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
556
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 <; D BCD d = 6.74 cm
(given) (base +s of isosceles D) (AD bisects BC, given)
` D ABD / D ACD ] SAS g
13.
2. x = 75c , y = 46c , z = 29c
(similarly)
27.36 m 2
7.
14.
+DAC = +ACB +BAC = +ACD
(alternate +s, AD < BC) (alternate +s, AB < DC)
Let ABCD be a square with diagonals AC and BD and +D = 90c (adjacent sides of square) AD = DC ` D ADC is isosceles (base angles of isosceles D) `+DAC = +DCA (angle sum of D) +DAC + +DCA = 90° ` +DAC = +DCA = 45° Similarly, +BAC = +BCA = 45°
AC is common ` D ABC / D ADC (AAS) (corresponding sides in congruent Ds) ` AB = DC Similarly, AD = BC ` opposite sides are equal 15. (a) 24 cm2 (b) 5 cm
16. 9
(other angles can be proved similarly)
8.
17. +BFG + +FGD = 109c - 3x + 3x + 71c = 180c These are supplementary cointerior +s. ` AB < CD 18. 57 cm2 19. +ACB = 180c - ] +A + +B g = 180c - x - y +ACD = 180c - +ACB z = 180c - (180c - x - y) = 180c - 180c + x + y =x+y 20. (a)
+A AC EF AB DE AC ` EF
= +E 2.97 = = 1.1 2.7 3.96 = = 1.1 3.6 AB = DE
(+sum of D) (straight +)
^ given h
So Δ ABC and Δ DEF are similar (two sides in proportion, included +s equal). (b) x = 4.3 cm
Let ABCD be a kite (given) AD = AB (given) DC = BC AC is common ` by SSS, D ADC / D ABC ` +DAC = +BAC (corresponding angles in congruent Ds) (given) (found)
AD = AB +DAE = +BAE AE is common
ANSWERS
` by SAS, D ADE / D ABE ` +DEA = +BEA (corresponding angles in congruent Ds) But +DEA + +BEA = 180c (DEB is a straight angle) ` +DEA = +BEA = 90c ` the diagonals are perpendicular 9.
(exterior angle of D MNZ) +MNY + 84c = (15c + 112c) ` +MNY = 43c (exterior angle of D XYZ) +XYZ + 69c = 112c ` +XYZ = 43c ` +MNY = +XYZ = 43c These are equal corresponding angles. ` MN < XY
10. x = 2.12 m
(b) 10 + 2 5 = 2 ^ 5 + 5 h m
11. (a) 6 m 2
12. x = 28.7 cm, y = 3.8 cm 14. (a)
13. x = 7.40 m, y = 4.19 m
(adjacent sides in square) AB = BC +ABE = +CBE = 45 (diagonals in square make 45c with sides)
EB is common. ` by SAS, D ABE / DCBE ` AE = CE (corresponding sides in congruent Ds) Since AB = BC and AE = CE, ABCE is a kite.
18. r = 1.55 19. x 2 1
20.
7 15
21. x =
4 ! 12 =2! 3 2
23. x = 4, y = 11 or x = -1, y = - 4 25. 7 28.
24. x = 2, y = -1
27. ] 2x - 1 g ^ 4x 2 + 2x + 1 h
26. 7.02 cm
6 15 + 2 6 43
1 49
22.
29. 7
30. $643.08
32. -2 10 + 3 5 - 2 2 + 3
31. 1.1
33. $83.57
34. x = 22c, y = 29c, w = z = 90c 35. 56.7 cm 2 36. a - 21 b 10 = 39. - x - 7
b 10 a 21
37. x 2 6, x 1 -2
40. x =
1 4
2 5
38.
41. x # -3, x $ 3
= =
48. x = 98c, y = 41c
2x 2 2x
50. (a) 12x - 8y
1 BD 2 2x = units 2
DE =
2. 2 ^ 5 + y h ^ x - y h
4
3. (a) x - 1
p =9
4.
6y - 10
7.
2 x= 7
9.
(given) +ABC = +EDC = 90° (vertically opposite angles) +ACB = +ECD (given) AB = ED ` by AAS D ABC / DEDC (corresponding sides in congruent triangles) ` AC = EC ` D ACE is isosceles
(b) x 3
6. x 3 + 2x 2 - 16x + 3
2 8. x-3
11. - 3
14. 3 10 - 4
(b) 2 31
2 17
47. x = 53c
1 3x + 2 (c)
x-3
(d) 3 2 + 1
x 2 - 3x + 9
y7 - ]x + 5g 11 3 (f) (g) x - 14 y 7 z -11 = 14 11 ]x + 1g]x - 1g 6 x z 3 1 (h) (i) 8 5 (j) 13 2 5a ] a + b g ] 1 + 2b g
1.
10. 231.3
49.
54
(e)
Practice assessment task set 1
25 + 5 2 23
1 6
(adjacent sides in rhombus) AB = BC (alternate +s, AD < BC) +DAC = +ACB (base +s of isosceles D ABC) +BAC = +ACB ` +DAC = +BAC ` diagonal AC bisects the angle it meets. Similarly, diagonal BD bisects the angle it meets.
x2 + x2
5.
42.
43. Given diagonal AC in rhombus ABCD:
44. ] x + 3 g-1 45. x 3 + 6x 2 + 12x + 8 46. (b) BD =
1 8
12. 135c
13. 7.33 # 10
15. 3.04 16. x + 3
-2
17. x = 1.78, -0.281
51. x = 2.7, y = 3.1
52. x = 25
53. r =
2 3
r
cm
54. 17.3 cm 55. Let +DEA = x (base +s of isosceles D) Then +EAD = x +CDA = x + x (exterior + of DEAD) = 2x ` +ABC = 2x (opposite +s of < gram are equal) ` +ABC = 2+DEA 56.
2 5
57. 5% 58. 2.2 # 10 8 kmh -1
60. 9xy y
59. k = 20
61. 147c 16l 62. 5.57 m 2
63. (a) 5 ] a + 2b - 2 g ^ a 2 - 4a - 2ab + 4b 2 + 4b + 4 h (b) ] 3a + 4b g ] a - 6b + 2c g
557
558
Maths In Focus Mathematics Preliminary Course
64. - 1 65.
BC < AD (ABCD is a < gram) BC < FE (BCEF is a < gram) ` AD < FE Also BC = AD ^ opposite sides of < gram h BC = FE ^ similarly h ` AD = FE Since AD and FE are both parallel and equal, AFED is a parallelogram.
66. b = 11.95 m 68.
Exercises 5.3
3 1 #x 15 4 8
67. (a)
34 cm
18 3 + 31 2 - 25 5 75
1.
(h) x-intercept - 3 5 , y-intercept 5 (i) x-intercept -3, no y-intercept (j) x-intercept !3, y-intercept 9
(b) 30 cm2 2.
69. 20 70. 32 m
71. BD bisects AC So AD = DC +BDC = +BDA = 90c (given) BD is common ` DBAD / DBCD (SAS) (corresponding sides in congruent ` AB = CB
3.
2
78. (d)
73. (b)
74. (c)
75. (a) 76. (b)
77. (b)
79. (d)
4.
5. 6.
Chapter 5: Functions and graphs
Exercises 5.1 1.
Yes
2. No
3. No
8.
Yes
9. Yes
10. No
14. No
4. Yes 5. Yes 11. Yes
6. Yes 7. No
12. No
8.
13. Yes
15. Yes
Exercises 5.2
9.
1.
f ] 1 g = 4, f ] -3 g = 0
3.
f ] 5 g = -25, f ] -1 g = -1, f ] 3 g = -9, f ] -2 g = -4
5.
-35
6. x = 9
2. h ] 0 g = -2, h ] 2 g = 2, h ] -4 g = 14
7. x = !5
8. x = -3
10. f ^ p h = 2p - 9, f ] x + h g = 2x + 2h - 9 11. g ] x - 1 g = x 2 + 2 13. t = -1; t = 2, -4
12. f ] k g = ] k - 1 g ^ k 2 + k + 1 h
f ] - x g = ] - x g2 - 1 = x2 - 1 = f (x) ` even function
7. f ] - x g = 4 ] - x g - ] - x g 3 = - 4x + x 3 = - ^ 4x - x 3 h = - f ]xg ` odd function
f ] -x g = ] -x g 4 + ] -x g 2 = x4 + x2 = f ]xg ` even function f ]xg - f ]- xg = 0 (a) Odd (b) Neither (c) Even (d) Neither (e) Neither
12. (a) (i) x 2 0
(ii) x 1 0
(iii) Even
(b) (i) x 1 2
(ii) x 2 2
(iii) Neither
(d) (i) All real x ! 0 (e) (i) None
1.
20. (a) 3 (b) x - 3 = 3 - 3 = 0 Denominator cannot be 0 so the function doesn’t exist for x = 3. (c) 4 22. 4x + 2h + 1
25. (a) 2 (b) 0
(ii) x 1 -2, x 2 2 (ii) None
(ii) All real x
(c) n 4 + n 2 + 2
(iii) Neither
(iii) Odd
(iii) Neither
Exercises 5.4
19. -28
23. 5] x - c g 24. 3k 2 + 5
f ] - x g = - x = -f ] x g ` odd function
11. (a) No value of n (b) Yes, when n is odd (1, 3, 5, …)
14. 0
21. f ] x + h g - f ] x g = 2xh + h 2 - 5h
g ] - x g = ] - x g8 + 3 ] - x g4 - 2 ] - x g 2 = x 8 + 3x 4 - 2x 2 = g (x) ` even function
(c) (i) -2 1 x 1 2
16. f ] 2 g - f ] -2 g + f ] -1 g = 0 - 4 + 1 = -3 18. 7
(d) Neither odd nor even
(b) Odd values i.e. n = 1, 3, 5, f
15. f ] 5 g = 125, f ] 1 g = 1, f ] -1 g = -1
17. 10
(b) 7 f ] x g A 2 = x 6 + 2x 3 + 1
10. (a) Even values i.e. n = 2, 4, 6, f
4. 14
9. z = 1, -4
(a) f ^ x 2 h = x 6 + 1
(c) f ] - x g = - x + 1
triangles)
x2 + y2
f ] -x g = ] -x g 2- 2 = x2 - 2 = f (x) ` even function
3
So triangle ABC is isosceles 72.
2 , y-intercept -2 3 (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 (a) x-intercept
(a) x-intercept 2, y-intercept -2 1 (b) x-intercept -1 , y-intercept 3 2 1 (c) x-intercept , y-intercept 1 2 (d) x-intercept -3, y-intercept 3 2 1 (e) x-intercept , y-intercept 3 3
ANSWERS
2.
(a)
y
(e) 5
5 4 3 2 1
4 3 2 1 1
-4 -3 -2 -1 -1
2
3 4
-2 -3 -4 -5 (b)
1 21
-4 -3 -2 -1 -1 -2 -3 -4 -5
x
y
(f)
y
2 3 4
5 4 3 2 1
5 4 3 2 1 -4 -3 -2 -1 -1
1
2 3
-4 -3 -2 -1 -1 -2 -3 -4 -5
x
4
-2 -3
x
1 2 3 4
-4 -5 (c)
(g) 5 4 3 2 2 - 1
y 5 4 3
3
2
-4 -3 -2 -1 -1 -2 -3 -4 -5
1 -4 -3 -2 -1 -1
1
2
3
x
4
-2 -3 -4
x
5 4 3 2 1
y 5 4 3 2 1 -4 -3 -2 -1 -1 -2 -3 -4 -5
1 2 3 4
y
(h)
-5 (d)
y
1 2 3 4
x
-4 -3 -2 -1 -1 -2 -3 -4 -5
1 2 3 4
x
559
560
Maths In Focus Mathematics Preliminary Course
Exercises 5.5
y
(i) 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)
4 3 2 1 1
-4 -3 -2 -1 -1
2
3
4
x
y 6 5 4 3 2 1
-2 -3 -4 -5 y
(j)
-4 -3 -2 -1 -1 -2 -3 -4 -5
5 4 3 2 1 -4 -3 -2 -1 -1
3.
4.
111 2
2 3
4
6
-2
5
-3
4
-4
3
-5
2 1
(a) " all real x ,, " all real y , (b) " all real x ,, " y: y = 2 , (c) ! x: x = -4 +, " all real y , (d) ! x: x = 2 +, " all real y , (e) ! all real x +, " y: y = 3 , (a) Odd
y
(b)
x
(b) Even
x
1 2 3 4 5
(c) Neither (d) Odd
-4 -3 -2 -1 -1
1
2
3
4
x
5
-2 -3
(e) Odd
-4
5.
y
-5
5
(c)
4 3 2 1 -4 -3 -2 -1 -1 -2 -3 -4 -5
(3, -1)
111 2 2
3
4
x
y 6 5 4 3 2 1 -4 -3 -2 -1 -1 -2 -3 -4 -5
1 2 3 4
5
x
ANSWERS
y
(d) 6 5 4 3 2 1
5 4 3 2 1
1 2 3 4
-4 -3 -2 -1 -1
5
-4 -3 -2 -1 -1 -2 -3 -4 -5 -6
x
-2 -3 -4 -5 y
(e)
1 2 3 4
x
5
y
(i) 5 4 3 2 1
6 5 4 3 2 1 -4 -3 -2 -1 -1 -2 -3 -4 -5
1 2 3 4
5
-4 -3 -2 -1 -1 -2
x
1112 2 3 4
5
1 2 3 4
5
x
-3 -4 -5 -6 y
(f)
y
(h)
y
(j)
12 10 8 6 4 2
5 4 3 2 1
1 2
-4 -3 -2 -1 -2 -4 -6 -8
3 4
5
-4 -3 -2 -1 -1 -2 -3 -4
x
x
-5 -6
-10 3.
5 4 3
(a) (i) x-intercepts 3, 4, y-intercept 12 (ii) {all real x}, 1 ( y: y $ - 2 4 (b) (i) x-intercepts 0, -4, y-intercept 0 (ii) {all real x}, " y: y $ -4 ,
2 1
(c) (i) x-intercepts -2, 4, y-intercept -8 " y: y $ - 9 ,
y
(g)
-4 -3 -2 -1 -1 -2 -3 -4 -5 -6
1 2 3 4
5
(ii) {all real x},
(d) (i) x-intercept 3, y-intercept 9 (ii) {all real x}, " y: y $ 0 ,
x
(e) (i) x-intercepts ! 2, y-intercept 4 " y: y # 4 , 4.
(a) {all real x}, " y: y $ -5 ,
(ii) {all real x},
(b) {all real x}, " y: y $ - 9 ,
561
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Maths In Focus Mathematics Preliminary Course
1 (c) {all real x}, ( y: y $ -2 2 4
5.
5
(a) 0 # y # 9
3
4
(b) 0 # y # 4
(c) -1 # y # 24 1 (e) -18 # y # 2 4
(a) (i) x 2 0 (ii) x 1 0
2 1
(b) (i) x 1 0 (ii) x 2 0
-4 -3 -2 -1 -1
(c) (i) x 2 0 (ii) x 1 0 (d) (i) x 1 2 (ii) x 2 2 (e) (i) x 2 -5 (ii) x 1 -5 7.
8.
f ] -x g = - ] -x g = -x2 = f (x) ` even
2.
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
-2 -3
2
-4 -5 y
(d)
(a) Even (b) Even (c) Even (d) Neither (e) Neither (f) Even (g) Neither (h) Neither (i) Neither (j) Neither
5 4 3
Exercises 5.6 1.
y
(c)
(e) {all real x}, " y: y $ 0 ,
(d) -4 # y # 21 6.
(d) {all real x}, " y: y # 0 ,
2 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 2 (g) x-intercept , y-intercept 2 3 4 (h) x-intercept - , y-intercept 4 5 1 (i) x-intercept , y-intercept 1 7 (j) No x-intercepts, y-intercept 9 (a)
-4 -3 -2 -1 -1 -2 -3 -4 -5
5 4 3 2
5
1
4
-4 -3 -2 -1 -1 -2
3 2 1 1
2
3
4
5
x
-4 -5 (f)
-3 -4
y 5
-5
4 3
y
2
5 4 3 2 1 -4 -3 -2 -1 -1 -2 -3 -4 -5
x
-3
-2
(b)
y
(e)
y
-4 -3 -2 -1 -1
x
1 -4 -3 -2 -1 -1 -2 1 2 3 4 5
x
-3 -4 -5
x
ANSWERS
3.
y
(g) 5 4 3 2
4.
