756

Maths In Focus Mathematics Extension 1 Preliminary Course

Answers
8.

o oo oo o (a) 0.83 (b) 0.07 (c) 0.13 (d) 0.16 o oo o or 0. 142857 (h) 1.18
(g) 0.142857

9.

(a)

8
9

(h)

Chapter 1: Basic arithmetic

13
60

Problem
5

Exercises 1.1
1.

2.

(a) Rational (b) Rational
(e) Rational (f) Irrational
(i) Rational (j) Irrational

(e) - 4.3

(a) 18 (b) 11 (c) 6 (d) 11
(h) 1

3.

(c) Rational
(g) Irrational

19
20

(i) 2

(j) 3

(d) Irrational
(h) Rational

(f) −1

(g) 2

7
15

1
3

(f) 0.17 (g) 0.36 (h) 1.20 (i) - 4.27
1300

8.

5. 950

600

16. 1.7

6. 3000

(j) 8.16

1.

7. 11 000

8.

17. 79 cents 18. 2.73 19. 1.1 20. 3.6 m

21. $281.93

22. 1.8 g

(b) 2

3
20
(d)

12. 0.73 13. 33 14. 3.248 15. 4.21

Exercises 1.2

6.

- 1.2

10. - 2
15. 5

3. - 56

4. 10

(a)

7. - 7.51

8. - 35.52

11. - 7

12. −23

13. 10

(b)

17. 1

14. 1

18. 60 19. −20 20. 9

51
1000

(c) 5

1
20

(d) 11

7
20
3
(e)
5

(a)

4.

7
18

(d) 2

6
11

(g)

7
45

oo
(e) 1.72
4
45

(e)

14. 17.5%

3
28

17
20

3. (a)

15. 41.7%

(b)

7
10

(c) 1

7. $65

179 cm 9. (a) 11.9 (b) 5.3 (c) 19 (d) 3.2 (e) 3.5
(f) 0.24 (g) 0.000 18 (h) 5720 (i) 0.0874 (j) 0.376

15. 402.5 g
19. 573

12. 1152.125 g

16. 41.175 m

13. $10.71

17. $30.92

20. $2898

5 minutes after 1 o’clock.
11

Exercises 1.5
1
64

1.

(a) 500

(b) 145

(c)

(a) 13.7

(b) 1.1

(c) 0.8

(a) a 17

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

5.

1
(a) 35% (b) 33 %
3

6.

(a) 124% (b) 70%

7.

(a) 0.52;

(d) 1.09; 1

(f)

1
20
7
4. $547.56 5. 714.3 g 6.
24

2.

(g) y 6

3.

13
25

8
11

(c)

3.

4
5

o
(a) 0.4 (b) 1.875 (c) 0.416
(b)

1
8

67
99

oo
(d) 0.68

2.

2.

1
50

o
(c) 0.73

14. 5.9%

5

9. 6.57

16. 3

16
25

(e)

37
495

11. 54.925 mL

5. - 4

Exercises 1.3
1.

7
9

Problem

2. - 11

4
15

(j) 1

13. 77.5%

18. 3.2 m

1

(d) 3

10. $52.50

23. $3.20

24. (a) 7.95 (b) 30.03 (c) 0.37 (d) 5.74 (e) 0.52 25. 0.2

1.

217
990

5
9

Exercises 1.4

9. $8 000 000 10. $34 600 000

11. 844 km

5
8

(c) 1

(b) 7.4

12. 74%

(a) 16.36 (b) 21.87 (c) 8.80 (d) 22.71 (e) - 13.20

4.

(i)

o
10. (a) 0.5
11. (a)

2
9

(b)

oo
(f) 0.15

o
(e) 0.6

3
8

(c)

1
1000

(d) 1

4.

(d) 0.1%

(e) 0.434;

(c) 0.168;
217
500

21
125

(f) 0.1225;

(h) p - 1
5.

49
400

(a) x14

(a) p5q15
(f) x4y10

y

(d) 2.7

(c) a - 4

(i) 4x 10

(d) w

(j) 81y - 8

(o) x -3

(n) p 5

(q) x - 5 y 2 or

(c) 40.5% (d) 127.94%
7
100

(h) x 21

(m) w 10

97
1000

2
(c) 226 %
3

(b) 0.07;

9
100

oo
(d) 0.63

(b) y 0 = 1

(d) 3

2

(e) 2
(e) - 2.6
(e) x 5
(k) a

(p) a - 2 b 3 or

(f) 0.5

(f) p 10
(l)

x 10

b3

y 45

a2

x5

(b) a -7

(c) m4 (d) k10 (e) a -8

(f) x

(g) mn2

(i) 9x22 (j) x21
(b)

a8
8

b
2k 23
(g)
27

(c)

64a 3 b 12

(d) 49a10b2 (e) 8m17

(h) 16y47 (i) a3 (j) 125x - 21 y 18

ANSWERS

6.

4

1
2

7. 324

8. 2

10
27

9. (a) a3b

3

1
25

(b)

-

2

(a) x 2

6.

5

5

(c) x 3

1
2

5.

(d) x 3

(e) x 4

(a) x + x 2 + 2x 2

(b) x

3

7
(b)
32

10. (a) pq r

2 2

14.

1
81

4
11.
9

1
108

15.

1
12.
18

1
12

16.

4
13.
27

5 22

17.

49
3888

18.

2 58

(d) x + x - 1 + 2
7.

Exercises 1.6
1.

2.

3.

(d)

1
1
1
1
1
(b)
(c)
(d)
(e)
(f) 1
4
27
343
10 000
256
1
1
1
1
1
1
(g)
(h)
(i)
(j)
(k)
(l)
(m) 1
7
64
9
32
81
81
1
1
1
1
(n)
(o)
(p)
(q)
(r) 1
36
125
100 000
128
1
1
(s)
(t)
64
64

1.

(e) x

1

- 3x

^ y - 3 h2

3

(e)

3 4 ^ x + y h5

1
2

1

(b)

a - 2b

-

-

3
2

+x

(c)

-

5
2

4

7

] 6a + 1 g4

6

7 9 ] 3x + 8 g2

(a) m - 3

(b) x - 1

-4

(h) 3y

(c) p - 7
−2

3t - 8
(j)
5

2.

]x + 1g
4

(h)

1
5

t
5
x7

(m)

(t)
1

(b)

(c)

6

x
1

(i)

1 y 3

]k - 3g

(d)

1 n 8

(e)

1

(k)

2 x (f)

(g)

1
(l)
8y + z

] x + 1 g6

(o) x5 (p) y10 (q)

^ 3x + 2y h x-y 3x + y 7 o (s)
(t) e x+y 2w - z

(r) ] a + b g2

9

3 m 4

(g) 7.6#10

3.

(a) 3 y

(b) 3 y 2 or _ 3 y i

(c)

(f) 3 6q + r

(g)

2

1

1

3

(a) t 2

(b) y 5

(c) x 2

-

(i) ] x - 2 g

-

2
3

1
2

2a 2
2
y - 1k
3

x

-

3
2

(d)
1

5

] x + 7 g2
1

1

(j)

1

(d) ] 9 - x g 3

(g) ^ 5x - y h

1

(l)

(e) 2 #10

-6

(h) 2.3#10

(c) 4 # 10 - 3

(f) 8#10 - 8
-1

(i) 8.5#10 - 3

(j) 7#10 - 11

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
(a) 6.61

1.305 # 10 10

(b) 0.686

(c) 8.25

(d) 1.30

7. 6.51 # 10 - 10

Exercises 1.9
1.

(a) 7 (b) 5 (c) 6 (d) 0 (e) 2 (f) 11 (g) 6 (h) 24
(i) 25 (j) 125 2. (a) 5 (b) −1 (c) 2 (d) 14 (e) 4
(f) −67 (g) 7 (h) 12 (i) −6 (j) 10 3. (a) 3 (b) 3
(c) 1 (d) 3 (e) 1 4. (a) a (b) - a (c) 0 (d) 3a
(e) −3a (f) 0 (g) a + 1 (h) -a - 1 (i) x - 2
(j) 2 - x
(a) | a + b | = 6
(b) | a + b | = 3
(c) | a + b | = 1
(d) | a + b | = 1
(e) | a + b | = 10

6.

(a)

x2 = | x | = 5

(b)

x2 = | x | = 2

(d)

x2 = | x | = 4

(e)

x2 = | x | = 9

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

(f) ] 2t + 3 g

-6

(b) 5.5 # 10 - 5

(j) 8 #10 4

(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

2

2.

1

-4

(i) 2 # 10 7

3.

p

(a) 9 (b) 3 (c) 4 (d) 2 (e) 7 (f) 10 (g) 2 (h) 8
(i) 4 (j) 1 (k) 3 (l) 2 (m) 0 (n) 5 (o) 7 (p) 2
1
1
(q) 4 (r) 27 (s)
(t)
2
16

3x - 1

(a) 5.7 # 10 - 2
(d) 6.2 #10

Exercises 1.7

(e)

(h) 1.376 # 10

5.

w

10

1

1

(n)

2

(f) 4.16 # 10 5

4

6.

5 ] a + 3b g
9

1
(j)
4n

8x 3

1

(e) 8.67 # 10 9

2

-7

- 11

(a)

(c) 6.19 # 10 4

5.

2x - 1
(k)
7

2y - 7
5m
(m)
(n) ] 3x + 4 g- 2 (o) ] a + b g- 8
2
3
(p) ] x - 2 g- 1 (q) ^ 5p + 1 h- 3 (r) 2 ] 4t - 9 g- 5

(s)

(b) 1.23#10 6

(g) 9 #10

(d) d −9 (e) d −5 (f) x - 2

1 z- 6
(i) z - 6 or
2
2

(a) 3.8 # 10 3
(d) 1.2 #10 7

(l)

4.

1

(c) p 2 + p - 1 + 2p 2

Exercises 1.8

1
11
1
(a) 1 (b) 16 (c) 1
(d) 1
(e) 1 (f) 125 (g) 1
2
25
3
3
13
19
1
(h) 49 (i) 3
(j) 32 (k) 2
(l) 1 (m) 1
(n) 1
8
3
36
81
5
16
7
(o) 1 (p) 16 (q) - 15
(r) (s) 1 (t)
8
23
25

-6

1.

1
3

2

(a)

(g) 2x

4.

(a)

2

(b) a 3 - b 3

2x + 5 or |a | + | b |= 6 ` | a + b | # | a | + | b |
|a | + | b |= 3 ` | a + b | # | a | + | b |
|a | + | b |= 5 ` | a + b | # | a | + | b |
|a | + | b |= 9 ` | a + b | # | a | + | b |
| a | + | b | = 10 ` | a + b | # | a | + | b |
(c)

x2 = | x | = 3

7.

(a) x + 5 for x 2 - 5 and - x - 5 for x 1 - 5
(b) b - 3 for b 2 3 and 3 - b for x 1 3
(c) a + 4 for a 2 - 4 and - a - 4 for a 1 - 4
(d) 2y - 6 for y 2 3 and 6 - 2y for y 1 3
(e) 3x + 9 for x 2 - 3 and - 3x - 9 for x 1 - 3
(f) 4 - x for x 1 4 and x - 4 for x 2 4
1
1
(g) 2k + 1 for k 2 - and - 2k - 1 for k 1 2
2
2
2
(h) 5x - 2 for x 2 and - 5x + 2 for x 1
5
5
(i) a + b for a 2 - b and - a - b for a 1 - b
(j) p - q for p 2 q and q - p for p 1 q

8.

x = !3

1
^ 5 x + 7 h2
1

(e) ] 4s + 1 g 2
5

(h) ] 3x + 1 g 2
1

1
^ y + 7 h 2 (k) 5 ] x + 4 g 3
2
3
3
4
(m) _ x 2 + 2 i
5

9. !1

10. !1, x ! 2

757

758

Maths In Focus Mathematics Extension 1 Preliminary Course

Test yourself 1
1.

(a)

9
20

(b) 0.14 (c) 0.625
2. (a)

(f) 73.3%
3.

Chapter 2: Algebra and surds

1
49

(b)

157
200
1
(c)
3

(d)

1
5

Exercises 2.1

(e) 1.2%

1.

(a) 8.83 (b) 1.55 (c) 1.12 (d) 342 (e) 0.303 4. (a) 1
(e) - 10 (f) - 1 (g) 4 5. (a) x 9
8x 18
29
(b) 25y 6 (c) a 11 b 6 (d)
(e) 1 6. (a)
27
40
1
1
1
(b) 3
(c) 12 (d) 2
(e) 12
7. (a) 4 (b) 6 (c) 19
7
2
2
1
1
(d)
(e) 4 (f) 3 (g)
(h) 2 (i) 1 (j) 4
7
64
(b) 1 (c) 39 (d) 2

8.

5

30 18

(a) a

(b) x y

(c) p

(g)

1 -3 x 2

36

11

(d) 16b

(d) ] x + 1 g

(b) x - 5 (c) ^ x + y h- 1
(f) 2x - 1

9

4

(e) 8x y

1
4

9. (a) n

1
2

1

a5

(b) 4 n

1
13.
192

12. 1

x+1

(d)

1

(c) ] x + 3 g 6

(b) y - 1

-

3
4

(b) 1.23 # 10 11

1

b 5
(c) c m a x

1
(c)
8

(b)

1
2a + 5

21. (a)

7
9

7

(e) y 3
(b)

41
330

23. 14 500

24. LHS = | -2 + - 5 | = 7, RHS = | -2 | + | -5 | = 7.
So | a + b | # | a | +| b | since 7 # 7.

1.

4

2. 1

4.
9.

18 h

11
18

1
1
53 % 5.
3
16

11.

6. 3.04 # 10

14

51 o , 0. 5
99

3271
7. 83% 8. 1
9990

19. 6x - 6y

15. 0

20. a - 3b

23. m 2 - 6m + 12
26. - 2ab + 10b

11. 9t

16. 5b

12. 10w

17. 11b

21. 4xy + 2y

24. p 2 - 2p - 6
27. 2bc - ac

29. x 3 - 2xy 2 + 3x 2 y + 2y 3

18. - 10x

22. - 6ab 2

25. 8x + 3y

28. 2a 5 - 9x 3 + 1

30. 3x 3 + x 2 - 7x - 6

10b

2. 8xy

3. 10p 2

15ab 6. 14xyz 7. 48abc 8. 12d 2

9.

12a3

10. - 27y3

12. 6a 2 b 3
19. - 14m

11. 32x10

13. - 10a 3 b 2

15. 5a 3 b 3

4. - 6wz

14. 21p 3 q 4

16. - 8n 10 17. k 3 p 3
20. 24x y

11

6

18. 81t 12

3

Exercises 2.3
1.

2. 2

6x

3. 4a 2

4. 8a

5. 4a

y

6.

2

7. 3p

ab
4
1
-2
9.
10. - 3x 3 11. 3a 12.
13.
qs
3y
2
3ab 2
4 7
6
2 a b
2
z b 14.
15.
16. 6p 4 q 17.
18.
4c
2a
3c 2 d
2x 2

8.

19. -

x3 z3
3y

a 13

20.

2b 6

2x - 8

5.

x 2 - 2x 6. 6a 2 - 16ab 7. 2a 2 b + ab 2 8. 5n 2 - 20n

2. 6h + 9

3. - 5a + 10

9.

3x3 y2 + 6x2 y3

10. 4k + 7

3
7

2
14. 6 %
3

1
1
15. when x 2 - 1, when x 1 - 1 16. 0.73 x-1 1-x
18 4.54 19. 4.14 # 10 - 20

20. | a + b | = | a | + | b | when a 2 0, b 2 0 or a 1 0, b 1 0;
| a + b | 1 | a | + | b | when a 2 0, b 1 0 or a 1 0, b 2 0;
` | a + b | # | a + b | for all a, b

12. 4y + 11y

13. - 5b - 6

16. 8h - 19 17. d - 6

2

19. 3x - 9x - 5
2

22. - 7y + 4

14. 8 - 2x

20. 2ab - 2a b + b
2

23. 2 b

4. 2xy + 3x

11. 2t - 17

15. - 3m + 1

10. 1.98

o
12. −24 35 13. - 0.34, 2, 1. 5, 0,

17. 0.6%

14. - x

10. 3k

1.
3. 0.502, 51%,

LHS = 2 ^ 2 k - 1 h + 2 k + 1
= 2k+1 - 2 + 2k+1
= 2:2 k + 1 - 2
= 2 ^ 2k+1 - 1 h
= RHS
` 2 ^ 2k - 1 h + 2k+1 = 2 ^ 2k+1 - 1 h

.

13. - m

9. 0

6. −3r

Exercises 2.4

Challenge exercise 1
278
303

8. −5x

5. 3b

1.

(e)

(d) ] 2x - 3 g- 11

20. (a) 1.3 # 10 - 5
3

1 x-y (j) m

7
14. 689 mL 15. (a) 6 h (b)
12

1

22. (a)

-y

4. 6a

16. $38 640 17. 70% 18. 6.3 # 10 23

(d) 33.3%
19. (a) x 2

(c)

7.

3. z

5.

(i) ] 5x + 3 g 7

(h) x 3

1
] 4t - 7 g4
1
1
(f) 5 a + b (g)
(h) 4 b 3 (i) 3 ] 2x + 3 g4 (j)
3
x x3 11. | a + b | = 2 | a | +| b | = 8 ` | a + b | # | a | + | b |
10. (a)

2. 3a

Exercises 2.2

1

(e) ] a + b g 7
9

7x

24. 5t - 6

18. a 2 - 2a + 4
21. 4x - 1

25. 2a + 26

Exercises 2.5
1.

a 2 + 7a + 10

2. x 2 + 2x - 3

4.

m 2 - 6m + 8

5. x 2 + 7x + 12

7.

2x 2 + x - 6

8. h 2 - 10h + 21

3. 2y 2 + 7y - 15
6. y 2 - 3y - 10
9. x 2 - 25

10. 15a 2 - 17a + 4 11. 8y 2 + 6y - 9 12. xy + 7x - 4y - 28
13. x 3 - 2x 2 + 3x - 6
16. 16 - 49y 2
20. y 2 - 36

14. n 2 - 4

17. a 2 - 4b 2

21. 9a 2 - 1

15. 4x 2 - 9

18. 9x 2 - 16y 2

22. 4z 2 - 49

19. x 2 - 9

ANSWERS

23. x 2 - 2xy + 11x - 18y + 18 24. 2ab + 2b 2 - 7b - 6a + 3

Exercises 2.8

25. x + 8

1.

]x + 4g]2 + bg

4.

]m - 2g]m + 3g

5. ] d - c g ] a + b g 6. ] x + 1 g ^ x 2 + 3 h

7.

] 5a - 3 g ] b + 2 g

8. ^ 2y - x h ^ x + y h

3

26. a - 27

27. a + 18a + 81

3

28. k - 8k + 16

29. x + 4x + 4

2

33. 9a + 24ab + 16b
35. 4a + 4ab + b
2

2

32. 4t 2 - 4t + 1

2

38. a - 2ab + b

30. y - 14y + 49

2

31. 4x 2 + 12x + 9

2

2

34. x - 10xy + 25y

2

2

36. a - b

2

2

39. a + b

2

3

3

10. ] x + 5 g ] x - 1 g

2

37. a + 2ab + b

2

2

40. a - b
3

t + 8t + 16

4.
7.

n 2 + 2n + 1

23. ] x - 2 g ^ 3x 2 - 5 h

5. q 2 + 6q + 9

6. k 2 - 14k + 49

25. ^ y + 7 h ] x - 4 g

9. 9 - 6x + x 2

y 2 + 16y + 64

10. 9y - 6y + 1

11. x + 2xy + y

2

2

13. 16d + 40de + 25e
2

21. 16a 2 - 1

24. x + 10x + 25
4

27. a 2 -

2

2

a2

29. 5 (y - 3) (1 + 2x)

Exercises 2.9

29. ] a + b g2 + 2 ] a + b g c + c 2 = a 2 + 2ab + b 2 + 2ac + 2bc + c 2

1.

]x + 3g]x + 1g

4.

] t + 4 g2

7.

2

28. x 2 - ^ y - 2 h2 = x 2 - y 2 + 4y - 4

1

28. 3 (a + 2b) (a + 3)
30. ] r + 2 g ] rr - 3 g

23. x 4 - 4
4
26. x + 4 + 2 x 2

26. (x - 4) (x 3 - 5)

19. 4a 2 - 9

22. 49 - 9x 2

25. 9a b - 16c

2

2

15. x 2 - 9

18. x 2 - 100

24. ] a - 3b g ] 4 + c g

27. (2x - 3) (2x + 4) = 2 (2x - 3) (x 2 + 2)

12. 9a - 6ab + b

2

17. ] x - 3 g ^ 7 - y h

20. ] a + 3 g ] 2 - b g

2

2

14. t - 16

2

17. r 2 - 36

20. x 2 - 25y 2

2

2

14. ^ a + b h ] ab - 4 g

22. ^ q - 3 h ^ p + q h

3. x - 2x + 1

2. z - 12z + 36

12. (m - 2) (1 - 2y)

2

18. ] d + 3 g ] 4 - e g 19. ] x - 4 g ^ 3 + y h

8. 4b 2 + 20b + 25

1.

