

Exercises 3.14

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

3. , ,a b c 4 2 7= - = = 4. , ,a b c 1 2 3= = = -

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

7. , , p q r 3 7 4= - = = 8. , ,x y z1 1 2= = - =

9. , ,h j k3 2 4= - = = - 10. , ,a b c 3 1 2= = - = -

Test yourself 3

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

(d) ,x x43

132# - - (e) p 4#

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

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

2

18= - = -

5. (a) 2x = (b)4

1 y =

6. (a) ,b 2 1 3

1= -

(b) , g 2 4

1=

(c) ,x x4 3$ #

7. (a) 36 A = (b) 12b = 8. ,x2

11=

9. 1 3 y 1 #-

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 714

12

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

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

17. , ,a b c 3 2 4= = = -

18. ,n n0 33

12 1 -

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

(c) 2x = (d) 2x = (e) ,x 3 15

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

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

(j) 1, 1x x# $- (k)6

5x = (l)

2

12b# #-

(m) No solutions (n) 23

1,

5

3t = (o) 1 3x1 1-

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

(s) 2 25

2n1 # (t) x

2

1

5

11 #-


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

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

6. ; ,x x x x x x3 3 2 2 4 3 22 !+ - - + + =] ] ] ^g g g h 7. ,x y 1 2= = and ,x y 1 0= - =

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

10. 1 1t 1 1- 11. x3 8# #- 12. 4

1x =

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

16. , y y 22

1

3

21# #- 17. 2247.36 P =

18. x3

4 102 !=

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

20. , y y 15

31 2-


12/95

767 ANSWERS

Chapter 4: Geometry 1

Exercises 4.1

1. (a) 47 y c= (b) 39x c= (c) 145m c= (d) 60 y c=

(e) 101b c= (f) 36x c= (g) 60a c= (h) 45x c=

(i) 40 y 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 c 43 137 101c c c= = =

(d) ,,a b d c 97 41 42c c c= = = =

(e) , ,a b c 68 152 28c c c= = = (f) ,a b10 150c c= =

7. 0x x x x

x

x

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 10

7 20 10

150

150 30180

c

c cc

+

+ +

= +

= +

=

+ = +

=

DBE` + is a straight angle

` AC and DE are straight lines

8.

AFC x

CD bisects

`

`

+ =

AFE+

( )

( )

( )

( )

DFB x

x

CFE x x

x

AFC 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 x

c

= - + +

=

So ABC+ is a straight angle.

AC is a straight line.

10. AEB BEC CED+ + ++ +

y y y 50 8 5 20 3 6090c

= - + - + +

=

So AED+ is a right angle.

Exercises 4.2

1. (a) ,a b e f c d g 148 32c c= = = = = = =

(b) ,x z y 70 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 c 95 85 32c c c= = =

(g) , ,a b c 27 72 81c c c= = =

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

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

2. (a) CGF

BFG CGF

180 121

59

59

( is a straight angle) FGH

`

c c

c

c

+

+ +

= -

=

= =

These are equal alternate angles.

AB CD` <

(b) BAC 360 292 68

(angle of revolution)

c c c+ = - =

BAC DCA 68 112

180` c c

c

+ ++ = +

=

These are supplementary cointerior angles.

AB CD` <

(c) 180 76

104104

BCD

ABC BCDc

c

+

+ +

= -

=

= =

( BCE+ is a straight angle)


AB CD` ;

(d) 180 128

5252

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=

42 BFD` c+ = (vertically opposite angles)

ABF BFD 138 42180

c cc

+ ++ = +

=


AB CD` ;


13/95


Exercises 4.3

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

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

(i) , ,a b c 75 27 46c c c= = =

(j) , ,a b c 36 126 23c c c= = =

(k) , ,x y z w 67 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 )

ACB

ABC

DEC ABC85

85


(angle sum of )

`

c

c c c

c

c

+

+

+ +

D

=

= - +

=

= =


AB DE` <

5.

