maths in focus - margaret grove - ans
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Mathematics Preliminary Course - 2nd EditionTRANSCRIPT
540 Maths In Focus Mathematics Preliminary Course
Answers Chapter 1: Basic arithmetic
Problem
5
Exercises 1.1
1. (a) Rational (b) Rational (c) Rational (d) Irrational (e) Rational (f) Irrational (g) Irrational (h) Rational (i) Rational (j) Irrational
2. (a) 18 (b) 11 (c) 6 (d) 11 (e) .4 3- (f) −1 (g) 2157
(h) 12019
(i) 2 (j) 331
3. (a) 16.36 (b) 21.87 (c) 8.80 (d) 22.71 (e) .13 20-
(f) 0.17 (g) 0.36 (h) 1.20 (i) .4 27- (j) 8.16
4. 1300 5. 950 6. 3000 7. 11 000
8. 600 9. $8 000 000 10. $34 600 000
11. 844 km 12. 0.73 13. 33 14. 3.248 15. 4.21
16. 1.7 17. 79 cents 18. 2.73 19. 1.1 20. 3.6 m
21. $281.93 22. 1.8 g 23. $3.20
24. (a) 7.95 (b) 30.03 (c) 0.37 (d) 5.74 (e) 0.52 25. 0.2
Exercises 1.2
1. 1 2. 11- 3. 56- 4. 10 5. 4-
6. .1 2- 7. .7 51- 8. .35 52- 9. 6.57
10. 2154
- 11. 7- 12. −23 13. 10 14. 1
15. 5 16. 3 17. 1 18. 60 19. −20 20. 9
Exercises 1.3
1. (a) 2516
(b) 100051
(c) 5201
(d) 1154
2. (a) 0.4 (b) 1.875 (c) .0 416o (d) .0 63oo
3. (a) 501
(b) 83
(c) 1000
1 (d) 1
100097
4. (a) 0.27 (b) 1.09 (c) 0.003 (d) 0.0623
5. (a) 35% (b) %3331
(c) %22632
(d) 0.1%
6. (a) 124% (b) 70% (c) 40.5% (d) 127.94%
7. (a) . ;0 522513
(b) . ;0 071007
(c) . ;0 16812521
(d) . ;1 09 11009
(e) . ;0 434500217
(f) . ;0 122540049
8. (a) .0 83o (b) .0 07oo (c) .0 13oo (d) .0 16o (e) .0 6o (f) .0 15oo
(g) .0 142857o o or 0.142857 (h) .1 18oo
9. (a) 98
(b) 92
(c) 195
(d) 397
(e) 9967
(f) 116
(g) 457
(h) 6013
(i) 990217
(j) 149537
10. (a) .0 5o (b) 7.4 (c) 0.73o (d) .0 68oo (e) .1 72oo
11. (a) 85
(b) 281
(c) 118
(d) 2187
(e) 454
12. 74% 13. 77.5% 14. 17.5% 15. 41.7%
Exercises 1.4
1. 203
2. 207
3. (a) 2017
(b) 107
(c) 1201
(d) 283
(e) 53
4. $547.56 5. 714.3 g 6. 247
7. $65
8. 179 cm 9. (a) 11.9 (b) 5.3 (c) 19 (d) 3.2 (e) 3.5 (f) 0.24 (g) 0.000 18 (h) 5720 (i) 0.0874 (j) 0.376
10. $52.50 11. 54.925 mL 12. 1152.125 g 13. $10.71
14. 5.9% 15. 402.5 g 16. 41.175 m 17. $30.92
18. 3.2 m 19. 573 20. $2898
Problem
5115 minutes after 1 o’clock.
Exercises 1.5
1. (a) 500 (b) 145 (c) 641
(d) 3 (e) 2
2. (a) 13.7 (b) 1.1 (c) 0.8 (d) 2.7 (e) .2 6- (f) 0.5
3. (a) a17 (b) y 10 = (c) a 4- (d) w (e) x5 (f) p10
(g) y6 (h) x21 (i) x4 10 (j) y81 8- (k) a (l) y
x45
10
(m) w10 (n) p5 (o) x 3- (p) a ba
bor2 3
2
3-
(q) x yx
yor5 2
5
2
-
4. (a) x 14 (b) a 7- (c) m 4 (d) k 10 (e) a 8- (f) x (g) mn 2
(h) p 1- (i) 9 x 22 (j) x 21
5. (a) p 5 q 15 (b) b
a8
8
(c) b
a6412
3
(d) 49 a 10 b 2 (e) 8 m 17
(f) x 4 y 10 (g) k27
2 23
(h) 16 y 47 (i) a 3 (j) x y125 21 18-
Answer S1-S5.indd 540 7/31/09 1:35:59 PM
541ANSWERS
6. 421
7. 324 8. 22710
9. (a) a 3 b (b) 251
10. (a) pq 2 r 2 (b) 327
11. 94
12. 181
13. 274
14. 811
15. 108
1 16.
121
17. 2
558
22
18. 388849
Exercises 1.6
1. (a) 271
(b) 41
(c) 3431
(d) 10 000
1 (e)
2561
(f) 1
(g) 321
(h) 811
(i) 71
(j) 811
(k) 641
(l) 91
(m) 1
(n) 361
(o) 125
1 (p)
100 0001
(q) 1281
(r) 1
(s) 641
(t) 641
2. (a) 1 (b) 16 (c) 121
(d) 12511
(e) 1 (f) 125 (g) 131
(h) 49 (i) 383
(j) 32 (k) 231
(l) 1 (m) 13613
(n) 18119
(o) 1 (p) 16 (q) 1585
- (r) 237
- (s) 1 (t) 2516
3. (a) m 3- (b) x 1- (c) p 7- (d) d −9 (e) k −5 (f) x 2-
(g) 2x 4- (h) 3 y −2 (i) 21
z 6- or 2
z 6-
(j) 5
3t 8-
(k) 7
2x 1-
(l) 2
5m 6-
(m) 3
2y 7-
(n) 3 4x 2+ -] g (o) a b 8+ -] g
(p) 2x 1- -] g (q) 5 1p 3+ -^ h (r) 2 4 9t 5- -] g
(s) 41x 11+ -] g
(t) 9
5 3a b 7+ -] g
4. (a) 1
t5 (b)
1
x6 (c)
1
y3 (d)
1
n8 (e)
1
w10 (f)
2x (g)
3
m4
(h) 5
x7 (i)
8
1
x3 (j)
41n
(k) 1
1
x 6+] g (l) 8
1y z+
(m) 3
1
k 2-] g ( n) 3 2
1
x y 9+^ h (o) x 5 (p) y 10 (q) 2
p
(r) a b 2+] g (s) x y
x y
+
- (t)
2
3
w z
x y 7
-
+e o
Exercises 1.7
1. (a) 9 (b) 3 (c) 4 (d) 2 (e) 7 (f) 10 (g) 2 (h) 8 (i) 4 (j) 1 (k) 3 (l) 2 (m) 0 (n) 5 (o) 7 (p) 2
(q) 4 (r) 27 (s) 21
(t) 161
2. (a) 2.19 (b) 2.60 (c) 1.53 (d) 0.60 (e) 0.90 (f) 0.29
3. (a) y3 (b) y yor23 32_ i (c)
1
x (d) 2 5x +
(e) 3 1
1
x - (f) 6q r3 + (g)
x x7
1
7
1or
25 5 2+ +] ^g h
4. (a) 2t1
(b) 5y1
(c) 2x3
(d) x9 31
-] g (e) 2s4 1+1] g
(f) 2t2 3+-
1] g (g) -
2x y5 -
3^ h (h) 2x3 1+5] g
(i) 3x 2--
2] g (j) 2y21
7+-
1^ h (k) -
3x5 4+1] g
(l) y32 1 2
1
--
2a k (m) x53 2 4
3
+-
2_ i
5. (a) x 23
(b) x 21
-
(c) x 32
(d) x 35
(e) x 45
6. (a) x x x2 23
+ +2 (b) a b32
32
- (c) p p p2 21
+ +1-2
(d) 2x x 1+ +- (e) x x x321
23
25
- +- - -
7. (a) 2
1
a b3 - (b)
3
1
y 23 -^ h (c) 6 1
4
a 47 +] g
(d) 3
1
x y 54 +^ h (e) 7 3 8
6
x 29 +] g
Exercises 1.8
1. (a) .3 8 103# (b) .1 23 106
# (c) .6 19 104#
(d) 1.2 107# (e) .8 67 109
# (f) .4 16 105#
(g) 9 102# (h) .1 376 104
# (i) 2 107# (j) 8 104
#
2. (a) .5 7 10 2#
- (b) .5 5 10 5#
- (c) 4 10 3#
-
(d) 6.2 10 4#
- (e) 2 10 6#
- (f) 8 10 8#
-
(g) 7.6 10 6#
- (h) 2.3 10 1#
- (i) 8.5 10 3#
- (j) 7 10 11#
-
3. (a) 36 000 (b) 27 800 000 (c) 9 250 (d) 6 330 000 (e) 400 000 (f) 0.072 3 (g) 0.000 097 (h) 0.000 000 038 (i) 0.000 007 (j) 0.000 5
4. (a) 240 000 (b) 9 200 000 (c) 11 000 (d) 0.36 (e) 1.3 (f) 9.0 (g) 16 (h) 320 (i) 2900 (j) 9.1
5. (a) 6.61 (b) 0.686 (c) 8.25 (d) 1.30
6. 1.305 1010# 7. 6.51 10 10
#-
Exercises 1.9
1. (a) 7 (b) 5 (c) 6 (d) 0 (e) 2 (f) 11 (g) 6 (h) 24 (i) 25 (j) 125 2. (a) 5 (b) −1 (c) 2 (d) 14 (e) 4 (f) −67 (g) 7 (h) 12 (i) −6 (j) 10 3. (a) 3 (b) 3 (c) 1 (d) 3 (e) 1 4. (a) a (b) a- (c) 0 (d) 3 a (e) −3 a (f) 0 (g) 1a + (h) a 1- - (i) 2x - (j) 2 x-
5. (a) | | 6a b+ = | | | |a b 6+ = | | | | | |a b a b` #+ + (b) | | 3a b+ = | | | |a b 3+ = | | | | | |a b a b` #+ + (c) | | 1a b+ = | | | |a b 5+ = | | | | | |a b a b` #+ + (d) | | 1a b+ = | | | |a b 9+ = | | | | | |a b a b` #+ + (e) | |a b 10+ = | | | |a b 10+ = | | | | | |a b a b` #+ +
6. (a) | | 5x x2 = = (b) | | 2x x2 = = (c) | | 3x x2 = =
(d) | | 4x x2 = = (e) | | 9x x2 = =
7. (a) x x x x5 5 5 5for and for2 1+ - - - - (b) b b b x3 3 3 3for and for2 1- - (c) a a a a4 4 4 4for and for2 1+ - - - - (d) y y y y2 6 3 6 2 3for and for2 1- - (e) 3 9 3 3 9 3x x x xfor and for2 1+ - - - - (f) 4 4 4 4x x x xfor and for1 2- -
(g) 2 1 2 1k k k k21
21
for and for2 1+ - - - -
(h) 5 2 5 2x x x x52
52
for and for2 1- - +
(i) a b a b a b a bfor and for2 1+ - - - - (j) p q p q q p p qfor and for2 1- -
8. x 3!= 9. 1! 10. , x1 2! !
Answer S1-S5.indd 541 7/31/09 1:36:00 PM
542 Maths In Focus Mathematics Preliminary Course
Test yourself 1
1. (a) 209
(b) 0.14 (c) 0.625 (d) 200157
(e) 1.2%
(f) 73.3% 2. (a) 491
(b) 51
(c) 31
3. (a) 8.83 (b) 1.55 (c) 1.12 (d) 342 (e) 0.303 4. (a) 1
(b) 1 (c) 39 (d) 2 (e) 10- (f) 1- (g) 4 5. (a) x9
(b) 25y6 (c) a b11 6 (d) 27
8x18
(e) 1 6. (a) 4029
(b) 371
(c) 12 (d) 221
(e) 1221
7. (a) 4 (b) 6 (c) 19
(d) 641
(e) 4 (f) 3 (g) 71
(h) 2 (i) 1 (j) 4
8. (a) a 5 (b) x 30 y 18 (c) p 9 (d) 16 b 36 (e) 8 x 11 y 9. (a) 2n1
(b) x 5- (c) x y 1+ -^ h (d) 4x 1+1] g (e) 7a b+
1] g
(f) 2x 1- (g) 21
x 3- (h) 3x4
(i) 7x5 3+9] g (j) 4m
-3
10. (a) 1
a5 (b) n4 (c) 1x + (d)
1x y-
(e) 4 7
1
t 4-] g
(f) a b5 + (g) 1
x3 (h) b34 (i) 2 3x 43 +] g (j)
1
x3
11. | | 2a b+ = | | | | 8a b+ = | | | | | |a b a b` #+ +
12. 1 13. 192
1 14. 689 mL 15. (a) 6 h (b)
127
(c) 81
(d) 33.3% 16. $38 640 17. 70% 18. 6.3 1023#
19. (a) 2x1
(b) y 1- (c) 6x 3+1] g (d) 2 3x 11- -] g (e) 3y
7
20. (a) 1.3 10 5#
- (b) 1.23 1011# 21. (a)
97
(b) 33041
22. (a) 1
x3 (b)
2 51
a + (c) a
b 5c m 23. 14 500
24. | | ,2 5 7LHS = - + - = | | | | .2 5 7RHS = - + - = So | | | | | |a b a b#+ + since .7 7#
Challenge exercise 1
1. 4303278
2. 11811
3. . , %, , .0 502 519951
0 5o
4. 5331
% 5. 161
6. 3.04 1014# 7. 83% 8. 1
99903271
9. 18 h 10. 1.98
11.
2 2 1 22 2 22 2 22 2 1
2 2 1 2 2 2 1
LHS
RHS
k k
k k
k
k
k k k
1
1 1
1
1
1 1
:
`
= - +
= - +
= -
= -
=
- + = -
+
+ +
+
+
+ +
^
^^ ^
h
hh h
12. −2 4 .3 5 13. . , , . , ,0 34 2 1 5 073
- o 14. 632
%
15. ,x
xx
x1
11
11
1when when2 1-
--
- 16. 0.73
17. 0.6% 18 4.54 19. 4.14 10 20#
-
20.
| | | | | | , , ;| | | | | | , , ;a b a b a b a ba b a b b ba
0 0 0 00 0 0 0
when orwhen ora
2 2 1 11 2 1 1 2
+ = +
+ +
| | | |a b a b` #+ + for all a , b
Chapter 2: Algebra and surds
Exercises 2.1
1. 7 x 2. 3 a 3. z 4. 6 a 5. 3 b 6. −3 r
7. y- 8. −5 x 9. 0 10. 3 k 11. 9 t 12. 10 w
13. m- 14. x- 15. 0 16. 5 b 17. 11 b 18. 10x-
19. 6 6x y- 20. 3a b- 21. 4 2xy y+ 22. 6ab2-
23. m m6 122 - + 24. 2 6p p2 - - 25. 8 3x y+
26. 2 10ab b- + 27. 2bc ac- 28. 2 9 1a x5 3- +
29. 2 3 2x xy x y y3 2 2 3- + + 30. 3 7 6x x x3 2+ - -
Exercises 2.2
1. b10 2. xy8 3. p10 2 4. wz6-
5. ab15 6. xyz14 7. abc48 8. d12 2
9. a12 3 10. y27 3- 11. x32 10
12. a b6 2 3 13. a b10 3 2- 14. p q21 3 4
15. a b5 3 3 16. h8 10- 17. k p33 18. t81 12
19. 14m11- 20. x y24 6 3
Exercises 2.3
1. 6 x 2. 2 3. 4a2 4. 8 a 5. 4 a 6. 2
y 7. 3 p
8. 2ab
9. 34y
10. 3x3- 11. 3 a 12. 3
1
ab2 13.
2qs-
14. 3
2
c d2 15.
x
z
2 2
2
16. 6p q4 17. 4c
a b4 7
18. 2ab6
19. 3y
x z3 3
- 20. 2b
a6
13
Exercises 2.4
1. x2 8- 2. h6 9+ 3. a5 10- + 4. xy x2 3+
5. x x22 - 6. a ab6 162 - 7. a b ab2 2 2+ 8. n n5 202 -
9. 3x y x y63 2 2 3+ 10. k4 7+ 11. t2 17-
12. y y4 112 + 13. b5 6- - 14. x8 2-
15. m3 1- + 16. h8 19- 17. d 6- 18. a a2 42 - +
19. x x3 9 52 - - 20. ab a b b2 2 2- + 21. x4 1-
22. y7 4- + 23. b2 24. t5 6- 25. a2 26+
Exercises 2.5
1. 7 10a a2 + + 2. 2 3x x2 + - 3. 2 7 15y y2 + -
4. 6 8m m2 - + 5. 7 12x x2 + + 6. 3 10y y2 - -
7. 2 6x x2 + - 8. 10 21h h2 - + 9. 25x2 -
10. 15 17 4a a2 - + 11. 8 6 9y y2 + - 12. 7 4 28xy x y+ - -
13. 2 3 6x x x3 2- + - 14. 4n2 - 15. 4 9x2 -
16. 16 49y2- 17. 4a b2 2- 18. 9 16x y2 2- 19. 9x2 -
20. 36y2 - 21. 9 1a2 - 22. 4 49z2 -
Answer S1-S5.indd 542 7/31/09 1:36:01 PM
543ANSWERS
23. 2 11 18 18x xy x y2 - + - + 24. 2 2 7 6 3ab b b a2+ - - +
25. x 83 + 26. 27a3 - 27. 18 81a a2 + +
28. 8 16k k2 - + 29. 4 4x x2 + + 30. 14 49y y2 - +
31. 4 12 9x x2 + + 32. 4 4 1t t2 - +
33. 9 24 16a ab b2 2+ + 34. 10 25x xy y2 2- +
35. 4 4a ab b2 2+ + 36. a b2 2- 37. 2a ab b2 2+ +
38. 2a ab b2 2- + 39. a b3 3+ 40. a b3 3-
Exercises 2.6
1. 8 16t t2 + + 2. 12 36z z2 - + 3. 2 1x x2 - +
4. 16 64y y2 + + 5. 6 9q q2 + + 6. 14 49k k2 - +
7. 2 1n n2 + + 8. 4 20 25b b2 + + 9. 9 6x x2- +
10. 9 6 1y y2 - + 11. 2x xy y2 2+ + 12. 9 6a ab b2 2- +
13. 16 40 25d de e2 2+ + 14. 16t2 - 15. x 92 -
16. 1p2 - 17. 36r2 - 18. 100x2 - 19. 4 9a2 -
20. 25x y2 2- 21. 16 1a2 - 22. 49 9x2- 23. 4x4 -
24. 10 25x x4 2+ + 25. 9 16a b c2 2 2- 26. 44
xx
2
2+ +
27. 1
aa
2
2- 28. 2 4 4x y x y y2 2 2 2- - = - + -^ h
29. 2 2 2 2a b a b c c a ab b ac bc c2 2 2 2 2+ + + + = + + + + +] ]g g
30. 1 2 1 2 1 2 2x x y y x x xy y y2 2 2 2+ - + + = + + - - +] ]g g
31. 12a 32. 32 z2- 33. 9 8 3x x2 + -
34. 3 2x xy y x2 2+ + - 35. 14 4n2 -
36. 12 48 64x x x3 2- + - 37. x2 38. 2x x y y4 2 2 4- +
39. 8 60 150 125a a a3 2+ + +
40. 4 16 15 4 4x x x x4 3 2+ + - -
Problem
2,a = 7,b = 9,c = 4,d = 3,e = 8,f = 0,g = 6,h = 1i =
Exercises 2.7
1. y2 3+^ h 2. x5 2-] g 3. m3 3-] g 4. x2 4 1+] g 5. y6 4 3-^ h 6. xx 2+] g 7. m m 3-] g 8. y y2 2+^ h 9. a a3 5 -] g 10. ab b 1+] g 11. xy x2 2 1-] g 12. mn n3 32 +^ h 13. xy x z2 4 -] g 14. a b a6 3 2+ -] g 15. x x y5 2- +^ h 16. q q3 22 3 -_ i 17. b b5 32 +] g 18. a b ab3 22 2 -] g 19. 5)( 7)(m x+ + 20. 1 2y y- -^ ^h h 21. 7 )(4 3 )( y x+ - 22. 2 6 5a x- +] ]g g 23. 2 1t x y+ -] ^g h 24. 3 2 2 3x a b c- + -] ]g g 25. 3 2 3x x2 +] g 26. 3 2q pq3 2 -_ i 27. ab a b3 5 13 2 +^ h 28. 4 6x x2 -] g 29. 5 7 5m n mn2 3 -^ h 30. 4 6 4ab ab2 3 +^ h 31. r r h2r +] g 32. 3 2x x- +] ]g g 33. ( ) ( )x y4 22+ +
34. 1a- +] g 35. ( ) ( )a ab1 4 32 + -
Exercises 2.8
1. 4 2x b+ +] ]g g 2. 3y a b- +^ ]h g 3. 5 2x x+ +] ]g g 4. 2 3m m- +] ]g g 5. d c a b- +] ]g g 6. 1 3x x2+ +] ^g h 7. 5 3 2a b- +] ]g g 8. 2y x x y- +^ ^h h 9. 1 1y a+ +^ ]h g 10. 5 1x x+ -] ]g g 11. 3)(1 )(y a+ + 12. 2)(1 2 )(m y- -
13. 5 2 3x y x y+ -^ ^h h 14. 4a b ab2+ -^ ]h g 15. 5 3x x- +] ]g g 16. 7)( 4)(x x3+ - 17. 3 7x y- -] ^g h 18. 3 4d e+ -] ]g g 19. 4 3x y- +] ^g h 20. 3 2a b+ -] ]g g 21. 3)( 6)(x x2- + 22. 3q p q- +^ ^h h 23. 2 3 5x x2- -] ^g h 24. 3 4a b c- +] ]g g 25. 7 4y x+ -^ ]h g 26. 4)( 5)(x x3- -
27. (2 3)(2 4) (2 3)( )x x x x2 22 2- + = - +
28. ( ) ( )a b a3 2 3+ + 29. 5( 3)(1 2 )y x- +
30. r r2 3r+ -] ]g g
Exercises 2.9
1. 3 1x x+ +] ]g g 2. 4 3y y+ +^ ^h h 3. 1m 2+] g 4. 4t 2+] g 5. 3 2z z+ -] ]g g 6. 1 6x x+ -] ]g g 7. 3 5v v- -] ]g g 8. 3t 2-] g 9. 10 1x x+ -] ]g g 10. 7 3y y- -^ ^h h 11. 6 3m m- -] ]g g 12. 12 3y y+ -^ ^h h 13. 8 3x x- +] ]g g 14. a 2 2-] g 15. 2 16x x- +] ]g g 16. 4 9y y+ -^ ^h h 17. 6 4n n- -] ]g g 18. x 5 2-] g
19. 9 1p p+ -^ ^h h 20. 2 5k k- -] ]g g 21. 4 3x x+ -] ]g g 22. 7 1m m- +] ]g g 23. 10 2q q+ +^ ^h h 24. 5 1d d- +] ]g g 25. 9 2l l- -] ]g g
Exercises 2.10
1. 2 1)( 5)( a a+ + 2. 5 2 1y y+ +^ ^h h 3. 3 7)( 1)( x x+ + 4. 3 2)( 2)( x x+ + 5. 2 3)( 1)( b b- -
6. 7 2)( 1)( x x- - 7. 3 1 2y y- +^ ^h h 8. 2 3 4x x+ +] ]g g 9. 5 2 3p p- +^ ^h h 10. 3 5 2 1x x+ +] ]g g 11. 2 1)( 6)( y y+ - 12. 5 1 2 1x x- +] ]g g 13. 4 1)(2 3)( t t- - 14. 3 4)(2 3)( x x+ -
15. 6 1 8y y- +^ ^h h 16. 4 3 2n n- -] ]g g 17. 4 1 2 5t t- +] ]g g 18. 3 2 4 5q q+ +^ ^h h 19. r r r r4 1 2 6 4 12 3- + - +=] ] ] ]g g g g 20. 2 5 2 3x x- +] ]g g 21. 6 1 2y y- -^ ^h h 22. 2 3 3 2p p- +^ ^h h 23. 8 7)( 3)( x x+ +
24. 3 4 4 9b b- -] ]g g 25. 6 1)( 9)( x x+ -
26. 3 5x 2+] g 27. 4 3y 2+^ h 28. 5 2k 2-] g 29. 6 1a 2-] g 30. 7 6m 2+] g
Answer S1-S5.indd 543 7/31/09 1:36:02 PM
544 Maths In Focus Mathematics Preliminary Course
Exercises 2.11
1. 1y 2-^ h 2. ( 3)x 2+ 3. ( 5)m 2+ 4. ( 2)t 2-
5. ( 6)x 2- 6. 2 3x 2+] g 7. 4 1b 2-] g 8. 3 2a 2+] g 9. 5 4x 2-] g 10. 7 1y 2+^ h 11. 3 5y 2-^ h 12. 4 3k 2-] g 13. 5 1x 2+] g 14. 9 2a 2-] g 15. 7 6m 2+] g 16.