2

9. ^ y + 1 h ] a + 1 g

15. ] 5 - x g ] x + 3 g 16. (x + 7) (x 3 - 4)

3

21. (x - 3) (x 2 + 6)

2

3. ] x + 5 g ] x + 2 g

11. (y + 3) (1 + a)

13. ^ x + 5y h ^ 2x - 3y h

2

Exercises 2.6

16. p 2 - 1

2. ^ y - 3 h ] a + b g

]v - 3g]v - 5g

2. ^ y + 4 h ^ y + 3 h

5. ] z + 3 g ] z - 2 g
8. ] t - 3 g

2

3. ] m + 1 g2

6. ] x + 1 g ] x - 6 g
9. ] x + 10 g ] x - 1 g

30. ] x + 1 g2 - 2 ] x + 1 g y + y 2 = x 2 + 2x + 1 - 2xy - 2y + y 2

10. ^ y - 7 h ^ y - 3 h

11. ] m - 6 g ] m - 3 g

12. ^ y + 12 h ^ y - 3 h

13. ] x - 8 g ] x + 3 g

31. 12a

14. ] a - 2 g

32. 32 - z

2

34. x 2 + 3xy + y 2 - 2x

33. 9x + 8x - 3
2

2

37. x

2

15. ] x - 2 g ] x + 16 g

16. ^ y + 4 h ^ y - 9 h

35. 14n 2 - 4

36. x - 12x + 48x - 64
3

2

38. x - 2x y + y
4

2

2

4

17. ] n - 6 g ] n - 4 g 18. ] x - 5 g 2

19. ^ p + 9 h ^ p - 1 h

20. ] k - 2 g ] k - 5 g 21. ] x + 4 g ] x - 3 g

39. 8a + 60a + 150a + 125

22. ] m - 7 g ] m + 1 g 23. ^ q + 10 h ^ q + 2 h

40. 4x + 16x + 15x - 4x - 4

24. ] d - 5 g ] d + 1 g 25. ] l - 9 g ] l - 2 g

Problem

Exercises 2.10

a = 2, b = 7, c = 9, d = 4, e = 3, f = 8, g = 0, h = 6, i = 1

1.

(2a + 1) (a + 5) 2. ^ 5y + 2 h ^ y + 1 h

3.

(3x + 7) (x + 1) 4. (3x + 2) (x + 2) 5. (2b - 3) (b - 1)

6.

(7x - 2) (x - 1) 7. ^ 3y - 1 h ^ y + 2 h

9.

^ 5p - 2 h ^ p + 3 h 10. ] 3x + 5 g ] 2x + 1 g

3

2

4

3

2

Exercises 2.7
2. 5 ] x - 2 g 3. 3 ] m - 3 g 4. 2 ] 4x + 1 g

1.

2^y + 3h

5.

6 ^ 4 - 3y h

9.

3a ] 5 - a g 10. ab ] b + 1 g 11. 2xy ] 2x - 1 g

6. x ] x + 2 g 7. m ] m - 3 g 8. 2y ^ y + 2 h

12. 3mn ^ n 2 + 3 h

13. 2xy ] 4x - z g 14. a ] 6b + 3 - 2a g

15. x ^ 5x - 2 + y h

17. 5b 2 ] b + 3 g

16. q 2 _ 3q 3 - 2 i

18. 3a b ] 2b - a g 19. (m + 5) (x + 7)
2

20. ^ y - 1 h ^ 2 - y h

2

21. (7 + y) (4 - 3x)

22. ] a - 2 g ] 6x + 5 g

23. ] 2t + 1 g ^ x - y h

24. ] 3x - 2 g ] a + 2b - 3c g

25. 3x ] 2x + 3 g
2

28. 4x 2 ] x - 6 g

26. 3q _ pq 2 - 2 i
3

29. 5m 2 n ^ 7mn 3 - 5 h

31. 2rr ] r + h g 32. ] x - 3 g ] x + 2 g
34. - ] a + 1 g

27. 3ab ^ 5a 3 b 2 + 1 h

35. (a 2 + 1) (4ab - 3)

30. 4ab 2 ^ 6ab 3 + 4 h

33. (x + 4) (y 2 + 2)

8. ] 2x + 3 g ] x + 4 g

11. (2y + 1) (y - 6)

12. ] 5x - 1 g ] 2x + 1 g

13. (4t - 1) (2t - 3)

14. (3x + 4) (2x - 3)

15. ^ 6y - 1 h ^ y + 8 h

16. ] 4n - 3 g ] n - 2 g

17. ] 4t - 1 g ] 2t + 5 g 18. ^ 3q + 2 h ^ 4q + 5 h
19. ] 4r - 1 g ] 2r + 6 g = 2 ] 4r - 1 g ] r + 3 g
20. ] 2x - 5 g ] 2x + 3 g

21. ^ 6y - 1 h ^ y - 2 h

22. ^ 2p - 3 h ^ 3p + 2 h

23. (8x + 7) (x + 3)

24. ] 3b - 4 g ] 4b - 9 g

25. (6x + 1) (x - 9)

26. ] 3x + 5 g2

27. ^ 4y + 3 h2

29. ] 6a - 1 g2

30. ] 7m + 6 g2

28. ] 5k - 2 g2

759

760

Maths In Focus Mathematics Extension 1 Preliminary Course

Exercises 2.11
1.

^y - 1h

5.

(x - 6)

9.

] 5x - 4 g2

Exercises 2.14

2. (x + 3)

2

6. ] 2x + 3 g

2

2

2

10. ^ 7y + 1 h2
2

13. ] 5x + 1 g

14. ] 9a - 2 g

2

2

2

12. ] 4k - 3 g

11. ^ 3y - 5 h

2

15. ] 7m + 6 g2

16. d t +

1 n 2

2

17. d x -

2

18. d 3y +

2 n 3

1 n 5

2 2
20. d 5k - n k (a + 2) (a - 2)

2. (x + 3) (x - 3)

4.

]x + 5g]x - 5g

5. (2x + 7) (2x - 7)

7.

(1 + 2z) (1 - 2z) 8. ] 5t + 1 g ] 5t - 1 g 9. ] 3t + 2 g ] 3t - 2 g

3. (y + 1) (y - 1)
6. (4y + 3) (4y - 3)

10. ] 3 + 4x g ] 3 - 4x g

11. (x + 2y) (x - 2y)

12. ^ 6x + y h ^ 6x - y h

13. ] 2a + 3b g ] 2a - 3b g

18. ] z + w + 1 g ] z - w - 1 g

1
1
19. d x + n d x - n
2
2

3

+ 1oe

3

10. 2 ] 3x - 2 g ] x + 2 g 11. ] m - 5 g ] 3 + n g

12. - 7 ] 2x + 1 g

14. ] x - 1 g ] x + 2 g ^ x 2 - 2x + 4 h

13. ^ y + 5 h ^ y + 4 h ^ y - 4 h

19. 3 ] 2 - b g ^ 4 + 2b + b 2 h

18. y (2xy + 1) (2xy - 1)

20. 3 ] 3x - 2 g ] 2x + 5 g 21. 3 ] x - 1 g2
23. z ] z + 3 g2

22. (x + 2) (x + 5) (x - 5)

24. ] x + 1 g ] x - 1 g ] 2x + 3 g ] 2x - 3 g
27. 5x ] 2 - x g ^ 4 + 2x + x 2 h

28. (a + 2) (a - 2) (a + 3) (a - 3)

29. 4k (k + 5) 2

30. 3 (x + 1) (x - 1) (x + 3)

Exercises 2.15

- 1 o 21. ^ x + 2y + 3 h ^ x - 2y + 1 h

25. (a 4 + 1) (a 2 + 1) (a + 1) (a - 1)

x 2 + 4x + 4 = ] x + 2 g2

x 2 - 10x + 25 = ] x - 5 g2

5.

m - 14m + 49 = ] m - 7 g

7.

x 2 + 2x + 1 = ] x + 1 g2

9.

23. _ 3x 3 + 2y i _ 3x 3 - 2y i 24. _ x 2 + 4y 2 i ^ x + 2y h ^ x - 2y h

1.
3.

22. (x 2 + 1) (x 2 - 1) = (x 2 + 1) (x + 1) (x - 1)

2. b 2 - 6b + 9 = ] b - 3 g2

x 2 - 20x + 100 = ] x - 10 g2

4. y 2 + 8y + 16 = ^ y + 4 h2

2

6. q 2 + 18x + 81 = ^ q + 9 h2

2

8. t 2 - 16t + 64 = ] t - 8 g2

10. w 2 + 44w + 484 = ] w + 22 g2

Exercises 2.13
2. ] x + 3 g ^ x 2 - 3x + 9 h

1.

(b - 2) (b 2 + 2b + 4)

3.

]t + 1g^t - t + 1h

5.

(1 - x) (1 + x + x )

7.

(y + 2z) (y 2 - 2yz + 4z 2)

9.

^ 2x + 3y h _ 4x 2 - 6xy + 9y 2 i 10. ] ab - 1 g ^ a 2 b 2 + ab + 1 h

11. x 2 - 32x + 256 = ] x - 16 g2

2

2

6. ^ 2 + 3y h _ 4 - 6y + 9y 2 i

13. x 2 - 7x +

12. d

x x 2 3x
- 3ne +
+ 9o
4
2
2

10 1 100 10
1
13. d
+ ne 2 + o a b ab b 2 a 15. ^ 5xy + 6z h _ 25x y - 30xyz + 36z i
2

15. x 2 + 9x +

81
9 2
= dx + n
4
2

16. y 2 -

17. k 2 -

11k 121
11
n
+
= dk 4
2
16

5y
2

1
1 2
= da + n
4
2

+

25
5 2
= dy - n
4
16

2

16. - 9 ^ a - a + 1 h

20. p 2 - 8pq + 16q 2 = ^ p - 4q h2

2

Exercises 2.16

2

x x x ne1 + + o
9
3
3

18. ^ x + y + 3 h _ y 2 - 3y - xy + 9 + 6x + x 2 i
19. ^ x + y - 1 h _ x 2 + 4x - xy + y 2 - 5y + 7 i
20. (2a + 6 - b) (4a + 24a + 2ab + 6b + b + 36)
2

14. a 2 + a +

3 2
9
= dy + n
4
2

18. x 2 + 6xy + 9y 2 = ^ x + 3y h2 19. a 2 - 4ab + 4b 2 = ] a - 2b g2

14. ^ x + 1 - y h _ x 2 + 2x + 1 + xy + y + y 2 i
2

49
7 2
= dx - n
4
2

8. (x - 5y) (x 2 + 5xy + 25y 2)

11. (10 + 2t) (100 - 20t + 4t 2)

2

12. y 2 + 3y +

4. (a - 4) (a + 4a + 16)

2

17. d 1 -

ab ] 3 + 2ab g ] 3 - 2ab g 9. x ] x + 1 g ] x - 1 g

26. 4a (a + 3) (a - 3)

15. ] 2a + 9b g ] 2a - 9b g
17. (a + b - 3) (a - b + 1)

20. e

8.

25. 2 ] x + 2 g ] x - 2 g ^ x + y h _ x 2 - xy + y 2 i

16. ^ x + 2 + y h ^ x + 2 - y h

y

5 ] a - 1 g2 6. - ] 2x - 3 g ] x - 4 g 7. 3z ] z + 5 g ] z + 4 g

16. x ] x + 2 g ] x - 5 g 17. ] x + 3 g (x - 3) 2

1.

y

5 ^ y - 1 h _ y 2 + y + 1 i 4. 2ab ^ a + 2b) (2a - 1 h

15. ] x + 1 g ^ x 2 - x + 1 h ] x - 1 g ^ x 2 + x + 1 h

Exercises 2.12

14. ^ x + 10y h ^ x - 10y h

2 ] x + 3 g ] x - 3 g 2. 3 ^ p + 3 h ^ p - 4 h

3.

2

8. ] 3a + 2 g

7. ] 4b - 1 g

2

1 2
19. c x + m x 4. (t - 2)

2

1.

5.

3. (m + 5)

2

2

1.

a+2

2. 2t - 1

6.

1 y-4 7.

10.
14.

p+5
3

2 ] b - 2a g a-3 11.

a+1 a+3 p-2
4p - 2p + 1
2

3.

15.

4y + 1
3

s-1 s+3 8.

12.

4.

4
2d - 1
9.

3+y x + 2x + 4

a+b
2a - b

2

5.

x
5x - 2

b2 + b + 1 b+1 13. x - 3

ANSWERS

Exercises 2.17
1.

2.

(a)

(a)

(d)
3.

5x
4

(b)

Exercises 2.20

13y + 3

b
2a - 1

(b)

a+8
12

(d)

^ p - 2 h _ q2 - q + 1 i

ab

a+b+3
(c)
a+b

-x + 2
(b)
x ]x - 1g

(a)

x - 13
6

2 _ 3y 2 + 14y + 13 i

^y + 2h^y + 3h^y - 1h x2 ] x + 2 g

(b)

8 _ y 2 - 3y + 9 i

1. 3 5

10 ] 2b - 1 g

(e)

3p 2 + 5pq - 2q 2

pq ^ p + q h ^ p - q h

10. 4 5

14.

15. 5 7

16.

1.

6.

2.

3. 3 6

15

8. 14

9. 60

16. 28

15y

18. - 2 105

17.

30

25. 2 3
3

31.

30.

2 2

27.

3 10
3

32.

2

1.

(a)

10 + 6

9
2 5

33.

29.

3 5
5

34.

2 2

6

1
2

2
3

35.

5
7

(c) 12 + 8 15

2. 47

3. - 7

3
4

(e) 0.6

4. 375

(m) 10 6 - 120

5. - 196

10. 51.935 11. - 1

8. 284 9. - 40
14.

(k) - 8 + 12 12 = - 8 + 24 3

(d) - 37.7

(c) 48.1

3
4

2.

(f) 15 - 15 + 18 10 - 6 6

(j) 3 6

(f) 10 2

(g) 4 3

(h) 5 3

(i) 4 2

(k) 4 7

(l) 10 3

(m) 8 2

(n) 9 3

(p) 6 3

(q) 3 11

(r) 5 5

(a) 6 3

(b) 20 5

(c) 28 2

(g) 72 5
20
117

176

(h)

98

(h) - 1

(i) - 12

(n) 7 + 2 10

(j) 43

(k) 3

(o) 11 - 4 6

(l) - 241
(p) 25 + 6 14

(r) 27 - 4 35

(d) 4 7

(s) 77 - 12 40 = 77 - 24 10

(t) 53 + 12 10

3.

(h) 30 2

(c)

(o) 4 3 - 12

(a) 10 + 3 6 + 3 5 + 9 3
(b) 10 - 35 - 2 + 14

(q) 57 + 12 15

(d) 5 2

(l) 210 - 14 15

(n) - 10 - 2 2

(c) 2 10 - 6 + 10 15 - 15 6
(d) 12 20 + 18 60 - 8 10 - 12 30 =
24 5 + 36 15 - 8 10 - 12 30

15. 15 16. 10

(c) 2 6

(h) 5 - 5 15

(j) 2 54 + 6 = 6 6 + 6

(e) 6 2

4.

2 5

1

(g) - 6 - 12 6

(i) 6 + 30

(b) 3 7

(g)

28.

8

24.

(f) 5 33 + 3 21

1.

160

1

23. 1

(e) - 6 + 4 18 = - 6 + 12 2

(m) - 6

(f)

22. 4 3

(b) 2 6 - 15

(g) 4

(b)

19. 18

Exercises 2.22

a 2 - 2ab - b 2 + 1
]a + bg]a - bg

Exercises 2.19

18

15. 2

21. 2 6

1

26.

(e) 52 - 13 10

(a)

6. 30

12 = 2 3

14. - 84

18. 23.987 19. 352.47 20. 93 21. 4

3.

10.

5. - 6 6

12. 15 28 = 30 7

20. 30 50 = 150 2

17. 2 3

(f) 8 14

4. 10 14

13. 2 20 = 4 5

- ] 5x + 22 g
(j)
]x + 4g]x - 4g]x + 3g

13. 1838.8

2.

7-5 2

24. - 2 - 2 3

11. 2 48 = 8 3

12. 22.4

(a) 2 3

21.

(d) 5 14 - 2 21

(b) - 6.9

5.5 7. 377

21

7. - 12 55

^x + yh^x - yh

(f) 2.3 (g) - 5.3

17. 13 6

2

20. 5 2 - 2 3

23. 7 6 + 3 5

12. 5 3

2

Exercise 2.21

2]x - 1g
(f)
]x + 1g]x - 3g

y ^x + y + 1h

(a) - 7.1

2

19. 47 3

11.

6. 3 6

25. - 17 5 + 10 2
2x
(d) x+2 Exercises 2.18
1.

5. - 3 5

9. - 4 2

22. - 2 3 - 4 5

^y + 2h^y + 1h

(d)

4. 3 3

8. 8 5

18. - 9 10

x 2 + 10x - 24
3b 2 - 5b - 10
(d)
(e) x
2 ]x - 3g]x - 4g
2b ] b + 1 g
3x - 13
3 - 5x
(a)
(b)
]x - 5g]x - 2g]x + 3g
]x + 2g]x - 2g
(c)

3. 6 3

2

7. - 7 2

(c)

5.

2.

13. - 3

b 2 ^ x + 2y h

a+2
(h)
] a + 1 g2

- 3x + 8
(g)
]x + 2g]x - 2g

4.

(e)

6

]x - 3g]x - 1g
(e)
]x - 5g]x - 2g

^ p + qh^ p - qh + 1 p2 - q2 + 1
=
(e) p+q p+q

(i)

4p + 3

(c)

q+1

x 2 - xy + y 2

5
(a)
x

(c)

15

(a) 18

(d) 19 + 6 2

4.

(a) a = 21, b = 80

5.

(a) a - 1

6.

k = 25

9.

a = 107, b = - 42

(o) 7 5

(e) 16 5

(i) 14 10

(j) 24 5

(d)

128

(e)

75

(i)

363

(j)

1008

(a) x = 45 (b) x = 12 (c) x = 63 (d) x = 50
(e) x = 44 (f) x = 147 (g) x = 304 (h) x = 828
(i) x = 775 (j) x = 960

(b) 108 2

(c) 432 2

(e) 9

(b) a = 19, b = - 7

(b) 2p - 1 - 2 p ^ p - 1 h
7. 2x - 3y - 5 xy

8. a = 17, b = 240

10. 9 + 5 units 2

761

762

Maths In Focus Mathematics Extension 1 Preliminary Course

Exercises 2.23
1.

(a)
(e)
(h)

2.

7
7

8.
6
4

(b)

3+ 6
3

(f)

3 14 - 4 7
14

(c)

2 15
5

(d)

12 - 5 2
2
(i)

6 14
3 14
=
5
10

(g)

(a) 4 3 - 4 2 = 4 ^ 3 - 2 h

(b)

(j)

4 15 - 2 10
35

6 15 - 9 6 + 2 10 - 6
2

3.

So rational
9.

1.

(j)
(l)
4.

(i)

2-1

=

28 - 2 6 - 7 3
13

(b) a = 1, b = 8

8
5
(d) a = - 1 , b = 9
9

=

(k)

+
2+1
2-1

2
2-1

2+1
2-1
^ 2 - 1h^ 2 - 1h

+
+

4
2

(a) 4 (b) 14

7.

3 5 - 2 - 15 - 3
3

#

4 2
2

^ 2 h2 - 1 2
2- 2- 2+1
=
+2 2
2-1
3-2 2
=
+2 2
1
=3-2 2+2 2
=3
So rational
6.

1
1
(c) a = - , b =
2
2

(e) a = 5, b = 32

4

#

10.

2.

2
2

(a) - 2y

(b) a + 4

b+4 b+4 b-4 (c) - 6k 5

5x + 3y

(g) 4 5

(d)

(e) 3a - 8b

15

(b) ] a + 3 g ] a - 1 g (c) 4ab ] b - 2 g

(a) ] x + 6 g ] x - 6 g

(e) 2 ^ 2n - p + 3 h

(d) (y - 3) (5 + x)

(f) (2 - x) (4 + 2x + x 2)
3.

(b) 2x 2 + 5x - 3

(a) 4b - 6

(d) 16x - 24x + 9
(g) 2 6 - 5 3

5.

V = 157.464

(f) - 1 - 7a

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

8

(a)

(c) 4m + 17

(e) p 2 - 25

2

4.

2 15 + 2 10 - 2 6 - 3 - 5
2

(a) a = 45, b = 10

5.

4 6+9 3
21

15 30 - 30 5 - 4 3
30

x = -^ 3 + 2h

(f) 6 2

20 12 + 19 6 + 25 3 - 6
19 6 + 65 3 - 6
(g)
=
15
15
6+9 2+2 3
6

2

Test yourself 2

(a) 2 2
(b) - ^ 2 + 6 - 3 2 + 3 3 h = - 2 - 6 + 3 2 - 3 3
22 5 + 14 2
(c)
39
^ 6 6 - 16 - 3 84 + 8 14 h
(d) 10
- 3 6 + 8 + 3 21 - 4 14
=
5
(e) - 4 (f) 4 2

(h)

2

6-4 2
+4 2
9-4#2
6-4 2
=
+4 2
1
=6-4 2+4 2
=6

(c)

(f)

2
3-2 2
8
=
#
+
#
3+2 2
3-2 2
2
2^3 - 2 2 h 8 2
+
=
2
2
32 - ^ 2 2 h
=

-^ 6 + 7 3 h
47

- ^ 2 15 - 4 18 h
- 2 ^ 15 - 6 2 h
=
19
19
- ^ 19 - 8 3 h
8 3 - 19
=
(d)
(e) 6 + 2 + 5 3 + 5 2
13
13

8

+

2

5 + 2 10
5

8 5 + 3 10
20

2
3+2 2

(b)

b 2 ^ a 2 + 3a + 9 h

6. (a) 17

15
] m - 2 g2
(b)

6 15 - 9
17

4x + 5
8. (a) 36 (b) - 2
]x + 3g]x - 2g
1
9. (a)
(b) 8 10. d = 11.25
5
2 3
2+ 6
11. (a)
(b)
15
2
7.