1 1
-4 -3 -2 -1 -1 -2
2
3
4
x
5
5.
-3
(a) {all real x}, " y: y $ 0 , (b) {all real x}, " y: y $ -8 , (c) {all real x}, " y: y $ 0 , (d) {all real x}, " y: y $ -3 , (e) {all real x}, " y: y # 0 , (a) (i) x 2 2 (ii) x 1 2 (b) (i) x 2 0 (ii) x 1 0 1 1 (d) (i) x 2 0 (ii) x 1 0 (c) (i) x 2 1 (ii) x 1 1 2 2 (e) (i) x 1 0 (ii) x 2 0 (a) 0 # y # 2
(b) - 8 # y # -4
(d) 0 # y # 11
-4 -5
6.
(a) x 2 -3 (e) x 1 -2
(b) x 1 0
7.
(a) x = !3
(b) x 2 1, x 1 -1
y
(h) 5
(c) 0 # y # 6
(e) -1 # y # 0 (c) x 2 9
(d) x 2 2
(c) -2 # x # 2
(d) x = -1, -3 (e) x = 3 (f) x = 1, 2 (g) -3 1 x 1 5
4
(h) - 4 # x # 2
(j) x # 2, x $ 4 1 (k) - 4 # x # 1 (l) x # 0, x $ 1 (m) x = 2, 2 (n) No solutions (o) x = 0 (p) x = 1 (q) x = 0, -2 1 (t) x = 0, 6 (r) No solutions (s) x = 3
3 2 1 1
-4 -3 -2 -1 -1
2
3
4
x
5
-2 -3
Exercises 5.7
-4
1.
-5
(i) x 2 4, x 1 0
(a) (i) {all real x: x ! 0}, {all real y: y ! 0} (ii) no y-intercept (iii)
y
y
(i)
5
5
4
4
3
3
2
2
1
1 1
-4 -3 -2 -1 -1
2
3
4
x
5
x
-2
2
3
4
-2
-3 -4
-3
-5
-4
(j)
1
-4 -3 -2 -1 -1
-5
y
(b) (i) {all real x: x ! 0}, {all real y: y ! 0} (ii) no y-intercept
5 4
(iii)
3
y
2
2
1 -4 -3 -2 -1 -1
1
2
3
4
5
x
1
-2 -3 -4 -5
-2
1
-1 -1
-2
2
x
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Maths In Focus Mathematics Preliminary Course
(c) (i) {all real x: x ! -1}, {all real y: y ! 0} (ii) 1
5
y
(iii)
y
(iii)
4
2
3 2
1
1
-2
x
x
-1
1
-4 -3 -2 -1 -1
2
1
2 3
4
5
-2
-1
-3 -4
-2
-5
1 (d) (i) {all real x: x ! 2}, {all real y: y ! 0} (ii) -1 2
(g) (i) {all real x: x ! 1}, {all real y: y ! 0} (ii) -4
y
(iii)
y
(iii)
5 4
5
3
4
2
3
1
2 1
x -4 -3 -2 -1 -1
1 2
3
4
5
-2
-4 -3 -2 -1 -1
-3
-2
-4
-3
-5
-4
(e) (i) {all real x: x ! -2}, {all real y: y ! 0} (ii)
x 1
2
3
4
5
-5
1 6
(h) (i) {all real x: x ! -1}, {all real y: y ! 0} (ii) -2
y
(iii)
y
(iii) 2
5 4
1
3
x -2
1
-1
2
-1
1
x -4 -3 -2 -1 -1
-2 (f) (i) {all real x: x ! 3}, {all real y: y ! 0} (ii)
2
-2 2 3
-3 -4 -5
1
2
3
4
5
ANSWERS
(i) (i) ' all real x: x !
Exercises 5.8
1 2 1, {all real y: y ! 0} (ii) 2 3
1.
y
(a) (i)
y
(iii)
3
2
1
1 2
-1
-2
-
-1
x 1
x
3
-3
2
2 3
-3
-2
(ii) ! x: -3 # x # 3 +, " y: -3 # y # 3 ,
(j) (i) {all real x: x ! -2}, {all real y: y ! 0} (ii) -3
y
(b) (i)
y
(iii)
4
5 4 3 2
x
4
-4
1
x -4 -3 -2 -1 -1
1
2
3
4
5
-2
-4
-3 -4
(ii) ! x: -4 # x # 4 +, " y: -4 # y # 4 ,
-5
2.
y
(c) (i)
2 f ] -x g = -x 2 =x = - f (x) ` odd function
5 4 3 2
3.
4.
1 (a) # y # 1 9
1 (b) # y # 1 3
3 (d) # y # 3 7
1 (e) - 2 # y # 8
(a) 1 # x # 3
(b) 1 # x # 4
(d) 1 # x # 4
(e) 1 # x # 2
1 1 (c) -2 # y # 2 2
(c) - 6 # x # 0
(2, 1)
1
x -4 -3 -2 -1 -1 -2 -3 -4 -5
1
2
3
4
565
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Maths In Focus Mathematics Preliminary Course
(ii) ! x: 0 # x # 4 +, " y: -1 # y # 3 ,
(iii) ! x: -5 # x # 5 +, " y: -5 # y # 0 , (b) (i) Above x-axis
y
(d) (i)
y
(ii) 5 4 3 1
2 1 -4 -3 -2 -1 -1
1
2
3
x
4
-2
x
1
-1
-3 (iii) ! x: -1 # x # 1 +, " y: 0 # y # 1 ,
-4 -5
(c) (i) Above x-axis
(ii) ! x: -4 # x # 2 +, " y: -3 # y # 3 , (e) (i)
y
(ii)
y 6
5 4 3 2 (-2, 1)
6
-6
x
1 1
-4 -3 -2 -1 -1
2
3
x
4
(iii) ! x: -6 # x # 6 +, " y: 0 # y # 6 , (d) (i) Below x-axis
-2
(ii)
y
(ii) ! x: -3 # x # -1 +, " y : 0 # y # 2 , 2.
(a) (i) Below x-axis (ii)
y
8
-8
-8
5
-5
-5
x
(iii) ! x: -8 # x # 8 +, " y: -8 # y # 0 ,
x
ANSWERS
6.
(e) (i) Below x-axis (ii)
y
(a) {y: - 9 # y # 3} (b) {y: 0 # y # 9} (c) {y: -8 # y # 1} 1 (d) ' y: # y # 1 1 (e) {y: 0 # y # 4} 5 (f) {y: -1 # y # 15} (g) {y: -1 # y # 0} (h) " y: - 1 # y # 8 , (i) {y: - 4 # y # 21} 1 (j) ' y: - 6 # y # 6 1 4
7
- 7
x
7.
(a) {all real x: x ! -1} (b) x-intercept: y = 0 3 0= x+1 0=3 This is impossible so there is no x-intercept (c) {all real y: y ! 0}
8.
(a) {all real x: x ! 0}
(b) {all real y: y ! !1}
9.
(a)
y
- 7 (iii) " x: - 7 # x # 3.
7 ,, # y: - 7 # y # 0 -
(a) Radius 10, centre (0, 0) (b) Radius
5 , centre (0, 0)
25
(c) Radius 4, centre (4, 5) (d) Radius 7, centre (5, -6) (e) Radius 9, centre (0, 3) 4.
20 15 10
(a) x 2 + y 2 = 16 (b) x - 6x + y - 4y - 12 = 0 (c) x 2 + 2x + y 2 - 10y + 17 = 0 (d) x 2 - 4x + y 2 - 6y - 23 = 0 (e) x 2 + 8x + y 2 - 4y - 5 = 0 (f) x 2 + y 2 + 4y + 3 = 0 (g) x 2 - 8x + y 2 - 4y - 29 = 0 (h) x 2 + 6x + y 2 + 8y - 56 = 0 (i) x 2 + 4x + y 2 - 1 = 0 (j) x 2 + 8x + y 2 + 14y + 62 = 0 2
2
5
x -4 -3 -2 -1 -5
5.
(a) {all real x}, {all real y} (b) {all real x}, {y: y = -4} (c) {x: x = 3}, {all real y} (d) {all real x}, {y: y $ -1} 1 (e) {all real x}, {all real y} (f) {all real x}, ' y: y # 12 1 4 (g) {x: -8 # x # 8}, {y: -8 # y # 8} (h) {all real t: t ! 4}, {all real f (t): f (t) ! 0} (i) {all real z: z ! 0}, {all real g ^ z h: g ^ z h ! 5} (j) {all real x}, {y: y $ 0}
(a) {x: x # - 1, x $ 2}
(b) {t: t # - 6, t $ 0}
1
2
3
4
5
8
1.
(a) x = 0, 5 (b) x = -3, 1, 2 (c) x = 0, 2, 4 (d) x = 0, ! 4 (e) x = !7 4. (a) -1 # x # 1 (b) {x: -1 # x # 1}
4
y
(b)
6
3.
3
-15
4
(a) {x: x $ 0}, {y: y $ 0} (b) {x: x $ 2}, {y: y $ 0} (c) {all real x}, {y: y $ 0} (d) {all real x}, {y: y $ -2} 1 (e) ' x: x $ -2 1, {y: y # 0} 2 (f) {all real x}, {y: y # 5} (g) {all real x}, {y: y 2 0} (h) {all real x}, {y: y 1 0} (i) {all real x: x ! 0}, {all real y: y ! 1} (j) {all real x: x ! 0}, {all real y: y ! 2}
2
-10
Exercises 5.9
2.
1
2
-4 -3 -2 -1 -2
x
-4 -6 -8 y
(c) 25 20 15 10 5
x -4 -3 -2 -1 -5 -10 -15
1
2
3
4
5
567
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Maths In Focus Mathematics Preliminary Course
(g)
y
(d)
y
8 6
3
4
2
2
1
x 1
-4 -3 -2 -1 -2
2
3
4
x 1
-1
-4
-1
-6 -8
10. (a) " x: x $ 1 , " y: y $ 0 , (b)
y
y
(e)
2 1
8
x
6
-1
4
1
2
3
2 1
-4 -3 -2 -1 -2
2
3
4
x
y
11. 6
-4
5
-6
4
-8
3 2 1
-1
y
(f)
1
x
12. (a) (i) {all real x}, {all real y} (ii) All x (iii) None (b) (i) {all real x}, " y: y 2 -2 , (ii) x 2 0 (iii) x 1 0 (c) (i) {all real x: x ! 0}, {all real y: y ! 0} (ii) None (iii) All x ! 0 (d) (i) {all real x}, {all real y} (ii) All x (iii) None (e) (i) {all real x}, " y: y 2 0 , (ii) All x (iii) None
10
-10
-1
10
x
13. (a) - 2 # x # 2 (b) (i) {x: - 2 # x # 2}, { y: 0 # y # 2} (ii) {x: - 2 # x # 2}, { y: - 2 # y # 0}
Exercises 5.10 1. -10
(b) -10 (c) 8 (d) 3 (e) 3 (f) 75 1 (j) 1 (k) - 7 (l) x 2 - 3x (h) - 6 (i) 4
(a) 21
(m) 2x 3 + 3x - 5
(n) 3c 2
(g) 0
ANSWERS
2.
3.
(a) Continuous (b) Discontinuous at x = - 1 (c) Continuous (d) Continuous (e) Discontinuous at x = !2
y
(b) 6 5
(a)
4 3 2 1
x -4 -3 -2 -1 -1
1
2
3
4
1
2
3
4
1
2
3
4
-2 -3 -4
(b) (c)
y 6 5 4 3 2 1
x -4 -3 -2 -1 -1
(c)
-2 -3 -4 y
(d) 6 5 4 3
Exercises 5.11 1.
2
(a)
y
1
6
-4 -3 -2 -1 -1
5
-2
4
-3
3
-4
2 1
x -4 -3 -2 -1 -1 -2 -3 -4
1
2
3
4
x
569
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Maths In Focus Mathematics Preliminary Course
(h)
y
(e) 6
y 6
y = x +1
5
5
4
4
3
3
2
2
1
1
x
x 1
-4 -3 -2 -1 -1
2 3
1
-4 -3 -2 -1 -1
4
-2
-2
-3
-3
-4
-4
y
-6
6
y
(i)
5
y = 2x -3
4
6
3
5
2
4
1
3
x -4 -3 -2 -1 -1
4
3
3x - y - 6 = 0
-5 (f)
2
1
2
3
2
x + 2y - 2 = 0
4
-2
1 1
-4 -3 -2 -1 -1
-3
2
3
x
4
-2 -3 -4
y
(g)
-5
6
-6
5 4
x+y = 1
2
6
1
5
x -4 -3 -2 -1 -1
y
(j)
3
1
2
3
4
4 3
-2
2
-3
1
-4
x -4 -3 -2 -1 -1
1
2
x=
1 2
-2 -3 -4 -5 -6
3
4
ANSWERS
2.
(a) x 2 -3 (e) y $ 2
3.
(b) y $ -2
(d)
(d) y 2 x 2 - 4
(c) y $ x + 1
y
x
5
y
(a)
3
5 4
2
y = x2 - 1
1
3
-4 -3 -2 -1
2 1
x 1
-1
2
3
4
5
3
4
-2 x
-4 -3 -2 -1 -1
y=x2
4
2
1
3
4
-3
5
-4
-2
-5
-3 -4
y
(e)
-5 8 6
y
(b)
4 2 3
x 2
1
-4 -3 -2 -1 -2 -4 -3
3
y = x3
x
4.
(a) y 1 3x - 2 (b) y 2 x 2 + 2 (c) x 2 + y 2 1 49 (d) x 2 + y 2 2 81 (e) x 1 5, y 2 2
5.
(a)
-3
y
(c)
-6 -8
y 5
1
4 3 2 1 1
-1
x
x -4 -3 -2 -1 -1 -2
-1
1
2
3
4
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Maths In Focus Mathematics Preliminary Course
y
(b)
y
(c)
5
6
4
5
3
4
2
3
y = 3x – 5
2
1
1
x -4 -3 -2 -1 -1
1
2
3
4
x -4 -3
-2
-2
1
-1 -1
2
3
4
-2 y
(c)
-3
5
-4
4
-5
3
-6
2
y
(d)
1
6
x -5 -4 -3 -2 -1 -1
1
2
3
4
5
y=x+1
4
-2
3
6.
2
y
(a)
1
6 5
-4
4
-3
3
-2
2
-3
1
1
-2 -1 -1
2
3
4
x
y=3–x
-4 x 2
1
-4 -3 -2 -1 -1
3
-5
4
-6
-2 -3
y
(e)
-4
3
y
(b) 6 5
y=1
4 3
-3
2
3
y=x -3
1
x -4
-3 -2 -1 -1 -2 -3 -4 -5 -6
1
2
3
4
-3
x
ANSWERS
y
(f)
y
(i)
2
1
1 1
x -2
-1
x
1
-1
2
-1
-2 x=–1 y
(g)
y
(j) y=4
5
6
4
5
3
4
2
x - y = -1
3
y = x2
1
2
x -4 -3 -2 -1 -1
1
2
3
4
x-y=2
1
5
x
-2
-4 -3 -2 -1 -1
-3
-2
-4
-3
-5
-4
1
2
3
4
-5 -6
y
(h) x = -2
8
y = x3
7.
6
5
2
-4 -3 -2 -1 -2 -4
y y = x2
y=3
4
(a)
4
x 1
2
3
4
3 2 1
-6 -8
x -4 -3 -2 -1 -1 -2 -3 -4 -5
1
2
3
4
5
573
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Maths In Focus Mathematics Preliminary Course
(e)
y
(b) 8
y
y = x3
6
2
4
y=1
1
2
x 1
-4 -3 -2 -1 -2
2
3
4
-4
-4
y=
-6
-3 -2
-1
1
2
3
x
4
-1
1 x+2
-2
-8 8.