21
t2
+d n
17. 32
x2
-d n 18. 351
y2
+d n 19. 1
x x
2
+c m 20. 52
kk
2
-d n
Exercises 2.12
1. 2)( 2)(a a+ - 2. 3)( 3)(x x+ - 3. 1)( 1)(y y+ -
4. 5 5x x+ -] ]g g 5. 2 7)(2 7)( x x+ - 6. 4 3)(4 3)( y y+ -
7. 1 2 )(1 2 )( z z+ - 8. 5 1 5 1t t+ -] ]g g 9. 3 2 3 2t t+ -] ]g g 10. 3 4 3 4x x+ -] ]g g 11. 2 )( 2 )(x y x y+ -
12. 6 6x y x y+ -^ ^h h 13. 2 3 2 3a b a b+ -] ]g g 14. 10 10x y x y+ -^ ^h h 15. 2 9 2 9a b a b+ -] ]g g 16. 2 2x y x y+ + + -^ ^h h 17. 3)( 1)(a b a b+ - - +
18. 1 1z w z w+ + - -] ]g g 19. 21
21
x x+ -d dn n
20. 3
13
1y y+ -e eo o 21. 2 3 2 1x y x y+ + - +^ ^h h
22. ( )( ) ( )( )( )x x x x x1 1 1 1 12 2 2+ - = + + -
23. 3 2 3 2x y x y3 3+ -_ _i i 24. 4 2 2x y x y x y2 2+ + -_ ^ ^i h h 25. 1)( 1)( 1)( 1)(a a a a4 2+ + + -
Exercises 2.13
1. 2)( 2 4)(b b b2- + + 2. 3 3 9x x x2+ - +] ^g h 3. 1 1t t t2+ - +] ^g h 4. 4)( 4 16)(a a a2- + +
5. 1 )(1 )( x x x2- + + 6. 2 3 4 6 9y y y2+ - +^ _h i 7. ( ) ( )y z y yz z2 2 42 2+ - + 8. 5 )( 5 25 )(x y x xy y2 2- + +
9. 2 3 4 6 9x y x xy y2 2+ - +^ _h i 10. 1 1ab a b ab2 2- + +] ^g h 11. 10 2 )(100 20 4 )( t t t+ - + 2 12.
23
4 23
9x x x2
- + +d en o 13.
10 1 100 10 1a b a ab b2 2
+ - +d en o 14. 1 2 1x y x x xy y y2 2+ - + + + + +^ _h i 15. xy z x y xyz z5 25 306 362 2 2+ - +^ _h i 16. a a 19 2 - +- ^ h 17. 1
31
3 9x x x2
- + +d en o 18. 3 3 9 6x y y y xy x x2 2+ + - - + + +^ _h i 19. 1 4 5 7x y x x xy y y2 2+ - + - + - +^ _h i 20. 2 6 )(4 24 2 6 36)( a b a a ab b b2 2+ - + + + + +
Exercises 2.14
1. x x2 3 3+ -] ]g g 2. p p3 3 4+ -^ ^h h 3. y y y5 1 12- + +^ _h i 4. ) (ab a b a2 2 2 1+ -^ h 5. 5 1a 2-] g 6. x x2 3 4- -- ] ]g g 7. z z z3 5 4+ +] ]g g 8. ab ab ab3 2 3 2+ -] ]g g 9. x xx 1 1+ -] ]g g 10. x x2 3 2 2- +] ]g g 11. 5 3m n- +] ]g g 12. x7 2 1- +] g
13. 5 4 4y y y+ + -^ ^ ^h h h 14. 1 2 2 4x x x x2- + - +] ] ^g g h 15. x x x x x x1 1 1 12 2+ - + - + +] ^ ] ^g h g h 16. x x x2 5+ -] ]g g 17. ( )x x3 3 2+ -] g
18. ( ) ( )xy xyy 2 1 2 1+ - 19. b b b3 2 4 2 2- + +] ^g h 20. x x3 3 2 2 5- +] ]g g 21. x3 1 2-] g 22. 2)( 5)( 5)(x x x+ + - 23. 3z z 2+] g 24. 1 1 2 3 2 3x x x x+ - + -] ] ] ]g g g g 25. x x x y x xy y2 2 2 2 2+ - + - +] ] ^ _g g h i 26. ( ) ( )a a a4 3 3+ - 27. x x xx 2 4 25 2- + +] ^g h 28. 2)( 2)( 3)( 3)(a a a a+ - + - 29. 4 ( 5)k k 2+
30. 3( 1) 1) 3)( (x x x+ - +
Exercises 2.15
1. 4 4 2x x x2 2+ + = +] g 2. 6 9 3b b b2 2- + = -] g 3. 10 25 5x x x2 2- + = -] g 4. 8 16 4y y y2 2+ + = +^ h
5. 14 49 7m m m2 2- + = -] g 6. 18 81 9q x q2 2+ + = +^ h
7. 2 1 1x x x2 2+ + = +] g 8. 16 64 8t t t2 2- + = -] g 9. 20 100 10x x x2 2- + = -] g 10. 44 484 22w w w2 2+ + = +] g 11. 32 256 16x x x2 2- + = -] g 12. 3
49
23
y y y22
+ + = +d n
13. 74
4927
x x x22
- + = -d n 14. 41
21
a a a22
+ + = +d n
15. 94
8129
x x x22
+ + = +d n 16. 5
yy
y1625
45
22
2
- + = -d n
17. 11
kk
k16
1214
112
22
- + = -d n
18. 6 9 3x xy y x y2 2 2+ + = +^ h 19. 4 4 2a ab b a b2 2 2- + = -] g 20. 8 16 4p pq q p q2 2 2- + = -^ h Exercises 2.16
1. 2a + 2. 2 1t - 3. 3
4 1y + 4.
2 14
d - 5.
5 2xx-
6. 4
1y -
7. ab a
322
-
-] g 8.
31
ss+
- 9.
11
bb b2
+
+ +
10. 3
5p + 11.
31
aa+
+ 12.
2 4
3
x x
y2 + +
+ 13. 3x -
14. 4 2 1
2
p p
p2 - +
- 15.
2a ba b
-
+
Answer S1-S5.indd 544 7/31/09 1:36:03 PM
545ANSWERS
Exercises 2.17
1. (a) 45x
(b) 15
13 3y + (c)
128a +
(d) 6
4 3p + (e)
613x -
2. (a) 2 1a
b-
(b) 1
2 1
q
p q q2
+
- - +^ _h i (c)
b
x yb
2 1
2
10
2
-
+
]^
gh
(d) ab
x xy y2 2- + (e)
5 2
3 1
x x
x x
- -
- -
] ]] ]
g gg g
3. (a) 5x (b)
xx
x 12
-
- +
] g (c) 3
a ba b
+
+ + (d)
22
xx+
(e) p q
p q p q
p q
p q 11 2 2
+
+ -=
+
- ++^ ^h h (f)
x x
x
1 3
12
+ -
-
] ]]g gg
(g) 2 23 8
x xx
+ -
- +
] ]g g (h) 1
2
a
a2+
+
] g
(i) y y y
y y
2 3 1
3 14 132 2
+ + -
+ +
^ ^ ^_h h h
i (j)
x x x
x
4 4 3
5 22
+ - +
+-
] ] ]]g g g
g
4. (a) y y
xx
3 9
2
8
2
2 - +
+
_]
ig
(b) 15
2 1
y
y y+ +^ ^h h
(c) x x
x x2 3 4
210 42
- -
-+
] ]g g (d) b
b bb 1
3 5 102
2
+
- -
] g (e) x
5. (a) 5 2 3
3 13x x x
x- - +
-
] ] ]g g g (b) 2 2
3 5x x
x+ -
-
] ]g g
(c) p q p q
p pq q
pq
3 5 22 2
+ -
+ -
^ ^h h (d) 2 1
a b a ba ab b2 2
+ -
- - +
] ]g g
(e) x y x y
x yy 1
+ -
+ +
^ ^^h h
h
Exercises 2.18
1. (a) 7.1- (b) 6.9- (c) 48.1 (d) 37.7- (e) 0.6
(f) 2.3 (g) 5.3- 2. 47 3. 7- 4. 375 5. 196-
6. 5.5 7. 377 8. 284 9. 40- 10. 51.935 11. 143
-
12. 22.4 13. 1838.8 14. 43
15. 15 16. 10
17. 2 312 = 18. 23.987 19. 352.47 20. 93 21. 4
Exercises 2.19
1. (a) 2 3 (b) 3 7 (c) 2 6 (d) 5 2 (e) 6 2
(f) 10 2 (g) 4 3 (h) 5 3 (i) 4 2 (j) 3 6
(k) 4 7 (l) 10 3 (m) 8 2 (n) 9 3 (o) 7 5
(p) 6 3 (q) 3 11 (r) 5 5
2. (a) 6 3 (b) 20 5 (c) 28 2 (d) 4 7 (e) 16 5
(f) 8 14 (g) 72 5 (h) 30 2 (i) 14 10 (j) 24 5
3. (a) 18 (b) 20 (c) 176 (d) 128 (e) 75
(f) 160 (g) 117 (h) 98 (i) 363 (j) 1008
4. (a) 45x = (b) 12x = (c) 63x = (d) 50x =
(e) 44x = (f) 147x = (g) 304x = (h) 828x =
(i) 775x = (j) 960x =
Exercises 2.20
1. 3 5 2. 2 3. 6 3 4. 3 3 5. 3 5- 6. 3 6
7. 7 2- 8. 8 5 9. 4 2- 10. 4 5 11. 2 12. 5 3
13. 3- 14. 2 15. 5 7 16. 2 17. 13 6
18. 9 10- 19. 47 3 20. 2 2 35 - 21. 7 5 2-
22. 2 3 4 5- - 23. 7 6 3 5+ 24. 2 2 3- -
25. 17 5 10 2- +
Exercise 2.21
1. 21 2. 15 3. 3 6 4. 10 14 5. 6 6- 6. 30
7. 12 55- 8. 14 9. 60 10. 12 2 3=
11. 2 48 8 3= 12. 15 28 30 7=
13. 2 20 4 5= 14. 84- 15. 2
16. 28 17. 30 18. 2 105- 19. 18
20 . 30 50 150 2= 21. 2 6 22. 4 3 23. 1 24. 6
8
25. 2 3 26. 3 10
1 27.
2 5
1 28.
3 5
1 29.
21
30. 2 2
3 31.
2
3 32.
2 5
9 33.
2 2
5 34.
32
35. 75
Exercises 2.22
1. (a) 10 6+ (b) 2 6 15- (c) 12 8 15+
(d) 5 14 2 21- (e) 6 4 18 6 12 2- + = - +
(f) 5 33 3 21+ (g) 6 12 6- - (h) 5 5 15-
(i) 6 30+ (j) 2 54 6 6 6 6+ = +
(k) 8 12 12 8 24 3- + = - + (l) 210 14 15-
(m) 10 6 120- (n) 10 2 2- - (o) 4 3 12-
2. (a) 10 3 6 3 5 9 3+ + +
(b) 10 35 2 14- - +
(c) 2 10 6 10 15 15 6- + -
(d) 12 18 8 1224 36 8 12
20 60 10 305 15 10 30
+ - - =
+ - -
(e) 52 13 10- (f) 15 15 18 10 6 6- + -
(g) 4 (h) 1- (i) 12- (j) 43 (k) 3 (l) 241-
(m) 6- (n) 7 2 10+ (o) 11 4 6- (p) 25 6 14+
(q) 57 12 15+ (r) 27 4 35-
(s) 77 12 40 77 24 10- = - (t) 53 12 10+
3. (a) 18 (b) 108 2 (c) 432 2 (d) 19 6 2+ (e) 9
4. (a) 21, 80a b= = (b) 19, 7a b= = -
5. (a) 1a - (b) p pp2 1 2 1- - -^ h 6. 25k = 7. 2 3 5x y xy- - 8. 17, 240a b= =
9. 107, 42a b= = - 10. 9 5 units2+
Answer S1-S5.indd 545 8/7/09 12:28:47 PM
546 Maths In Focus Mathematics Preliminary Course
Exercises 2.23
1. (a) 77
(b) 46
(c) 5
2 15 (d)
106 14
53 14
=
(e) 3
3 6+ (f)
212 5 2-
(g) 5
5 2 10+
(h) 14
3 14 4 7- (i)
208 5 3 10+
(j) 35
4 15 2 10-
2. (a) 4 4 3 243 2- -= ^ h (b) 47
6 7 3+- ^ h
(c) 19
2 15 4 1819
15 6 22-=
-- -^ ^h h
(d) 13
19 8 313
8 3 19-=
-- ^ h (e) 6 2 5 3 5 2+ + +
(f) 2
6 15 9 6 2 10 6- + -
3. (a) 2 2
(b) 2 3 32 6 3 2 3 3 6 2 3- + - + = - - + -^ h
(c) 39
22 5 14 2+
(d)
106 6 16 3 84 8 14
6 21 145
3 8 3 4
- - - +
=- + + -
^ h
(e) 4- (f) 4 2
(g) 15
20 12 19 6 25 3 615
19 6 65 3 6+ + -=
+ -
(h) 6
6 9 2 2 3+ + (i)
214 6 9 3+
(j) 415
30330 30 5- -
(k) 13
28 2 6 7 3- -
(l) 2
2 15 2 10 2 6 3 5+ - - -
4. (a) 45, 10a b= = (b) 1, 8a b= = (c) 21
,21
a b= - =
(d) 195
,98
a b= - = - (e) 5, 32a b= =
5.
2
2
3 2 23
2 1
2 1
2
4
2 1
2 1
2 1
2 1
2
4
2
2
2 1
2 1 2 12
4 2
2 12 2 2 1
2
13 2 2
2
2 2
2 2
# #
+
-+
=+
-
-
-+
=-
- -+
=-
- - ++
=-
+
= - +
=
^^ ^
hh h
So rational
6. (a) 4 (b) 14 (c) 16
7. 3
3 5 2 15 3- - -
8. 3 2 2
2
2
8
3 2 2
2
3 2 2
3 2 2
2
8
2
2
3 2 2
2 3 2 22
8 2
9 4 26 4 2
4 2
16 4 2
4 2
6 4 2 4 26
2 2
# #
#
++
=+ -
-+
=-
-+
=-
-+
=-
+
= - +
=
^^
hh
So rational
9. x 3 2= - +^ h 10. 4
4 4b
b b-
+ +
Test yourself 2
1. (a) 2y- (b) 4a + (c) 6k5- (d) 15
5 3x y+ (e) 3 8a b-
(f) 6 2 (g) 4 5
2. (a) 6 6x x+ -] ]g g (b) 3 1a a+ -] ]g g (c) ab b4 2-] g (d) 3)(5 )(y x- + (e) n p2 32 - +^ h (f) 2 )(4 2 )( x x x2- + +
3. (a) 4 6b - (b) 2 5 3x x2 + - (c) 4 17m +
(d) 16 24 9x x2 - + (e) 25p2 - (f) 1 7a- -
(g) 2 6 5 3- (h) 3 3 6 21 2 7- + -
4. (a) a ab 3 9
822 + +^ h (b)
2
15
m 2-] g
5. 157.464V = 6. (a) 17 (b) 17
6 15 9-
7. 3 2
4 5x x
x+ -
+
] ]g g 8. (a) 36 (b) 2- (c) 2 (d) 216 (e) 2
9. (a) 5
1 (b) 8 10. 11.25d =
11. (a) 15
2 3 (b)
22 6+
12. (a) 3 6 6 4 3 4 2- - + (b) 11 4 7+
13. (a) 3( 3)( 3)x x- + (b) x x6 3 1- +] ]g g (c) y y y5 2 2 42+ - +^ _h i
14. (a) 3y
x4
3
(b) 3 1
1x -
15. (a) 99 (b) 24 3
16. (a) a b2 2- (b) 2a ab b2 2+ + (c) 2a ab b2 2- +
17. (a) a b 2-] g (b) a b a ab b2 2- + +] ^g h 18.
23 3 1+
19. (a) 4 3
abb a+
(b) 10
3 11x -
20. 7
21 5 46 2- -
Answer S1-S5.indd 546 7/31/09 1:36:05 PM
547ANSWERS
21. (a) 6 2 (b) 8 6- (c) 2 3 (d) 3
4 (e) 30a b2
(f) 3n
m4 (g) 2 3x y-
22. (a) 2 6 4+ (b) 10 14 5 21 6 10 3 15- - +
(c) 7 (d) 43 (e) 65 6 14-
23. (a) 7
3 7 (b)
156
(c) 5 1
2+
(d) 15
12 2 6- (e)
5320 3 15 4 10 3 6+ + +
24. (a) 10
10x + (b)
2117 15a -
(c) ( 1)( 1)x x
x3 2+
-
-
(d) 1
1k -
(e) 3
15 6 15 3 15 2- - -
25. (a) 48n = (b) 175n = (c) 392n =
(d) n 5547= (e) 1445n =
26. 312171
27. (b), (c) 28. (d) 29. (a), (d) 30. (c)
31. (c) 32. (b) 33. (a) 34. (d) 35. (b)
Challenge exercise 2
1. (a) 2 8 6a b ab a2 2 3- + (b) 4y4 -
(c) 8 60 150 125x x x3 2- + -
2. 17
17 3 2 5 20+ + 3.
2 2
142
or
4. ab
xa
bx
ab
x4 2
2
2
2 2
+ + += d n
5. (a) x x4 9+ +] ]g g (b) ( ) ( )x y x y x y x y x y3 2 3 3 22 2 2- + = + - +_ _ _i i i (c) 5 7 25 35 49x x x2+ - +] ^g h (d) 2 2 2b a a- + -] ] ]g g g
6. 4 12 9 2 3x x x2 2+ + = +] g 7. x
y
1
1
2 -
+
] g 8. 2 5
9. 1
1
a a
a2
2
- +
+] g 10.
2 2x b
ax b
a+ -d dn n
11. x x x
x x x xy y
3 3 2
3 6 3 643 2
- + -
- + + -
] ] ]g g g
12. (a) 8 12 6 1x x x3 2- + - (b) 2 1
3 4
x
x2-
+
] g
13. x x x 97 153 2 + --
14. 13
66 6 4 2 15 4 5 65 3- + - + -
15. xx
x32
91
312
2
+ + = +d n 16. 2x =
17. 10
400 59 5- 18. (a) 3
12171
(b) ,a b2317
2314
= = -
19. 121
i = 20. 4
r4
3 3
r r
r= =
21. 2 6 3s = +
Chapter 3: Equations
Exercises 3.1
1. 5t = - 2. 5.6z = - 3. 1y = 4. 6.7w = 5. 12x =
6. 4x = 7. 151
y = 8. 35b = 9. 16n = - 10. 4r =
11. 9y = 12. 6k = 13. 2d = 14. 5x = 15. 15y =
16. 20x = 17. 20m = 18. 4x = 19. 7a = - 20. 3y =
21. 4b = - 22. 3x = 23. 132
a = - 24. 4t = -
25. 1.2x = 26. 1.6a = 27. 81
b = 28. 39t =
29. 5p = 30. 4.41x Z
Exercises 3.2
1. 331
b = 2. 35x = 3. 494
y = 4. 1359
x = 5. 585
k =
6. 36x = 7. 0.6t = 8. 3x = - 9. 1.2y = - 10. 69x =
11. 13w = 12. 30t = 13. 14x = 14. 1x = -
15. 0.4x = - 16. 3p = 17. 8.2t = 18. 9.5x = -
19. 22q = 20. 3x = - 21. 0.8b = 22. 0.375a = -
23. 3x = 24. 1y = 25. 132
t = -
Exercises 3.3
1. 8.5t = 2. 122l = 3. 8b = 4. 41a = 5. 4y =
6. 6.68r = 7. 6.44x = 8. 15n = 9. 332
y1 =
10. 3.7h = 11. (a) 25.39BMI = (b) 69.66w =
(c) 1.94h = 12. 0.072r = 13. 9x1 = - 14. 2.14t =
15. x 2!= 16. 2.12r = 17. 10.46r = 18. 1.19x =
19. 5.5x = 20. 3.3r =
Exercises 3.4
1. (a) x 32
-4 -3 -2 -1 0 1 2 3 4
(b) y 4#
-4 -3 -2 -1 0 1 2 3 4
2. (a) 7t 2 (b) x 3$ (c) 1p 2 - (d) x 2$ - (e) 9y 2 -
(f) a 1$ - (g) y 221
$ - (h) x 21 - (i) a 6# -
(j) y 121 (k) b 181 - (l) 30x 2 (m) x 343
#
(n) m 1432
2 (o) b 1641
$ (p) r 9# - (q) 8z 2
(r) w 254
1 (s) x 35$ (t) t 9$ - (u) 6q52
2 -
(v) 1x32
2 - (w) b 1141
# -
Answer S1-S5.indd 547 7/31/09 1:36:06 PM
548 Maths In Focus Mathematics Preliminary Course
3. (a) x1 71 1
0 1 2 3 4 5 6 7 8
(b) p2 51#-
-3 -2 -1 0 1 2 3 4 5
(c) x1 41 1
-3 -2 -1 0 1 2 3 4 5
(d) y3 5# #-
-3 -2 -1 0 1 2 3 4 5
(e) y61
132
1 1
-3 -2 -1 0 1 2 3 4 5
Exercises 3.5
1. (a) x 5!= (b) y 8!= (c) 4 4a1 1-
(d) ,k k1 1$ # - (e) 6, 6x x 12 -
(f) p10 10# #- (g) 0x = (h) ,a a14 142 1 -
(i) 12 12y1 1- (j) ,b b20 20$ # -
2. (a) ,x 5 9= - (b) ,n 4 2= - (c) ,a a2 212 -
(d) x4 6# # (e) ,x 3 6= - (f) ,x 5 475
= -
(g) 3 2y211 1- (h) ,x x9 6$ # - (i) x 12!=
(j) a2 10# #
3. (a) 141
x = (b) ,a 331
= - (c) 231
b =
(d) No solutions (e) 272
y = - (f) 7x = (g) ,m 5 132
=
(h) ,d 221
143
= - (i) ,y54
2= - (j) No solutions
4. (a) ,x 221
= - (b) 3, 231
y = (c) 10, 153
a = -
(d) ,x 4 731
= - (e) ,d 4 5= -
5. (a) ,t 3 152
= - (b) 1 3t521 1-
-3 -2 -1 0 1 2 3 4 5
Exercises 3.6
1. (a) 3x = (b) y 8!= (c) 2n != (d) 2x 5!=
(e) 10p = (f) x 5!= (g) y 3!= (h) 2w =
(i) n 4!= (j) 2q = -
2. (a) 6.71p != (b) 4.64x = (c) 2.99n = (d) 5.92x !=
(e) 1.89y = (f) .d 2 55!= (g) 4.47k != (h) 2.22x =
(i) .y 3 81!= (j) 3.01y =
3. (a) 27n = (b) 16t = (c) 32x = (d) 8t =
(e) 243p = (f) 625m = (g) 216b = (h) 27y =
(i) 128a = (j) 81t =
4. (a) 51
x = (b) 21
a = (c) 21
y = (d) x71
!=
(e) 32
n = (f) 2a = (g) 2x != (h) 9b =
(i) x32
!= (j) b 121
!=
5. (a) x512
1= (b) 6
41
x = (c) 811
a = (d) 625
1k =
(e) x81
!= (f) 4x = (g) y 8!= (h) n 73219
=
(i) 8b = (j) 1216127
m =
6. (a) 4n = (b) 5y = (c) 9m = (d) 5x = (e) 0m =
(f) 3x = (g) 2x = (h) 2x = (i) 1x = (j) k 2=
7. (a) 2x = (b) 1x = (c) 2x = - (d) 2n = (e) 0x =
(f) 6x = (g) y31
= (h) 2x = (i) 2x = (j) a 0=
8. (a) 21
m = (b) 31
x = (c) 31
x = (d) 21
k = -
(e) 32
k = - (f) 43
n = (g) 121
x = (h) 32
n =
(i) 61
k = - (j) 132
x =
9. (a) x 1= - (b) x 131
= - (c) k 4= - (d) n 3=
(e) x 212
= - (f) x32
= - (g) x 421
= - (h) x 1117
= -
(i) x 154
= (j) x 18=
10. (a) 41
m = (b) 243
k = - (c) 283
x = (d) 121
k =
(e) 181
n = (f) 21
n = - (g) 54
x = (h) 361
b = -
(i) 171
x = - (j) 5m =
Puzzle
1. All months have 28 days. Some months have more days as well. 2. 10 3. Bottle $1.05; cork 5 cents
4. 16 each time 5. Friday
Exercises 3.7
1. ,y 0 1= - 2. ,b 2 1= - 3. ,p 3 5= - 4. ,t 0 5=
5. ,x 2 7= - - 6. q 3!= 7. x 1!= 8. ,a 0 3= -
Answer S1-S5.indd 548 7/31/09 1:36:07 PM
549ANSWERS
9. ,x 0 4= - 10. x21
!= 11. ,x 1 131
= - -
12. ,y 1 121
= - 13. ,b43
21
= 14. ,x 5 2= - 15. ,x 032
=
16. ,x 1 221
= 17. 0, 5x = 18. 1, 2y = - 19. ,n 53=
20. 3, 4x = 21. 6, 1m = - 22. , ,x 0 1 2= - -
23. , ,y 1 5 2= - - 24. ,x 5 7= - 25. ,m 8 1= -
Exercises 3.8
1. (a) x 5 2!= - (b) 3a 7!= + (c) 4y 23!= +
(d) 1x 13!= - (e) p 44 7 2 11 7! != - = -
(f) x 28 5 2 7 5! != + = +
(g) 510y 88 2 22 2210 2! ! ! -= - = - = ^ h (h) 1x 2!= + (i) 12n 137!= -
(j) 3
y25!