124 68

ACB

CBA

CBA

CBA ACB

ABC

180 12456

68 124

5656

is isosceles

( is a straight a ngle)

(exterior angle of )

DCB

`

`

c c

c

c c

c c

c

c

+

+

+

+ +

D

D

= -

=

+ =

= -

=

= =

6. 38 y c=

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

(d) ,a b29 70c c= =

8. 180 (35 25 )

120180 120

60

180 (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 c

c

c c

c

c c c

c

c c

c

+

+

+

+

D

D

D

= - +

=

= -

=

= - +

=

= - +

=

Since 60 , IJL JIL ILJ c+ + += = =

IJLD is equilateral

( )

( )

( )

KJL

JLK

KJI

JKL

180 60

120

180 30 120

30

is a straight angle

angle sum of

c c

c

c c c

c

+

+ 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+ += =


AB ED` ;

10. 180

32

OQP 75 73

c

+ = - +

=

] g (angle sum of triangle) MNO OQP 32` c+ += =


MN QP ` ;

Exercises 4.4

1. (a) Yes

5 AB EF cm= = (given)

6 BC DF cm= = (given)

8 AC DE cm= = (given)

ABC DEF ` /D D (SSS )

(b)Yes

4.7 XY 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 KL

B L

BC JL

ABC JKL

438

5by SAS

(given)

(given)

(given)

`

c+ +

/D D

= =

= =

= =

(b)

,

Z B XY AC

YZ BC

RHS XYZ ABC

907

2

by

(given)

(given)

(given)

`

c+ +

/D D

= =

= =

= =

(c)

,

MN QRNO PR

MO PQ

MNO PQR

8

85

by SSS

(given)

(given)

(given)

` /D D

= =

= =

= =

(d)

.

Y T

Z S

XY TR XYZ STR

90

35

1 3by AAS,

(given)

(given)

(given)

`

c

c

+ +

+ +

/D D

= =

= =

= =

(e)

,

BC DE

C E AC EF

ABC DEF

4

907

by SAS

(given)

(given)

(given)

`

c+ +

/D D

= =

= =

= =

3. (a) B C BDA CDA

AD

ABD ACD

90

is common

by AAS,

(base anglesof isosceles )

(given)

`

c

+ +

+ +

/D D

D=

= =


14/95

769 ANSWERS

(b) BD DC AD BCbisects

(corresponding sides in congruent s)`

`

D=

4. , ) AB CD ABD BDC ernate angles+ +


15/95


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

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

8. ( , )

( , )

( )

ABF BEC

CBE BFA

C A

s AB CD

BC AD

s

alternate angle

similarly

angle sum of `

+ +

+ +

+ +

zz

D

=

=

=

`

since 3 pairs of angles are equal, | CEBD

|| ABF D

9. A+ is common

..

..

AB

AD

AC

AE

AB

AD

AC

AE

3

1 20 4

2

0 80 4

`

= =

= =

=

Since two pairs of sides are in proportion and their

included angles are equal, | , . ABC m 4 25D =|| AEDD

10..

.

. .

.

..