(a)
y
y
(c)
y = x2 2
y=5
5 4 3
2 1
x 1
-2
2
-4 -3 -2 -1
1
-1
2
3
4
x
5
-2 -2
-3
x=2
-4
x=1
-5 (d)
y y
(b) 6
2
5 4
1
3
1
-1
2
3
4
x
2
2 y= x
1 -2
-4 -3 -2 -1 -1 -2 -3 -4 -5 y=x-2
-6
1
2
3
x
4
y = -1
x=3
ANSWERS
Test yourself 5
y
(c)
y = 2x + 1
6 5
1.
(a) f ] - 2 g = 6
2.
(a)
4 3 2
2x - 3y = 6
1
x -4 -3 -2 -1 -1 -2
2
1
3
4
(b)
-3 -4 -5 -6
y
(d)
(c) 3
y=2
x = -3
x
3
-3
(d)
-3 (e) y
(e) 6
y = |x|
5 4
(f)
3
y=3
2 1
-4 -3 -2 -1 -1
1
2
3
-2 -3
x=2
4
x
(b) f ] a g = a 2 - 3a - 4
(c) x = 4, -1
575
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Maths In Focus Mathematics Preliminary Course
9.
(g)
(h) 10.
3.
4.
1 4 (b) Domain: all real x; range: all real y (c) Domain: - 1 # x # 1; range: - 1 # y # 1 (d) Domain: - 1 # x # 1; range: 0 # y # 1 (e) Domain: - 1 # x # 1; range: - 1 # y # 0 (f) Domain: all real x ! 0; range: all real y ! 0 (g) Domain: all real x; range: all real y (h) Domain: all real x; range: y $ 0 (a) Domain: all real x; range: y $ - 6
15
5. (a) 4 (b) 5
(c) 9
(d) 3
11. (a) y # 3
(b) y 2 x + 2
(c) y $ - x 2, y # 0
12. (a) Domain: all real x ! 3, range: all real y ! 0 (b)
(e) 2
6.
13. (a) 7.
(b) (i) x = 2, -4 (ii) - 4 1 x 1 2 (iii) x 2 2, x 1 - 4 14. (a) 2 8.
(b) x = 3
2 3
(c) 1
1 3
15. (a) x-intercept -10, y-intercept 4 (b) x-intercepts - 2, 7, y-intercept -14 16. (a) i
(b) iii (c) ii
(d) i
(e) iii
ANSWERS
17. (a) 4 (b)
2 5
(c) - 1
1 2
(d) 3
3.
18.
4.
19. (a) Domain: x $ 2, range: y $ 0 (b)
5.
f ] 3 g = 9, f ] -4 g = 16, f ] 0 g = 1
6.
Domain: all real x ! ! 1; range: y # - 1, y 2 0
20. (a)
f ( x) = x 4 + 3 x 2 - 1 f (- x) = ] - x g4 + 3 ] - x g2 - 1 = x 4 + 3x 2 - 1 = f (x) So f ] x g is even.
f (x) = x 3 - x f (- x) = ] - x g3 - (- x) = - x3 + x = - (x 3 - x) = - f (x) So f ] x g is odd. (b)
Challenge exercise 5 1. 2.
2 b=- ,3 3
7.
577
578
Maths In Focus Mathematics Preliminary Course
8.
Domain: x $ 0; range: y $ 0
9. x = 0, 3, - 2
16. (a)
10.
1 x+3 2]x + 3g 1 = + x+3 x+3 2x + 6 + 1 = x+3 2x + 7 = x+3 = LHS 2x + 7 1 =2+ ` x+3 x+3 RHS = 2 +
(b) Domain: all real x ! - 3; range: all real y ! 2 (c)
11. h ] 2 g + h ] -1 g - h ] 0 g = - 3 + 0 - ] -1 g = - 2
17.
12.
18.
13.
19. Domain: x $ 3; range: y $ 0 20. Domain: - 2 # x # 2 21.
14. f ^ (-a) 2 h = 2 (-a 2) - 1 = 2a 2 - 1 = f (a 2) 15. x =
1 ! 41 4
ANSWERS
Chapter 6: Trigonometry
Exercises 6.1 cos i =
2.
3 5 4 sin b = , cot b = , sec b = 5 4 3
3.
sin b =
4.
cos x =
5 , tan x = 9
5.
cos i =
3 4 , sin i = 5 5
6.
5 5 3 tan i = , sec i = , sin i = 2 2 3
7.
cos i =
35 , tan i = 6
8.
tan i =
51 51 , sin i = 7 10
9.
(a)
2
10. (a)
3
7 74
7 , cos b = 5
, tan b =
(a) 17c 20l (b) 34c 20l (c) 34c 12l
1.6 m
5.
(a) 18.4 cm
7.
47.4 mm 8. 20.3 m (c) 9.0 cm
9 56
3. 20.3 cm
4. 13.9 m
(b) 13.8 cm
6. 10 cm and 10.5 cm
9. (a) 7.4 cm
(b) 6.6 cm
1 10. (a) 6.8 cm
35
(b) 6.5 cm
1.
(a) x = 39c 48l (b) a = 35c 06l (c) i = 37c 59l (d) a = 50c 37l (e) a = 38c 54l (f) b = 50c 42l (g) x = 44c 50l (h) i = 30c 51l (i) a = 29c 43l (j) i = 45c 37l (k) a = 57c 43l (l) i = 43c 22l (m) i = 37c 38l (n) i = 64c 37l (o) b = 66c 16l (p) a = 29c 56l (q) i = 54c 37l (r) a = 35c 58l (s) i = 59° 2l (t) c = 56c 59l
2.
37c 57l 3. 22c 14l
3 1 1 , cos 30c = , tan 30c = 2 2 3 3
6.
(a) 11.4 cm
13. tan 48c = cot 42c = 1.11
14. (a) 2 cos 61c or 2 sin 29c
8.
(a) 13 m
(e) 2
15. x = 80c
16. y = 22c
19. t = 20c
20. k = 15c
11. 38 cm
Exercises 6.4
3 1 , cos 60c = , tan 60c = 2 2
(d) 1
(e) 1.393
2.
74
12. sec 82c = cosec 8c = 7.19
(c) 0
(d) 0.928
(a) x = 6.3 (b) y = 5.6 (c) b = 3.9 (d) x = 5.6 (e) m = 2.9 (f) x = 13.5 (g) y = 10.0 (h) p = 3.3 (i) x = 5.1 (j) t = 28.3 (k) x = 3.3 cm (l) x = 2.9 cm (m) x = 20.7 cm (n) x = 20.5 mm (o) y = 4.4 m (p) k = 20.6 cm (q) h = 17.3 m (r) d = 1.2 m (s) x = 17.4 cm (t) b = 163.2 m
11. sin 67c = cos 23c = 0.92
(b) 0
(c) 0.339
1. 5
(b) 45c 1 1 (c) sin 45c = , cos 45c = , tan 45c = 1 2 2
(c) sin 60c =
6.
(b) 0.697
Exercises 6.3
56 , cosec x = 5
(b) sin 30c =
(a) 0.635
(d) 46c 34l (e) 79c 10l
5 12 12 , sin i = , tan i = 5 13 13
1.
5.
4. 36c52l 5. 50c
(b) 37c 52l 7. a = 31c 58l, b = 45c 44l
(b) 65c 17l 9. (a) 11c 19l (b) 26 cm
10. 4.96 cm and 17.3 cm
17. p = 31c
18. b = 25c
11. (a) 12.9 m
(b) 56c 34l
Exercises 6.5 1.
(a)
North
Exercises 6.2 1.
(a) 47c
2.
(a) 47c 13l (b) 81c 46l (c) 19c 26l
(b) 82c
(c) 19c
(d) 77c
(e) 52c
(d) 76c 37l (e) 52c 30l 3.
4.
(a) 77.75c
(b) 65.5c
(c) 24.85c
(d) 68.35c
(e) 82.517c
(a) 59c 32l (b) 72c 14l (c) 85c 53l (d) 46c 54l (e) 73c 13l
Beach house
100c
Boat
579
580
Maths In Focus Mathematics Preliminary Course
(b)
(f) North
North
Farmhouse
Jamie
12c
Campsite
Dam (g)
North
320c
(c)
North
House 160c
Jetty
200c
Mohammed (h)
North
Seagull (d) North
Alistair Mine shaft 80c Town 50c
(i)
Bus stop
(e)
North Yvonne
North
Plane
349c
B Hill 285c
School
ANSWERS
(j)
North
6.
(a) 1st
1 (f) 2
1 (e) 2 8.
2.
(a) 248c
3.
080c
4. 210c
7.
21 m
8. 126.9 m 9. 72c48l
9.
5. 160c 6. 10.4 m
12. 1.8 km
15. 035c
28. 217c
3+1 2
(g) 1
1 4
1 3
(k) 0
3 2 2
(a) x =
3.
60c
7.
(a) 6 2 m
4.
5.
(b) 4 m
9 3 2
3m
(e)
4 3 3
2 ^ 3 + 1h 4
(i)
(f)
(r) -
1 2
(s) 6
2 3 3
(n) 2 3
(t)
6 2- 3 2
(c) p = 2 3 6.
9.
16. tan x =
19. sin i = 20. (a) sin i
5^3 + 3 h m 3
1.
(a) 1st, 4th (b) 1st, 3rd (c) 1st, 2nd (e) 3rd, 4th (f) 2nd, 3rd (g) 3rd (h) 3rd (i) 2nd (j) 4th
2.
(a) 3rd
4.
(a) 2nd (b) - 3
3. (a) 4th (b) -
5. (a) 2nd
(b)
(d) 1
(i)
3 3 2
(i) −1
3
(d)
(j)
2 1 2
3 2
(i) 3 2
(e) -
(j) -
(e) -
(j) -
1 2 1 2
3 2
1 2
1 2
(f)
3
1 2
2 1 2
5 21
89 5
, cot x = -
2 21
21 2
, tan x = -
4 65
, sec i =
9 65
55 8 8 , sec x = - , cosec x = 3 3 55 3 10
(b) cos x = -
91 3 , tan x = 10 91
61 61 , cosec a = 5 6
51 7 , cot i = 10 51 (b) cos x
(c) tan b
(d) - sin a
(e) - tan i
(h) - tan x
1.
(a) i = 20c 29l, 159c 31l (b) i = 120c, 240c (c) i = 135c, 315c (d) i = 60c, 120c (e) i = 150c, 330c (f) i = 30c, 330c (g) i = 30c, 120c, 210c, 300c ] 0c # 2i # 720c g (h) i = 70c, 110c, 190c, 230c, 310c, 350c ] 0c # 3i # 1080c g (i) i = 30c, 150c, 210c, 330c (j) i = 15c, 45c, 75c, 105c, 135c, 165c, 195c, 225c, 255c, 285c, 315c, 345c
2.
(a) i = !79c 13l (b) i = 30c, 150c
(d) 2nd, 4th
1
, cosec x = -
Exercises 6.8
Exercises 6.7
1 2
1
(c)
5 18. cot a = - , sec a = 6
10. 100 3 m
(b) -
3
(c)
(f) - sin i (g) cos a
10 3 m 3
8. 0.9 m
3 2
2 1
(d)
7 74 5 74 , sin x = 74 74
17. (a) sin x =
3
(m) 2 ^ 2 - 1 h
(q) 2 3
(b) y =
2m
(d) 4
(l) 1
(p) 3 - 2 2
2.
2
6+ 2 = 4
(h)
(j) - ^ 2 + 3 h (o) 1
89
15. tan i = -
(b) 1 (c)
(h) -
3
2
8
12. cos x =
14. cos x = -
Exercises 6.6 (a)
3 (g) 2
(c) - 3
33 4 , tan i = 7 33
(b) 7.2 km 30. (a) 13.1 m (b) 50c26l
29. (a) 1.2 km
(b)
(h) -
1
(h)
13. cosec x = -
23. 34.6 m 24. 149c
27. 9.2 m
2
(b)
11. cos i = -
16. 9.2 m 17. 171 m 18. 9.8 km 19. 51c 41l 20. 2.6 m
25. 198 m 26. 4.8 km
1
1 2
1
7. (a) 1
3 4 10. sin i = - , cos i = 5 5
13. 12 m 14. 242c
22. 1931.9 km
(g)
3 2
(a) (g)
(b) 145c (c) 080c (d) 337c (e) 180c
10. (a) 1056.5 km (b) 2265.8 km (c) 245c
1.
2
Island
280c
21. 9c21l
(b) -
(f) - 3
Boat ramp
11. 83.1 m
1
(a) -
3 2
(b)
(d) i = - 60c, -120c
(c) i = 45c, -135c
(e) i = 150c, -30c
(f) i = !30c, !150c (g) i = 22c 30l, 112c 30l, -67c 30l, -157c 30l
581
582
Maths In Focus Mathematics Preliminary Course
(h) i = !15c, !45c, !75c, !105c, !135c, !165c (i) i = 135c, -45c
17.
(j) i = !30c, !60c, !120c, !150c
3.
Exercises 6.9 4.
-1
5.
1.
(a) cos i (b) - tan i (c) cos i (d) tan i (e) - sec a
2.
(a) sin i (b) sec i (c) cosec x 2
(f) cosec x (j) sin2 x 3.
6.
x = 0c, 180c, 360c
9.
x = 0c, 360c
7. - 1
8. 1
10.
2
(e) sin a
(h) tan i (i) 5 cosec 2 i
(g) sec x
(k) 1
(d) cos2 x
2
(l) sin i cos i
(a) LHS = cos 2 x - 1 = 1 - sin 2 x - 1 = - sin 2 x = RHS So cos 2 x - 1 = -sin 2 x (b) LHS = sec i + tan i sin i 1 = + cos i cos i 1 + sin i = cos i = RHS 1 + sin i So sec i + tan i = cos i (c) LHS = 3 + 3 tan 2 a = 3 (1 + tan 2 a ) = 3 sec 2 a 3 = cos 2 a 3 = 1 - sin 2 a = RHS
11. 0
12. x = 270c
14. x = 0c, 180c, 360c 16.
13. x = 0c, 180c, 360c 15. x = -270c, 90c
So 3 + 3 tan 2 a =
3 1 - sin 2 a
(d) LHS = sec 2 x - tan 2 x = tan 2 x + 1 - tan 2 x =1 = cosec 2 x - cot 2 x = RHS So sec 2 x - tan 2 x = cosec 2 x - cot 2 x (e) LHS = ] sin x - cos x g 3 = ] sin x - cos x g ] sin x - cos x g 2 = ] sin x - cos x g ^ sin 2 x - 2 sin x cos x + cos 2 x h = ] sin x - cos x g ] 1 - 2 sin x cos x g = sin x - 2 sin 2 x cos x - cos x + 2 sin x cos 2 x = RHS So ] sin x - cos x g 3 = sin x - 2 sin 2 x cos x - cos x + 2 sin x cos 2 x
ANSWERS
1 - sin 2 i + 2 sin i sin i cos i cos 2 i + 2 sin i = sin i cos i 2 sin i cos 2 i = + sin i cos i sin i cos i cos i 2 = + sin i cos i = cot i + 2 sec i = LHS
(f) RHS =
So cot i + 2 sec i =
RHS = =
=
1 - sin 2 i + 2 sin i sin i cos i
So
1 - sin 2 i cos 2 i
So 4.