=+
2. (a) . , .x 3 45 1 45= - (b) . , .x 4 59 7 41= - -
(c) . , .q 0 0554 18 1= - (d) . , .x 4 45 0 449= -
(e) . , .b 4 26 11 7= - - (f) . , .x 17 7 6 34=
(g) . , .r 22 3 0 314= - (h) . , .x 0 683 7 32= - -
(i) . , .a 0 162 6 16= - (j) . , .y 40 1 0 0749= -
Exercises 3.9
1. (a) . , .y 0 354 5 65= - - (b) , .x 1 1 5=
(c) . , .b 3 54 2 54= - (d) , .x 1 0 5= -
(e) . , .x 0 553 0 678= - (f) . , .n 0 243 8 24= -
(g) ,m 2 5= - - (h) ,x 0 7= (i) ,x 1 6= -
(j) . , .y 2 62 0 382=
2. (a) x2
1 17!=
- (b) x
65 13!
=
(c) q2
4 282 7
!!= =
(d) h8
12 1282
3 2 2! !=
-=
-
(e) s6
8 403
4 10! != =
(f) x2
11 133!=
- (g) d
125 73!
=-
(h) x2
2 321 2 2
!!= = (i) t
21 5!
=
(j) x4
7 41!=
Exercises 3.10
1. 3 0x 11- 2. 0 4y1 1 3. ,n n0 1# $
4. ,x x2 2# $- 5. ,n n1 11 2- 6. n5 3# #-
7. ,c c1 21 2- 8. x4 2# #- - 9. 4 5x1 1
10. ,b b221
# $- - 11. ,a a131
1 2-
12. ,y y121
21 2- 13. ,x x32
1# $
14. ,b b352
1 2- 15. x121
31
# #- -
16. y4 3# #- 17. ,x x4 41 2- 18. a1 1# #-
19. 2 3x 11- 20. ,x x1 3# $- 21. 0 2x1 1
22. a1 121
# # 23. ,y y254
# $-
24. ,m m132
121
1 2- 25. x1 131
# #
Exercises 3.11
1. ,a b1 3= = 2. ,x y2 1= = 3. ,p q2 1= = -
4. ,x y6 17= = 5. ,x y10 2= - = 6. ,t v3 1= =
7. ,x y3 2= - = 8. ,x y64 39= - = - 9. ,x y3 4= = -
10. ,m n2 3= = 11. ,w w1 51 2= - = 12. ,a b0 4= =
13. ,p q4 1= - = 14. ,x x1 11 2= = -
15. ,x y1 4= - = - 16. ,s t2 1= = -
17. ,a b2 0= - = 18. ,k h4 1= - =
19. ,v v2 41 2= - = 20. , .x y2 1 41Z=
Problem
23 adults and 16 children.
Exercises 3.12
1. ,x y0 0= = and ,x y1 1= =
2. ,x y0 0= = and ,x y2 4= - =
3. ,x y0 3= = and ,x y3 0= =
4. ,x y4 3= = - and ,x y3 4= = - 5. ,x y1 3= - = -
6. ,x y3 9= = 7. ,t x2 4= - = and ,t x1 1= =
8. ,m n4 0= - = and ,m n0 4= = -
9. ,x y1 2= = and ,x y1 2= - = -
10. ,x y0 0= = and ,x y1 1= =
11. ,x y2 1= = and ,x y1 2= - = - 12. ,x y0 1= =
13. ,x y1 5= = and ,x y4 11= =
14. ,x y41
4= = and ,x y1 1= - = - 15. ,t h21
41
= - =
16. ,x y2 0= =
17. ,x y0 0= = and ,x y2 8= - = - and ,x y3 27= =
18. ,x y0 0= = and ,x y1 1= = and ,x y1 1= - =
19. ,x y21
243
= = 20. 135
,1312
x y= - = -
Exercises 3.13
1. , ,x y z2 8 1= - = - = - 2. , ,a b c2 1 2= - = - =
3. , ,a b c4 2 7= - = = 4. , ,a b c1 2 3= = = -
5. , ,x y z5 0 2= = = - 6. , ,x y z0 5 4= = - =
7. , ,p q r3 7 4= - = = 8. , ,x y z1 1 2= = - =
9. , ,h j k3 2 4= - = = - 10. , ,a b c3 1 2= = - = -
Answer S1-S5.indd 549 7/31/09 1:36:08 PM
550 Maths In Focus Mathematics Preliminary Course
Test yourself 3
1. (a) 10b = (b) 116a = - (c) 7x = - (d) p 4#
2. (a) 1262.48A = (b) 8558.59P =
3. (a) x x x8 16 42 2- + = -] g (b) k k k4 4 22 2+ + = +] g 4. (a) ,x y2 5= - = (b) ,x y4 1= = and ,x y
21
8= - = -
5. (a) 2x = (b) 41
y =
6. (a) ,b 2 131
= - (b) ,g 241
= (c) ,x x4 3$ #
7. (a) 36A = (b) 12b = 8. ,x21
1=
9. 1 3y1 #-
10. (a) . , .x 0 298 6 70= - - (b) . , .y 4 16 2 16= -
(c) . , .n 0 869 1 54= -
11. (a) 764.5V = (b) 2.9r = 12. x 7141
2
13. ,x x2 91 2 14. . , .x y2 4 3 2= = 15. (a) 2100V =
(b) 3.9r = 16. (a) ii (b) i (c) ii (d) iii (e) iii
17. , ,a b c3 2 4= = = -
18. ,n n0 331
2 1 -
19. 4x = - 20. 2x = - 21. (a) 3y 2 (b) n3 0# #-
(c) 2x = (d) 2x = (e) ,x 3 152
= - (f) 1, 2t t$ # -
(g) 4 2x# #- (h) 3x = - (i) ,y y2 22 1 -
(j) 1, 1x x# $- (k) 65
x = (l) 21
2b# #-
(m) No solutions (n) 231
,53
t = (o) 1 3x1 1-
(p) ,m m3 2# $-
Challenge exercise 3
1. 1y = 2. ,x a x a1 2-
3. ,a b3 2!= = 4. . , .x 2 56 1 56= -
5. ; ,x x x x x x3 3 2 2 4 3 22!+ - - + + =] ] ] ^g g g h
6. ,x y1 2= = and ,x y1 0= - =
7. ; . , .b x4 17 4 8 12 0 123! Z= = + - 8. x 1!=
9. 1 1t1 1- 10. x3 8# #- 11. 41
x =
12. 2.31r = 13. No solutions 14. x b a a2!= + +
15. 2247.36P = 16. x3
4 102 !=^ h
17. ,y y153
1 2-
Chapter 4 : Geometry 1
Exercises 4.1
1. (a) 47y c= (b) 39x c= (c) 145m c= (d) 60y c= (e) 101b c= (f) 36x c= (g) 60a c= (h) 45x c=
(i) 40y c= (j) 80x c= 2. (a) 121c (b) 72 29c l (c) 134 48c l 3. (a) 42c (b) 55 37c l (c) 73 3c l
4. (a) (i) 47c (ii) 137c (b) (i) 9c (ii) 99c (c) (i) 63c (ii) 153c (d) (i) 35c (ii) 125c (e) (i) 52c (ii) 142c (f) (i) 15c7l (ii) 105c7l (g) (i) 47c36l (ii) 137c36l (h) (i) 72c21l (ii) 162c21l (i) (i) 26c 11 l (ii) 116c 11 l (j) (i) 38c 15 l (ii) 128c 15 l 5. (a) 49x c= (b) 41c (c) 131c 6. (a) ,y x z15 165c c= = =
(b) , ,x y z142 48 28c c c= = =
(c) , ,a b c43 137 101c c c= = =
(d) ,,a b d c97 41 42c c c= = = =
(e) , ,a b c68 152 28c c c= = = (f) ,a b10 150c c= =
7. 0x x x x
xx
8 10 2 10 10 7 10 36
18 36020
- + - + + + + =
=
=
(angleof revolution)
( )
( )
ABE x
EBC x
ABE EBC
8 108 20 101502 102 20 1030150 30180
c
cc cc
+
+
+ +
= -= -
== -= -
=+ = +
=
ABC`+ is a straight angle
( )
DBC x
DBC EBC
7 107 20 10150150 30180
cc cc
+
+ +
= +
= +
=
+ = +
=
DBE`+ is a straight angle AC and DE are straight lines
8.
AFC x
CD bisects
`
`
+ =
AFE+
( )
( )
( )
( )
DFB x
x
CFE x x
xAFC CFE
AFB
AFB
180 180
180 180 2
is a straight angle
(vertically opposite angles)
is a straight angle
`
c c
c c
+
+
+ +
+
+
= - -
=
= - + -
=
=
9. ABD DBC+ ++
110 3 3 70180
x xc
= - + +
=
So ABC+ is a straight angle. AC is a straight line.
10. AEB BEC CED+ + ++ +
y y y50 8 5 20 3 60
90c= - + - + +
=
So AED+ is a right angle.
Exercises 4.2
1. (a) ,a b e f c d g148 32c c= = = = = = =
(b) ,x z y70 110c c= = =
(c) , ,x y z55 36 89c c c= = = (d) ,y x z125 55c c= = =
(e) ,n e g a c z x 98c= = = = = = = 82o m h f b d y w c= = = = = = = =
Answer S1-S5.indd 550 8/1/09 6:50:34 PM
551ANSWERS
(f) , ,a b c95 85 32c c c= = =
(g) , ,a b c27 72 81c c c= = =
(h) , , ,x y z a b56 124 116 64c c c c= = = = =
(i) 61x c= (j) 37y c=
2. (a) CGF
BFG CGF
180 121
5959
( is a straight angle)FGH
`
c c
cc
+
+ +
= -
=
= =
These are equal alternate angles. AB CD` < (b) BAC 360 292 68
(angle of revolution)c c c+ = - =
BAC DCA 68 112180
` c cc
+ ++ = +
=
These are supplementary cointerior angles.
AB CD` <
(c) 180 76104
104
BCD
ABC BCDc
c
+
+ +
= -
=
= =
( BCE+ is a straight angle)
These are equal alternate angles.
AB CD` ;
(d) 180 12852
52
CEF
CEF ABEc
c
+
+ +
= -
=
= =
( CED+ is a straight angle)
These are equal corresponding angles.
AB CD` ;
(e) 180 23 115CFH+ = - +] g ( EFG+ is a straight angle)
42c=
42BFD` c+ = (vertically opposite angles)
ABF BFD 138 42
180c cc
+ ++ = +
=
These are supplementary cointerior angles. AB CD` ;
Exercises 4.3
1. (a) 60x c= (b) 36y c= (c) 71m c= (d) 37x c=
(e) 30x c= (f) 20x c= (g) 67x c= (h) 73a c=
(i) , ,a b c75 27 46c c c= = =
(j) , ,a b c36 126 23c c c= = =
(k) , ,x y z w67 59 121c c c= = = =
2. All angles are equal. Let them be x . x x x 180Then (angle sum of )D+ + =
xx
3 18060
=
=
So all angles in an equilateral triangle are 60 .c
3. x90 c-] g
4. 50
180 (50 45 )ACBABC
DEC ABC85
85
(vertically opposite angles)(angle sum of )
`
cc c cc
c
++
+ +
D
=
= - +
=
= =
These are equal alternate angles.
AB DE` <
5.
124 68
ACB
CBACBA
CBA ACBABC
180 12456
68 124
5656
is isosceles
( is a straight angle)
(exterior angle of )
DCB
`
`
c cc
c cc cc
c
+
++
+ +D
D
= -
=
+ =
= -
=
= =
6. 38y c=
7. (a) x 64c= (b) ,x y64 57c c= = (c) 63x c=
(d) ,a b29 70c c= =
8. 180 (35 25 )120180 12060180 (90 30 )60180 (60 60 )60
HJI
IJL
JIL
ILJ
(angle sum of )
( is a straight angle)
(angle sum of )
(angle sum of )
HJI
HJL
IKL
JIL
c c ccc ccc c cc
c cc
+
+
+
+
D
D
D
= - +
=
= -
=
= - +
=
= - +
=
Since 60 ,IJL JIL ILJ c+ + += = = IJLD is equilateral
( )
( )
( )
KJL
JLK
KJI
JKL
180 60120180 30 12030
is a straight angle
angle sum of
c cc
c c cc
+
+ D
= -
=
= - +
=
°JLK JKL 30`+ += =
JKL` D is isosceles
9. BC BD=
BDC 46` c+ = (base angles of isosceles triangle)
CBD 180 2 4688
#c
+ = -=
CBD BDE 88` c+ += = These are equal alternate angles.
AB ED` ;
10. 18032
OQP 75 73c
+ = - +
=
] g (angle sum of triangle)
MNO OQP 32` c+ += =
These are equal alternate angles.
MN QP` ;
Exercises 4.4
1. (a) Yes
5AB EF cm= = (given)
6BC DF cm= = (given)
8AC DE cm= = (given)
ABC DEF` /D D ( SSS )
(b)Yes
4.7XY BC m= = (given)
XYZ BCA 110c+ += = (given)
2.3YZ AC m= = (given)
XYZ ABC` /D D ( SAS )
(c) No
Answer S1-S5.indd 551 7/31/09 1:36:09 PM
552 Maths In Focus Mathematics Preliminary Course
(d) Yes
PQR SUT 49c+ += = (given)
PRQ STU 52c+ += = (given)
8QR TU cm= = (given)
PQR STU` /D D ( AAS )
(e) No
2. (a)
,
AB KLB L
BC JLABC JKL
438
5by SAS
(given)(given)(given)
`
c+ +
/D D
= =
= =
= =
(b)
,
Z BXY ACYZ BC
RHS XYZ ABC
9072
by
(given)(given)(given)
`
c+ +
/D D
= =
= =
= =
(c)
,
MN QRNO PRMO PQ
MNO PQR
885
by SSS
(given)(given)(given)
` /D D
= =
= =
= =
(d)
.
Y TZ S
XY TRXYZ STR
90351 3
by AAS,
(given)(given)(given)
`
cc
+ ++ +
/D D
= =
= =
= =
(e)
,
BC DEC E
AC EFABC DEF
490
7by SAS
(given)(given)(given)
`
c+ +
/D D
= =
= =
= =
3. (a) B CBDA CDA
ADABD ACD
90is common
by AAS,
(base angles of isosceles )(given)
`
c+ +
+ +
/D D
D=
= =
(b) BD DCAD BCbisects
(corresponding sides in congruent s)`
`
D=
4. , )AB CDABD BDC ernate angles+ + <= (alt
( , )ADB DBCBD
ABD CDBAD BC
AD BCis common
by AAS,
alternate angles
(corresponding sides in congruent s)
`
`
+ +
/
<
D D
D
=
=
5. (a) OA OC= (equal radii)
OB OD= (similarly)
AOB COD+ += (vertically opposite angles)
AOB COD` /D D ( SAS )
(b) AB CD= (corresponding sides in congruent
triangles)
6. (a) AB AD= (given)
BC DC= (given)
AC is common
ABC ADC` /D D ( SSS )
(b) ABC ADC+ += (corresponding angles in congruent
triangles)
7. (a) OA OC= (equal radii)
OB is common
AOB COB 90c+ += = (given)
OAB OBC` /D D ( SAS )
(b) OCB OBC+ += (base angles of OBC, an isosceles
right angled triangle)
But OCB OBC 90c+ ++ = (angle sum of triangle)
So OCB OBC 45c+ += =
Similarly 45OBA c+ =
45 45 90OBA OBC` c c c+ ++ = + =
So ABC+ is right angled
8. (a) 90AEF BDC c+ += = (given)
AF BC= (given)
FE CD= (given)
AFE BCD` /D D ( RHS )
(b) AFE BCD+ += (corresponding angles in
congruent triangles)
9. (a) OA OC= (equal radii)
OB is common
AB BC= (given)
OAB OBC` /D D ( SSS )
(b) OBA OBC+ += (corresponding angles in
congruent triangles)
But 180OBA OBC c+ ++ = ( ABC is a straight angle)
So 90OBA OBC c+ += =
OB is perpendicular to AC.
10. (a) AD BC= (given)
ADC BCD 90c+ += = (given) DC is common ADC BCD` /D D ( SAS )
(b) AC BD= (corresponding sides in congruent
triangles)
Exercises 4.5
1. (a) .x 15 1= (b) 4.4x = (c) 6.6m =
(d) , ,76 23 81c c ca i b= = = (e) 4.5b =
(f) , , .x y115 19 3 2c ca = = = (g) 9.7p =
2. . , .a b1 81 5 83= =
3. ( , )
( )
BAC EDCABC DECACB ECD
AB EDalternate angles(similarly)vertically opposite angles
+ ++ ++ +
<=
=
=
since 3 pairs of angles are equal, | CDED||ABCD
Answer S1-S5.indd 552 7/31/09 1:36:10 PM
553ANSWERS
4.
.
..
..
.
GFE EFD
EFGF
DFEF
EFGF
DFEF
2 71 5
0 5
4 862 7
0 5
(given)
`
+ +=
= =
= =
=
o
o
Since two pairs of sides are in proportion and their included angles are equal, then | FGED||DEFD
5. ..
.
..
.
..
.
DEAB
DFAC
EFBC
DEAB
DFAC
EFBC
1 821 3
0 714
5 884 2
0 714
6 864 9
0 714
`
= =
= =
= =
= =
Since three pairs of sides are in proportion, | DEFD||ABCD
y 41c=
6. (a) OA OBOC OD
ODOA
OCOB
AOB COD
(equal radii)(similarly)
(vertically opposite angles)
`
+ +
=
=
=
=
Since two pairs of sides are in proportion and their included angles are equal, | OCD3||OAB3
(b) 5.21AB cm=
7. (a) A+ is common
( , )ABC ADE
ACB AEDBC DEcorresponding angles
(similarly)+ ++ +
<=
=
since 3 pairs of angles are equal, | ADED||ABCD
(b) . , .x y2 17 2 25= =
8. ( , )( , )( )
ABF BECCBE BFA
C A
s AB CDBC AD
s
alternate anglesimilarlyangle sum of`
+ ++ ++ +
z
z
D
=
=
=
since 3 pairs of angles are equal, | CEBD||ABFD
9. A+ is common
..
..
ABAD
ACAE
ABAD
ACAE
31 2
0 4
20 8
0 4
`
= =
= =
=
Since two pairs of sides are in proportion and their included angles are equal, | , .ABC m 4 25D =||AEDD
10. .
.
..
..
.
CDAB
ACBC
ADAC
CDAB
ACBC
ADAC
2 62
0 769
3 93
0 769
5 073 9
0 769
`
= =
= =
= =
= =
Since three pairs of sides are in proportion,
,c| ,ACD x y109 47cD = =||ABCD
11. (a) 7.8x = (b) . , .m p4 0 7 2= = (c) 6.5x =
(d) . , .x y6 2 4 4= = (e) . , .x y1 4 9 2= =
12. (a) BCAB
DEAD
DEAD
FGAF
BCAB
FGAF
Also
`
=
=
=
(b)ACAB
AEAD
AEAD
AGAF
ACAB
AGAF
Also
`
=
=
=
(c) CEBD
AEAD
AEAD
EGDF
CEBD
EGDF
Also
`
=
=
=
13. . , .a b4 8 6 9= = 14. 0.98y = 15. . , .x y3 19 1 64= =
Exercises 4.6
1. (a) 6.4x = (b) 6.6y = (c) 5.7b = (d) 6.6m =
2. (a) 61p = (b) 58t = (c) 65x = (d) 33y =
3. .s 6 2 m= 4. .CE 15 3 cm=
5. 81, 144, 225AB CB CA2 2 2= = =
AB CB
CA
81 144225
2 2
2
+ = +
=
=
ABC` D is right angled
6. 1XY YZ= = XYZ` D is isosceles
,YZ XY XZYZ XY
XZ
1 21 12
2 2 2
2 2
2
= = =
+ = +
=
=
XYZ` D is right angled
7. AC AB BC
BCBC
BCBC
AC
BC
2 34 311
22 12
2 2 2
2 2 2
2
2
`
#
= +
= +
= +
=
=
=
=
=
^ h
8. (a) 5AC =
(b) , ,AC CDAD
25 144169
2 2
2
= =
=
25 144169
AC CD
AD
2 2
2
+ = +
=
=
ACD` D has a right angle at ACD+ AC` is perpendicular to DC
Answer S1-S5.indd 553 7/31/09 1:36:10 PM
554 Maths In Focus Mathematics Preliminary Course
9. AB b3= 10. xx y2 2+
11. d t tt t t t
t t
20 3 15 2400 120 9 225 60 413 180 625
2 2 2
2 2
2
= - + -
= - + + - +
= - +
] ]g g
12. 1471 mm 13. 683 m 14. 12.6 m 15. 134.6 cm
16. 4.3 m 17. 42.7 cm
18. 1.3 1.1 2.9 1.5 2.25and2 2 2+ = =
. . .1 3 1 1 1 52 2 2!+ so the triangle is not right angled the property is not a rectangle
19. No. The diagonal of the boot is the longest available space and it is only 1.4 m.
20. (a) 6 4BC2 2 2= - 20= 20BC = 6AO cm= (equal radii) So 6 4AC2 2 2= - 20= 20AC = Since ,BC AC= OC bisects AB
(b) OCA OCB 90c+ += = (given) OA OB= (equal radii) OC is common OAC OBC` /D D ( RHS ) So AC BC= (corresponding sides in congruent triangles) OC bisects AB
Exercises 4.7
1. (a) x 94c= (b) y 104c= (c) x 111c= (d) x 60c= (e) y 72c= (f) °, °x y102 51= = (g) °, °x y43 47= =
2. ABED is isosceles.
( s )
( )
B ECBE DEB
76180 76104
base equal
straight s
` cc cc
+ ++ +
+
+
= =
= = -
=
D
DD
62 104 104 360270 360
90
(angle sum of quadrilateral)c c c cc c
c
++
+
+ + + =
+ =
=
CD is perpendicular to AD
3. (a)
( )( , )
( , )
( , )
D x
C x
xx
A C xB xB D x
A D AB DC
C D AD BC
B C AB DC
180
180 180
180 180
180180
and cointerior angles
and cointerior angles
and cointerior angles`
`
c
c c
c c
cc
+
+
+ +++ +
+ +
+ +
+ +
<
<
<
= -
= - -
= - +
=
= =
= -
= = -
(b) x x x x180 180360
Angle sum c cc
= + + - + -
=
4. ,a b150 74c c= =
5. (a) 5 , 3 , 108 , 72a b x z ym m c c= = = = = (b) , ,x y z53 56 71c c c= = = (c) 5 , 68x y cm ca b= = = =
(d) , ,121 52 77c c ca b i= = = (e) 60x c= (f) ,x y3 7= =
6. ( ), ),
ADB CDBCDB ABDADB DBCABD DBC
BD ABC
BD ADCAB DCAD BC
bisects
bisects(alternate angles(alternate angles )
`
`
+ ++ ++ ++ +
+
+
<
<
=
=
=
=
7. (a) ..
AD BCAB DC
3 85 3
cmcm
(given)(given)
= =
= =
Since two pairs of opposite sides are equal, ABCD is a parallelogram.
(b) AB DCAB DC
7cm (given)
(given)<
= =
Since one pair of opposite sides is both equal and parallel, ABCD is a parallelogram.