CD

AB

AC

BC

AD

AC

CD

AB

AC

BC

AD

AC

2 6

20 769

3 9

3

0 769

5 07

3 90 769

`

= =

= =

= =

= =

Since three pairs of sides are in proportion,

,c| , ACD x y 109 47cD = =|| ABCD

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

(d) . , .x y 6 2 4 4= = (e) . , .x y 1 4 9 2= =

12. (a) BC

AB

DE

AD

DE AD

FG AF

BC

AB

FG

AF

Also

`

=

=

=

(b) AC

AB

AE

AD

AE AD

AG AF

AC

AB

AG

AF

Also

`

=

=

=

(c)CE

BD

AE

AD

AE

AD

EG

DF

CE

BD

EG

DF

Also

`

=

=

=

13. . , .a b4 8 6 9= = 14. 0.98 y = 15. . , .x y 3 19 1 64= =

Exercises 4.6

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

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

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

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

AB CB

CA

81 144

225

2 2

2

+ = +

=

=

ABC` D is right angled

6. 1 XY YZ = =

XYZ ` D is isosceles

,YZ XY XZ

YZ XY

XZ

1 2

1 1

2

2 2 2

2 2

2

= = =

+ = +

=

=

XYZ ` D is right angled

7. AC AB BC

BC

BC

BC BC

AC

BC

2 3

4 3

11

2

2 1

2

2 2 2

2 2 2

2

2

`

#

= +

= +

= +

=

=

=

=

=

^ h

8. (a) 5 AC =

(b) , , AC CD

AD

25 144

169

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. x

x y 2 2+

11. d t t

t t t t

t t

20 3 15 2

400 120 9 225 60 4

13 180 625

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 4 BC2 2 2= -

20=

20 BC =

6 AO cm= (equal radii)

So 6 4 AC2 2 2= -

20=

20 AC =

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 y 102 51= = (g) °, °x y 43 47= =


16/95

771 ANSWERS

2. ABED is isosceles.

( s )

( )

B E

CBE DEB76

180 76

104

base equal

straight s

` c

c c

c

+ +

+ +

+

+

= =

= = -

=

D

D D

62 104 104 360

270 360

90

(angle sum of quadrilateral)c c c c

c c

c

+

+

+

+ + + =

+ =

=

` CD is perpendicular to AD ̀

3. (a)

( )( , )

( , )

( , )

D x

C x

x

x

A C x B x

B D x

A D AB DC

C D AD BC

B C AB DC

180

180 180

180 180

180180

and cointerior angles



`

`

c

c c

c c

c

c

+

+

+ +

+

+ +

+ +

+ +

+ +

<

<

<

= -

= - -

= - +

=

= =

= -

= = -

(b) x x x x180 180360

Angle sum c cc

= + + - + -

=

4. ,a b150 74c c= =

5. (a) 5 , 3 , 108 , 72a b x z y m 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 y 3 7= =

6. ( )

, )

,

ADB CDB

CDB ABD ADB DBC

ABD DBC

BD ABC

BD ADC

AB DC

AD BC

bisects

bisects

(alternate angles

(alternate angles )

`

`

+ +

+ +

+ +

+ +

+

+

<

<

=

=

=

=

7. (a) ..

AD BC AB DC

3 85 3

cmcm

(given)(given)

= =

= =

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

parallelogram.

(b) AB DC

AB 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 c

c

+ ++ = +

=


XY MN ` <

XM YN Also, (given)<

` XMNY is a parallelogram

(d) AE EC

DE EB56

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 y 3 5cm cm= = (d) ,x y 58 39c c= =

(e) x 12cm=

9. 6.4 cm 10. 59 , 31 , 59 ECB 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 n

n n

n

n

145

145 180 36035

10 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

6

720

6

720

120

2 180

2 180

#

#

c

c

c

c

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)

ED A 120 3090

c

c

+ = -

=

Similarly, DE B 90c+ =

So ED DE A 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

8

1080

2 180

2 180#

#

c

c

c

= -

= -

=

] g

AHG8

1080

135

` c

c

+ =

=

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


17/95


HAG2

180 135

22 30

` c

c

+ =-

= l


GAC 135 2 22 3090

# 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

72 EAC 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 35m 2 (b) .21 855 cm2 (c) .18 75 mm2 (d) 45 m2

(e) 57cm2 (f) 81 m2 (g) .28 27 cm2 2. .4 83 m2

3. (a) .42 88 cm2 (b) .2 9 5 m2 (c) .32 5cm2 (d) .14 32 m2

(e) .100 53cm2 4. (a) 25 m2 (b) .101 85 cm2

(c) .2 9 4 m2 (d) .10 39 cm2 (e) 45cm2

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. w 12 units2

Test yourself 4

1. (a) , ,x y z43 137 147c c c= = = (b) 36x c=

(c) , ,a b c 79 101 48c c c= = = (d) 120x c=

(e) 7.2r cm= (f) 5.6 , 8.5x y cm cm= = (g) 45ci =

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

AGF CFESo + + i= =

These are equal corresponding .s+

AB CD` <

3. 118.28 cm2

4. (a)

( )

DAE BAC ADE ABC

AED ACB

ABC ADE AAAand are similar

(common)

