8/20/2019 Answers for maths in focus 3u
1/95
756 Maths In Focus Mathematics Extension 1 Preliminary Course
Answers
Chapter 1: Basic arithmetic
Problem
5
Exercises 1.1
1. (a) Rational (b) Rational (c) Rational (d) Irrational
(e) Rational (f) Irrational (g) Irrational (h) Rational
(i) Rational (j) Irrational
2. (a) 18 (b) 11 (c) 6 (d) 11 (e) .4 3- (f) −1 (g) 215
7
(h) 1
20
19 (i) 2 (j) 3
3
1
3. (a) 16.36 (b) 21.87 (c) 8.80 (d) 22.71 (e) .13 20-
(f) 0.17 (g) 0.36 (h) 1.20 (i) .4 27- (j) 8.16
4. 1300 5. 950 6. 3000 7. 11 000
8. 600 9. $8 000 000 10. $34 600 000
11. 844 km 12. 0.73 13. 33 14. 3.248 15. 4.21
16. 1.7 17. 79 cents 18. 2.73 19. 1.1 20. 3.6 m
21. $281.93 22. 1.8 g 23. $3.20
24. (a) 7.95 (b) 30.03 (c) 0.37 (d) 5.74 (e) 0.52 25. 0.2
Exercises 1.2
1. 1 2. 11- 3. 56- 4. 10 5. 4-
6. .1 2- 7. .7 51- 8. .35 52- 9. 6.57
10. 215
4- 11. 7- 12. −23 13. 10 14. 1
15. 5 16. 3 17. 1 18. 60 19. −20 20. 9
Exercises 1.3
1. (a)2516 (b)
100051 (c) 5
201 (d) 11
54
2. (a) 0.4 (b) 1.875 (c) .0 416o (d) .0 63oo
3. (a)50
1 (b)
8
3 (c)
1000
1 (d) 1
1000
97
4. (a) 0.27 (b) 1.09 (c) 0.003 (d) 0.0623
5. (a) 35% (b) %333
1 (c) %226
3
2 (d) 0.1%
6. (a) 124% (b) 70% (c) 40.5% (d) 127.94%
7. (a) . ;0 5225
13 (b) . ;0 07
100
7 (c) . ;0 168
125
21
(d) . ;1 09 1 100
9
(e) . ;0 434 500
217
(f) . ;0 1225 400
49
8. (a) .0 83o (b) .0 07oo (c) .0 13oo (d) .0 16o (e) .0 6o (f) .0 15oo
(g) .0 142857o o or 0.142857 (h) .1 18oo
9. (a)9
8 (b)
9
2 (c) 1
9
5 (d) 3
9
7 (e)
99
67 (f)
11
6 (g)
45
7
(h)60
13 (i)
990
217 (j) 1
495
37
10. (a) .0 5o (b) 7.4 (c) 0.73o (d) .0 68oo (e) .1 72oo
11. (a)8
5 (b) 2
8
1 (c)
11
8 (d) 2
18
7 (e)
45
4
12. 74% 13. 77.5% 14. 17.5% 15. 41.7%
Exercises 1.4
1.20
3 2.
20
7 3. (a)
20
17 (b)
10
7 (c) 1
20
1
(d)28
3 (e)
5
3 4. $547.56 5. 714.3 g 6.
24
7 7. $65
8. 179 cm 9. (a) 11.9 (b) 5.3 (c) 19 (d) 3.2 (e) 3.5
(f) 0.24 (g) 0.000 18 (h) 5720 (i) 0.0874 (j) 0.376
10. $52.50 11. 54.925 mL 12. 1152.125 g 13. $10.71
14. 5.9% 15. 402.5 g 16. 41.175 m 17. $30.92
18. 3.2 m 19. 573 20. $2898
Problem
511
5 minutes after 1 o’clock.
Exercises 1.5
1. (a) 500 (b) 145 (c)64
1 (d) 3 (e) 2
2. (a) 13.7 (b) 1.1 (c) 0.8 (d) 2.7 (e) .2 6- (f) 0.5
3. (a) a17 (b) y 10 = (c) a 4- (d) w (e) x5 (f) p10
(g) y 6 (h) x21 (i) x4 10 (j) y 81 8- (k) a (l) y
x45
10
(m) w 10 (n) p5 (o) x 3- (p) a ba
bor2 3
2
3-
(q) x y x
y or5 2
5
2
-
4. (a) x 14 (b) a 7- (c) m 4 (d) k 10 (e) a 8- (f) x (g) mn 2
(h) p 1- (i) 9x 22 (j) x 21
5. (a) p 5 q 15 (b)b
a8
8
(c)b
a6412
3
(d) 49a 10 b 2 (e) 8m 17
(f)x 4 y
10 (g)k
27
2 23 (h) 16
y 47 (i)
a 3 (j)
x y 125 21 18-
8/20/2019 Answers for maths in focus 3u
2/95
757 ANSWERS
6. 42
1 7. 324 8. 2
27
10 9. (a) a 3 b (b)
25
1
10. (a) pq 2 r 2 (b)32
7 11.
9
4 12.
18
1 13.
27
4
14.81
1 15.
108
1 16.
12
1 17.
2
558
22
18. 3888
49
Exercises 1.6
1. (a)27
1 (b)
4
1 (c)
343
1 (d)
10 000
1 (e)
256
1 (f) 1
(g)32
1 (h)
81
1 (i)
7
1 (j)
81
1 (k)
64
1 (l)
9
1 (m) 1
(n)36
1 (o)
125
1 (p)
100 000
1 (q)
128
1 (r) 1
(s)64
1 (t)
64
1
2. (a) 1 (b) 16 (c) 12
1 (d) 1
25
11 (e) 1 (f) 125 (g) 1
3
1
(h) 49 (i) 383 (j) 32 (k) 2
31 (l) 1 (m) 1
3613 (n) 1
8119
(o) 1 (p) 16 (q) 158
5- (r)
23
7- (s) 1 (t)
25
16
3. (a) m 3- (b) x 1- (c) p 7- (d) d −9 (e) d −5 (f) x 2-
(g) 2x 4- (h) 3 y −2 (i)2
1z 6- or
2
z 6- (j)
5
3t 8- (k)
7
2x 1-
(l)2
5m 6- (m)
3
2 y 7-
(n) 3 4x 2+ -] g (o) a b 8+ -] g (p) 2x 1- -] g (q) 5 1 p 3+ -^ h (r) 2 4 9t 5- -] g (s)
4
1x 11+ -] g (t)
9
5 3a b 7+ -] g
4. (a)1
t 5 (b)1
x6 (c)1
y 3 (d)1
n8 (e)1
w 10 (f)2x (g)
3
m4
(h)5
x7 (i)
8
1
x3 (j)
4
1
n (k)
1
1
x 6+] g (l) 81
y z+
(m)3
1
k 2-] g (n) 3 21
x y 9+^ h (o) x 5 (p) y 10 (q)
2
p
(r) a b 2+] g (s)x y
x y
+
-
(t)2
3
w z
x y 7
-
+e o
Exercises 1.7
1. (a) 9 (b) 3 (c) 4 (d) 2 (e) 7 (f) 10 (g) 2 (h) 8
(i) 4 (j) 1 (k) 3 (l) 2 (m) 0 (n) 5 (o) 7 (p) 2
(q) 4 (r) 27 (s)
2
1 (t)
16
1
2. (a) 2.19 (b) 2.60 (c) 1.53 (d) 0.60 (e) 0.90 (f) 0.29
3. (a) y 3 (b) y y or23 32_ i (c) 1
x (d) 2 5x +
(e)3 1
1
x - (f) 6q r 3 + (g)
x x7
1
7
1or
25 5 2+ +] ^g h
4. (a) 2t 1
(b) 5 y 1
(c) 2x3
(d) x9 31
-] g (e) 2s4 1+ 1] g (f) 2t 2 3+
-1] g (g) - 2x y 5 - 3^ h (h) 2x3 1+ 5] g
(i) 3x 2--
2] g (j) 2 y 2
17+
-1^ h (k) - 3x5 4+ 1] g
(l) y 3
21
21
-
-2
a k (m) x
5
32
43
+
-2
_ i
5. (a) x23
(b) x 21
-
(c) x32
(d) x35
(e) x45
6. (a) x x x2 23
+ +2 (b) a b3
2
3
2
- (c) p p p2 21
+ +1-2
(d) 2x x 1+ +- (e) x x x321
2
3
25
- +
- - -
7. (a)
2
1
a b3
-
(b)
3
1
y 2
3 -^ h (c)
6 1
4
a47
+] g
(d)3
1
x y 54 +^ h (e) 7 3 86
x 29 +] g
Exercises 1.8
1. (a) .3 8 103# (b) .1 23 106# (c) .6 19 104#
(d) 1.2 107# (e) .8 67 109# (f) .4 16 105#
(g) 9 102# (h) .1 376 104# (i) 2 107# (j) 8 104#
2. (a) .5 7 10 2# - (b) .5 5 10 5# - (c) 4 10 3# -
(d) 6.2 10 4# - (e) 2 10 6# - (f) 8 10 8# -
(g) 7.6 10 6# - (h) 2.3 10 1# - (i) 8.5 10 3# - (j) 7 10 11# -
3. (a) 36 000 (b) 27 800 000 (c) 9 250 (d) 6 330 000
(e) 400 000 (f) 0.072 3 (g) 0.000 097
(h) 0.000 000 038 (i) 0.000 007 (j) 0.000 5
4. (a) 240 000 (b) 9 200 000 (c) 11 000 (d) 0.36
(e) 1.3 (f) 9.0 (g) 16 (h) 320 (i) 2900 (j) 9.1
5. (a) 6.61 (b) 0.686 (c) 8.25 (d) 1.30
6. 1.305 1010# 7. 6.51 10 10# -
Exercises 1.9
1. (a) 7 (b) 5 (c) 6 (d) 0 (e) 2 (f) 11 (g) 6 (h) 24
(i) 25 (j) 125 2. (a) 5 (b) −1 (c) 2 (d) 14 (e) 4
(f) −67 (g) 7 (h) 12 (i) −6 (j) 10 3. (a) 3 (b) 3
(c) 1 (d) 3 (e) 1 4. (a) a (b) a- (c) 0 (d) 3a
(e) −3a (f) 0 (g) 1a + (h) a 1- - (i) 2x -
(j) 2 x-
5. (a) | | 6a b+ = | | | |a b 6+ = | | | | | |a b a b` #+ +
(b) | | 3a b+ = | | | |a b 3+ = | | | | | |a b a b` #+ +
(c) | | 1a b+ = | | | |a b 5+ = | | | | | |a b a b` #+ +
(d) | | 1a b+ = | | | |a b 9+ = | | | | | |a b a b` #+ +
(e) | |a b 10+ = | | | |a b 10+ = | | | | | |a b a b` #+ +
6. (a) | | 5x x2 = = (b) | | 2x x2 = = (c) | | 3x x2 = =
(d) | | 4x x2 = = (e) | | 9x x2 = =
7. (a) x x x x5 5 5 5for and for2 1+ - - - -
(b) b b b x3 3 3 3for and for2 1- -
(c) a a a a4 4 4 4for and for2 1+ - - - -
(d) y y y y 2 6 3 6 2 3for and for2 1- -
(e) 3 9 3 3 9 3x x x xfor and for2 1+ - - - -
(f) 4 4 4 4x x x xfor and for1 2- -
(g) 2 1 2 1k k k k2
1
2
1for and for2 1+ - - - -
(h) 5 2 5 2x x x x5
2
5
2for and for2 1- - +
(i) a b a b a b a bfor and for2 1+ - - - -
(j) p q p q q p p qfor and for2 1- -
8. x 3!= 9. 1! 10. , x1 2! !