(c) 54 126180
X M c cc
+ ++ = +
=
These are supplementary cointerior angles. XY MN` <
XM YNAlso, (given)<
XMNY is a parallelogram
(d) AE ECDE EB
56
cmcm
(given)(given)
= =
= =
Since the diagonals bisect each other, ABCD is a parallelogram.
8. (a) ,x 5 66cm ci= = (b) , ,90 25 65c c ca b c= = = (c) ,x y3 5cm cm= = (d) ,x y58 39c c= = (e) x 12 cm=
9. 6.4 cm 10. 59 , 31 , 59ECB EDC ADEc c c+ + += = =
11. 4 2 cm 12. 57x y c= =
Exercises 4.8
1. (a) 540c (b) 720c (c) 1080c (d) 1440c (e) 1800c (f) 2880c 2. (a) 108c (b) 135c (c) 150c (d) 162c (e) 156c 3. (a) 60c (b) 36c (c) 45c (d) 24c
4. 128 34c l 5. (a) 13 (b) 152 18c l 6. 16 7. 3240c
8. 2340c 9. 168 23c l
10. ( )
.
n nn n
nn
145145 180 360
3510 3
2 180Sum # c= = -
= -
=
=
But n must be a positive integer. no polygon has interior angles of 145 .c
11. (a) 9 (b) 12 (c) 8 (d) 10 (e) 30
12. (a) ABCDEF is a regular hexagon. AF BC= (equal sides) FE CD= (equal sides) AFE BCD+ += (equal interior angles) AFE BCD` /D D ( SAS )
Answer S1-S5.indd 554 7/31/09 1:36:11 PM
555ANSWERS
(b) ( )
S n
AFE
6720
6720
120
2 1802 180#
#
cc
cc
c
+
= -
= -
=
=
=
] g
Since ,AF FE= triangle AFE is isosceles. So FEA FAE+ += (base angles in isosceles triangle)
FEA2
180 120
30
`c
c
+ =-
=
(angle sum of triangle)
EDA 120 3090
cc
+ = -
=
Similarly, DEB 90c+ =
So ED DEA B 180c+ ++ = These are supplementary cointerior angles AE BD` <
13. A regular octagon has equal sides and angles. AH AB= (equal sides)
GH BC= (equal sides) AHG ABC+ += (equal interior angles)
AHG ABC` /D D ( SAS )
So AG AC= (corresponding sides in congruent triangles)
( )S n
81080
2 1802 180#
#
cc
c
= -
= -
=
] g
AHG
81080
135
`c
c
+ =
=
HGA HAG+ += (base angles in isosceles triangle)
HAG2
180 135
22 30
`c
c
+ =-
= l
(angle sum of triangle)
GAC 135 2 22 30
90# c
c+ = -
=
l
We can similarly prove all interior angles are 90c and adjacent sides equal . So ACEG is a square .
14. EDC5
5
108
2 180# c
c
+ =-
=
] g
ED CD= (equal sides in regular pentagon)
So EDC is an isosceles triangle. DEC ECD`+ += (base angles in isosceles triangle)
36
DEC2
180 108c
c
+ =-
=
(angle sum of triangle)
108 3672
AEC cc
+ = -
=
Similarly, using triangle ABC , we can prove that 72EAC c+ = So EAC is an isosceles triangle. (Alternatively you could prove EDC and ABC congruent triangles and then AC EC= are corresponding sides in congruent triangles.)
15. (a) p
360
(b) Each interior angle:
180360
180 360
180 360
180 2
p
p
p
p
p
p
p
p
-
= -
=-
=-^ h
Exercises 4.9
1. (a) .26 35 m2 (b) .21 855 cm2 (c) .18 75 mm2 (d) 45 m2 (e) 57 cm2 (f) 81 m2 (g) .28 27 cm2 2. .4 83 m2
3. (a) .42 88 cm2 (b) .29 5 m2 (c) .32 5 cm2 (d) .14 32 m2 (e) .100 53 cm2 4. (a) 25 m2 (b) .101 85 cm2 (c) .29 4 m2 (d) .10 39 cm2 (e) 45 cm2
5. 7 51 98 7 51 14 cm2+ = +^ h 6. .22 97 cm2
7. $621.08 8. (a) .161 665 m2 (b) 89 m2 (c) 10.5 m
9. (a) 48 cm (b) 27 cm 10. w12 units2
Test yourself 4
1. (a) , ,x y z43 137 147c c c= = = (b) 36x c= (c) , ,a b c79 101 48c c c= = = (d) 120x c= (e) 7.2r cm= (f) 5.6 , 8.5x ycm cm= = (g) 45ci =
2. )AGF HGB(vertically opposite+ +i=
AGF CFESo+ + i= =
These are equal corresponding .s+ AB CD` <
3. 118.28 cm 2
4. (a)
( )
DAE BACADE ABCAED ACB
ABC ADE AAAand are similar
(common)(corresponding angles, DE BC)(similarly)
`
+ ++ ++ +
<
D D
=
=
=
(b) 3.1 , 5.2x ycm cm= =
5. 162c 6. 1020.7 cm 3 7. 36 m
8. (a) AB ADBC DC
(adjacent sides in kite)(similarly)
=
=
AC is common Δ ABC and Δ ADC are congruent (SSS)
(b) AO COBO DO
AOB COD
(equal radii)(similarly)(vertically opposite angles)+ +
=
=
=
Δ AOB and Δ COD are congruent (SAS)
9. 73.5 cm 2
10. 6 2 7 36 28 64 82 2 2+ = + = =^ h ` Δ ABC is right angled (Pythagoras)
Answer S1-S5.indd 555 7/31/09 1:36:11 PM
556 Maths In Focus Mathematics Preliminary Course
11. AGAF
AEAD
AEAD
ACAB
AGAF
ACAB
(equal ratios on intercepts)
(similarly)
`
=
=
=
12. (a) (base s of isosceles+ D)( , )
AB ACB C
BD DC AD BC
(given)
bisects given+ +
=
=
=
ABD ACD SAS` /D D ] g (b)
180ADB ADC
ADB ADCBut(corresponding s in congruent s)
(straight )c+ ++ +
+
+
D=
+ =
So 90ADB ADC c+ += =
So AD and BC are perpendicular.
13.
34˚ 34
( )( )
ACBCAD
CAD ADC
6868 34
base s of isoscelesexterior of
`
cc cc
c
++
+ +
+
+
D
D
=
= -
=
= =
So Δ ACD is isosceles base s equal+^ h 14.
( , )
, )DAC ACBBAC ACD
AD BCs AB DC
alternate s(alternate
+ ++ +
+
+
<
<
=
=
AC is common
ABC ADC
AB DC(AAS)
(corresponding sides in congruent s)`
`
/D D
D=
Similarly, AD BC= opposite sides are equal
15. (a) 24 cm 2 (b) 5 cm 16. 9
17. BFG FGD x x109 3 3 71180c cc
+ ++ = - + +
=
These are supplementary cointerior .s+ AB CD` <
18. 57 cm 2
19. (
(( )
)
)
ACB A Bx y
ACD ACBz x y
x yx y
180180180180 180180 180
sum of
straight
cccc cc c
+ + +
+ +
+
+
D= - +
= - -
= -
= - - -
= - + +
= +
] g
20. (a)
..
.
..
.
A E
EFAC
DEAB
EFAC
DEAB
2 72 97
1 1
3 63 96
1 1
given
`
+ +=
= =
= =
=
^ h
So Δ ABC and Δ DEF are similar (two sides in proportion, included s+ equal).
(b) 4.3x cm=
Challenge exercise 4
1. 94c 2. , ,x y z75 46 29c c c= = = 3. ,1620 32 44c c l
4. , )
( )
BAD DBCABD BDCADB DCB
AB DC(given)(alternate anglesangle sum of`
+ ++ ++ +
<
D
=
=
=
since 3 pairs of angles are equal, BCDD;ABD <D
6.74d cm=
5. AB DCA D 131 49
180
(given)c cc
+ +=
+ = +
=
A+ and D+ are supplementary cointerior angles AB DC` <
Since one pair of opposite sides are both parallel and equal, ABCD is a parallelogram.
6. .27 36 m2
7.
Let ABCD be a square with diagonals AC and BD and
D
AD DC90
(adjacent sides of square)c+ =
=
°
°°
ADCDAC DCA
DAC DCADAC DCABAC BCA
904545
is isosceles
Similarly,
(base angles of isosceles )(angle sum of )
(other angles can be proved similarly)
`
`
`
+ ++ +
+ ++ +
D
D
D
=
+ =
= =
= =
8.
Let ABCD be a kite
AD ABDC BC
(given)(given)
=
=
AC is common
, ADC ABCDAC BAC
AD ABDAE BAE
by SSS
(corresponding angles in congruent s)(given)(found)
`
` + +
+ +
/D D
D
=
=
=
AE is common
Answer S1-S5.indd 556 8/1/09 8:50:48 PM
557ANSWERS
,(
( )
ADE ABEDEA BEADEA BEADEA BEA
DEB18090
by SAS
But
the diagonals are perpendicular
corresponding angles in congruent s)is a straight angle
`
`
`
`
cc
+ ++ ++ +
/D D
D=
+ =
= =
9. 84 (15 112 ) )
( )
MNYMNY
XYZXYZMNY XYZ
MNZ
XYZ43
69 11243
43
(exterior angle of
exterior angle of`
`
`
c c cc
c cc
c
++
+++ +
D
D
+ = +
=
+ =
=
= =
These are equal corresponding angles. MN XY` <
10. .x 2 12 m= 11. (a) 6 m2 (b) 10 2 5 2 5 5 m+ = +^ h
12. . , .x y28 7 3 8cm cm== 13. 7.40 , 4.19x ym m= =
14. (a) AB BCABE CBE 45
(adjacent sides in square)
(diagonals in square make 45 with sides)c
+ +=
= =
EB is common.
, ABE CBE
AE CEby SAS
(corresponding sides in congruent s)`
`
/D D
D=
Since AB BC= and ,AE CE= ABCE is a kite.
(b) BD x x
xx
DE BD
x
22
21
22
units
2 2
2
= +
=
=
=
=
Practice assessment task set 1
1. 9p = 2. 2 5 y x y+ -^ ^h h 3. (a) x 1- (b) 3x4
4. 6 10y - 5. 23
25 5 2+ 6. 2 16 3x x x3 2+ - +
7. 72
x = 8. 3
2x -
9. °ABC EDCACB ECD
AB EDABC EDC
AC ECACE
90
by AAS
is isosceles
(given)(vertically opposite angles)(given)
(corresponding sides in congruent triangles)`
`
`
+ ++ +
/D D
D
= =
=
=
=
10. 231.3 11. 3- 12. 135c 13. 7.33 10 2#
-
14. 3 10 4- 15. 3.04 16. 3x + 17. . , .x 1 78 0 281= -
18. 1.55r = 19. x 12
20. 157
21. x2
42 3
12!!= = 22.
491
23. 4, 11 1, 4x y x yor= = = - = - 24. ,x y2 1= = -
25. 7 26. 7.02 cm 27. 2 1 4 2 1x x x2- + +] ^g h
28. 43
6 15 2 6+ 29. 7 30. $643.08 31. 1.1
32. 2 10 3 5 2 2 3- + - + 33. $83.57
34. , ,x y w z22 29 90c c c= = = = 35. 56.7 cm2
36. a ba
b21 10
21
10
=- 37. ,x x6 252
2 1 - 38. 81
39. x 7- - 40. 41
x = 41. ,x x3 3# $- 42. 61
43. Given diagonal AC in rhombus ABCD :
)
)
AB BCDAC ACBBAC ACBDAC BAC
AD BCABC
(adjacent sides in rhombus)(alternate s,(base s of isosceles
`
+ ++ ++ +
+
+
<
D
=
=
=
=
` diagonal AC bisects the angle it meets. Similarly, diagonal BD bisects the angle it meets.
44. x 3 1+ -] g 45. 6 12 8x x x3 2+ + + 46. 2
517
4
47. 53x c=
48. ,x y98 41c c= = 49. 3 2
1
x +
50. (a) 12 8x y- (b) 2 31 (c) 3 9
3
x x
x2 - +
- (d) 3 2 1+
(e) 1 1
5
x x
x
+ -
- +
] ]]g g
g (f)
611 3
(g) x y zx z
y14 7 11
14 11
7
=- -
(h) 5 1 2
3a a b b+ +] ]g g (i) 8 5 (j) 13
21
51. . , .x y2 7 3 1= = 52. 25x = 53. r2
cm3 r
=
54. 17.3 cm
55. DEA xEAD xCD x x
xABC xABC DEA
A222
LetThen (base s of isosceles )
(exterior of )
(opposite s of gram are equal)
EAD
`
`
+++
++ +
+
+
+ <
D
D
=
=
= +
=
=
=
56. 52
57. 5% 58. 2.2 10 kmh8 1#
- 59. 20k =
60. 9xy y 61. 147 16c l 62. 5.57 m2
63. (a) a b a a ab b b5 2 2 4 2 4 4 42 2+ - - - + + +] ^g h (b) 3 4 6 2a b a b c+ - +] ]g g
Answer S1-S5.indd 557 7/31/09 1:36:13 PM
558 Maths In Focus Mathematics Preliminary Course
64. x181
543
1#-
65. (BCEF is a gram)<
(BC AD ABCDBC FEAD FE
is a gram)
`
< <
<
<
BC ADBC FEAD FE
Also opposite sides of gram
similarly`
<=
=
=
^^
hh
Since AD and FE are both parallel and equal, AFED is a parallelogram.
66. 11.95b m= 67. (a) 34 cm (b) 30 cm 2
68. 75
18 3 31 2 25 5+ - 69. 20 70. 32 m
71. BD bisects AC So AD DC= 90BDC BDA c+ += = (given) BD is common BAD BCD` /D D ( SAS ) AB CB` = (corresponding sides in congruent
triangles) So triangle ABC is isosceles
72. 2
x y2 2+ 73. (b) 74. (c) 75. (a) 76. (b) 77. (b)
78. (d) 79. (d)
Chapter 5 : Functions and graphs
Exercises 5.1
1. Yes 2. No 3. No 4. Yes 5. Yes 6. Yes 7. No
8. Yes 9. Yes 10. No 11. Yes 12. No 13. Yes
14. No 15. Yes
Exercises 5.2
1. 4, 0f f1 3= - =] ]g g 2. , ,h h h0 2 2 2 4 14= - = - =] ] ]g g g
3. 25, 1, 9, 4f f f f5 1 3 2= - - = - = - - = -] ] ] ]g g g g 4. 14
5. 35- 6. 9x = 7. x 5!= 8. x 3= - 9. ,z 1 4= -
10. 2 9, 2 2 9f p p f x h x h= - + = + -^ ]h g
11. 1 2g x x2- = +] g 12. f k k k k1 12= - + +] ] ^g g h 13. ; ,t t1 2 4= - = - 14. 0
15. 125, 1, 1f f f5 1 1= = - = -] ] ]g g g
16. 0 4 1 3f f f2 2 1- - + - = - + = -] ] ]g g g
17. 10 18. 7 19. 28-
20. (a) 3 (b) 3 3 3 0x - = - = Denominator cannot be 0 so the function doesn’t exist for .x 3= (c) 4
21. 2 5f x h f x xh h h2+ - = + -] ]g g 22. 4 2 1x h+ +
23. x c5 -] g 24. 3 5k2 + 25. (a) 2 (b) 0 (c) 2n n4 2+ +
Exercises 5.3
1. (a) x -intercept 32
, y -intercept -2
(b) x -intercept -10, y -intercept 4 (c) x -intercept 12, y -intercept 4 (d) x -intercepts 0, -3, y -intercept 0 (e) x -intercepts 2! , y -intercept -4 (f) x -intercepts -2, -3, y -intercept 6 (g) x -intercepts 3, 5, y -intercept 15
(h) x -intercept 53- , y -intercept 5 (i) x -intercept -3, no y -intercept (j) x -intercept ,3! y -intercept 9
2. 2
( )
f x xxf x
2
even function
2
`
- = - -
= -
=
2] ]g g
3. (a) 1f x x2 6= +^ h (b) f x x x2 12 6 3= + +] g7 A
(c) 1f x x3- = - +] g (d) Neither odd nor even
4.
( )
g x x x xx x xg x
3 23 2
even function
8 4 2
8 4 2
`
- = - + - - -
= + -
=
] ] ] ]g g g g
5. f x x f x- = - = -] ]g g odd function
6. 1
( )
f x xxf x
1
even function
2
2
`
- = - -
= -
=
] ]g g 7. f x x xx xx x
f x
444
odd function
3
3
3
`
- = - - -
= - +
= - -
= -
] ] ]^]
g g gh
g
8. f x x xx xf x
even function
4 2
4 2
`
- = - + -
= +
=
] ] ]]
g g gg
0f x f x- - =] ]g g
9. (a) Odd (b) Neither (c) Even (d) Neither (e) Neither
10. (a) Even values i.e. , , ,n 2 4 6 f=
(b) Odd values i.e. , , ,n 1 3 5 f=
11. (a) No value of n (b) Yes, when n is odd (1, 3, 5, …)
12. (a) (i) x 02 (ii) x 01 (iii) Even
(b) (i) x 21 (ii) x 22 (iii) Neither
(c) (i) x2 21 1- (ii) ,x x2 21 2- (iii) Neither
(d) (i) All real x 0! (ii) None (iii) Odd
(e) (i) None (ii) All real x (iii) Neither
Exercises 5.4
1. (a) x -intercept 2, y -intercept -2
(b) x -intercept 121
- , y -intercept 3
(c) x -intercept 21
, y -intercept 1
(d) x -intercept -3, y -intercept 3
(e) x -intercept 32
, y -intercept 31
-
Answer S1-S5.indd 558 7/31/09 1:36:13 PM
559ANSWERS
2. (a)
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(b) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(c) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(d) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2-1
1
(e) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2-1
112
(f) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2-1
1
(g) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2-1
1
23
-
(h) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
Answer S1-S5.indd 559 7/31/09 1:36:14 PM
560 Maths In Focus Mathematics Preliminary Course
(i) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(j) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-1111
2
3. (a) ,x yall real all real" ", , (b) :,x y y 2all real =" ", , (c) : ,x x y4 all real= -! "+ , (d) : ,x x y2 all real=! "+ , (e) , :x y y 3all real =! "+ ,
4. (a) Odd (b) Even (c) Neither (d) Odd (e) Odd
5. y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-1111
2
(3, -1)
Exercises 5.5
1. (a) x -intercepts 0, -2, y -intercept 0 (b) x -intercepts 0, 3, y -intercept 0 (c) x -intercepts ! 1, y -intercept -1 (d) x -intercepts -1, 2, y -intercept -2 (e) x -intercepts 1, 8, y -intercept 8
2. (a) y
x-4
-5
-3 -2-1 2 3 4 5
2
1
3
4
5
6
-3-4
-2-1
1
(b) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2
-11
(c) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2-1
1
Answer S1-S5.indd 560 7/31/09 1:36:15 PM
561ANSWERS
(d) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2
-1 1
(e) y
x-4
-5
-3 -2 -1 2 3 4 5
21
3
4
5
6
-3
-4
-2
-11
(f) y
x-4
-10
-3 -2 -1 2 3 4 5
4
6
8
2
10
12
-6
-8
-4-2
1
(g) y
x-4
-5
-3 -2 -1 2 3 4 5
21
3
4
5
-3
-4
-6
-2
-11
(h) y
x
-5
-3-4 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-6
-2
-1 1
(i) y
x-4
-5
-3 -2 -1 3 4 5
2
1
3
4
5
-3
-4
-6
-2-1 2111
2
(j) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3-4
-6
-2-1
1
3. (a) (i) x -intercepts 3, 4, y -intercept 12 (ii) {all real x },
:y y41
$ -( 2 (b) (i) x -intercepts 0, -4, y -intercept 0 (ii) {all real x }, :y y 4$ -" , (c) (i) x -intercepts -2, 4, y -intercept -8 (ii) {all real x }, : 9y y $ -" , (d) (i) x -intercept 3, y -intercept 9 (ii) {all real x }, :y y 0$" , (e) (i) x -intercepts ,2! y -intercept 4 (ii) {all real x }, :y y 4#" ,
4. (a) {all real x }, :y y 5$ -" , (b) {all real x }, :y y 9$ -" ,
Answer S1-S5.indd 561 7/31/09 1:36:16 PM
562 Maths In Focus Mathematics Preliminary Course
(c) {all real x }, :y y 241
$ -( 2 (d) {all real x }, :y y 0#" , (e) {all real x }, : 0y y $" ,
5. (a) y0 9# # (b) y0 4# # (c) y1 24# #-
(d) y4 21# #- (e) y18 241
# #-
6. (a) (i) x 02 (ii) x 01 (b) (i) x 01 (ii) x 02
(c) (i) x 02 (ii) x 01 (d) (i) x 21 (ii) x 22 (e) (i) x 52 - (ii) x 51 -
7.
( )
f x xx
f xeven
2
2
`
- = - -
= -
=
] ]g g
8. (a) Even (b) Even (c) Even (d) Neither (e) Neither (f) Even (g) Neither (h) Neither (i) Neither (j) Neither
Exercises 5.6
1. (a) x -intercept 0, y -intercept 0 (b) No x -intercepts, y -intercept 7 (c) x -intercepts ,2! y -intercept -2 (d) x -intercept 0, y -intercept 0 (e) x -intercepts ,3! y -intercept 3 (f) x -intercept -6, y -intercept 6
(g) x -intercept 32
, y -intercept 2
(h) x -intercept 54
- , y -intercept 4
(i) x -intercept 71
, y -intercept 1
(j) No x -intercepts, y -intercept 9
2. (a) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(b) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(c) y
-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(d) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(e) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(f) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
Answer S1-S5.indd 562 7/31/09 1:36:16 PM
563ANSWERS
(g) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(h) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(i) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(j) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
3. (a) {all real x }, :y y 0$" , (b) {all real x }, :y y 8$ -" , (c) {all real x }, :y y 0$" , (d) {all real x }, :y y 3$ -" , (e) {all real x }, :y y 0#" ,
4. (a) (i) x 22 (ii) x 21 (b) (i) x 02 (ii) x 01
(c) (i) x 121
2 (ii) x 121
1 (d) (i) x 02 (ii) x 01
(e) (i) x 01 (ii) x 02
5. (a) 0 2y# # (b) y8 4# #- - (c) 0 6y# #
(d) 0 11y# # (e) y1 0# #-
6. (a) x 32 - (b) x 01 (c) x 92 (d) x 22 (e) x 21 -
7. (a) x 3!= (b) ,x x1 12 1 - (c) x2 2# #-
(d) ,x 1 3= - - (e) 3x = (f) ,x 1 2= (g) x3 51 1-
(h) x4 2# #- (i) ,x x4 02 1 (j) ,x x2 4# $
(k) x4 1# #- (l) ,x x0 1# $ (m) ,x 221
= -
(n) No solutions (o) 0x = (p) 1x = (q) ,x 0 2= -
(r) No solutions (s) 31
x = ( t) 0, 6x =
Exercises 5.7
1. (a) (i) {all real x : x ! 0}, {all real y : y ! 0} (ii) no y -intercept
(iii) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(b) (i) {all real : },x x 0! {all real :y y 0! } (ii) no y -intercept
(iii) y
x-2 -1 2
2
1
-2
-1
1
Answer S1-S5.indd 563 7/31/09 1:36:17 PM
564 Maths In Focus Mathematics Preliminary Course
(c) (i) {all real :x x 1! - }, {all real : 0y y ! } (ii) 1
(iii) y
x-2 -1 2
2
1
-2
-1
1
(d) (i) {all real :x x 2! }, {all real : 0y y ! } (ii) 121
-
(iii) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(e) (i) {all real :x x 2! - }, {all real : 0y y ! } (ii) 61
(iii) y
x-2 -1 2
2
1
-2
-1
1
(f) (i) {all real :x x 3! }, {all real :y y 0! } (ii) 32
(iii) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(g) (i) {all real : 1x x ! }, {all real : 0y y ! } (ii) -4
(iii) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(h) (i) {all real : 1x x ! - }, {all real : 0y y ! } (ii) -2
(iii) y
x-4
-5
-3 -2-1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
Answer S1-S5.indd 564 7/31/09 1:36:18 PM
565ANSWERS
(i) (i) :x x21
all real !' 1 , {all real : 0y y ! } (ii) 32
-
(iii) y
x-2 -1 2
2
1
-2
-1
1
23
-
12
(j) (i) {all real :x x 2! - }, {all real :y y 0! } (ii) -3
(iii) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
2.