12. (a) 3 6 - 6 - 4 3 + 4 2

(c) 2

(d) 216

(b) 11 + 4 7

(b) 6 ] x - 3 g ] x + 1 g

13. (a) 3 (x - 3) (x + 3)

(c) 5 ^ y + 2 h _ y 2 - 2y + 4 i

14. (a)

x3

(b)

3y 4

15. (a) 99

1
3x - 1

(b) 24 3

16. (a) a 2 - b 2

(b) a 2 + 2ab + b 2

17. (a) ] a - b g2

(b) ] a - b g ^ a 2 + ab + b 2 h

18.

3 3+1
2

20.

21 5 - 46 - 2
7

(c) 16

19. (a)

4b + 3a ab (c) a 2 - 2ab + b 2

(b)

3x - 11
10

(e) 2

ANSWERS

21. (a) 6 2
(f)

(b) - 8 6

m

24. (a)

(d)

3 7
7

6
15

5+1
2

(c)

(e)

x + 10
10

17a - 15
21

1 k-1 (b)

(e)

71
121

20 + 3 15 + 4 10 + 3 6
53
(c)

3 - 2x
(x + 1) (x - 1)

15 - 6 - 15 3 - 15 2
3

(b) n = 175

(d) n = 5547

27. (b), (c)

29. (a), (d)

34. (d)

30. (c)

35. (b)

2.
4.

x2 +

(b) y 4 - 4

4x 2 + 12x + 9 = ] 2x + 3 g2
]a + 1g

a2 - a + 1

10. d

9
35

5. k = 5

5
8

24. y = 1

14. x = - 1

17. t = 8.2

20. x = - 3

21. b = 0.8
2
25. t = - 1
3

7.

(d) ] b - 2 g ] a + 2 g ] a - 2 g y+1 2]x - 1g

8. 2 5

18. x = - 9.5
22. a = - 0.375

12. r = 0.072

16. r = 2.12

19. x = 5.5

9. y 1 = 3

17. r = 10.46

3
4 r

=

3 r
4r

(b) a =

18. x = 1.19

Exercises 3.4
1.

2.

(a) x 2 3
-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

4

1
2

(g) y $ - 2

(k) b 1 - 18

(l) x 2 30

(n) m 2 14
17
14
, b=23
23

4

(a) t 2 7 (b) x $ 3 (c) p 2 - 1 (d) x $ - 2 (e) y 2 - 9

(j) y 1 12

16. x = 2
71
121

14. t = 2.14

20. r = 3.3

-4

(a) 3

2
3

(b) w = 69.66
13. x 1 = - 9

(f) a $ - 1

2x 1
1 2
+ = dx + n
9
3
3

21. s = 2 + 6 3

5. y = 4

(b) y # 4

- 66 6 + 4 2 - 15 + 4 5 - 65 3
13

20. r =

4. a = 41

8. n = 15

11. (a) BMI = 25.39

15. x = ! 2

-4

13. x 3 - 7x 2 + 15x - 9

18.

3. b = 8

7. x = 6.44

(c) h = 1.94

3x + 4
(b)
] 2x - 1 g2

2

400 - 59 5
10

2. l = 122

10. h = 3.7

2 a 2 a
+ nd - n x b x b

12. (a) 8x - 12x + 6x - 1

1
2

4. x = 1

13. x = 14

16. p = 3

r = 6.68

]x - 3g]x + 3g]x - 2g

19. i = 1

12. t = 30

t = 8.5

3x 3 - 6x 2 + 3x + 4xy - 6y

3

17.

11. w = 13
15. x = - 0.4

6.

(a) ] x + 4 g ] x + 9 g

15. x 2 +

4
9

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

1.

2
3.
or
4
2 2

b b2 b 2 n x + 2 = dx + a 2a
4a

2

14.

2. x = 35 3. y = 4

1

(c) ] 5x + 7 g ^ 25x 2 - 35x + 49 h

11.

1
3

b =3

Exercises 3.3

(b) _ x 2 - 3y i _ x 2 + 2y i = (x + 3 y) (x - 3 y) _ x 2 + 2y i

9.

30. x Z 4.41

2

17 3 + 2 5 + 20
17

6.

29. p = 5

19. q = 22

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

5.

16. x = 20 17. m = 20 18. x = 4 19. a = - 7 20. y = 3
2
21. b = - 4 22. x = 3 23. a = - 1
24. t = - 4
3
1
25. x = 1.2 26. a = 1.6 27. b =
28. t = 39
8

23. x = 3

(a) 2a 2 b - 8ab 2 + 6a 3
3

2. z = - 5.6 3. y = 1 4. w = 6.7 5. x = 12
1
8. b = 35 9. n = - 16 10. r = 4
6. x = 4 7. y =
15
11. y = 9 12. k = 6 13. d = 2 14. x = 5 15. y = 15

6.

Challenge exercise 2
1.

t = -5

1.

(c) n = 392

28. (d)

33. (a)

1.

Exercises 3.2

(e) n = 1445

32. (b)

Chapter 3: Equations

Exercises 3.1

12 - 2 6
15

31. (c)

(e) 30a 2 b

3

(b) 10 14 - 5 21 - 6 10 + 3 15

(b)

25. (a) n = 48

26. 3

4

(d) 43 (e) 65 - 6 14

(c) 7

(d)

(d)

(g) 2x - 3y

3n 4

22. (a) 2 6 + 4

23. (a)

(c) 2 3

2
3

1
4

(o) b $ 16

(h) x 1 - 2

(i) a # - 6

(m) x # 3

(p) r # - 9

3
4

(q) z 2 8

763

764

Maths In Focus Mathematics Extension 1 Preliminary Course

(r) w 1 2

4
5

(s) x $ 35
2
3

(v) x 2 - 1
3.

(t) t $ - 9

(w) b # - 11

(u) q 2 - 6

(e) y = 1.89 (f) d = ! 2.55 (g) k = ! 4.47 (h) x = 2.22

2
5

(i) y = ! 3.81

1
4

3.

1

3

4

5

6

7

8

0

2

-2

1

2

3

4

5

-1

-2

0

-1

1

2

3

4

1

2

3

0

-1

-2

4

1

2

3

(a) x = ! 5

(b) y = ! 8

4

(f) - 10 # p # 10

(g) x = 0

(i) - 12 1 y 1 12

1
7

(f) a = 2 (g) x = ! 2 (h) b = 9
2
3

(j) b = ! 1

(d) x = !

1
2

(b) y = 5

(c) m = 9

(g) x = 2

(h) x = 2

(a) x = 2

(b) x = 1 (c) x = - 2 (d) n = 2 (e) x = 0
1
(g) y =
(h) x = 2 (i) x = 2 (j) a = 0
3

(a) x = 5, - 9

(b) n = 4, -2

(f) x = 6
8.

(c) a 2 2, a 1 - 2

(h) x $ 9, x # - 6

1
4

1
3

(c) b = 2

(d) No solutions (e) y = - 2
3
1
(h) d = 2 , -1
4
2
(a) x = 2, -

1
2

(d) x = 4, -7

1
3

2
(a) t = 3, -1
5

2
7

(b) y = 3, 2

1
3

1
3
2
3

(c) x =

3
4

(f) n =

1
6

(j) x = 1

1
3

1
2

(h) n =

2
3

1
2

2
3
1
3

(c) k = - 4

(f) x = -

2
3

(g) x = - 4

1
2

(e) m = 0
(j) k = 2

(d) k = -

(b) x = - 1

(a) x = - 1

4
5
1
4

(b) k = - 2

(e) n =

1
18

(f) n = -

(d) n = 3
1
2

(h) x = - 1

7
11

(j) x = 18

(i) x = 1

3
5

(e) d = 4, -5

(i) x = - 1

2
(b) - 1 1 t 1 3
5

(d) x = 5
(i) x = 1

(g) x = 1

10. (a) m =

(j) No solutions

(c) a = - 10, 1

1
3

2
3

(e) x = - 2

(f) x = 7 (g) m = 5, 1

4
, -2
5

(i) y =

(b) x =

(i) k = -

(i) x = ! 12
9.

(b) a = 3, -

1
2

(a) m =

(e) k = -

5
(f) x = 5, -4
7

(e) x = 3, -6

1
1y 12
2

(a) x =

(h) a 2 14, a 1 - 14

(j) 2 # a # 10

5.

2
3

(c) y =

(a) n = 4

7.

(j) b $ 20, b # - 20

(d) 4 # x # 6

4.

(b) a =

6.

5

(e) x 2 6, x 1 - 6

(a) x = 1

1
5

1
1
1
1
(b) x = 6
(c) a =
(d) k =
4
512
81
625
19
1
(e) x =
(f) x = 4 (g) y = 8 (h) n = 7
8
32
127
(i) b = 8 (j) m = 1
216

5

(c) - 4 1 a 1 4

(d) k $ 1, k # - 1

3.

(a) x =

5

Exercises 3.5

(g) - 3

1
2

(h) y = 27

(f) x = 3

0

-2
-1
1
2
(e) 1 y 1 1
6
3

2.

1
2

(g) b = 216

5.

-3

1.

(j) t = 81

(i) x = !

(d) - 3 # y # 5

-3

(f) m = 625

(d) t = 8

(e) n =

4.

(c) 1 1 x 1 4
-3

(c) x = 32

(i) a = 128

(b) - 2 # p 1 5
-3

(b) t = 16

(e) p = 243

(a) 1 1 x 1 7
0

(a) n = 27

(j) y = 3.01

1
7

3
4

1
2

(c) x = 2
(g) x =

4
5

3
8

(d) k = 1

1
2

(h) b = - 3

1
6

(j) m = 5

Puzzle
1.

-3

-2

0

1

2

3

4

(a) x = 3

(b) y = ! 8

(e) p = 10
(i) n = ! 4
2.

(f) x = ! 5

(c) n = ! 2
(g) y = ! 3

16 each time

5

Exercises 3.6
1.

All months have 28 days. Some months have more days as well. 2. 10 3. Bottle $1.05; cork 5 cents

4.

-1

(d) x = ! 2 5
(h) w = 2

(j) q = - 2

(a) p = ! 6.71 (b) x = 4.64 (c) n = 2.99 (d) x = ! 5.92

5. Friday

Exercises 3.7
1.

y = 0, -1

5.

x = -2, -7

2. b = 2, -1
6. q = !3

3. p = 3, -5
7. x = !1

4. t = 0, 5

8. a = 0, -3

ANSWERS

9.

x = 0, - 4

12. y = 1, -1
16. x = 1, 2

10. x = !
1
2

20. x = 3, 4

11. x = -1, -1

3 1
,
4 2

13. b =

1
2

1
2

17. x = 0, 5

10. x 1 2, x 2 2

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

21. m = - 6, 1

23. y = 1, -5, -2

1
3
2
3

19. n = 3, 5

22. x = 0, -1, -2

24. x = 5, -7

25. m = 8, -1

(b) a = ! 7 + 3

(d) x = ! 13 - 1

(c) y = ! 23 + 4

(e) p = ! 44 - 7 = ! 2 11 - 7

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

(g) y = ! 88 - 10 = ! 2 22 - 10 = 2 ^ ! 22 - 5 h
(h) x = ! 2 + 1

2.

(i) n = ! 137 - 12

! 5+3
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

(d) x = 1, - 0.5
(f) n = 0.243, -8.24

(h) x = 0, 7

- 1 ! 17
2

(c) q =

(b) x =

(i) x = 1, - 6

5 ! 13
6

(f) x =

- 11 ! 133
2

7.

11x 11

1
4

3. n # 0, n $ 1

5. n 1 - 1, n 2 1

6. - 5 # n # 3

8. - 4 # x # - 2
1
2

2
5

9. 4 1 x 1 5

11. a 1 - 1, a 2
13. x #
15. - 1

1
3

2
,x $1
3

1
1
#x #2
3

(g) d =

17. x 1 - 4, x 2 4

18. - 1 # a # 1

20. x # - 1, x $ 3

21. 0 1 x 1 2

- 5 ! 73
12
(i) t =

1
2

23. y # - 2, y $

2
1
24. m 1 - 1 , m 2 1
3
2
26. 0 1 x 1

1! 5
2

1
2

4
5

30. x # - 8, x 2 - 5

4
5

25. 1 # x # 1

27. x 1 0, x $

28. y 1 - 1, y 2 -

1
2

29. 3 1 n # 3
31. x 1

1
3

1
2

3
2
,x 2
5
7

1
32. x # 4 , x 2 5
5
2. 0 1 x 1

1
2

3
5. x 1 - , x > 0
5
8. - 3

3. x 1 0, x 2 1

1
2

6. - 2 # b 1 0

1
1 z 1 -3
5

9. 2 1 x # 2

3
4

1
33. x # - 1 , x 2 - 1
4

34. x 1 - 3, x 2 2

Exercises 3.10

2
7

c 1 - 1, c 2 2

22. 1 # a # 1

7 ! 41
4

01m #

2
4
1x 17
3

19. - 2 1 x 1 3

2 ! 32
=1!2 2
2

4.

29. x 1 - 4, -

16. - 4 # y # 3

8 ! 40
4 ! 10
=
6
3

y 2 1, y 1 0

x # - 2, x $ 2

14. b 1 - 3, b 2

- 12 ! 128
-3 ! 2 2
(d) h =
=
8
2
(e) s =

27. x $ 3, - 1 # x 1 2

2. 0 1 y 1 4

10. b # - 2, b $ -

4 ! 28
= 2! 7
2

1.

-3 1 x 1 0

1
12. y 1 - 1 , y 2 2
2

(a) x =

25. x # - 1, 3 1 x # 4

1
,11x #7
2

7.

(b) x = 1, 1.5

(j) y = 2.62, 0.382

(j) x =

1
2
#x 1
2
3

4.

(g) m = - 2, - 5

23. x 2 5, - 3 1 x 1 0

24. m # - 2, - 1 1 m # 6

1.

(e) x = - 0.553, 0.678

1
2
2
19. t # , t 2 2
5
5
3

Exercises 3.11

Exercises 3.9
(c) b = 3.54, - 2.54

1
1 p 1 26
2

8
1 m 1 0 21. x 1 - 5, 0 1 x 1 1
9

30. x #

(j) y = 40.1, - 0.0749

(a) y = - 0.354, - 5.65

17. 4

22. 0 1 n # 2, n $ 4

(h) x = - 0.683, -7.32

(i) a = 0.162, - 6.16

(h) x =

15. y 1 - 2, y 2 - 1

28. n 1 - 1, 3 1 n 1 5

(f) x = 17.7, 6.34

(g) r = 22.3, - 0.314

2.

20. -

26. x # - 2,

(e) b = - 4.26, -11.7

1.

5
1
1x 1
9
2

5
# x 1 -4
9

1
1
13. a 1 - 3 , a 2 - 2
4
2

1
7
1x 11
3
15

18. x # - 1, x 2 -

(a) x = ! 5 - 2

(j) y =

14.

11. - 4

7
16. x # - , x 2 4
8

Exercises 3.8
1.

12. 1

1
6

35. -

3
3
#x 14
5

765

766

Maths In Focus Mathematics Extension 1 Preliminary Course

3.

1.

a = 1, b = 3

4.

x = 6, y = 17

7.

2. x = 2, y = 1

x = - 3, y = 2

10. m = 2, n = 3

5. x = - 10, y = 2
11. w 1 = - 1, w 2 = 5

9. x = 3, y = - 4

15. x = - 1, y = - 4 16. s = 2, t = - 1
18. k = - 4, h = 1

19. v 1 = - 2, v 2 = 4

5.

(a) x = 2

(a) b = 2, -1
(a) A = 36

20. x = 2, y Z 1.41

9.

-1 1 y # 3

1
3

1
4

(b) g = 2,

(b) b = 12

10. (a) x = - 0.298, -6.70

Problem

(b) k 2 + 4k + 4 = ] k + 2 g2

1
(b) x = 4, y = 1 and x = - , y = - 8
2

(b) y =

12. a = 0, b = 4

14. x 1 = 1, x 2 = - 1

17. a = - 2, b = 0

(a) x = - 2, y = 5

6. t = 3, v = 1

8. x = - 64, y = - 39

13. p = - 4, q = 1

4.

7.

3. p = 2, q = - 1

(a) x 2 - 8x + 16 = ] x - 4 g2

6.

Exercises 3.12

1
4

(c) x $ 4, x # 3

8. x =

1
,1
2

(b) y = 4.16, -2.16

(c) n = 0.869, -1.54
11. (a) V = 764.5

Exercises 3.13

(b) r = 2.9

13. x 1 2, x 2 9

23 adults and 16 children.

14. x = 2.4, y = 3.2

(b) r = 3.9

1.

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

2.

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

3.

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

4.

x = 4, y = - 3 and x = 3, y = - 4

6.

x = 3, y = 9

8.

m = - 4, n = 0 and m = 0, n = - 4

9.

12. x 2 71

16. (a) ii (b) i

(c) ii

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

18. n 2 0, n 1 - 3
5. x = - 1, y = - 3

19. x = - 4

(d) x = 2

(g) - 4 # x # 2

1
1
15. t = - , h =
4
2

(m) No solutions

(k) x =

(e) iii

(s) 2 1 n # 2

5
12
,y =13
13

2
5

2
5

(b) - 3 # n # 0
(f) t $ 1, t # - 2

(i) y 2 2, y 1 - 2

5
6

(l) -

1 3
(n) t = 2 ,
3 5

(p) m # - 3, m $ 2

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

(e) x = 3, -1

(h) x = - 3

(j) x # - 1, x $ 1

17. x = 0, y = 0 and x = - 2, y = - 8 and x = 3, y = 27
3
1
,y =2
4
2

21. (a) y 2 3

12. x = 0, y = 1

16. x = 2, y = 0

19. x =

(d) iii

1
3

20. x = - 2

(c) x = 2

10. x = 0, y = 0 and x = 1, y = 1
13. x = 1, y = 5 and x = 4, y = 11
1
14. x = , y = 4 and x = - 1, y = - 1
4

15. (a) V = 2100

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

7. t = - 2, x = 4 and t = 1, x = 1

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

1
4

1
#b #2
2

(o) - 1 1 x 1 3

(q) t 1 - 1, t 2 0

(t) -

(r) 1 1 y 1 3

1
1
1x #
5
2

Challenge exercise 3

Exercises 3.14

1.
2. a = - 2, b = - 1, c = 2

y =1

4.

x = 2.56, -1.56

2. x 1 - a, x 2 a

3. a = 3, b = !2

1.

x = - 2, y = - 8, z = - 1

3.

a = - 4, b = 2, c = 7

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

6.

5.

x = 5, y = 0, z = - 2

6. x = 0, y = - 5, z = 4

] x + 3 g ] x - 3 g ] x - 2 g ^ x 2 + 2x + 4 h; x = ! 3, 2

7.

7.

p = - 3, q = 7, r = 4

8. x = 1, y = - 1, z = 2

x = 1, y = 2 and x = - 1, y = 0

8.

9.

h = - 3, j = 2, k = - 4

b = 4; x = ! 17 + 4 Z 8.12, - 0.123

10. a = 3, b = - 1, c = - 2

10. - 1 1 t 1 1

Test yourself 3

13. r = 2.31

1.

16. y # - 2,

2.

(a) b = 10

(b) a = - 116 (c) x = - 7
1
(d) x # - 4 , x 2 - 3 (e) p # 4
3
(a) A = 1262.48

(b) P = 8558.59

18. x =

5. y # - 2, 0 1 y # 3

11. - 3 # x # 8

14. No solutions
1
2
#y 1
2
3

2 ^ 4 ! 10 h
3

20. y 1 -1, y 2

3
5

9. x = ! 1
1
12. x =
4

15. x = ! b + a 2 + a

17. P = 2247.36

19. x 1 - 4, - 2.2 1 x 1 0.7

ANSWERS

Chapter 4: Geometry 1

10. +AEB + +BEC + +CED
= 50 - 8y + 5y - 20 + 3y + 60
= 90c

Exercises 4.1
1.

4.

7.

So +AED is a right angle.

(a) y = 47c (b) x = 39c (c) m = 145c (d) y = 60c
(e) b = 101c (f) x = 36c (g) a = 60c (h) x = 45c
(i) y = 40c (j) x = 80c 2. (a) 121c (b) 72c 29l
(c) 134c 48l 3. (a) 42c (b) 55c 37l (c) 73c 3l
(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 l l l l
(h) (i) 72c21 (ii)162c21 (i) (i) 26c11 (ii) 116c11 l l
(j) (i) 38c51 (ii) 128c51 5. (a) x = 49c (b) 41c
(c) 131c 6. (a) y = 15c, x = z = 165c
(b) x = 142c, y = 48c, z = 28c
(c) a = 43c, b = 137c, c = 101c
(d) a = 97c, b = d = 41c c = 42c
,
(e) a = 68c b = 152c c = 28c (f) a = 10c, b = 150c
,
,
8x - 10 + 2x - 10 + x + 10 + 7x + 10 = 360
(angle of revolution)

18x = 360 x = 20
+ABE = 8x - 10
= 8 (20) - 10
= 150c
+EBC = 2x - 10
= 2 (20) - 10
= 30c
+ABE + +EBC = 150c + 30c
= 180c
` +ABC is a straight angle
+DBC = 7x + 10
= 7 (20) + 10
= 150c
+DBC + +EBC = 150c + 30c
= 180c
` +DBE is a straight angle
` AC and DE are straight lines
8.

+DFB = 180c - (180 - x) c
(+AFB is a straight angle)

=x
` +AFC = x

(vertically opposite angles)

+CFE = 180c - (x + 180c - 2x)
(+AFB is a straight angle)

=x
` +AFC = +CFE
` CD bisects +AFE
9.

+ABD + +DBC
= 110 - 3x + 3x + 70
= 180c
So +ABC is a straight angle.
AC is a straight line.

Exercises 4.2
1.