8/20/2019 Answers for maths in focus 3u
3/95
758 Maths In Focus Mathematics Extension 1 Preliminary Course
Test yourself 1
1. (a)20
9 (b) 0.14 (c) 0.625 (d)
200
157 (e) 1.2%
(f) 73.3% 2. (a)49
1 (b)
5
1 (c)
3
1
3. (a) 8.83 (b) 1.55 (c) 1.12 (d) 342 (e) 0.303 4. (a) 1
(b) 1 (c) 39 (d) 2 (e) 10- (f) 1- (g) 4 5. (a) x9
(b) 25 y 6 (c) a b11 6 (d)27
8x18 (e) 1 6. (a)
40
29
(b) 37
1 (c) 12 (d) 2
2
1 (e) 12
2
1 7. (a) 4 (b) 6 (c) 19
(d)64
1 (e) 4 (f) 3 (g)
7
1 (h) 2 (i) 1 (j) 4
8. (a) a 5 (b) x 30 y 18 (c) p 9 (d) 16b 36 (e) 8x 11 y 9. (a) 2n1
(b) x 5- (c) x y 1+ -^ h (d) 4x 1+ 1] g (e) 7a b+ 1] g (f) 2x 1- (g)
2
1x 3- (h) 3x
4
(i) 7x5 3+9] g (j) 4m- 3
10. (a) 1a5
(b) n4 (c) 1x + (d) 1x y -
(e)4 7
1t 4-] g
(f) a b5 + (g)1
x3 (h) b34 (i) 2 3x 43 +] g (j) 1
x3
11. | | 2a b+ = | | | | 8a b+ = | | | | | |a b a b` #+ +
12. 1 13. 192
1 14. 689 mL 15. (a) 6 h (b)
12
7 (c)
8
1
(d) 33.3% 16. $38 640 17. 70% 18. 6.3 1023#
19. (a) 2x1
(b) y 1- (c) 6x 3+1] g (d) 2 3x 11- -] g (e) 3 y 7
20. (a) 1.3 10 5# - (b) 1.23 1011# 21. (a) 9
7 (b)
330
41
22. (a)
1
x3 (b) 2 5
1
a + (c) a
b 5
c m 23. 14 500 24. | | ,2 5 7LHS = - + - = | | | | .2 5 7RHS = - + - =
So | | | | | |a b a b#+ + since .7 7#
Challenge exercise 1
1. 4303
278 2. 1
18
11 3. . , %, , .0 502 51
99
510 5o
4. 533
1% 5.
16
1 6. 3.04 1014# 7. 83% 8. 1
9990
3271
9. 18 h 10. 1.98
11.
2 2 1 2
2 2 22 2 2
2 2 1
2 2 1 2 2 2 1
LHS
RHS
k k
k k
k
k
k k k
1
1 1
1
1
1 1
:
`
= - +
= - +
= -
= -
=
- + = -
+
+ +
+
+
+ +
^
^^ ^
h
hh h
12. −24.35 13. . , , . , ,0 34 2 1 5 07
3-
o 14. 63
2%
15. ,x
xx
x1
11
1
11when when2 1
--
-- 16. 0.73
17. 0.6% 18 4.54 19. 4.14 10 20# -
20. | | | | | | , , ;
| | | | | | , , ;
a b a b a b a b
a b a b b ba
0 0 0 0
0 0 0 0
when or
when ora
2 2 1 1
1 2 1 1 2
+ = +
+ +
| | | |a b a b`
#+ +
for all a , b
Chapter 2: Algebra and surds
Exercises 2.1
1. 7x 2. 3a 3. z 4. 6a 5. 3b 6. −3r
7. y - 8. −5x 9. 0 10. 3k 11. 9t 12. 10w
13. m- 14. x- 15. 0 16. 5b 17. 11b 18. 10x-
19. 6 6x y - 20. 3a b- 21. 4 2xy y + 22. 6ab2-
23. m m6 122 - + 24. 2 6 p p2 - - 25. 8 3x y +
26. 2 10ab b- + 27. 2bc ac - 28. 2 9 1a x5 3- +
29. 2 3 2x xy x y y 3 2 2 3- + + 30. 3 7 6x x x3 2+ - -
Exercises 2.2
1. b10 2. xy 8 3. p10 2 4. wz6-
5. ab15 6. xyz14 7. abc 48 8. d 12 2
9. a12 3 10. y 27 3- 11. x32 10
12. a b6 2 3 13. a b10 3 2- 14. p q21 3 4
15. a b5 3 3 16. n8 10- 17. k p33 18. t 81 12
19. 14m11- 20. x y 24 6 3
Exercises 2.3
1. 6x 2. 2 3. 4a2 4. 8a 5. 4a 6. 2
y 7. 3 p
8.2
ab 9.
3
4
y 10. 3x3- 11. 3a 12.
3
1
ab2 13.
2qs
-
14.3
2
c d 2 15.
x
z
2 2
2
16. 6 p q4 17. 4c
a b4 7 18.
2a
b6
19.3 y
x z3 3- 20.
2b
a6
13
Exercises 2.4
1. x2 8- 2. h6 9+ 3. a5 10- + 4. xy x2 3+
5. x x22 - 6. a ab6 162 - 7. a b ab2 2 2+ 8. n n5 202 -
9. 3x y x y 63 2 2 3+ 10. k4 7+ 11. t 2 17-
12. y y 4 112 + 13. b5 6- - 14. x8 2-
15. m3 1- + 16. h8 19- 17. d 6- 18. a a2 42 - +
19. x x3 9 52 - - 20. ab a b b2 2 2- + 21. x4 1-
22. y 7 4- + 23. b2 24. t 5 6- 25. a2 26+
Exercises 2.5
1. 7 10a a2 + + 2. 2 3x x2 + - 3. 2 7 15 y y 2 + -
4. 6 8m m2 - + 5. 7 12x x2 + + 6. 3 10 y y 2 - -
7. 2 6x x2 + - 8. 10 21h h2 - + 9. 25x2 -
10. 15 17 4a a2 - + 11. 8 6 9 y y 2 + - 12. 7 4 28xy x y + - -
13. 2 3 6x x x3 2- + - 14. 4n2 - 15. 4 9x2 -
16. 16 49 y 2- 17. 4a b2 2- 18. 9 16x y 2 2- 19. 9x2 -
20. 36 y 2 - 21. 9 1a2 - 22. 4 49z2 -
8/20/2019 Answers for maths in focus 3u
4/95
759 ANSWERS
23. 2 11 18 18x xy x y 2 - + - + 24. 2 2 7 6 3ab b b a2+ - - +
25. x 83 + 26. 27a3 - 27. 18 81a a2 + +
28. 8 16k k2 - + 29. 4 4x x2 + + 30. 14 49 y y 2 - +
31. 4 12 9x x2 + + 32. 4 4 1t t 2 - +
33. 9 24 16a ab b2 2+ + 34. 10 25x xy y 2 2- +
35. 4 4a ab b2 2+ + 36. a b2 2- 37. 2a ab b2 2+ +
38. 2a ab b2 2- + 39. a b3 3+ 40. a b3 3-
Exercises 2.6
1. 8 16t t 2 + + 2. 12 36z z2 - + 3. 2 1x x2 - +
4. 16 64 y y 2 + + 5. 6 9q q2 + + 6. 14 49k k2 - +
7. 2 1n n2 + + 8. 4 20 25b b2 + + 9. 9 6x x2- +
10. 9 6 1 y y 2 - + 11. 2x xy y 2 2+ + 12. 9 6a ab b2 2- +
13. 16 40 25d de e2 2+ + 14. 16t 2 - 15. x 92 -
16. 1 p2 - 17. 36r 2 - 18. 100x2 - 19. 4 9a2 -
20. 25x y 2 2- 21. 16 1a2 - 22. 49 9x2- 23. 4x4 -
24. 10 25x x4 2+ + 25. 9 16a b c 2 2 2- 26. 44
xx
2
2+ +
27.1
aa
2
2- 28. 2 4 4x y x y y 2 2 2 2- - = - + -^ h
29. 2 2 2 2a b a b c c a ab b ac bc c 2 2 2 2 2+ + + + = + + + + +] ]g g 30. 1 2 1 2 1 2 2x x y y x x xy y y 2 2 2 2+ - + + = + + - - +] ]g g 31. 12a 32. 32 z2- 33. 9 8 3x x2 + -
34. 3 2x xy y x2 2+ + - 35. 14 4n2 -
36. 12 48 64x x x3 2- + - 37. x2 38. 2x x y y 4 2 2 4- +
39. 8 60 150 125a a a3 2+ + +
40. 4 16 15 4 4x x x x4 3 2+ + - -
Problem
2,a = 7,b = 9,c = 4,d = 3,e = 8,f = 0, g = 6,h = 1i =
Exercises 2.7
1. y 2 3+^ h 2. x5 2-] g 3. m3 3-] g 4. x2 4 1+] g 5. y 6 4 3-^ h 6. xx 2+] g 7. m m 3-] g 8. y y 2 2+^ h 9. a a3 5 -] g 10. ab b 1+] g 11. xy x2 2 1-] g 12. mn n3 32 +^ h 13. xy x z2 4 -] g 14. a b a6 3 2+ -] g 15. x x y 5 2- +^ h 16. q q3 2
2 3
-_ i 17. b b5 32
+] g 18. a b ab3 22 2 -] g 19. 5)( 7)(m x+ + 20. 1 2 y y - -^ ^h h 21. 7 )(4 3 )( y x+ - 22. 2 6 5a x- +] ]g g 23. 2 1t x y + -] ^g h 24. 3 2 2 3x a b c - + -] ]g g 25. 3 2 3x x2 +] g 26. 3 2q pq3 2 -_ i 27. ab a b3 5 13 2 +^ h 28. 4 6x x2 -] g 29. 5 7 5m n mn2 3 -^ h 30. 4 6 4ab ab2 3 +^ h 31. r r h2r +] g 32. 3 2x x- +] ]g g 33. ( ) ( )x y 4 22+ + 34. 1a- +] g 35. ( ) ( )a ab1 4 32 + -
Exercises 2.8
1. 4 2x b+ +] ]g g 2. 3 y a b- +^ ]h g 3. 5 2x x+ +] ]g g 4. 2 3m m- +] ]g g 5. d c a b- +] ]g g 6. 1 3x x2+ +] ^g h 7. 5 3 2a b- +] ]g g 8. 2 y x x y - +^ ^h h 9. 1 1 y a+ +^ ]h g 10. 5 1x x+ -] ]g g 11. 3)(1 )( y a+ + 12. 2)(1 2 )(m y - - 13. 5 2 3x y x y + -^ ^h h 14. 4a b ab2+ -^ ]h g 15. 5 3x x- +] ]g g 16. 7)( 4)(x x3+ - 17. 3 7x y - -] ^g h 18. 3 4d e+ -] ]g g 19. 4 3x y - +] ^g h 20. 3 2a b+ -] ]g g 21. 3)( 6)(x x2- + 22. 3q p q- +^ ^h h 23. 2 3 5x x2- -] ^g h 24. 3 4a b c - +] ]g g 25. 7 4 y x+ -^ ]h g 26. 4)( 5)(x x3- - 27. (2 3)(2 4) (2 3)( )x x x x2 22 2- + = - +
28. ( )( )a b a3 2 3+ + 29. 5( 3)(1 2 ) y x- +
30. r r 2 3r+ -] ]g g
Exercises 2.91. 3 1x x+ +] ]g g 2. 4 3 y y + +^ ^h h 3. 1m 2+] g 4. 4t 2+] g 5. 3 2z z+ -] ]g g 6. 1 6x x+ -] ]g g 7. 3 5v v - -] ]g g 8. 3t 2-] g 9. 10 1x x+ -] ]g g 10. 7 3 y y - -^ ^h h 11. 6 3m m- -] ]g g 12. 12 3 y y + -^ ^h h 13. 8 3x x- +] ]g g 14. a 2 2-] g 15. 2 16x x- +] ]g g 16. 4 9 y y + -^ ^h h 17. 6 4n n- -] ]g g 18. x 5 2-] g 19. 9 1 p p+ -^ ^h h 20. 2 5k k- -] ]g g 21. 4 3x x+ -] ]g g 22. 7 1m m- +] ]g g 23. 10 2q q+ +^ ^h h 24. 5 1d d - +] ]g g 25. 9 2l l- -] ]g g Exercises 2.10
1. 2 1)( 5)( a a+ + 2. 5 2 1 y y + +^ ^h h 3. 3 7)( 1)( x x+ + 4. 3 2)( 2)( x x+ + 5. 2 3)( 1)( b b- -
6. 7 2)( 1)( x x- - 7. 3 1 2 y y - +^ ^h h 8. 2 3 4x x+ +] ]g g 9. 5 2 3 p p- +^ ^h h 10. 3 5 2 1x x+ +] ]g g 11. 2 1)( 6)( y y + - 12. 5 1 2 1x x- +] ]g g 13. 4 1)(2 3)( t t - - 14. 3 4)(2 3)( x x+ -
15. 6 1 8 y y - +^ ^h h 16. 4 3 2n n- -] ]g g 17. 4 1 2 5t t - +] ]g g 18. 3 2 4 5q q+ +^ ^h h 19. r r r r 4 1 2 6 4 12 3- + - +=] ] ] ]g g g g 20. 2 5 2 3x x- +] ]g g 21. 6 1 2 y y - -^ ^h h 22. 2 3 3 2 p p- +^ ^h h 23. 8 7)( 3)( x x+ + 24. 3 4 4 9b b- -] ]g g 25. 6 1)( 9)( x x+ - 26. 3 5x 2+] g 27. 4 3 y 2+^ h 28. 5 2k 2-] g 29. 6 1a 2-] g 30. 7 6m 2+] g
8/20/2019 Answers for maths in focus 3u
5/95
760 Maths In Focus Mathematics Extension 1 Preliminary Course
Exercises 2.11
1. 1 y 2-^ h 2. ( 3)x 2+ 3. ( 5)m 2+ 4. ( 2)t 2- 5. ( 6)x 2- 6. 2 3x 2+] g 7. 4 1b 2-] g 8. 3 2a 2+] g 9. 5 4x 2-] g 10. 7 1 y 2+^ h 11. 3 5 y 2-^ h 12. 4 3k 2-] g 13. 5 1x 2+] g 14. 9 2a 2-] g 15. 7 6m 2+] g 16.