(j) LHS = =
1 + cot b
- cos b cosec b 1 + cot b - cos b cosec b
= =
5.
cosec b 1 + cot b - cot b
cosec b 1 = cosec b = sin b
1 sin b
sin 2 b + cos 2 b sin b cos b sec b
1 + cot b cosec b
- cos b = sin b
LHS = x 2 + y 2 = ] 2 cos i g 2 + ] 2 sin i g 2 = 4 cos 2 i + 4 sin 2 i = 4 (cos 2 i + sin 2 i) = 4 ]1g =4 = RHS
LHS = x 2 + y 2 = ] 9 cos i g 2 + ] 9 sin i g 2 = 81 cos 2 i + 81 sin 2 i = 81 (cos 2 i + sin 2 i) = 81 ] 1 g = 81 = RHS So x 2 + y 2 = 81
Exercises 6.10 1.
cosec b 1 + cot b - cos b #
cos b sin b sec b
So x 2 + y 2 = 4
= tan 2 i + cos 2 i
cos 2 i
cos b
LHS = RHS
1 - sin 2 i cos 2 i
cos 2 i sin 2 i cos 2 i 1 = 2 cos i cos 2 i 2 2 = sec i - sin i = tan 2 i + 1 - (1 - cos 2 i) = tan 2 i + 1 - 1 + cos 2 i = tan 2 i + cos 2 i = RHS
+
1 sin b cos b cos b sin b = sec b # 1 cos b sin b 1 = # 1 cos b = sin b
So ] cosec x + cot x g ] cosec x - cot x g = 1
(i) LHS =
sin b
=
(g) LHS = cos 2 ] 90c - i g cot i = sin 2 i cot i cos i = sin 2 i # sin i = sin i cos i = RHS So cos 2 ] 90c - i g cot i = sin i cos i (h) LHS = ] cosec x + cot x g ] cosec x - cot x g = cosec 2 x - cot 2 x = 1 + cot 2 x - cot 2 x =1 = RHS
sec b tan b + cot b sec b
(a) x = 8.9
(b) y = 9.4 cm
(d) b = 10.7 m 2.
(c) a = 10.0
(e) d = 8.0
(a) i = 51c 50l (b) a = 61c 23l (c) x = 43c 03l (d) a = 87c 04l (e) i = 150c 56l
3.
126c 56l 4. (a) 13.5 mm
5.
(a) 1.8 m
7.
(a) 10.3 m
9.
(a) 14.1 cm (b) 15.6 cm
(b) 2.7 m (b) 9.4 m
(b) 25 mm
6. 5.7 cm 8. (a) 60c 22l (b) 57c 9l
10. (a) 54.7 mm (b) 35.1 mm
583
584
Maths In Focus Mathematics Preliminary Course
Exercises 6.11 1.
5.
(a) m = 5.8
(b) b = 10.4 m
(d) n = 16.4 2.
(c) h = 7.4 cm
2 cos 2 i 1 - sin i 2 ^ 1 - sin 2 i h = 1 - sin i 2 ] 1 + sin i g ] 1 - sin i g = 1 - sin i = 2 (1 + sin i) = 2 + 2 sin i = RHS
LHS =
(e) y = 9.3
(a) i = 54c 19l (b) i = 60c 27l (c) x = 57c 42l (d) b = 131c 31l (e) i = 73c 49l
3.
32.94 mm
4. 11.2 cm and 12.9 cm
5.
(a) 11.9 cm (b) 44c 11l (c) 82c 13l
6.
+XYZ = +XZY = 66c 10l, +YXZ = 47c 40l
7.
(a) 18.1 mm (b) 80c49l 8. (a) 6.2 cm
9.
12.9 cm 10. (i) 11 cm
So
(b) 12.7 cm
(ii) 30c
2 cos 2 i = 2 + 2 sin i 1 - sin i
6.
b = 40c
7. (a)
8.
x = 120c, 240c
1
3 2
(b) -
2
(c) - 3
9.
Exercises 6.12 1.
12.5 cm and 4.7 cm 2. (a) 040c (b) 305c 3. 16.4 m
4.
103c
7.
(a) 1.21 km
5. 1.97 m 6. 11c (b) 1 minute 8. 32 m 9. 107 m
10. (a) AC =
5.8 sin 42c 29l (b) i = 74c 50l sin 101c 36l
11. h = 8.5
12. 7.7 km
14. 1841 km
x = 90c, 270c 10. 122 km
13. 5.7 km and 5.4 km
11. 5 3
12. (a) 6.3 cm
13. (a) i = 65c 5l (b) i = 84c 16l (c) i = 39c 47l
15. 35.8 m 16. 89c 52l 17. 9.9 km
14. 65.3 cm 2
18. 163.5 km 19. 64.1 m 20. 3269 km
15. (a) x = !60c, !120c
(b) x = 15c, 105c, -75c, -165c
21. (a) 11.3 cm (b) 44c 40l 22. 141c
(c) x = 0c, !180c, 30c, -150c
23. (a) 11.6 cm (b) 73c 14l 24. (a) 265.5 km (b) 346c 33l
3 4 16. sin i = - , cot i = 5 3
25. (a) 35c 5l (b) (i) 4.5 m
18. (a) AD =
(ii) 0.55 m
Exercises 6.13
20 sin 39c sin 99c
17. (a) 209c (b) 8.5 m
(b) 029c
19. 2951 km
Challenge exercise 6
1.
(a) 7.5 cm 2 (b) 32.3 units 2 (c) 9.9 mm 2 (d) 30.2 units 2 (e) 6.3 cm 2
2.
15 3 2 m 2
6.
1.2 m 2
9.
(a) 7.8 cm (b) 180.8 cm 2
3. 7.5 cm
7. 42 cm 2
4. 15.5 cm
2
2
5. 34.8 cm
8. 247.7 mm 2
10. (a) 5.6 cm (b) 18.5 cm 2
(c) 19.1 cm 2
1.
92c 58l 2. 50.2 km
4.
(a) AC =
25.3 sin 39c 53l (b) h = 25.2 cm sin 41c 21l
6.
- cos x
7. 16 3 cm 2
9.
x = 22c 30l, 112c 30l, 202c 30l, 292c 30l 10. i = 75c 45l
2
11. 5.4 m 14. -
Test yourself 6 cos i =
5
2.
(a) cos x
(b) 2
3.
(a) 0.64
(b) 1.84 (c) 0.95
4.
(a) i = 46c 3l (b) i = 73c 23l (c) i = 35c 32l
, sin i =
56 9
3. x = 12.7 cm
8.
12. i = 110c, 230c 15. 31 m
5. 4.1 km
1 2
13. 6.43 km
16. LHS =
cos i ] sin i + cos i g
= cos i
3
1.
34
(b) 8.7 m
1 - sin 2 i ] sin i + cos i g
cos 2 i sin i + cos i = cos i = tan i + 1 = RHS
34
(c) cosec A
17. x 2 + y 2 + 4y - 5 = 0
ANSWERS
8.
Chapter 7: Linear functions
2.
(a) 5 (b) 10
(c) 13
(c)
52 = 2 13
85
(d)
(a)
13
(b)
65
3.
(a) 9.85 (b) 6.71 (c) 16.55 4. 12 units
5.
Two sides =
6.
Show AB = BC =
7.
Show points are
8.
Radius = 3 units, equation x 2 + y 2 = 9
9.
Distance of all points from ^ 0, 0 h is x + y = 11 2
2
34 , 1 side =
(b) XY = AC =
85
10. a = 3
37 , QP = MN =
15. BD = AC =
98
11. x 2 + y 2 = 4
11 , equation
29 , BC =
20 , so parallelogram
16. (a) AB = AC = 17. 2 101
18.
1.
116 , AC =
145
3.
20. XY =
40 , BC = 4
130 , XZ =
(b) 1
(g) - 4
1 2
x = 1.8
(c) - 1
(h) -
2 3
1 3
(i) 2
4. x = 9
(d) - 2 1 4
2 5
2 3
(e)
(j) - 2
2. y 1 = 21
5. (a) Show m 1 = m 2 =
(3, 4)
4 3 (-2, 1)
(7, 2)
2 1
-3 -2 -1 -1
1
-2
6.
2 3 4 (2, -1)
6
5
Gradient of AB = gradient of CD = 1
1 2
Gradient of BC = gradient of AD = 0
30.2
7.
Exercises 7.2
(b) a = - 5, b = 6
(c) a = -1, b = - 2 (d) a = -1, b = - 2
3.
3 + ]-3g -4 + 4 = 0, =0 2 2
5.
^ 4, 3 h 6. x = 3 is the vertical line through midpoint ^ 3, 2 h.
7.
1 1 Midpoint of AC = midpoint of BD = d 2 , 3 n . 2 2 Diagonals bisect each other
4. P = Q = ^ 2, -1 h
3 4
8.
Gradient of AC = 1, gradient of BD = -1
9.
(a) Show AB 2 + BC 2 = AC 2 (b) Gradient of AB =
(e) a = 6, b = 1
1 3
1 Gradient of AC = - 5 , 2 1 gradient of BD = 2
1 1 n (h) d 1 , 1 n 2 2
1 1 1 (i) d , 2 n (j) d 0, 5 n 2 2 2
Gradient of AB = gradient of CD = -1 Gradient of BC = gradient of AD =
(a) ^ 2, 4 h (b) ^ 1, -1 h (c) ^ - 2, 1 h (d) ^ - 3, 2 h (e) ^ -1, 1 h (f) ^ - 3, 2 h (g) d 3,
(f) -
y
Problem
(a) a = 9, b = - 3
1 3
65
XY 2 + XZ 2 = 65 + 65 = 130 = YZ 2 So triangle XYZ is right angled. (Pythagoras’ theorem)
2.
2
(b) Lines are parallel.
Since XY = YZ, triangle XYZ is isosceles.
1.
(a) 2
2
65 , YZ =
2 , AB = 2
12. x 2 + y 2 = 1
61 units
AB + BC = 29 + 116 = 145 = AC 2 So triangle ABC is right angled (Pythagoras’ theorem) 2
34 ; YZ =
34 , 2
40 = 2 10 ; XZ =
Exercises 7.3
11. a = ! 6 - 2
14. MQ = NP =
(b) OC = OB = 2
10 , BC =
128
17 units from ^ 7, -3 h
9. ^ - 8, 13 h
1 1 1 1 10. (a) X = d - , 3 n , Y = d , n , Z = ^ 1, 1 h 2 2 2 2
12. All 3 sides are 2 units. 13. a = 10, - 2
19. AB =
125 , midpoint AC = midpoint
1 BD = d 4, - n ; rectangle 2
Exercises 7.1 1.
AC = BD =
gradient of BC = -
5 , 4
4 5
10. (a) F = ^ 1, - 2 h, G = d 4,
1 n 2
(b) Gradient of FG = gradient of BC =
5 6
7
3 5
1 8
585
586
Maths In Focus Mathematics Preliminary Course
11. 4x - 3y - 11 = 0 12. Gradient of ^ 2, - 4 h and ^ 3, -1 h = gradient of ^ 3, -1 h and ^ 5, 5 h = 3 14. 0.93 15. 21 16. 50c 12l 17. 108c 26l
13. 1 18. (a) 19.
2.
1
(b)
3
(c) - 3
3
-5 - ] -2 g = 7-4 -3 = 3 = -1 = tan i = tan i = 180c - 45c ^ 2nd quadrant h = 135c
m
m -1 ` i
(a) (i) 3 (ii) 5
(b) (i) 2
(ii) 3
(g) (i) - 2
(ii) 6
(h) (i) -1
(ii) 1 (i) (i) 9
(f) (i) 1
(j) (i) 1
2 3
(ii)
(d) - 2
(e) -1
(f) - 3
3.
2 3
(g) 2
3 7 # = -1 7 3
3. (a) 4 (h) -
(b) 5x - y - 8 = 0
12. 7x + 6y - 24 = 0
15. 2x - 3y + 18 = 0
Exercises 7.7 1.
(b) - 2 (i) 1
(a) ^ 2, - 4 h
(b) ^ -1, - 3 h
(e) ^ 5, -1 h
(f) ^ -1, 1 h
(c) ^ 4, 4 h
(g) ^ 3, 7 h
1 2 3.
(h) ^ 4, 0 h
^ 2, 5 h, ^ 4, 1 h and ^ -1, -1 h
at ^ 2, -3 h
5.
All lines meet at ^ - 5, 0 h
7.
5x + 6y - 27 = 0
9.
6. 11x + 6y = 0
8. 4x + 7y + 23 = 0
13. x - y + 1 = 0
14. x - 3y + 2 = 0
(g) y = x - 1
15. 3x + y - 7 = 0
(b) x = -1
19. 2x - y - 1 = 0
(d) 3x + 4y - 25 = 0
4. 4x + y - 8 = 0
Exercises 7.8
10. 3x + 8y - 15 = 0
Exercises 7.6 (a) - 3 (h)
1 3
(b)
(i)
1 3 1 3
(c)
(j)
3 4 1 5
(d) 1
1 2
(e) 1
(f) -
5 6
18. 3x - y - 14 = 0
20. 3x - y - 11 = 0
21. 5x - y + 17 = 0
5. (a) y = 3
7. 3x - 4y - 12 = 0
9. x = - 4
16. x + 5y + 13 = 0
17. 27x - 5y - 76 = 0
(b) 3x - 4y + 4 = 0
6. y = - 2x
2x + y - 3 = 0
2. x + y - 8 = 0
(i) ^ 41, 26 h
4. All lines intersect
12. x - 2y - 3 = 0
(c) y = 5x
(d) ^ 0, - 2 h
1 7 n 2. Substitute ^ 3, - 4 h into both lines (j) d , 19 19
(d) y = 4x + 20 (e) 3x + y - 3 = 0 (f) 4x - 3y - 12 = 0
(a) 4x - 3y + 7 = 0
8. m 1 = m 2 = 4
(c) 2x + y + 2 = 0
10. 2x + y - 2 = 0
(h) y = x + 5
2 3
7. k = -
x+y-1=0
(c) 4x - 5y + 13 = 0
1.
m1 # m2 = -
11. x + y - 3 = 0
(a) y = 4x - 1 (b) y = - 3x + 4
(e) x - 2y + 2 = 0
8.
1 5
13. x + y - 3 = 0 14. 2x - y - 5 = 0
Exercises 7.5 1.
1 # 5 = - 1 so perpendicular 5
(d) 2x - 3y + 16 = 0
(ii) 0
1 4 3 1 2 1 1 2 (j) 1 (k) (l) (m) (n) (o) 4 5 7 5 3 2 3 1 1 1 (p) (q) 15 (r) - 1 (s) (t) 6 14 2 8 (c) 0
m1 m2 = -
(ii) - 2
(j) (i) 5 (ii) - 2 2. (a) (i) - 2 (ii) 3 (b) (i) - 5 (ii) - 6 1 (c) (i) 6 (ii) -1 (d) (i) 1 (ii) 4 (e) (i) - 2 (ii) 2 1 1 4 (f) (i) 3 (ii) 1 (g) (i) (ii) - 2 (h) (i) (ii) 2 5 2 3 1 2
4.
11. (a) y = - x
(ii) -7
(ii) 1 (c) (i) 6
(e) (i) - 4
(ii) -
m 1 = m 2 = 3 so parallel
5 AB < CD _ m 1 = m 2 = 3 i and BC < AD d m 1 = m 2 = - n 8 1 10. Gradient of AC: m 1 = , gradient of BD: m 2 = - 2, 2 1 m 1 # m 2 = # - 2 = -1 2
(ii) 0
1 2
(g) 3x + 4y + 13 = 0
9.
(d) (i) -1
(i) (i) 3
(f) x + 3y - 1 = 0 3.
6.
Exercises 7.4 1.
(e) x - 2y + 4 = 0
5. m 1 = m 2 = 1
2^ 3 + 3h 3
20. x =
(a) x - y + 1 = 0 (b) x - 3y + 16 = 0 (c) x + y - 5 = 0 (d) x + 2y + 5 = 0
(g)
1 3
3 13
1.
(a) 2.6
2.
(a) 3.48
3.
(a)
4.
d1 = d2 = d3 = 1
(b) 1
7 13 13
(c) 2.5
(b) 1.30 (b)
5
(d) 2.4
(c) 0.384 (c)
4 205 205
(e)
(d) 5.09 (d)
8 13 (e) 1.66
5 26 13
(e)
14 13 13
ANSWERS
5.
14
A: d =
5
, B: d =
13. Solving simultaneously, x - y - 4 = 0 and
-3
2x + y + 1 = 0 have point of intersection ^ 1, - 3 h .
5
Substitute ^ 1, - 3 h in 5x - 3y - 14 = 0:
Opposite signs so points lie on opposite sides of the line 6.
7.
