Maths in Focus - Margaret Grove - Ans
August 13, 2017 | Author: Sam Scheding | Category: N/A
Short Description
Descripción: Mathematics Preliminary Course - 2nd Edition...
Description
540
Maths In Focus Mathematics Preliminary Course
Answers Chapter 1: Basic arithmetic
8.
o (b) 0.07 oo (c) 0.13 oo (d) 0.16 o (a) 0.83 o oo o (g) 0.142857 or 0. 142857 (h) 1.18
9.
(a)
8 9
(h)
13 60
Problem 5
Exercises 1.1 1.
2.
(a) Rational (b) Rational (e) Rational (f) Irrational (i) Rational (j) Irrational
(e) - 4.3
(a) 18 (b) 11 (c) 6 (d) 11 (h) 1
3.
(c) Rational (g) Irrational
19 20
(i) 2
(j) 3
8.
600
5. 950
7 15
o 10. (a) 0.5 11. (a)
(c) 8.80 (d) 22.71 (e) - 13.20
6. 3000
(j) 8.16
1.
7. 11 000
8.
17. 79 cents 18. 2.73 19. 1.1 20. 3.6 m 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
(b) 2
1.
1
6.
- 1.2
10. - 2 15. 5
3. - 56
4. 10
(a)
7. - 7.51
8. - 35.52
9. 6.57
11. - 7
12. −23
13. 10
16. 3
16 25
(b)
17. 1
14. 1
18. 60 19. −20 20. 9
51 1000
(c) 5
1 20
(d) 11
4 5
7 20 3 (e) 5
3.
(a)
4.
(a) 0.27 (b) 1.09 (c) 0.003 (d) 0.0623
5.
1 (a) 35% (b) 33 % 3
6.
(a) 124% (b) 70% (c) 40.5% (d) 127.94%
7.
(a) 0.52;
13 25
(d) 1.09; 1
(e)
o (c) 0.73
oo (d) 0.68
8 11
7 18
(c)
67 99
(f)
6 11
(g)
7 45
(d) 2
oo (e) 1.72 4 45
(e)
14. 17.5%
15. 41.7%
1 20 7 4. $547.56 5. 714.3 g 6. 24
2. 3 28
17 20
3. (a)
(b)
7 10
(c) 1
7. $65
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
14. 5.9%
15. 402.5 g 19. 573
12. 1152.125 g
16. 41.175 m
13. $10.71
17. $30.92
20. $2898
3 8
(c)
1 1000
9 100
(d) 1
7 100
(e) 0.434;
Exercises 1.5 (a) 500
(b) 145
(c)
2.
(a) 13.7
(b) 1.1
(c) 0.8
3.
(a) a 17
(c) 0.168; 217 500
(m) w 10
97 1000
4.
21 125
(f) 0.1225;
(b) y 0 = 1 (h) x 21
49 400
(d) w
(j) 81y - 8
(o) x -3
(e) 2 (e) - 2.6 (e) x 5 (k) a
(p) a - 2 b 3 or
(f) 0.5
(f) p 10 (l)
x 10
b3
y 45
a2
x5 (c) m4 (d) k10 (e) a -8
(f) x
(g) mn2
(i) 9x22 (j) x21
(a) p5q15 (b) (f) x4y10
(d) 2.7
2
(a) x14 (b) a -7 (h) p - 1
5.
y
(d) 3
(c) a - 4
(i) 4x 10
(n) p 5
(q) x - 5 y 2 or
(d) 0.1%
1 64
1.
oo (d) 0.63
2 (c) 226 % 3
(b) 0.07;
5 minutes after 1 o’clock. 11
(g) y 6
o (a) 0.4 (b) 1.875 (c) 0.416 (b)
7 9
37 495
11. 54.925 mL
5
2.
1 50
(j) 1
10. $52.50
5. - 4
Exercises 1.3 1.
(d) 3
Problem
2. - 11
4 15
1 8
5 9
13. 77.5%
18. 3.2 m
Exercises 1.2
217 990
(b) 7.4
3 20 (d)
12. 0.73 13. 33 14. 3.248 15. 4.21
21. $281.93
5 8
(c) 1
Exercises 1.4
9. $8 000 000 10. $34 600 000
11. 844 km 16. 1.7
(g) 2
(i)
12. 74%
(f) 0.17 (g) 0.36 (h) 1.20 (i) - 4.27 1300
(f) −1
1 3
(a) 16.36 (b) 21.87
4.
(d) Irrational (h) Rational
2 9
(b)
oo (f) 0.15
o (e) 0.6
a8 8
b 2k 23 (g) 27
(c)
64a 3 b 12
(h) 16y47
(d) 49a10b2 (e) 8m17 (i) a3 (j) 125x - 21 y 18
ANSWERS
6.
4
1 2
7. 324
8. 2
10 27
9. (a) a3b
3
1 25
(b)
-
1 2
5.
(a) x 2
6.
(a) x + x 2 + 2x 2
(b) x
2
5
5
(c) x 3
(d) x 3
(e) x 4
3
7 (b) 32
10. (a) pq r
2 2
14.
1 81
4 11. 9
1 108
15.
1 12. 18
1 12
16.
4 13. 27
5 22
17.
49 3888
18.
2 58
(d) x + x - 1 + 2 7.
Exercises 1.6 1.
2.
3.
(d)
1 1 1 1 1 (b) (c) (d) (e) (f) 1 4 27 343 10 000 256 1 1 1 1 1 1 (g) (h) (i) (j) (k) (l) (m) 1 7 64 9 32 81 81 1 1 1 1 (n) (o) (p) (q) (r) 1 36 125 100 000 128 1 1 (s) (t) 64 64
1.
(e) x
1
1 2
- 3x
^ y - 3 h2
3
(e)
3 4 ^ x + y h5
-
1
(b)
a - 2b
-
3 2
+x
(c)
-
5 2
4
7
] 6a + 1 g4
6
7 9 ] 3x + 8 g2
(a) m - 3
(b) x - 1
-4
(h) 3y
(c) p - 7 −2
2.
(d) d −9 (e) k −5 (f) x - 2 3t - 8 (j) 5
1 z- 6 (i) z - 6 or 2 2
2x - 1 (k) 7
2y - 7 5m (m) (n) ] 3x + 4 g- 2 (o) ] a + b g- 8 2 3 (p) ] x - 2 g- 1 (q) ^ 5p + 1 h- 3 (r) 2 ] 4t - 9 g- 5 ]x + 1g 4
(a) (h)
1 5
t 5 x7
(m)
(t) 1
(b)
(c)
6
x 1
(i)
5 ] a + 3b g 9 1 y
3
1 n
8
(e)
1 w
(k)
2 x
(f)
10
1
(g)
1 (l) 8y + z
] x + 1 g6
1
(n)
]k - 3g
(d)
1 (j) 4n
8x 3
1
(o) x5 (p) y10 (q)
^ 3x + 2y h x-y 3x + y 7 o (s) (t) e x+y 2w - z
2
(r) ] a + b g2
9
3 m
4
(a) 2.19 (b) 2.60 (c) 1.53 (d) 0.60 (e) 0.90 (f) 0.29
3.
(a) 3 y
(b) 3 y 2 or _ 3 y i
(c)
(f) 3 6q + r
(g)
2
1
1
3
(a) t 2
(b) y 5
(c) x 2
(f) ] 2t + 3 g
-
(i) ] x - 2 g
-
2 3
1 2
2a 2 2 y - 1k 3
x
-
3 2
(d) 1
5
] x + 7 g2 1
1
(j)
1
(d) ] 9 - x g 3
(g) ^ 5x - y h
1
(l)
(e) 8.67 # 10 9
(f) 4.16 # 10 5
(h) 1.376 # 10
2
(a) 5.7 # 10 - 2 -4
-6
4
(i) 2 # 10 7
(b) 5.5 # 10 - 5 (e) 2 #10
-6
(h) 2.3#10
(j) 8 #10 4
(c) 4 # 10 - 3
(f) 8#10 - 8 -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
6.
1.305 # 10 10
(b) 0.686
(c) 8.25
(d) 1.30
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) a + 1 (h) -a - 1 (i) x - 2 (j) 2 - x
5.
(a) | a + b | = 6 (b) | a + b | = 3 (c) | a + b | = 1 (d) | a + b | = 1 (e) | a + b | = 10
6.
(a)
x2 = | x | = 5
(b)
x2 = | x | = 2
(d)
x2 = | x | = 4
(e)
x2 = | x | = 9
2
2.
1
(d) 1.2 #10 7
(g) 7.6#10
p
(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 1 1 (q) 4 (r) 27 (s) (t) 2 16
3x - 1
(c) 6.19 # 10 4
(d) 6.2 #10
Exercises 1.7
(e)
(b) 1.23#10 6
-7
- 11
(s)
(a) 3.8 # 10 3 (g) 9 #10
(l)
4.
1
(c) p 2 + p - 1 + 2p 2
Exercises 1.8
1 11 1 (a) 1 (b) 16 (c) 1 (d) 1 (e) 1 (f) 125 (g) 1 2 25 3 3 13 19 1 (h) 49 (i) 3 (j) 32 (k) 2 (l) 1 (m) 1 (n) 1 8 3 36 81 5 16 7 (o) 1 (p) 16 (q) - 15 (r) (s) 1 (t) 8 23 25
-6
1.
1 3
2
(a)
(g) 2x
4.
(a)
2
(b) a 3 - b 3
2x + 5 or
|a | + | b |= 6 ` | a + b | # | a | + | b | |a | + | b |= 3 ` | a + b | # | a | + | b | |a | + | b |= 5 ` | a + b | # | a | + | b | |a | + | b |= 9 ` | a + b | # | a | + | b | | a | + | b | = 10 ` | a + b | # | a | + | b | (c)
x2 = | x | = 3
7.