2
1t
2
+d n 17. 3
2x
2
-d n 18. 35
1 y
2
+d n 19.
1x
x
2
+c m 20. 5 2kk
2
-d n Exercises 2.12
1. 2)( 2)(a a+ - 2. 3)( 3)(x x+ - 3. 1)( 1)( y y + -
4. 5 5x x+ -] ]g g 5. 2 7)(2 7)( x x+ - 6. 4 3)(4 3)( y y + - 7. 1 2 )(1 2 )( z z+ - 8. 5 1 5 1t t + -] ]g g 9. 3 2 3 2t t + -] ]g g 10. 3 4 3 4x x+ -] ]g g 11. 2 )( 2 )(x y x y + - 12. 6 6x y x y + -^ ^h h 13. 2 3 2 3a b a b+ -] ]g g 14. 10 10x y x y + -^ ^h h 15. 2 9 2 9a b a b+ -] ]g g 16. 2 2x y x y + + + -^ ^h h 17. 3)( 1)(a b a b+ - - + 18. 1 1z w z w + + - -] ]g g 19.
2
1
2
1x x+ -d dn n
20.3
13
1 y y
+ -e eo o 21. 2 3 2 1x y x y + + - +^ ^h h 22. ( )( ) ( ) ( ) ( )x x x x x1 1 1 1 12 2 2+ - = + + -
23. 3 2 3 2x y x y 3 3+ -_ _i i 24. 4 2 2x y x y x y 2 2+ + -_ ^ ^i h h 25. 1)( 1)( 1)( 1)(a a a a4 2+ + + -
Exercises 2.13
1. 2)( 2 4)(b b b2- + + 2. 3 3 9x x x2+ - +] ^g h 3. 1 1t t t 2+ - +] ^g h 4. 4)( 4 16)(a a a2- + + 5. 1 )(1 )( x x x2- + + 6. 2 3 4 6 9 y y y 2+ - +^ _h i 7. ( )( ) y z y yz z2 2 42 2+ - + 8. 5 )( 5 25 )(x y x xy y 2 2- + +
9. 2 3 4 6 9x y x xy y 2 2+ - +^ _h i 10. 1 1ab a b ab2 2- + +] ^g h 11. 10 2 )(100 20 4 )( t t t + - + 2 12.
23
4 2
39
x x x2- + +d en o
13.10 1 100 10 1a b a ab b2 2
+ - +d en o 14. 1 2 1x y x x xy y y 2 2+ - + + + + +^ _h i 15. xy z x y xyz z5 25 306 362 2 2+ - +
^ _
h i 16. a a 19 2 - +- ^ h
17. 13
13 9
x x x2- + +d en o
18. 3 3 9 6x y y y xy x x2 2+ + - - + + +^ _h i 19. 1 4 5 7x y x x xy y y 2 2+ - + - + - +^ _h i 20. 2 6 )(4 24 2 6 36)( a b a a ab b b2 2+ - + + + + +
Exercises 2.14
1. x x2 3 3+ -] ]g g 2. p p3 3 4+ -^ ^h h 3. y y y 5 1 12- + +^ _h i 4. )(ab a b a2 2 2 1+ -^ h 5. 5 1a 2-] g 6. x x2 3 4- -- ] ]g g 7. z z z3 5 4+ +] ]g g 8. ab ab ab3 2 3 2+ -] ]g g 9. x xx 1 1+ -] ]g g 10. x x2 3 2 2- +] ]g g 11. 5 3m n- +] ]g g 12. x7 2 1- +] g 13. 5 4 4 y y y + + -^ ^ ^h h h 14. 1 2 2 4x x x x2- + - +] ] ^g g h 15. x x x x x x1 1 1 12 2+ - + - + +] ^ ] ^g h g h 16. x x x2 5+ -] ]g g 17. ( )x x3 3 2+ -] g 18. ( )( )xy xy y 2 1 2 1+ - 19. b b b3 2 4 2 2- + +] ^g h 20. x x3 3 2 2 5- +] ]g g 21. x3 1 2-] g 22. 2)( 5)( 5)(x x x+ + - 23. 3z z 2+] g 24. 1 1 2 3 2 3x x x x+ - + -] ] ] ]g g g g 25. x x x y x xy y 2 2 2 2 2+ - + - +] ] ^ _g g h i 26. ( ) ( )a a a4 3 3+ - 27. x x xx 2 4 25 2- + +] ^g h 28. 2)( 2)( 3)( 3)(a a a a+ - + - 29. 4 ( 5)k k 2+
30. 3( 1) 1) 3)( (x x x+ - +
Exercises 2.15
1. 4 4 2x x x2 2+ + = +] g 2. 6 9 3b b b2 2- + = -] g 3. 10 25 5x x x2 2- + = -] g 4. 8 16 4 y y y 2 2+ + = +^ h 5. 14 49 7m m m2 2- + = -] g 6. 18 81 9q x q2 2+ + = +^ h 7. 2 1 1x x x2 2+ + = +] g 8. 16 64 8t t t 2 2- + = -] g 9. 20 100 10x x x2 2- + = -] g 10. 44 484 22w w w 2 2+ + = +] g 11. 32 256 16x x x2 2- + = -] g 12. 3
4
9
2
3 y y y 2
2
+ + = +d n
13. 74
49
2
7x x x2
2
- + = -d n 14. 4
1
2
1a a a2
2
+ + = +d n
15. 94
81
2
9x x x2
2
+ + = +d n 16. 5 y y y 16
25
4
5
22
2
- + = -d n
17.11
k k
k16
121
4
11
22
2
- + = -d n 18. 6 9 3x xy y x y 2 2 2+ + = +^ h 19. 4 4 2a ab b a b2 2 2- + = -] g 20. 8 16 4 p pq q p q2 2 2- + = -^ h
Exercises 2.16
1. 2a + 2. 2 1t - 3. 3
4 1 y + 4.
2 1
4
d - 5.
5 2x
x
-
6.4
1
y - 7.
a
b a
3
22
-
-] g 8.
3
1
s
s
+
- 9.
1
1
b
b b2
+
+ +
10.3
5 p + 11.
3
1
a
a
+
+ 12.
2 4
3
x x
y
2+ +
+
13. 3x -
14.4 2 1
2
p p
p
2- +
-
15.2a b
a b
-
+
8/20/2019 Answers for maths in focus 3u
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761 ANSWERS
Exercises 2.17
1. (a)4
5x (b)
15
13 3 y + (c)
12
8a + (d)
6
4 3 p + (e)
6
13x -
2. (a)2 1a
b
- (b)
1
2 1
q
p q q2
+
- - +^ _h i (c)
b
x y b
2 1
2
10
2
-
+
]^
gh
(d)ab
x xy y 2 2
- + (e)
5 2
3 1x x
x x
- -
- -
] ]] ]g gg g
3. (a)5x
(b)x
x
x 1
2
-
- +
] g (c)3
a b
a b
+
+ + (d)
2
2
x
x
+
(e) p q
p q p q
p q
p q 11 2 2
+
+ -
=+
- ++^ ^h h (f)
x x
x
1 3
12
+ -
-
] ]]g g
g
(g)2 2
3 8
x x
x
+ -
- +
] ]g g (h) 12
a
a2
+
+
] g
(i) y y y
y y
2 3 1
3 14 132 2
+ + -
+ +
^ ^ ^_
h h hi
(j)x x x
x
4 4 3
5 22
+ - +
+-
] ] ]]g g g
g
4. (a) y y
xx
3 9
2
8
2
2- +
+
_ ] ig (b) 152 1
y
y y + +
^ ^h h (c)
x x
x x
2 3 4
10 242
- -
+ -
] ]g g (d) bb b
b 1
3 5 10
2
2
+
- -
] g (e) x 5. (a)
5 2 3
3 13
x x x
x
- - +
-
] ] ]g g g (b) 2 23 5
x x
x
+ -
-
] ]g g
(c) p q p q
p pq q
pq
3 5 22 2
+ -
+ -
^ ^h h (d)2 1
a b a b
a ab b2 2
+ -
- - +
] ]g g
(e)x y x y
x y y 1
+ -
+ +
^ ^^
h hh
Exercises 2.18
1. (a) 7.1- (b) 6.9- (c) 48.1 (d) 37.7- (e) 0.6
(f) 2.3 (g) 5.3- 2. 47 3. 7- 4. 375 5. 196-
6. 5.5 7. 377 8. 284 9. 40- 10. 51.935 11. 14
3-
12. 22.4 13. 1838.8 14. 4
3 15. 15 16. 10
17. 2 3 18. 23.987 19. 352.47 20. 93 21. 4
Exercises 2.19
1. (a) 2 3 (b) 3 7 (c) 2 6 (d) 5 2 (e) 6 2
(f) 10 2 (g) 4 3 (h) 5 3 (i) 4 2 (j) 3 6
(k) 4 7 (l) 10 3 (m) 8 2 (n) 9 3 (o) 7 5
(p) 6 3 (q) 3 11 (r) 5 5
2. (a) 6 3 (b) 20 5 (c) 28 2 (d) 4 7 (e) 16 5
(f) 8 14 (g) 72 5 (h) 30 2 (i) 14 10 (j) 24 5
3. (a) 18 (b) 20 (c) 176 (d) 128 (e) 75
(f) 160 (g) 117 (h) 98 (i) 363 (j) 1008
4. (a) 45x = (b) 12x = (c) 63x = (d) 50x =
(e) 44x = (f) 147x = (g) 304x = (h) 828x =
(i) 775x = (j) 960x =
Exercises 2.20
1. 3 5 2. 2 3. 6 3 4. 3 3 5. 3 5- 6. 3 6
7. 7 2- 8. 8 5 9. 4 2- 10. 4 5 11. 2 12. 5 3
13. 3- 14. 2 15. 5 7 16. 2 17. 13 6
18. 9 10- 19. 47 3 20. 2 2 35 - 21. 7 5 2-
22. 2 3 4 5- - 23. 7 6 3 5+ 24. 2 2 3- -
25. 17 5 10 2- +
Exercise 2.21
1. 21 2. 15 3. 3 6 4. 10 14 5. 6 6- 6. 30
7. 12 55- 8. 14 9. 60 10. 12 2 3=
11. 2 48 8 3= 12. 15 28 30 7=
13. 2 20 4 5= 14. 84- 15. 2
16. 28 17. 30 18. 2 105- 19. 18
20. 30 50 150 2= 21. 2 6 22. 4 3 23. 1 24.