(a) a = b = e = f = 148c , c = d = g = 32c
(b) x = z = 70c , y = 110c
(c) x = 55c , y = 36c , z = 89c

(d) y = 125c , x = z = 55c

(e) n = e = g = a = c = z = x = 98c, o = m = h = f = b = d = y = w = 82c
(f) a = 95c , b = 85c , c = 32c
(g) a = 27c , b = 72c , c = 81c
(h) x = 56c , y = 124c , z = a = 116c , b = 64c
(i) x = 61c
2.

(a)

(j) y = 37c

+CGF = 180c - 121c

(FGH is a straight angle)

= 59c
` +BFG = +CGF = 59c
These are equal alternate angles.
` AB < CD
(b) +BAC = 360c - 292c = 68c
(angle of revolution)

` +BAC + +DCA = 68c + 112c
= 180c
These are supplementary cointerior angles.
` AB < CD
(c) +BCD = 180 - 76
(+BCE is a straight angle)
= 104c
+ABC = +BCD = 104c
These are equal alternate angles.
` AB ; CD
(d) +CEF = 180 - 128
(+CED is a straight angle)
= 52c
+CEF = +ABE = 52c
These are equal corresponding angles.
`AB ; CD
(e) +CFH = 180 - ] 23 + 115 g (+EFG is a straight angle)
= 42c
` BFD = 42c (vertically opposite angles)
+
+ABF + +BFD = 138c + 42c
= 180c
These are supplementary cointerior angles.
` AB ; CD

767

768

Maths In Focus Mathematics Extension 1 Preliminary Course

Exercises 4.3
1.

(a) x = 60c

(b) y = 36c

(c) m = 71c

(e) x = 30c

(f) x = 20c

(g) x = 67c

(d) x = 37c
(h) a = 73c

10. +OQP = 180 - ] 75 + 73 g (angle sum of triangle)
= 32c
` +MNO = +OQP = 32c
These are equal alternate angles.

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

` MN ; QP

(j) a = 36c , b = 126c , c = 23c
(k) x = 67c , y = z = 59c , w = 121c
2.

Exercises 4.4

All angles are equal. Let them be x.
(angle sum of D)
Then x + x + x = 180
3x = 180 x = 60

1.

(a) Yes

4.

(given)
(given)

So all angles in an equilateral triangle are 60c.
3.

AB = EF = 5cm
BC = DF = 6 cm
AC = DE = 8 cm

(given)

] 90 - x g c

` D ABC / DDEF

(SSS)

(b)Yes

(vertically opposite angles)
+ACB = 50c
+ABC = 180c - (50c + 45c) (angle sum of D)
= 85c
` +DEC = +ABC = 85c
These are equal alternate angles.

XY = BC = 4.7 m

(given)

+XYZ = +BCA = 110c (given)

5.

YZ = AC = 2.3 m

(given)

` AB < DE

` D XYZ / DABC

(SAS)

+ACB = 180c - 124c
= 56c
+CBA + 68c = 124c
+CBA = 124c - 68c
= 56c
` +CBA = +ACB = 56c
` D ABC is isosceles

(c) No

6.

(a) x = 64c

(d) Yes

(exterior angle of D)

+PQR = +SUT = 49c
+PRQ = +STU = 52c

(b) x = 64c , y = 57c

+HJI = 180c - (35c + 25c)
= 120c
+IJL = 180c - 120c
= 60c
+JIL = 180c - (90c + 30c)
= 60c
+ILJ = 180 - (60c + 60c)
= 60c

2.
(angle sum of D HJI)

(angle sum of D IKL)
(angle sum of D JIL)

(given)
(d) +Y = +T = 90c
(given)
+Z = +S = 35c
(given)
XY = TR = 1.3
` by AAS, D XYZ / D STR

(angle sum of D JKL)

(given)
(e) BC = DE = 4
(given)
+C = +E = 90c
(given)
AC = EF = 7
` by SAS, D ABC / D DEF

BC = BD

` AB ; ED

(given)
(a) AB = KL = 4
(given)
+B = +L = 38c
(given)
BC = JL = 5
` by SAS, D ABC / D JKL

(given)
(c) MN = QR = 8
(given)
NO = PR = 8
(given)
MO = PQ = 5
` by SSS, D MNO / D PQR

(KJI is a straight angle)

` BDC = 46c (base angles of isosceles triangle)
+
+CBD = 180 - 2 # 46
= 88c
` CBD = +BDE = 88c
+
These are equal alternate angles.

(AAS)

(given)
(b) +Z = +B = 90c
(given)
XY = AC = 7
(given)
YZ = BC = 2
` by RHS, D XYZ / D ABC

(HJL is a straight angle)

` +JLK = +JKL = 30°
` D JKL is isosceles
9.

(given)

(e) No

(c) x = 63c

Since +IJL = +JIL = +ILJ = 60c,
D IJL is equilateral
+KJL = 180c - 60c
= 120c
+JLK = 180c - (30c + 120c)
= 30c

(given)

`DPQR / DSTU

(d) a = 29c , b = 70c
8.

(given)

QR = TU = 8 cm

y = 38c

7.

(DCB is a straight angle)

3.

(a)

+B = +C
(base angles of isosceles D)
+BDA = +CDA = 90c (given)
AD is common
` by AAS, D ABD / D ACD

ANSWERS

(b) ` BD = DC (corresponding sides in congruent Ds)
` AD bisects BC
4.

But +OBA + +OBC = 180c

+ABD = +BDC

OB is perpendicular to AC.

(corresponding sides in congruent Ds)

5.

OA = OC

(similarly)

10. (a) AD = BC
+ADC = +BCD = 90c
DC is common
`DADC / DBCD
(b) AC = BD

+AOB = +COD

(SAS)

(b) AB = CD

1.

(corresponding sides in congruent

`DABC / DADC

(corresponding angles in congruent

+BAC = +EDC
+ABC = +DEC
+ACB = +ECD

4.

(equal radii)

OB is common
+AOB = +COB = 90c (given)
`DOAB / DOBC

(SAS)

(b) +OCB = +OBC

(base angles of OBC, an isosceles right angled triangle)
(angle sum of triangle)

5.

So +OCB = +OBC = 45c
Similarly +OBA = 45c
` +OBA + +OBC = 45c + 45c = 90c
So +ABC is right angled
(a) +AEF = +BDC = 90c

(given)

AF = BC

(given)

FE = CD

(given)

`DAFE / DBCD

(RHS)

(b) +AFE = +BCD

(corresponding angles in

(a) OA = OC

(alternate angles, AB < ED)
(similarly)
(vertically opposite angles)

(given)
+GFE = +EFD
1.5
GF o =
= 0.5
EF
2.7
2.7
EF o =
= 0.5
DF
4.86
GF
EF
`
=
EF
DF
Since two pairs of sides are in proportion and their included angles are equal, then DDEF ||| DFGE

1.3
AB
=
= 0.714
DE
1.82
4.2
AC
=
= 0.714
DF
5.88
4.9
BC
=
= 0.714
EF
6.86
AC
BC
AB
=
=
`
DE
DF
EF
Since three pairs of sides are in proportion,
D ABC ||| D DEF y = 41c

6.

congruent triangles)

9.

(e) b = 4.5
(g) p = 9.7

` since 3 pairs of angles are equal, DABC ||| DCDE

triangles)

8.

(c) m = 6.6

a = 1.81, b = 5.83

(SSS)

(b) +ABC = +ADC

But +OCB + +OBC = 90c

(b) x = 4.4

3.

(given)

(a) OA = OC

(corresponding sides in congruent

(f) a = 115c , x = 19c , y = 3.2

AC is common

7.

(a) x = 15.1

2.

(given)

BC = DC

(SAS)

(d) a = 76c , i = 23c , b = 81c

triangles)

(a) AB = AD

(given)

Exercises 4.5

(vertically opposite angles)

`DAOB / DCOD

6.

(given)

triangles)

(equal radii)

OB = OD

(a)

So +OBA = +OBC = 90c

(alternate angles, AB < CD)

(alternate angles, AD < BC)
+ADB = +DBC
BD is common
` by AAS, D ABD / D CDB
` AD = BC

(ABC is a straight angle)

(equal radii)

OB is common

(a) OA = OB
OC = OD
OA
OB
`
=
OD
OC
+AOB = +COD

(equal radii)
(similarly)

(vertically opposite angles)

AB = BC

(given)

Since two pairs of sides are in proportion and their included angles are equal, 3 OAB ||| 3 OCD

`DOAB / DOBC

(SSS)

(b) AB = 5.21 cm

(b) +OBA = +OBC

(corresponding angles in congruent triangles)

7.

(a) +A is common
+ABC = +ADE
+ACB = +AED

(corresponding angles, BC < DE)
(similarly)

769

770

Maths In Focus Mathematics Extension 1 Preliminary Course

6.

` since 3 pairs of angles are equal, D ABC ||| D ADE
(b) x = 2.17, y = 2.25
8.

+ABF = +BEC
+CBE = +BFA
` +C = +A

YZ 2 = XY 2 = 1, XZ 2 = 2
YZ 2 + XY 2 = 1 + 1
=2
= XZ 2
` D XYZ is right angled

(alternate angles, AB z CD)
(similarly, BC z AD)
(angle sum of Ds)

` since 3 pairs of angles are equal, D ABF ||| DCEB
9.

+A is common
1.2
AD
=
= 0.4
AB
3
0.8
AE
=
= 0.4
AC
2
AD
AE
`
=
AB
AC
Since two pairs of sides are in proportion and their included angles are equal, D AED ||| D ABC, m = 4.25

AB
10.
CD
BC
AC
AC
AD
AB
`
CD

=
=
=
=

XY = YZ = 1
` D XYZ is isosceles

7.

AC 2 = AB 2 + BC 2
2
2 2 = ^ 3 h + BC 2
4
1
`1
AC

8.

2
= 0.769
2.6
3
= 0.769
3.9
3.9
= 0.769
5.07
BC
AC
=
AD
AC

= 3 + BC 2
= BC 2
= BC
=2
=2#1
= 2BC

(a) AC = 5
(b) AC 2 = 25, CD 2 = 144,
AD 2 = 169
AC 2 + CD 2 = 25 + 144
= 169
= AD 2
` D ACD has a right angle at +ACD
` AC is perpendicular to DC

9.

D ABC ||| D ACD, x = 109c, y = 47c

11. d 2 = ] 20 - 3t g 2 + ] 15 - 2t g 2
= 400 - 120t + 9t 2 + 225 - 60t + 4t 2
= 13t 2 - 180t + 625

11. (a) x = 7.8

(b) m = 4.0, p = 7.2

(d) x = 6.2, y = 4.4
AB
AD
12. (a)
=
DE
BC
AD
AF
Also
=
DE
FG
AB
AF
`
=
BC
FG
(c)

(c) x = 6.5

(e) x = 1.4, y = 9.2

Also
`

16. 4.3 m

14. y = 0.98

(a) x = 6.4

2.

(a) p =

3.

s = 6.2 m

5.

AB 2 = 81, CB 2 = 144, CA 2 = 225
AB 2 + CB 2 = 81 + 144
= 225
= CA 2
` D ABC is right angled

61

(c) b = 5.7

(b) t =

(c) x =

58

15. 134.6 cm

17. 42.7 cm

20. (a) BC 2 = 6 2 - 4 2
= 20
BC = 20
AO = 6 cm
(equal radii)
So AC 2 = 6 2 - 4 2
= 20
AC = 20
Since BC = AC, OC bisects AB
(b) +OCA = +OCB = 90c
(given)
OA = OB
(equal radii)
OC is common
` DOAC / DOBC
(RHS)
So AC = BC (corresponding sides in congruent triangles)
` OC bisects AB

(d) m = 6.6
65

14. 12.6 m

19. No. The diagonal of the boot is the longest available space and it is only 1.4 m.

15. x = 3.19, y = 1.64

(b) y = 6.6

13. 683 m

18. 1.3 + 1.1 2 = 2.9 and 1.5 2 = 2.25
1.3 2 + 1.1 2 ! 1.5 2 so the triangle is not right angled
` the property is not a rectangle

Exercises 4.6
1.

10.

2

BD
AD
=
AE
CE
AD
DF
Also
=
AE
EG
BD
DF
`
=
CE
EG

13. a = 4.8, b = 6.9

3b

12. 1471 mm

AB
AD
=
AE
AC
AD
AF
=
AE
AG
AB
AF
=
AC
AG

(b)

AB =

x2 + y2 x Since three pairs of sides are in proportion,

(d) y =

33

4. CE = 15.3 cm

Exercises 4.7
1.

(a) x = 94c (b) y = 104c (c) x = 111c (d) x = 60c
(e) y = 72c (f) x = 102°, y = 51° (g) x = 43°, y = 47°

ANSWERS

2.

9.

D ABE is isosceles.
` +B = +E = 76c
(base +s equal)
+CBE = +DEB = 180c - 76c
= 104c (straight +s)
+D + 62c + 104c + 104c = 360c (angle sum of quadrilateral)
+D + 270c = 360c
+D = 90c
` CD is perpendicular to AD`

3.

(a)

+D = 180c - x
(+A and +D cointerior angles, AB < DC)

+C = 180c - (180c - x)

(+C and +D cointerior angles, AD < BC)

= 180c - 180c + x
=x
`+A = +C = x
+B = 180c - x (+B and +C cointerior angles, AB < DC)
`+B = +D = 180c - x
(b) Angle sum = x + x + 180c - x + 180c - x
= 360c
4.

a = 150c , b = 74c

5.

(a) a = 5 m, b = 3 m, x = z = 108c, y = 72c
(b) x = 53c, y = 56c, z = 71c
(c) x = y = 5 cm, a = b = 68c
(d) a = 121c, b = 52c, i = 77c
(e) x = 60c (f) x = 3, y = 7

6.

+ADB = +CDB
+CDB = +ABD
+ADB = +DBC
` +ABD = +DBC
` BD bisects +ABC

7.

(a) AD = BC = 3.8 cm
AB = DC = 5.3 cm

(BD bisects +ADC)
(alternate angles, AB < DC)
(alternate angles, AD < BC)

(given)
(given)

Since two pairs of opposite sides are equal, ABCD is a parallelogram. (b) AB = DC = 7cm
(given)
AB < DC
(given)
Since one pair of opposite sides is both equal and parallel, ABCD is a parallelogram.
(c) +X + +M = 54c + 126c
= 180c
These are supplementary cointerior angles.
` XY < MN
Also, XM < YN

(given)

` XMNY is a parallelogram
(d) AE = EC = 5 cm
DE = EB = 6 cm

(given)
(given)

Since the diagonals bisect each other, ABCD is a parallelogram. 8.

(a) x = 5 cm, i = 66c (b) a = 90c, b = 25c, c = 65c
(c) x = 3 cm, y = 5 cm (d) x = 58c, y = 39c
(e) x = 12 cm

6.4 cm

11. 4 2 cm

10. +ECB = 59c, +EDC = 31c, +ADE = 59c
12. x = y = 57c

Exercises 4.8
1.

(a) 540c (b) 720c (c) 1080c (d) 1440c (e) 1800c
(f) 2880c 2. (a) 108c (b) 135c (c) 150c (d) 162c
(e) 156c 3. (a) 60c (b) 36c (c) 45c (d) 24c

4.

128c34l 5. (a) 13

8.

2340c

(b) 152c18l 6. 16

7. 3240c

9. 168c23l

10. Sum = 145n = (n - 2) # 180c
145n = 180n - 360
= 35n
10.3 = n
But n must be a positive integer.
` no polygon has interior angles of 145c.
11. (a) 9

(b) 12

(c) 8

(d) 10

(e) 30

12. (a) ABCDEF is a regular hexagon.
AF = BC
(equal sides)
FE = CD
(equal sides)
+AFE = +BCD
(equal interior angles)
` D AFE / D BCD
(SAS)

S = ] n - 2 g # 180c
= (6 - 2) # 180c
= 720c
720c
+AFE =
6
= 120c
Since AF = FE, triangle AFE is isosceles.
So +FEA = +FAE
(base angles in isosceles triangle)
180 - 120c
` +FEA =
(angle sum of triangle)
2
= 30c
+AED = 120 - 30c
= 90c
Similarly, +BDE = 90c
(b)

So +AED + +BDE = 180c
These are supplementary cointerior angles
`AE < BD
13. A regular octagon has equal sides and angles.
AH = AB
(equal sides)
GH = BC
(equal sides)
+AHG = +ABC
(equal interior angles)
` D AHG / D ABC
(SAS)
So AG = AC
(corresponding sides in congruent triangles)

S = ] n - 2 g # 180c
= (8 - 2) # 180c
= 1080c
1080c
` +AHG =
8
= 135c
+HGA = +HAG

(base angles in isosceles triangle)

771

772

Maths In Focus Mathematics Extension 1 Preliminary Course

180 - 135c
(angle sum of triangle)
2
= 22c30l
+GAC = 135 - 2 # 22c30l
= 90c
We can similarly prove all interior angles are 90c and adjacent sides equal.
So ACEG is a square.
`+HAG =

2.

3.

118.28 cm2

4.

(common)
(a) +DAE = +BAC
(corresponding angles, DE < BC)
+ADE = +ABC
(similarly)
+AED = +ACB
` D ABC and D ADE are similar (AAA)

] 5 - 2 g # 180c
5
= 108c

(b) x = 3.1 cm, y = 5.2 cm
162c

8.

(a) AB = AD
BC = DC

(adjacent sides in kite)
(similarly)

AC is common
` Δ ABC and Δ ADC are congruent (SSS)
(b)

AO = CO
BO = DO
+AOB = +COD

(equal radii)
(similarly)
(vertically opposite angles)

` Δ AOB and Δ COD are congruent (SAS)
9.

11.

73.5 cm2

AF
AD
=
AE
AG
AD
AB
=
AE
AC
AF
AB
`
=
AG
AC

12. (a) AB = AC
+B = +C
BD = DC

Exercises 4.9
1.

3.

(a) 42.88 cm 2 (b) 29.5 m 2 (c) 32.5 cm 2 (d) 14.32 m 2
(e) 100.53 cm 2 4. (a) 25 m 2 (b) 101.85 cm 2
(c) 29.4 m 2 (d) 10.39 cm 2 (e) 45 cm 2

5.

7 51 + 98 = 7 ^ 51 + 14 h cm 2

7.

$621.08

9.

(b) 89 m 2

So AD and BC are perpendicular.
13.

+ACB = 68c
+CAD = 68c - 34c
= 34c
` ˚+CAD = +ADC = 34c
So Δ ACD is isosceles

14.

(a) x = 43c, y = 137c, z = 147c (b) x = 36c
(c) a = 79c, b = 101c, c = 48c (d) x = 120c
(e) r = 7.2 cm (f) x = 5.6 cm, y = 8.5 cm (g) i = 45c

(given)
(base +s of isosceles D)
(AD bisects BC, given)

So +ADB = +ADC = 90c

(c) 10.5 m

Test yourself 4

(similarly)

(b) +ADB = +ADC
(corresponding +s in congruent Ds)
(straight +)
But + ADB + +ADC = 180c

(a) 48 cm (b) 27 cm 10. 12w units 2

6. 22.97 cm 2

(equal ratios on intercepts)

` D ABD / D ACD ] SAS g

(a) 26.35 m 2 (b) 21.855 cm 2 (c) 18.75 mm 2 (d) 45 m 2
(e) 57 cm 2 (f) 81 m 2 (g) 28.27 cm 2 2. 4.83 m 2

1.

6. 1020.7 cm3 7. 36 m

5.

2
10. 6 2 + ^ 2 7 h = 36 + 28 = 64 = 8 2
` ΔABC is right angled (Pythagoras)

360 p (b) Each interior angle:
360
180 p
180p 360
=
p p 180p - 360
=
p
180 ^ p - 2 h
=
p

15. (a)

8. (a) 161.665 m 2

(vertically opposite +HGB)

So +AGF = +CFE = i
These are equal corresponding +s.
` AB < CD

14. +EDC =

ED = CD
(equal sides in regular pentagon)
So EDC is an isosceles triangle.
`+DEC = +ECD
(base angles in isosceles triangle)
180 - 108c
+DEC =
(angle sum of triangle)
2
= 36c
+AEC = 108 - 36c
= 72c
Similarly, using triangle ABC, we can prove that
+EAC = 72c
So EAC is an isosceles triangle.
(Alternatively you could prove EDC and ABC congruent triangles and then AC = EC are corresponding sides in congruent triangles.)

+AGF = i

(base +s of isosceles D)
(exterior + of D)

^ base +s equal h

ANSWERS

+DAC = +ACB
+BAC = +ACD

(alternate +s, AD < BC)
(alternate +s, AB < DC)

7.

AC is common
` D ABC / D ADC (AAS)
(corresponding sides in congruent Ds)
`
AB = DC
Similarly, AD = BC
` opposite sides are equal
15. (a) 24 cm2 (b) 5 cm

Let ABCD be a square with diagonals AC and BD and
+D = 90c
(adjacent sides of square)
AD = DC
` D ADC is isosceles
(base angles of isosceles D)
`+DAC = +DCA
(angle sum of D)
+DAC + +DCA = 90°
`
+DAC = +DCA = 45°
Similarly, +BAC = +BCA = 45°

16. 9

17. +BFG + +FGD = 109c - 3x + 3x + 71c
= 180c
These are supplementary cointerior +s.
` AB < CD
18. 57 cm2

(other angles can be proved similarly)

19. +ACB = 180c - ] +A + +B g
= 180c - x - y
+ACD = 180c - +ACB z = 180c - (180c - x - y)
= 180c - 180c + x + y
=x+y
20. (a)

+A
AC
EF
AB
DE
AC
`
EF

= +E
2.97
=
= 1.1
2.7
3.96
=
= 1.1
3.6
AB
=
DE

(+sum of D)
(straight +)

8.