^ 2, - 3 h: d =
13 10
5
, ^ 9, 2 h: d =
LHS = 5 # 1 - 3 # - 3 - 14 = 0 = RHS
10
` point lies on 5x - 3y - 14 = 0:
Same signs so points lie on the same side of the line
Substitute ^ 1, - 3 h in 3x - 2y - 9 = 0:
^ - 3, 2 h : d = - 4 , ^ 4 , 1 h : d = 2
LHS = 3 # 1 - 2 # - 3 - 9 = 0 = RHS
1 5
` point lies on 3x - 2y - 9 = 0:
Opposite signs so points lie on opposite sides of the line 8. 9.
` lines are concurrent
d 1 = d 2 = 2 so the point is equidistant from both lines ^ 8, - 3 h: d =
55 37
14. - 0.499
9
, ^ 1, 1 h: d =
18. ^ - 2, 1 h: d =
37
Same signs so points lie on same side of the line 10. ^ - 3, 2 h: d =
-6 5
, ^ 4, 1 h: d =
19. x - y - 4 = 0
5
11. d 1 = d 2 = 4 so same distance 12. 14. 4.2 15. x = 9
17. m = - 1
2 or -17 3
8 5 units 5 16. b = 3
1 1 or -1 4 12
2 1 or -18 3 3
4 1 20. (a) ^ 3, -1 h, d 3 , n, ^ - 2, 2 h 7 7
(b)
2 10 13 5 26 34 , , 5 5 119
1.
1 6.4 units 2. d 2 , - 2 n 2
3.
(a) - 1
4.
(a) 7x - y - 11 = 0
(d)
2 13
20. 3x - 7y - 14 = 0
2. x - 3 y - 3 3 = 0
3. 10x 2 + 10y 2 = 81
k = -2
4.
Show AC and BD have the same midpoint ^ 1, 2 h and m AC # m BD = -1
5.
Show distance of all points from ^ 0, 0 h is 3; radius 3; equation x 2 + y 2 = 9
6.
4 13 13
8.
12 13 13
3
15.
3 5
(b) 5x + y - 6 = 0
(c) 3x + 2y = 0
(e) x - 3y - 3 = 0
5.
6 5 units 5
6.
1 m 1 = - , m 2 = 4 so m 1 m 2 = -1 4 ` lines are perpendicular.
7.
x-intercept 5, y-intercept - 2
8.
(a) 2x + y - 1 = 0
7. +OBA = 45c; a = b (sides of isosceles D) 9. 113c12l 10. 2x + 3y + 13 = 0
9.
m 1 = m 2 = 5, so lines are parallel
1 2
12. a = 6, b = 1
(c)
5 units 2 10. 3x - 4y = 0
3x +y + 3 - 2 3 =0
Exercises 8.1
12. ^ 3, - 5 h
14. 2x + 5y + 14 = 0 16. x - y + 6 = 0
Chapter 8: Introduction to calculus
1.
(b)
4 5
1.
13. a = 2, b = 3
1
(b) 2 (c)
(d) 3x + 5y - 14 = 0
11. ^ -1, 1 h
, ^ 6, 3 h: d =
11. BC = AC = 18 , AB = 6, so D is isosceles; m BC # m AC = -1, so D is right angled.
Test yourself 7
1 5
13
17. x = 1
Challenge exercise 7
18. Show distance between ^ 0, 0 h and the line is 5 19. Show distance between ^ 0, 0 h and the line is greater than 1
-8
16. y = 3
Opposite signs so points lie on opposite sides of the line
7
Opposite signs so points lie on opposite sides of the line
13. 1
15. c = -13, - 65
587
588
Maths In Focus Mathematics Preliminary Course
2.
8.
9. 3.
10. 4.
5.
Exercises 8.2 2. Yes, x = x 1
1.
Yes, x = 0
5.
Yes, x = x 1, x = x 2
8.
Yes, x = 2
3. No
6. Yes, x = 0
9. Yes, x = - 2, 3
11. Yes, x = 90c, 270c
4. Yes, x = 0 7. Yes, x = - 3
10. Yes, -1 # x 1 0
12. Yes, x = 0
13. No
14. No
15. Yes, x = !3 6.
Exercises 8.3 1.
(a) 3 (b) -7 (c) 3 (d) 8 (h) -1 (i) 10 (j) -1
2.
(a) x 2 - 2x - 4 (b) 2x 3 + x - 1 (c) - 7x - 1 (d) 4x 4 - x 2 (e) - 4x + 3 (f) 2x 2 + 6 (g) - 2x (h) 4x 2 (i) 3x - 1 (j) x 2 - 2x + 9
(e) 2
(f) - 3
(g) 2
Exercises 8.4
7.
1.
(a) 4.06
2.
(a) 13.61
4.
(a) f ] x + h g = x 2 + 2xh + h 2
(b) 3.994
(c) 4
(b) 13.0601
(c) 12.9401
(d) 13
2 2 2 (b) f (x + h) - f (x) = x + 2xh + h - x 2 = 2xh + h
(c)
f ]x + hg - f ]xg h
2xh + h 2 h h ] 2x + h g = h = 2x + h =
3. 6
ANSWERS
f ]x + hg - f ]xg
(d) f l(x) = lim
13. (a) 0.252
h "0
h = lim (2x + h)
(b) 0.25
14. (a) - 0.04008
(c) 0.2498
(b) - 0.03992
(c) - 0.04
15. -1
h "0
= 2x 5.
(a) f (x + h) = 2 ] x + h g2 - 7 (x + h) + 3 = 2 (x 2 + 2xh + h 2) - 7x - 7h + 3 = 2x 2 + 4xh + 2h 2 - 7x - 7h + 3 2 2 (b) f (x + h) - f (x) = (2x + 4xh + 2h - 7x - 7h + 3) 2 - (2x - 7x + 3) = 2x 2 + 4xh + 2h 2 - 7x - 7h + 3 - 2x 2 + 7x - 3 = 4xh + 2h 2 - 7h
(c) f ] x + h g - f ] x g h
4xh + 2h - 7h = h h ] 4x + 2h - 7 g = h = 4x + 2h - 7 2
h
(e) f l] 2 g = 5
9.
2.
(a) 4x + 1 (b) 8x - 12 (e) 6x 2 + 6x - 3
3.
(a)
6.
(b) f ] 2 + h g = h 2 + 5h + 11
(a) f ] 2 g = 11
(d) f ] 2 + h g - f ] 2 g
8.
(a) 1 (b) 5 (c) 2x + 3 (d) 10x - 1 (e) 3x 2 + 4x - 7 (f) 6x 2 - 14x + 7 (g) 12x 3 - 4x + 5 (h) 6x 5 - 25x 4 - 8x 3 (i) 10x 4 - 12x 2 + 2x - 2 (j) 40x 9 - 63x 8
4.
(c) f ] 2 + h g - f ] 2 g = h 2 + 5h
7.
1.
h 2 + 5h h h ]h + 5 g = h =h+5 =
(a) f ] 3 g = 8
(c) f l] 3 g = 6
(b) f ] 3 + h g - f ] 3 g = 6h + h 2
(a) f l] 1 g = - 13
11.
dV = 4rr 2 dr
= 2x + 2
11. (a) 2 (b) 5 (c) -12 (b)
(c) f l] x g = 8x - 4
(g)
dy dy dx
= 3x 2
dy dx
2.
3.
9.
12. 3
(b) x = ! 2
dv = 30t dt
10.
13. (a) 5
(b) - 5
dh = 40 - 4t dt (c) x = 4
15. 18
(c) 11
(d) -18
(g) 11
(h) 136
(i) - 4
(j) 149
(a) -
= 2x + 5 dy dx
(b)
(g) -
(a) (i) 6
(e) (i) 11
1 25
1 71
(ii) -
(c) (h)
1 6
1 24 1 (ii) 11
(ii) -
1 20
1 20
(d) (i) -
(b) (i) 8 (d) (i) - 8
1 8
(e) 18
1 43
(e)
(j) -
(f) 27
1 10
1 5
1 8 1 (ii) 8
(ii) -
4.
(a) 27x - y - 47 = 0 (b) 7x - y - 1 = 0 (c) 4x + y + 17 = 0 (d) 36x - y - 47 = 0 (e) 44t - v - 82 = 0
5.
(a) x + 24y - 555 = 0 (c) x - 17y - 516 = 0 (e) x + 2y - 9 = 0
6.
(a) (i) 7x - y + 4 = 0 (ii) x + 7y - 78 = 0 (b) (i) 10x - y + 36 = 0 (ii) x + 10y - 57 = 0 (c) (i) 10x + y - 6 = 0 (ii) x - 10y - 41 = 0 (d) (i) 2x + y + 2 = 0 (ii) x - 2y - 19 = 0 (e) (i) 2x - y + 2 = 0 (ii) x + 2y - 9 = 0
7.
x = !3
= 10x - 1
(f) f l] x g = 6x 2 + 5 (h) f l] x g = - 6x 2
1 7
1 26
(c) (i) 24
(d) 15 (e) - 9
(d)
= 3x - 4 x + 3 2
(e)
ds = 10t - 20 dt
7.
(b) -13
(f)
y + dy = ] x + dx g2 + 2 (x + dx) = x 2 + 2 xd x + d x 2 + 2 x + 2 d x Since y = x 2 + 2x d y = 2x d x + d x 2 + 2 d x dy 2x d x + d x 2 + 2d x (b) = dx dx d x ] 2x + d x + 2 g = dx = 2x + dx + 2
dx
(d) 4x
5. - 56
(a) 72
(b) 17
Substitute _ x + dx, y + dy i:
(e)
8x 7 - 6x 5 3
= 60x 9 - 40x 7 + 35x 4 - 3
gl] x g = - 20x - 5
10. (a) y = x 2 + 2x
12. (a) f l] x g = 2x
dy dx
8.
1. (c) 12
dy
(c)
Exercises 8.6
(b) f ] -1 + h g - f ] -1 g = 4h 3 - 12h 2 + 12h
dx
(b) 2x 3 - x 2
f l] x g = 16x - 7
14. (a) 12
(a) f ] -1 g = -7
(c)
x -1 3
(d) 16x 3 - 24x
(c) 2x
(f) 2x 2 - 2x + 2
(d) f l] x g = 4x - 7 6.
Exercises 8.5
(b) x - 8y + 58 = 0 (d) x - 45y + 3153 = 0
8. (1, 2) and (-1, 0)
9. (- 5, -7)
1 4
589
590
Maths In Focus Mathematics Preliminary Course
3 15 n 12. d - 1 , - 4 4 16
10. (0, 1) 11. (1, 2) 13. (a) (1, -1)
(k) -
(b) 6x - y - 7 = 0 (l)
14. 10t - h - 7 = 0
15. 4x - 2y - 19 = 0
Exercises 8.7
x
2 2-x - 2 ] 5x + 3 g
+ 2-x =
] 2x - 1 g2
+
3. 1264
4.
2.
26
6.
10x - y - 9 = 0
8.
x=
4 - 3x 2 2-x
5 11 =2x - 1 ] 2x - 1 g2 1
7+
7
8
=
5. 176
7
1
1.
(a) - 3x - 4 (e) x
2.
-
(a) (f) (j) -
3.
1 27
8.
1 8
1 2
+ 3x - 2
1
(b)
x2 1
2x 3
(f) x 5
-
2 3
(g) 6x
(c)
2 x (g) -
2 x3 1
(c) 1.2x - 0.8
(b) 1.4x 0.4
1
x7
-
10
15
(e)
x6
(i) -
5.
x4
1
(b) -
2 x3
3x 2
1.
6. −3 7. 2x + 3 x + 1
1 16
13. (9, 3)
(w) 2.
9
(u) -
^ 4x 3 - 9 x 2 + 3 h
^ x 4 - 3x 3 + 3x h2
3
4 ] 3x - 1 g3
(v) -
16 3 4x + 1 (x) 3
4. (4, 1) 5. x = 2, -1
1 2
(y)
(t)
15 ] x + 5 g2
(c)
- x + 14x 2
(e)
x4 -6 ] x - 2 g2
(h)
= (i)
x 4 - 12x 2 ^x - 4h
2
2
- x + 14 x3 - 34 ] 4x - 3 g2
=
x 2 ^ x 2 - 12 h ^ x 2 - 4 h2
(f)
11 ] x + 3 g2
(j)
-14 ] 3x + 1 g2
(l)
2
2x 3 + 9x 2 + 7 ] x + 3 g2
(p)
3x 2 + 8x - 5 ] 3x + 4 g2
x 4 - 2x 3 - 4x 2 - 1 ^ x 2 - x - 1 h2
1 2
-
2]x + 5g - x ]x + 5g (r) x+5
1 2
] 2x - 9 g2 ] 20x + 51 g 6 ] 5x + 1 g ] 2x - 9 g2 - 5 ] 2x - 9 g3 = 2 ] 5x + 1 g ] 5x + 1 g2
] 7x + 2 g4 - 28 ] x - 1 g ] 7x + 2 g3
(u)
] 7x + 2 g8
=
- 21x + 30 ] 7x + 2 g5
15 ] 2x - 5 g3 ] 3x + 4 g4 - 6 ] 3x + 4 g5 ] 2x - 5 g2 ] 2x - 5 g6 4 ] ] g 3 3x + 4 4x - 33 g = ] 2x - 5 g4 3x + 1
3 x+1 -2 x+1 3x + 5 (v) = x+1 2 ] x + 1 g3 2x - 3
27
(w)
5
4 4 ] 7 - x g9
2 x-1 -2 x-1 - 2x + 1 = ] 2x - 3 g2 2 x - 1 ] 2x - 3 g2
x ] x - 9 g2
6. 8x + y + 7 = 0
(a) 8x 3 + 9x 2 (b) 12x - 1 (c) 30x + 21 (d) 72x 5 - 16x 3 (e) 30x 4 - 4x (f) x ] 5x + 2 g ] x + 1 g2 (g) 8 ] 9x - 1 g ] 3x - 2 g4 (h) 3x 3 ] 16 - 7x g ] 4 - x g 2 (i) ] 10x + 13 g ] 2x + 5 g3 4 5 (j) 10x ^ x 3 + 5x 2 - 3 h ^ x 2 + 1 h + ^ 3x 2 + 10x h ^ x 2 + 1 h 4 3 2 2 ^ h ^ h = x 13x + 60x + 3x - 20 x + 1
(b)
- 3x 2 - 6x - 7
(s)
2 ] 2x + 7 g10
Exercises 8.9 1.
^ 2x 2 - x h2
(q)
5
3. 40
- 2x 2
(o)
2 2 15. d 5, n, d - 5, - n 5 5
2 ] 4 + x g3
9. 34x - y + 29 = 0
4x ] x - 3 g 4x 2 - 12x = ] 2x - 3 g2 ] 2x - 3 g2 ^ 3x 2 - 7 h 2x 2 ] x + 6 g - 18x 2x 3 + 12x 2 = (m) (n) 2 2 2 ] g ] x + 4 g2 ^x - 5h x+4
6 (a) 4 ] x + 3 g3 (b) 6 ] 2x - 1 g2 (c) 70x ^ 5x 2 - 4 h 5 4 (d) 48 ] 8x + 3 g (e) - 5 ] 1 - x g (f) 135 ] 5x + 9 g8 3 (g) 4 ] x - 4 g (h) 4 ^ 6x 2 + 3 h ^ 2x 3 + 3x h 7 2 ^ h (i) 8 ] 2x + 5 g x + 5x - 1 1 3 5 (j) 6 ^ 6x 5 - 4x h ^ x 6 - 2x 2 + 3 h (k) ] 3x - 1 g 2 2 2 5 -4 (l) 2 ] 4 - x g- 3 (m) - 6x ^ x 2 - 9 h (n) ] 5x + 4 g 3 3 1 3^ 2 3 4 3x - 14x + 1 h ^ x 3 - 7x 2 + x h (o) (p) 4 2 3x + 4 8x 5 2 (q) (r) (s) 3 ] 5x - 2 g2 ^ x 2 + 1 h5 7 - 3x
(t) -
16 ] 5x + 1 g2
(k)
Exercises 8.8 1.