6
8
25. 2 3 26.3 10
1 27.
2 5
1 28.
3 5
1 29.
2
1
30.2 2
3 31.
2
3 32.
2 5
9 33.
2 2
5 34.
3
2 35.
7
5
Exercises 2.22
1. (a) 10 6+ (b) 2 6 15- (c) 12 8 15+
(d) 5 14 2 21- (e) 6 4 18 6 12 2- + = - +
(f) 5 33 3 21+ (g) 6 12 6- - (h) 5 5 15-
(i) 6 30+ (j) 2 54 6 6 6 6+ = +
(k) 8 12 12 8 24 3- + = - + (l) 210 14 15-
(m) 10 6 120- (n) 10 2 2- - (o) 4 3 12-
2. (a) 10 3 6 3 5 9 3+ + +
(b) 10 35 2 14- - +
(c) 2 10 6 10 15 15 6- + -
(d) 12 18 8 12
24 36 8 12
20 60 10 30
5 15 10 30
+ - - =
+ - -
(e) 52 13 10- (f) 15 15 18 10 6 6- + -
(g) 4 (h) 1- (i) 12- (j) 43 (k) 3 (l) 241-
(m) 6- (n) 7 2 10+ (o) 11 4 6- (p) 25 6 14+
(q) 57 12 15+ (r) 27 4 35-
(s) 77 12 40 77 24 10- = - (t) 53 12 10+
3. (a) 18 (b) 108 2 (c) 432 2 (d) 19 6 2+ (e) 9
4. (a) 21, 80a b= = (b) 19, 7a b= = -
5. (a) 1a - (b) p p p2 1 2 1- - -^ h 6. 25k = 7. 2 3 5x y xy - - 8. 17, 240a b= =
9. 107, 42a b= = - 10. 9 5 units2+
8/20/2019 Answers for maths in focus 3u
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762 Maths In Focus Mathematics Extension 1 Preliminary Course
Exercises 2.23
1. (a)7
7 (b)
4
6 (c)
5
2 15 (d)
10
6 14
5
3 14=
(e)3
3 6+ (f)
2
12 5 2- (g)
5
5 2 10+
(h) 14
3 14 4 7- (i) 20
8 5 3 10+ (j) 35
4 15 2 10-
2. (a) 4 4 3 243 2- -= ^ h (b)47
6 7 3+- ^ h
(c)19
2 15 4 18
19
15 6 22-=
-- -^ ^h h
(d)13
19 8 3
13
8 3 19-=
-- ^ h (e) 6 2 5 3 5 2+ + +
(f)2
6 15 9 6 2 10 6- + -
3. (a) 2 2
(b) 2 3 32 6 3 2 3 3 6 2 3- + - + = - - + -
^ h (c)39
22 5 14 2+
(d)
10
6 6 16 3 84 8 14
6 21 14
5
3 8 3 4
- - - +
=- + + -
^ h
(e) 4- (f) 4 2
(g)15
20 12 19 6 25 3 6
15
19 6 65 3 6+ + -=
+ -
(h)6
6 9 2 2 3+ + (i)
21
4 6 9 3+
(j)415
30
330 30 5- - (k) 13
28 2 6 7 3- -
(l)2
2 15 2 10 2 6 3 5+ - - -
4. (a) 45, 10a b= = (b) 1, 8a b= = (c)2
1,
2
1a b= - =
(d) 19
5,
9
8a b= - = - (e) 5, 32a b= =
5.
2
2
3 2 2
3
2 1
2 1
2
4
2 1
2 1
2 1
2 1
2
4
2
2
2 1
2 1 2 1
2
4 2
2 1
2 2 2 12
1
3 2 22
2 2
2 2
# #
+
-+
=
+
-
-
-+
=
-
- -+
=-
- - ++
=-
+
= - +
=
^^ ^
hh h
So rational
6. (a) 4 (b) 14 (c) 16
7.3
3 5 2 15 3- - -
8.3 2 2
2
2
8
3 2 2
2
3 2 2
3 2 2
2
8
2
2
3 2 2
2 3 2 2
2
8 2
9 4 2
6 4 24 2
1
6 4 24 2
6 4 2 4 26
2 2
# #
#
+
+
=
+ -
-+
=
-
-+
=-
-+
=-
+
= - +
=
^
^
h
h
So rational
9. x 3 2= - +^ h 10.4
4 4
b
b b
-
+ +
Test yourself 2
1. (a) 2 y - (b) 4a + (c) 6k5- (d)15
5 3x y + (e) 3 8a b-
(f) 6 2 (g) 4 5
2. (a) 6 6x x+ -] ]g g (b) 3 1a a+ -] ]g g (c) ab b4 2-] g (d) 3)(5 )( y x- + (e) n p2 32 - +^ h (f) 2 )(4 2 )( x x x2- + +
3. (a) 4 6b - (b) 2 5 3x x2 + - (c) 4 17m +
(d) 16 24 9x x2 - + (e) 25 p2 - (f) 1 7a- -
(g) 2 6 5 3- (h) 3 3 6 21 2 7- + -
4. (a)a ab 3 9
822
+ +^ h (b) 215
m 2-] g
5. 157.464V = 6. (a) 17 (b)17
6 15 9-
7.3 2
4 5
x x
x
+ -
+
] ]g g 8. (a) 36 (b) 2- (c) 2 (d) 216 (e) 29. (a)
5
1 (b) 8 10. 11.25d =
11. (a)15
2 3 (b)
2
2 6+
12. (a) 3 6 6 4 3 4 2- - + (b) 11 4 7+
13. (a) 3( 3)( 3)x x- + (b) x x6 3 1- +] ]g g (c) y y y 5 2 2 42+ - +^ _h i
14. (a)3 y x
4
3
(b)3 1
1x -
15. (a) 99 (b) 24 3
16. (a) a b2 2- (b) 2a ab b2 2+ + (c) 2a ab b2 2- +
17. (a) a b 2-] g (b) a b a ab b2 2- + +] ^g h 18.
2
3 3 1+ 19. (a)
4 3
ab
b a+ (b)
10
3 11x -
20.7
21 5 46 2- -
8/20/2019 Answers for maths in focus 3u
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763 ANSWERS
21. (a) 6 2 (b) 8 6- (c) 2 3 (d)3
4 (e) 30a b2
(f)3n
m4 (g) 2 3x y -
22. (a) 2 6 4+ (b) 10 14 5 21 6 10 3 15- - +
(c) 7 (d) 43 (e) 65 6 14-
23. (a)7
3 7 (b)
15
6 (c)
5 1
2
+
(d)15
12 2 6- (e)
53
20 3 15 4 10 3 6+ + +
24. (a)10
10x + (b)
21
17 15a - (c)
( 1)( 1)x x
x3 2
+
-
-
(d)1
1
k - (e)
3
15 6 15 3 15 2- - -
25. (a) 48n = (b) 175n = (c) 392n =
(d) n 5547= (e) 1445n =
26. 3121
71 27. (b), (c) 28. (d) 29. (a), (d) 30. (c)
31. (c) 32. (b) 33. (a) 34. (d) 35. (b)
Challenge exercise 2
1. (a) 2 8 6a b ab a2 2 3- + (b) 4 y 4 -
(c) 8 60 150 125x x x3 2- + -
2.17
17 3 2 5 20+ + 3.
2 2
1
4
2or
4.ab
x
a
bx
a
bx
42
2
2
2 2
+ + += d n 5. (a) x x4 9+ +] ]g g
(b) ( )( )x y x y x y x y x y 3 2 3 3 22 2 2- + = + - +_ _ _i i i (c) 5 7 25 35 49x x x2+ - +] ^g h (d) 2 2 2b a a- + -] ] ]g g g
6. 4 12 9 2 3x x x2 2+ + = +] g 7. x
y
1
1
2 -
+
] g 8. 2 5
9.1
1
a a
a2
2
- +
+] g 10.
2 2x b
ax b
a+ -d dn n
11.x x x
x x x xy y
3 3 2
3 6 3 643 2
- + -
- + + -
] ] ]g g g
12. (a) 8 12 6 1x x x3 2- + - (b)2 13 4xx 2
-
+
] g 13. x x x 97 153 2 + --
14.13
66 6 4 2 15 4 5 65 3- + - + -
15. x x
x3
2
9
1
3
122
+ + = +d n 16. 2x =
17.10
400 59 5- 18. (a) 3
121
71 (b) ,a b
23
17
23
14= = -
19. 12
1i = 20.