^ given h

Let ABCD be a kite
(given)
AD = AB
(given)
DC = BC

So Δ ABC and Δ DEF are similar (two sides in proportion, included +s equal).
(b) x = 4.3 cm

AC is common
` by SSS, D ADC / D ABC
`
+DAC = +BAC

Challenge exercise 4
1.

94c

2. x = 75c , y = 46c , z = 29c

4.

+BAD = +DBC
+ABD = +BDC
` +ADB = +DCB

(corresponding angles in congruent Ds)
(given)
(found)

3. 1620c 32c 44l
,

AD = AB
+DAE = +BAE

(given)
(alternate angles, AB < DC)
(angle sum of D)

AE is common
` by SAS, D ADE / D ABE
` +DEA = +BEA
(corresponding angles in congruent Ds)
But +DEA + +BEA = 180c
(DEB is a straight angle)
` +DEA = +BEA = 90c
` the diagonals are perpendicular

` since 3 pairs of angles are equal,
D ABD < D BCD
;
d = 6.74 cm
5.

AB = DC
(given)
+A + +D = 131c + 49c
= 180c
+A and +D are supplementary cointerior angles
` AB < DC
Since one pair of opposite sides are both parallel and equal, ABCD is a parallelogram.

6.

27.36 m 2

9.

(exterior angle of D MNZ)
+MNY + 84c = (15c + 112c)
`
+MNY = 43c
(exterior angle of D XYZ)
+XYZ + 69c = 112c
`
+XYZ = 43c
`
+MNY = +XYZ = 43c
These are equal corresponding angles.
` MN < XY

10. x = 2.12 m

11. (a) 6 m 2

12. x = 28.7 cm, y = 3.8 cm

(b) 10 + 2 5 = 2 ^ 5 + 5 h m
13. x = 7.40 m, y = 4.19 m

773

774

Maths In Focus Mathematics Extension 1 Preliminary Course

14. (a)

(adjacent sides in square)
AB = BC
+ABE = +CBE = 45
(diagonals in square make 45c with sides)

34. x = 22c, y = 29c, w = z = 90c 35. 56.7 cm 2
36. a - 21 b 10 =

EB is common.

b 10 a 21

37. x 2 6, x 1 -2

2
5

1
8

38.

` by SAS, D ABE / DCBE
`
AE = CE (corresponding sides in congruent Ds)

39. - x - 7

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

43. Given diagonal AC in rhombus ABCD:

40. x =

1
4

41. x # -3, x $ 3

42.

1
6

(adjacent sides in rhombus)
AB = BC
(alternate +s, AD < BC)
+DAC = +ACB
(base +s of isosceles D ABC)
+BAC = +ACB
` +DAC = +BAC
` diagonal AC bisects the angle it meets.
Similarly, diagonal BD bisects the angle it meets.

44. ] x + 3 g-1 45. x 3 + 6x 2 + 12x + 8 46.
(b) BD =

x2 + x2

=
=

2x 2
2x

48. x = 98c, y = 41c

1
BD
2
2x
= units 2

DE =

51. (a) 12x - 8y

2. 2 ^ 5 + y h ^ x - y h

p =9

25 + 5 2
5.
23

(b) x 3

7.

x=

9.

(given)
+ABC = +EDC = 90°
(vertically opposite angles)
+ACB = +ECD
(given)
AB = ED
` by AAS D ABC / DEDC
(corresponding sides in congruent triangles)
` AC = EC
` D ACE is isosceles

10. 231.3

14. 3 10 - 4

6. x 3 + 2x 2 - 16x + 3

2 x-3 11. - 3

12. 135c

13. 7.33 # 10 - 2

15. 3.04 16. x + 3

17. x = 1.78, -0.281

52. x = 2.7, y = 3.1

21. x =

4 ! 12
=2! 3
2

28.

26. 7.02 cm

6 15 + 2 6
43

22.

1
49

24. x = 2, y = -1

27. ] 2x - 1 g ^ 4x 2 + 2x + 1 h
29. 7

(d) 3 2 + 1

53. x = 25

54. r =

2
3

r

cm

57.

2
5

58. 5%

61. 9xy y

59. 2.2 # 10 8 kmh -1

60. k = 20

62. 147c 16l 63. 5.57 m 2

64. (a) 5 ] a + 2b - 2 g ^ a 2 - 4a - 2ab + 4b 2 + 4b + 4 h
(b) ] 3a + 4b g ] a - 6b + 2c g

23. x = 4, y = 11 or x = -1, y = - 4
25. 7

x-3 x 2 - 3x + 9

56. Let +DEA = x
(base +s of isosceles D)
Then +EAD = x
+CDA = x + x (exterior + of DEAD)
= 2x
`
+ABC = 2x (opposite +s of < gram are equal)
`
+ABC = 2+DEA

65. - 1
7
15

3x + 2

55. 17.3 cm

18. r = 1.55 19. x 2 1

20.

1

50.

4

3. (a) x - 1

6y - 10

8.

(c)

47. x = 53c

y7
- ]x + 5g
11 3
(f)
(g) x - 14 y 7 z -11 = 14 11
]x + 1g]x - 1g
6
x z
3
1
(h)
(i) 8 5 (j) 13
2
5a ] a + b g ] 1 + 2b g

4.

2
7

(b) 2 31

2 17

(e)

Practice assessment task set 1
1.

49. 0 1 x 1 5

54

30. $643.08 31. 1.1

32. -2 10 + 3 5 - 2 2 + 3

66.

3
1
#x 15
4
8

BC < AD
(ABCD is a < gram)
BC < FE
(BCEF is a < gram)
` AD < FE
Also BC = AD ^ opposite sides of < gram h
BC = FE
^ similarly h
` AD = FE
Since AD and FE are both parallel and equal, AFED is a parallelogram. 67. b = 11.95 m
33. $83.57

68. (a)

34 cm

(b) 30 cm2

ANSWERS

69.

18 3 + 31 2 - 25 5
75

70. 20 71. 32 m

(g) x-intercepts 3, 5, y-intercept 15

72. BD bisects AC
So AD = DC
+BDC = +BDA = 90c (given)
BD is common
` DBAD / DBCD
(SAS)
` AB = CB
(corresponding sides in congruent

(h) x-intercept - 3 5 , y-intercept 5
(i) x-intercept -3, no y-intercept
(j) x-intercept !3, y-intercept 9
2.

triangles)

3.

So triangle ABC is isosceles
73.

x2 + y2
2

79. (d)

74. (b)

75. (c)

76. (a) 77. (b)

78. (b)

4.

Chapter 5: Functions and graphs
5.

Exercises 5.1
Yes

2. No

3. No

8.

Yes

9. Yes

4. Yes

10. No

5. Yes

11. Yes

6. Yes

12. No

7. No

6.

13. Yes

14. No 15. Yes

7.

Exercises 5.2
1.

f ] 1 g = 4, f ] -3 g = 0

3.

f ] 5 g = -25, f ] -1 g = -1, f ] 3 g = -9, f ] -2 g = -4

5.

-35

6. x = 9

2. h ] 0 g = -2, h ] 2 g = 2, h ] -4 g = 14

7. x = !5

8. x = -3

4. 14

9. z = 1, -4

8.

10. f ^ p h = 2p - 9, f ] x + h g = 2x + 2h - 9
11. g ] x - 1 g = x 2 + 2

12. f ] k g = ] k - 1 g ^ k 2 + k + 1 h

13. t = -1; t = 2, -4

14. 0

9.

15. f ] 5 g = 125, f ] 1 g = 1, f ] -1 g = -1

18. 7

f ] - x g = - x = -f ] x g
` odd function f ] - x g = ] - x g2 - 1
= x2 - 1
= f (x)
` even function f ] -x g = 4 ] -x g - ] -x g 3
= - 4x + x 3
= - ^ 4x - x 3 h
= - f ]xg
` odd function f ] -x g = ] -x g 4 + ] -x g 2
= x4 + x2
= f ]xg
` even function f ]xg - f ]- xg = 0
(a) Odd (b) Neither (c) Even (d) Neither (e) Neither

(b) Yes, when n is odd (1, 3, 5, …)

12. (a) (i) x 2 0

21. f ] x + h g - f ] x g = 2xh + h 2 - 5h

(ii) x 1 0

(iii) Even

(b) (i) x 1 2

(ii) x 2 2

(iii) Neither

(c) (i) -2 1 x 1 2
(d) (i) All real x ! 0

22. 4x + 2h + 1

25. (a) 2 (b) 0

Exercises 5.3
2
, y-intercept -2
3
(b) x-intercept -10, y-intercept 4
(c) x-intercept 12, y-intercept 4
(d) x-intercepts 0, -3, y-intercept 0
(e) x-intercepts !2, y-intercept -4
(f) x-intercepts -2, -3, y-intercept 6
(a) x-intercept

g ] - x g = ] - x g + 3 ] - x g4 - 2 ] - x g 2
= x 8 + 3x 4 - 2 x 2
= g (x)
` even function

11. (a) No value of n

20. (a) 3 (b) x - 3 = 3 - 3 = 0
Denominator cannot be 0 so the function doesn’t exist for x = 3. (c) 4

1.

(d) Neither odd nor even

8

(b) Odd values i.e. n = 1, 3, 5, f

19. -28

23. 5] x - c g 24. 3k 2 + 5

(b) 7 f ] x g A 2 = x 6 + 2x 3 + 1

10. (a) Even values i.e. n = 2, 4, 6, f

16. f ] 2 g - f ] -2 g + f ] -1 g = 0 - 4 + 1 = -3
17. 10

(a) f ^ x 2 h = x 6 + 1

(c) f ] - x g = - x 3 + 1

80. (d)

1.

f ] -x g = ] -x g 2- 2
= x2 - 2
= f (x)
` even function

(c) n 4 + n 2 + 2

(e) (i) None

(ii) x 1 -2, x 2 2
(ii) None

(ii) All real x

(iii) Odd

(iii) Neither

Exercises 5.4
1.

(iii) Neither

(a) x-intercept 2, y-intercept -2
1
(b) x-intercept -1 , y-intercept 3
2
1
(c) x-intercept , y-intercept 1
2
(d) x-intercept -3, y-intercept 3
2
1
(e) x-intercept , y-intercept 3
3

775

776

Maths In Focus Mathematics Extension 1 Preliminary Course

2.

(a)

y

(e)
5

5
4
3
2
1

4
3
2
1
1

-4 -3 -2 -1
-1

2

3 4

-2
-3
-4
-5

2 3 4

x

y

(f)

y

(b)

1
21

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

5
4
3
2
1

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

1

2 3

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

x

4

-2
-3

x

1 2 3 4

-4
-5

5
4
3
2
2
- 1

y

(c)
5
4
3

3

2

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

1
-4 -3 -2 -1
-1

1

2

3

x

4

-2
-3
-4

x

5
4
3
2
1

y
5
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
-5

1 2 3 4

y

(h)

-5
(d)

y

(g)

1 2 3 4

x

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

1 2 3 4

x

ANSWERS

Exercises 5.5

y

(i)

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.

5

(a)

4
3
2
1
1

-4 -3 -2 -1
-1

2

3

4

x

6
5
4
3
2
1

-2
-3
-4
-5
y

(j)

y

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

5
4
3
2
1
111
2

-4 -3 -2 -1
-1

2 3

4

y

(b)

x

6

-2

4

-4

3

-5

4.

5

-3

3.

2
1

(a) " all real x ,, " all real y , (b) " all real x ,, " y: y = 2 ,
(c) ! x: x = -4 +, " all real y , (d) ! x: x = 2 +, " all real y ,
(e) ! all real x +, " y: y = 3 ,
(a) Neither

(b) Even

5.

(c) Neither

(d) Odd

-4 -3 -2 -1
-1

2
1

-4
-5

4

x

5

-3

y

(c)

3

-3

3

-4

4

-2

2

-5

5

-4 -3 -2 -1
-1

1

-2

(e) Odd

y

(3, -1)

x

1 2 3 4 5

111 2
2

3

4

x

6
5
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
-5

1 2 3 4

5

x

777

778

Maths In Focus Mathematics Extension 1 Preliminary Course

y

(d)
6
5
4
3
2
1

5
4
3
2
1

1 2 3 4

-4 -3 -2 -1
-1

5

-4 -3 -2 -1
-1
-2
-3
-4
-5
-6

x

-2
-3
-4
-5
y

(e)

1 2 3 4

x

5

y

(i)
5
4
3
2
1

6
5
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
-5

1 2 3 4

5

111 2 3 4
2

-4 -3 -2 -1
-1
-2

x

5

1 2 3 4

5

x

-3
-4
-5
-6
y

(f)

y

(h)

y

(j)

12
10
8
6
4
2

5
4
3
2
1

1 2

-4 -3 -2 -1
-2
-4
-6
-8

3 4

5

-4 -3 -2 -1
-1
-2
-3
-4

x

x

-5
-6

-10
3.

5
4
3

(a) (i) x-intercepts 3, 4, y-intercept 12 (ii) {all real x},
1
( y: y $ - 2
4
(b) (i) x-intercepts 0, -4, y-intercept 0 (ii) {all real x},
" y: y $ -4 ,

2
1

(c) (i) x-intercepts -2, 4, y-intercept -8
" y: y $ - 9 ,

y

(g)

-4 -3 -2 -1
-1
-2
-3
-4
-5
-6

1 2 3 4

5

(ii) {all real x},

(d) (i) x-intercept 3, y-intercept 9 (ii) {all real x}, " y: y $ 0 ,

x

(e) (i) x-intercepts ! 2, y-intercept 4
" y: y # 4 ,
4.

(a) {all real x}, " y: y $ -5 ,

(ii) {all real x},

(b) {all real x}, " y: y $ - 9 ,

ANSWERS

1
(c) {all real x}, ( y: y $ -2 2
4

(d) {all real x}, " y: y # 0 ,

(e) {all real x}, " y: y $ 0 ,
5.

(a) 0 # y # 9

6.

5
4

(b) 0 # y # 4

(c) -1 # y # 24
1
(e) -18 # y # 2
4

(d) -4 # y # 21

(a) (i) x 2 0 (ii) x 1 0

3
2
1

(b) (i) x 1 0 (ii) x 2 0

(c) (i) x 2 0 (ii) x 1 0 (d) (i) x 1 2 (ii) x 2 2
(e) (i) x 2 -5 (ii) x 1 -5
7.

8.

y

(c)

1

2

3

4

5

1

2

3

4

5

1

-4 -3 -2 -1
-1

2

3

4

5

1

2

3

4

5

-2

f ] -x g = - ] -x g 2
= -x2
= f (x)
` even

-3
-4
-5

(a) Even (b) Even (c) Even (d) Neither
(e) Neither (f) Even (g) Neither
(h) Neither (i) Neither (j) Neither

y

(d)
5
4
3

Exercises 5.6
1.

2.

2

(a) x-intercept 0, y-intercept 0
(b) No x-intercepts, y-intercept 7
(c) x-intercepts ! 2, y-intercept -2
(d) x-intercept 0, y-intercept 0
(e) x-intercepts ! 3, y-intercept 3
(f) x-intercept -6, y-intercept 6
2
(g) x-intercept , y-intercept 2
3
4
(h) x-intercept - , y-intercept 4
5
1
(i) x-intercept , y-intercept 1
7
(j) No x-intercepts, y-intercept 9

1

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

y

(e)
5
4

y

(a)

3

5

2

4

1

3

-4 -3 -2 -1
-1
-2

2
1
-4 -3 -2 -1
-1

1

2

3

4

5

-4
-5

-3

(f)

-4

y
5

-5

4

y

3
2

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

x

-3

x

-2

(b)

x

1
-4 -3 -2 -1
-1

1 2 3 4 5

x

-2
-3
-4
-5

x

779

780

Maths In Focus Mathematics Extension 1 Preliminary Course

y

(g)

3.

5
4
3
2

(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.
1

-4 -3 -2 -1
-1
-2

2

3

4

(a) (i) x 2 2 (ii) x 1 2 (b) (i) x 2 0 (ii) x 1 0
1
1
(d) (i) x 2 0 (ii) x 1 0
(c) (i) x 2 1 (ii) x 1 1
2
2
(e) (i) x 1 0 (ii) x 2 0

5.

1

(a) 0 # y # 2

x

5

-3

(b) - 8 # y # -4

(d) 0 # y # 11

-4
-5

6.

(a) x 2 -3
(e) x 1 -2

(b) x 1 0

7.

(a) x = !3

(b) x 2 1, x 1 -1

y

(h)
5

(c) 0 # y # 6

(e) -1 # y # 0
(c) x 2 9

(d) x 2 2

(c) -2 # x # 2

(d) x = -1, -3 (e) x = 3 (f) x = 1, 2 (g) -3 1 x 1 5

4

(h) - 4 # x # 2

(j) x # 2, x $ 4
1
(k) - 4 # x # 1 (l) x # 0, x $ 1 (m) x = 2, 2
(n) No solutions (o) x = 0 (p) x = 1 (q) x = 0, -2
1
(t) x = 0, 6
(r) No solutions (s) x =
3

3
2
1
1

-4 -3 -2 -1
-1

2

3

4

x

5

-2
-3

Exercises 5.7

-4

1.

(i) x 2 4, x 1 0

-5

(a) (i) {all real x: x ! 0}, {all real y: y ! 0} (ii) no y-intercept y (iii) y (i)

5

5

4

4

3

3

2

2

1

1
1

-4 -3 -2 -1
-1

2

3

4

x

5

1

-4 -3 -2 -1
-1

-2

2

3

x

4

-2

-3
-4

-3

-5

-4
-5

y

(j)

(b) (i) {all real x: x ! 0}, {all real y: y ! 0}
(ii) no y-intercept

5
4

y

(iii)

3
2

2

1
-4 -3 -2 -1
-1

1

2

3

4

5

x

1

-2
-3
-4
-5

-2

1

-1
-1

-2

2

x

ANSWERS

(c) (i) {all real x: x ! -1}, {all real y: y ! 0} (ii) 1 y (iii)

y

(iii)
5
4

2

3
2

1

1

-2

-1

1

x

2

-4 -3 -2 -1
-1

1

2 3

4

5

x

-2

-1

-3
-4

-2

-5

1
(d) (i) {all real x: x ! 2}, {all real y: y ! 0} (ii) -1
2

(g) (i) {all real x: x ! 1}, {all real y: y ! 0} (ii) -4

y

(iii)

y

(iii)

5
4

5

3

4

2

3

1

2

-4 -3 -2 -1
-1

1 2

3

4

1

x

5

-2

-4 -3 -2 -1
-1

-3
-4

-3

-5

x

-2
-4

(e) (i) {all real x: x ! -2}, {all real y: y ! 0} (ii)

2

3

4

5

-5

1
6

(h) (i) {all real x: x ! -1}, {all real y: y ! 0} (ii) -2

y

(iii)

1

y

(iii)
2

5
4

1

3

-2

1

-1

2

-1

2
1

-4 -3 -2 -1
-1

-2
(f) (i) {all real x: x ! 3}, {all real y: y ! 0} (ii)

x

-2
2
3

-3
-4
-5

1

2

3

4

5

x

781

782

Maths In Focus Mathematics Extension 1 Preliminary Course

(i) (i) ' all real x: x !

1
2
1, {all real y: y ! 0} (ii) 2
3

Exercises 5.8
1.

y

(iii)

y

(a) (i)

2

3

1

1
2

-1

-2

-

-1

1

2

x

3

-3

x

2
3

-3
-2
(ii) ! x: -3 # x # 3 +, " y: -3 # y # 3 ,
(j) (i) {all real x: x ! -2}, {all real y: y ! 0} (ii) -3

y

(b) (i) y (iii)

4

5
4
3
2
1

-4 -3 -2 -1
-1

1

2

3

4

5

x

4

-4 x -2
-4

-3
-4
-5

2.

(ii) ! x: -4 # x # 4 +, " y: -4 # y # 4 ,

f ] -x g =

2
-x
2
=x
= - f (x)
` odd function

5
4
3

4.

(a)

1
#y #1
9

(b)

(d)

3.

y

(c) (i)

1
#y #1
3

3
#y #3
7

(e) - 2 # y # -

(a) 1 # x # 3

(b) 1 # x # 4

(d) 1 # x # 4

(e) 1 # x # 2

(c) -2

1
1
#y #2
2

1
8

(c) - 6 # x # 0

2

(2, 1)

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

1

2

3

4

x

ANSWERS

(ii) ! x: 0 # x # 4 +, " y: -1 # y # 3 ,

(iii) ! x: -5 # x # 5 +, " y: -5 # y # 0 ,

y

(d) (i)

(b) (i) Above x-axis y (ii)
5
4
3
1

2
1
-4 -3 -2 -1
-1

1

2

3

x

4

-2

x

1

-1

-3
(iii) ! x: -1 # x # 1 +, " y: 0 # y # 1 ,

-4
-5

(c) (i) Above x-axis

(ii) ! x: -4 # x # 2 +, " y: -3 # y # 3 ,

y

(ii)

y

(e) (i)

6

5
4
3
2
(-2, 1)

6

-6

x

1
1

-4 -3 -2 -1
-1

2

3

x

4

(iii) ! x: -6 # x # 6 +, " y: 0 # y # 6 ,
(d) (i) Below x-axis

-2

y

(ii)

(ii) ! x: -3 # x # -1 +, " y : 0 # y # 2 ,
2.

(a) (i) Below x-axis
(ii)

y

8

-8

-8

5

-5

-5

x

(iii) ! x: -8 # x # 8 +, " y: -8 # y # 0 ,

x

783

784

Maths In Focus Mathematics Extension 1 Preliminary Course

6.