(a) x + 5 for x 2 - 5 and - x - 5 for x 1 - 5 (b) b - 3 for b 2 3 and 3 - b for x 1 3 (c) a + 4 for a 2 - 4 and - a - 4 for a 1 - 4 (d) 2y - 6 for y 2 3 and 6 - 2y for y 1 3 (e) 3x + 9 for x 2 - 3 and - 3x - 9 for x 1 - 3 (f) 4 - x for x 1 4 and x - 4 for x 2 4 1 1 (g) 2k + 1 for k 2 - and - 2k - 1 for k 1 2 2 2 2 (h) 5x - 2 for x 2 and - 5x + 2 for x 1 5 5 (i) a + b for a 2 - b and - a - b for a 1 - b (j) p - q for p 2 q and q - p for p 1 q
8.
x = !3
1 ^ 5 x + 7 h2 1
(e) ] 4s + 1 g 2 5
(h) ] 3x + 1 g 2 1
1 ^ y + 7 h 2 (k) 5 ] x + 4 g 3 2 3 3 4 (m) _ x 2 + 2 i 5
9. !1
10. !1, x ! 2
541
542
Maths In Focus Mathematics Preliminary Course
Test yourself 1 1.
(a)
9 20
(b) 0.14 (c) 0.625 2. (a)
(f) 73.3% 3.
Chapter 2: Algebra and surds
1 49
(b)
157 200 1 (c) 3
(d)
1 5
Exercises 2.1
(e) 1.2%
(a) 8.83 (b) 1.55 (c) 1.12 (d) 342 (e) 0.303 4. (a) 1 (e) - 10 (f) - 1 (g) 4 5. (a) x 9 8x 18 29 (b) 25y 6 (c) a 11 b 6 (d) (e) 1 6. (a) 27 40 1 1 1 (b) 3 (c) 12 (d) 2 (e) 12 7. (a) 4 (b) 6 (c) 19 7 2 2 1 1 (d) (e) 4 (f) 3 (g) (h) 2 (i) 1 (j) 4 7 64 (b) 1 (c) 39 (d) 2
8.
5
30 18
(a) a
(b) x y
(c) p
(g)
1 -3 x 2
36
11
(d) 16b
(d) ] x + 1 g
(b) x - 5 (c) ^ x + y h- 1 (f) 2x - 1
9
4
(e) 8x y
1 4
9. (a) n
9
-
3 4
1 ] 4t - 7 g4 1 1 (f) 5 a + b (g) (h) 4 b 3 (i) 3 ] 2x + 3 g4 (j) 3 x x3 11. | a + b | = 2 | a | +| b | = 8 ` | a + b | # | a | + | b | 10. (a)
1
a5
(b) 4 n
1 13. 192
12. 1
x+1
(d)
1
(c) ] x + 3 g 6
(b) y - 1
(d) ] 2x - 3 g- 11
20. (a) 1.3 # 10 - 5
(b) 1.23 # 10 11
1
b 5 (c) c m a
x
3
(e)
7 14. 689 mL 15. (a) 6 h (b) 12
1
22. (a)
1 (c) 8
(b)
1 2a + 5
21. (a)
7 9
7
(e) y 3 (b)
41 330
23. 14 500
24. LHS = | -2 + - 5 | = 7, RHS = | -2 | + | -5 | = 7. So | a + b | # | a | +| b | since 7 # 7.
1.
4
4.
1 1 53 % 5. 3 16
9.
18 h
11.
2. 1
11 18
3. 0.502, 51%,
6. 3.04 # 10
14
51 o , 0. 5 99
3271 7. 83% 8. 1 9990
10. 1.98
LHS = 2 ^ 2 k - 1 h + 2 k + 1 = 2k+1 - 2 + 2k+1 = 2:2 k + 1 - 2 = 2 ^ 2k+1 - 1 h = RHS ` 2 ^ 2k - 1 h + 2k+1 = 2 ^ 2k+1 - 1 h
.
o , 0, 12. −24 35 13. - 0.34, 2, 1. 5
3 7
2 14. 6 % 3
1 1 15. when x 2 - 1, when x 1 - 1 16. 0.73 x-1 1-x 17. 0.6%
7.
-y
3. z
8. −5x
13. - m
9. 0
14. - x
19. 6x - 6y
4. 6a
10. 3k
15. 0
20. a - 3b
23. m 2 - 6m + 12 26. - 2ab + 10b
5. 3b
6. −3r
11. 9t
16. 5b
12. 10w
17. 11b
21. 4xy + 2y
24. p 2 - 2p - 6 27. 2bc - ac
29. x 3 - 2xy 2 + 3x 2 y + 2y 3
18. - 10x
22. - 6ab 2
25. 8x + 3y
28. 2a 5 - 9x 3 + 1
30. 3x 3 + x 2 - 7x - 6
1.
10b
2. 8xy
3. 10p 2
5.
15ab 6. 14xyz 7. 48abc 8. 12d 2
9.
12a3
10. - 27y3
12. 6a 2 b 3
19. - 14m
11. 32x10
13. - 10a 3 b 2
15. 5a 3 b 3
4. - 6wz
14. 21p 3 q 4
16. - 8h 10 17. k 3 p 3 20. 24x y
11
6
18. 81t 12
3
Exercises 2.3 1.
2. 2
6x
3. 4a 2
4. 8a
5. 4a
y
6.
2
7. 3p
ab 4 1 -2 9. 10. - 3x 3 11. 3a 12. 13. qs 3y 2 3ab 2 4 7 6 2 a b 2 z b 14. 15. 16. 6p 4 q 17. 18. 4c 2a 3c 2 d 2x 2
8.
19. -
x3 z3 3y
a 13
20.
2b 6
Exercises 2.4
Challenge exercise 1 278 303
2. 3a
16. $38 640 17. 70% 18. 6.3 # 10 23
(d) 33.3% 19. (a) x 2
(c)
1 x-y
(j) m
7x
Exercises 2.2
1
(e) ] a + b g 7
(i) ] 5x + 3 g 7
(h) x 3
1 2
1.
18 4.54 19. 4.14 # 10 - 20
20. | a + b | = | a | + | b | when a 2 0, b 2 0 or a 1 0, b 1 0; | a + b | 1 | a | + | b | when a 2 0, b 1 0 or a 1 0, b 2 0; ` | a + b | # | a + b | for all a, b
1.
2x - 8
5.
x 2 - 2x 6. 6a 2 - 16ab 7. 2a 2 b + ab 2 8. 5n 2 - 20n
2. 6h + 9
3. - 5a + 10
9.
3x3 y2 + 6x2 y3
10. 4k + 7
11. 2t - 17
12. 4y + 11y
13. - 5b - 6
15. - 3m + 1
16. 8h - 19 17. d - 6
2
19. 3x - 9x - 5 2
22. - 7y + 4
14. 8 - 2x
20. 2ab - 2a b + b 2
23. 2 b
4. 2xy + 3x
24. 5t - 6
18. a 2 - 2a + 4 21. 4x - 1
25. 2a + 26
Exercises 2.5 1.
a 2 + 7a + 10
2. x 2 + 2x - 3
4.
m 2 - 6m + 8
5. x 2 + 7x + 12
7.
2x 2 + x - 6
8. h 2 - 10h + 21
3. 2y 2 + 7y - 15 6. y 2 - 3y - 10 9. x 2 - 25
10. 15a 2 - 17a + 4 11. 8y 2 + 6y - 9 12. xy + 7x - 4y - 28 13. x 3 - 2x 2 + 3x - 6 16. 16 - 49y 2 20. y 2 - 36
14. n 2 - 4
17. a 2 - 4b 2
21. 9a 2 - 1
15. 4x 2 - 9
18. 9x 2 - 16y 2
22. 4z 2 - 49
19. x 2 - 9
ANSWERS
23. x 2 - 2xy + 11x - 18y + 18 24. 2ab + 2b 2 - 7b - 6a + 3
Exercises 2.8
25. x + 8
1.
]x + 4g]2 + bg
4.
]m - 2g]m + 3g
5. ] d - c g ] a + b g 6. ] x + 1 g ^ x 2 + 3 h
7.
] 5a - 3 g ] b + 2 g
8. ^ 2y - x h ^ x + y h
3
26. a - 27
27. a + 18a + 81
3
28. k - 8k + 16
29. x + 4x + 4
2
33. 9a + 24ab + 16b 35. 4a + 4ab + b 2
2
32. 4t 2 - 4t + 1
2
38. a - 2ab + b
30. y - 14y + 49
2
31. 4x 2 + 12x + 9
2
2
34. x - 10xy + 25y
2
2
36. a - b
2
2
39. a + b
2
3
3
10. ] x + 5 g ] x - 1 g
2
37. a + 2ab + b
2
2
40. a - b 3
1.
t + 8t + 16
4.
y 2 + 16y + 64
7.
n 2 + 2n + 1
5. q 2 + 6q + 9
6. k 2 - 14k + 49
25. ^ y + 7 h ] x - 4 g
8. 4b 2 + 20b + 25
9. 9 - 6x + x 2
10. 9y - 6y + 1
11. x + 2xy + y 2
13. 16d + 40de + 25e 2
21. 16a 2 - 1
24. x + 10x + 25 4
27. a 2 -
2
2
2
28. x 2 - ^ y - 2 h2 = x 2 - y 2 + 4y - 4
1 a2
28. 3 (a + 2b) (a + 3)
29. 5 (y - 3) (1 + 2x)
30. ] r + 2 g ] rr - 3 g
23. x 4 - 4 4 26. x + 4 + 2 x
2
26. (x - 4) (x 3 - 5)
19. 4a 2 - 9
22. 49 - 9x 2
25. 9a b - 16c
2
2
15. x 2 - 9
18. x 2 - 100
24. ] a - 3b g ] 4 + c g
27. (2x - 3) (2x + 4) = 2 (2x - 3) (x 2 + 2)
12. 9a - 6ab + b
2
17. ] x - 3 g ^ 7 - y h
20. ] a + 3 g ] 2 - b g
2
2
14. t - 16
2
17. r 2 - 36
20. x 2 - 25y 2
2
2
14. ^ a + b h ] ab - 4 g
22. ^ q - 3 h ^ p + q h
23. ] x - 2 g ^ 3x 2 - 5 h
2
12. (m - 2) (1 - 2y)
2
18. ] d + 3 g ] 4 - e g 19. ] x - 4 g ^ 3 + y h
3. x - 2x + 1
2. z - 12z + 36 2
9. ^ y + 1 h ] a + 1 g
15. ] 5 - x g ] x + 3 g 16. (x + 7) (x 3 - 4)
3
21. (x - 3) (x 2 + 6)
2
3. ] x + 5 g ] x + 2 g
11. (y + 3) (1 + a)
13. ^ x + 5y h ^ 2x - 3y h
2
Exercises 2.6
16. p 2 - 1
2. ^ y - 3 h ] a + b g
29. ] a + b g2 + 2 ] a + b g c + c 2 = a 2 + 2ab + b 2 + 2ac + 2bc + c 2
Exercises 2.9 1.