4r
4
3 3
r r
r= =
21. 2 6 3s = +
Chapter 3: Equations
Exercises 3.1
1. 5t = - 2. 5.6z = - 3. 1 y = 4. 6.7w = 5. 12x =
6. 4x = 7. 15
1 y = 8. 35b = 9. 16n = - 10. 4r =
11. 9 y = 12. 6k = 13. 2d = 14. 5x = 15. 15 y =
16. 20x = 17. 20m = 18. 4x = 19. 7a = - 20. 3 y =
21. 4b = - 22. 3x = 23. 13
2a = - 24. 4t = -
25. 1.2x = 26. 1.6a = 27. 8
1b = 28. 39t =
29. 5 p = 30. 4.41x Z
Exercises 3.2
1. 33
1b = 2. 35x = 3. 4
9
4 y = 4. 1
35
9x = 5. 5
8
5k =
6. 36x = 7. 0.6t = 8. 3x = - 9. 1.2 y = - 10. 69x =
11. 13w = 12. 30t = 13. 14x = 14. 1x = -
15. 0.4x = - 16. 3 p = 17. 8.2t = 18. 9.5x = -
19. 22q = 20. 3x = - 21. 0.8b = 22. 0.375a = -
23. 3x = 24. 1 y = 25. 13
2t = -
Exercises 3.3
1. 8.5t = 2. 122l = 3. 8b = 4. 41a = 5. 4 y =
6. 6.68r = 7. 6.44x = 8. 15n = 9. 33
2 y
1=
10. 3.7h = 11. (a) 25.39 BMI = (b) 69.66w =
(c) 1.94h = 12. 0.072r = 13. 9x1
= - 14. 2.14t =
15. x 2!= 16. 2.12r = 17. 10.46r = 18. 1.19x =
19. 5.5x = 20. 3.3r =
Exercises 3.4
1. (a) x 32
-4 -3 -2 -1 0 1 2 3 4
(b) y 4#
-4 -3 -2 -1 0 1 2 3 4
2. (a) 7t 2 (b) x 3$ (c) 1 p 2 - (d) x 2$ - (e) 9 y 2 -
(f) a 1$ - (g) y 22
1$ - (h) x 21 - (i) a 6# -
(j) y 121 (k) b 181 - (l) 30x 2 (m) x 34
3#
(n) m 143
22 (o) b 16
4
1$ (p) r 9# - (q) 8z 2
8/20/2019 Answers for maths in focus 3u
9/95
764 Maths In Focus Mathematics Extension 1 Preliminary Course
(r) w 25
41 (s) x 35$ (t) t 9$ - (u) 6q
5
22 -
(v) 1x3
22 - (w) b 11
4
1# -
3. (a) x1 71 1
0 1 2 3 4 5 6 7 8
(b) p2 51#-
-3 -2 -1 0 1 2 3 4 5
(c) x1 41 1
-3 -2 -1 0 1 2 3 4 5
(d) y 3 5# #-
-3 -2 -1 0 1 2 3 4 5
(e) y 6
11
3
21 1
-3 -2 -1 0 1 2 3 4 5
Exercises 3.5
1. (a) x 5!= (b) y 8!= (c) 4 4a1 1-
(d) ,k k1 1$ # - (e) 6, 6x x 12 -
(f) p10 10# #- (g) 0x = (h) ,a a14 142 1 -
(i) 12 12 y 1 1- (j) ,b b20 20$ # -
2. (a) ,x 5 9= - (b) ,n 4 2= - (c) ,a a2 212 -
(d) x4 6# # (e) ,x 3 6= - (f) ,x 5 47
5= -
(g) 3 2 y 2
11 1-
(h) ,x x9 6$ # -
(i) x 12!=
(j) a2 10# #
3. (a) 14
1x = (b) ,a 3
3
1= - (c) 2
3
1b =
(d) No solutions (e) 27
2 y = - (f) 7x = (g) ,m 5 1
3
2=
(h) ,d 22
11
4
3= - (i) , y
5
42= - (j) No solutions
4. (a) ,x 22
1= - (b) 3, 2
3
1 y = (c) 10, 1
5
3a = -
(d) ,x 4 7 3
1= - (e) ,d 4 5= -
5. (a) ,t 3 15
2= - (b) 1 3t
5
21 1-
-3 -2 -1 0 1 2 3 4 5
Exercises 3.6
1. (a) 3x = (b) y 8!= (c) 2n != (d) 2x 5!=
(e) 10 p = (f) x 5!= (g) y 3!= (h) 2w =
(i) n 4!= (j) 2q = -
2. (a) 6.71 p !=
(b) 4.64x =
(c) 2.99n =
(d) 5.92x !=
(e) 1.89 y = (f) .d 2 55!= (g) 4.47k != (h) 2.22x =
(i) . y 3 81!= (j) 3.01 y =
3. (a) 27n = (b) 16t = (c) 32x = (d) 8t =
(e) 243 p = (f) 625m = (g) 216b = (h) 27 y =
(i) 128a = (j) 81t =
4. (a)5
1x = (b)
2
1a = (c)
2
1 y = (d) x
7
1!=
(e)3
2n = (f) 2a = (g) 2x != (h) 9b =
(i) x3
2!= (j) b 1
2
1!=
5. (a) x512
1= (b) 6
4
1x = (c)
81
1a = (d)
625
1k =
(e) x8
1= (f) 4x = (g) 8 y = (h) n 7
32
19=
(i) 8b = (j) 1216
127m =
6. (a) 4n = (b) 5 y = (c) 9m = (d) 5x = (e) 0m =
(f) 3x = (g) 2x = (h) 2x = (i) 1x = (j) k 2=
7. (a) 2x = (b) 1x = (c) 2x = - (d) 2n = (e) 0x =
(f) 6x = (g) y 3
1= (h) 2x = (i) 2x = (j) a 0=
8. (a)2
1m = (b)
3
1x = (c)
3
1x = (d)
2
1k = -
(e)3
2k = - (f)
4
3n = (g) 1
2
1x = (h)
3
2n =
(i)6
1k = - (j) 1
3
2x =
9. (a) x 1= - (b) x 13
1= - (c) k 4= - (d) n 3=
(e) x 21
2= - (f) x
3
2= - (g) x 4
2
1= - (h) x 1
11
7= -
(i) x 15
4= (j) x 18=
10. (a)4
1m = (b) 2
4
3k = - (c) 2
8
3x = (d) 1
2
1k =
(e)18
1n = (f)
2
1n = - (g)
5
4x = (h) 3
6
1b = -
(i) 17
1x = - (j) 5m =
Puzzle
1. All months have 28 days. Some months have more days
as well. 2. 10 3. Bottle $1.05; cork 5 cents
4. 16 each time 5. Friday
Exercises 3.7
1. , y 0 1= - 2. ,b 2 1= - 3. , p 3 5= - 4. ,t 0 5=
5. ,x 2 7= - - 6. q 3!= 7. x 1!= 8. ,a 0 3= -
8/20/2019 Answers for maths in focus 3u
10/95
765 ANSWERS
9. ,x 0 4= - 10. x2
1!= 11. ,x 1 1
3
1= - -
12. , y 1 12
1= - 13. ,b
4
3
2
1= 14. ,x 5 2= - 15. ,x 0
3
2=
16. ,x 1 22
1= 17. 0, 5x = 18. 1, 2 y = - 19. ,n 53=
20. 3, 4x = 21. 6, 1m = - 22. , ,x 0 1 2= - -
23. , , y 1 5 2= - - 24. ,x 5 7= - 25. ,m 8 1= -
Exercises 3.8
1. (a) x 5 2!= - (b) 3a 7!= + (c) 4 y 23!= +
(d) 1x 13!= - (e) p 44 7 2 11 7! != - = -
(f) x 28 5 2 7 5! != + = +
(g) 510 y 88 2 22 2210 2! ! ! -= - = - = ^ h (h) 1x 2!= + (i) 12n 137!= -
(j)3
y 2
5!=
+
2.
(a) . , .x 3 45 1 45= -
(b) . , .x 4 59 7 41= - -
(c) . , .q 0 0554 18 1= - (d) . , .x 4 45 0 449= -
(e) . , .b 4 26 11 7= - - (f) . , .x 17 7 6 34=
(g) . , .r 22 3 0 314= - (h) . , .x 0 683 7 32= - -
(i) . , .a 0 162 6 16= - (j) . , . y 40 1 0 0749= -
Exercises 3.9
1. (a) . , . y 0 354 5 65= - - (b) , .x 1 1 5=
(c) . , .b 3 54 2 54= - (d) , .x 1 0 5= -
(e) . , .x 0 553 0 678= - (f) . , .n 0 243 8 24= -
(g) ,m 2 5= - - (h) ,x 0 7= (i) ,x 1 6= -
(j) . , . y 2 62 0 382=
2. (a) x2
1 17!=
- (b) x
6
5 13!=
(c) q2
4 282 7
!!= =
(d) h8
12 128
2
3 2 2! !=
-=
-
(e) s6
8 40
3
4 10! != =
(f) x2
11 133!=
- (g) d
12
5 73!=
-
(h) x2
2 321 2 2
!!= = (i) t
2
1 5!=
(j) x4
7 41!=
Exercises 3.10
1. , y y 1 02 1 2. x02
11 1 3. 0, 1x x
2
121
4. m07
21 # 5. ,x x
5
30>1 - 6. b2 01#-
7. x1 14
11 1 8. z3
5
131 1- - 9. x2 2
4
31 #
10. ,x x2 26
11 2 11. x4
9
541#- -
12. x13
1115
71 1 13. ,a a3
4
122
11 2- -
14. x2
1
9
51 1 15. , y y 2 11 2- -
16. ,x x87 42# - 17. 4 26 p
21 11
18. ,x x15
12# - - 19. ,t t
5
223
22#
20. 0m9
81 1- 21. 5, 0 1x x1 1 1-
22. ,n n0 2 41 # $ 23. ,x x5 3 01 12 -
24. 2, 1 6m m1# #- - 25. ,x x1 3 41# #-
26. ,x x22
1
3
21# #- 27. ,x x3 1 21$ #-
28. ,n n1 3 51 1 1- 29. ,x x4 3
2
7
41 1 1- - -
30. ,x x2
11 71# #
Exercises 3.11
1. 3 0x 11- 2. 0 4 y 1 1 3. ,n n0 1# $
4. ,x x2 2# $- 5. ,n n1 11 2- 6. n5 3# #-
7. ,c c 1 21 2- 8. x4 2# #- - 9. 4 5x1 1
10. ,b b22
1# $- - 11. ,a a1
3
11 2-
12. , y y 1 2
121 2- 13. ,x x3
21# $
14. ,b b35
21 2- 15. x1
2
1
3
1# #- -
16. y 4 3# #- 17. ,x x4 41 2- 18. a1 1# #-
19. 2 3x 11- 20. ,x x1 3# $- 21. 0 2x1 1
22. a1 12
1# # 23. , y y 2
5
4# $-
24. ,m m13
212
11 2- 25. x1 1
3
1# #
26. 0 x2
111 27. ,x x0
2
11 $
28. , y y 15
41 2- - 29. 3 3n
2
11 #
30. ,x x8 52# - - 31. ,x x5
2
7
31 2
32. ,x x45
152# 33. ,x x1
4
112# - -
34. ,x x3 21 2- 35. x4
3
5
31#- -
8/20/2019 Answers for maths in focus 3u
11/95
766 Maths In Focus Mathematics Extension 1 Preliminary Course
Exercises 3.12
1. ,a b1 3= = 2. ,x y 2 1= = 3. , p q2 1= = -
4. ,x y 6 17= = 5. ,x y 10 2= - = 6. ,t v 3 1= =
7. ,x y 3 2= - = 8. ,x y 64 39= - = - 9. ,x y 3 4= = -
10. ,m n2 3= = 11. ,w w 1 51 2
= - = 12. ,a b0 4= =
13. , p q4 1= - = 14. ,x x1 11 2= = - 15. ,x y 1 4= - = - 16. ,s t 2 1= = -
17. ,a b2 0= - = 18. ,k h4 1= - =
19. ,v v 2 41 2
= - = 20. , .x y 2 1 41Z=
Problem
23 adults and 16 children.
Exercises 3.13
1. ,x y 0 0= = and ,x y 1 1= =
2. ,x y 0 0= = and ,x y 2 4= - =
3. ,x y 0 3= = and ,x y 3 0= =
4. ,x y 4 3= = - and ,x y 3 4= = - 5. ,x y 1 3= - = -
6. ,x y 3 9= = 7. ,t x2 4= - = and ,t x1 1= =
8. ,m n4 0= - = and ,m n0 4= = -
9. ,x y 1 2= = and ,x y 1 2= - = -
10. ,x y 0 0= = and ,x y 1 1= =
11. ,x y 2 1= = and ,x y 1 2= - = - 12. ,x y 0 1= =
13. ,x y 1 5= = and ,x y 4 11= =
14. ,x y 4
14= = and ,x y 1 1= - = - 15. ,t h
2
1
4
1= - =
16. ,x y 2 0= =
17. ,x y 0 0= = and ,x y 2 8= - = - and ,x y 3 27= =
18. ,x y 0 0= = and ,x y 1 1= = and ,x y 1 1= - =
19. ,x y 2
12
4
3= = 20.