(e) (i) Below x-axis y (ii)

(a) {y: - 9 # y # 3}
(b) {y: 0 # y # 9} (c) {y: -8 # y # 1}
1
(d) ' y: # y # 1 1 (e) {y: 0 # y # 4}
5
(f) {y: -1 # y # 15} (g) {y: -1 # y # 0}
(h) " y: - 1 # y # 8 , (i) {y: - 4 # y # 21}
1
(j) ' y: - 6 # y # 6 1
4

7

- 7

x

7.

(a) {all real x: x ! -1}
(b) x-intercept: y = 0
3
0= x+1 0=3
This is impossible so there is no x-intercept
(c) {all real y: y ! 0}

8.

(a) {all real x: x ! 0}

(b) {all real y: y ! !1}

9.

(a)

y

- 7
(iii) " x: - 7 # x #
3.

7 , , # y: - 7 # y # 0 -

(a) Radius 10, centre (0, 0) (b) Radius

5 , centre (0, 0)

25

(c) Radius 4, centre (4, 5) (d) Radius 7, centre (5, -6)
(e) Radius 9, centre (0, 3)
4.

20
15
10

(a) x 2 + y 2 = 16
(b) x - 6x + y - 4y - 12 = 0
(c) x 2 + 2x + y 2 - 10y + 17 = 0
(d) x 2 - 4x + y 2 - 6y - 23 = 0
(e) x 2 + 8x + y 2 - 4y - 5 = 0
(f) x 2 + y 2 + 4y + 3 = 0
(g) x 2 - 8x + y 2 - 4y - 29 = 0
(h) x 2 + 6x + y 2 + 8y - 56 = 0
(i) x 2 + 4x + y 2 - 1 = 0
(j) x 2 + 8x + y 2 + 14y + 62 = 0
2

2

5

-4 -3 -2 -1
-5

1

2

3

4

1

2

3

4

x

5

-10
-15
y

(b)
8

Exercises 5.9

6

1.

4

2.

3.

5.

(a) {all real x}, {all real y} (b) {all real x}, {y: y = -4}
(c) {x: x = 3}, {all real y} (d) {all real x}, {y: y $ -1}
1
(e) {all real x}, {all real y} (f) {all real x}, ' y: y # 12 1
4
(g) {x: -8 # x # 8}, {y: -8 # y # 8}
(h) {all real t: t ! 4}, {all real f (t): f (t) ! 0}
(i) {all real z: z ! 0}, {all real g ^ z h: g ^ z h ! 5}
(j) {all real x}, {y: y $ 0}
(a) {x: x $ 0}, {y: y $ 0} (b) {x: x $ 2}, {y: y $ 0}
(c) {all real x}, {y: y $ 0} (d) {all real x}, {y: y $ -2}
1
(e) ' x: x $ -2 1, {y: y # 0}
2
(f) {all real x}, {y: y # 5} (g) {all real x}, {y: y 2 0}
(h) {all real x}, {y: y 1 0}
(i) {all real x: x ! 0}, {all real y: y ! 1}
(j) {all real x: x ! 0}, {all real y: y ! 2}
(a) x = 0, 5 (b) x = -3, 1, 2 (c) x = 0, 2, 4
(d) x = 0, ! 4 (e) x = !7 4. (a) -1 # x # 1
(b) {x: -1 # x # 1}
(a) {x: x # - 1, x $ 2}

(b) {t: t # - 6, t $ 0}

2

-4 -3 -2 -1
-2

x

-4
-6
-8 y (c)
25
20
15
10
5

-4 -3 -2 -1
-5
-10
-15

1

2

3

4

5

x

ANSWERS

y

(d)

(g)

y

8
6

3

4

2

2
1

-4 -3 -2 -1
-2

2

3

4

1

x

-4

x

1

-1
-1

-6
-8

10. (a) " x: x $ 1 , " y: y $ 0 , y (b) y (e)

2
1

8
6

-1

4

1

2

2
1

-4 -3 -2 -1
-2

2

3

4

x

x

3

y

11.
6

-4

5

-6

4

-8

3
2
1

-1

y

(f)

1

x

12. (a) (i) {all real x}, {all real y} (ii) All x (iii) None
(b) (i) {all real x}, " y: y 2 -2 , (ii) x 2 0 (iii) x 1 0
(c) (i) {all real x: x ! 0}, {all real y: y ! 0} (ii) None
(iii) All x ! 0
(d) (i) {all real x}, {all real y} (ii) All x (iii) None
(e) (i) {all real x}, " y: y 2 0 , (ii) All x (iii) None
13. (a) - 2 # x # 2 (b) (i) {x: - 2 # x # 2}, { y: 0 # y # 2}
(ii) {x: - 2 # x # 2}, { y: - 2 # y # 0}

10

-10

-1

10

x

Exercises 5.10
1.

-10

(b) -10 (c) 8 (d) 3 (e) 3 (f) 75
1
(j) 1 (k) - 7 (l) x 2 - 3x
(h) - 6 (i)
4
(a) 21

(m) 2x 3 + 3x - 5
2.

(g) 0

(n) 3c 2

(a) Continuous (b) Discontinuous at x = - 1
(c) Continuous (d) Continuous (e) Discontinuous at x = !2

785

786

Maths In Focus Mathematics Extension 1 Preliminary Course

3.

5.

(a)

(a)

(b)

(b)

(c)

(c)

(d)

Exercises 5.11
1.

(a) 0 (b) 0 (c) 0 (d) 2 (e) 1
(h) 0 (i) 5x (j) 3

2.

(a) RHS = 1 +
=

(f) 6 (g)

3
1
+ 2 x x x2 + x + 3

x2
= LHS
(b) 1 from above (c) 1 from below
3.

(a) 2 from below (b) 2 from above

4.

(a)

x
3

(b)

5x 2
4

2
3

(e)

ANSWERS

(f)

7.

21x #2

9.

x#

2
5

2
,x 21
3

8. x 1 - 6, x 2 - 3
10. - 2

2
# x 1 -2
3

Exercises 5.13
1.

y

(a)
6

(g)

5
4
3
2
1
1

-4 -3 -2 -1
-1

2

3

4

x

-2

(h)

-3
-4

y

(b)
6
5

(i)

4
3
2
1
1

-4 -3 -2 -1
-1

2

3

4

1

2

3

4

x

-2
(j)

-3
-4
y

(c)
6
5
4
3
2
1

Exercises 5.12
1.

-

1
1x 10
2

2. 0 1 x 1

1
3

4.

-

1
#x 10
2

5. 1 1 x 1 1

3. 0 1 x # 1
1
3

6. x $ -1, x 1 - 2

-4 -3 -2 -1
-1
-2
-3
-4

x

787

788

Maths In Focus Mathematics Extension 1 Preliminary Course

y

(d)

y

(g)

6

6

5

5

4

4

3

x+y = 1

2

3
2
1

1

-4 -3 -2 -1
-1

2

1

3

x

4

1

-2

4

2

3

4

x

-3

-4

3

-2

-3

2

1

-4 -3 -2 -1
-1

-4 y (e)

y

(h)

6

6

y = x +1

5

5

4

4

3

3

2

2

1

1
1

-4 -3 -2 -1
-1

2 3

x

4

-4 -3 -2 -1
-1

-2

-2

-3

-3

-4

x

-4
3x - y - 6 = 0

-5

y

(f)

-6

6

y

(i)

5

y = 2x -3

4

6

3

5

2

4

1

-4 -3 -2 -1
-1
-2
-3

1

2

3

4

3

x x + 2y - 2 = 0

2
1

-4 -3 -2 -1
-1
-2
-3
-4
-5
-6

1

2

3

4

x

ANSWERS

y

(j)

y

(c)

6
1

5
4
3
2
1

-4 -3 -2 -1
-1

1

2

3

x

1

-1 x 4

-2

-1

-3
-4
-5

x=

-6
2.

(a) x 2 -3
(e) y $ 2

3.

(b) y $ -2

1
2

y

(d)
5

(d) y 2 x 2 - 4

(c) y $ x + 1

y=x2

4

x

3

y

(a)

2
1

5
4

-4 -3 -2 -1

y = x2 - 1

3

-1

2

3

4

5

3

4

x

-2

2

-3

1

-4 -3 -2 -1
-1

1

1

2

3

4

5

-4

x

-5

-2

y

(e)

-3
-4

8

-5

6
4

y

(b)

2

-4 -3 -2 -1
-2

3

-4 y = x3

-3

3

-3

x
4.

(a) y 1 3x - 2
(b) y 2 x 2 + 2
(c) x 2 + y 2 1 49
(d) x 2 + y 2 2 81
(e) x 1 5, y 2 2

-6
-8

1

2

x

789

790

Maths In Focus Mathematics Extension 1 Preliminary Course

5.

y

(a)

y

(b)

5

6

4

5

3

4

2

3
2

1
1

-4 -3 -2 -1
-1

2

3

4

x

y=x -3

1

-4

-2

1

-3 -2 -1
-1

2

3

x

4

-2
-3

y

(b)

-4

5

-5

4

-6

3
2

y

(c)

1

-4 -3 -2 -1
-1

1

2

3

4

6

x

5

y = 3x – 5

4

-2

3
2

y

(c)

1

5
4

-4 -3

3

1

-1
-1

2

3

4

x

-2

2

-3

1

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

1

2

3

4

-4

x

-5
-6

-2
6.

-2

y

(d)

y

(a)

6

6

5

5
4

3

3

2

2

1

1

-4 -3 -2 -1
-1
-2
-3
-4

y=x+1

4

1

2

3

4

x

-4

-3

-2 -1
-1
-2
-3
-4
-5
-6

1

2

3

4

y=3–x

x

ANSWERS

y

(e)

y

(h) x = -2

3

8

y = x3

6

y=3

4
2

y=1

-3

-4 -3 -2 -1
-2

x

3

1

2

3

x

4

-4
-6
-8

-3

y

(f)

y

(i)

2

1

1
1

-2

-1

x

2

-2

x

1

-1

-1

x=–1 y (g)

y

(j)
5

6

4

5

3

y=4

4

2

y = x2

3

1

-4 -3 -2 -1
-1

x - y = -1

2
1

2

3

4

5

x

x-y=2

1

-2

-4 -3 -2 -1
-1

-3

-2

-4

-3

-5

-4
-5
-6

1

2

3

4

x

791

792

Maths In Focus Mathematics Extension 1 Preliminary Course

7.

y

(a)

y

(d)

y = x2
5
4

2

3
2

1

1

-4 -3 -2 -1
-1

1

2

3

4

x

5

1

-1

2

3

x

4

2 y= x

-2

-2

-3
-4
-5
(e)

y

y

(b)
8

y = x3

2

6
4

y=1

1

2
1

-4 -3 -2 -1
-2

2

3

x

4

-4 y= -4

-3 -2

-1

1

2

3

x

4

-1

1 x+2 -2

-6
-8
8. y (c)

(a)

y y = x2

y=5

5

2

4
3

2

1

-2

2

1

x
-4 -3 -2 -1

-1

1

2

3

-2
-3

-2

-4 x=1 -5

x=2

4

5

x

ANSWERS

(e)

y

(b)

y

6

6

5

y = |x|

5

4

4

3

3

2

2

1

1

y=3

x
-4 - 3 - 2 - 1
-1
-2

1

2

3

x
-4 -3 -2 -1
-1

4

y = -1

-3

2

-3

-5

3

4

x=2

Test yourself 5

-6

1.

(a) f ] - 2 g = 6

2.

y

(c)

1

-2

x=3

-4 y=x-2 793

(b) f ] a g = a 2 - 3a - 4

(a)

(c) x = 4, -1

y = 2x + 1

6
5
4
3
2

2x - 3y = 6

1

(b)

x

-4 -3 -2 -1
-1
-2

1

2

3

4

-3
-4
-5
-6
(c) y (d)

3

y=2

(d)

-3

3

x

(e) x = -3

Answer S1-S5.indd 793

-3

8/11/09 11:31:52 AM

794

Maths In Focus Mathematics Extension 1 Preliminary Course

8.

(f)

9.
(g)

(h)
10.

3.

4.

1
4
(b) Domain: all real x; range: all real y
(c) Domain: - 1 # x # 1; range: - 1 # y # 1
(d) Domain: - 1 # x # 1; range: 0 # y # 1
(e) Domain: - 1 # x # 1; range: - 1 # y # 0
(f) Domain: all real x ! 0; range: all real y ! 0
(g) Domain: all real x; range: all real y
(h) Domain: all real x; range: y $ 0
(a) Domain: all real x; range: y $ - 6

15

5. (a) 4 (b) 5

(c) 9

(d) 3

11. (a) y # 3

(b) y 2 x + 2

(c) y $ - x 2, y # 0

12. (a) Domain: all real x ! 3, range: all real y ! 0
(b)

(e) 2

6.

13. (a)

7.

(b) (i) x = 2, -4 (ii) - 4 1 x 1 2 (iii) x 2 2, x 1 - 4

ANSWERS

14. (a) 2

(b) x = 3

2
3

1
3

(c) 1

y

(b)

15. (a) x-intercept -10, y-intercept 4
(b) x-intercepts - 2, 7, y-intercept -14 y 16. (a) i (b) iii (c) ii
17. (a) 4

(b)

2
5

(c) - 1

(d) i
1
2

(e) iii

(d) 3
-1

18.

x

1

y

(c)

19. (a) Domain: x $ 2, range: y $ 0

2

(b)

y

(d) f (x) = x 4 3x 2 1 f ( x) = ] - x g4 3 ] x g2
= x 4 3x 2 1
3x
= f (x)
So f ] x g is even.

x

4

-4

20. (a)

1
1
x

f (x) = x 3 x f ( x) = ] - x g3 ( )
= - x3 + x
= -( 3
)
= - f (x)
So f ] x g is odd.
(b)

y

(e)

21. (a) a y

2

1

1 x x
-4 -3 -2 -1
-1
4

-2

1

2

3

4

5

795

796

Maths In Focus Mathematics Extension 1 Preliminary Course

6.

1.

f ] 3 g = 9, f ] -4 g = 16, f ] 0 g = 1

7.

Challenge exercise 5

Domain: all real x ! ! 1; range: y # - 1, y 2 0

2 b=- ,3
3

2.

y

3.

8.

-2

2

x

9.

Domain: x $ 0; range: y $ 0

10. x = 0, 3, - 2

11.

4.

12. h ] 2 g + h ] -1 g - h ] 0 g = - 3 + 0 - ] -1 g = - 2
5.

ANSWERS

13.

18.

14.

19.

15. f ^ (-a) h = 2 (-a ) - 1
= 2a 2 - 1
= f (a 2)
2

16. x =

17. (a)

20. Domain: x $ 3; range: y $ 0 21. Domain: - 2 # x # 2

2

22.

1 ! 41
4

1 x+3 2]x + 3g
1
=
+
x+3 x+3 2x + 6 + 1
=
x+3
2x + 7
=
x+3
= LHS
2x + 7
1
=2+
`
x+3 x+3 RHS = 2 +

23. (a) 0
(b)

(b) Domain: all real x ! - 3; range: all real y ! 2
(c)
24.

797

798

Maths In Focus Mathematics Extension 1 Preliminary Course

5.

Exercises 6.1 cos i =

2.

3
5
4 sin b = , cot b = , sec b =
5
4
3

3.

sin b =

4.

cos x =

5
, tan x =
9

5.

cos i =

3
4
, sin i =
5
5

6.

5
5
3 tan i =
, sec i = , sin i =
2
2
3

7.

cos i =

35
, tan i =
6

8.

tan i =

51
51
, sin i =
7
10

9.

(a)

2

10. (a)

3

7
74

(a) x = 6.3 (b) y = 5.6 (c) b = 3.9 (d) x = 5.6
(e) m = 2.9 (f) x = 13.5 (g) y = 10.0 (h) p = 3.3
(i) x = 5.1 (j) t = 28.3 (k) x = 3.3 cm (l) x = 2.9 cm
(m) x = 20.7 cm (n) x = 20.5 mm (o) y = 4.4 m
(p) k = 20.6 cm (q) h = 17.3 m (r) d = 1.2 m
(s) x = 17.4 cm (t) b = 163.2 m
1.6 m

5.

(a) 18.4 cm

7.

47.4 mm 8. 20.3 m
(c) 9.0 cm

5
74

56
, cosec x =
5

9
56

3. 20.3 cm

10. (a) 6.8 cm

35

(b) 13.8 cm

6. 10 cm and 10.5 cm

9. (a) 7.4 cm

(b) 6.6 cm

(b) 6.5 cm

1.

(a) x = 39c 48l (b) a = 35c 06l (c) i = 37c 59l
(d) a = 50c 37l (e) a = 38c 54l (f) b = 50c 42l l (g) x = 44c 50l (h) i = 30c 51 (i) a = 29c 43l
(j) i = 45c 37l (k) a = 57c 43l (l) i = 43c 22l
(m) i = 37c 38l (n) i = 64c 37l (o) b = 66c 16l
(p) a = 29c 56l (q) i = 54c 37l (r) a = 35c 58l
(s) i = 59° 2l (t) c = 56c 59l

2.

37c 57l 3. 22c 14l

3
1
1
, cos 30c =
, tan 30c =
2
2
3
3

6.

(a) 11.4 cm

13. tan 48c = cot 42c = 1.11

14. (a) 2 cos 61c or 2 sin 29c

8.

(a) 13 m

(e) 2

15. x = 80c

16. y = 22c

19. t = 20c

11. 38 cm

Exercises 6.4

3
1
, cos 60c = , tan 60c =
2
2

(d) 1

4. 13.9 m

1

12. sec 82c = cosec 8c = 7.19

(c) 0

(e) 1.393

1.

11. sin 67c = cos 23c = 0.92

(b) 0

(d) 0.928

Exercises 6.3

(b) 45c
1
1
(c) sin 45c =
, cos 45c =
, tan 45c = 1
2
2

(c) sin 60c =

(c) 0.339

2.

7
, cos b =
5

, tan b =

(b) sin 30c =

(a) 17c 20l (b) 34c 20l (c) 34c 12l

(b) 0.697

(d) 46c 34l (e) 79c 10l

5
12
12
, sin i =
, tan i =
5
13
13

1.

(a) 0.635

6.

Chapter 6: Trigonometry

4. 36c52l 5. 50c

(b) 37c 52l 7. a = 31c 58l, b = 45c 44l

(b) 65c 17l 9. (a) 11c 19l (b) 26 cm

10. 4.96 cm and 17.3 cm

17. p = 31c

18. b = 25c

20. k = 15c

11. (a) 12.9 m

(b) 56c 34l

Exercises 6.5
1.

(a)

North

Exercises 6.2
1.

(a) 47c

2.

(a) 47c 13l (b) 81c 46l (c) 19c 26l

(b) 82c

(c) 19c

(d) 77c

(e) 52c

(d) 76c 37l (e) 52c 30l
3.

(b) 65.5c

(d) 68.35c
4.

(a) 77.75c

(c) 24.85c

(e) 82.517c

(a) 59c 32l (b) 72c 14l (c) 85c 53l
(d) 46c 54l (e) 73c 13l

Beach house 100c

Boat

ANSWERS

North

(b)

(f) North

Farmhouse

Jamie

12c

Campsite

Dam
(g)

320c

North

North

(c)

House 160c

Jetty

200c

Mohammed
(h)

North

Seagull
(d) North

Alistair
Mine shaft
80c
Town
50c

(i)

Bus stop

Yvonne

North

North

(e)

Plane

349c

B Hill
285c

School

799

800

Maths In Focus Mathematics Extension 1 Preliminary Course

North

(j)

4.

(a) 2nd

6.

(a) 1st
(e) -

8.

Boat ramp
Island

280c

2.

(a) 248c

3.

080c

7.

4. 210c 5. 160c 6. 10.4 m

12. 1.8 km

3
2

1
2

13. 12 m 14. 242c

15. 035c

25. 198 m 26. 4.8 km 27. 9.2 m 28. 217c

3+1
2

(g) 1

1
4

6+ 2
=
4

(h)

(j) - ^ 2 + 3 h
1
3

2

(k) 0

(d) 4

2 ^ 3 + 1h
4

(l) 1

(q) 2 3

3 2
2

9 3
2

(a) x =

3.

60c

7.

(a) 6 2 m

4.

(b) y =

2m

5.

3m

4 3
3
(i)

(f)

(r) -

1
2

(s) 6

2 3
3

5
21

(n)
2
3

(t)

6
2- 3
2

(c) p = 2 3
6.

10 3 m 3

(b) 4 m 8. 0.9 m

9.

5^3 + 3 h m 3

10. 100 3 m

1.

(a) 1st, 4th (b) 1st, 3rd (c) 1st, 2nd
(e) 3rd, 4th (f) 2nd, 3rd (g) 3rd
(h) 3rd (i) 2nd (j) 4th

2.

(a) 3rd

(b) -

1
2

3. (a) 4th (b) -

1
2

(d) 2nd, 4th

65

1

(d)

(j)

2
1
2

3
2

(i) 3
2

1
2

(j) -

(e) -

(j) -

(e) -

(d)

1
2

3
2

1
2

1
2

(f)

3

1
2

, cot x = -

, sec i =

3
10

20. (a) sin i

2
21

21
2

, tan x = -

9
65

(b) cos x = -

5
18. cot a = - , sec a =
6
19. sin i =

89
5

55
8
8
, sec x = - , cosec x = 3
3
55

17. (a) sin x =

91
3
, tan x = 10
91

61
61
, cosec a = 5
6

51
7
, cot i = 10
51
(b) cos x

(f) - sin i (g) cos a

(c) tan b

(d) - sin a

(e) - tan i

(h) - tan x

Exercises 6.8
1.

Exercises 6.7

4

15. tan i = 16. tan x =

3

(m) 2 ^ 2 - 1 h

(p) 3 - 2 2

2.