-2 ] 2x - 1 g2
(a)
(g)
10. x - y + 9 = 0
12. x + 16y - 16 = 0
- 6 ! 30 3
Exercises 8.10
2
(d) 1 32
7. 69x - y - 129 = 0
3 2
x5
9. 3x + 16y - 8 = 0
14. x = 4
(h) x
(d) -
6 6 x5
3 x (h) 2
3
1 4
12
-
4. −3
11. (a) -
-
1 -2 x 2
(d)
(x)
x2 + 1
5 9
- 2 ] x - 9 g x2 + 1 ] x - 9 g4
2.
1 8
6.
x - 18y + 8 = 0
3. - 1
4. x = 0, 1
=
- x 2 - 9x - 2
x2 + 1 ] x - 9 g3
5. x = - 9, 3
7. 17x - 25y - 19 = 0
ANSWERS
Test yourself 8
17. 9
1.
20.
(a)
18. 12x + y - 4 = 0
19.
ds 1 = u + at, t = 5 dt
7 10
Challenge exercise 8
(b)
1.
f ] 1 g = - 3, f l] 1 g = - 36
3.
dx = 8t 3 + 300t 2; t = 0, - 37.5 dt
4.
2x + y = 0, 3x - y - 3 = 0, 6x - y + 12 = 0
5.
^ 2, 2 h, ^ - 2, -14 h, x + 12y - 26 = 0, x + 12y + 170 = 0
6.
3 4
7.
5 ] 5x + 1 g3 ] x - 9 g4 + 15 ] x - 9 g5 ] 5x + 1 g2 = 10 ] 5x + 1 g2 ] x - 9 g4 (4x - 13)
8.
2.
dy dx (b) (d)
3. (a)
= 10x - 3 dy dx dy dx dy
=
11 ] 2x + 1 g2
(c)
(f)
dy dx
4.
dv = 4t - 3 dt
5. (a) 1
8.
(a) x = - 2
(b) x = 1
9.
(a) f l] x g = 32 ] 4x + 9 g3 dy dx
= 42x 5 - 9x 2 + 2x - 8
dy dx
= 9 (2x + 4) (x 2 + 4x - 2)8
= 40x ] 2x - 1 g3 + 5 ] 2x - 1 g4 = 5 ] 2x - 1 g3 (10x - 1)
5 x3 (e) = 2 dx
(c)
dy dx
=-
- 6 ! 204 - 3 ! 51 = 6 12
x3
16. n = 8
(c) x = 2 dy dx
18. x =
dy dx
1 1 1 1 ! 13 n 13. x = , 12. P = d - 2 , 6 4 16 3 3 10
11 3 3 o 17. e 1 , , 12x - 12 3 y + 31 = 0 12 2
1 1 3 , -1 , 1 2 2 5
19. (a) x = 90c, 270c
y
(b)
=-
1 27
10. 2x + y - 25 = 0
15. 3x - y + 5 = 0, Q = ^ 0, 5 h, PQ =
1 2
= ] 9x - 1 g ] 3x - 1 g (d)
(e) f l] x g =
x=
11. a = 14.
13 18
2 ] 4x - 9 g4 - 16 ] 2x + 1 g ] 4x - 9 g3 ] 4x - 9 g8 - 2 ] 12x + 17 g = ] 4x - 9 g5
10
(b) 20 6. 10 7. 42
(b)
9.
2. -
5 ] x - 3 g2
=-
4 x2
1 5 5 x4
10.
1 90c
y
20. ^ - 4, -73 h
180c
270c
21. 3x - 9y - 14 = 0
360c
22.
23. (a) 16x + 32y + 1 = 0, 4x - 2y - 1 = 0 1 (b) m 1 $ m 2 = - # 2 2 = -1 11. 9x - y - 7 = 0
12. (2, 3)
13.
So perpendicular
dS = 8rr dr
14. (- 2, 71), (5, - 272) 15. 4x - y - 6 = 0
16. 3525
x
3 ] 4 - 5x g 4x 4 3x - 2
591
592
Maths In Focus Mathematics Preliminary Course
24. x = 0, 2, 6 27. p = 1
1 2
25. a = - 14, b = 7
28.
8r 3 dV = 3 dr
31. 4x - y - 13 = 0
32. -
26.
29. k = 4 1 48
5 22 22
14.
30. x - y - 4 = 0
33. a = -1, b = 2, c = 4
34. S = 8rr - 8r + 2rrh 35. (a) 6x 2 - 5 ] 3x - 1 g ] 3x - 5 g3 36. x =
(b)
- ] 5x + 6 g
] x - 3 g4 2x + 1
4 ! 13 6 15.
37. (a) x + 7y - 80 = 0 1 1 n (b) Q = d - 4 , 12 7 49
Practice assessment task set 2 2. 1 3. 5x + 2y - 1 = 0
4. ^ 2, - 2 h
1.
- 0.77
5.
- 0.309
7.
3 8 1 m 1 m 2 = # - = -1; A = d -1, 1 n 4 6 2
9.
7 12
6. (a)
3 cm 2
(b) AC =
13 cm, BD = 1 cm 8. x = 15c
10.
16. sin 4 i 17. 2 units 19. i = 120c, 240c 23. y = 16.5 26. 7
18. x - 8y + 15 = 0
20. - 1
2 3
21. 2
24. 3x + y - 5 = 0
27. x = 3
22. a = 115c 56l
25. 1
2 1x 13 3
28. - 3
29. Show perpendicular distance from ^ 0, 0 h to the line is 2 units, or solving simultaneous equations gives only one solution. 30. (a) g ] 2 g = 1, g ] - 3 g = - 6 (b)
11.
31. 3x 2 - 4x
32. -
34. x = - 2, y = -17
1 2
33. 17.5 m
35. (a) AB = 7.0 m
36. 3 cos i 37. (a) 2x - y + 4 = 0
(b) 27.8 m2
(b) P ^ - 2, 0 h, Q ^ 0, 4 h
2
(c) 4 units 1 12. 45c 49’ 13. Domain: all real x ! ; range: all 2 real y ! 0
38. 127 m
39. 15 units2 40. f (- x) = ] - x g6 - ] - x g2 - 3 = x6 - x2 - 3 = f (x)
41. 16x 2 ^ 2x 2 + 1 h + ^ 2x 2 + 1 h = ^ 18x 2 + 1 h ^ 2x 2 + 1 h 3
4
3
ANSWERS
42. - 4
1 #y#9 3
43. -
44. (a) 3x - y - 4 = 0
8 units 13
47.
1
50.
11. (a) x = -3; (-3, -12)
(d) R = ^ -10, 0 h
46. Domain: all x ! - 4; range: all y ! 0 48. 4.9 km 49. 8x - 7 - 10x 51. 2x - 3
53. x + 6y - 56 = 0 55. a = 2, b = - 9
52.
-17 - 2x x 2 + 5x
=
-3
- ] 17 + 2x g x 2 + 5x
54. f ] - 2 g = - 45, f l] - 2 g = 48 56. 7x - 5y + 9 = 0
57. 47x - y + 109 = 0 58. x = - 0.25 1 range: y $ 0 2 (b) domain: all real x ! -7 range: all real y ! 0
59. (a) domain: x $
(c) domain: - 2 # x # 2
range: - 2 # y # 0
60. (a) (1, 1) (b) 2 13 units
(c) - 1
1 1 1 (c) x = 1 ; d 1 , 3 n 4 4 8
1 1 1 (d) x = -1 ; d -1 , -13 n 4 4 4
62. (b), (d)
12. (a) (i) x = -1 (b) (i) x = 1
64. (c)
(iii) (-1, -3)
(ii) 1
(iii) (1, 1)
13. (a) Minimum (-1, 0) (b) Minimum (4, -23) (c) Minimum (-2, -7) (d) Minimum (1, -1) (e) Minimum (2, -11) 1 1 (f) Minimum d - , -3 n 4 8 (g) Maximum (-1, 6) (h) Maximum (2, 11) 1 3 (i) Maximum d , 7 n 2 4
14. (a) (i) -2 (iii)
63. (a)
(ii) -3
(j) Maximum (1, -3)
1 2
(d) 3x + 2y - 5 = 0 61. (a)
(b) x = -4; (-4, 17)
(e) x = -3; ^ -3, -23 h
2x - 7 5 ] x + 1 g2
Minimum value 3.75, no solutions
10. Minimum value 0, 1 solution
(b) x - y - 2 = 0
(c) x + 3y + 10 = 0 45.
9.
3 x2
(ii) Minimum 0 y
65. (c) 5 4 3
Chapter 9: The quadratic function
2
Exercises 9.1 1.
1 x
Axis of symmetry x = - 1, minimum value - 1
-4 -3 -2 -1 -1
2
1
-2 -3 (b) (i) -1, 3
(ii) Minimum -4 y
(iii) 5 4 3
2.
Axis of symmetry x = - 1.5, minimum value - 7.5
3.
Axis of symmetry x = - 1.5, minimum value - 0.25
2 1
x -4 -3 -2 -1 -1
4.
Axis of symmetry x = 0, minimum value - 4
5.
Axis of symmetry x =
6.
Axis of symmetry x = 1, maximum value -6
-4
7.
Axis of symmetry x = - 1, maximum point ^ - 1, 7 h
-5
8.
Minimum value -1, 2 solutions
3 7 3 n , minimum point d , 8 8 16
-2 -3
1
2
3
4
593
594
Maths In Focus Mathematics Preliminary Course
(c) (i) 5.83, 0.17 (ii) Minimum -8 (iii)
y
(iii)
y
5
10
4
8
3
6
2
4
1
2 1
-4 -3 -2 -1 -2
2
3
4
5
x
6
1
-4 -3 -2 -1 -1
-4
-2
-6
-2 1 -3 12 -4
-8 -10
3
2
4
x
5
2 3
-5 -6
(d) (i) -2, 0 (ii) Minimum -1 y
(iii)
(g) (i) 1.65, -3.65
5
(ii) Maximum 7 y
(iii)
4
7
3
6
2
5
1
4
x 2
1
-4 -3 -2 -1 -1
3 2
-2
1
-3
x (e) (i) ! 3
3
4
4
5
5
-2
y
(iii)
2
1
-4 -3 -2 -1 -1
(ii) Minimum -18
-3
2 1 -4 -3 -2 -1 -2
1
-4 -6
2
3
4
5
x (h) (i) 1.3, -2.3
(ii) Maximum 3
1 4
y
(iii)
-8
5
-10
4
-12
3
-14
2
-16
1
3
1 4
-18 2 (f) (i) -1, 3
1 (ii) Minimum - 2 12
-4 -3 -2 -1 -1 -2 -3
x 1
2
3
ANSWERS
(i) (i) 0.56, -3.56 (ii) Minimum 4
1 4
16. (a) None
y
(iii)
(b) 6
3 4
(c)
y
41 4
5
14
4
12
3
10
2
8
1
6 x
-4 -3 -2 -1 -1
1
2
3
4
5
4
2
-2 -3
2
1
-4 -3 -2 -1 -1
3
4
x
5
-2 -3
(j) (i) 2.87, -0.87 (ii) Maximum 7 y
(iii)
17. (a) - 3
7
7 8
(b) None y
(c)
6 5
2
4
1
3 2
1
-4 -3 -2 -1 -2
1 1
-4 -3 -2 -1 -1
2
3
4
5
2
3
4
x
5
-4
x
-6 -8
-2
-10
-3
-12 -14 15. (a) 4 (b) None
-16
(c)
-18
y 7
18. (a)
y
6
8
5
6
4 3
4
2
2
1 -4 -3 -2 -1 -1
x 1
2
3
4
5
x
-4 -3 -2 -1 -1
1
-2
-2
-3
-3 (b) x 1 2, x 2 3
(c) 2 # x # 3
2
3
4
5
595
596
Maths In Focus Mathematics Preliminary Course
19.
22.
y
y
8
2
6
1
4
-4 -3 -2 -1 -1
x
2 1
-3 -2 -1 -2
3
2
4
5
-3
5
-4
-4
-5
-6
-6 -7
Graph is always above the x-axis so y 2 0 for all x ` 3x 2 - 2x + 4 2 0 for all x 20.
4
-2 x
-4
3
2
1
Graph is always below the x-axis so y 1 0 for all x ` - 5x 2 + 4x -1 1 0 for all x
y 8
Exercises 9.2
6
1.
x 1 -3, x 2 3
2. - 1 # n # 0
4.
x 1 - 2, x 2 2
5. 0 # y # 6
7.
x 1 - 4, x 2 2
8. p # - 3, p $ - 1
4 2
x 1
-4 -3 -2 -1 -2
2
3
4
5
10. x # - 3, x $ 2
-4
13. - 2
-6
Graph is always above the x-axis so y 2 0 for all x ` x 2 + x + 2 2 0 for all x 21.
6. 0 1 t 1 2
1 11. 1 1 h 1 2 2
9. m 1 2, m 2 4
12. - 4 # x # 5
14. q 1 3, q 2 6
15. All real x
16. n # - 4, n $ 3
17. - 3 1 x 1 5
18. - 6 # t # 2
1 19. y 1 - , y 2 5 3
20. x # - 2, x $ 4
Exercises 9.3
y 4 2
1.
(a) 20 (b) -47 (c) -12 (h) 64 (i) 17 (j) 0
2.
(a) 17 unequal real irrational roots (b) -39 no real roots (c) 1 unequal real rational roots (d) 0 equal real rational roots (e) 33 unequal real irrational roots (f) -16 no real roots (g) 49 unequal real rational roots (h) -116 no real roots (i) 1 unequal real rational roots (j) 48 unequal real irrational roots
3.
p = 1 4. k = ! 2 5. b # -
8.
a =320
x -4 -3 -2 -1 -2
1 #k #7 2
3. a # 0, a $ 2
1
2
3
4
5
-4 -6 -8 -10 -12 -14 -16 -18 Graph is always below the x-axis so y 1 0 for all x ` - x 2 + 2x - 7 1 0 for all x
(d) 49
(e) 9
(f) -16
(g) 0
1 7 6. p 2 2 7. k 2 - 2 12 8
b 2 - 4ac = ] - 1 g2 - 4 ] 3 g ] 7 g = - 83 10 So 3x 2 - x + 7 2 0 for all x
9.
k # - 5, k $ 3
12. k # - 1, k $ 1
10. 0 1 k 1 4 13. p 1 -
1 3
11. m 1 - 3, m 2 3 14. 0 # b # 2
1 2
ANSWERS
15. p # - 2, p $ 6
2.
m = 2, p = - 5, q = 2
16. Solving simultaneously: y = 2x + 6
3.
x 2 - 4x + 5 = x ] x - 2 g - 2 ] x + 1 g + 3 + 4
y =x +3 2
(1)
4.
(2)
Substitute (2) in (1): x 2 + 3 = 2x + 6 x 2 - 2x - 3 = 0 b 2 - 4ac = ] - 2 g2 - 4 ] 1 g ] - 3 g = 16 20 So there are 2 points of intersection 17. 3x + y - 4 = 0 y = x 2 + 5x + 3 From (1): y = - 3x + 4 Substitute (2) in (3): x 2 + 5x + 3 = - 3x + 4 x 2 + 8x - 1 = 0 b 2 - 4ac = 8 2 - 4 ] 1 g ] - 1 g = 68 20 So there are 2 points of intersection
(1) (2)
18. y = - x - 4 y = x2 Substitute (2) in (1): x2 = - x - 4 x2 + x + 4 = 0 b 2 - 4ac = 1 2 - 4 ] 1 g ] 4 g = - 15 10 So there are no points of intersection
(1) (2)
19. y = 5x - 2 y = x 2 + 3x - 1 Substitute (2) in (1): x 2 + 3x - 1 = 5 x - 2 x 2 - 2x + 1 = 0 b 2 - 4ac = ] - 2 g2 - 4 ] 1 g ] 1 g =0 So there is 1 point of intersection ` the line is a tangent to the parabola
(1) (2)
20. p = 3
1 4
5.
A = 1, B = 5 , C = - 6
7.