( )
f x x
xf x
2
2
odd function`
- =-
= -
= -
] g
3. (a) 1y91
## (b) 1y31# # (c) y2
21
21
# #- -
(d) 3y73
## (e) 2 y81
# #- -
4. (a) 1 3x# # (b) 1 4x# # (c) 6 0x# #-
(d) 1 4x# # (e) 1 2x# #
Exercises 5.8
1. (a) (i) y
x-3
3
3
-3
(ii) : , :x x y y3 3 3 3# # # #- -! "+ , (b) (i) y
x-4
4
4
-4
(ii) : , :x x y y4 4 4 4# # # #- -! "+ , (c) (i)
(2, 1)
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
y
x
Answer S1-S5.indd 565 7/31/09 1:36:18 PM
566 Maths In Focus Mathematics Preliminary Course
(ii) : 0 4 , : 1 3x x y y# # ## -! "+ , (d) (i)
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
y
x
(ii) : , :x x y y4 2 3 3# # # #- -! "+ , (e) (i)
-4 -3 -2 -1 2 3 4
2
1
3
4
5
-2
-1
(-2, 1)
1
y
x
(ii) : , :x x y y3 1 0 2# # # #- -! "+ , 2. (a) (i) Below x -axis
(ii) y
x-5 5
-5
(iii) : , :x x y y5 5 5 0# # # #- -! "+ , (b) (i) Above x -axis
(ii) y
x-1
1
1
(iii) : , :x x y y1 1 0 1# # # #-! "+ , (c) (i) Above x -axis
(ii) y
x-6
6
6
(iii) : , :x x y y6 6 0 6# # # #-! "+ , (d) (i) Below x -axis
(ii) y
x-8 8
-8
(iii) : , :x x y y8 8 8 0# # # #- -! "+ ,
Answer S1-S5.indd 566 7/31/09 1:36:19 PM
567ANSWERS
(e) (i) Below x -axis
(ii) y
x- 7
- 7
7
(iii) : , :x x y y7 7 7 0# # # #- -" #, - 3. (a) Radius 10, centre (0, 0) (b) Radius 5 , centre (0, 0)
(c) Radius 4, centre (4, 5) (d) Radius 7, centre (5, -6) (e) Radius 9, centre (0, 3)
4. (a) 16x y2 2+ =
(b) 6 4 12 0x x y y2 2- + - - =
(c) 2 10 17 0x x y y2 2+ + - + =
(d) 4 6 23 0x x y y2 2- + - - =
(e) 8 4 5 0x x y y2 2+ + - - =
(f) 4 3 0x y y2 2+ + + =
(g) 8 4 29 0x x y y2 2- + - - =
(h) 6 8 56 0x x y y2 2+ + + - =
(i) 4 1 0x x y2 2+ + - =
(j) 8 14 62 0x x y y2 2+ + + + =
Exercises 5.9
1. (a) {all real x }, {all real y } (b) {all real x }, {y: y = -4} (c) {x: x = 3}, {all real y } (d) {all real x }, { y : y $ -1 }
(e) {all real x }, {all real y } (f) {all real x }, : 1241
y y #' 1 (g) { : 8 8}, { : 8 8}x x y y# # # #- -
(h) {all real :t t 4! }, {all real ( ): ( )f t f t 0! }
(i) {all real : 0!z z }, {all real :g g 5!zz^ ^h h }
(j) {all real x }, { :y y 0$ }
2. (a) { x : 0x $ }, { y : y 0$ } (b) { x : x 2$ }, { y : y 0$ } (c) {all real x }, { y : y 0$ } (d) {all real x }, { y : y 2$ - }
(e) : 221
, { : }x x y y 0$ #-' 1
(f) {all real x }, { :y y 5# } (g) {all real x }, { : }y y 02
(h) {all real x }, { : }y y 01
(i) {all real :x x 0! }, {all real :y y 1! } (j) {all real :x x 0! }, {all real :y y 2! }
3. (a) ,x 0 5= (b) , ,x 3 1 2= - (c) , ,x 0 2 4=
(d) ,x 0 4!= (e) x 7!= 4. (a) x1 1# #-
(b) { : }x x1 1# #-
5. (a) { : , }x x x1 2# $- (b) { : , }t t t6 0# $-
6. (a) { y : y9 3# #- }
(b) { y : y0 9# # } (c) { y : y8 1# #- }
(d) :51
1y y# #' 1 (e) { y : 0 4y# # }
(f) { y : y1 15# #- } (g) { y : y1 0# #- }
(h) :y y1 8# #-" , (i) { y : 4 21y# #- }
(j) :y y61
64
# #-' 1 7. (a) {all real :x x 1! - }
(b) x -intercept: 0y =
01
3x
=+
0 3= This is impossible so there is no x -intercept (c) {all real :y y 0! }
8. (a) {all real :x x 0! } (b) {all real :y y 1!! }
9. (a) y
x-4 -3 -2 -1 2 3 4 5
10
5
15
20
25
-15
-10
-51
(b) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-8
-4
-21
(c) y
x-4 -3 -2 -1 2 3 4 5
10
5
15
20
25
-15
-10
-51
Answer S1-S5.indd 567 7/31/09 1:36:20 PM
568 Maths In Focus Mathematics Preliminary Course
(d) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-8
-4
-21
(e) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-8
-4
-21
(f) y
x-10 10
10
-10
(g) y
x-1
1
2
3
-1
1
10. (a) : : 0x x y y1$ $" ", , (b) y
x2 3
2
1
-11
11. y
x-1
4
3
2
1
5
6
-1 1
12. (a) (i) {all real x }, {all real y } (ii) All x (iii) None (b) (i) {all real x }, :y y 22 -" , (ii) x 02 (iii) x 01 (c) (i) {all real :x x 0! }, {all real : 0y y ! } (ii) None (iii) All 0x ! (d) (i) {all real x }, {all real y } (ii) All x (iii) None (e) (i) {all real x }, :y y 02" , (ii) All x (iii) None
13. (a) 2 2x ##- (b) (i) { x : 2 2x# #- }, { y: 0 2y# # } (ii) { x : 2 2x# #- }, { y: 2 0y# #- }
Exercises 5.10
1. (a) 21 (b) 10- (c) 8 (d) 3 (e) 3 (f) 75 (g) 0
(h) 6- (i) 41
(j) 1 (k) 7- (l) 3x x2 -
(m) 2 3 5x x3 + - (n) 3c2
Answer S1-S5.indd 568 7/31/09 1:36:21 PM
569ANSWERS
2. (a) Continuous (b) Discontinuous at 1x = - (c) Continuous (d) Continuous (e) Discontinuous at x 2!=
3. (a)
(b)
(c)
Exercises 5.11
1. (a) y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-4
-2
-11
(b) y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-4
-2
-11
(c) y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-4
-2
-11
(d) y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-4
-2
-11
Answer S1-S5.indd 569 7/31/09 1:36:21 PM
570 Maths In Focus Mathematics Preliminary Course
(e)
y = x +1
y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-4
-2
-11
(f)
y = 2x-3
y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-2
-11
(g)
x + y = 1
y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-2
-11
-4
(h)
3x - y - 6 = 0
y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-2
-11
-4
-5
-6
(i)
x + 2y - 2 = 0
y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-2
-11
-4
-5
-6
(j)
x-4 -3 -2 -1 2 3 41
y
2
1
3
4
5
6
-3
-2
-1
-4
-5
-6
x =12
Answer S1-S5.indd 570 7/31/09 1:36:22 PM
571ANSWERS
2. (a) x 32 - (b) y 2$ - (c) y x 1$ + (d) y x 422 -
(e) y 2x$
3. (a) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-2
-11
-4
-5
y = x2 - 1
(b)
-3 3
3
-3
y
x
(c) y
x-1 1
1
-1
(d)
x-3-4 -2 -1 2 3 4 51
y = x 2
y
1
2
3
4
5
-3
-2
-1
-4
-5
(e) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-8
-4
-21
y = x3
4. (a) y x3 21 - (b) y x 222 +
(c) x y 492 21+
(d) x y 812 22+
(e) ,x y5 21 2
5. (a) y
x-4 -3 -2 -1 2 3 4
3
1
2
4
5
-2
-11
Answer S1-S5.indd 571 7/31/09 1:36:23 PM
572 Maths In Focus Mathematics Preliminary Course
(b) y
x-4 -3 -2 -1 2 3 4
3
1
2
4
5
-2
-11
(c) y
x-4-5 -3 -2 -1 2 3 4
3
1
2
4
5
-2
-11
6. (a) y
x-4 -3 -1-2 2 3 4
3
1
2
4
5
6
-2
-3
-4
-11
(b) y
x-4 -3 -1-2 2 3 4
3
1
2
4
5
6
-2
-3
-4
-5
-6
-11
y = x - 3
(c) y
x-4 -3 -1-2 2 3 4
3
1
2
4
5
6
-2
-3
-4
-5
-11
y = 3x – 5
-6
(d) y
x-4 -3 -1-2 2 3 4
3
1
2
4
5
6
-2
-3
-4
-5
-6
-11
y = x + 1
y = 3 – x
(e) y
x-3 3
3
-3
y = 1
Answer S1-S5.indd 572 8/1/09 8:13:02 PM
573ANSWERS
(f) y
x-1-2 2
1
2
-2
x = – 1
(g) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-2
-11
-4
-5
y = x2
y = 4
(h) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-4
-21
-8
y = x3
y = 3
x = -2
(i) y
x-1 1
1
1
-1
(j) y
x-4 -3 -1-2 2 3 4
3
1
2
4
5
6
-2
-3
-4
-5
-6
-11
x - y = 2
x - y = -1
7. (a) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-2
-11
-4
-5
y = x2
Answer S1-S5.indd 573 7/31/09 1:36:25 PM
574 Maths In Focus Mathematics Preliminary Course
(b) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-4
-21
-8
y = x3
y = 1
(c) y
x1-2 2
2
-2
x = 1
(d)
1-1 2 3 4
1
2
-2
y
x
y =2x
(e)
-1 2 3-2-3-4 1 4
1
2
-1
-2y =
1x + 2
x
y
8. (a)
x2 3 4 51-1-3-4 -2
y
y = x2
y = 5
x = 2
3
2
1
4
5
-2
-1
-3
-4
-5
(b)
x2 3 41-1-3-4 -2
y
x = 3
y = -1
y = x - 2
3
2
1
4
5
6
-2
-1
-3
-4
-5
-6
Answer S1-S5.indd 574 7/31/09 1:36:26 PM
575ANSWERS
(c)
x2 3 41-1-3-4 -2
y
y = 2x + 1
2x - 3y = 6
3
2
1
4
5
6
-2
-1
-3
-4
-5
-6
(d)
-3 3
3
-3
x
x = -3
y = 2
y
(e)
x2 3 41-1-3-4 -2
y
y = 3
y = |x |
x = 2
3
2
1
4
5
6
-2
-1
-3
Test yourself 5
1. (a) f 2 6- =] g (b) f a a a3 42= - -] g (c) ,x 4 1= -
2. (a)
(b)
(c)
(d)
(e)
(f)
Answer S1-S5.indd 575 7/31/09 1:36:26 PM
576 Maths In Focus Mathematics Preliminary Course
(g)
(h)
3. (a) Domain: all real x ; range: y 641
$ -
(b) Domain: all real x ; range: all real y (c) Domain: 1 1;x# #- range: 1 1y# #- (d) Domain: 1 1;x# #- range: 0 1y# # (e) Domain: 1 1;x# #- range: 1 0y ##- (f) Domain: all real ;x 0! range: all real y 0! (g) Domain: all real x ; range: all real y (h) Domain: all real x ; range: y 0$
4. 15 5. (a) 4 (b) 5 (c) 9 (d) 3 (e) 2
6.
7.
8.
9.
10.
11. (a) y 3# (b) y x 22 + (c) ,y x y 02$ #-
12. (a) Domain: all real ,x 3! range: all real y 0!
(b)
13. (a)
(b) (i) ,x 2 4= - (ii) 4 2x 11- (iii) ,x x2 42 1 -
14. (a) 2 (b) 332
x = (c) 131
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
Answer S1-S5.indd 576 7/31/09 1:36:28 PM
577ANSWERS
17. (a) 4 (b) 52
(c) 121
- (d) 3
18.
19. (a) Domain: 2,x $ range: 0y $
(b)
20. (a) ( ) 3 1( )
( )
f x x xf x x x
x xf x
3 13 1
4 2
4 2
4 2
= + -
- = - + - -
= + -
=
] ]g g
So f x] g is even.
(b) ( )( ) ( )
( )( )
f x x xf x x x
x xx x
f x
3
3
3
3
= -
- = - - -
= - +
= - -
= -
] g
So f x] g is odd.
Challenge exercise 5
1. ,b32
3= -
2.
3.
4.
5. , ,f f f3 9 4 16 0 1= - = =] ] ]g g g
6. Domain: all real ;x 1!! range: ,y y1 02# -
7.
Answer S1-S5.indd 577 7/31/09 1:36:29 PM
578 Maths In Focus Mathematics Preliminary Course
8. Domain: ;x 0$ range: y 0$ 9. , ,x 0 3 2= -
10.
11. h h h2 1 0 3 0 1 2+ - - = - + - - = -] ] ] ]g g g g
12.
13.
14. ( ) ( )
( )
f a aa
f a
2 12 1
2 2
2
2
- = - -
= -
=
^ h
15. x4
1 41!=
16. (a) 2
2
x
xx
x
xx
xx
xx
x
31
32 3
31
32 6 1
32 7
32 7
31
RHS
LHS
`
= ++
=+
++
+
=+
+ +
=+
+
=
+
+= +
+
] g
(b) Domain: all real ;x 3! - range: all real y 2!
(c)
17.
18.
19. Domain: ;x 3$ range: 0y $ 20. Domain: x2 2# #-
21.
Answer S1-S5.indd 578 7/31/09 1:36:30 PM
579ANSWERS
Chapter 6: Trigonometry
Exercises 6.1
1. , ,cos sin tan135
1312
512
i i i= = =
2. , ,sin cot sec54
43
35
b b b= = =
3. , ,sin tan cos74
757
74
5b b b= = =
4. , ,cos tan cosecx x x95
556
56
9= = =
5. ,cos sin53
54
i i= =
6. , ,tan sec sin25
23
35
i i i= = =
7. ,cos tan635
35
1i i= =
8. ,tan sin751
1051
i i= =
9. (a) 2 (b) 45c
(c) , ,sin cos tan452
145
2
145 1c c c= = =
10. (a) 3 (b) , ,sin cos tan3021
3023
303
1c c c= = =
(c) , ,sin cos tan6023
6021
60 3c c c= = =
11. .sin cos67 23 0 92c c= = 12. .sec cosec82 8 7 19c c= =
13. .tan cot48 42 1 11c c= = 14. (a) 2 61 2 29cos sinorc c
(b) 0 (c) 0 (d) 1 (e) 2
15. 80x c= 16. 22y c= 17. 31p c= 18. 25b c=
19. 20t c= 20. 15k c=
Exercises 6.2
1. (a) 47c (b) 82c (c) 19c (d) 77c (e) 52c
2. (a) 47 13c l (b) 81 46c l (c) 19 26c l
(d) 76 37c l (e) 52 30c l
3. (a) 77.75c (b) 65.5c (c) 24.85c
(d) 68.35c (e) 82.517c
4. (a) 59 32c l (b) 72 14c l (c) 85 53c l
(d) 46 54c l (e) 73 13c l
5. (a) 0.635 (b) 0.697 (c) 0.339 (d) 0.928 (e) 1.393
6. (a) 17 20c l (b) 34 20c l (c) 34 12c l
(d) 46 34c l (e) 79 10c l
Exercises 6.3
1. (a) 6.3x = (b) 5.6y = (c) 3.9b = (d) 5.6x = (e) 2.9m = (f) 13.5x = (g) 10.0y = (h) 3.3p = (i) 5.1x = (j) 28.3t = (k) 3.3x cm= (l) 2.9x cm= (m) 20.7x cm= (n) 20.5x mm= (o) 4.4y m= (p) 20.6k cm= (q) 17.3h m= (r) 1.2d m= (s) 17.4x cm= (t) 163.2b m=
2. 1.6 m 3. 20.3 cm 4. 13.9 m
5. (a) 18.4 cm (b) 13.8 cm 6. 10 cm and 10.5 cm
7. 47.4 mm 8. 20.3 m 9. (a) 7.4 cm (b) 6.6 cm (c) 9.0 cm
10. (a) 6.8 cm (b) 6.5 cm 11. 38 cm
Exercises 6.4
1. (a) x 39 48c= l (b) 35 06ca = l (c) 37 59ci = l (d) 50 37ca = l (e) 38 54ca = l (f) 50 42cb = l (g) x 44 50c= l (h) 3 10 5ci = l (i) 29 43ca = l (j) 45 37ci = l (k) 57 43ca = l (l) 43 22ci = l (m) 37 38ci = l (n) 64 37ci = l (o) 66 16cb = l (p) 29 56ca = l (q) 54 37ci = l (r) 35 58ca = l (s) °59 2i = l (t) 56 59cc = l
2. 37 57c l 3. 22 14c l 4. 36 52c l 5. 50c
6. (a) 11.4 cm (b) 37 52c l 7. ,31 58 45 44c ca b= =l l
8. (a) 13 m (b) 65 17c l 9. (a) 11 19c l (b) 26 cm
10. 4.96 cm and 17.3 cm 11. (a) 12.9 m (b) 56 34c l
Exercises 6.5
1. (a)
100c
Boat
Beachhouse
North
AnswerS6.indd 579 7/31/09 11:07:53 AM
580 Maths In Focus Mathematics Preliminary Course
(b)
320c
Campsite
Jamie
North
(c)
200c
Seagull
Jetty
North
(d)
50c
Alistair
Bus stop
North
(e)
B Hill285c
Plane
North
(f)
12c
Dam
FarmhouseNorth
(g)
160cHouse
Mohammed
North
(h)
80c
Town
Mine shaft
North
(i)
349cSchool
YvonneNorth
AnswerS6.indd 580 7/31/09 11:07:54 AM
581ANSWERS
(j)
Island
Boat ramp
280c
North
2. (a) 248c (b) 145c (c) 080c (d) 337c (e) 180c
3. 080c 4. 210c 5. 160c 6. 10.4 m
7. 21 m 8. 126.9 m 9. 72 48c l
10. (a) 1056.5 km (b) 2265.8 km (c) 245c
11. 83.1 m 12. 1.8 km 13. 12 m 14. 242c 15. 035c
16. 9.2 m 17. 171 m 18. 9.8 km 19. 51 41c l 20. 2.6 m
21. 9 21c l 22. 1931.9 km 23. 34.6 m 24. 149c
25. 198 m 26. 4.8 km 27. 9.2 m 28. 217c
29. (a) 1.2 km (b) 7.2 km 30. (a) 13.1 m (b) 50 26c l
Exercises 6.6
1. (a) 2
3 1+ (b) 1 (c) 2 (d) 4 (e)
34 3
(f) 3
2 3
(g) 141
(h) 4
6 24
2 3 1+=
+^ h (i) 3
(j) 2 3- +^ h (k) 0 (l) 1 (m) 2 2 1-^ h (n) 6
(o) 131
(p) 3 2 2- (q) 2 3 (r) 21
- (s) 632
(t) 2 3
2-
2. (a) 2
3 2x = (b)
29 3
y = (c) 2 3p =
3. 60c 4. 2 m 5. 3 m 6. 3
10 3m
7. (a) 6 2 m (b) 4 m 8. 0.9 m 9. 3
5 3 3m
+^ h
10. 100 3 m
Exercises 6.7
1. (a) 1 st , 4 th (b) 1 st , 3 rd (c) 1 st , 2 nd (d) 2 nd , 4 th (e) 3 rd , 4 th (f) 2 nd , 3 rd (g) 3 rd (h) 3 rd (i) 2 nd (j) 4 th
2. (a) 3 rd (b) 21
- 3. (a) 4 th (b) 2
1-
4. (a) 2 nd (b) 3- 5. (a) 2 nd (b) 2
1
6. (a) 1 st (b) 23
7. (a) 1 (b) 2
1 (c) 3- (d)
21
(e) 21
- (f) 21
- (g) 23
(h) 3
1- (i)
23
- (j) 2
1-
8. (a) 2
1- (b)
23
- (c) 3 (d) 23
- (e) 23
-
(f) 3- (g) 21
(h) 3
1- (i)
2
1 (j)
2
1-
9. (a) 23
- (b) 3 (c) 23
(d) 21
(e) 21
- (f) 3
(g) 2
1 (h)
2
1 (i) −1 (j)
21
10. ,sin cos53
54
i i= - = -
11. ,cos tan733
33
4i i= - = -
12. ,cos cosecx x89
8589
= = -
13. , ,cosec cot tanx x x21
5
21
2221
= - = - = -
14. ,cos sinx x74
7 7474
5 74= - = -
15. ,tan sec65
4
65
9i i= - =
16. , ,tan sec cosecx x x355
38
55
8= = - = -
17. (a) 103
sinx = (b) 1091
,91
3cos tanx x= - = -
18. , ,cot sec cosec65
561
661
a a a= - = = -
19. ,sin cot1051
51
7i i= = -
20. (a) sin i (b) cos x (c) tan b (d) sin a- (e) tan i-
(f) sin i- (g) cos a (h) tan x-
Exercises 6.8
1. (a) ,20 29 159 31c ci = l l (b) ,120 240c ci = (c) ,135 315c ci = (d) ,60 120c ci = (e) ,150 330c ci = (f) ,30 330c ci =
(g) , , ,30 120 210 300 0 2 720c c c c c c# #i i= ] g (h) 70 , 110 , 190 , 230 , 310 , 350
0 3 1080c c c c c c
c c# #
i
i
=
] g
(i) , , ,30 150 210 330c c c ci = (j) , , , , , , , ,
, , ,15 45 75 105 135 165 195 225255 285 315 345c c c c c c c c
c c c c
i =
2. (a) 79 13! ci = l (b) ,30 150c ci = (c) ,45 135c ci = -
(d) ,60 120c ci = - - (e) ,150 30c ci = -
(f) ,30 150! !c ci =
(g) , , ,22 30 112 30 67 30 157 30c c c ci = - -l l l l
AnswerS6.indd 581 7/31/09 11:07:55 AM
582 Maths In Focus Mathematics Preliminary Course
(h) , , , , ,15 45 75 105 135 165! ! ! ! ! !c c c c c ci =
(i) ,135 45c ci = - (j) , , ,30 60 120 150! ! ! !c c c ci =
3.
4. 1-
5.
6. , ,x 0 180 360c c c= 7. 1- 8. 1
9. ,x 0 360c c=
10.
11. 0 12. 270x c= 13. , ,x 0 180 360c c c=
14. , ,x 0 180 360c c c= 15. ,x 270 90c c= -
16.
17.
Exercises 6.9
1. (a) cos i (b) tan i- (c) cos i (d) tan i (e) sec a-
2. (a) sin i (b) sec i (c) cosec x (d) cos 2 x (e) sin a
(f) cosec 2 x (g) sec 2 x (h) tan2 i (i) cosec5 2 i
(j) sin 2 x (k) 1 (l) sin cosi i
3. (a) 1cos xLHS 2= -
sin
sinx
x1 1
RHS
2
2
= - -
= -
=
So cos sinx x12 2- = -
(b) sec tanLHS i i= +
cos cossin
cossin
1
1
RHS
i i
i
i
i
= +
=+
=
So sec tancos
sin1i i
i
i+ =
+
(c) 3 3 tanLHS 2 a= +
( )tansec
cos
sin
3 13
3
1
3
RHS
2
2
2
2
a
a
a
a
= +
=
=
=-
=
So tansin
3 31
32
2a
a+ =
-
(d) sec tantan tan
cosec cot
x xx x
x x
11
LHS
RHS
2 2
2 2
2 2
= -
= + -
=
= -
=
So sec tan cosec cotx x x x2 2 2 2- = -
(e) sin cossin cos sin cossin cos sin sin cos cossin cos sin cos
sin sin cos cos sin cos
x xx x x xx x x x x xx x x x
x x x x x x
21 2
2 2
LHS
RHS
2 2
2 2
= -
= - -
= - - +
= - -
= - - +
=
3
2
]] ]] ^] ]
gg gg hg g
So sin cos sin sin cos cossin cos
x x x x x xx x2
2
2
2
- = - -
+
3] g
AnswerS6.indd 582 7/31/09 11:07:55 AM
583ANSWERS
(f) sin cossin sin
sin coscos sin
sin coscos
sin cossin
sincos
coscot sec
1 2
2
2
2
2
RHS
LHS
2
2
2
i i
i i
i i
i i
i i
i
i i
i
i
i
ii i
=- +
=+
= +
= +
= +
=
So cot secsin cossin sin
21 22
i ii i
i i+ =
- +
(g) cos cotsin cot
sinsincos
sin cos
90LHS
RHS
2
2
2#
c i i
i i
ii
i
i i
= -
=
=
=
=
] g
So 90cos cot sin cos2 c i i i i- =] g
(h) cosec cot cosec cotcosec cot
cot cot
x x x xx x
x x11
LHS
RHS
2 2
2 2
= + -
= -
= + -
=
=
] ]g g
So cosec cot cosec cotx x x x 1+ - =] ]g g
(i)
( )
cos
sin cos
cos cos
sin cos
sec sintan costan costan cos
1
1
1 11 1
LHS
RHS
2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
i
i i
i i
i i
i i
i i
i i
i i
=-
= -
= -
= + - -
= + - +
= +
=
So cos
sin costan cos
12
2 22 2
i
i ii i
-= +
(j) cosec
cotcos
cosec
cot cos cosec
cosec
cot cossin
cosec
cot cot
cosecsin
1
1
11
1
1
LHS
#
b
bb
b
b b b
b
b bb
b
b b
b
b
=+
-
=+ -
=
+ -
=+ -
=
=
tan cot
sec
cos
sin
sin
cos
sec
sin cos
sin cos
sec
sin cos
sec
seccos sin
cos
cos sin
sin
1
1
11
RHS
2 2
#
#
b b
b
b
b
b
b
b
b b
b b
b
b b
b
bb b
b
b b
b
=+
=
+
=+
=
=
=
=
LHS RHS=
So cosec
cotcos sin
1
b
bb b
+- =
4.