]x + 3g]x + 1g
4.
] t + 4 g2
7.
]v - 3g]v - 5g
2. ^ y + 4 h ^ y + 3 h
5. ] z + 3 g ] z - 2 g 8. ] t - 3 g
2
3. ] m + 1 g2
6. ] x + 1 g ] x - 6 g 9. ] x + 10 g ] x - 1 g
30. ] x + 1 g2 - 2 ] x + 1 g y + y 2 = x 2 + 2x + 1 - 2xy - 2y + y 2
10. ^ y - 7 h ^ y - 3 h
11. ] m - 6 g ] m - 3 g
12. ^ y + 12 h ^ y - 3 h
13. ] x - 8 g ] x + 3 g
31. 12a
14. ] a - 2 g
32. 32 - z
2
34. x 2 + 3xy + y 2 - 2x
33. 9x + 8x - 3 2
35. 14n 2 - 4
36. x - 12x + 48x - 64 3
2
37. x
2
38. x - 2x y + y 4
2
2
4
2
15. ] x - 2 g ] x + 16 g
16. ^ y + 4 h ^ y - 9 h
17. ] n - 6 g ] n - 4 g 18. ] x - 5 g 2
19. ^ p + 9 h ^ p - 1 h
20. ] k - 2 g ] k - 5 g 21. ] x + 4 g ] x - 3 g
39. 8a + 60a + 150a + 125
22. ] m - 7 g ] m + 1 g 23. ^ q + 10 h ^ q + 2 h
40. 4x + 16x + 15x - 4x - 4
24. ] d - 5 g ] d + 1 g 25. ] l - 9 g ] l - 2 g
Problem
Exercises 2.10
a = 2, b = 7, c = 9, d = 4, e = 3, f = 8, g = 0, h = 6, i = 1
1.
(2a + 1) (a + 5) 2. ^ 5y + 2 h ^ y + 1 h
3.
(3x + 7) (x + 1) 4. (3x + 2) (x + 2) 5. (2b - 3) (b - 1)
6.
(7x - 2) (x - 1) 7. ^ 3y - 1 h ^ y + 2 h
9.
^ 5p - 2 h ^ p + 3 h 10. ] 3x + 5 g ] 2x + 1 g
3
2
4
3
2
Exercises 2.7 2. 5 ] x - 2 g 3. 3 ] m - 3 g 4. 2 ] 4x + 1 g
1.
2^ y + 3h
5.
6 ^ 4 - 3y h
9.
3a ] 5 - a g 10. ab ] b + 1 g 11. 2xy ] 2x - 1 g
6. x ] x + 2 g 7. m ] m - 3 g 8. 2y ^ y + 2 h
12. 3mn ^ n 2 + 3 h
13. 2xy ] 4x - z g 14. a ] 6b + 3 - 2a g
15. x ^ 5x - 2 + y h
17. 5b 2 ] b + 3 g
16. q 2 _ 3q 3 - 2 i
18. 3a b ] 2b - a g 19. (m + 5) (x + 7) 2
20. ^ y - 1 h ^ 2 - y h
2
21. (7 + y) (4 - 3x)
22. ] a - 2 g ] 6x + 5 g
23. ] 2t + 1 g ^ x - y h
24. ] 3x - 2 g ] a + 2b - 3c g
25. 3x ] 2x + 3 g 2
28. 4x 2 ] x - 6 g
26. 3q _ pq 2 - 2 i 3
29. 5m 2 n ^ 7mn 3 - 5 h
31. 2rr ] r + h g 32. ] x - 3 g ] x + 2 g 34. - ] a + 1 g
27. 3ab ^ 5a 3 b 2 + 1 h
35. (a 2 + 1) (4ab - 3)
30. 4ab 2 ^ 6ab 3 + 4 h
33. (x + 4) (y 2 + 2)
8. ] 2x + 3 g ] x + 4 g
11. (2y + 1) (y - 6)
12. ] 5x - 1 g ] 2x + 1 g
13. (4t - 1) (2t - 3)
14. (3x + 4) (2x - 3)
15. ^ 6y - 1 h ^ y + 8 h
16. ] 4n - 3 g ] n - 2 g
17. ] 4t - 1 g ] 2t + 5 g 18. ^ 3q + 2 h ^ 4q + 5 h 19. ] 4r - 1 g ] 2r + 6 g = 2 ] 4r - 1 g ] r + 3 g 20. ] 2x - 5 g ] 2x + 3 g
21. ^ 6y - 1 h ^ y - 2 h
22. ^ 2p - 3 h ^ 3p + 2 h
23. (8x + 7) (x + 3)
24. ] 3b - 4 g ] 4b - 9 g
25. (6x + 1) (x - 9)
26. ] 3x + 5 g2
27. ^ 4y + 3 h2
29. ] 6a - 1 g2
30. ] 7m + 6 g2
28. ] 5k - 2 g2
543
544
Maths In Focus Mathematics Preliminary Course
Exercises 2.11 1.
^y - 1h
5.
(x - 6)
9.
] 5x - 4 g2
3. (m + 5)
2
6. ] 2x + 3 g
2
18. d 3y +
2
12. ] 4k - 3 g2
11. ^ 3y - 5 h2
14. ] 9a - 2 g2
15. ] 7m + 6 g2 1 2 19. c x + m x
1 2 n 5
2
8. ] 3a + 2 g
2
10. ^ 7y + 1 h2
2 2 n 3
4. (t - 2)
2
7. ] 4b - 1 g
2
13. ] 5x + 1 g2 17. d x -
Exercises 2.14
2. (x + 3)
2
16. d t + 20. d 5k -
2
1 n 2
2 2 n k
Exercises 2.12 1.
(a + 2) (a - 2)
2. (x + 3) (x - 3)
3. (y + 1) (y - 1)
4.
]x + 5g]x - 5g
5. (2x + 7) (2x - 7)
7.
(1 + 2z) (1 - 2z) 8. ] 5t + 1 g ] 5t - 1 g 9. ] 3t + 2 g ] 3t - 2 g
6. (4y + 3) (4y - 3)
10. ] 3 + 4x g ] 3 - 4x g
11. (x + 2y) (x - 2y)
12. ^ 6x + y h ^ 6x - y h
13. ] 2a + 3b g ] 2a - 3b g 17. (a + b - 3) (a - b + 1)
18. ] z + w + 1 g ] z - w - 1 g
1 1 19. d x + n d x - n 2 2
y 3
+ 1oe
y 3
22. (x 2 + 1) (x 2 - 1) = (x 2 + 1) (x + 1) (x - 1) 23. _ 3x 3 + 2y i _ 3x 3 - 2y i 24. _ x 2 + 4y 2 i ^ x + 2y h ^ x - 2y h 25. (a 4 + 1) (a 2 + 1) (a + 1) (a - 1)
Exercises 2.13 2. ] x + 3 g ^ x 2 - 3x + 9 h
1.
(b - 2) (b 2 + 2b + 4)
3.
]t + 1g^t - t + 1h
5.
(1 - x) (1 + x + x )
7.
(y + 2z) (y - 2yz + 4z 2)
9.
^ 2x + 3y h _ 4x 2 - 6xy + 9y 2 i 10. ] ab - 1 g ^ a 2 b 2 + ab + 1 h
4. (a - 4) (a + 4a + 16) 2
2
2
6. ^ 2 + 3y h _ 4 - 6y + 9y i 2
8. (x - 5y) (x 2 + 5xy + 25y 2)
11. (10 + 2t) (100 - 20t + 4t 2)
12. d
x x 2 3x - 3ne + + 9o 4 2 2
10 1 100 10 1 13. d + ne 2 + o a b ab b 2 a
17. d 1 -
5.
5 ] a - 1 g2 6. - ] 2x - 3 g ] x - 4 g 7. 3z ] z + 5 g ] z + 4 g
8.
ab ] 3 + 2ab g ] 3 - 2ab g 9. x ] x + 1 g ] x - 1 g
10. 2 ] 3x - 2 g ] x + 2 g 11. ] m - 5 g ] 3 + n g
12. - 7 ] 2x + 1 g
14. ] x - 1 g ] x + 2 g ^ x 2 - 2x + 4 h
13. ^ y + 5 h ^ y + 4 h ^ y - 4 h
15. ] x + 1 g ^ x 2 - x + 1 h ] x - 1 g ^ x 2 + x + 1 h 16. x ] x + 2 g ] x - 5 g 17. ] x + 3 g (x - 3) 2 19. 3 ] 2 - b g ^ 4 + 2b + b 2 h
18. y (2xy + 1) (2xy - 1)
20. 3 ] 3x - 2 g ] 2x + 5 g 21. 3 ] x - 1 g2 23. z ] z + 3 g2
22. (x + 2) (x + 5) (x - 5)
26. 4a (a + 3) (a - 3)
27. 5x ] 2 - x g ^ 4 + 2x + x 2 h
28. (a + 2) (a - 2) (a + 3) (a - 3)
29. 4k (k + 5) 2
Exercises 2.15 1.
x 2 + 4x + 4 = ] x + 2 g2
3.
x 2 - 10x + 25 = ] x - 5 g2
16. - 9 ^ a 2 - a + 1 h
2. b 2 - 6b + 9 = ] b - 3 g2
5.
m - 14m + 49 = ] m - 7 g
7.
x 2 + 2x + 1 = ] x + 1 g2
9.
x 2 - 20x + 100 = ] x - 10 g2
4. y 2 + 8y + 16 = ^ y + 4 h2
2
6. q 2 + 18x + 81 = ^ q + 9 h2
2
8. t 2 - 16t + 64 = ] t - 8 g2
10. w 2 + 44w + 484 = ] w + 22 g2 11. x 2 - 32x + 256 = ] x - 16 g2
12. y 2 + 3y +
13. x 2 - 7x +
49 7 2 = dx - n 4 2
14. a 2 + a +
15. x 2 + 9x +
81 9 2 = dx + n 4 2
16. y 2 -
17. k 2 -
14. ^ x + 1 - y h _ x 2 + 2x + 1 + xy + y + y 2 i 15. ^ 5xy + 6z h _ 25x 2 y 2 - 30xyz + 36z 2 i
5 ^ y - 1 h _ y 2 + y + 1 i 4. 2ab ^ a + 2b) (2a - 1 h
30. 3 (x + 1) (x - 1) (x + 3)
- 1 o 21. ^ x + 2y + 3 h ^ x - 2y + 1 h
2
3.