K = 1, L = 6, M = 7.5 8. 12 ] x + 5 g + ] 2x - 3 g2 - 65 - 2
9.
a = 0, b = - 4, c = - 21
10. (a) y = x 2 - x - 5 (3)
1.
(a) a = 1, b = 2, c = -6 (b) a = 2, b = -11, c = 15 (c) a = 1, b = 1, c = - 2 (d) a = 1, b = 7, c = 18 (e) a = 3, b = -11, c = -16 (f) a = 4, b = 17, c = 11 (g) a = 2, b = -12, c = -9 (h) a = 3, b = - 8, c = 2 (i) a = - 1, b = 10, c = - 24 (j) a = - 2, b = 0, c = - 1
6. a = 2, b = 1, c = - 1
(b) y = x 2 - 3x
(c) y = 2x 2 - 3x + 7
(d) y = x 2 + 4x - 9
(e) y = - x - 2x + 1 2
Exercises 9.5 1.
(a) a + b = - 2, ab = 1 (b) a + b = 1.5, ab = - 3 (c) a + b = 0.2, ab = - 1.8 (d) a + b = - 7, ab = 1 2 (e) a + b = 2 , ab = 1 3
2.
(a) 3
3.
(a) x 2 + 3x - 10 = 0 (b) x 2 - 4x - 21 = 0 (c) x 2 + 5x + 4 = 0 (d) x 2 - 8x + 11 = 0 (e) x 2 - 2x - 27 = 0
4.
m = 0.5
5. k = - 32
9.
k = -5
10. m = ! 3
(b) - 6
13. p = 2, r = - 7 16. ab = 1 ` b = 17. (a) k = - 1
(c) - 0.5
(d) 21
6. b = 4
7. k = 1
11. k = - 1
14. b = - 6, c = 8
8. p = 13
12. n = - 1, 3 15. a = 0, b = - 1
1 a
(b) k = - 1, 0
(c) k = - 1.8
(d) k = 3
(e) k # - 1, k $ 0 18. (a) p = ! 2 3 (c) p = !
21. (c) and (d)
Exercises 9.4
RHS = a ] x - 2 g ] x + 3 g + b ] x - 2 g + c = 1 ] x - 2 g ] x + 3 g + 1 ] x - 2 g + 17 = x 2 + 3x - 2x - 6 + x - 2 + 17 = x 2 + 2x + 9 = RHS ` true
(b) p # - 2 3 , p $ 2 3
3 3 2
19. (a) k = 2
(b) k = - 3
20. (a) m = 1
(b) m 1
(c) k = 2
3 - 10 3 + 10 ,m2 2 2
(c) m = - 3
Exercises 9.6 1.
(a) x = -1, - 4 (b) y = 2, 5 (c) x = - 4, 2 (d) n = - 1, 4 (e) a = - 3, 5 (f) p = 3, 4 (g) x = 2, - 4 (h) k = 5, 12 (i) t = 6, - 4 (j) b = -12, - 4
597
598
Maths In Focus Mathematics Preliminary Course
2.
(a) x = - 2, 3
(b) x = 2, 3
(c) x = 4, 5
1 (e) x = 1 , 4 2 3.
(a) x = ! 3
(b) y = ! 2, ! 2
(d) x = 1.37, - 4.37, 0.79, - 3.79
(d) x = 3, 5
1! 5 (c) x = 2 (e) a = - 2, - 2 ! 6
10. 3 ] x - 2 g2 + 12 ] x + 3 g - 41 12. (a) k = 3
(c) iii
16. (a) i
5.
x = ! 1, !2
17. (a) iii
7.
x = ! 2.19, !0.46, !1.93, !0.52
8.
(a) x = 0c , 90c , 180c , 360c (b) x = 90c , 180c , 270c (c) x = 90c , 210c , 330c (d) x = 60c , 90c , 270c , 300c (e) x = 0c , 180c , 270c , 360c
x+3+
2 =5 x+3
2 # (x + 3) = 5 # (x + 3) (x + 3) # (x + 3) + ]x + 3g ] x + 3 g2 + 2 = 5 ] x + 3 g ] x + 3 g2 - 5 ] x + 3 g + 2 = 0 Let u = x + 3 u 2 - 5u + 2 = 0 b 2 - 4ac = ] - 5 g2 - 4 ] 1 g ] 2 g = 17 20 So u has 2 real irrational roots. ` x + 3 and so x has 2 real irrational roots
Test yourself 9 1.
(a) 0 # x # 3
(b) n 1 - 3, n 2 3
2.
a = 1, b = - 9, c = 14
4.
a =120 D = b 2 - 4ac = ] -2 g 2 -4 # 1 # 7 = - 24 10 ` positive definite
3. (a) x = 2
5.
(a) 6 (b) 3
7.
(a) iv (b) ii
8.
a = -1 1 0 D = b 2 - 4ac = 3 2 - 4 # (-1) # (- 4) = -7 10
(c) 2 (d) 18 (c) iii
(d) ii
9.
1 (b) 6 8
(b) i
(d) i
(c) i
(e) k = 2
15. x = 0, 2
(e) ii
(d) ii 1 a c = a k = k = RHS = 1
(c) - 2 # y # 2 (b) - 3
ab 1 a LHS a
∴ roots are reciprocals for all x. 19. (a) x 2 + 3x - 28 = 0
(b) x 2 - 10x + 18 = 0
20. x = 1, 3
Challenge exercise 9 1.
D = ] k - 4 g2 $ 0 and a perfect square ∴ real, rational roots
2.
y = x 2 - 5x + 4
5.
11
9.
x = !1
3. a = 4, b = - 3, c = 7
6. n = - 2.3375
7. p 2 0.75
4. x = ! 2
8. Show D = 0
10. A = 2, B = - 19, C = 67 or A = - 2, B = 13, C = - 61 11.
4x + 1 x2 - x - 2
12. k #
=
3 1 + x-2 x+1
1 - 21 1 + 21 ,k$ 2 2
13. x = 30c , 90c , 150c
14. x = 1,
15. x = 60c , 90c , 270c , 300c
3! 5 2
16. - 23
Chapter 10: Locus and the parabola Exercises 10.1 2 1 (e) 30 6. x = 1 , 3 3 (e) i
` - 4 + 3x - x 2 1 0 for all x 1 (a) x = 4
(b) i
9 16
(d) k = 3
18. For reciprocal roots b =
(a) x = 0c , 45c , 180c , 225c , 360c (b) x = 0c , 180c , 360c (c) x = 0c , 30c , 150c , 180c , 360c (d) x = 45c , 60c ,135c , 120c , 225c , 240c , 315c , 300c (e) x = 30c , 60c , 120c , 150c , 210c , 240c , 300c , 330c
10.
(c) k = 3
14. m 1 -
(a) x = 0, 3 (b) p = 1 (c) x = 1 (d) x = 1 (e) x = 1, 3
9.
(b) k = 1
1 13. x = - , 3 2
4.
6. x = - 1
1 4
11. x = 30c , 150c , 270c
1.
A circle
2. A straight line parallel to the ladder.
3.
An arc
6.
The straight line - 2 1 x 1 2 or | x | 1 2
7.
A circle, centre the origin, radius 2 (equation x2 + y2 = 4 i
8.
lines y = !1
4. A (parabolic) arc
5. A spiral
9. lines x = !5
10. line y = 2
11. Circle x 2 + y 2 = 1 (centre origin, radius 1) 12. Circle, centre ^ 1, -2 h, radius 4
13. y = -5
ANSWERS
14. Circle, centre (1, 1), radius 3
15. x = -7
16. x = 3
4.
x 2 + 4x + y 2 + 4y - 8 = 0
6.
x 2 + 6x + y 2 - 16y + 69 = 0
19. Circle, centre ^ -2, 4 h, radius 6
7.
x 2 - 10x + y 2 + 4y + 27 = 0
20. Circle, centre ^ -4, 5 h, radius 1
9.
x 2 - 2x + y 2 - 10y + 25 = 0
17. y = !8
18. x = !4
Exercises 10.2 x +y =1
2. x + 2x + y + 2y - 79 = 0
3.
x - 10x + y + 4y + 25 = 0
5.
12x - 26y - 1 = 0
7.
3x 2 - 32x + 3y 2 - 50y + 251 = 0
8.
5x 2 - 102x + 5y 2 + 58y - 154 = 0
9.
x 2 - 4x + 20y - 36 = 0
2
2
8. x 2 + y 2 - 9 = 0
10. x 2 + 12x + y 2 - 2y + 1 = 0
1.
2
5. x 2 - 2x + y 2 - 48 = 0
2
2
2
11. y 2 + 8x - 32 = 0 13. x 2 + 12y = 0
4. 8x - 6y + 13 = 0
6. y = ! x
10. x 2 - 20y = 0
12. x 2 - 2x + 8y - 7 = 0
14. x 2 - 5x + y 2 - 2y - 11 = 0
11. x 2 - 8x + y 2 - 6y + 22 = 0
12. x 2 + y 2 + 6y + 1 = 0
13. (a) Radius 3, centre (2, 1) (b) Radius 5, centre (−4, 2) (c) Radius 1, centre (0, 1) (d) Radius 6, centre (5, −3) (e) Radius 1, centre (−1, 1) (f) Radius 6, centre (6, 0) (g) Radius 5, centre (−3, 4) (h) Radius 8, centre (−10, 2) (i) Radius 5, centre (7, −1) (j) Radius 10 , centre (−1, −2) 14. Centre ^ 3, -1 h , radius 4
15. Centre ^ 2, 5 h , radius 5
16. Centre ^ - 1, -6 h , radius 7
17. Centre (4, 7), radius 8
1 1 18. Centre d - 1 , 1 n , radius 2 2 2 19.
15. x 2 + 3x + y 2 - y - 4 = 0 16. x 2 + x + y 2 - 2y - 17 = 0 17. 2x 2 + 4x + 2y 2 - 6y + 47 = 0 18. 2x 2 + 2x + 2y 2 + 4y + 27 = 0 19. 3x + 4y + 25 = 0, 3x + 4y - 15 = 0 20. 12x - 5y - 14 = 0, 12x - 5y + 12 = 0 21. x - 2y - 3 ! 5 5 = 0
20. Show perpendicular distance from the line to ^ 4, -2 h is 5 units, or solve simultaneous equations.
22. x - 7y + 9 = 0, 7x + y - 5 = 0 23. 7x - 4y - 30 = 0, 32x + 56y - 35 = 0 24. xy - 16x - 7y + 40 = 0
22. x 2 + 2x + y 2 + 2y - 23 = 0
25. x 2 - 6x - 3y 2 - 12y + 9 = 0
Problem 12x + 5y - 40 = 0, 12x + 5y + 38 = 0
Exercises 10.3 1.
(a) Radius 10, centre (0, 0) (b) Radius 5 , centre (0, 0) (c) Radius 4, centre (4, 5) (d) Radius 7, centre (5, −6) (e) Radius 9, centre (0, 3)
2.
(a) x + y = 16 (b) x - 6x + y - 4y - 12 = 0 (c) x 2 + 2x + y 2 - 10y + 17 = 0 (d) x 2 - 4x + y 2 - 6y - 23 = 0 (e) x 2 + 8x + y 2 - 4y - 5 = 0 (f) x 2 + y 2 + 4y + 3 = 0 (g) x 2 - 8x + y 2 - 4y - 29 = 0 (h) x 2 + 6x + y 2 + 8y - 56 = 0 (i) x 2 + 4x + y 2 - 1 = 0 (j) x 2 + 8x + y 2 + 14y + 62 = 0
3.
2
2
2
x 2 - 18x + y 2 + 8y + 96 = 0
21. (a) Both circles have centre ^ 1, -2 h (b) 1 unit
2
23.
34 units
24. (a) 5 units (b) 3 units and 2 units (c) XY is the sum of the radii. The circles touch each other at a single point, ^ 0, 1 h . 25. Perpendicular distance from centre ^ 0, 0 h to the line is equal to the radius 2 units; perpendicular distance from centre ^ -1, 2 h to the line is equal to the radius 3 units. 26. (a) x 2 + 2x + y 2 - 6y - 15 = 0 (b) ^ 2, 7 h, ^ -1, -2 h (c) Z = ^ -1, 8 h 1 (d) m zx # m yx = - # 3 3 = -1 ` +ZXY = 90c 27. (a) 4 units
(b) x 2 - 4x + y 2 + 10y + 13 = 0
599
600
Maths In Focus Mathematics Preliminary Course
Exercises 10.4 1.
(b) x = 36y
(c) x = 4y
(e) x 2 = 40y
(f) x 2 = 12y
(g) x 2 = 24y
(i) x = 8y 2
2.
Exercises 10.5
(a) x = 20y 2
2
2
(d) x = 16y 2
1.
(a) y 2 = 8x (b) y 2 = 20x (c) y 2 = 56x (d) y 2 = 36x (e) y 2 = 32x (f) y 2 = 24x (g) y 2 = 28x (h) y 2 = 12x (i) y 2 = 16x (j) y 2 = 4x
2.
(a) y 2 = -36x (b) y 2 = - 16x (c) y 2 = -40x (d) y 2 = -24x (e) y 2 = - 8x (f) y 2 = -48x (g) y 2 = - 44x (h) y 2 = -20x (i) y 2 = -12x (j) y 2 = -28x
3.
(a) (i) (2, 0) (c) (i) (4, 0) (e) (i) (7, 0) (g) (i) (6, 0) 1 (i) (i) c , 0 m 4
4.
(a) (i) (−2, 0) (ii) x = 2 (b) (i) (−3, 0) (ii) x = 3 (c) (i) (−7, 0) (ii) x = 7 (d) (i) (−1, 0) (ii) x = 1 (e) (i) (−6, 0) (ii) x = 6 (f) (i) (−13, 0) (ii) x = 13 1 1 (g) (i) (−15, 0) (ii) x = 15 (h) (i) c - , 0 m (ii) x = 2 2 1 1 1 1 (i) (i) c - 6 , 0 m (ii) x = 6 (j) (i) c - 1 , 0 m (ii) x = 1 4 4 2 2
5.
(a) y 2 = 20x
(h) x 2 = 44y
(j) x = 48y
(a) x 2 = -4y
2
(b) x 2 = -12y
(d) x 2 = -28y
(e) x 2 = -24y
(g) x 2 = -32y
(h) x 2 = -8y
(c) x 2 = -16y (f) x 2 = -36y (i) x 2 = -60y
(j) x = -52y 2
3.
4.
(a) (i) (0, 1) (ii) y = -1 (b) (i) (0, 7) (ii) y = -7 (c) (i) (0, 4) (ii) y = -4 (d) (i) (0, 9) (ii) y = -9 (e) (i) (0, 10) (ii) y = -10 (f) (i) (0, 11) (ii) y = -11 1 1 (g) (i) (0, 3) (ii) y = -3 (h) (i) c (0, 1 m (ii) y = -1 2 2 3 1 1 (i) (i) c 0, 2 m (ii) y = -2 (j) (i) c 0, 3 m 4 2 2 3 (ii) y = -3 4 (a) (i) (0, −1) (ii) y = 1 (b) (i) (0, −6) (ii) y = 6 (c) (i) (0, −2) (ii) y = 2 (d) (i) (0, −12) (ii) y = 12 (e) (i) (0, −5) (ii) y = 5 (f) (i) (0, −4) (ii) y = 4 (g) (i) (0, −8) (ii) y = 8 (h) (i) (0, −10) (ii) y = 10 1 1 (i) (i) c 0, - m (ii) y = 2 2
5.
6.
1 1 (j) (i) c 0, -5 m (ii) y = 5 2 2
(a) x 2 = 28y (b) x 2 = 44y (c) x 2 = -24y (d) x 2 = 8y (e) x 2 = !12y (f) x 2 = !32y 1 (g) x 2 = 32y (h) x 2 = y 7 (a) Focus ^ 0, 2 h, directrix y = -2, focal length 2 (b) Focus ^ 0, 6 h, directrix y = -6, focal length 6
(c) Focus ^ 0, -3 h, directrix y = 3, focal length 3 1 1 1 (d) Focus d 0, n, directrix y = - , focal length 2 2 2 3 3 3 (e) Focus d 0, -1 n, directrix y = 1 , focal length 1 4 4 4 1 1 1 (f) Focus d 0, n, directrix y = - , focal length 8 8 8
7.
y =2
8. ^ 4, 4 h
9.