( )
cos sincos sincos sin
x y2 2
4 444 14
LHS
RHS
2 2
2 2
2 2
2 2
i i
i i
i i
= +
= +
= +
= +
=
=
=
] ]
]
g g
g
So 4x y2 2+ =
5.
( )
cos sincos sincos sin
x y9 9
81 818181 181
LHS
RHS
2 2
2 2
2 2
i i
i i
i i
= +
= +
= +
= +
=
=
=
2 2] ]
]
g g
g
So 81x y2 2+ =
Exercises 6.10
1. (a) 8.9x = (b) 9.4y cm= (c) 10.0a =
(d) 10.7b m= (e) 8.0d =
2. (a) 51 50ci = l (b) c61 23a = l (c) x 43 03c= l
(d) 87 04ca = l (e) 150 56ci = l
3. 126 56c l 4. (a) 13.5 mm (b) 25 mm
5. (a) 1.8 m (b) 2.7 m 6. 5.7 cm
7. (a) 10.3 m (b) 9.4 m 8. (a) 60 22c l (b) 57 9c l
9. (a) 14.1 cm (b) 15.6 cm
10. (a) 54.7 mm (b) 35.1 mm
AnswerS6.indd 583 8/7/09 12:40:20 PM
584 Maths In Focus Mathematics Preliminary Course
Exercises 6.11
1. (a) 5.8m = (b) 10.4b m= (c) 7.4h cm=
(d) 16.4n = (e) 9.3y =
2. (a) 54 19ci = l (b) 60 27ci = l (c) x 57 42c= l
(d) 131 31cb = l (e) 73 49ci = l
3. 32.94 mm 4. 11.2 cm and 12.9 cm
5. (a) 11.9 cm (b) 44 11c l (c) 82 13c l
6. ,XYZ XZY YXZ66 10 47 40c c+ + += = =l l
7. (a) 18.1 mm (b) 80 49c l 8. (a) 6.2 cm (b) 12.7 cm
9. 12.9 cm 10. (i) 11 cm (ii) 30c
Exercises 6.12
1. 12.5 cm and 4.7 cm 2. (a) 040c (b) 305c 3. 16.4 m
4. 103c 5. 1.97 m 6. 11c
7. (a) 1.21 km (b) 1 minute 8. 32 m 9. 107 m
10. (a) .sin
sinAC
101 365 8 42 29
c
c=
l
l (b) 74 50ci = l
11. 8.5h = 12. 7.7 km 13. 5.7 km and 5.4 km
14. 1841 km 15. 35.8 m 16. 89 52c l 17. 9.9 km
18. 163.5 km 19. 64.1 m 20. 3269 km
21. (a) 11.3 cm (b) 444 0c l 22. 141c
23. (a) 11.6 cm (b) 73 14c l
24. (a) 265.5 km (b) 346 33c l
25. (a) 35 5c l (b) (i) 4.5 m (ii) 0.55 m
Exercises 6.13
1. (a) 7.5 cm2 (b) 32.3 units2 (c) 9.9 mm2 (d) 30.2 units2 (e) 6.3 cm2
2. 2
15 3m2 3. 7.5 cm2 4. 15.5 cm2 5. 34.8 cm2
6. 1.2 m2 7. 42 cm2 8. 247.7 mm2
9. (a) 7.8 cm (b) 180.8 cm2
10. (a) 5.6 cm (b) 18.5 cm2 (c) 19.1 cm2
Test yourself 6
1. ,cos sin34
5
34
3i i= =
2. (a) cos x (b) 2 (c) cosec A
3. (a) 0.64 (b) 1.84 (c) 0.95
4. (a) 46 3ci = l (b) 73 23ci = l (c) 35 32ci = l
5.
( )
sincos
sinsin
sinsin sin
sinsin
12
12 1
12 1 1
2 12 2
LHS
RHS
2
2
i
i
i
i
i
i i
ii
=-
=-
-
=-
+ -
= +
= +
=
^] ]
hg g
2 2sin
cossin
12
So2
i
ii
-= +
6. b 40c= 7. (a) 2
1 (b)
23
- (c) 3-
8. ,x 120 240c c=
9.
,x 90 270c c=
10. 122 km 11. 5 3 12. (a) 6.3 cm (b) 8.7 m
13. (a) 65 5ci = l (b) 84 16ci = l (c) 39 47ci = l
14. 65.3 cm2 15. (a) ,x 60 120!! c c=
(b) , , ,x 15 105 75 165c c c c= - -
(c) , , ,x 0 180 30 150!c c c c= -
16. ,sin cot53
34
i i= - = 17. (a) 209c (b) 029c
18. (a) sin
sinAD
9920 39
c
c= (b) 8.5 m 19. 2951 km
Challenge exercise 6
1. 92 58c l 2. 50.2 km 3. 12.7x cm=
4. (a) .sin
sinAC
41 2125 3 39 53
c
c=
l
l (b) 25.2h cm= 5. 4.1 km
6. cos x- 7. 16 3 cm2 8. 2
1
9. , , ,x 22 30 112 30 202 30 292 30c c c c= l l l l 10. 75 45ci = l
11. 5.4 m 12. ,110 230c ci = 13. 6.43 km
14. 956
- 15. 31 m 16. sin
cos sin cos
coscos
sin cos
cossin cos
tan
1
1
LHS
RHS
2
2
i
i i i
ii
i i
i
i i
i
=-
+
=+
=+
= +
=
]]
gg
17. 4 5 0x y y2 2+ + - =
AnswerS6.indd 584 8/4/09 1:22:03 PM
585ANSWERS
Chapter 7: Linear functions
Exercises 7.1
1. (a) 5 (b) 10 (c) 13 2. (a) 13 (b) 65
(c) 85 (d) 52 2 13=
3. (a) 9.85 (b) 6.71 (c) 16.55 4. 12 units
5. , 134 128Two sides side= =
6. Show 85AB BC= =
7. Show points are 17 units from ,7 3-^ h 8. 3 , 9x yRadius units equation 2 2= + =
9. Distance of all points from ,0 0^ h is 11, equation
11x y2 2+ = 10. 3a = 11. a 6 2!= -
12. All 3 sides are 2 units. 13. ,a 10 2= -
14. , ,MQ NP QP MN37 20= = = = so parallelogram
15. 98BD AC= = 16. (a) ,AB AC BC40 4= = =
(b) OC OB 2= = 17. 2 101 18. 61 units
19. 29, 116, 145AB BC AC= = =
AB BC
AC
29 116145
2 2
2
+ = +
=
=
So triangle ABC is right angled (Pythagoras’ theorem)
20. , ,XY YZ XZ65 130 65= = =
Since XY YZ= , triangle XYZ is isosceles.
XY XZ
YZ
65 65130
2 2
2
+ = +=
=
So triangle XYZ is right angled. (Pythagoras’ theorem)
Problem
30.2
Exercises 7.2
1. (a) ,2 4^ h (b) ,1 1-^ h (c) ,2 1-^ h (d) ,3 2-^ h (e) ,1 1-^ h (f) ,3 2-^ h (g) ,3
21d n (h) ,1
21
1d n (i) ,
21
221d n (j) ,0 5
21d n
2. (a) ,a b9 3= = - (b) ,a b5 6= - =
(c) ,a b1 2= - = - (d) ,a b1 2= - = -
(e) ,a b6 1= =
3. ,2
3 30
24 4
0+ -
=- +
=] g
4. ,P Q 2 1= = -^ h 5. ,4 3^ h 6. 3x = is the vertical line through
midpoint ,3 2^ h . 7. Midpoint of , .AC BD 2
21
321
midpoint of= = d n
Diagonals bisect each other
8. 125,AC BD= = midpoint AC midpoint=
,BD 421
= - ;d n rectangle 9. ,8 13-^ h
10. (a) , , ,X Y Z21
321
21
21
1 1= - = =, ,d d ^n n h
(b) , ; ,XY BC XZ10 40 2 10234
= = = =
; ,AC YZ AB3422
2= = =
11. 4x y2 2+ = 12. 1x y2 2+ =
Exercises 7.3
1. (a) 2 (b) 131
(c) 131
- (d) 252
- (e) 32
(f) 81
-
(g) 421
- (h) 32
- (i) 241
(j) 2- 2. 21y1 =
3. 1.8x = 4. 9x = 5. (a) Show 53
m m1 2= =
(b) Lines are parallel .
y
-3 -2 -1 3 4 5 6 7
1
3
4
-2
-1 2(2, -1)
(-2, 1)
(7, 2)
(3, 4)
1
2
6. Gradient of 1AB CD21
gradient of= =
Gradient of 0BC ADgradient of= =
7. Gradient of 1AB CD31
gradient of= = -
Gradient of BC AD43
gradient of= =
Gradient of ,AC 521
= -
gradient of 21
BD = -
8. Gradient of 1,AC = gradient of BD 1= -
9. (a) Show AB BC AC2 2 2+ =
(b) Gradient of 45
,AB =
gradient of 54
BC = -
10. (a) , , ,F G1 2 421
= - =^ dh n (b) Gradient of FG BC
65
gradient of= =
Answer S7-S8.indd 585 7/31/09 12:08:49 PM
586 Maths In Focus Mathematics Preliminary Course
11. 4 3 11 0x y- - = 12. Gradient of ,2 4-^ h and , ,3 1 3 1gradient of- = -^ ^h h and ,5 5 3=^ h
13. 1 14. 0.93 15. 21 16. 50 12c l 17. 108 26c l
18. (a) 3 (b) 3
1 (c) 3-
19.
tantan
m
m
7 45 2
33
1
1180 45 2135
nd quadrant` c c
c
ii
i
=-
- - -
=-
= -
=
- =
= -
=
]
^
g
h
20. 3
2 3 3x =
+^ h
Exercises 7.4
1. (a) (i) 3 (ii) 5 (b) (i) 2 (ii) 1 (c) (i) 6 (ii) 7-
(d) (i) 1- (ii) 0 (e) (i) 4- (ii) 3 (f) (i) 1 (ii) 2-
(g) (i) 2- (ii) 6 (h) (i) 1- (ii) 1 (i) (i) 9 (ii) 0
(j) (i) 5 (ii) 2- 2. (a) (i) 2- (ii) 3 (b) (i) 5- (ii) 6-
(c) (i) 6 (ii) 1- (d) (i) 1 (ii) 4 (e) (i) 2- (ii) 21
(f) (i) 3 (ii) 121
(g) (i) 31
- (ii) 2- (h) (i) 54
- (ii) 2
(i) (i) 321
(ii) 21
- (j) (i) 132
(ii) 32
3. (a) 4 (b) 2-
(c) 0 (d) 2- (e) 1- (f) 3- (g) 2 (h) 41
- (i) 121
(j) 141
(k) 32
(l) 21
(m) 51
(n) 72
(o) 53
-
(p) 141
- (q) 15 (r) 121
- (s) 61
(t) 83
-
Exercises 7.5
1. (a) 4 1y x= - (b) y x3 4= - + (c) 5y x=
(d) 4 20y x= + (e) 3 3 0x y+ - = (f) x y4 3 12 0- - =
(g) 1y x= - (h) 5y x= + 2. 8 0x y+ - =
3. (a) 4 3 7 0x y- + = (b) 3 4 4 0x y- + =
(c) 4 5 13 0x y- + = (d) 3 4 25 0x y+ - =
(e) 2 2 0x y- + = 4. 4 8 0x y+ - = 5. (a) 3y =
(b) x 1= - 6. y x2= - 7. 3 4 12 0x y- - =
8. 2 3 0x y+ - = 9. 4x = - 10. 3 8 15 0x y+ - =
Exercises 7.6
1. (a) 3- (b) 31
(c) 43
(d) 121
(e) 1 (f) 65
- (g) 3
1
(h) 31
(i) 3
1 (j)
51
2. (a) 1 0x y- + = (b) 3 16 0x y- + = (c) 5 0x y+ - =
(d) 2 5 0x y+ + = (e) 2 4 0x y- + =
(f) 3 1 0x y+ - = (g) 3 4 13 0x y+ + =
3. 3m m1 2= = so parallel
4. m m51
5 11 2 #= - = - so perpendicular
5. 151
m m1 2= =
6. m m37
73
11 2# #= - = - 7. 32
k = - 8. 4m m1 2= =
9. AB CD m m 31 2< = =_ i and BC AD m m85
1 2< = = -d n 10. Gradient of : ,AC m
21
1 = gradient of BD : 2,m2 = -
m m21
2 11 2# #= - = -
11. (a) y x= - (b) 5 8 0x y- - = (c) 2 2 0x y+ + =
(d) 2 3 16 0x y- + = 12. 7 6 24 0x y+ - =
13. 3 0x y+ - = 14. 2 5 0x y- - =
15. 2 3 18 0x y- + =
Exercises 7.7
1. (a) ,2 4-^ h (b) ,1 3- -^ h (c) ,4 4^ h (d) ,0 2-^ h (e) ,5 1-^ h (f) ,1 1-^ h (g) ,3 7^ h (h) ,4 0^ h (i) ,41 26^ h (j) ,
191
197
-d n 2. Substitute ,3 4-^ h into both lines
3. , , ,2 5 4 1^ ^h h and ,1 1- -^ h 4. All lines intersect
at ,2 3-^ h 5. All lines meet at ,5 0-^ h 6. 11 6 0x y+ =
7. 5 6 27 0x y+ - = 8. x y4 7 23 0+ =+
9. 1 0x y+ - = 10. 2 2 0x y+ - =
11. 3 0x y+ - = 12. 2 3 0x y- - =
13. x y 1 0- + = 14. 3 2 0x y- + =
15. 3 7 0x y+ - = 16. 5 13 0x y+ + =
17. 27 5 76 0x y- - = 18. 3 14 0x y- - =
19. 2 1 0x y- - = 20. 3 11 0x y- - =
21. 5 17 0x y- + =
Exercises 7.8
1. (a) 2.6 (b) 1133
(c) 2.5 (d) 2.4 (e) 138
2. (a) 3.48 (b) 1.30 (c) 0.384 (d) 5.09 (e) 1.66
3. (a) 13
7 13 (b) 5 (c)
2054 205
(d) 13
5 26 (e)
1314 13
4. d d d 11 2 3= = =
Answer S7-S8.indd 586 7/31/09 12:08:50 PM
587ANSWERS
5. : , :A d B d5
14
5
3= =
-
Opposite signs so points lie on opposite sides of the line
6. , : , , :d d2 310
139 2
10
5- = =^ ^h h
Same signs so points lie on the same side of the line
7. , : , , :d d3 2 4 4 1 251
- = - =^ ^h h
Opposite signs so points lie on opposite sides of the line
8. 2d d1 2= = so the point is equidistant from both lines
9. , : , , :d d8 337
551 1
37
9- = =^ ^h h
Same signs so points lie on same side of the line
10. , : , , :d d3 25
64 1
5
7- =
-=^ ^h h
Opposite signs so points lie on opposite sides of the line
11. 4d d1 2= = so same distance 12. 5
8 5 units
13. 1 14. 4.2 15. 9 17x32
or= - 16. 3 1b41
121
or= -
17. m 1 1832
31
or= - -
18. Show distance between ,0 0^ h and the line is 5
19. Show distance between ,0 0^ h and the line is greater than 1
20. (a) , , , , ,3 1 374
71
2 2- -^ d ^h n h (b) , ,5
2 105
13 5119
26 34
Test yourself 7
1. 6.4 units 2. ,221
2-d n
3. (a) 151
- (b) 2 (c) 3
1 (d)
53
4. (a) 7 11 0x y- - = (b) 5 6 0x y+ - = (c) 3 2 0x y+ =
(d) 3 5 14 0x y+ - = (e) 3 3 0x y- - =
5. 5
6 5units
6. ,m m41
41 2= - = so m m 11 2 = -
` lines are perpendicular.
7. x -intercept 5, y -intercept 2-
8. (a) 2 1 0x y+ - = (b) 21
(c) 25
units
9. 5,m m1 2= = so lines are parallel 10. 3 4 0x y- =
11. ,1 1-^ h 12. ,a b6 1= =
13. Solving simultaneously, 4 0x y- - = and
2 1 0x y+ + = have point of intersection , .1 3-^ h
Substitute ,1 3-^ h in 5 3 14 0:x y- - =
5 1 3 3 14 0LHS RHS# #= - - - = =
point lies on 5 3 14 0:x y- - =
Substitute ,1 3-^ h in 3 2 9 0:x y- - =
3 1 2 3 9 0LHS RHS# #= - - - = =
point lies on 3 2 9 0:x y- - =
lines are concurrent
14. 0.499- 15. ,c 13 65= - - 16. 3y = 17. 154
x =
18. , : , , :d d2 113
86 3
13
2- =
-=^ ^h h
Opposite signs so points lie on opposite sides of the line
19. 4 0x y- - = 20. 3 7 14 0x y- - =
Challenge exercise 7
1. 2k = - 2. 3 3 3 0x y- - = 3. 10 10 81x y2 2+ =
4. Show AC and BD have the same midpoint ,1 2^ h and m m 1AC BD# = -
5. Show distance of all points from ,0 0^ h is 3; radius 3; equation 9x y2 2+ =
6. 13
4 13 7. 45 ; ( )OBA a b sides of isoscelesc+ D= =
8. 13
12 13 9. 113 12c l 10. 2 3 13 0x y+ + =
11. .angled
, , ;,
BC AC ABm m
18 61
so is isoscelesso is rightBC AC#
D
D
= = =
= -
12. ,3 5-^ h
13. ,a b2 3= = 14. 2 5 14 0x y+ + =
15. 3 3 2 3 0x y+ + - = 16. 6 0x y- + =
Chapter 8: Introduction to calculus
Exercises 8.1
1.
Answer S7-S8.indd 587 7/31/09 12:08:51 PM
588 Maths In Focus Mathematics Preliminary Course
2.
3.
4.
5.
6.
7.
8.
9.
10.