25. 2 ] x + 2 g ] x - 2 g ^ x + y h _ x 2 - xy + y 2 i
16. ^ x + 2 + y h ^ x + 2 - y h
20. e
2 ] x + 3 g ] x - 3 g 2. 3 ^ p + 3 h ^ p - 4 h
24. ] x + 1 g ] x - 1 g ] 2x + 3 g ] 2x - 3 g
15. ] 2a + 9b g ] 2a - 9b g
14. ^ x + 10y h ^ x - 10y h
1.
11k 121 11 n + = dk 4 2 16
5y 2
3 2 9 = dy + n 4 2
1 1 2 = da + n 4 2
+
25 5 2 = dy - n 4 16
2
18. x 2 + 6xy + 9y 2 = ^ x + 3y h2 19. a 2 - 4ab + 4b 2 = ] a - 2b g2 20. p 2 - 8pq + 16q 2 = ^ p - 4q h2
2
x x x ne1 + + o 9 3 3
18. ^ x + y + 3 h _ y - 3y - xy + 9 + 6x + x i 2
Exercises 2.16
2
19. ^ x + y - 1 h _ x 2 + 4x - xy + y 2 - 5y + 7 i 20. (2a + 6 - b) (4a 2 + 24a + 2ab + 6b + b 2 + 36)
1.
a+2
2. 2t - 1
6.
1 y-4
7.
10. 14.
p+5 3
2 ] b - 2a g a-3
11.
a+1 a+3
p-2 4p - 2p + 1 2
3.
15.
4y + 1 3
s-1 s+3
8.
12.
4.
4 2d - 1 9.
3+y x + 2x + 4
a+b 2a - b
2
5.
x 5x - 2
b2 + b + 1 b+1 13. x - 3
ANSWERS
Exercises 2.17 1.
2.
(a)
(a)
(d) 3.
5x 4
(b)
Exercises 2.20
13y + 3
b 2a - 1
(b)
a+8 12
(d)
^ p - 2 h _ q2 - q + 1 i
ab
a+b+3 (c) a+b
-x + 2 (b) x ]x - 1g
(a)
x - 13 6
b 2 ^ x + 2y h
10 ] 2b - 1 g
2 _ 3y 2 + 14y + 13 i
^y + 2h^y + 3h^y - 1h x2 ] x + 2 g
(b)
8 _ y 2 - 3y + 9 i
1. 3 5
(e)
3p 2 + 5pq - 2q 2
pq ^ p + q h ^ p - q h
9. - 4 2
10. 4 5
13. - 3
14.
15. 5 7
16.
- ] 5x + 22 g (j) ]x + 4g]x - 4g]x + 3g
1.
5.5 7. 377
12. 22.4 17.
15y
2. 47
14.
3 4
8. 14
14. - 84
16. 28
18. - 2 105
17.
30
25. 2 3
26.
3
31.
2 2
(e) 0.6
4. 375
3 10 3
32.
2
(a)
10 + 6
(m) 10 6 - 120
5. - 196 3 4
2.
2.
(i) 4 2 (n) 9 3
(p) 6 3
(q) 3 11
(r) 5 5
(a) 6 3
(b) 20 5
(c) 28 2
(f) 8 14 3.
4.
(a)
18
(f)
160
(g) 72 5 (b) (g)
20 117
(d) 4 7
(h) 30 2
(c)
176
(h)
98
2 5
33.
5
34.
2 2
2 3
35.
5 7
(c) 12 + 8 15
21. 4
(e) 16 5 (j) 24 5
(d)
128
(e)
75
(i)
363
(j)
1008
(a) x = 45 (b) x = 12 (c) x = 63 (d) x = 50 (e) x = 44 (f) x = 147 (g) x = 304 (h) x = 828 (i) x = 775 (j) x = 960
(h) 5 - 5 15
(l) 210 - 14 15
(n) - 10 - 2 2
(o) 4 3 - 12
(c) 2 10 - 6 + 10 15 - 15 6 (d) 12 20 + 18 60 - 8 10 - 12 30 = 24 5 + 36 15 - 8 10 - 12 30 (f) 15 - 15 + 18 10 - 6 6
(h) - 1
(i) - 12
(n) 7 + 2 10
(j) 43
(k) 3
(o) 11 - 4 6
(l) - 241 (p) 25 + 6 14
(r) 27 - 4 35
(s) 77 - 12 40 = 77 - 24 10
(t) 53 + 12 10
3.
(a) 18
(d) 19 + 6 2
4.
(a) a = 21, b = 80
5.
(a) a - 1
6.
k = 25
9.
a = 107, b = - 42
(o) 7 5
(i) 14 10
3 5
6
1 2
(a) 10 + 3 6 + 3 5 + 9 3 (b) 10 - 35 - 2 + 14
(j) 3 6
(h) 5 3
9
29.
(g) - 6 - 12 6
(q) 57 + 12 15
(m) 8 2
2 5
1
(j) 2 54 + 6 = 6 6 + 6
(e) 6 2
(g) 4 3
28.
8
24.
(f) 5 33 + 3 21
1.
(l) 10 3
1
23. 1
(e) - 6 + 4 18 = - 6 + 12 2
(m) - 6
(k) 4 7
22. 4 3
(b) 2 6 - 15
(g) 4
(f) 10 2
19. 18
(d) 5 14 - 2 21
Exercises 2.19 (d) 5 2
27.
6. 30
15. 2
21. 2 6
1
5. - 6 6
12 = 2 3
12. 15 28 = 30 7
15. 15 16. 10
(c) 2 6
10.
11. 2 48 = 8 3
1.
18. 23.987 19. 352.47 20. 93
(b) 3 7
4. 10 14
9. 60
(e) 52 - 13 10
(a) 2 3
7-5 2
Exercises 2.22
10. 51.935 11. - 1
8. 284 9. - 40
3. 3 6
15
(k) - 8 + 12 12 = - 8 + 24 3
(d) - 37.7
3. - 7
21.
24. - 2 - 2 3
13. 2 20 = 4 5
30.
a 2 - 2ab - b 2 + 1 ]a + bg]a - bg
(c) 48.1
13. 1838.8
12 = 2 3
2.
(i) 6 + 30
(f) 2.3 (g) - 5.3 6.
21
7. - 12 55
^x + yh^x - yh
(b) - 6.9
12. 5 3
2
17. 13 6
2
20. 5 2 - 2 3
23. 7 6 + 3 5
20. 30 50 = 150 2
y ^x + y + 1h
(a) - 7.1
2
19. 47 3
11.
6. 3 6
Exercise 2.21
2]x - 1g (f) ]x + 1g]x - 3g
Exercises 2.18 1.
5. - 3 5
25. - 17 5 + 10 2 2x (d) x+2
^y + 2h^y + 1h
(d)
4. 3 3
8. 8 5
22. - 2 3 - 4 5
x 2 + 10x - 24 3b 2 - 5b - 10 (d) (e) x 2 ]x - 3g]x - 4g 2b ] b + 1 g 3x - 13 3 - 5x (a) (b) ]x - 5g]x - 2g]x + 3g ]x + 2g]x - 2g (c)
3. 6 3
2
7. - 7 2
(c)
5.
2.
18. - 9 10
a+2 (h) ] a + 1 g2
- 3x + 8 (g) ]x + 2g]x - 2g
4.
(e)
6
]x - 3g]x - 1g (e) ]x - 5g]x - 2g
^ p + qh^ p - qh + 1 p2 - q2 + 1 = (e) p+q p+q
(i)
4p + 3
(c)
q+1
x 2 - xy + y 2
5 (a) x
(c)
15
(b) 108 2
(c) 432 2
(e) 9
(b) a = 19, b = - 7
(b) 2p - 1 - 2 p ^ p - 1 h 7. 2x - 3y - 5 xy
8. a = 17, b = 240
10. 9 + 5 units 2
545
546
Maths In Focus Mathematics Preliminary Course
Exercises 2.23 1.
(a) (e) (h)
2.
7 7
8. 6 4
(b)
3+ 6 3
(f)
3 14 - 4 7 14
(c)
2 15 5
(d)
12 - 5 2 2 (i)
6 14 3 14 = 5 10
(g)
(a) 4 3 - 4 2 = 4 ^ 3 - 2 h
(b)
(j)
4 15 - 2 10 35
6 15 - 9 6 + 2 10 - 6 2
3.
So rational 9.
1.
2.
(j) (l) 4.
(i)
2-1
=
28 - 2 6 - 7 3 13
(b) a = 1, b = 8
8 5 (d) a = - 1 , b = 9 9
=
(k)
2 15 + 2 10 - 2 6 - 3 - 5 2
(a) a = 45, b = 10
5.
4 6+9 3 21
15 30 - 30 5 - 4 3 30
+ 2+1 2-1
2 2-1
2+1 2-1 ^ 2 - 1h^ 2 - 1h
+ +
4 2
(a) 4 (b) 14
7.
3 5 - 2 - 15 - 3 3
#
4 2 2
^ 2 h2 - 1 2 2- 2- 2+1 = +2 2 2-1 3-2 2 = +2 2 1 =3-2 2+2 2 =3 So rational 6.
1 1 (c) a = - , b = 2 2
(e) a = 5, b = 32
4
#
x = -^ 3 + 2h
10.
2 2
(a) - 2y
(b) a + 4
b+4 b+4 b-4
(f) 6 2
(g) 4 5
(c) - 6k 5
(d)
5x + 3y
(e) 3a - 8b
15
(b) ] a + 3 g ] a - 1 g (c) 4ab ] b - 2 g
(a) ] x + 6 g ] x - 6 g
(e) 2 ^ 2n - p + 3 h
(d) (y - 3) (5 + x)
(f) (2 - x) (4 + 2x + x 2) 3.