13
5,
13
12x y = - = -
Exercises 3.14
1. , ,x y z2 8 1= - = - = - 2. , ,a b c 2 1 2= - = - =
3. , ,a b c 4 2 7= - = = 4. , ,a b c 1 2 3= = = -
5. , ,x y z5 0 2= = = - 6. , ,x y z0 5 4= = - =
7. , , p q r 3 7 4= - = = 8. , ,x y z1 1 2= = - =
9. , ,h j k3 2 4= - = = - 10. , ,a b c 3 1 2= = - = -
Test yourself 3
1. (a) 10b = (b) 116a = - (c) 7x = -
(d) ,x x43
132# - - (e) p 4#
2. (a) 1262.48 A = (b) 8558.59 P =
3. (a) x x x8 16 42 2- + = -] g (b) k k k4 4 22 2+ + = +] g 4. (a) ,x y 2 5= - = (b) ,x y 4 1= = and ,x y
2
18= - = -
5. (a) 2x = (b)4
1 y =
6. (a) ,b 2 1 3
1= -
(b) , g 2 4
1=
(c) ,x x4 3$ #
7. (a) 36 A = (b) 12b = 8. ,x2
11=
9. 1 3 y 1 #-
10. (a) . , .x 0 298 6 70= - - (b) . , . y 4 16 2 16= -
(c) . , .n 0 869 1 54= -
11. (a) 764.5V = (b) 2.9r = 12. x 714
12
13. ,x x2 91 2 14. . , .x y 2 4 3 2= = 15. (a) 2100V =
(b) 3.9r = 16. (a) ii (b) i (c) ii (d) iii (e) iii
17. , ,a b c 3 2 4= = = -
18. ,n n0 33
12 1 -
19. 4x = - 20. 2x = - 21. (a) 3 y 2 (b) n3 0# #-
(c) 2x = (d) 2x = (e) ,x 3 15
2= - (f) 1, 2t t $ # -
(g) 4 2x# #- (h) 3x = - (i) , y y 2 22 1 -
(j) 1, 1x x# $- (k)6
5x = (l)
2
12b# #-
(m) No solutions (n) 23
1,
5
3t = (o) 1 3x1 1-
(p) ,m m3 2# $- (q) ,t t 1 01 2- (r) 1 3 y 1 1
(s) 2 25
2n1 # (t) x
2
1
5
11 #-
Challenge exercise 3
1. 1 y = 2. ,x a x a1 2- 3. ,a b3 2!= =
4. . , .x 2 56 1 56= - 5. , y y 2 0 31# #-
6. ; ,x x x x x x3 3 2 2 4 3 22 !+ - - + + =] ] ] ^g g g h 7. ,x y 1 2= = and ,x y 1 0= - =
8. ; . , .b x4 17 4 8 12 0 123! Z= = + - 9. x 1!=
10. 1 1t 1 1- 11. x3 8# #- 12. 4
1x =
13. 2.31r = 14. No solutions 15. x b a a2!= + +
16. , y y 22
1
3
21# #- 17. 2247.36 P =
18. x3
4 102 !=
^ h 19. , . .x x4 2 2 0 71 1 1- -
20. , y y 15
31 2-
8/20/2019 Answers for maths in focus 3u
12/95
767 ANSWERS
Chapter 4: Geometry 1
Exercises 4.1
1. (a) 47 y c= (b) 39x c= (c) 145m c= (d) 60 y c=
(e) 101b c= (f) 36x c= (g) 60a c= (h) 45x c=
(i) 40 y c= (j) 80x c= 2. (a) 121c (b) 72 29c l
(c) 134 48c l 3. (a) 42c (b) 55 37c l (c) 73 3c l
4. (a) (i) 47c (ii) 137c (b) (i) 9c (ii) 99c (c) (i) 63c
(ii) 153c (d) (i) 35c (ii) 125c (e) (i) 52c (ii)142c
(f) (i) 15c7l (ii) 105c7l (g) (i) 47c36l (ii) 137c36l
(h) (i) 72c21l (ii)162c21l (i) (i) 26c 11 l (ii) 116c 11 l
(j) (i) 38c 15 l (ii) 128c 15 l 5. (a) 49x c= (b) 41c
(c) 131c 6. (a) , y x z15 165c c= = =
(b) , ,x y z142 48 28c c c= = =
(c) , ,a b c 43 137 101c c c= = =
(d) ,,a b d c 97 41 42c c c= = = =
(e) , ,a b c 68 152 28c c c= = = (f) ,a b10 150c c= =
7. 0x x x x
x
x
8 10 2 10 10 7 10 36
18 36020
- + - + + + + =
=
=
(angleof revolution)
( )
( )
ABE x
EBC x
ABE EBC
8 108 20 101502 102 20 1030150 30180
c
cc cc
+
+
+ +
= -
= -
=
= -
= -
=
+ = +
=
ABC` + is a straight angle
( )
DBC x
DBC EBC
7 10
7 20 10
150
150 30180
c
c cc
+
+ +
= +
= +
=
+ = +
=
DBE` + is a straight angle
` AC and DE are straight lines
8.
AFC x
CD bisects
`
`
+ =
AFE+
( )
( )
( )
( )
DFB x
x
CFE x x
x
AFC CFE
AFB
AFB
180 180
180 180 2
is a straight angle
(vertically opposite angles)
is a straight angle
`
c c
c c
+
+
+ +
+
+
= - -
=
= - + -
=
=
9. ABD DBC+ ++
110 3 3 70180
x x
c
= - + +
=
So ABC+ is a straight angle.
AC is a straight line.
10. AEB BEC CED+ + ++ +
y y y 50 8 5 20 3 6090c
= - + - + +
=
So AED+ is a right angle.
Exercises 4.2
1. (a) ,a b e f c d g 148 32c c= = = = = = =
(b) ,x z y 70 110c c= = =
(c) , ,x y z55 36 89c c c= = = (d) , y x z125 55c c= = =
(e) ,n e g a c z x 98c= = = = = = =
82o m h f b d y w c= = = = = = = =
(f) , ,a b c 95 85 32c c c= = =
(g) , ,a b c 27 72 81c c c= = =
(h) , , ,x y z a b56 124 116 64c c c c= = = = =
(i) 61x c= (j) 37 y c=
2. (a) CGF
BFG CGF
180 121
59
59
( is a straight angle) FGH
`
c c
c
c
+
+ +
= -
=
= =
These are equal alternate angles.
AB CD` <
(b) BAC 360 292 68
(angle of revolution)
c c c+ = - =
BAC DCA 68 112
180` c c
c
+ ++ = +
=
These are supplementary cointerior angles.
AB CD` <
(c) 180 76
104104
BCD
ABC BCDc
c
+
+ +
= -
=
= =
( BCE+ is a straight angle)
These are equal alternate angles.
AB CD` ;
(d) 180 128
5252
CEF
CEF ABEc
c
+
+ +
= -
=
= =
( CED+ is a straight angle)
These are equal corresponding angles.
AB CD` ;
(e) 180 23 115CFH + = - +] g ( EFG+ is a straight angle)42c=
42 BFD` c+ = (vertically opposite angles)
ABF BFD 138 42180
c cc
+ ++ = +
=
These are supplementary cointerior angles.
AB CD` ;
8/20/2019 Answers for maths in focus 3u
13/95
768 Maths In Focus Mathematics Extension 1 Preliminary Course
Exercises 4.3
1. (a) 60x c= (b) 36 y c= (c) 71m c= (d) 37x c=
(e) 30x c= (f) 20x c= (g) 67x c= (h) 73a c=
(i) , ,a b c 75 27 46c c c= = =
(j) , ,a b c 36 126 23c c c= = =
(k) , ,x y z w 67 59 121c c c= = = =
2. All angles are equal. Let them be x .
x x x 180Then (angle sum of )D+ + =
xx
3 18060
=
=
So all angles in an equilateral triangle are 60 .c
3. x90 c-] g 4. 50
180 (50 45 )
ACB
ABC
DEC ABC85
85
(vertically opposite angles)
(angle sum of )
`
c
c c c
c
c
+
+
+ +
D
=
= - +
=
= =
These are equal alternate angles.
AB DE` <
5.
124 68
ACB
CBA
CBA
CBA ACB
ABC
180 12456
68 124
5656
is isosceles
( is a straight a ngle)
(exterior angle of )
DCB
`
`
c c
c
c c
c c
c
c
+
+
+
+ +
D
D
= -
=
+ =
= -
=
= =
6. 38 y c=
7. (a) x 64c= (b) ,x y 64 57c c= = (c) 63x c=
(d) ,a b29 70c c= =
8. 180 (35 25 )
120180 120
60
180 (90 30 )
60180 (60 60 )
60
HJI
IJL
JIL
ILJ
(angle sum of )
( is a straight angle)
(angle sum of )
(angle sum of )
HJI
HJL
IKL
JIL
c c c
c
c c
c
c c c
c
c c
c
+
+
+
+
D
D
D
= - +
=
= -
=
= - +
=
= - +
=
Since 60 , IJL JIL ILJ c+ + += = =
IJLD is equilateral
( )
( )
( )
KJL
JLK
KJI
JKL
180 60
120
180 30 120
30
is a straight angle
angle sum of
c c
c
c c c
c
+
+ D
= -
=
= - +
=
° JLK JKL 30` + += =
JKL` D is isosceles
9. BC BD=
BDC 46` c+ = (base angles of isosceles triangle)
CBD 180 2 4688
#
c+ = -
=
CBD BDE 88` c+ += =
These are equal alternate angles.
AB ED` ;
10. 180
32
OQP 75 73
c
+ = - +
=
] g (angle sum of triangle) MNO OQP 32` c+ += =
These are equal alternate angles.
MN QP ` ;
Exercises 4.4
1. (a) Yes
5 AB EF cm= = (given)
6 BC DF cm= = (given)
8 AC DE cm= = (given)
ABC DEF ` /D D (SSS )
(b)Yes
4.7 XY BC m= = (given)
XYZ BCA 110c+ += = (given)
2.3YZ AC m= = (given)
XYZ ABC` /D D (SAS )
(c) No
(d) Yes
PQR SUT 49c+ += = (given)
PRQ STU 52c+ += = (given)
8QR TU cm= = (given)
PQR STU ` /D D ( AAS )
(e) No
2. (a)
,
AB KL
B L
BC JL
ABC JKL
438
5by SAS
(given)
(given)
(given)
`
c+ +
/D D
= =
= =
= =
(b)
,
Z B XY AC
YZ BC
RHS XYZ ABC
907
2
by
(given)
(given)
(given)
`
c+ +
/D D
= =
= =
= =
(c)
,
MN QRNO PR
MO PQ
MNO PQR
8
85
by SSS
(given)
(given)
(given)
` /D D
= =
= =
= =
(d)
.
Y T
Z S
XY TR XYZ STR
90
35
1 3by AAS,
(given)
(given)
(given)
`
c
c
+ +
+ +
/D D
= =
= =
= =
(e)
,
BC DE
C E AC EF
ABC DEF
4
907
by SAS
(given)
(given)
(given)
`
c+ +
/D D
= =
= =
= =
3. (a) B C BDA CDA
AD
ABD ACD
90
is common
by AAS,
(base anglesof isosceles )
(given)
`
c
+ +
+ +
/D D
D=
= =
8/20/2019 Answers for maths in focus 3u
14/95
769 ANSWERS
(b) BD DC AD BCbisects
(corresponding sides in congruent s)`
`
D=
4. , ) AB CD ABD BDC ernate angles+ +
8/20/2019 Answers for maths in focus 3u
15/95
770 Maths In Focus Mathematics Extension 1 Preliminary Course
̀ since 3 pairs of angles are equal, | ADED|| ABCD
(b) . , .x y 2 17 2 25= =
8. ( , )
( , )
( )
ABF BEC
CBE BFA
C A
s AB CD
BC AD
s
alternate angle
similarly
angle sum of `
+ +
+ +
+ +
zz
D
=
=
=
`
since 3 pairs of angles are equal, | CEBD
|| ABF D
9. A+ is common
..
..
AB
AD
AC
AE
AB
AD
AC
AE
3
1 20 4
2
0 80 4
`
= =
= =
=
Since two pairs of sides are in proportion and their
included angles are equal, | , . ABC m 4 25D =|| AEDD
10..
.
. .
.
..