(corresponding angles, DE BC)

(similarly)

`

+ +

+ +

+ +

<

D D

=

=

=

(b) 3.1 , 5.2x y cm cm= =

5. 162c 6. 1020.7 cm3 7. 36 m

8. (a) AB AD BC DC

(adjacent sides in kite)

(similarly)

=

=

AC is common

` ∆ ABC and ∆ ADC are congruent (SSS)

(b) AO CO BO DO

AOB COD

(equal radii)

(similarly)

(vertically opposite angles)+ +

=

=

=

` ∆ AOB and ∆ COD are congruent (SAS)

9. 73.5 cm2

10. 6 2 7 36 28 64 822 2

+ = + = =^ h ` ∆ ABC is right angled (Pythagoras)

11. AG

AF

AE

AD

AE

AD

AC

AB

AG

AF

AC

AB

(equal ratios on intercepts)

(similarly)

`

=

=

=

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

( , )

AB AC

B C

BD DC AD BC

(given)

bisects given

+ +

=

=

=

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

180 ADB ADC

ADB ADCBut(corresponding s in congruent s)

(straight )c

+ +

+ +

+

+

D=

+ =

So 90 ADB ADC c+ += =

So AD and BC are perpendicular.

13.

34˚ 34

( )

( )

ACBCAD

CAD ADC

6868 34

base s of isosceles

exterior of

`

c

c c

cc

+

+

+ +

+

+

D

D

=

= -

=

= =

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


18/95

773 ANSWERS

( , )

, )

DAC ACB BAC ACD

AD BC

s AB DC

alternate s

(alternate

+ +

+ +

+

+

<

<

=

=

AC is common

ABC ADC AB DC

(AAS)(corresponding sides in congruent s)

`

`

/D DD=

Similarly, AD BC=

` opposite sides are equal

15. (a) 24 cm2 (b) 5 cm 16. 9

17. BFG FGD x x109 3 3 71

180

c c

c

+ ++ = - + +

=

These are supplementary cointerior .s+

AB CD` <

18. 57 cm2

19. (

(

( )

)

)

ACB A B

x y ACD ACB

z x y

x y x y

180

180

180

180 180

180 180

sum of

straight

c

c

c

c c

c c

+ + +

+ +

+

+

D= - +

= - -

= -

= - - -

= - + +

= +

] g

20. (a)

.

..

.

..

A E

EF

AC

DE

AB

EF

AC

DE

AB

2 7

2 971 1

3 6

3 961 1

given

`

+ +=

= =

= =

=

^ h

So ∆ ABC and ∆ 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 DBC

ABD BDC

ADB DCB

AB DC

(given)

(alternate angles

angle sum of `

+ +

+ +

+ +

<

D

=

=

=

` since 3 pairs of angles are equal,

BCDD; ABD


19/95


14. (a) AB BC ABE CBE 45

(adjacent sides in square)

(diagonals in square make 45 with sides)c

+ +

=

= =

EB is common.

, ABE CBE AE CE

by SAS(corresponding sides in congruent s)

`

`

/D D

D=

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

(b) BD x x

x

x

DE BD

x

2

2

2

1

2

2units

2 2

2

= +

=

=

=

=

Practice assessment task set 1

1. 9 p = 2. 2 5 y x y + -^ ^h h 3. (a) x 1- (b) 3x 4 4. 6 10 y - 5.

23

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

7.7

2x = 8.

3

2

x -

9. ° ABC EDC ACB ECD

AB ED

ABC EDC AC EC

ACE

90

by AAS

is isosceles

(given)


(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. x

24 2 312! != = 22.

491

23. 4, 11 1, 4x y x y or= = = - = - 24. ,x y 2 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 1021

10

=- 37. ,x x6 2

5

22 1 - 38.

8

1

39. x 7- - 40. 4

1x = 41. ,x x3 3# $- 42.