20 12 + 19 6 + 25 3 - 6 19 6 + 65 3 - 6 (g) = 15 15 6+9 2+2 3 6
2
Test yourself 2
(a) 2 2 (b) - ^ 2 + 6 - 3 2 + 3 3 h = - 2 - 6 + 3 2 - 3 3 22 5 + 14 2 (c) 39 ^ (d) - 6 6 - 16 - 3 84 + 8 14 h 10 - 3 6 + 8 + 3 21 - 4 14 = 5 (e) - 4 (f) 4 2
(h)
2
6-4 2 +4 2 9-4#2 6-4 2 = +4 2 1 =6-4 2+4 2 =6
(c)
(f)
2 3-2 2 8 = # + # 3+2 2 3-2 2 2 2^3 - 2 2 h 8 2 + = 2 2 32 - ^ 2 2 h =
-^ 6 + 7 3 h 47
- ^ 2 15 - 4 18 h - 2 ^ 15 - 6 2 h = 19 19 - ^ 19 - 8 3 h 8 3 - 19 = (d) (e) 6 + 2 + 5 3 + 5 2 13 13
8
+
2
5 + 2 10 5
8 5 + 3 10 20
2 3+2 2
(b) 2x 2 + 5x - 3
(a) 4b - 6
(d) 16x - 24x + 9 (g) 2 6 - 5 3 (a)
5.
V = 157.464
(f) - 1 - 7a
(h) 3 3 - 6 + 21 - 2 7
8
4.
(c) 4m + 17
(e) p 2 - 25
2
(b)
b 2 ^ a 2 + 3a + 9 h
6. (a) 17
15 ] m - 2 g2 (b)
6 15 - 9 17
4x + 5 8. (a) 36 (b) - 2 ]x + 3g]x - 2g 1 9. (a) (b) 8 10. d = 11.25 5 2 3 2+ 6 11. (a) (b) 15 2 7.
12. (a) 3 6 - 6 - 4 3 + 4 2
(c) 2
(d) 216
(b) 11 + 4 7
(b) 6 ] x - 3 g ] x + 1 g
13. (a) 3 (x - 3) (x + 3)
(c) 5 ^ y + 2 h _ y 2 - 2y + 4 i
14. (a)
x3
(b)
3y 4
15. (a) 99
1 3x - 1
(b) 24 3
16. (a) a 2 - b 2
(b) a 2 + 2ab + b 2
17. (a) ] a - b g2
(b) ] a - b g ^ a 2 + ab + b 2 h
18.
3 3+1 2
20.
21 5 - 46 - 2 7
(c) 16
19. (a)
4b + 3a ab
(c) a 2 - 2ab + b 2
(b)
3x - 11 10
(e) 2
ANSWERS
21. (a) 6 2 (f)
(b) - 8 6
m
24. (a)
(d)
3 7 7
6 15
5+1 2
(c)
(e)
x + 10 10
17a - 15 21
1 k-1
(b)
(e)
71 121
20 + 3 15 + 4 10 + 3 6 53 (c)
3 - 2x (x + 1) (x - 1)
15 - 6 - 15 3 - 15 2 3
(b) n = 175
(d) n = 5547
27. (b), (c)
(c) n = 392
28. (d)
33. (a)
29. (a), (d)
34. (d)
30. (c)
35. (b)
(b) y 4 - 4
2. 4.
x2 +
2 3. or 4 2 2
b b2 b 2 n x + 2 = dx + a a 2 4a
4x 2 + 12x + 9 = ] 2x + 3 g2 ]a + 1g
a2 - a + 1
10. d
11. w = 13
7.
(d) ] b - 2 g ] a + 2 g ] a - 2 g y+1
2]x - 1g
8. 2 5
13. x = 14
16. p = 3
5. k = 5
5 8
14. x = - 1
17. t = 8.2
18. x = - 9.5
20. x = - 3 24. y = 1
1.
t = 8.5
6.
r = 6.68
21. b = 0.8 2 25. t = - 1 3
22. a = - 0.375
2x 1 1 2 + = dx + n 9 3 3
20. r =
21. s = 2 + 6 3
4. a = 41
3 4 r
=
71 121 3 r 4r
(b) a =
8. n = 15
11. (a) BMI = 25.39 12. r = 0.072
19. x = 5.5
20. r = 3.3
5. y = 4
9. y 1 = 3
2 3
(b) w = 69.66 13. x 1 = - 9
17. r = 10.46
14. t = 2.14
18. x = 1.19
Exercises 3.4 1.
2.
16. x = 2
(a) 3
7. x = 6.44
16. r = 2.12
(a) x 2 3
-4
- 66 6 + 4 2 - 15 + 4 5 - 65 3 13
18.
3. b = 8
-3
-2
-1
0
1
2
3
-2
-1
0
1
2
3
4
(b) y # 4
3x + 4 (b) ] 2x - 1 g2
2
400 - 59 5 10
2. l = 122
15. x = ! 2
-4
13. x 3 - 7x 2 + 15x - 9
1 2
12. t = 30
15. x = - 0.4
2 a 2 a + nd - n x b x b
12. (a) 8x - 12x + 6x - 1
19. i = 1
9 35
x = 36 7. t = 0.6 8. x = - 3 9. y = - 1.2 10. x = 69
]x - 3g]x + 3g]x - 2g 3
17.
4. x = 1
6.
3x 3 - 6x 2 + 3x + 4xy - 6y
15. x 2 +
2. x = 35 3. y = 4
4 9
b =3
(c) h = 1.94
(a) ] x + 4 g ] x + 9 g
2
14.
1 3
1.
10. h = 3.7
(c) ] 5x + 7 g ^ 25x 2 - 35x + 49 h
11.
30. x Z 4.41
1
(b) _ x 2 - 3y i _ x 2 + 2y i = (x + 3 y) (x - 3 y) _ x 2 + 2y i
9.
29. p = 5
2
17 3 + 2 5 + 20 17
6.
16. x = 20 17. m = 20 18. x = 4 19. a = - 7 20. y = 3 2 21. b = - 4 22. x = 3 23. a = - 1 24. t = - 4 3 1 25. x = 1.2 26. a = 1.6 27. b = 28. t = 39 8
Exercises 3.3
(c) 8x - 60x + 150x - 125
5.
2. z = - 5.6 3. y = 1 4. w = 6.7 5. x = 12 1 8. b = 35 9. n = - 16 10. r = 4 6. x = 4 7. y = 15 11. y = 9 12. k = 6 13. d = 2 14. x = 5 15. y = 15
23. x = 3
(a) 2a 2 b - 8ab 2 + 6a 3 3
t = -5
19. q = 22
Challenge exercise 2 1.
1.
Exercises 3.2
(e) n = 1445
32. (b)
Chapter 3: Equations
Exercises 3.1
12 - 2 6 15
31. (c)
(e) 30a 2 b
3
(b) 10 14 - 5 21 - 6 10 + 3 15
(b)
25. (a) n = 48
26. 3
4
(d) 43 (e) 65 - 6 14
(c) 7
(d)
(d)
(g) 2x - 3y
3n 4
22. (a) 2 6 + 4
23. (a)
(c) 2 3
17 14 , b=23 23
-3
4
(a) t 2 7 (b) x $ 3 (c) p 2 - 1 (d) x $ - 2 (e) y 2 - 9 1 2
(f) a $ - 1
(g) y $ - 2
(j) y 1 12
(k) b 1 - 18
(l) x 2 30
(n) m 2 14
2 3
1 4
(r) w 1 2
4 5
(v) x 2 - 1
(o) b $ 16 (s) x $ 35
2 3
(h) x 1 - 2
(w) b # - 11
(m) x # 3
(p) r # - 9
(t) t $ - 9 1 4
(i) a # - 6 3 4
(q) z 2 8
(u) q 2 - 6
2 5
547
548
Maths In Focus Mathematics Preliminary Course
3.
(a) 1 1 x 1 7 1
0
3.
2
3
4
5
6
7
8
0
1
2
3
4
5
(b) - 2 # p 1 5 -3
-2
-1
4.
(c) 1 1 x 1 4 -3
-2
0
-1
1
2
3
4
-2 -1 1 2 (e) 1 y 1 1 6 3 -1
-2
0
1
2
3
4
0
1
2
3
5
5.
5
4
1.
2.
(a) x = ! 5
(b) y = ! 8
(e) x 2 6, x 1 - 6
(f) - 10 # p # 10
(g) x = 0
(i) - 12 1 y 1 12
(j) b $ 20, b # - 20
(a) x = 5, - 9
8.
5 (f) x = 5, -4 7
(h) x $ 9, x # - 6
(i) x = ! 12
(j) 2 # a # 10 3.
(a) x = 1
1 4
(b) a = 3, -
1 3
(c) b = 2
(d) No solutions (e) y = - 2 3 1 (h) d = 2 , -1 4 2 4.
(a) x = 2, -
1 2
(a) t = 3, -1
-3
-2
(b) y = 3, 2
1 3
(e) d = 4, -5
2 5
(b) - 1
-1
9.
(f) x = 7 (g) m = 5, 1
4 , -2 5
(i) y =
1 3
(d) x = 4, -7 5.
2 7
1 3 2 3
2.
(a) x = 3
2 3
(f) a = 2 (g) x = ! 2 (h) b = 9 2 3
0
1
2
3
4
5
(b) y = ! 8
(e) p = 10
(f) x = ! 5
(i) n = ! 4
(j) q = - 2
(c) n = ! 2 (g) y = ! 3
(d) x = ! 2 5
1 512 1 (e) x = ! 8
(e) y = 1.89 (f) d = ! 2.55 (g) k = ! 4.47 (h) x = 2.22 (j) y = 3.01
(b) x = 6
1 7
1 2
1 4
(f) x = 4
(j) m = 1
(d) x = !
(c) a =
1 81
1 625 19 (h) n = 7 32 (d) k =
(g) y = ! 8
127 216
(a) n = 4
(b) y = 5
(c) m = 9
(f) x = 3
(g) x = 2
(h) x = 2
(a) x = 2
(b) x = 1 (c) x = - 2 (d) n = 2 (e) x = 0 1 (g) y = (h) x = 2 (i) x = 2 (j) a = 0 3
1 2
(a) m =
(b) x =
1 3 3 4
(e) k = -
2 3
(f) n =
(i) k = -
1 6
(j) x = 1
(a) x = - 1
(c) x =
(d) x = 5 (i) x = 1
1 3
(g) x = 1
(d) k = -
1 2
(h) n =
2 3
1 2
2 3
(b) x = - 1
1 3
(c) k = - 4
(f) x = -
2 3
(g) x = - 4
1 2
(j) x = 18
10. (a) m =
1 4
(b) k = - 2
(e) n =
1 18
(f) n = 1 7
(e) m = 0 (j) k = 2
3 4
1 2
(c) x = 2 (g) x =
4 5
3 8
(d) n = 3 1 2
(h) x = - 1
(d) k = 1
1 2
(h) b = - 3
1 6
7 11
(j) m = 5
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
(h) w = 2
(a) p = ! 6.71 (b) x = 4.64 (c) n = 2.99 (d) x = ! 5.92 (i) y = ! 3.81
(j) b = ! 1
(a) x =
(i) x = - 1
Exercises 3.6 1.