(e)

3
2

(i) −1

(d) (i)

3

(c)

3

(c) - 3

7 74
5 74
, sin x = 74
74

14. cos x = -

(b) 1 (c)

1

, cosec x = -

(b) 7.2 km 30. (a) 13.1 m (b) 50c26l

(a)

(o) 1

89

13. cosec x = -

Exercises 6.6
1.

3

3

2

8

12. cos x =

22. 1931.9 km 23. 34.6 m 24. 149c

29. (a) 1.2 km

(c)

2
1

2

33
4
, tan i = 7
33

11. cos i = -

l
16. 9.2 m 17. 171 m 18. 9.8 km 19. 51c 41 20. 2.6 m
21. 9c21l

(h) -

(h) -

1

(h)

3
2

3
2

1
2

(b)

1

(b)

1

3
4
10. sin i = - , cos i = 5
5

10. (a) 1056.5 km (b) 2265.8 km (c) 245c
11. 83.1 m

(g)

(b)

(g)

(b) -

2

5. (a) 2nd

7. (a) 1

1
2

(f) -

1

(a) (g)

8. 126.9 m 9. 72c48l

21 m

1
2

(a) -

3
2

(b)

(f) - 3

9.

(b) 145c (c) 080c (d) 337c (e) 180c

(b) - 3

l
(a) i = 20c 29l, 159c 31 (b) i = 120c, 240c
(c) i = 135c, 315c (d) i = 60c, 120c (e) i = 150c, 330c
(f) i = 30c, 330c
(g) i = 30c, 120c, 210c, 300c ] 0c # 2i # 720c g
(h) i = 70c, 110c, 190c, 230c, 310c, 350c
] 0c # 3i # 1080c g

ANSWERS

(i) i = 30c, 150c, 210c, 330c
(j) i = 15c, 45c, 75c, 105c, 135c, 165c, 195c, 225c,
255c, 285c, 315c, 345c
2.

(a) i = !79c 13l (b) i = 30c, 150c
(d) i = - 60c, -120c

16.

(c) i = 45c, -135c

(e) i = 150c, -30c

(f) i = !30c, !150c
(g) i = 22c 30l, 112c 30l, -67c 30l, -157c 30l
(h) i = !15c, !45c, !75c, !105c, !135c, !165c
(i) i = 135c, -45c

(j) i = !30c, !60c, !120c, !150c

3.
17.

4.

-1

5.

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
(f) cosec x
2

(j) sin2 x
3.

6.

x = 0c, 180c, 360c

9.

7. - 1

8. 1

x = 0c, 360c

(g) sec x

(k) 1

2

(l) sin i cos i

(a) LHS = cos x - 1
= 1 - sin 2 x - 1
= - sin 2 x
= RHS
So cos 2 x - 1 = -sin 2 x
2

(b) LHS = sec i + tan i sin i
1
=
+
cos i cos i
1 + sin i
=
cos i
= RHS
1 + sin i
So sec i + tan i = cos i

10.

(c) LHS = 3 + 3 tan 2 a

11. 0

12. x = 270c

14. x = 0c, 180c, 360c

13. x = 0c, 180c, 360c
15. x = -270c, 90c

= 3 (1 + tan 2 a )
= 3 sec 2 a
3
= cos 2 a
3
=
1 - sin 2 a
= RHS
So 3 + 3 tan 2 a =

(d) cos2 x

(e) sin a

(h) tan i (i) 5 cosec 2 i

2

3
1 - sin 2 a

801

802

Maths In Focus Mathematics Extension 1 Preliminary Course

(d) LHS = sec 2 x - tan 2 x
= tan 2 x + 1 - tan 2 x
=1
= cosec 2 x - cot 2 x
= RHS

(j) LHS =
=

=

(e) LHS = ] sin x - cos x g 3
= ] sin x - cos x g ] sin x - cos x g 2
= ] sin x - cos x g ^ sin 2 x - 2 sin x cos x + cos 2 x h
= ] sin x - cos x g ] 1 - 2 sin x cos x g
= sin x - 2 sin 2 x cos x - cos x + 2 sin x cos 2 x
= RHS
So ] sin x - cos x g 3 = sin x - 2 sin 2 x cos x - cos x
+ 2 sin x cos 2 x

=

RHS =

1 - sin 2 i + 2 sin i sin i cos i cos 2 i + 2 sin i
=
sin i cos i
2 sin i cos 2 i
=
+ sin i cos i sin i cos i cos i
2
=
+
sin i cos i
= cot i + 2 sec i
= LHS

=

=

(g) LHS = cos 2 ] 90c - i g cot i
= sin 2 i cot i cos i
= sin 2 i # sin i
= sin i cos i
= RHS
So cos 2 ] 90c - i g cot i = sin i cos i

So
4.

So

cos 2 i

= tan 2 i + cos 2 i

sin b cos b
+
cos b sin b sec b sin 2 b + cos 2 b sin b cos b sec b

1 + cot b cosec b

- cos b = sin b

LHS = x 2 + y 2
= ] 2 cos i g 2 + ] 2 sin i g 2
= 4 cos 2 i + 4 sin 2 i
= 4 (cos 2 i + sin 2 i)
= 4 ]1g
=4
= RHS
So x 2 + y 2 = 4

1 - sin 2 i cos 2 i

1 - sin 2 i cos 2 i

tan b + cot b sec b

LHS = RHS

So ] cosec x + cot x g ] cosec x - cot x g = 1 cos 2 i sin 2 i cos 2 i
=
cos 2 i cos 2 i
= sec 2 i - sin 2 i
= tan 2 i + 1 - (1 - cos 2 i)
= tan 2 i + 1 - 1 + cos 2 i
= tan 2 i + cos 2 i
= RHS

sec b

1 sin b cos b cos b sin b
= sec b #
1
cos b sin b
1
=
#
1 cos b
= sin b

1 - sin i + 2 sin i sin i cos i

1

cosec b
1 + cot b - cot b

=

2

(i) LHS =

cosec b

cosec b
1
= cosec b
= sin b

(f) RHS =

(h) LHS = ] cosec x + cot x g ] cosec x - cot x g
= cosec 2 x - cot 2 x
= 1 + cot 2 x - cot 2 x
=1
= RHS

- cos b cosec b
1 + cot b - cos b cosec b
1 + cot b - cos b #

So sec 2 x - tan 2 x = cosec 2 x - cot 2 x

So cot i + 2 sec i =

1 + cot b

5.

LHS = x 2 + y 2
= ] 9 cos i g 2 + ] 9 sin i g 2
= 81 cos 2 i + 81 sin 2 i
= 81 (cos 2 i + sin 2 i)
= 81 ] 1 g
= 81
= RHS
So x 2 + y 2 = 81

1 sin b

ANSWERS

Exercises 6.10
1.

(a) x = 8.9

Exercises 6.13
1.

2.

15 3 2 m 2
1.2 m 2

9.

(c) a = 10.0

(a) 7.5 cm 2 (b) 32.3 units 2 (c) 9.9 mm 2
(d) 30.2 units 2 (e) 6.3 cm 2

6.

(b) y = 9.4 cm

(d) b = 10.7 m
2.

(a) 7.8 cm

(e) d = 8.0

(a) i = 54c 57l (b) a = 61c 23l (c) x = 43c 03l
(d) a = 87c 04l (e) i = 150c 56l

3.

126c 56l 4. (a) 13.5 mm

5.

(a) 1.8 m

7.

(a) 10.3 m

9.

(a) 14.1 cm (b) 15.6 cm

(b) 2.7 m 6. 5.7 cm
(b) 9.4 m 8. (a) 60c 22l (b) 57c 9l

2.

8. 247.7 mm 2

(b) 180.8 cm 2

10. (a) 5.6 cm

(b) 18.5 cm 2

(c) 19.1 cm 2

1.

(e) y = 9.3

(a) i = 51c 50l (b) i = 60c 27l (c) x = 57c 42l

l
(a) 2 m (b) 2.2 m (c) 65c 21 2. (a) 1.9 m (b) 49c 46l
(a) 109 cm 2

5.

(a) 9 m

7.

(c) h = 7.4 cm

l
(d) b = 131c 31 (e) i = 73c 49l

(a) 48 m

(b) 128.6 m

9.

16c 50l

10. 11c 10l

(b) 16c 20l 4. 65c 9l

(b) 25c 7l 6. (a) 56 m

3.

32.94 mm 4. 11.2 cm and 12.9 cm l (a) 11.9 cm (b) 44c 11 (c) 82c 13l

1.

(b) 89.7 m

(c) 97.7 m 8. 84 m

Exercises 6.15

5.
6.
7.
9.

+XYZ = +XZY = 66c 10l, +YXZ = 47c 40l
(a) 18.1 mm (b) 80c49l 8. (a) 6.2 cm

(a) sin a cos b - cos a sin b
(c)

(b) 12.7 cm
(e)

12.9 cm 10. (a) 11 cm

(b) 30c

tan a + tan b
1 - tan a tan b

(b) cos p cos q - sin p sin q

(d) sin x cos 20c + cos x sin 20c

tan 48 + tan x
1 - tan 48c tan x

(f) cos 2i cos a + sin 2i sin a

(g) cos x cos 75c - sin x sin 75c

(h)

Exercises 6.12
1.

12.5 cm and 4.7 cm 2. (a) 040c (b) 305c 3. 16.4 m

4.

103c

7.

(a) 1.21 km

5. 1.97 m 6. 11c

(i) sin 4a cos b - cos 4a sin b (j)
2.

(b) 1 minute 8. 32 m 9. 107 m

11. h = 8.5

12. 7.7 km

13. 5.7 km and 5.4 km

14. 1841 km

15. 35.8 m 16. 89c 52l 17. 9.9 km

18. 163.5 km

(a) sin ] a + b g (b) tan 65c
(e) tan 2i (f) sin 32c
(i) 2 sin x sin y

5.8 sin 42c 29l
10. (a) AC =
(b) i = 74c 50l sin 101c 36l
3.

(a)

19. 64.1 m 20. 3269 km

21. (a) 11.3 cm (b) 44c 40l 22. 141c
23. (a) 11.6 cm (b) 73c 14l

(c)

(d)

(e)

24. (a) 265.5 km (b) 346c 33l l 25. (a) 35c 5 (b) (i) 4.5 m

S6.indd 803

5. 34.8 cm 2

3.
(b) b = 10.4 m

(d) n = 16.4

7. 42 cm 2

4. 15.5 cm 2

Exercises 6.14

Exercises 6.11
(a) m = 5.8

3. 7.5 cm 2

(b) 25 mm

10. (a) 54.7 mm (b) 35.1 mm

1.

803

(ii) 0.55 m

(g)

1+ 3

=

2 2
1+ 3

2 2

(c) cos 55c

(d) sin ^ 2x + 3y h

(g) 2 sin a cos b

(h) 2 cos x sin y

(j) 2 cos m cos n
2+ 6
4

(b)

1+ 3

2+ 6
4

=

2 2

=

-^2 3 + 4h
= -^ 3 + 2h
2

=

2 2
1+ 3

1 + tan a tan 3b

2 3+4
=
2

1- 3
1- 3

tan a - tan 3b

=

3-1
1+ 3

tan 5x - tan 7y
1 + tan 5x tan 7y

=

2- 6
4

(f)

3+2

3-1
2 2

=

6- 2
4

2+ 6
4

8/11/09 11:38:03 AM

804

Maths In Focus Mathematics Extension 1 Preliminary Course

(h)

1+ 3
1- 3

=

15. (a) sin 6x

-^4 + 2 3 h
= -^2 + 3 h
2

(e)

3-1
1+ 3 o + cos x e o (i) sin x e
2
2
(j)

4.

2
2

cos y =

(c)

2 cos y

5. (a)

6 + 35
3 5-2 7

6 + 35
12

=

3 5+2 7
12

(b)

(g)

32 5 + 27 7
17

6.

(a) 2 sin i cos i (b) cos 2 i - sin 2 i (c)

7.

2 tan i

(a) 3 sin i cos 2 i - sin 3 i
(b) cos 3 i - 3 sin 2 i cos i (c)

8.
9.

1 - tan 2 i

3 tan i - tan 3 i

3
(c) 2
11. (a)

4
5

(b)

3
(d) 2
12
13

(c) -

(e)
33
65

10. (a)

1
2

18. (a)

(b)

(d)

12
5

(e) - 3

(b) -2 sin x sin y

21.

15
16

(c) 2 cos x sin y

(f)

2 tan y ^ tan 2 x + 1 h
1 - tan 2 x tan 2 y

2 tan i
1 - tan 2 i

(c) cos 2 i - sin 2 i

(d) sin x cos 2y + cos x sin 2y
= sin x _ cos 2 y - sin 2 y i + 2 cos x sin y cos y
(e) cos 2a cos b - sin 2a sin b
= ^ cos 2 a - sin 2 a h cos b - 2 sin a cos a sin b
(f)

tan x + tan 2y

1 - tan x tan 2y tan x - tan x tan 2 y + 2 tan y
=
1 - tan 2 y - 2 tan x tan y
(g) sin 2i cos d - cos 2i sin d
= 2 sin i cos i cos d - cos 2 i sin d + sin 2 i sin d
(h) cos i cos 2c + sin i sin 2c
= cos i _ cos 2 c - sin 2 c i + 2 sin i sin c cos c tan x - tan 2z tan x - tan x tan 2 z - 2 tan z
=
1 + tan x tan 2z
1 - tan 2 z + 2 tan x tan z
(j) sin 2x cos 2y - cos 2x sin 2y
= 2 sin x cos x _ cos 2 y - sin 2 y i
- 2 sin y cos y ^ cos 2 x - sin 2 x h

(i)

(b)

(h) 1

2

63
65

(b)

1
2

(c)
1

(i)

2 2

1

(d)

3
(j) -

1
2

5 39
7
, sin 2x =
32
32
7
25

(c)

120
169

(d) -

33
56

(b)

1
2+ 3

=2- 3

2-1

(d) cos x cos y - sin x sin y + sin x cos y - cos x sin y

(b)

1

2
4

3

22. (a)

13. (a) 2 sin x cos y

14. (a) 2 sin b cos b

2 2

=

17. cos 2x = -

1
[cos 115c + cos ] - 15c g]
2

1 - tan 2 x tan 2 y

1

20. (a) tan x

3

(b)

2 tan x _ 1 + tan 2 y i

(g) cos 6a

1

12. (a) 2 cos x cos y

(e)

1 sin 12i (f) 1 + sin 2x
2

19. 4 sin i cos i ^ cos 2 i - sin 2 i h =
4 sin i cos 3 i - 4 sin 3 i cos i

1 - 3 tan 2 i

(a) tan 4i (b) sin 7i cos 3i - cos 7i sin 3i cos 2x cos 7x - sin 2x sin 7x

(c) tan 10i (d) cos 2y
(h) cos 80c

(i) tan 2b (j) 1 - sin 6x
16. (a)

tan 2x

(b) cos 14y

(b)

1 sin 2i tan i
2
1
= (2 sin i cos i) tan i
2
sin i
= sin i cos i cos i
= sin 2 i
= LHS
1
2
` sin i = sin 2i tan i
2
RHS =

RHS =

1 - cos i sin i

i i - sin 2 n
2
2
=
i i cos
2 sin
2
2 i i
1 - cos 2 + sin 2
2
2
=
i i cos
2 sin
2
2 i i sin 2 + sin 2
2
2
=
i i 2 sin cos 2
2
i
2 sin 2
2
= i i
2 sin cos 2
2
i sin 2
=
i cos 2 i = tan
2
= LHS i 1 - cos i
` tan =
2
sin i
1 - d cos 2

1
2

(e)

3

(f)

3
2

ANSWERS

23. RHS = sin 11i sin 3i
= sin (7i + 4i) sin (7i - 4i)
= (sin 7i cos 4i + cos 7i sin 4i)
(sin 7i cos 4i - cos 7i sin 4i)
= sin 2 7i cos 2 4i - cos 2 7i sin 2 4i
= sin 2 7i (1 - sin 2 4i) - (1 - sin 2 7i) sin 2 4i
= sin 2 7i - sin 2 7i sin 2 4i - sin 2 4i + sin 2 7i sin 2 4i
= sin 2 7i - sin 2 4i
= LHS
` sin 2 7i - sin 2 4i = sin 11i sin 3i

5.

(a)

1 + t2
2t

(i)

1

4.

LHS =

=

1
2

1 - t2

(c)

1+t
1-t

(f)

8.

5 sin ] i - 63c 26lg

29 sin ] i - 21c 48lg

10 cos ] i - 18c 26lg 9. 2 cos ] i + 60cg

10. (a)

85 sin ] i + 12c 32lg

(b)

85 cos ] i - 77c 28lg

Exercises 6.17
1.

(a) x = 45c, 225c

(b) x = 30c, 210c

(f) x = 0c, 60c, 300c, 360c

(i) x = 30c, 135c, 150c, 315c

2

2.

(d)

1 + t2

3 - 3t + 8t
1+t

2

(h)

3.

+

(e) i = 180n - 45c

(f) b = 360n ! 45c
(h) i = 180n + 30c

l
(i) i = 360n ! 75c 49l (j) a = 180n + ] -1 gn # 23c 31

2

1 + t2
1 + t 2 + 2t + 1 - t 2

1 + sin i - cos i
=t
1 + sin i + cos i

(b) a = 180n + 60c

(d) x = 180n - ] -1 gn # 30c

(g) c = 180n ! 60c

1 - t2

2

(a) i = 180n + ] - 1 gn # 30c
(c) i = 360n ! 30c

1 t ^1 + t h

2

(d) i = 180c, 270c

(i) i = 51c 2l, 190c 54l (j) i = 160c 32l, 270c

2t + 1 - t 2

2 2

2t

(a) i = 126c 52l, 306c52l (b) i = 35c58l, 189c16l

(g) i = 90c, 340c 32l (h) i = 56c 34l, 176c 34l

2

(g)

(h) x = 0c, 180c, 360c

(j) x = 0c, 360c

(d) cos 50c

4t ^ 1 - t 2 h

1+t
2t 2 + 2t
=
2 + 2t
2t ] t + 1 g
=
2]1 + t g
=t
= RHS
`

34 sin ] i + 59c 2lg

l
(e) i = 240c 43l, 327c 21 (f) i = 90c, 180c

1 - t2
2t

2

1+t
1+t
1 + t 2 + 2t - 1 + t 2

=

2 sin ] i - 45cg (b)

(e)

(d) 0

1 + sin i - cos i
1 + sin i + cos i
2t
1 - t2
1+
2
1+t
1 + t2
1+

(a)

(c) i = 60c, 240c

1 + t2

2

(j)

(j)

(e) x = 90c, 210c, 330c

(c)

2

1 - t + 2t
1+t
1-t

41 sin ] i + 38c 40lg

l
65 sin ] i + 60c 15 g

(g) x = 0c, 45c, 180c, 225c, 360c

(b)

1-t

(h)

10 sin ] i + 18c 26lg

(f)

(d) x = 0c, 45c, 180c, 225c, 360c

2

(e)

13 sin ] i + 56c 19lg

29 sin ] i + 21c 48lg

(c) 2 sin ] i - 60cg (d) 2 sin ] i - 30cg

(e) sin 2i (f) cos i

3.

17 sin ] i + 14c 2lg

(b) 2 sin ] i + 60cg

(c) x = 0c, 60c, 180c, 300c, 360c

(a) tan i (b) cos i (c) tan 20c

(b)

2 sin ] i + 45cg (d)

(i)

Exercises 6.16

2.

5 sin ] i + 26c 34lg

(g)

25. 3 sin x - 4 sin 3 x

3
(a)
2

(a)

(e)

` cos 3i = 4 cos 3 i - 3 cos i

1.

^ 1 + t 2 h2

(c)

6.

7.

24. LHS = cos 3i
= cos (2i + i)
= cos 2i cos i - sin 2i sin i
= (cos 2 i - sin 2 i) cos i - 2 sin 2 i cos i
= cos 3 i - sin 2 i cos i - 2 sin 2 i cos i
= cos 3 i - 3 sin 2 i cos i
= cos 3 i - 3 (1 - cos 2 i) cos i
= cos 3 i - 3 cos i + 3 cos 3 i
= 4 cos 3 i - 3 cos i
= RHS

4t - 4t 3 - 1 + 6t 2 - t 4

4.

x = 52c 30l, 82c 30l, -97c 30l, -127c 30l

5.

x = 180n + ] - 1 gn # 30c, 360n ! 90c

6.

x = -180c, 0c, 90c, 180c

7.

(a) i = 180n

(b) x = 360n

(d) i = 180n + (-1) 270c n 8.

(a) (i) x = 30c, 150c

(c) x = 180n
(e) 360n ! 90c

(ii) x = 180n + ] - 1 gn ! 30c

l l l
(b) (i) x = 41c 25 , 318c 35 (ii) x = 360n ! 41c 25
(c) (i) x = 71c 34’, 251c 34l (ii) x = 180n + 71c 34l
(d) (i) x = 161c 34l, 341c 34l (ii) x = 180n - 18c 26l
(e) (i) x = 45c

(ii) x = 180n + (-1) n 90c - 45c

805

806

Maths In Focus Mathematics Extension 1 Preliminary Course

9.

10. (a) x = 0c, 120c, 240c, 360c

cos i =

5

2.

(a) cos x

(b) 2

34

, sin i =

(b)

34

(c) cosec A

(d) cos i

(e) cos 20i

4.

(a) i = 46c 3l (b) i = 73c 23l (c) i = 35c 32l

5.

LHS =

1
(d) 2
8.

(b) -

2 2

2 ^ 3 + 1h
4

=

2 ^1 - 3 h
4

(c)

1
2 2

=

2
4

(b) x = 180n + 45c

24. a = 51c 40l

23. i = 0c, 120c, 360c

(b) cos ] x + x g = cos x cos x - sin x sin x
= cos 2 x - sin 2 x
= ^ 1 - sin 2 x h - sin 2 x
= 1 - 2 sin 2 x

Challenge exercise 6
1.