CD
AB
AC
BC
AD
AC
CD
AB
AC
BC
AD
AC
2 6
20 769
3 9
3
0 769
5 07
3 90 769
`
= =
= =
= =
= =
Since three pairs of sides are in proportion,
,c| , ACD x y 109 47cD = =|| ABCD
11. (a) 7.8x = (b) . , .m p4 0 7 2= = (c) 6.5x =
(d) . , .x y 6 2 4 4= = (e) . , .x y 1 4 9 2= =
12. (a) BC
AB
DE
AD
DE AD
FG AF
BC
AB
FG
AF
Also
`
=
=
=
(b) AC
AB
AE
AD
AE AD
AG AF
AC
AB
AG
AF
Also
`
=
=
=
(c)CE
BD
AE
AD
AE
AD
EG
DF
CE
BD
EG
DF
Also
`
=
=
=
13. . , .a b4 8 6 9= = 14. 0.98 y = 15. . , .x y 3 19 1 64= =
Exercises 4.6
1. (a) 6.4x = (b) 6.6 y = (c) 5.7b = (d) 6.6m =
2. (a) 61 p = (b) 58t = (c) 65x = (d) 33 y =
3. .s 6 2 m= 4. .CE 15 3cm=
5. 81, 144, 225 AB CB CA2 2 2= = =
AB CB
CA
81 144
225
2 2
2
+ = +
=
=
ABC` D is right angled
6. 1 XY YZ = =
XYZ ` D is isosceles
,YZ XY XZ
YZ XY
XZ
1 2
1 1
2
2 2 2
2 2
2
= = =
+ = +
=
=
XYZ ` D is right angled
7. AC AB BC
BC
BC
BC BC
AC
BC
2 3
4 3
11
2
2 1
2
2 2 2
2 2 2
2
2
`
#
= +
= +
= +
=
=
=
=
=
^ h
8. (a) 5 AC =
(b) , , AC CD
AD
25 144
169
2 2
2
= =
=
25 144169
AC CD
AD
2 2
2
+ = +
=
=
ACD` D has a right angle at ACD+
AC` is perpendicular to DC
9. AB b3= 10. x
x y 2 2+
11. d t t
t t t t
t t
20 3 15 2
400 120 9 225 60 4
13 180 625
2 2 2
2 2
2
= - + -
= - + + - +
= - +
] ]g g
12. 1471 mm 13. 683 m 14. 12.6 m 15. 134.6 cm
16. 4.3 m 17. 42.7 cm
18. 1.3 1.1 2.9 1.5 2.25and2 2 2+ = = . . .1 3 1 1 1 52 2 2!+ so the triangle is not right angled
` the property is not a rectangle
19. No. The diagonal of the boot is the longest available
space and it is only 1.4 m.
20. (a) 6 4 BC2 2 2= -
20=
20 BC =
6 AO cm= (equal radii)
So 6 4 AC2 2 2= -
20=
20 AC =
Since , BC AC= OC bisects AB
(b) OCA OCB 90c+ += = (given)
OA OB= (equal radii)
OC is common
OAC OBC` /D D ( RHS )
So AC BC= (corresponding sides in congruent triangles)
` OC bisects AB
Exercises 4.7
1. (a) x 94c= (b) y 104c= (c) x 111c= (d) x 60c=
(e) y 72c= (f) °, °x y 102 51= = (g) °, °x y 43 47= =
8/20/2019 Answers for maths in focus 3u
16/95
771 ANSWERS
2. ABED is isosceles.
( s )
( )
B E
CBE DEB76
180 76
104
base equal
straight s
` c
c c
c
+ +
+ +
+
+
= =
= = -
=
D
D D
62 104 104 360
270 360
90
(angle sum of quadrilateral)c c c c
c c
c
+
+
+
+ + + =
+ =
=
` CD is perpendicular to AD ̀
3. (a)
( )( , )
( , )
( , )
D x
C x
x
x
A C x B x
B D x
A D AB DC
C D AD BC
B C AB DC
180
180 180
180 180
180180
and cointerior angles
and cointerior angles
and cointerior angles
`
`
c
c c
c c
c
c
+
+
+ +
+
+ +
+ +
+ +
+ +
<
<
<
= -
= - -
= - +
=
= =
= -
= = -
(b) x x x x180 180360
Angle sum c cc
= + + - + -
=
4. ,a b150 74c c= =
5. (a) 5 , 3 , 108 , 72a b x z y m m c c= = = = =
(b) , ,x y z53 56 71c c c= = =
(c) 5 , 68x y cm ca b= = = =
(d) , ,121 52 77c c ca b i= = =
(e) 60x c= (f) ,x y 3 7= =
6. ( )
, )
,
ADB CDB
CDB ABD ADB DBC
ABD DBC
BD ABC
BD ADC
AB DC
AD BC
bisects
bisects
(alternate angles
(alternate angles )
`
`
+ +
+ +
+ +
+ +
+
+
<
<
=
=
=
=
7. (a) ..
AD BC AB DC
3 85 3
cmcm
(given)(given)
= =
= =
Since two pairs of opposite sides are equal, ABCD is a
parallelogram.
(b) AB DC
AB DC
7cm (given)
(given)<
= =
Since one pair of opposite sides is both equal and
parallel, ABCD is a parallelogram.
(c) 54 126180
X M c c
c
+ ++ = +
=
These are supplementary cointerior angles.
XY MN ` <
XM YN Also, (given)<
` XMNY is a parallelogram
(d) AE EC
DE EB56
cmcm
(given)
(given)
= =
= =
Since the diagonals bisect each other, ABCD is a
parallelogram.
8. (a) ,x 5 66cm ci= = (b) , ,90 25 65c c ca b c= = =
(c) ,x y 3 5cm cm= = (d) ,x y 58 39c c= =
(e) x 12cm=
9. 6.4 cm 10. 59 , 31 , 59 ECB EDC ADEc c c+ + += = =
11. 4 2 cm 12. 57x y c= =
Exercises 4.8
1. (a) 540c (b) 720c (c) 1080c (d) 1440c (e) 1800c
(f) 2880c 2.
(a) 108c (b) 135c (c) 150c (d) 162c (e) 156c 3. (a) 60c (b) 36c (c) 45c (d) 24c
4. 128 34c l 5. (a) 13 (b) 152 18c l 6. 16 7. 3240c
8. 2340c 9. 168 23c l
10. ( )
.
n n
n n
n
n
145
145 180 36035
10 3
2 180Sum # c= = -
= -
=
=
But n must be a positive integer.
` no polygon has interior angles of 145 .c
11. (a) 9 (b) 12 (c) 8 (d) 10 (e) 30
12. (a) ABCDEF is a regular hexagon.
AF BC= (equal sides)
FE CD= (equal sides)
AFE BCD+ += (equal interior angles)
AFE BCD` /D D (SAS )
(b)
( )
S n
AFE
6
720
6
720
120
2 180
2 180
#
#
c
c
c
c
c
+
= -
= -
=
=
=
] g
Since , AF FE= triangle AFE is isosceles.
So FEA FAE+ += (base angles in isosceles triangle)
FEA2
180 120
30
` c
c
+ =-
=
(angle sum of triangle)
ED A 120 3090
c
c
+ = -
=
Similarly, DE B 90c+ =
So ED DE A B 180c+ ++ =
These are supplementary cointerior angles
AE BD` <
13. A regular octagon has equal sides and angles.
AH AB= (equal sides)
GH BC= (equal sides)
AHG ABC+ += (equal interior angles) AHG ABC` /D D (SAS )
So AG AC=
(corresponding sides in congruent triangles)
( )
S n
8
1080
2 180
2 180#
#
c
c
c
= -
= -
=
] g
AHG8
1080
135
` c
c
+ =
=
HGA HAG+ += (base angles in isosceles triangle)
8/20/2019 Answers for maths in focus 3u
17/95
772 Maths In Focus Mathematics Extension 1 Preliminary Course
HAG2
180 135
22 30
` c
c
+ =-
= l
(angle sum of triangle)
GAC 135 2 22 3090
# c
c
+ = -
=
l
We can similarly prove all interior angles are 90c and
adjacent sides equal.
So ACEG is a square.
14. EDC5
5
108
2 180# c
c
+ =-
=
] g
ED CD= (equal sides in regular pentagon)
So EDC is an isosceles triangle.
DEC ECD`+ += (base angles in isosceles triangle)
36
DEC2
180 108c
c
+ =-
=
(angle sum of triangle)
108 3672
AEC cc
+ = -
=
Similarly, using triangle ABC , we can prove that
72 EAC c+ =
So EAC is an isosceles triangle.
(Alternatively you could prove EDC and ABC congruent
triangles and then AC EC= are corresponding sides in
congruent triangles.)
15. (a) p
360
(b) Each interior angle:
180360
180 360
180 360
180 2
p
p
p
p
p
p
p
p
-
= -
=
-
=
-^ h
Exercises 4.9
1. (a) .26 35m 2 (b) .21 855 cm2 (c) .18 75 mm2 (d) 45 m2
(e) 57cm2 (f) 81 m2 (g) .28 27 cm2 2. .4 83 m2
3. (a) .42 88 cm2 (b) .2 9 5 m2 (c) .32 5cm2 (d) .14 32 m2
(e) .100 53cm2 4. (a) 25 m2 (b) .101 85 cm2
(c) .2 9 4 m2 (d) .10 39 cm2 (e) 45cm2
5. 7 51 98 7 51 14 cm2+ = +^ h 6. .22 97 cm2 7. $621.08 8. (a) .161 665 m2 (b) 89 m2 (c) 10.5 m
9. (a) 48 cm (b) 27 cm 10. w 12 units2
Test yourself 4
1. (a) , ,x y z43 137 147c c c= = = (b) 36x c=
(c) , ,a b c 79 101 48c c c= = = (d) 120x c=
(e) 7.2r cm= (f) 5.6 , 8.5x y cm cm= = (g) 45ci =
2. ) AGF HGB(vertically opposite+ +i=
AGF CFESo + + i= =
These are equal corresponding .s+
AB CD` <
3. 118.28 cm2
4. (a)
( )
DAE BAC ADE ABC
AED ACB
ABC ADE AAAand are similar
(common)
(corresponding angles, DE BC)
(similarly)
`
+ +
+ +
+ +
<
D D
=
=
=
(b) 3.1 , 5.2x y cm cm= =
5. 162c 6. 1020.7 cm3 7. 36 m
8. (a) AB AD BC DC
(adjacent sides in kite)
(similarly)
=
=
AC is common
` ∆ ABC and ∆ ADC are congruent (SSS)
(b) AO CO BO DO
AOB COD
(equal radii)
(similarly)
(vertically opposite angles)+ +
=
=
=
` ∆ AOB and ∆ COD are congruent (SAS)
9. 73.5 cm2
10. 6 2 7 36 28 64 822 2
+ = + = =^ h ` ∆ ABC is right angled (Pythagoras)
11. AG
AF
AE
AD
AE
AD
AC
AB
AG
AF
AC
AB
(equal ratios on intercepts)
(similarly)
`
=
=
=
12. (a)(base s of isosceles+ D)
( , )
AB AC
B C
BD DC AD BC
(given)
bisects given
+ +
=
=
=
ABD ACD SAS` /D D ] g (b)
180 ADB ADC
ADB ADCBut(corresponding s in congruent s)
(straight )c
+ +
+ +
+
+
D=
+ =
So 90 ADB ADC c+ += =
So AD and BC are perpendicular.
13.
34˚ 34
( )
( )
ACBCAD
CAD ADC
6868 34
base s of isosceles
exterior of
`
c
c c
cc
+
+
+ +
+
+
D
D
=
= -
=
= =
So ∆ ACD is isosceles base s equal+^ h 14.
8/20/2019 Answers for maths in focus 3u
18/95
773 ANSWERS
( , )
, )
DAC ACB BAC ACD
AD BC
s AB DC
alternate s
(alternate
+ +
+ +
+
+
<
<
=
=
AC is common
ABC ADC AB DC
(AAS)(corresponding sides in congruent s)
`
`
/D DD=
Similarly, AD BC=
` opposite sides are equal
15. (a) 24 cm2 (b) 5 cm 16. 9
17. BFG FGD x x109 3 3 71
180
c c
c
+ ++ = - + +
=
These are supplementary cointerior .s+
AB CD` <
18. 57 cm2
19. (
(
( )
)
)
ACB A B
x y ACD ACB
z x y
x y x y
180
180
180
180 180
180 180
sum of
straight
c
c
c
c c
c c
+ + +
+ +
+
+
D= - +
= - -
= -
= - - -
= - + +
= +
] g
20. (a)
.
..
.
..
A E
EF
AC
DE
AB
EF
AC
DE
AB
2 7
2 971 1
3 6
3 961 1
given
`
+ +=
= =
= =
=
^ h
So ∆ ABC and ∆ DEF are similar (two sides in proportion,
included s+ equal).
(b) 4.3x cm=
Challenge exercise 4
1. 94c 2. , ,x y z75 46 29c c c= = = 3. ,1620 32 44c c l
4.