(e) n =
(c) y =
4 5
2 1t 13 5
1 2
(h) y = 27
(b) a =
(i) x = 1
3 5
(g) b = 216
1 5
(e) x = - 2
(j) No solutions
(c) a = - 10, 1
1 2
(d) t = 8
(a) x =
(f) x = 6
(c) a 2 2, a 1 - 2
(e) x = 3, -6
1 1y 12 2
7.
(h) a 2 14, a 1 - 14
(b) n = 4, -2
(d) 4 # x # 6
6.
(c) - 4 1 a 1 4
(d) k $ 1, k # - 1
(g) - 3
(f) m = 625 (j) t = 81
(i) b = 8
Exercises 3.5
(c) x = 32
(i) a = 128
(i) x = !
-3
(b) t = 16
(e) p = 243
5
(d) - 3 # y # 5
-3
(a) n = 27
Exercises 3.7 1.
y = 0, -1
5.
x = -2, -7
2. b = 2, -1 6. q = !3
3. p = 3, -5 7. x = !1
4. t = 0, 5
8. a = 0, -3
ANSWERS
9.
x = 0, - 4
12. y = 1, -1 16. x = 1, 2 20. x = 3, 4
10. x = ! 1 2
1 2
1 2
11. x = -1, -1
3 1 , 4 2
13. b =
17. x = 0, 5
14. x = 5, -2 15. x = 0, 18. y = - 1, 2
21. m = - 6, 1
23. y = 1, -5, -2
1 3
1 12. y 1 - 1 , y 2 2 2 2 3
19. n = 3, 5
24. x = 5, -7
25. m = 8, -1
15. - 1
(a) x = ! 5 - 2
(b) a = ! 7 + 3
(c) y = ! 23 + 4
(e) p = ! 44 - 7 = ! 2 11 - 7
18. - 1 # a # 1
19. - 2 1 x 1 3
20. x # - 1, x $ 3
21. 0 1 x 1 2
22. 1 # a # 1
1 2
23. y # - 2, y $
(f) x = ! 28 + 5 = ! 2 7 + 5
1. 4.
x = 6, y = 17
(h) x = ! 2 + 1
7.
x = - 3, y = 2
! 5+3 (j) y = 2
a = 1, b = 3
10. m = 2, n = 3
(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
4 5
25. 1 # x # 1
1 3
Exercises 3.11
(g) y = ! 88 - 10 = ! 2 22 - 10 = 2 ^ ! 22 - 5 h (i) n = ! 137 - 12
1 1 #x #2 3
17. x 1 - 4, x 2 4
2 1 24. m 1 - 1 , m 2 1 3 2
(d) x = ! 13 - 1
2.
2 5
2 ,x $1 3
16. - 4 # y # 3
22. x = 0, -1, -2
Exercises 3.8 1.
14. b 1 - 3, b 2
13. x #
2. x = 2, y = 1
3. p = 2, q = - 1
5. x = - 10, y = 2
6. t = 3, v = 1
8. x = - 64, y = - 39 11. w 1 = - 1, w 2 = 5
13. p = - 4, q = 1
9. x = 3, y = - 4 12. a = 0, b = 4
14. x 1 = 1, x 2 = - 1
15. x = - 1, y = - 4 16. s = 2, t = - 1 17. a = - 2, b = 0
18. k = - 4, h = 1
19. v 1 = - 2, v 2 = 4
20. x = 2, y Z 1.41
Problem Exercises 3.9 1.
23 adults and 16 children.
(a) y = - 0.354, - 5.65 (c) b = 3.54, - 2.54
(d) x = 1, - 0.5
(e) x = - 0.553, 0.678 (g) m = - 2, - 5
(b) x = 1, 1.5 (f) n = 0.243, -8.24
(h) x = 0, 7
(i) x = 1, - 6
(j) y = 2.62, 0.382 2.
(a) x =
- 1 ! 17 2
(c) q =
4 ! 28 = 2! 7 2
(b) x =
5 ! 13 6
- 12 ! 128 -3 ! 2 2 (d) h = = 8 2
- 5 ! 73 (g) d = 12
2 ! 32 (h) x = =1!2 2 2 (j) x =
1.
x = 0, y = 0 and x = 1, y = 1
2.
x = 0, y = 0 and x = - 2, y = 4
3.
x = 0, y = 3 and x = 3, y = 0
4.
x = 4, y = - 3 and x = 3, y = - 4
6.
x = 3, y = 9
8.
m = - 4, n = 0 and m = 0, n = - 4
9.
x = 1, y = 2 and x = - 1, y = - 2
5. x = - 1, y = - 3
7. t = - 2, x = 4 and t = 1, x = 1
10. x = 0, y = 0 and x = 1, y = 1
8 ! 40 4 ! 10 = (e) s = 6 3 - 11 ! 133 (f) x = 2
Exercises 3.12
1! 5 (i) t = 2
7 ! 41 4
11. x = 2, y = 1 and x = - 1, y = - 2
12. x = 0, y = 1
13. x = 1, y = 5 and x = 4, y = 11 1 14. x = , y = 4 and x = - 1, y = - 1 4
1 1 15. t = - , h = 4 2
16. x = 2, y = 0 17. x = 0, y = 0 and x = - 2, y = - 8 and x = 3, y = 27 18. x = 0, y = 0 and x = 1, y = 1 and x = - 1, y = 1 19. x =
Exercises 3.10
3 1 ,y =2 4 2
20. x = -
5 12 ,y =13 13
Exercises 3.13
1.
-3 1 x 1 0
2. 0 1 y 1 4
4.
x # - 2, x $ 2
7.
c 1 - 1, c 2 2
10. b # - 2, b $ -
3. n # 0, n $ 1
5. n 1 - 1, n 2 1 8. - 4 # x # - 2 1 2
6. - 5 # n # 3 9. 4 1 x 1 5
11. a 1 - 1, a 2
1 3
1.
x = - 2, y = - 8, z = - 1
2. a = - 2, b = - 1, c = 2
3.
a = - 4, b = 2, c = 7
4. a = 1, b = 2, c = - 3
5.
x = 5, y = 0, z = - 2
6. x = 0, y = - 5, z = 4
7.
p = - 3, q = 7, r = 4
8. x = 1, y = - 1, z = 2
9.
h = - 3, j = 2, k = - 4
10. a = 3, b = - 1, c = - 2
549
550
Maths In Focus Mathematics Preliminary Course
Test yourself 3 1.
(a) b = 10
(b) a = - 116
2.
(a) A = 1262.48
3.
(a) x 2 - 8x + 16 = ] x - 4 g2
4.
(a) x = - 2, y = 5
5.
(a) x = 2
6.
(a) b = 2, -1
7.
(a) A = 36
9.
-1 1 y # 3
(d) p # 4
(b) P = 8558.59
(b) y = 1 3
(c) x = - 7
(i) y = 40c (j) x = 80c 2. (a) 121c (b) 72c 29l (c) 134c 48l 3. (a) 42c (b) 55c 37l (c) 73c 3l 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) 26c11l (ii) 116c11l (j) (i) 38c51l (ii) 128c51l 5. (a) x = 49c (b) 41c (c) 131c 6. (a) y = 15c, x = z = 165c (b) x = 142c, y = 48c, z = 28c (c) a = 43c, b = 137c, c = 101c (d) a = 97c, b = d = 41c, c = 42c (e) a = 68c, b = 152c, c = 28c (f) a = 10c, b = 150c
7.
8x - 10 + 2x - 10 + x + 10 + 7x + 10 = 360
(b) k 2 + 4k + 4 = ] k + 2 g2
1 (b) x = 4, y = 1 and x = - , y = - 8 2 1 4
(b) g = 2,
(b) b = 12
1 4
(c) x $ 4, x # 3
8. x =
1 ,1 2
(angle of revolution)
10. (a) x = - 0.298, -6.70
18x = 360 x = 20 +ABE = 8x - 10 = 8 (20) - 10 = 150c +EBC = 2x - 10 = 2 (20) - 10 = 30c +ABE + +EBC = 150c + 30c = 180c ` +ABC is a straight angle +DBC = 7x + 10 = 7 (20) + 10 = 150c +DBC + +EBC = 150c + 30c = 180c ` +DBE is a straight angle ` AC and DE are straight lines
(b) y = 4.16, -2.16
(c) n = 0.869, -1.54 11. (a) V = 764.5
(b) r = 2.9
13. x 1 2, x 2 9
14. x = 2.4, y = 3.2
(b) r = 3.9
16. (a) ii
12. x 2 71
(b) i
(c) ii
1 4
15. (a) V = 2100 (d) iii
(e) iii
17. a = 3, b = 2, c = - 4 18. n 2 0, n 1 - 3 19. x = - 4
1 3
20. x = - 2
(c) x = 2
(d) x = 2
(g) - 4 # x # 2
21. (a) y 2 3 (e) x = 3, -1
(h) x = - 3
(j) x # - 1, x $ 1
2 5
(b) - 3 # n # 0 (f) t $ 1, t # - 2
(i) y 2 2, y 1 - 2
5 (k) x = 6
8.
1 3 (m) No solutions (n) t = 2 , 3 5
=x ` +AFC = x
(vertically opposite angles)
(o) - 1 1 x 1 3
+CFE = 180c - (x + 180c - 2x) (+AFB is a straight angle)
(p) m # - 3, m $ 2
=x ` +AFC = +CFE ` CD bisects +AFE
Challenge exercise 3 1.
y =1
3.
a = 3, b = !2
+DFB = 180c - (180 - x) c (+AFB is a straight angle)
1 (l) - # b # 2 2
2. x 1 - a, x 2 a
9.
4. x = 2.56, -1.56
+ABD + +DBC
] x + 3 g ] x - 3 g ] x - 2 g ^ x 2 + 2x + 4 h; x = ! 3, 2
= 110 - 3x + 3x + 70 = 180c
6.
x = 1, y = 2 and x = - 1, y = 0
7.
b = 4; x = ! 17 + 4 Z 8.12, - 0.123
So +ABC is a straight angle. AC is a straight line.