Exercises 8.2
1. Yes, 0x = 2. Yes, x x1= 3. No 4. Yes, 0x =
5. Yes, ,x x x x1 2= = 6. Yes, 0x = 7. Yes, x 3= -
8. Yes, 2x = 9. Yes, ,x 2 3= - 10. Yes, x1 01#-
11. Yes, ,x 90 270c c= 12. Yes, 0x = 13. No 14. No
15. Yes, x 3!=
Exercises 8.3
1. (a) 3 (b) 7- (c) 3 (d) 8 (e) 2 (f) 3- (g) 2 (h) 1- (i) 10 (j) 1-
2. (a) 2 4x x2 - - (b) 2 1x x3 + - (c) 7 1x- - (d) 4x x4 2- (e) 4 3x- + (f) 2 6x2 + (g) 2x- (h) 4x2 (i) 3 1x - (j) 2 9x x2 - +
Exercises 8.4
1. (a) 4.06 (b) 3.994 (c) 4
2. (a) 13.61 (b) 13.0601 (c) 12.9401 (d) 13 3. 6
4. (a) 2f x h x xh h2 2+ = + +] g
(b) ( ) ( )f x h f x x xh h xxh h
22
2 2 2
2
+ - = + + -
= +
(c) h
f x h f x
hxh h
hh x h
x h
2
2
2
2+ -=
+
=+
= +
] ]
]
g g
g
Answer S7-S8.indd 588 7/31/09 12:08:51 PM
589ANSWERS
(d) ( )
( )
lim
lim
f xh
f x h f x
x h
x
2
2
h
h
0
0
=+ -
= +
=
"
"
l] ]g g
5. (a) ( ) ( )( )
f x h x h x hx xh h x h
x xh h x h
2 7 32 2 7 7 32 4 2 7 7 3
2
2 2
2 2
+ = + - + +
= + + - - +
= + + - - +
] g
(b) ( ) ( ) ( )( )
f x h f x x xh h x hx x
x xh h x hx x
xh h h
2 4 2 7 7 32 7 3
2 4 2 7 7 32 7 3
4 2 7
2 2
2
2 2
2
2
+ - = + + - - +
- - +
= + + - - +
- + -= + -
(c)
h
f x h f x
hxh h h
hh x h
x h
4 2 7
4 2 7
4 2 7
2+ -=
+ -
=+ -
= + -
] ]
]
g g
g
(d) f x x4 7= -l] g
6. (a) f 2 11=] g (b) 2 5 11f h h h2+ = + +] g
(c) f h f h h2 2 52+ - = +] ]g g
(d)
h
f h f
hh h
hh h
h
2 2 5
5
5
2+ -=
+
=+
= +
] ]
]
g g
g
(e) f 2 5=l] g
7. (a) f 1 7- = -] g
(b) f h f h h h1 1 4 12 123 2- + - - = - +] ]g g (c) 12
8. (a) f 3 8=] g (b) f h f h h3 3 6 2+ - = +] ]g g (c) f 3 6=l] g
9. (a) f 1 13= -l] g (b) 17
10. (a) 2y x x2= +
Substitute ,x x y yd d+ +_ i :
( )
2
y y x x x xx x x x x x
y x xy x x x x
22 2 2
2 2Since
2
2 2
2
2
d d d
d d d
d d d d
+ = + + +
= + + + += +
= + +
] g
(b) x
y
xx x x x
x
x x x
x x
2 2
2 2
2 2
2
d
d
d
d d d
d
d d
d
=+ +
=+ +
= + +
] g
(c) 2 2dx
dyx= +
11. (a) 2 (b) 5 (c) 12- (d) 15 (e) 9-
12. (a) f x x2=l] g (b) 2 5dx
dyx= +
(c) f x x8 4= -l] g (d) 10 1dx
dyx= -
(e) 3dx
dyx2= (f) f x x6 52= +l] g
(g) 3 4 3dx
dyx x2= - + (h) x xf 6 2= -l] g
13. (a) 0.252 (b) 0.25 (c) 0.2498
14. (a) 0.04008- (b) 0.03992- (c) 0.04- 15. 1-
Exercises 8.5
1. (a) 1 (b) 5 (c) 2 3x + (d) 10 1x - (e) 3 4 7x x2 + - (f) 6 14 7x x2 - + (g) 12 4 5x x3 - + (h) 6 25 8x x x5 4 3- - (i) 10 12 2 2x x x4 2- + - (j) 40 63x x9 8-
2. (a) 4 1x + (b) 8 12x - (c) 2 x (d) 16 24x x3 - (e) 6 6 3x x2 + -
3. (a) x3
1- (b) x x2 3 2- (c) 3
86
xx
75- (d) 4 x (e)
41
(f) 2 2 2x x2 - +
4. f x x16 7= -l] g 5. 56-
6. 60 40 35 3dx
dyx x x9 7 4= - + - 7. 10 20
dtds
t= -
8. g x x20 5= - -l] g 9. 30dtdv
t= 10. 40 4dtdh
t= -
11. drd
rV
4 2r= 12. 3 13. (a) 5 (b) 5- (c) 4x =
14. (a) 12 (b) x 2!= 15. 18
Exercises 8.6
1. (a) 72 (b) 13- (c) 11 (d) 18- (e) 18 (f) 27
(g) 11 (h) 136 (i) 4- (j) 149
2. (a) 261
- (b) 251
(c) 201
(d) 431
- (e) 101
(f) 71
(g) 711
- (h) 201
(i) 81
- (j) 51
-
3. (a) (i) 6 (ii) 61
- (b) (i) 8 (ii) 81
-
(c) (i) 24 (ii) 241
- (d) (i) 8- (ii) 81
(e) (i) 11 (ii) 111
-
4. (a) 27 47 0x y- - = (b) 7 1 0x y- - = (c) 4 17 0x y+ + = (d) 36 47 0x y- - = (e) 44 82 0t v- - =
5. (a) x y24 555 0+ - = (b) 8 58 0x y- + = (c) 17 516 0x y- - = (d) 45 3153 0x y- + = (e) 2 9 0x y+ - =
6. (a) (i) 7 4 0x y- + = (ii) 7 78 0x y+ - = (b) (i) 10 36 0x y- + = (ii) 10 57 0x y+ - = (c) (i) 10 6 0x y+ - = (ii) 10 41 0x y- - = (d) (i) 2 2 0x y+ + = (ii) 2 19 0x y- - = (e) (i) 2 2 0x y- + = (ii) 2 9 0x y+ - =
7. x 3!= 8. (1, 2) and ( 1- , 0) 9. ( 5- , 7- )
Answer S7-S8.indd 589 7/31/09 12:08:52 PM
590 Maths In Focus Mathematics Preliminary Course
10. (0, 1) 11. (1, 2) 12. ,143
41615
- -d n 13. (a) (1, 1- ) (b) 6 7 0x y- - =
14. 10 7 0t h- - = 15. x y4 2 19 0- - =
Exercises 8.7
1. (a) 3x 4- - (b) 1.4x0.4 (c) 1.2x 0.8- (d) 2x21 -
1
(e) 2x x3 2+-
-
1
(f) 3x-
2
(g) 4x6-
1
(h) 2x-
3
2. (a) x
12
- (b) 2
5
x (c)
6
1
x56 (d)
10
x6- (e)
15
x4
(f) 2
1
x3- (g)
3
x7- (h)
23 x
(i) 3
2
x2-
(j) 2
1 12
x x3 5- -
3. 271
4. −3 5. 321
6. −3 7. 2 3 1x x+ +
8. 81
9. 3 16 8 0x y+ - = 10. 9 0x y- + =
11. (a) 2
1
x3- (b)
161
- 12. x y16 016+ - = 13. (9, 3)
14. 4x = 15. , , ,552
552
- -d dn n
Exercises 8.8
1. (a) 4 3x 3+] g (b) 6 2 1x 2-] g (c) 70 5 4x x2 6-^ h
(d) 48 8 3x 5+] g (e) 5 1 x 4- -] g (f) 135 5 9x 8+] g (g) x4 4-] g (h) 4 6 3 2 3x x x2 3 3
+ +^ ^h h (i) 8 2 5 5 1x x x2 7
+ + -] ^g h (j) 6 6 4 2 3x x x x5 6 2 5
- - +^ ^h h (k) 2x23
3 1--
1] g
(l) 2 4 x 3- -] g (m) 6 9x x2 4- -
-^ h (n) -
3x35
5 4+2] g
(o) -
4x x x x x43
3 14 1 72 3 2- + - +
1^ ^h h (p) 2 3 4
3
x +
(q) 5 2
5
x 2-
-] g (r) 1
8
x
x2 5
-+^ h (s)
7 3
2
x3-
-
(t) 2 4
5
x 3-
+] g (u) 4 3 1
3
x 3-
-] g (v) 2 2 7
27
x 10-
+] g
(w) 3 3
4 9 3
x x x
x x4 3 2
3 2
-- +
- +
^^
hh (x)
316 4 1x3 +
(y) 4 7
5
x 94 -] g
2. 9 3. 40 4. (4, 1) 5. ,x 2 121
= - 6. 8 7 0x y+ + =
Exercises 8.9
1. (a) 8 9x x3 2+ (b) 12 1x - (c) 30 21x +
(d) 72 16x x5 3- (e) 30 4x x4 -
(f) 5 2 1x x x 2+ +] ]g g (g) 8 9 1 3 2x x 4- -] ]g g (h) x x x3 16 7 4 23 - -] ]g g (i) 10 13 2 5x x 3+ +] ]g g (j) x x x x x x x
x x x x x
10 5 3 1 3 10 1
13 60 3 20 1
3 2 2 4 2 2 5
3 2 2 4
+ - + + + +
= + + - +
^ ^ ^ ^^ ^
h h h hh h
(k) x
xx
x
x
2 22
2 2
4 3-
-+ - =
-
-
(l) x
xx x2 1
2 5 32 1
5
2 1
112 2-
- ++
-= -
-]]
]gg
g
2. 26 3. 1264 4. 77
1
7
8+ = 5. 176
6. 10 9 0x y- - = 7. 69 129 0x y- - =
8. x3
6 30!=
- 9. 34 29 0x y- + =
Exercises 8.10
1. (a) x2 1
22-
-
] g (b) 5
15
x 2+] g (c) x
x x
x
x x
4
12
4
122 2
4 2
2 2
2 2
-
-=
-
-
^ ^^
h hh
(d) 5 1
16
x 2+] g (e) 14 14
x
x x
x
x4
2
3
- +=
- + (f)
3
11
x 2+] g
(g) 2
2
x x
x2 2
2
-
-
^ h (h) 2
6
x 2-
-
] g (i) x4 3
342-
-
] g (j) x3 1
142+
-
] g
(k) 3 7
3 6 7
x
x x2 2
2
-
- - -
^ h (l) x
x x
x
xx
2 3
4 12
2 3
342
2
2-
-=
-
-
] ]]
g gg
(m) x
x
5
182 2-
-
^ h (n) x
x x
x
x x
4
2 12
4
2 62
3 2
2
2
+
+=
+
+
] ]]
g gg
(o) x
x x
3
2 9 72
3 2
+
+ +
] g (p) 3 4
3 8 5
x
x x2
2
+
+ -
] g
(q) x x
x x x
1
2 4 12 2
4 3 2
- -
- - -
^ h (r)
-2 2
xx x x
52 5 5
+
+ - +
1 1
] ]g g
(s)
(t) 28
x
x x x
x
x
7 2
7 1 7
7 2
21 302 28 5
4 3
+
-=
+
- ++ - +
]] ] ]
]gg g g
g
(u) x
x x x x
x
x x2 5
15 2 5 3 4 6 3 4 2 5
2 5
3 3 4 4 33
6
3 4 5 2
4
4
-
- + - + -
=-
+ -
]] ] ] ]
]] ]
gg g g g
gg g
(v) x
x x
x
x
x1
1 2 1
3 1
2 1
3 53+
+ +
+
=+
+-3
] g
(w) x
x
x
x
x x
x
2 3
2 1
2 3
1
2 1 2 3
2 12 2-
-
-
-=
- -
- +2-
] ]g g
(x) x
x
x x
x x
x x
x x
9
1
9
9 1
1 9
9 24
2
2
2
2 3
2
-
+
-
- +=
+ -
- - -2-
]
]]
]g
gg
g
2. 81
3. 195
- 4. 0, 1x = 5. 9, 3x = -
6. 18 8 0x y- + = 7. 17 25 19 0x y- - =
x
x x x
x
x x
5 1
6 5 1 2 9 5 2 9
5 1
2 9 20 512
2 3
2
2
+
+ - - -=
+
- +
]] ] ]
]] ]
gg g g
gg g
Answer S7-S8.indd 590 7/31/09 12:08:52 PM
591ANSWERS
Test yourself 8
1. (a)
(b)
2. 10 3dx
dyx= - 3. (a) 42 9 2 8
dx
dyx x x5 2= - + -
(b) 2 1
11dx
dy
x 2=
+] g (c) 8( ) ( )dx
dyx x x9 2 4 4 22= + + -
(d) 40 5 5 (10 1)dx
dyx x x x x2 1 2 1 2 13 4 3= - + - = - -] ] ]g g g
(e) 2
5dx
dy x3
= (f) 10
dx
dy
x3= -
4. dtdv
t4 3= - 5. (a) 1 (b) 20 6. 10 7. 42
8. (a) 2x = - (b) 1x = (c) 2x =
9. (a) 32 4 9f x x 3= +l] ]g g (b) 3
5dx
dy
x 2= -
-] g
(c) dx
dyx x9 1 3 1= - -] ]g g (d)
4dx
dy
x2= -
(e) f xx5
145
=l] g
10. y
11. 9 7 0x y- - = 12. (2, 3) 13. drdS
r8r=
14. ( 2- , 71), (5, 272- ) 15. 4 6 0x y- - = 16. 3525
17. 9 18. x y12 4 0+ - = 19. ,51
dtds
u at t= + =
20. 107
Challenge exercise 8
1. ,f f1 3 1 36= - = -l] ]g g 2. 1813
-
3. ; , .dtdx
t t t8 300 0 37 53 2= + = -
4. , ,x y x y x y2 0 3 3 0 6 12 0+ = - - = - + =
5. , , , , 12 26 0, 12 170 0x y x y2 2 2 14- - + - = + + =^ ^h h 6.
43
7. 5 5 1 9 15 9 5 110 5 1 9 (4 13)x x x x
x x x
3 4 5 2
2 4+ - + - +
= + - -
] ] ] ]] ]g g g gg g
8. x
x x x
x
x4 9
2 4 9 16 2 1 4 9
4 9
2 12 17
8
4 3
5
-
- - + -
=-
- +
]] ] ]
]]
gg g g
gg
9. x12
6 2046
3 51! !=
-=
- 10. 2 25 0x y+ - =
11. 271
a = - 12. ,P 241
6161
= -d n 13. ,x31
31 13!
=
14. 21
15. , , ,x y Q PQ3 5 0 0 5 10- + = = =^ h
16. 8n = 17. , , x y11211
23 3
12 3 012 31- + =e o
18. , ,x21
121
153
= - 19. (a) ,x 90 270c c=
(b) y
x1
90c 180c 270c 360c
20. ,4 73- -^ h 21. 3 9 14 0x y- - = 22. x x
x
4 3 2
4 534 -
-] g
23. (a) ,x y x y16 32 1 0 4 2 1 0+ + = - - =
(b) 2m m21
1
1 2$ #= -
= -
So perpendicular
Answer S7-S8.indd 591 7/31/09 12:08:53 PM
592 Maths In Focus Mathematics Preliminary Course
24. 0, 2, 6x = 25. ,a b14 7= - = 26. 22
5 22
27. 121
p = 28. drdV
38 3r
= 29. 4k = 30. 4 0x y- - =
31. 4 13 0x y- - = 32. 481
- 33. , ,a b c1 2 4= - = =
34. 8 28S r rhr r r= +-
35. (a) 6 5 3 1 3 5x x x2 3- - -] ]g g (b) x x
x
3 2 1
5 64- +
+-
]]g
g
36. x6
4 13!=
37. (a) 7 80 0x y+ - =
(b) ,Q 471
12491
= -d n
Practice assessment task set 2
1. 0.77- 2. 1 3. 5 2 1 0x y+ - = 4. ,2 2-^ h 5. 0.309- 6. (a) 3 cm2 (b) , 1AC BD13 cm cm= =
7. 1; ,m m A43
68
1 121
1 2 #= - = - = -d n 8. x 15c=
9. 127
10.
11.
12. ’45 49c 13. Domain: all real ;x21
! range: all
real y 0!
14.
15.
16. sin4 i 17. 2 units 18. 8 15 0x y- + =
19. ,120 240c ci = 20. 132
- 21. 2 22. 11 565ca = l
23. .y 16 5= 24. 3 5 0x y+ - = 25. x132
31 1
26. 7 27. 3x = 28. 3-
29. Show perpendicular distance from ,0 0^ h to the line is 2 units, or solving simultaneous equations gives only one solution.
30. (a) ,g g2 1 3 6= - = -] ]g g
(b)
31. 3 4x x2 - 32. 2
1- 33. 17.5 m
34. ,x y2 17= - = - 35. (a) 7.0AB m= (b) 27.8 m 2
36. cos3 i 37. (a) 2 4 0x y- + = (b) , ,,P Q2 0 0 4-^ ^h h (c) 4 units 2
38. 127 m 39. 15 units 2 40. ( )
( )
f x x xx xf x
33
6 2
6 2
- = - - - -
= - -=
] ]g g
41. 16x x x x x1 1 18 1 2 12 22 2 2 2 2 33 4+ + + ++ =^ ^ ^ ^h h h h
Answer S7-S8.indd 592 7/31/09 12:08:54 PM
593ANSWERS
42. y431
9# #- 43. 3
x2-
44. (a) 3 4 0x y- - = (b) 2 0x y- - =
(c) 3 10 0x y+ + = (d) ,R 10 0= -^ h 45.
138
units 46. Domain: all ;x 4!- range: all y 0!
47. 2 7
1
x - 48. 4.9 km 49. 8 7 10x x 3- - -
50. 1
5
x 2+] g 51. 2 3x - 52. x x
x
x x
x
5
17 2
5
17 22 2+
- -=
+
+- ] g
53. 6 56 0x y+ - = 54. ,f f2 45 2 48- = - - =l] ]g g
55. ,a b2 9= = - 56. 7 5 9 0x y- + =
57. 47 109 0x y- + = 58. 0.25x = -
59. (a) domain: x21
$ range: y 0$
(b) domain: all real x 7!- range: all real y 0!
(c) domain: x2 2# #- range: y2 0# #-
60. (a) (1, 1) (b) 2 13 units (c) 121
-
(d) 3 2 5 0x y+ - =
61. (a) 62. (b), (d) 63. (a) 64. (c) 65. (c)
Chapter 9: The quadratic function
Exercises 9.1
1. Axis of symmetry 1,x = - minimum value 1-
2. Axis of symmetry 1.5,x = - minimum value 7.5-
3. Axis of symmetry 1.5,x = - minimum value 0.25-
4. Axis of symmetry 0,x = minimum value 4-
5. Axis of symmetry 83
,x = minimum point ,83
167d n
6. Axis of symmetry 1,x = maximum value 6-
7. Axis of symmetry 1,x = - maximum point ,1 7-^ h 8. Minimum value ,1- 2 solutions
9. Minimum value 3.75, no solutions
10. Minimum value 0, 1 solution
11. (a) ;x 3= - (-3, -12) (b) ;x 4= - (-4, 17)
(c) ; ,x 141
141
381
= d n (d) ; ,x 141
141
1341
= - - -d n (e) ; ,x 3 3 23= - - -^ h
12. (a) (i) x 1= - (ii) -3 (iii) (-1, -3)
(b) (i) 1x = (ii) 1 (iii) (1, 1)
13. (a) Minimum (-1, 0) (b) Minimum (4, -23) (c) Minimum (-2, -7) (d) Minimum (1, -1) (e) Minimum (2, -11)
(f) Minimum ,41
381
- -d n (g) Maximum (-1, 6)
(h) Maximum (2, 11)
(i) Maximum , 721
43d n
(j) Maximum (1, -3)
14. (a) (i) -2 (ii) Minimum 0 (iii) y
x-4 -3 -2 -1 2
3
2
1
4
5
-2
-3
-11
(b) (i) -1, 3 (ii) Minimum -4
(iii) y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
-3
-2
-11
-4
-5
Answer S7-S8.indd 593 7/31/09 12:08:55 PM
594 Maths In Focus Mathematics Preliminary Course
(c) (i) 5.83, 0.17 (ii) Minimum -8
(iii) y
x-4 -3 -2 -1 2 3 4 5 6
4
2
6
8
10
-6
-4
-21
-8
-10
(d) (i) -2, 0 (ii) Minimum -1
(iii) y
x-4 -3 -2 -1 2
2
1
3
4
5
-3
-2
-11
(e) (i) 3! (ii) Minimum -18
(iii) y
x-2-3-4 -1 1 2 5
1
2
-6
-8
-10
-12
-14
-16
-18
-4
-243
(f) (i) -1, 32
(ii) Minimum 21
12-
(iii) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-5
-6
-2
-1
-2
1
112
23
(g) (i) 1.65, -3.65 (ii) Maximum 7
(iii) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
7
6
-3
-2
-11
(h) (i) 1.3, -2.3 (ii) Maximum 341
(iii) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-2
-11
3 14
Answer S9-S10.indd 594 8/1/09 8:43:34 PM
595ANSWERS
(i) (i) 0.56, -3.56 (ii) Minimum 441
(iii) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-2
-11
4 14
(j) (i) 2.87, -0.87 (ii) Maximum 7
(iii) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
6
7
-3
-2
-11
15. (a) 4 (b) None
(c) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
6
7
-3
-2
-11
16. (a) None (b) 643
(c) y
x-4 -3 -2 -1 2 3 4 5
4
2
6
8
10
12
14
-3
-2
-11
17. (a) 387
- (b) None
(c) y
x-4 -3 -2 -1 2 3 4 5
1
2
-18
-16
-14
1
-12
-10
-8
-6
-4
-2
18. (a) y
x-4 -3 -2 -1 2 3 4 5
4
2
6
8
-3
-2
-11
(b) ,x x2 31 2 (c) x2 3# #
Answer S9-S10.indd 595 8/1/09 8:43:44 PM
596 Maths In Focus Mathematics Preliminary Course
19.
Graph is always above the x-axis so y 02 for all xx x3 2 4 02
` 2- + for all x
20. y
x-4 -3 -2 -1 2 3 4 5
4
2
6
8
-6
-4
-21
Graph is always above the x -axis so y 02 for all x x x 2 02` 2+ + for all x
21. y
x-4 -3 -2 -1 2 3 4 5
2
4
-18
-10
-12
-14
-16
-8
-6
-4
-21
Graph is always below the x -axis so y 01 for all x x x2 7 02` 1- + - for all x
22. y
x-4 -3 -2 -1 2 3 4 5
1
2
-5
-6
-7
-4
-3
-2
-11
Graph is always below the x -axis so y 01 for all x x x5 4 1 02` 1- + - for all x
Exercises 9.2
1. ,x x3 31 2- 2. 1 0n ##- 3. 0, 2a a# $
4. ,x x2 21 2- 5. y0 6# # 6. 0 2t 11
7. 4, 2x x1 2- 8. 3, 1p p# $- - 9. ,m m2 41 2
10. 3, 2x x# $- 11. h121
21 1 12. 4 5x ##-
13. 2 7k21# #- 14. ,q q 631 2 15. All real x
16. ,n n4 3# $- 17. x3 51 1- 18. t6 2# #-
19. ,y y31
51 2- 20. ,x x2 4# $-
Exercises 9.3
1. (a) 20 (b) -47 (c) -12 (d) 49 (e) 9 (f) -16 (g) 0 (h) 64 (i) 17 (j) 0
2. (a) 17 unequal real irrational roots (b) -39 no real roots (c) 1 unequal real rational roots (d) 0 equal real rational roots (e) 33 unequal real irrational roots (f) -16 no real roots (g) 49 unequal real rational roots (h) -116 no real roots (i) 1 unequal real rational roots (j) 48 unequal real irrational roots
3. 1p = 4. k 2!= 5. b87
# - 6. p 22 7. k 2121
2 -
8. a 3 02=
b ac4 1 4 3 7
830
2 2
1
- = - -
= -
] ] ]g g g
So x x3 7 02 2- + for all x
9. ,k k5 3$# - 10. k0 41 1 11. ,m m3 31 2-
12. ,k k1 1# $- 13. 3
p1
1 - 14. b0 221
# #
y
x-4 -3 -2 -1 2 3 4 5
4
2
6
8
-6
-4
-21
Answer S9-S10.indd 596 8/1/09 8:41:30 PM
597ANSWERS
15. ,p p2 6# $-
16. Solving simultaneously: 2 6y x= + (1)
3y x2= + (2)
Substitute (2) in (1):
x xx xb ac
3 2 62 3 04 2 4 1 3
160
2
2
2 2
2
+ = +
- - =
- = - - -
=
] ] ]g g g
So there are 2 points of intersection
17. 3 4 0x y+ - = (1) 5 3y x x2= + + (2) From (1): 3 4y x= - + (3) Substitute (2) in (3):
5 3 3 48 1 0
4 8 4680
x x xx x
b ac 1 1
2
2
2 2
2
+ + = - +
+ - =
- = - -
=
] ]g g
So there are 2 points of intersection
18. 4y x= - - (1) y x2= (2) Substitute (2) in (1):
44 0
4 1 415
0
x xx x
b ac 1 4
2
2
2 2
1
= - -
+ + =
- = -
= -
] ]g g
So there are no points of intersection
19. 5 2y x= - (1) 3 1y x x2= + - (2) Substitute (2) in (1):
x x xx x
b ac
3 1 5 22 1 0
4 2 4 1 10
2
2
2 2
+ - = -
- + =
- = - -
=
] ] ]g g g
So there is 1 point of intersection the line is a tangent to the parabola
20. 341
p =
21. (c) and (d)
Exercises 9.4
1. (a) , ,a b c1 2 6= = = - (b) , ,a b c2 11 15= = - = (c) , ,a b c1 1 2= = = - (d) , ,a b c1 7 18= = = (e) , ,a b c3 11 16= = - = - (f) , ,a b c4 17 11= = = (g) , ,a b c2 12 9= = - = - (h) , ,a b c3 8 2= = - = (i) , ,a b c1 10 24= - = = - (j) , ,a b c2 0 1= - = = -
2. , ,m p q2 5 2= = - =
3. 4 5 2 2 1 3 4x x x x x2 - + = - - + + +] ]g g
4. a x x b x cx x x
x x x xx x
2 3 21 2 3 1 2 17
3 2 6 2 172 9
RHS
RHS
2
2
= - + + - +
= - + + - +
= + - - + - +
= + +
=
] ] ]] ] ]
g g gg g g
true
5. , ,A B C1 5 6= = = - 6. , ,a b c2 1 1= = = -
7. , , .K L M1 6 7 5= = = 8. 12 5 2 3 65 2x x 2+ + - - -] ]g g
9. , ,a b c0 4 21= = - = -
10. (a) 5y x x2= - - (b) 3y x x2= -
(c) 2 3 7y x x2= - + (d) 4 9y x x2= + -
(e) 2 1y x x2= - - +
Exercises 9.5
1. (a) ,2 1a b ab+ = - = (b) . ,1 5 3a b ab+ = = - (c) . , .0 2 1 8a b ab+ = = - (d) ,7 1a b ab+ = - =
(e) ,232
1a b ab+ = =
2. (a) 3 (b) 6- (c) 0.5- (d) 21
3. (a) 3 10 0x x2 + - = (b) 4 21 0x x2 - - = (c) 5 4 0x x2 + + = (d) x x 08 112 - + = (e) 2 27 0x x2 - - =
4. 0.5m = 5. 32k = - 6. 4b = 7. 1k = 8. 13p =
9. 5k = - 10. m 3!= 11. 1k = - 12. ,n 1 3= -
13. ,p r2 7= = - 14. ,b c6 8= - = 15. ,a b0 1= = -
16. 11
`ab ba
= =
17. (a) 1k = - (b) 1, 0k = - (c) 1.8k = - (d) 3k =
(e) ,k k1 0# $-
18. (a) p 2 3!= (b) ,p p2 3 2 3# $-
(c) p2
3 3!=
19. (a) k 2= (b) 3k = - (c) 2k =
20. (a) 1m = (b) ,m m2
3 102
3 101 2
- +
(c) 3m = -
Exercises 9.6
1. (a) ,x 1 4= - - (b) 2, 5y = (c) 4, 2x = - (d) 1, 4n = - (e) 3, 5a = - (f) 3, 4p = (g) ,x 2 4= - (h) 5, 12k = (i) ,t 6 4= - (j) ,b 12 4= - -
Answer S9-S10.indd 597 8/1/09 6:52:33 PM
598 Maths In Focus Mathematics Preliminary Course
2. (a) 2, 3x = - (b) 2, 3x = (c) 4, 5x = (d) 3, 5x =
(e) 121
x = , 4
3. (a) x 3!= (b) ,y 2 2! != (c) x2
1 5!=
(d) . , . , . , .x 1 37 4 37 0 79 3 79= - - (e) ,a 2 2 6!= - -
4. (a) 0, 3x = (b) 1p = (c) 1x = (d) 1x = (e) 1, 3x =
5. 2,x 1! != 6. 1x = -
7. . , . , . , .x 2 19 0 46 1 93 0 52! ! ! !=
8. (a) , , ,x 0 90 180 360c c c c= (b) , ,x 90 180 270c c c= (c) , ,x 90 210 330c c c= (d) , , ,x 60 90 270 300c c c c= (e) , , ,x 0 180 270 360c c c c=
9. (a) , , , ,x 0 45 180 225 360c c c c c= (b) , ,x 0 180 360c c c= (c) , , , ,x 0 30 150 180 360c c c c c= (d) 45 , 60 ,135 , 120 , 225 , 240 , 315 , 300x c c c cc c c c= (e) 30 , 60 , 120 , 150 , 210 , 240 , 300 , 330x c c c c c c c c=
10.