, )
( )
BAD DBC
ABD BDC
ADB DCB
AB DC
(given)
(alternate angles
angle sum of `
+ +
+ +
+ +
<
D
=
=
=
` since 3 pairs of angles are equal,
BCDD; ABD
8/20/2019 Answers for maths in focus 3u
19/95
774 Maths In Focus Mathematics Extension 1 Preliminary Course
14. (a) AB BC ABE CBE 45
(adjacent sides in square)
(diagonals in square make 45 with sides)c
+ +
=
= =
EB is common.
, ABE CBE AE CE
by SAS(corresponding sides in congruent s)
`
`
/D D
D=
Since AB BC= and , AE CE= ABCE is a kite.
(b) BD x x
x
x
DE BD
x
2
2
2
1
2
2units
2 2
2
= +
=
=
=
=
Practice assessment task set 1
1. 9 p = 2. 2 5 y x y + -^ ^h h 3. (a) x 1- (b) 3x 4 4. 6 10 y - 5.
23
25 5 2+ 6. 2 16 3x x x3 2+ - +
7.7
2x = 8.
3
2
x -
9. ° ABC EDC ACB ECD
AB ED
ABC EDC AC EC
ACE
90
by AAS
is isosceles
(given)
(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. x
24 2 312! != = 22.
491
23. 4, 11 1, 4x y x y or= = = - = - 24. ,x y 2 1= = -
25. 7 26. 7.02 cm 27. 2 1 4 2 1x x x2- + +] ^g h
28.43
6 15 2 6+ 29. 7 30. $643.08 31. 1.1
32. 2 10 3 5 2 2 3- + - + 33. $83.57
34. , ,x y w z22 29 90c c c= = = = 35. 56.7 cm2
36. a ba
b21 1021
10
=- 37. ,x x6 2
5
22 1 - 38.
8
1
39. x 7- - 40. 4
1x = 41. ,x x3 3# $- 42.
6
1
43. Given diagonal AC in rhombus ABCD :
)
)
AB BC DAC ACB
BAC ACB
DAC BAC
AD BC
ABC
(adjacent sides in rhombus)
(alternate s,
(base s of isosceles
`
+ +
+ +
+ +
+
+
<
D
=
=
=
=
` diagonal AC bisects the angle it meets.
Similarly, diagonal BD bisects the angle it meets.
44. x 3 1+ -] g 45. 6 12 8x x x3 2+ + + 46. 2
517
4
47. 53x c=
48. ,x y 98 41c c= = 49. x0 51 1 50. 3 2
1
x +
51. (a) 12 8x y -
(b) 2 31 (c) 3 9
3
x x
x
2 - +
-
(d) 3 2 1+
(e)1 1
5
x x
x
+ -
- +
] ]]g g
g (f)
6
11 3 (g) x y z
x z
y 14 7 11
14 11
7
=- -
(h)5 1 2
3
a a b b+ +] ]g g (i) 8 5 (j) 13 21
52. . , .x y 2 7 3 1= = 53. 25x = 54. r 2
cm3 r
=
55. 17.3 cm
56. DEA x EAD x
CD x x
x ABC x
ABC DEA
A
222
LetThen (base s o f isosceles )
(exterior of )
(opposite s of gram are equal)
EAD
`
`
+
+
+
+
+ +
+
+
+ <
D
D
=
=
= +
=
=
=
57.5
2 58. 5% 59. 2.2 10 kmh8 1# - 60. 20k =
61. 9xy y 62. 147 16c l 63. 5.57 m2
64. (a) a b a a ab b b5 2 2 4 2 4 4 42 2+ - - - + + +] ^g h (b) 3 4 6 2a b a b c + - +] ]g g
65. x18
15
4
31#-
66.( BCEF is a gram)<( BC AD ABCD
BC FE
AD FE
is a gram)
`
<
8/20/2019 Answers for maths in focus 3u
20/95
775 ANSWERS
69.75
18 3 31 2 25 5+ - 70. 20 71. 32 m
72. BD bisects AC
So AD DC=
90 BDC BDA c+ += = (given)
BD is common
BAD BCD` /D D (SAS )
AB CB` = (corresponding sides in congruent
triangles)
So triangle ABC is isosceles
73.2
x y 2 2+ 74. (b) 75. (c) 76. (a) 77. (b) 78. (b)
79. (d) 80. (d)
Chapter 5: Functions and graphs
Exercises 5.1
1. Yes 2. No 3. No 4. Yes 5. Yes 6. Yes 7. No
8. Yes 9. Yes 10. No 11. Yes 12. No 13. Yes
14. No 15. Yes
Exercises 5.2
1. 4, 0f f 1 3= - =] ]g g 2. , ,h h h0 2 2 2 4 14= - = - =] ] ]g g g 3. 25, 1, 9, 4f f f f 5 1 3 2= - - = - = - - = -] ] ] ]g g g g 4. 145. 35- 6. 9x = 7. x 5!= 8. x 3= - 9. ,z 1 4= -
10. 2 9, 2 2 9f p p f x h x h= - + = + -^ ]h g 11. 1 2 g x x2- = +] g 12. f k k k k1 12= - + +] ] ^g g h 13. ; ,t t 1 2 4= - = - 14. 0
15. 125, 1, 1f f f 5 1 1= = - = -] ] ]g g g 16. 0 4 1 3f f f 2 2 1- - + - = - + = -] ] ]g g g 17. 10 18. 7 19. 28-
20. (a) 3 (b) 3 3 3 0x - = - =
Denominator cannot be 0 so the function doesn’t exist
for .x 3= (c) 4
21. 2 5f x h f x xh h h2+ - = + -] ]g g 22. 4 2 1x h+ + 23. x c 5 -] g 24. 3 5k2 + 25. (a) 2 (b) 0 (c) 2n n4 2+ + Exercises 5.3
1. (a) x -intercept3
2 , y -intercept -2
(b) x -intercept -10, y -intercept 4
(c) x -intercept 12, y -intercept 4
(d) x -intercepts 0, -3, y -intercept 0
(e) x -intercepts 2! , y -intercept -4
(f) x -intercepts -2, -3, y -intercept 6
(g) x -intercepts 3, 5, y -intercept 15
(h) x -intercept 53- , y -intercept 5
(i) x -intercept -3, no y -intercept
(j) x -intercept ,3! y -intercept 9
2. 2
( )
f x x
x
f x
2
even function
2
`
- = - -
= -
=
2] ]g g
3. (a) 1f x x2 6= +^ h (b) f x x x2 12 6 3= + +] g7 A (c) 1f x x3- = - +] g (d) Neither odd nor even
4.
( )
g x x x x
x x x g x
3 2
3 2
even function
8 4 2
8 4 2
`
- = - + - - -
= + -
=
] ] ] ]g g g g
5. f x x f x- = - = -] ]g g ` odd function
6. 1
( )
f x x
xf x 1
even function
2
2
`
- = - -
= -
=
] ]g g
7. f x x x
x x
x x
f x
4
4
4
odd function
3
3
3
`
- = - - -
= - +
= - -
= -
] ] ]^]
g g gh
g
8. f x x x
x x
f x
even function
4 2
4 2
`
- = - + -
= +
=
] ] ]]
g g gg
0f x f x- - =] ]g g 9. (a) Odd (b) Neither (c) Even (d) Neither (e) Neither
10. (a) Even values i.e. , , ,n 2 4 6 f=
(b) Odd values i.e. , , ,n 1 3 5 f=
11. (a) No value of n (b) Yes, when n is odd (1, 3, 5, …)
12. (a) (i) x 02 (ii) x 01 (iii) Even
(b) (i) x 21 (ii) x 22 (iii) Neither
(c) (i) x2 21 1- (ii) ,x x2 21 2- (iii) Neither
(d) (i) All real x 0! (ii) None (iii) Odd
(e) (i) None (ii) All real x (iii) Neither
Exercises 5.4
1. (a) x -intercept 2, y -intercept -2
(b) x -intercept 12
1- , y -intercept 3
(c) x -intercept2
1 , y -intercept 1
(d) x -intercept -3, y -intercept 3
(e) x -intercept3
2 , y -intercept
3
1-
8/20/2019 Answers for maths in focus 3u
21/95
776 Maths In Focus Mathematics Extension 1 Preliminary Course
2. (a)
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(b) y
x
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-1 1
(c) y
x
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-1 1
(d) y
x
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(e) y
x
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-1 1
1
2
(f) y
x
-4
-5
-3 -2-1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(g) y
x
-4
-5
-3-2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
2
3-
(h) y
x
-4
-5
-3-2-1 2 3 4
2
1
3
4
5
-3
-4
-2
-1 1
8/20/2019 Answers for maths in focus 3u
22/95
777 ANSWERS
(i) y
x
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-1 1
(j) y
x
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-1 11
1
2
3. (a) ,x y all real all real" ", , (b) :,x y y 2all real =" ", , (c) : ,x x y 4 all real= -! "+ , (d) : ,x x y 2 all real=! "+ , (e) , :x y y 3all real =
! "+ ,
4. (a) Neither (b) Even (c) Neither (d) Odd (e) Odd
5. y
x
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-111
1
2
(3, -1)
Exercises 5.5
1. (a) x -intercepts 0, -2, y -intercept 0
(b) x -intercepts 0, 3, y -intercept 0
(c) x -intercepts ! 1, y -intercept -1
(d) x -intercepts -1, 2, y -intercept -2
(e) x -intercepts 1, 8, y -intercept 8
2. (a) y
x
-4
-5
-3 -2-1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2
-11
(b) y
x
-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2
-1 1
(c) y
x
-4
-5
-3-
2-
1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2
-1 1
8/20/2019 Answers for maths in focus 3u
23/95
778 Maths In Focus Mathematics Extension 1 Preliminary Course
(d) y
x
-4
-5
-3-2-1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2
-1 1
(e) y
x
-4
-5
-3-2-1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2
-1 1
(f) y
x
-4
-10
-3-2-1 2 3 4 5
4
6
8
2
10
12
-6
-8
-4
-21
(g) y
x
-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-6
-2
-1 1
(h) y
x
-5
-3-4 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-6
-2
-1 1
(i) y
x
-4
-5
-3 -2 -1 3 4 5
2
1
3
4
5
-3
-4
-6
-2
-1 211
1
2
(j) y
x
-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-6
-2
-1 1
3. (a) (i) x -intercepts 3, 4, y -intercept 12 (ii) {all real x },: y y
4
1$ -( 2
(b) (i) x -intercepts 0, -4, y -intercept 0 (ii) {all real x },
: y y 4$ -" , (c) (i) x -intercepts -2, 4, y -intercept -8 (ii) {all real x },
: 9 y y $ -" , (d) (i) x -intercept 3, y -intercept 9 (ii) {all real x }, : y y 0$" , (e) (i) x -intercepts ,2! y -intercept 4 (ii) {all real x },
: y y 4#" , 4. (a) {all real x }, : y y 5$ -" , (b) {all real x }, : y y 9$ -" ,
8/20/2019 Answers for maths in focus 3u
24/95
779 ANSWERS
(c) {all real x }, : y y 24
1$ -( 2 (d) {all real x }, : y y 0#" ,
(e) {all real x }, : 0 y y $" ,
5. (a) y 0 9# # (b) y 0 4# # (c) y 1 24# #-
(d) y 4 21# #- (e) y 18 24
1# #-
6. (a) (i) x 02 (ii) x 01 (b) (i) x 01 (ii) x 02
(c) (i) x 02 (ii) x 01 (d) (i) x 21 (ii) x 22
(e) (i) x 52 - (ii) x 51 -
7.
( )
f x x
x
f x
even
2
2
`
- = - -
= -
=
] ]g g
8. (a) Even (b) Even (c) Even (d) Neither
(e) Neither (f) Even (g) Neither
(h) Neither (i) Neither (j) Neither
Exercises 5.6
1. (a) x -intercept 0, y -intercept 0
(b) No x -intercepts, y -intercept 7
(c) x -intercepts ,2! y -intercept -2
(d) x -intercept 0, y -intercept 0
(e) x -intercepts ,3! y -intercept 3
(f) x -intercept -6, y -intercept 6
(g) x -intercept3
2 , y -intercept 2
(h