9.
-1 1 t 1 1
5.
12. r = 2.31
10. - 3 # x # 8
13. No solutions
15. P = 2247.36
16. x =
8. x = ! 1 1 11. x = 4
10. +AEB + +BEC + +CED = 50 - 8y + 5y - 20 + 3y + 60 = 90c
14. x = ! b + a 2 + a
2 ^ 4 ! 10 h 3
17. y 1 -1, y 2
So +AED is a right angle.
3 5
Exercises 4.2 Chapter 4: Geometry 1
Exercises 4.1 1.
(a) y = 47c (b) x = 39c (c) m = 145c (d) y = 60c (e) b = 101c (f) x = 36c (g) a = 60c (h) x = 45c
1.
(a) a = b = e = f = 148c , c = d = g = 32c (b) x = z = 70c , y = 110c (c) x = 55c , y = 36c , z = 89c
(d) y = 125c , x = z = 55c
(e) n = e = g = a = c = z = x = 98c, o = m = h = f = b = d = y = w = 82c
ANSWERS
(f) a = 95c , b = 85c , c = 32c
5.
(g) a = 27c , b = 72c , c = 81c (h) x = 56c , y = 124c , z = a = 116c , b = 64c (i) x = 61c 2.
(a)
(j) y = 37c
+CGF = 180c - 121c
(FGH is a straight angle)
= 59c ` +BFG = +CGF = 59c These are equal alternate angles. ` AB < CD (b) +BAC = 360c - 292c = 68c
+ACB = 180c - 124c = 56c +CBA + 68c = 124c +CBA = 124c - 68c = 56c ` +CBA = +ACB = 56c ` D ABC is isosceles
6.
y = 38c
7.
(a) x = 64c
(DCB is a straight angle) (exterior angle of D)
(b) x = 64c , y = 57c
(c) x = 63c
(d) a = 29c , b = 70c
(angle of revolution)
` +BAC + +DCA = 68c + 112c = 180c These are supplementary cointerior angles.
8.
` AB < CD (+BCE is a straight angle) (c) +BCD = 180 - 76 = 104c +ABC = +BCD = 104c These are equal alternate angles.
+KJL = 180c - 60c = 120c +JLK = 180c - (30c + 120c) = 30c
(d) +CEF = 180 - 128 (+CED is a straight angle) = 52c +CEF = +ABE = 52c These are equal corresponding angles. 9.
1.
(a) x = 60c
(b) y = 36c
(c) m = 71c
(e) x = 30c
(f) x = 20c
(g) x = 67c
(d) x = 37c
2.
3. 4.
These are equal alternate angles. ` MN ; QP
Exercises 4.4 1.
(a) Yes AB = EF = 5cm
(given)
BC = DF = 6 cm
(given)
AC = DE = 8 cm
(given)
So all angles in an equilateral triangle are 60c.
` D ABC / DDEF
(SSS)
] 90 - x g c
(b)Yes
` AB < DE
(angle sum of D JKL)
BC = BD
All angles are equal. Let them be x. Then x + x + x = 180 (angle sum of D) 3x = 180 x = 60
(vertically opposite angles) +ACB = 50c +ABC = 180c - (50c + 45c) (angle sum of D) = 85c ` +DEC = +ABC = 85c These are equal alternate angles.
(KJI is a straight angle)
10. +OQP = 180 - ] 75 + 73 g (angle sum of triangle) = 32c ` +MNO = +OQP = 32c
(i) a = 75c , b = 27c , c = 46c (k) x = 67c , y = z = 59c , w = 121c
(angle sum of D JIL)
` AB ; ED
(h) a = 73c
(j) a = 36c , b = 126c , c = 23c
(angle sum of D IKL)
`+BDC = 46c (base angles of isosceles triangle) +CBD = 180 - 2 # 46 = 88c `+CBD = +BDE = 88c These are equal alternate angles.
= 42c
Exercises 4.3
(HJL is a straight angle)
` +JLK = +JKL = 30° ` D JKL is isosceles
(e) +CFH = 180 - ] 23 + 115 g (+EFG is a straight angle) `+BFD = 42c (vertically opposite angles) +ABF + +BFD = 138c + 42c = 180c These are supplementary cointerior angles. ` AB ; CD
(angle sum of D HJI)
Since +IJL = +JIL = +ILJ = 60c, D IJL is equilateral
` AB ; CD
`AB ; CD
+HJI = 180c - (35c + 25c) = 120c +IJL = 180c - 120c = 60c +JIL = 180c - (90c + 30c) = 60c +ILJ = 180 - (60c + 60c) = 60c
XY = BC = 4.7 m
(given)
+XYZ = +BCA = 110c (given) YZ = AC = 2.3 m
(given)
` D XYZ / DABC
(SAS)
(c) No
551
552
Maths In Focus Mathematics Preliminary Course
(b) +ABC = +ADC
(d) Yes +PQR = +SUT = 49c
(given)
+PRQ = +STU = 52c
(given)
triangles)
7.
(given)
`DPQR / DSTU
(AAS)
+AOB = +COB = 90c (given)
(a) AB = KL = 4 (given) (given) +B = +L = 38c (given) BC = JL = 5 ` by SAS, D ABC / D JKL
(given) (c) MN = QR = 8 (given) NO = PR = 8 (given) MO = PQ = 5 ` by SSS, D MNO / D PQR
(given) (e) BC = DE = 4 (given) +C = +E = 90c (given) AC = EF = 7 ` by SAS, D ABC / D DEF
(a)
(alternate angles, AD < BC) +ADB = +DBC BD is common ` by AAS, D ABD / D CDB ` AD = BC
(b) +OCB = +OBC
(base angles of OBC, an isosceles
Similarly +OBA = 45c ` +OBA + +OBC = 45c + 45c = 90c So +ABC is right angled 8.
(a) +AEF = +BDC = 90c
(given)
AF = BC
(given)
FE = CD
(given)
`DAFE / DBCD
(RHS)
(b) +AFE = +BCD
(corresponding angles in congruent triangles)
9.
(a) OA = OC
(equal radii)
OB is common AB = BC
(given)
`DOAB / DOBC
(SSS)
(b) +OBA = +OBC
(corresponding angles in congruent triangles)
But +OBA + +OBC = 180c
(a)
OB is perpendicular to AC.
10. (a) AD = BC +ADC = +BCD = 90c DC is common `DADC / DBCD (b) AC = BD
(equal radii)
Exercises 4.5
OB = OD
(similarly)
(given) (SAS) (corresponding sides in congruent
1.
(a) x = 15.1
(b) x = 4.4
(c) m = 6.6
(vertically opposite angles)
(d) a = 76c , i = 23c , b = 81c
(e) b = 4.5
`DAOB / DCOD
(SAS)
(f) a = 115c , x = 19c , y = 3.2
(g) p = 9.7
(b) AB = CD
(corresponding sides in congruent
2.
a = 1.81, b = 5.83
3.
+BAC = +EDC +ABC = +DEC +ACB = +ECD
triangles)
6.
(given)
triangles)
OA = OC
+AOB = +COD
(ABC is a straight angle)
So +OBA = +OBC = 90c
(corresponding sides in congruent Ds)
5.
(angle sum of triangle)
So +OCB = +OBC = 45c
(b) ` BD = DC (corresponding sides in congruent Ds) ` AD bisects BC (alternate angles, AB < CD)
(SAS)
But +OCB + +OBC = 90c
+B = +C (base angles of isosceles D) +BDA = +CDA = 90c (given) AD is common ` by AAS, D ABD / D ACD
+ABD = +BDC
`DOAB / DOBC
right angled triangle)
(given) (d) +Y = +T = 90c (given) +Z = +S = 35c (given) XY = TR = 1.3 ` by AAS, D XYZ / D STR
4.
(equal radii)
QR = TU = 8 cm
(given) (b) +Z = +B = 90c (given) XY = AC = 7 (given) YZ = BC = 2 ` by RHS, D XYZ / D ABC
3.
(a) OA = OC OB is common
(e) No 2.
(corresponding angles in congruent
(a) AB = AD
(given)
BC = DC
(given)
` since 3 pairs of angles are equal, DABC ||| DCDE
AC is common `DABC / DADC
(alternate angles, AB < ED) (similarly) (vertically opposite angles)
(SSS)
ANSWERS
4.
(given) +GFE = +EFD GF 1.5 o = = 0.5 EF 2.7 2.7 EF o = = 0.5 DF 4.86 GF EF ` = EF DF Since two pairs of sides are in proportion and their included angles are equal, then DDEF ||| DFGE
1.3 AB = = 0.714 5. DE 1.82 4.2 AC = = 0.714 DF 5.88 4.9 BC = = 0.714 EF 6.86 AC BC AB = = ` DE DF EF Since three pairs of sides are in proportion, D ABC ||| D DEF y = 41c 6.
(a) OA = OB OC = OD OA OB ` = OD OC +AOB = +COD
(equal radii) (similarly)
D ABC ||| D ACD, x = 109c, y = 47c 11. (a) x = 7.8
(b) AB = 5.21 cm
12. (a)
(c)
+ABF = +BEC +CBE = +BFA ` +C = +A
AB 10. CD BC AC AC AD AB ` CD
= = = =
2 = 0.769 2.6 3 = 0.769 3.9 3.9 = 0.769 5.07 BC AC = AD AC
Also `
BD AD = AE CE AD DF Also = AE EG BD DF ` = CE EG 14. y = 0.98
2.
(a) p =
3.
s = 6.2 m
5.
AB 2 = 81, CB 2 = 144, CA 2 = 225 AB 2 + CB 2 = 81 + 144 = 225 = CA 2 ` D ABC is right angled
6.
XY = YZ = 1 ` D XYZ is isosceles
15. x = 3.19, y = 1.64
61
(b) y = 6.6
(c) b = 5.7
(b) t =
(c) x =
58
4. CE = 15.3 cm
YZ 2 = XY 2 = 1, XZ 2 = 2 YZ 2 + XY 2 = 1 + 1 =2 = XZ 2 ` D XYZ is right angled
(alternate angles, AB z CD) (similarly, BC z AD) (angle sum of Ds)
+A is common 1.2 AD = = 0.4 AB 3 0.8 AE = = 0.4 2 AC AD AE ` = AB AC Since two pairs of sides are in proportion and their included angles are equal, D AED ||| D ABC, m = 4.25
AB AD = AE AC AD AF = AE AG AB AF = AC AG
(b)
(a) x = 6.4
` since 3 pairs of angles are equal, D ABF ||| DCEB 9.