3 1 X = d -1 , - n 2 8
10. ^ 4, -2 h and ^ -4, -2 h ; 8 units 11. (a) x 2 = - 12y
(b) y = 3
(c) 33
1 units 3
14. (a) Substitute Q into the equation of the parabola. (b) _ q 2 - 1 i x - 2qy + 2aq = 0 (c) Equation of latus rectum is y = a. Solving with x 2 = 4ay gives two endpoints A ^ -2a, a h, B ^ 2a, a h . Length of AB = 4a.
(b) y 2 = 4x
(e) y 2 = !36x 6.
(b) (i) (3, 0) (ii) x = -3 (d) (i) (1, 0) (ii) x = -1 (f) (i) (8, 0) (ii) x = -8 (h) (i) (9, 0) (ii) x = -9 1 1 (j) (i) c 4 , 0 m (ii) x = -4 2 2
(c) y 2 = -16x
(f) y 2 = !8x
(d) y 2 = 12x
(g) y 2 = 12x
(h) y 2 =
1 x 2
(a) Focus ^ 2, 0 h, directrix x = - 2, focal length 2
(b) Focus ^ 1, 0 h, directrix x = - 1, focal length 1 (c) Focus ^ -3, 0 h, directrix x = 3, focal length 3
1 1 1 (d) Focus d 1 , 0 n, directrix x = - 1 , focal length 1 2 2 2 1 1 1 (e) Focus d -1 , 0 n, directrix x = 1 , focal length 1 4 4 4 (f) Focus d
1 1 1 , 0 n, directrix x = - , focal length 12 12 12
7.
x = 4 (latus rectum)
9.
^ 9, - 6 h, ^ 81, 18 h
10. (a) 5x - 12y - 25 = 0 (d) 4
2 units 13
8. 12, ^ 3, 6 h, ^ 3, -6 h
5 1 (b) d - 5, - 4 n (c) 10 units2 6 12
(e) 11.7 units2
Exercises 10.6 1.
(a) ] x - 3 g2 = 8 ^ y + 3 h
(b) ] x - 5 g2 = 4 ^ y + 6 h
(c) ] x - 1 g = 4 ^ y + 3 h (d) ] x - 4 g2 = -12 ^ y - 3 h (e) ] x - 6 g2 = 8 ^ y + 7 h (f) ] x + 7 g2 = -16 ^ y - 3 h (g) ] x - 2 g2 = -4 ^ y - 5 h (h) ] x + 9 g2 = 12 ^ y + 6 h
12. (a) Substitute the point into the equation. 3 (b) 3x + 4y - 3 = 0 (c) d 2, - n 4 13. (a) x - 4y + 2 = 0 (b) ^ 0, 1 h does not lie on the line (c) x 2 - 4x + y 2 - 2y + 1 = 0 (d) Substitute ^ 0, 1 h into the equation of the circle.
= -2 = -4 = -7 = -6 1 (ii) x = 4
(ii) x (ii) x (ii) x (ii) x
2
(i) ] x + 1 g2 = 8 ^ y + 1 h
2.
(j) ] x - 3 g2 = - 4 ^ y - 2 h
(a) ^ y - 4 h2 = 4 ] x + 4 g (b) ^ y - 1 h2 = 8 ] x + 2 g (c) ^ y + 2 h2 = 12 ] x + 1 g (d) ^ y - 10 h2 = - 4 ] x - 29 g (e) ^ y + 3 h2 = - 16 ] x - 1 g
(f) ^ y - 6 h2 = 8 ] x + 4 g
(g) ^ y + 5 h = - 24 ] x - 2 g (h) ^ y + 12 h2 = 4 ] x + 36 g 2
(i) ^ y - 2 h2 = - 20 ] x - 1 g (j) ^ y + 4 h2 = - 8 ] x - 2 g
ANSWERS
3.
(a) x 2 + 2x - 8y + 9 = 0
(b) x 2 + 8x - 4y + 16 = 0
(c) x - 4x - 8y - 12 = 0 (e) x 2 + 4x - 16y + 20 = 0
(f) x 2 + 2x + 16y + 1 = 0
(g) x - 8x + 20y - 24 = 0
(h) x + 10x + 8y + 1 = 0
(i) x 2 + 6x + 12y + 45 = 0
(j) x 2 + 4y + 24 = 0
2
(k) y - 6y - 12x - 3 = 0
2
(n) y 2 + 4y - 16x - 12 = 0
(o) y + 2y - 8x - 7 = 0
(p) y + 8y + 12x + 4 = 0
2
2
(q) y 2 - 2y + 4x - 11 = 0 (s) y - 4y + 2x + 5 = 0 4.
2
(a) (i) (3, −2)
(ii) y = -4
(b) (i) (1, 1)
(ii) y = - 3
(c) (i) (−2, 0)
(ii) y = -2
(d) (i) (4, 2)
(ii) y = - 4
(e) (i) (−5, −1)
5.
(r) y 2 - 6y + 16x + 25 = 0 (t) y - 2y + 2x - 6 = 0
2
(ii) y = -5
(f) (i) (3, 1)
(g) (i) (−1, 0)
(ii) y = 4
(h) (i) (2, 0)
(i) (i) (4, −2)
(ii) y = 4
(j) (i) (−2, −3)
(ii) y = 3
(ii) y = 2 (ii) y = 5
(a) (i) (0, −1) (ii) x = -2 (b) (i) (2, 4) (ii) x = - 4 (c) (i) (0, 3) (ii) x = -4 (d) (i) (3, −2) (ii) x = -5 (e) (i) (7, 1) (ii) x = -5 (f) (i) (1, −5) (ii) x = 5 (g) (i) (11, −7) (ii) x = 13 (h) (i) (−3, 6) (ii) x = 7 1 1 (i) (i) (−7, 2) (ii) x = 9 (j) (i) c -10 , -3 m (ii) x = 9 2 2
6.
x 2 - 12y + 36 = 0
7.
x 2 + 4x - 8y - 4 = 0, x 2 + 4x + 8y + 12 = 0
8.
x 2 - 2x - 4y - 19 = 0
9. y 2 - 12y + 12x + 12 = 0
10. x 2 - 2x - 16y + 1 = 0
11. x 2 - 2x - 28y + 29 = 0
12. y 2 + 4y + 24x - 44 = 0
13. y 2 - 6y - 32x + 9 = 0
14. x - 6x + 8y - 15 = 0
15. y + 2y - 16x + 49 = 0
2
16. x 2 + 6x + 4y - 7 = 0
24. x 2 + 4x + 8y - 20 = 0
25. 0.3 m
2
(l) y - 8y - 4x + 8 = 0
2
(m) y 2 - 8x + 32 = 0
3 1 (b) d -1, -8 n, y = -9 4 4
(d) x 2 - 6x - 8y + 41 = 0
2
Exercises 10.7 1. 5.
18. y + 2y + 16x - 95 = 0
dy dx
1 3
=x
2. m = -4
3. m = -1
6. x - y - 2 = 0
4. m =
1 2
7. x - 2y + 12 = 0
8.
x + y - 6 = 0, x - y - 18 = 0
9.
x - 2y - 2 = 0, 2x + y - 9 = 0
7 1 10. 4x + y - 8 = 0, M = d 1 , n 8 2 11. x + y - 9 = 0, P = ^ - 18, 27 h 12. Q = ^ 33, 60.5 h 13. x + 4y + 144 = 0, 4x + 2y + 9 = 0, ^ 18, -40.5 h ; show the point lies on the parabola by substituting it into the equation of the parabola 14. x - y - 4 = 0, R = ^ 4, 0 h 15. (a) Substitute P into the equation of the parabola (b) x + py - 2p - p 3 = 0 (c) Substitute ^ 0, 1 h into the equation of the normal. 0 + p - 2p - p 3 = 0 0 = p3 + p = p (p 2 + 1) 2 Since p ! 0, p + 1 = 0
Test yourself 10 1.
8x + 6y - 29 = 0
3.
Centre ^ 3, 1 h, radius 4
5.
x 2 + y 2 = 25
7.
x 2 + x + y 2 - 3y - 10 = 0
9.
(a) (i) ^ 1, 1 h
2
17. x 2 - 4x - 12y - 8 = 0
m=
2. x 2 - 4x - 8y - 4 = 0 4. (a) ^ 1, - 3 h
6. (a) y = 2
(b) ^ 4, - 3 h
(b) ^ 0, - 2 h
2
19. (a) Vertex ^ - 2, 1 h, focus ^ - 2, 3 h, directrix y = -1 (b) Vertex ^ 3, 2 h, focus ^ 3, 5 h, directrix y = -1
(ii) ^ 1, 2 h
(c) Vertex ^ 1, -1 h, focus ^ 1, - 2 h, directrix y = 0
10. 2x + 3y + 6 = 0
(e) Vertex ^ 0, - 2 h, focus ^ 6, - 2 h, directrix x = -6
12. y 2 = - 24x
(d) Vertex ^ 3, 4 h, focus ^ 7, 4 h, directrix x = -1
(f) Vertex ^ - 5, 0 h, focus ^ - 7, 0 h, directrix x = -3 20. Vertex ^ - 1, 4 h, focus ^ -1, -3 h , directrix y = 11, axis x = - 1, maximum value 4 21. x 2 - 4x - 8y + 12 = 0 or x 2 - 4x + 8y - 36 = 0 22. (a) 8x 2 + 9y - 72 = 0 23. (a)
(b) d 0, 7
23 9 n, y = 8 32 32
8. x 2 - 8x + 16y - 16 = 0
(b) y = 0
11. 14 units
13. x 2 - 8y + 16 = 0
14. 4x - 3y - 16 = 0, 4x - 3y + 14 = 0 15. y = x, y = - x
16. y 2 = 20x
18. (a) x - 4y + 72 = 0
17. (a) -
1 2
(b) 2
1 (b) d 9, 20 n 4
19. Sub ^ 0, 4 h : LHS = 7 # 0 - 3 # 4 + 12 = 0 = RHS 2 20. d , -7 n 9 21. (a) x - y - 3 = 0 (b) R = ^ 0, -3 h (c) F = ^ 0, 3 h FP = FR = 6
601
602
Maths In Focus Mathematics Preliminary Course
Challenge exercise 10
8.
3x - 4y - 14 = 0, 3x - 4y + 16 = 0
1.
(a) 8x + 6y - 29 = 0 (b) Midpoint of AB lies on line; m 1 m 2 = -1
9.
Vertex ^ - 4, -17 h , focus ^ - 4, -16.75 h
2.
(a) x 2 - 2x + y 2 - 6y - 15 = 0 (b) Put y = 0 into equation
3.
1 d 2 , -3 n 2
4.
(a) 4x - 2y + 9 = 0; x + 2y - 24 = 0 (b) m 1 m 2 = - 1 (c) X = ^ 3, 10.5 h (d) 3x - 4y + 8 = 0; focus ^ 0, 2 h lies on the line
10. x = 0, 3
11. x + 2y + 2 = 0
12. b $ -2
13. x 2 + y 2 = 16, circle centre ^ 0, 0 h and radius 4 14. x 2 + 4x + y 2 + 6y - 12 = 0 15. x 2 - 3x + y 2 - 6y - 17 = 0 16. - 0.75
17. 5x 2 - 54x + 5y 2 + 20y - 79 = 0
18. a = 2, b = 1, c = 0
5.
^ 0, 0 h
6.
(a) 2x - 4y - 1 = 0; 2x + y + 4 = 0 (b) Point lies on line y = - 1
7.
y = - 2 x + 4x - 2 2
8. 3x + y + 2 = 0
19. -
x 9 - x2
20. x 2 - 4x - 16y + 20 = 0 21.
9.
`
AC = BC and CD = CE (given) AC BC = CD CE +ACB = +ECD (vertically opposite angles)
` since two sides are in proportion and their included angles are equal, ΔABC is similar to ΔCDE y = 5.3 cm 22. x - y - 4 = 0 23. x 2 + 2x - 16y - 15 = 0 25.
10. (a) x 2 + 4x + y 2 - 10y + 21 = 0 (b) ] x + 2 g2 + ^ y - 5 h2 = 8; centre ^ -2, 5 h; radius = 11. -
8 =2 2
24. x = 0, 2
a 10 D = b 2 - 4ac = 1 2 - 4 (- 1) (- 9) = - 35 10 Since a 1 0 and D 1 0, - x 2 + x - 9 1 0 for all x
26. 8 (3x - 1) (2x + 5) 3 + 3 (2x + 5) 4 = ] 30x + 7 g (2x + 5) 3 27. sec x cosec x
2 3 3
12. (a) y 2 + 4y - 16x + 52 = 0 13. 4 2 units
(b) 2x - y - 6 = 0
14. x 2 + y 2 - 2y - 2 = 0
15. 696 mm from the vertex 16. 141x + 127y + 32 = 0; 219x + 23y + 58 = 0
28. Centre ^ -5, 3 h, radius 2 29. a 2 0 D = b 2 - 4ac = ] -1 g2 - 4 (1) (3) = -11 10 Since a 2 0 and D 1 0, x 2 - x + 3 2 0 for all x 30. k = 1
31. 3x + 2y - 9 = 0
Practice assessment task set 3 2. 4x + 3y - 16 = 0
1.
m ≤ 2, m ≥ 3
3.
Centre ^ - 3, 5 h, radius 7
4.
(a)
5.
Focus ^ 0, -2 h, directrix y = 2
6.
x = - 5 or - 6
2 3
(b) -
1 3
(c) 1
1 9
7. k = - 1
32. (a) 217 km
(b) 153c
33. a = 3, b = - 18, c = - 34
34. x 2 4, x 1 3
35. i = 95c 44’ 36. T = 361 ^ 2 0 and a perfect square h 37. x + 2y + 9 = 0
38. k # 3
ANSWERS
39. 5x - 4y - 41 = 0
40.
3 6 - 10 + 3 3 - 5 22
41. x = 4.9 cm, y = 11.1 cm 44. 4.5 m 45.
128 2187
47. 2x + 3y - 3 = 0
42. x = 1
43. 8.25 units
46. x = 60°, 120°, 240°, 300° 1 1 48. y = 1 , 3 2
55. 1.8 units
56. tan i
58.
1 4
3
2
4
59. x 2 + 2x + y 2 - 3y - 25 = 0
60. Focus (2, 1), directrix y = 5 61. x - 2y - 36 = 0 62. Distance from centre ^ 0, 0 h to line is d=
| ax 1 + by 1 + c |
a2 + b2 40 = 10 =4 = radius ` line is tangent
65. Radius 3; x 2 + y 2 = 9
68. 54. -
1 31
+ACB +ABC AC ` by AAS, DABC
69. 46 m2
57. 8x ] 2x + 5 g (x - 1) + 2 (x - 1) = 2 (x 2 - 1) 3 (9x 2 + 20x - 1) 2
64. x 1 - 2, x 2 2
67. Domain: all real x; range: y $ - 3
1 1 51. x = - 1, y = 2 or x = - , y = 4 4 4 53. x = 43
1 2
66. a = 3, b = -14, c = 9
49. 162c
50. x = 45°, 135°, 225°, 315°
52. ] a - 2b g ^ a 2 + 2ab + 4b 2 h
63. k = -2
= +ECD ^ vertically opposite angles h = +CED (alternate angles AB||ED) = CD ^ given h / DCDE
70. x + y - 3 = 0
71. x 2 - 12x + 36 = ] x - 6 g2 72. y $ 2.5, y # - 6.5 73. (a) 9x - y + 16 = 0
(b) x + 9y + 20 = 0
(c) Q = ^ - 20, 0 h 74. (a) x - 8y + 129 = 0
1 1 n (b) R = d 7 , 17 64 8
75. a = 1, b = - 3, c = -1 76. (c)
77. (d)
78. (b)
79. (a)
80. (c)
81. (c)
603