( ) ( ) ( ) ( )
xx
x xx
x x
x xx x
33
25
3 33
23 5 3
3 2 5 33 5 3 2 0
2
2
# # #
+ ++
=
+ + ++
+ = +
+ + = +
+ - + + =
]] ]
] ]g
g gg g
Let 3u x= +
u ub ac
5 2 04 5 4 1 2
170
2
2 2
2
- + =
- = - -
=
] ] ]g g g
So u has 2 real irrational roots. x 3` + and so x has 2 real irrational roots
Test yourself 9
1. (a) x0 3# # (b) ,n n3 31 2- (c) 2 2y ##-
2. , ,a b c1 9 14= = - = 3. (a) 2x = (b) 3-
4. ab ac1 0
42 4 1 724
0positive definite
2
# #
`
2
1
D
=
= -
= - -
= -
2] g
5. (a) 6 (b) 3 (c) 2 (d) 18 (e) 30 6. ,x 132
31
=
7. (a) iv (b) ii (c) iii (d) ii (e) i
8.
( ) ( )
ab ac
1 04
3 4 1 47
0
2
2# #
1
1
D
= -
= -
= - - -
= -
x x4 3 02` 1- + - for all x
9. (a) 41
x = - (b) 681
10. 3 2 12 3 41x x2- + + -] ]g g 11. , ,x 30 150 270c c c=
12. (a) 341
k = (b) 1k = (c) 3k = (d) 3k = (e) 2k =
13. ,x21
3= - 14. m169
1 - 15. ,x 0 2=
16. (a) i (b) i (c) iii (d) i (e) ii
17. (a) iii (b) i (c) i (d) ii
18.
ac
kk
1
1
1
For reciprocal roots
LHS RHS
ba
ab
aa
=
=
=
= =
∴ roots are reciprocals for all x .
19. (a) 3 28 0x x2 + - = (b) 10 18 0x x2 - + =
20. 1, 3x =
Challenge exercise 9
1. k 4 02$D= -] g and a perfect square ∴ real, rational roots
2. y x x5 42= - + 3. , ,a b c4 3 7= = - = 4. x 2!=
5. 11 6. 2.3375n = - 7. .p 0 752 8. Show 0D =
9. x 1!=
10. 2, 19, 67 2, 13, 61A B C A B Cor= = - = = - = = -
11. 2
4 12
31
1
x x
xx x2 - -
+=
-+
+
12. ,k k2
1 212
1 21# $
- +
13. , ,x 30 90 150c c c= 14. ,x 12
3 5!=
15. , , ,x 60 90 270 300c c c c= 16. 23-
Chapter 10: Locus and the parabola
Exercises 10.1
1. A circle 2. A straight line parallel to the ladder.
3. An arc 4. A (parabolic) arc 5. A spiral
6. The straight line 2 2 | | 2x xor1 1 1-
7. A circle, centre the origin, radius 2 (equation 4x y2 2+ = i
8. lines y 1!= 9. lines x 5!= 10. line 2y =
11. Circle 1x y2 2+ = (centre origin, radius 1)
12. Circle, centre , ,1 2-^ h radius 4 13. 5y = -
Answer S9-S10.indd 598 8/1/09 6:52:45 PM
599ANSWERS
14. Circle, centre (1, 1), radius 3 15. x 7= - 16. 3x =
17. y 8!= 18. x 4!=
19. Circle, centre , ,2 4-^ h radius 6
20. Circle, centre , ,4 5-^ h radius 1
Exercises 10.2
1. x y 12 2+ = 2. 2 2 79 0x x y y2 2+ + + - =
3. 10 4 25 0x x y y2 2- + + + = 4. 8 6 13 0x y- + =
5. 12 26 1 0x y- - = 6. y x!=
7. 3 32 3 50 251 0x x y y2 2- + - + =
8. 5 102 5 58 154 0x x y y2 2- + + - =
9. 4 20 36 0x x y2 - + - = 10. 20 0x y2 - =
11. 8 32 0y x2 + - = 12. 2 8 7 0x x y2 - + - =
13. 12 0x y2 + = 14. 5 2 11 0x x y y2 2- + - - =
15. 3 4 0x x y y2 2+ + - - =
16. 2 17 0x x y y2 2+ + - - =
17. 2 4 2 6 47 0x x y y2 2+ + - + =
18. 2 2 2 4 27 0x x y y2 2+ + + + =
19. 3 4 25 0, 3 4 15 0x y x y+ + = + - =
20. ,x y x y12 5 14 0 12 5 12 0- - = - + =
21. x y2 3 5 5 0!- - =
22. 7 9 0, 7 5 0x y x y- + = + - =
23. 7 4 30 0, 32 56 35 0x y x y- - = + - =
24. 16 7 40 0xy x y- - + =
25. 6 3 12 9 0x x y y2 2- - - + =
Problem
,x y x y12 5 40 0 12 5 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) 16x y2 2+ = (b) 6 4 12 0x x y y2 2- + - - = (c) 2 10 17 0x x y y2 2+ + - + = (d) 4 6 23 0x x y y2 2- + - - = (e) 8 4 5 0x x y y2 2+ + - - = (f) 4 3 0x y y2 2+ + + = (g) 8 4 29 0x x y y2 2- + - - = (h) 6 8 56 0x x y y2 2+ + + - = (i) 4 1 0x x y2 2+ + - = (j) 8 14 62 0x x y y2 2+ + + + =
3. 18 8 96 0x x y y2 2- + + + =
4. 4 4 8 0x x y y2 2+ + + - = 5. 2 48 0x x y2 2- + - =
6. 6 16 69 0x x y y2 2+ + - + =
7. 10 4 27 0x x y y2 2- + + + = 8. 9 0x y2 2+ - =
9. 2 10 25 0x x y y2 2- + - + =
10. 12 2 1 0x x y y2 2+ + - + =
11. 8 6 22 0x x y y2 2- + - + = 12. 6 1 0x y y2 2+ + + =
13. (a) Radius 3, centre (2, 1) (b) Radius 5, centre (−4, 2) (c) Radius 1, centre (0, 1) (d) Radius 6, centre (5, −3) (e) Radius 1, centre (−1, 1) (f) Radius 6, centre (6, 0) (g) Radius 5, centre (−3, 4) (h) Radius 8, centre (−10, 2) (i) Radius 5, centre (7, −1) (j) Radius 10 , centre (−1, −2)
14. Centre ,3 1-^ h , radius 4 15. Centre ,2 5^ h , radius 5
16. Centre ,1 6- -^ h , radius 7 17. Centre (4, 7), radius 8
18. Centre ,121
1-d n , radius 221
19.
20. Show perpendicular distance from the line to ,4 2-^ h is 5 units, or solve simultaneous equations.
21. (a) Both circles have centre ,1 2-^ h (b) 1 unit
22. 2 2 23 0x x y y2 2+ + + - = 23. 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) 2 6 15 0x x y y2 2+ + - - = (b) , ,,2 7 1 2- -^ ^h h (c) ,Z 1 8= -^ h (d) m m
31
3
1
zx yx# #= -
= -
ZXY 90` c+ =
27. (a) 4 units (b) 4 10 13 0x x y y2 2- + + + =
Answer S9-S10.indd 599 8/1/09 6:52:57 PM
600 Maths In Focus Mathematics Preliminary Course
Exercises 10.4
1. (a) 20x y2 = (b) 36x y2 = (c) 4x y2 = (d) 16x y2 =
(e) 40x y2 = (f) 12x y2 = (g) 24x y2 = (h) 44x y2 =
(i) 8x y2 = (j) 48x y2 =
2. (a) x y42 = - (b) 12x y2 = - (c) 16x y2 = -
(d) 28x y2 = - (e) 24x y2 = - (f) 36x y2 = -
(g) 32x y2 = - (h) 8x y2 = - (i) 60x y2 = -
(j) 52x y2 = -
3. (a) (i) (0, 1) (ii) y 1= - (b) (i) (0, 7) (ii) 7y = - (c) (i) (0, 4) (ii) 4y = - (d) (i) (0, 9) (ii) 9y = - (e) (i) (0, 10) (ii) 10y = - (f) (i) (0, 11) (ii) 11y = -
(g) (i) (0, 3) (ii) 3y = - (h) (i) (0, 121c m (ii) 1y
21
= -
(i) (i) 0, 221c m (ii) 2y
21
= - (j) (i) 0, 343c m
(ii) 3y43
= -
4. (a) (i) (0, −1) (ii) 1y = (b) (i) (0, −6) (ii) 6y = (c) (i) (0, −2) (ii) 2y = (d) (i) (0, −12) (ii) 12y = (e) (i) (0, −5) (ii) 5y = (f) (i) (0, −4) (ii) 4y = (g) (i) (0, −8) (ii) 8y = (h) (i) (0, −10) (ii) 10y =
(i) (i) 0,21
-c m (ii) 21
y = (j) (i) 0, 521
-c m (ii) 521
y =
5. (a) 28x y2 = (b) 44x y2 = (c) 24x y2 = - (d) 8x y2 = (e) 12x y2
!= (f) 32x y2!=
(g) 32x y2 = (h) 71
x y2 =
6. (a) Focus , ,0 2^ h directrix 2,y = - focal length 2 (b) Focus , ,0 6^ h directrix 6,y = - focal length 6
(c) Focus , ,0 3-^ h directrix 3,y = focal length 3
(d) Focus , ,021d n directrix ,y
21
= - focal length 21
(e) Focus , ,0 143
-d n directrix 143
,y = focal length 143
(f) Focus , ,081d n directrix ,y
81
= - focal length 81
7. 2y = 8. ,4 4^ h 9. ,X 121
83
= - -d n 10. 4, 2-^ h and 4, 2- -^ h ; 8 units
11. (a) 12x y2 = - (b) 3y = (c) 3331
units
12. (a) Substitute the point into the equation.
(b) 3 4 3 0x y+ - = (c) ,243
-d n 13. (a) 4 2 0x y- + = (b) 0, 1^ h does not lie on the line
(c) 4 2 1 0x x y y2 2- + - + = (d) Substitute ,0 1^ h into the equation of the circle.
14. (a) Substitute Q into the equation of the parabola. (b) 1 2 2 0q x qy aq2 - - + =_ i (c) Equation of latus rectum is .y a= Solving with 4x ay2 = gives two endpoints , , ,A a a B a a2 2-^ ^h h . Length of 4AB a= .
Exercises 10.5
1. (a) 8y x2 = (b) 20y x2 = (c) 56y x2 = (d) 36y x2 = (e) 32y x2 = (f) 24y x2 = (g) 28y x2 = (h) 12y x2 = (i) 16y x2 = (j) 4y x2 =
2. (a) y x362 = - (b) 16y x2 = - (c) 40y x2 = - (d) y x242 = - (e) 8y x2 = - (f) 48y x2 = - (g) 44y x2 = - (h) y x202 = - (i) 12y x2 = - (j) 28y x2 = -
3. (a) (i) (2, 0) (ii) x 2= - (b) (i) (3, 0) (ii) 3x = - (c) (i) (4, 0) (ii) 4x = - (d) (i) (1, 0) (ii) 1x = - (e) (i) (7, 0) (ii) 7x = - (f) (i) (8, 0) (ii) 8x = - (g) (i) (6, 0) (ii) 6x = - (h) (i) (9, 0) (ii) x 9= -
(i) (i) 41
, 0c m (ii) 41
x = - (j) (i) 421
, 0c m (ii) 4x21
= -
4. (a) (i) (−2, 0) (ii) 2x = (b) (i) (−3, 0) (ii) 3x = (c) (i) (−7, 0) (ii) 7x = (d) (i) (−1, 0) (ii) 1x = (e) (i) (−6, 0) (ii) 6x = (f) (i) (−13, 0) (ii) 13x =
(g) (i) (−15, 0) (ii) 15x = (h) (i) 21
, 0-c m (ii) 21
x =
(i) (i) 621
, 0-c m (ii) 621
x = (j) (i) 141
, 0-c m (ii) 141
x =
5. (a) 20y x2 = (b) 4y x2 = (c) 16y x2 = - (d) 12y x2 =
(e) 36y x2!= (f) 8y x2
!= (g) 12y x2 = (h) 21
y x2 =
6. (a) Focus , ,2 0^ h directrix 2,x = - focal length 2
(b) Focus , ,1 0^ h directrix 1,x = - focal length 1
(c) Focus , ,3 0-^ h directrix 3,x = focal length 3
(d) Focus , ,121
0d n directrix 121
,x = - focal length 121
(e) Focus , ,141
0-d n directrix 141
,x = focal length 141
(f) Focus , ,121
0d n directrix ,x121
= - focal length 121
7. 4x = (latus rectum) 8. , , ,,12 3 6 3 6-^ ^h h 9. , ,,9 6 81 18-^ ^h h 10. (a) 5 12 25 0x y- - = (b) ,5 4
61
- -d n (c) 10125
units 2
(d) 4132
units (e) 11.7 units 2
Exercises 10.6
1. (a) yx 3 8 32- = +] ^g h (b) 5 4 6x y2- = +] ^g h (c) x y1 4 32 = +-] ^g h (d) 12x y4 32- = - -] ^g h (e) 6 8 7x y2- = +] ^g h (f) 16x y7 32+ = - -] ^g h (g) 4x y2 52- = - -] ^g h (h) 9 12 6x y2+ = +] ^g h (i) 1 8 1x y2+ = +] ^g h (j) x y3 4 22- = - -] ^g h
2. (a) 4 4 4y x2- = +^ ]h g (b) 1 8 2y x2- = +^ ]h g (c) y x2 12 12+ = +^ ]h g (d) 10 4 29y x2- = - -^ ]h g (e) 3 16 1y x2+ = - -^ ]h g (f) 6 8 4y x2- = +^ ]h g (g) 5 24 2y x2+ = - -^ ]h g (h) 12 4 36y x2+ = +^ ]h g (i) y x2 20 12- = - -^ ]h g (j) 4 8 2y x2+ = - -^ ]h g
AnswerS10.indd 600 7/31/09 2:55:56 PM
601ANSWERS
3. (a) 2 8 9 0x x y2 + - + = (b) x x y8 4 16 02 + - + =
(c) 4 8 12 0x x y2 - - - = (d) 6 8 41 0x x y2 - - + =
(e) 4 16 20 0x x y2 + - + = (f) 2 16 1 0x x y2 + + + =
(g) x x y8 20 4 022 - + - = (h) 10 8 1 0x x y2 + + + =
(i) 6 12 45 0x x y2 + + + = (j) x y4 24 02 + + =
(k) 6 12 3 0y y x2 - - - = (l) 8 4 8 0y y x2 - - + =
(m) 8 32 0y x2 - + = (n) y y x4 16 0122 + - =-
(o) 2 8 7 0y y x2 + - - = (p) y y x8 12 042 + + + =
(q) 2 4 11 0y y x2 - + - = (r) 6 16 25 0y y x2 - + + =
(s) 4 2 5 0y y x2 - + + = (t) y y x2 2 062 - + =-
4. (a) (i) (3, −2) (ii) 4y = - (b) (i) (1, 1) (ii) y 3= -
(c) (i) (−2, 0) (ii) 2y = - (d) (i) (4, 2) (ii) 4y = -
(e) (i) (−5, −1) (ii) 5y = - (f) (i) (3, 1) (ii) 3y =
(g) (i) (−1, 0) (ii) 4y = (h) (i) (2, 0) (ii) 2y =
(i) (i) (4, −2) (ii) 4y = (j) (i) (−2, −3) (ii) 5y =
5. (a) (i) (0, −1) (ii) 2x = - (b) (i) (2, 4) (ii) 4x = - (c) (i) (0, 3) (ii) 4x = - (d) (i) (3, −2) (ii) x 5= - (e) (i) (7, 1) (ii) 5x = - (f) (i) (1, −5) (ii) 5x = (g) (i) (11, −7) (ii) 13x = (h) (i) (−3, 6) (ii) 7x =
(i) (i) (−7, 2) (ii) 9x = (j) (i) 1021
, 3- -c m (ii) 921
x =
6. 12 36 0x y2 - + =
7. ,x x y x x y4 8 4 0 4 8 12 02 2+ - - = + + + =
8. 2 4 19 0x x y2 - - - = 9. 12 12 12 0y y x2 - + + =
10. 2 16 1 0x x y2 - - + = 11. 2 28 29 0x x y2 - - + =
12. 4 24 44 0y y x2 + + - = 13. 6 32 9 0y y x2 - - + =
14. 6 8 15 0x x y2 - + - = 15. 2 16 49 0y y x2 + - + =
16. 6 4 7 0x x y2 + + - = 17. 4 12 8 0x x y2 - - - =
18. 2 16 95 0y y x2 + + - =
19. (a) Vertex ,2 1-^ h , focus ,2 3-^ h , directrix 1y = -
(b) Vertex ,3 2^ h , focus ,3 5^ h , directrix 1y = -
(c) Vertex ,1 1-^ h , focus ,1 2-^ h , directrix 0y =
(d) Vertex ,3 4^ h , focus ,7 4^ h , directrix x 1= -
(e) Vertex ,0 2-^ h , focus ,6 2-^ h , directrix 6x = -
(f) Vertex ,5 0-^ h , focus ,7 0-^ h , directrix x 3= -
20. Vertex ,1 4-^ h , focus 1, 3- -^ h , directrix 11,y = axis 1,x = - maximum value 4
21. 4 8 12 0x x y2 - - + = or 4 8 36 0x x y2 - + - =
22. (a) 8 9 72 0x y2 + - = (b) , , y0 73223
8329
=d n
23. (a)
(b) , , y1 843
941
- - = -d n
24. 4 8 20 0x x y2 + + - = 25. 0.3 m
Exercises 10.7
1. 31
m = 2. m 4= - 3. m 1= - 4. 21
m =
5. dx
dyx= 6. 2 0x y- - = 7. 2 12 0x y- + =
8. 6 0, 18 0x y x y+ - = - - =
9. 2 2 0, 2 9 0x y x y- - = + - =
10. ,,x y M 187
21
4 8 0+ - = = d n 11. , ,x y P9 0 18 27+ - = = -^ h 12. 33, 60.5Q = ^ h 13. , ,x y x y4 144 0 4 2 9 0+ + = + + = , .18 40 5-^ h ; show
the point lies on the parabola by substituting it into the equation of the parabola
14. , ,x y R4 0 4 0- - = = ^ h 15. (a) Substitute P into the equation of the parabola
(b) 2 0x py p p3+ - - = (c) Substitute 0, 1^ h into the equation of the normal.
( )Since 0, 1 0
p p pp pp p
p p
0 2 00
1
3
3
2
2!
+ - - =
= +
= +
+ =
Test yourself 10
1. 8 6 29 0x y+ - = 2. 4 8 4 0x x y2 - - - =
3. Centre , ,3 1^ h radius 4 4. (a) ,1 3-^ h (b) 4, 3-^ h 5. 25x y2 2+ = 6. (a) 2y = (b) ,0 2-^ h 7. 3 10 0x x y y2 2+ + - - = 8. 8 16 16 0x x y2 - + - =
9. (a) (i) ,1 1^ h (ii) ,1 2^ h (b) 0y =
10. 2 3 6 0x y+ + = 11. 14 units
12. 24y x2 = - 13. 8 16 0x y2 - + =
14. ,x y x y4 3 16 0 4 3 14 0- - = - + =
15. ,y x y x= = - 16. 20y x2 = 17. (a) 21
- (b) 2
18. (a) 4 72 0x y- + = (b) ,9 2041d n
19. Sub ,0 4^ h: 7 0 3 4 12 0LHS RHS# #= - + = =
20. , 792
-d n 21. (a) 3 0x y- - = (b) ,R 0 3= -^ h
(c) ,F FP FR0 3 6= = =^ h
AnswerS10.indd 601 7/31/09 2:55:57 PM
602 Maths In Focus Mathematics Preliminary Course
Challenge exercise 10
1. (a) 8 6 29 0x y+ - = (b) Midpoint of AB lies on line; m m 11 2 = -
2. (a) 2 6 15 0x x y y2 2- + - - = (b) Put 0y = into equation
3. ,221
3-d n 4. (a) ;x y x y4 2 9 0 2 24 0- + = + - =
(b) 1m m1 2 = - (c) , .X 3 10 5= ^ h (d) 3 4 8 0;x y- + = focus ,0 2^ h lies on the line
5. ,0 0^ h 6. (a) ;x y x y2 4 1 0 2 4 0- - = + + =
(b) Point lies on line 1y = -
7. 2 4 2y x x2= - + - 8. 3 2 0x y+ + =
9.
10. (a) 4 10 21 0x x y y2 2+ + - + =
(b) 2 5 8;x y2 2+ + - =] ^g h centre , ;2 5-^ h 28 2radius = =
11. 3
2 3-
12. (a) 4 16 52 0y y x2 + - + = (b) 2 6 0x y- - =
13. 4 2 units 14. 2 2 0x y y2 2+ - - =
15. 696 mm from the vertex
16. ;x y x y141 127 32 0 219 23 58 0+ + = + + =
Practice assessment task set 3
1. ≤ , ≥m m2 3 2. 4 3 16 0x y+ - =
3. Centre , ,3 5-^ h radius 7
4. (a) 32
(b) 31
- (c) 191
5. Focus , ,0 2-^ h directrix 2y =
6. 5x = - or 6- 7. 1k = -
8. ,x y x y3 4 14 0 3 4 16 0- - = - + =
9. Vertex ,4 17- -^ h , focus , .4 16 75- -^ h 10. ,x 0 3= 11. 2 2 0x y+ + = 12. b 2$ -
13. 16,x y2 2+ = circle centre ,0 0^ h and radius 4
14. 4 6 12 0x x y y2 2+ + + - =
15. x x y y3 6 17 02 2- + - - =
16. 0.75- 17. 5 54 5 20 79 0x x y y2 2- + + - =
18. , ,a b c2 1 0= = =
19. 9 x
x2
--
20. 4 16 20 0x x y2 - - + =
21. and (given)
(vertically opposite angles)
AC BC CD CE
CDAC
CEBC
ACB ECD
`
+ +
= =
=
=
since two sides are in proportion and their included angles are equal, Δ ABC is similar to Δ CDE 5.3 cmy =
22. 4 0x y- - =
23. 2 16 15 0x x y2 + - - = 24. ,x 0 2=
25. 04
1 4( 1)( 9)35
0
ab ac2
2
1
1
D = -
= - - -
= -
Since a 01 and 0, 9 0x x21 1D - + - for all x
26. ( )( ) ( )x x x8 3 2 5 3 2 51 3 4+ + +- ( )x x30 7 2 5 3= + +] g
27. sec cosecx x
28. Centre , ,5 3-^ h radius 2
29.
( )( )
ab ac0
41 4 1 3
110
2
2
2
1
D = -
= - -
= -
] g
Since 0a 2 and ,01D x x 3 02 2- + for all x
30. 1k = 31. 3 2 9 0x y+ - =
32. (a) 217 km (b) 153c
33. , ,a b c3 18 34= = - = - 34. ,x x4 32 1
35. ’95 44ci =
36. 361 0 and a perfect squareT 2= ^ h 37. 2 9 0x y+ + = 38. k 3#
AnswerS10.indd 602 7/31/09 2:55:58 PM
603ANSWERS
39. 5 4 41 0x y- - = 40. 22
3 6 10 3 3 5- + -
41. 4.9 , 11.1x ycm cm= = 42. 1x = 43. 8.25 units
44. 4.5 m 45. 2187128
46. °, °, °, °x 60 120 240 300=
47. 2 3 3 0x y+ - = 48. ,y 131
21
= - 49. 162c
50. °, °, °, °x 45 135 225 315=
51. 1, 2 or , 4x y x y41
41
= - = = - =
52. a b a ab b2 2 42 2+- +] ^g h 53. 43x = 54. 311
-
55. 1.8 units 56. tan i
57. 8 2 5 ( 1) 2( 1)x x x x2 3 2 4+ - + -] g ( ) ( )x x x2 1 9 20 12 3 2= - + -
58. 41
59. 2 3 25 0x x y y2 2+ + - - =
60. Focus (2, 1), directrix 5y =
61. 2 36 0x y- - =
62. Distance from centre ,0 0^ h to line is
| |d
a b
ax by c
1040
4radius
line is tangent
2 2
1 1
`
=+
+ +
=
=
=
63. k 221
= -
64. ,x x2 21 2-
65. Radius 3; 9x y2 2+ =
66. , ,a b c3 14 9= = - =
67. Domain: all real x ; range: y 3$ -
68. )|ED(
,
ACB ECDABC CED
AC CDABC CDEby AAS
vertically opposite anglesalternate angles
given
AB|
`
+ ++ +
/D D
=
=
=
^^
hh
69. 46 m 2 70. 3 0x y+ - =
71. x x x12 36 62 2- + = -] g 72. . , .y y2 5 6 5$ # -
73. (a) x y9 16 0- + = (b) x y9 20 0+ + =
(c) ,Q 20 0= -^ h 74. (a) 8 129 0x y- + = (b) ,R 7
81
17641
= d n 75. , ,a b c1 3 1= = - = -
76. (c) 77. (d) 78. (b) 79. (a) 80. (c) 81. (c)
AnswerS10.indd 603 8/1/09 6:27:02 PM
AnswerS10.indd 604 7/31/09 2:56:00 PM