(e) x = 1.4, y = 9.2
1.
(b) x = 2.17, y = 2.25 8.
(c) x = 6.5
Exercises 4.6
(corresponding angles, BC < DE) (similarly)
` since 3 pairs of angles are equal, D ABC ||| D ADE
AB AD = DE BC AD AF Also = DE FG AB AF ` = BC FG
13. a = 4.8, b = 6.9
(a) +A is common +ABC = +ADE +ACB = +AED
(b) m = 4.0, p = 7.2
(d) x = 6.2, y = 4.4
(vertically opposite angles)
Since two pairs of sides are in proportion and their included angles are equal, 3 OAB ||| 3 OCD
7.
Since three pairs of sides are in proportion,
7.
AC 2 = AB 2 + BC 2 2 2 2 = ^ 3 h + BC 2 4 1 `1 AC
8.
= 3 + BC 2 = BC 2 = BC =2 =2#1 = 2BC
(a) AC = 5 (b) AC 2 = 25, CD 2 = 144, AD 2 = 169 AC 2 + CD 2 = 25 + 144 = 169 = AD 2 ` D ACD has a right angle at +ACD ` AC is perpendicular to DC
(d) m = 6.6 65
(d) y =
33
553
554
Maths In Focus Mathematics Preliminary Course
9.
AB =
3b
11. d 2 = ] 20 - 3t g 2 + ] 15 - 2t g 2 = 400 - 120t + 9t 2 + 225 - 60t + 4t 2 = 13t 2 - 180t + 625 12. 1471 mm
(d) a = 121c, b = 52c, i = 77c (e) x = 60c (f) x = 3, y = 7
x2 + y2 x
10.
6.
+ADB = +CDB +CDB = +ABD +ADB = +DBC ` +ABD = +DBC ` BD bisects +ABC
7.
(a) AD = BC = 3.8 cm AB = DC = 5.3 cm
13. 683 m 14. 12.6 m 15. 134.6 cm
16. 4.3 m 17. 42.7 cm 18. 1.3 2 + 1.1 2 = 2.9 and 1.5 2 = 2.25 1.3 2 + 1.1 2 ! 1.5 2 so the triangle is not right angled ` the property is not a rectangle
20. (a) BC 2 = 6 2 - 4 2 = 20 BC = 20 AO = 6 cm (equal radii) So AC 2 = 6 2 - 4 2 = 20 AC = 20 Since BC = AC, OC bisects AB
Since one pair of opposite sides is both equal and parallel, ABCD is a parallelogram. (c) +X + +M = 54c + 126c = 180c These are supplementary cointerior angles. ` XY < MN Also, XM < YN
(b) +OCA = +OCB = 90c (given) OA = OB (equal radii) OC is common ` DOAC / DOBC (RHS) So AC = BC (corresponding sides in congruent triangles) ` OC bisects AB
1.
(a) x = 94c (b) y = 104c (c) x = 111c (d) x = 60c (e) y = 72c (f) x = 102°, y = 51° (g) x = 43°, y = 47°
2.
D ABE is isosceles. ` +B = +E = 76c (base +s equal) +CBE = +DEB = 180c - 76c = 104c (straight +s) +D + 62c + 104c + 104c = 360c (angle sum of quadrilateral) +D + 270c = 360c +D = 90c ` CD is perpendicular to AD`
3.
(a)
+D = 180c - x (+A and +D cointerior angles, AB < DC)
+C = 180c - (180c - x)
(+C and +D cointerior angles, AD < BC)
= 180c - 180c + x =x `+A = +C = x +B = 180c - x (+B and +C cointerior angles, AB < DC) `+B = +D = 180c - x (b) Angle sum = x + x + 180c - x + 180c - x = 360c 4.
a = 150c , b = 74c
5.
(a) a = 5 m, b = 3 m, x = z = 108c, y = 72c (b) x = 53c, y = 56c, z = 71c (c) x = y = 5 cm, a = b = 68c
(given) (given)
Since two pairs of opposite sides are equal, ABCD is a parallelogram. (b) AB = DC = 7cm (given) AB < DC (given)
19. No. The diagonal of the boot is the longest available space and it is only 1.4 m.
Exercises 4.7
(BD bisects +ADC) (alternate angles, AB < DC) (alternate angles, AD < BC)
(given)
` XMNY is a parallelogram (d) AE = EC = 5 cm DE = EB = 6 cm
(given) (given)
Since the diagonals bisect each other, ABCD is a parallelogram. 8.
(a) x = 5 cm, i = 66c (b) a = 90c, b = 25c, c = 65c (c) x = 3 cm, y = 5 cm (d) x = 58c, y = 39c (e) x = 12 cm
9.
6.4 cm
11. 4 2 cm
10. +ECB = 59c, +EDC = 31c, +ADE = 59c 12. x = y = 57c
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.
128c34l 5. (a) 13
8.
2340c
(b) 152c18l 6. 16
7. 3240c
9. 168c23l
10. Sum = 145n = (n - 2) # 180c 145n = 180n - 360 = 35n 10.3 = n But n must be a positive integer. ` no polygon has interior angles of 145c. 11. (a) 9
(b) 12
(c) 8
(d) 10
(e) 30
12. (a) ABCDEF is a regular hexagon. (equal sides) AF = BC FE = CD (equal sides) +AFE = +BCD (equal interior angles) ` D AFE / D BCD (SAS)
ANSWERS
S = ] n - 2 g # 180c = (6 - 2) # 180c = 720c 720c +AFE = 6 = 120c Since AF = FE, triangle AFE is isosceles. So +FEA = +FAE (base angles in isosceles triangle) 180 - 120c ` +FEA = (angle sum of triangle) 2 = 30c +AED = 120 - 30c = 90c Similarly, +BDE = 90c
(b)
So +AED + +BDE = 180c These are supplementary cointerior angles `AE < BD 13. A regular octagon has equal sides and angles. (equal sides) AH = AB GH = BC (equal sides) +AHG = +ABC (equal interior angles) ` D AHG / D ABC (SAS) So AG = AC (corresponding sides in congruent triangles)
S = ] n - 2 g # 180c = (8 - 2) # 180c = 1080c 1080c ` +AHG = 8 = 135c +HGA = +HAG
360 p (b) Each interior angle: 360 180 p 180p 360 = p p 180p - 360 = p 180 ^ p - 2 h = p
15. (a)
Exercises 4.9 1.
(a) 26.35 m 2 (b) 21.855 cm 2 (c) 18.75 mm 2 (d) 45 m 2 (e) 57 cm 2 (f) 81 m 2 (g) 28.27 cm 2 2. 4.83 m 2
3.
(a) 42.88 cm 2 (b) 29.5 m 2 (c) 32.5 cm 2 (d) 14.32 m 2 (e) 100.53 cm 2 4. (a) 25 m 2 (b) 101.85 cm 2 (c) 29.4 m 2 (d) 10.39 cm 2 (e) 45 cm 2
5.
7 51 + 98 = 7 ^ 51 + 14 h cm 2
7.
$621.08
9.
(a) 48 cm
8. (a) 161.665 m 2 (b) 27 cm
6. 22.97 cm 2
(b) 89 m 2
(c) 10.5 m
10. 12w units 2
Test yourself 4
(base angles in isosceles triangle)
180 - 135c `+HAG = (angle sum of triangle) 2 = 22c30l +GAC = 135 - 2 # 22c30l = 90c We can similarly prove all interior angles are 90c and adjacent sides equal. So ACEG is a square.
1.
(a) x = 43c, y = 137c, z = 147c (b) x = 36c (c) a = 79c, b = 101c, c = 48c (d) x = 120c (e) r = 7.2 cm (f) x = 5.6 cm, y = 8.5 cm (g) i = 45c
2.
+AGF = i
So +AGF = +CFE = i These are equal corresponding +s. ` AB < CD 3.
118.28 cm2
4.
(common) (a) +DAE = +BAC (corresponding angles, DE < BC) +ADE = +ABC (similarly) +AED = +ACB ` D ABC and D ADE are similar (AAA)
] 5 - 2 g # 180c 5 = 108c
14. +EDC =
ED = CD (equal sides in regular pentagon) So EDC is an isosceles triangle. (base angles in isosceles triangle) `+DEC = +ECD 180 - 108c +DEC = (angle sum of triangle) 2 = 36c +AEC = 108 - 36c = 72c Similarly, using triangle ABC, we can prove that +EAC = 72c 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.)
(vertically opposite +HGB)
(b) x = 3.1 cm, y = 5.2 cm 5.
162c
6. 1020.7 cm3
8.
(a) AB = AD BC = DC
7. 36 m (adjacent sides in kite) (similarly)
AC is common ` Δ ABC and Δ ADC are congruent (SSS) (b)
AO = CO BO = DO +AOB = +COD
(equal radii) (similarly) (vertically opposite angles)
` Δ AOB and Δ COD are congruent (SAS) 9.
73.5 cm2
2 10. 6 2 + ^ 2 7 h = 36 + 28 = 64 = 8 2 ` ΔABC is right angled (Pythagoras)
555
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Maths In Focus Mathematics Preliminary Course
11.
AF AD = AE AG AD AB = AE AC AF AB ` = AG AC
12. (a) AB = AC +B = +C BD = DC
(equal ratios on intercepts)
Challenge exercise 4 1.
94c
4.
+BAD = +DBC +ABD = +BDC ` +ADB = +DCB
3. 1620c, 32c 44l
(b) +ADB = +ADC (corresponding +s in congruent Ds) (straight +) But + ADB + +ADC = 180c
5.
So +ADB = +ADC = 90c
(base +s of isosceles D) (exterior + of D)
6.
^ base +s equal h
So Δ ACD is isosceles
AB = DC (given) +A + +D = 131c + 49c = 180c +A and +D are supplementary cointerior angles ` AB < DC Since one pair of opposite sides are both parallel and equal, ABCD is a parallelogram.
So AD and BC are perpendicular. +ACB = 68c +CAD = 68c - 34c = 34c ` ˚+CAD = +ADC = 34c
(given) (alternate angles, AB < DC) (angle sum of D)
` since 3 pairs of angles are equal, D ABD
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