6

1

43. Given diagonal AC in rhombus ABCD :

)

)

AB BC DAC ACB

BAC ACB

DAC BAC

AD BC

ABC

(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 y 98 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

x

2 - +

-

(d) 3 2 1+

(e)1 1

5

x x

x

+ -

- +

] ]]g g

g (f)

6

11 3 (g) x y z

x z

y 14 7 11

14 11

7

=- -

(h)5 1 2

3

a a b b+ +] ]g g (i) 8 5 (j) 13 21

52. . , .x y 2 7 3 1= = 53. 25x = 54. r 2

cm3 r

=

55. 17.3 cm

56. DEA x EAD x

CD x x

x ABC x

ABC DEA

A

222

LetThen (base s o f isosceles )

(exterior of )

(opposite s of gram are equal)

EAD

`

`

+

+

+

+

+ +

+

+

+ <

D

D

=

=

= +

=

=

=

57.5

2 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. x18

15

4

31#-

66.( BCEF is a gram)<( BC AD ABCD

BC FE

AD FE

is a gram)

`

<


20/95

775 ANSWERS

69.75

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

72. BD bisects AC

So AD DC=

90 BDC 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 y 2 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 f 1 3= - =] ]g g 2. , ,h h h0 2 2 2 4 14= - = - =] ] ]g g g 3. 25, 1, 9, 4f f f f 5 1 3 2= - - = - = - - = -] ] ] ]g g g g 4. 145. 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 2 g x x2- = +] g 12. f k k k k1 12= - + +] ] ^g g h 13. ; ,t t 1 2 4= - = - 14. 0

15. 125, 1, 1f f f 5 1 1= = - = -] ] ]g g g 16. 0 4 1 3f f f 2 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 c 5 -] g 24. 3 5k2 + 25. (a) 2 (b) 0 (c) 2n n4 2+ + Exercises 5.3

1. (a) x -intercept3

2 , 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 x

x

f 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 x

x x x g x

3 2

3 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 x

xf x 1

even function

2

2

`

- = - -

= -

=

] ]g g

7. f x x x

x x

x x

f x

4

4

4

odd function

3

3

3

`

- = - - -

= - +

= - -

= -

] ] ]^]

g g gh

g

8. f x x x

x x

f 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 12

1- , y -intercept 3

(c) x -intercept2

1 , y -intercept 1

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

(e) x -intercept3

2 , y -intercept

3

1-


21/95


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

-1 1

(c) y

x

-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-1 1

(d) y

x

-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

(e) y

x

-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-1 1

1

2

(f) y

x

-4

-5

-3 -2-1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

(g) y

x

-4

-5

-3-2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-11

2

3-

(h) y

x

-4

-5

-3-2-1 2 3 4

2

1

3

4

5

-3

-4

-2

-1 1


22/95

777 ANSWERS

(i) y

x

-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-1 1

(j) y

x

-4

-5

-3 -2 -1 2 3 4

2

1

3

4

5

-3

-4

-2

-1 11

1

2

3. (a) ,x y all real all real" ", , (b) :,x y y 2all real =" ", , (c) : ,x x y 4 all real= -! "+ , (d) : ,x x y 2 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

-111

1

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

-11

(b) y

x

-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

6

-3

-4

-2

-1 1

(c) y

x

-4

-5

-3-

2-

1 2 3 4 5

2

1

3

4

5

6

-3

-4

-2

-1 1


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(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

2

1

3

4

5

6

-3

-4

-2

-1 1

(f) y

x

-4

-10

-3-2-1 2 3 4 5

4

6

8

2

10

12

-6

-8

-4

-21

(g) y

x

-4

-5

-3 -2 -1 2 3 4 5

2

1

3

4

5

-3

-4

-6

-2

-1 1

(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

-1 211

1

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 y

4

1$ -( 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 },

: 9 y 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$ -" ,


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779 ANSWERS

(c) {all real x }, : y y 24

1$ -( 2 (d) {all real x }, : y y 0#" ,

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

5. (a) y 0 9# # (b) y 0 4# # (c) y 1 24# #-

(d) y 4 21# #- (e) y 18 24

1# #-

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 x

x

f x

even

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 -intercept3

2 , y -intercept 2

(h

Documents

Answers for maths in focus 3u