(a) AC =

l
25.3 sin 39c 53
(b) h = 25.2 cm l sin 41c 21

6.
3
2

92c 58l 2. 50.2 km

4.

- cos x

7. 16 3 cm 2

9.

2 cos 2 i
So
= 2 + 2 sin i
1 - sin i

2

=

25. (a) cos ^ x + y h

2 cos 2 i
1 - sin i
2 ^ 1 - sin 2 i h
=
1 - sin i
2 ] 1 + sin i g ] 1 - sin i g
=
1 - sin i
= 2 (1 + sin i)
= 2 + 2 sin i
= RHS

1

2 2

3+1

22. (a) x = 360n ! 60c

(b) 1.84 (c) 0.95

7. (a)

1- 3

(b) 8.5 m

(c) x = 180n + ] - 1 gn # 60c

(a) 0.64

b = 40c

21. (a)

3

3.

6.

20 sin 39c sin 99c

(b) 360n ! 120c, 360n

Test yourself 6
1.

19. (a) AD =
20. 2951 km

x = 180n + ] -1 g n # 30c, 180n + (-1) n 270c

l x = 22c 30l, 112c 30l, 202c 30l, 292c 30l 10. i = 75c 45

3. x = 12.7 cm

8.

(c) - 3

140
(e) 221

11. 5.4 m

x = 120c, 240c

14. -

56
9

12. i = 110c, 230c
15. 31 m

1
2

13. 6.43 km

16. LHS =

9.

5. 4.1 km

cos i ] sin i + cos i g

= cos i

1 - sin 2 i
] sin i + cos i g

cos 2 i sin i + cos i
=
cos i
= tan i + 1
= RHS
17. x 2 + y 2 + 4y - 5 = 0 x = 90c, 270c
10. 122 km

11. 5 3

19. (a) 52c 37l (b) 9 m
12. (a) 6.3 cm

(b) 8.7 m

l
13. (a) i = 65c 5 (b) i = 84c 16l (c) i = 39c 47l
14. 65.3 cm 2

15. (a) x = !60c, !120c

(b) x = 15c, 105c, -75c, -165c
(c) x = 0c, !180c, 30c, -150c
3
4
16. sin i = - , cot i =
5
3

17. (a) 209c

18. i = 180n + ] -1 gn 30c + 53c 8l

(b) 029c

18. (a) 65 m

(b) 27c 42l

20. 30c 8l

21. LHS = cos 6i cos 4i - sin 6i sin 4i
= cos (6i + 4i)
= cos 10i
= cos 2 5i - sin 2 5i
= cos 2 5i - (1 - cos 2 5i)
= 2 cos 2 5i - 1
= RHS
` cos 6i cos 4i - sin 6i sin 4i = 2 cos 2 5i - 1
22. 30.1 m, 0.5 ms - 1

23. i = 30°, 150°, 270°

24. i = 180n + (-1) n 270c

25. - t

ANSWERS

Chapter 7: Linear functions

7.

1 1
Midpoint of AC = midpoint of BD = d 2 , 3 n .
2 2
Diagonals bisect each other

8.

AC = BD =

Exercises 7.1
1.

2.

(a) 5 (b) 10

(c) 13

(c)

52 = 2 13

85

(d)

(a)

13

(b)

65

3.

Two sides =

6.

Show AB = BC =

7.

Show points are

8.

Radius = 3 units, equation x 2 + y 2 = 9

9.

Distance of all points from ^ 0, 0 h is

1
BD = d 4, - n ; rectangle
2

(a) 9.85 (b) 6.71 (c) 16.55 4. 12 units

5.

x 2 + y 2 = 11

34 , 1 side =

AC =

11 , equation

11. a = ! 6 - 2

12. All 3 sides are 2 units. 13. a = 10, - 2
37 , QP = MN =

15. BD = AC =

98

(b) OC = OB = 2
19. AB =

29 , BC =

(b) XY =

17 units from ^ 7, -3 h

14. MQ = NP =

20 , so parallelogram

16. (a) AB = AC =
17. 2 101

18.

116 , AC =

65 , YZ =

130 , XZ =

1.

61 units
3.

12. x 2 + y 2 = 1

1
2

x = 1.8

1
3

(c) - 1

(h) -

2
3

4. x = 9

(d) - 2
1
4

2
5

2
3

(e)

(j) - 2

2. y 1 = 21

(3, 4)

3
(-2, 1)

(7, 2)

2
1

-3 -2 -1
-1

Exercises 7.2

1

2 3 4
(2, -1)

6

5

Gradient of AB = gradient of CD = 1

1
2

Gradient of BC = gradient of AD = 0
7.

1
1
(e) ^ -1, 1 h (f) ^ - 3, 2 h (g) d 3, n (h) d 1 , 1 n
2
2
1 1
1
(i) d , 2 n (j) d 0, 5 n
2 2
2

Gradient of AB = gradient of CD = -1
Gradient of BC = gradient of AD =

1
3

3
4

1
Gradient of AC = - 5 ,
2
1 gradient of BD = 2

(b) a = - 5, b = 6

(c) a = -1, b = - 2 (d) a = -1, b = - 2
8.

3.

3 + ]-3g
-4 + 4
= 0,
=0
2
2

5.

^ 4, 3 h 6. x = 3 is the vertical line through midpoint ^ 3, 2 h.

4. P = Q = ^ 2, -1 h

(f) -

5. (a) Show m 1 = m 2 =

-2

(e) a = 6, b = 1

1
3

(i) 2

4

(Pythagoras’ theorem)

(a) ^ 2, 4 h (b) ^ 1, -1 h (c) ^ - 2, 1 h (d) ^ - 3, 2 h

2

y

6.

(a) a = 9, b = - 3

(b) 1

65

30.2

2.

2
, AB =
2

(b) Lines are parallel.

Problem

1.

(a) 2
(g) - 4

Since XY = YZ, triangle XYZ is isosceles.
XY 2 + XZ 2 = 65 + 65
= 130
= YZ 2
So triangle XYZ is right angled.

34 ; YZ =

34
,
2

40 = 2 10 ; XZ =

Exercises 7.3

40 , BC = 4

145

10 , BC =

11. x 2 + y 2 = 4

AB 2 + BC 2 = 29 + 116
= 145
= AC 2
So triangle ABC is right angled (Pythagoras’ theorem)
20. XY =

9. ^ - 8, 13 h

1 1
1 1
10. (a) X = d - , 3 n , Y = d , n , Z = ^ 1, 1 h
2 2
2 2

128

85

10. a = 3

125 , midpoint AC = midpoint

Gradient of AC = 1, gradient of BD = -1

9.

(a) Show AB 2 + BC 2 = AC 2
(b) Gradient of AB = gradient of BC = -

4
5

5
,
4

7

3
5

1
8

807

808

Maths In Focus Mathematics Extension 1 Preliminary Course

10. (a) F = ^ 1, - 2 h, G = d 4,

Exercises 7.6

1 n 2

(b) Gradient of FG = gradient of BC =

1.

5
6

11. 4x - 3y - 11 = 0 12. Gradient of ^ 2, - 4 h and
^ 3, -1 h = gradient of ^ 3, -1 h and ^ 5, 5 h = 3

18. (a)
19.

(h)

14. 0.93 15. 21 16. 50c 12l 17. 108c 26l

13. 1

3

1

(b)

3

-5 - ] -2 g
7-4
-3
=
3
= -1 m = tan i
-1 = tan i
` i = 180c - 45c ^ 2nd quadrant h
= 135c

3
4

(j)

1
5

1

(i)

3

(d) 1

1
2

(d) x + 2y + 5 = 0

(b) (i) 2

(ii) -7

(ii) 1 (c) (i) 6

(e) (i) - 4

(ii) 3

(f) (i) 1

(ii) - 2

(ii) 6 (h) (i) -1

(ii) 1

(ii) -

1
2

(j) (i) 1

2
3

(ii)

2
3

3. (a) 4

1
(c) 0 (d) - 2 (e) -1 (f) - 3 (g) 2 (h) 4
3
1
2
1
1
2
(j) 1
(k)
(l)
(m)
(n)
(o) 4
5
7
5
3
2
3
1
1
1
(p) (q) 15 (r) - 1
(s)
(t) 6
14
2
8

(b) - 2

1
# 5 = - 1 so perpendicular
5
1
5

m1 # m2 = -

3
7
# = -1
7
3

7. k = -

2
3

8. m 1 = m 2 = 4

5
AB < CD _ m 1 = m 2 = 3 i and BC < AD d m 1 = m 2 = - n
8
1
10. Gradient of AC: m 1 = , gradient of BD: m 2 = - 2,
2
1 m 1 # m 2 = # - 2 = -1
2
11. (a) y = - x

(b) 5x - y - 8 = 0

(d) 2x - 3y + 16 = 0

(c) 2x + y + 2 = 0

12. 7x + 6y - 24 = 0

13. x + y - 3 = 0 14. 2x - y - 5 = 0
15. 2x - 3y + 18 = 0

Exercises 7.7
(a) ^ 2, - 4 h

(b) ^ -1, - 3 h

(e) ^ 5, -1 h

1.

1
(i) 1
2

(f) ^ -1, 1 h

(j) d
3.

(c) ^ 4, 4 h

(g) ^ 3, 7 h

(d) ^ 0, - 2 h

(h) ^ 4, 0 h

^ 2, 5 h, ^ 4, 1 h and ^ -1, -1 h

at ^ 2, -3 h

4. All lines intersect

All lines meet at ^ - 5, 0 h

7.

5x + 6y - 27 = 0

(d) y = 4x + 20 (e) 3x + y - 3 = 0 (f) 4x - 3y - 12 = 0

9.

x+y-1=0

10. 2x + y - 2 = 0

(g) y = x - 1

11. x + y - 3 = 0

12. x - 2y - 3 = 0

13. x - y + 1 = 0

14. x - 3y + 2 = 0

(a) y = 4x - 1 (b) y = - 3x + 4
(h) y = x + 5

(a) 4x - 3y + 7 = 0

2. x + y - 8 = 0

(b) 3x - 4y + 4 = 0

(c) 4x - 5y + 13 = 0
(e) x - 2y + 2 = 0

(c) y = 5x

(d) 3x + 4y - 25 = 0

4. 4x + y - 8 = 0

6. y = - 2x

5. (a) y = 3

7. 3x - 4y - 12 = 0

9. x = - 4

10. 3x + 8y - 15 = 0

(i) ^ 41, 26 h

1
7
n 2. Substitute ^ 3, - 4 h into both lines
,19
19

5.

Exercises 7.5

2x + y - 3 = 0

3

(g) 3x + 4y + 13 = 0

m1 m2 = -

(i) (i) 9 (ii) 0

(j) (i) 5 (ii) - 2 2. (a) (i) - 2 (ii) 3 (b) (i) - 5 (ii) - 6
1
(c) (i) 6 (ii) -1 (d) (i) 1 (ii) 4 (e) (i) - 2 (ii)
2
1
1
4
(f) (i) 3 (ii) 1
(g) (i) (ii) - 2 (h) (i) (ii) 2
5
2
3

8.

1

m 1 = m 2 = 3 so parallel

6.

(ii) 0

(b) x = -1

(g)

(e) x - 2y + 4 = 0

5. m 1 = m 2 = 1

(g) (i) - 2

3.

5
6

(a) x - y + 1 = 0 (b) x - 3y + 16 = 0 (c) x + y - 5 = 0

4.

(d) (i) -1

1.

(f) -

(e) 1

9.

(a) (i) 3 (ii) 5

1
2

(c)

1
3

3.

2^ 3 + 3h
3

(i) (i) 3

1
3

(b)

(f) x + 3y - 1 = 0

Exercises 7.4
1.

2.

(c) - 3

m=

20. x =

(a) - 3

15. 3x + y - 7 = 0

21. 5x - y + 17 = 0

8. 4x + 7y + 23 = 0

16. x + 5y + 13 = 0

17. 27x - 5y - 76 = 0
19. 2x - y - 1 = 0

6. 11x + 6y = 0

18. 3x - y - 14 = 0

20. 3x - y - 11 = 0

ANSWERS

Exercises 7.8

Exercises 7.10
3
13

8
13

1.

(a) 2.6 (b) 1

2.

(a) 3.48 (b) 1.30 (c) 0.384 (d) 5.09 (e) 1.66

3.

(a)

4.

d1 = d2 = d3 = 1

5.

7 13
13

A: d =

14
5

(b)

(c) 2.5 (d) 2.4

5

, B: d =

(c)

4 205
205

(e)

(d)

5 26
13

1.

(e)

9
1
1
2
1
(d) d 4 , -1 n (e) d 2 , - 2 n (f) d - 5, 2 n
7
7
4
2
10
14 13
13

^ 2, - 3 h: d =

13
10

6 6
6 4
4
1 n (i) d - , 1 n
(g) d 2 , 7 n (h) d - 3 , -1
7 7
7 7
11
11
2
2
(j) d 1 , -1 n
3
3

-3
5

Opposite signs so points lie on opposite sides of the line
6.

2.

5

, ^ 9, 2 h: d =

^ - 3, 2 h : d = - 4 , ^ 4 , 1 h : d = 2

8.

d 1 = d 2 = 2 so the point is equidistant from both lines

9.

^ 8, - 3 h: d =

3.

37

9

, ^ 1, 1 h: d =

10. ^ - 3, 2 h: d =

5

, ^ 4, 1 h: d =

A
(3, 2)

5.

3 1
P = d 1 , n , Q = ^ 16, -19 h, PQ = 24 units
5 5

6.

3
4
2
B = d 9 , -12 n 7. p = 4 , q = 20
5
5
5

8.

2 2
2 2
(a) d , 1 n (b) Each ratio gives d , 1 n . This means
3 3
3 3 that the intersection of the medians divides each median in the ratio 2:1.

9.

a = 8, b = 18

7
5

8 5 units 5

11. d 1 = d 2 = 4 so same distance 12.
14. 4.2 15. x = 9

17. m = - 1

2 or -17
3

2
2
(a) E = d , 2 n (b) F = d 1 , 2 n
3
3

4.

Opposite signs so points lie on opposite sides of the line

13. 1

(j) ^ 10, 13 h

(c) EF = 1, AC = 3 ` AC = 3EF

37

Same signs so points lie on same side of the line
-6

1
(d) d 12, 5 n
2

4
1
2
(f) d 9, -1 n (g) d - 6, - n (h) d 9, 1 n
7
2
3

(i) ^ - 58, 30 h

1
5

Opposite signs so points lie on opposite sides of the line

55

1
4
(a) d - 4, 3 n (b) d 6 , 2 n (c) ^ 19, 25 h
5
2
(e) ^ 40, 12 h

10

Same signs so points lie on the same side of the line
7.

3 2
1 3
4 8
(a) d - , 1 n (b) d 2 , 3 n (c) d - 2 , 1 n
5 5
5 5
9 9

16. b = 3

1
1
or -1
4
12

2
1
or -18
3
3

(1 2 , 3 1 )
3
3

18. Show distance between ^ 0, 0 h and the line is 5

1.

6.4 units

4 1
20. (a) ^ 3, -1 h, d 3 , n, ^ - 2, 2 h
7 7

3.

(a) - 1

2 8
10. P = d , 3 n
9 9

Test yourself 7

19. Show distance between ^ 0, 0 h and the line is greater than 1

B
(-1, 6)

( 1, 4 2 )
3
3

(b)

2 10 13 5 26 34
,
,
5
5
119

4.

Exercises 7.9
1.

l
(a) 149c 2l (b) 119c 45 (c) 143c 58l (d) 172c 14l
(e) 135c

3. 12c 20l

1
3

21c 2l, 120c 58l, 38c

7.

m = - 5.4, 1.53

9.

(a) +A = +C = 63c 26l, +B = +D = 116c 34l

6. m = 3, -

8. k Z -1.64, 0.095

l
(b) 124c 31 10. +A = 61c 56l, +B = +C = 59c 2l

(d)

3

(a) 7x - y - 11 = 0

3
5

(b) 5x + y - 6 = 0

(c) 3x + 2y = 0

(e) x - 3y - 3 = 0

6 5 units 5

6.

1 m 1 = - , m 2 = 4 so m 1 m 2 = -1
4
` lines are perpendicular.

7.

x-intercept 5, y-intercept - 2

8.

(a) 2x + y - 1 = 0

9.

m 1 = m 2 = 5, so lines are parallel

4. 53c 58l

5.

1

(c)

5.

(i) 74c 56l (j) 36c 52l
2.

(b) 2

(d) 3x + 5y - 14 = 0

l
(a) 18c 26l (b) 29c 45 (c) 82c 52l (d) 26c 34l l (e) 10c 29l (f) 41c 49l (g) 72c 15 (h) 18c 26l

1
5

1
2. d 2 , - 2 n
2

(b)

1
2

(c)

5 units 2
10. 3x - 4y = 0

809

810

Maths In Focus Mathematics Extension 1 Preliminary Course

11. ^ -1, 1 h

12. a = 6, b = 1

22. (a) AB: 7x + 5y + 14 = 0

13. 66c 48l

^ -7, 7 h lies on the line (show by substitution)

14. Solving simultaneously, x - y - 4 = 0 and

(b) -1:2 or 1:- 2

2x + y + 1 = 0 have point of intersection ^ 1, - 3 h .
Substitute ^ 1, - 3 h in 5x - 3y - 14 = 0:

2
23. x = 16 , y = -17
3

LHS = 5 # 1 - 3 # - 3 - 14 = 0 = RHS

2 1
1 1
25. (a) P = d 1 , 3 n (b) Q = d 4 , 3 n
3 3
3 3

` point lies on 5x - 3y - 14 = 0:
Substitute ^ 1, - 3 h in 3x - 2y - 9 = 0:

(c) PQ has gradient m 1 = 0

LHS = 3 # 1 - 2 # - 3 - 9 = 0 = RHS

AC has gradient m 2 = 0

` point lies on 3x - 2y - 9 = 0:

Since m 1 = m 2, PQ < AC

` lines are concurrent
5
1
15. d 2 , -1 n 16. - 0.499
9
3
19. ^ 4, 7 h

18. y = 3

22. ^ - 2, 1 h: d =

-8
13

1
(d) R = d 6 , 0 n
3

17. c = -13, - 65
4
5
2

(e) PR has gradient m 1 = -

21. 93c22l

20. x = 1

, ^ 6, 3 h: d =

BC has gradient m 2 = -

13

24. x - y - 4 = 0

Chapter 8: Introduction to calculus

2. x - 3 y - 3 3 = 0

3. 10x 2 + 10y 2 = 81

1.

k = -2

4.

Show AC and BD have the same midpoint ^ 1, 2 h and m AC # m BD = -1

5.

6.

4 13
13

8.

12 13
13

Exercises 8.1

Show distance of all points from ^ 0, 0 h is 3; radius 3; equation x 2 + y 2 = 9

9. 113c12l 10. 2x + 3y + 13 = 0

14. 2x + 5y + 14 = 0

3x +y + 3 - 2 3 =0

1
18. b = 2 , - 21
3
m1 - m2

20.

`

1 + m1 m2 m1 - m2

12. ^ 3, - 5 h

2.

15. 45c

17. x - y + 6 = 0

1
1
2
2
19. d 2 , - 2 n, d 1 , - 3 n
3
3
3
3
=1

=1
1 + m1 m2 m1 m2 + 1 = m1 - m2 m1 m2 = m1 - m2 - 1 m1 - m2 or = -1
1 + m1 m2 m 1 - m 2 = -1 - m 1 m 2 m1 m2 = m2 - m1 - 1

21. P = f

1.

7. +OBA = 45c; a = b (sides of isosceles D)

11. BC = AC = 18 , AB = 6, so D is isosceles; m BC # m AC = -1, so D is right angled.

16.

- 4p - 1 7p - 3 p , p -1 p -1

5
7

25. 3x - 7y - 14 = 0

Challenge exercise 7

13. a = 2, b = 3

5
7

Since m 1 = m 2, PR < BC

Opposite signs so points lie on opposite sides of the line
23. 63c 26l

24. m = - 0.059, - 9.2

3.

ANSWERS

4.

5.

10.

Exercises 8.2
2. Yes, x = x 1

1.

Yes, x = 0

5.

Yes, x = x 1, x = x 2

8.

Yes, x = 2

3. No

6. Yes, x = 0

9. Yes, x = - 2, 3

11. Yes, x = 90c, 270c

4. Yes, x = 0
7. Yes, x = - 3

10. Yes, -1 # x 1 0

12. Yes, x = 0

13. No

14. No

15. Yes, x = !3
6.

Exercises 8.3
1.

(a) 3 (b) -7 (c) 3 (d) 8
(h) -1 (i) 10 (j) -1

2.

(a) x 2 - 2x - 4 (b) 2x 3 + x - 1 (c) - 7x - 1
(d) 4x 4 - x 2 (e) - 4x + 3 (f) 2x 2 + 6 (g) - 2x
(h) 4x 2 (i) 3x - 1 (j) x 2 - 2x + 9

(e) 2

(f) - 3

(g) 2

Exercises 8.4
1.

(a) 4.06

2.

(a) 13.61

4.

7.

(a) f ] x + h g = x 2 + 2xh + h 2

(b) 3.994

(c) 4

(b) 13.0601

(c) 12.9401

(d) 13

3. 6

2
2
2
(b) f (x + h) - f (x) = x + 2xh + h - x
2
= 2xh + h

(c)

f ]x + hg - f ]xg h 8.
(d) f l(x) = lim

2xh + h 2 h h ] 2x + h g
=
h
= 2x + h f ]x + hg - f ]xg
=

h "0 h = lim (2x + h) h "0

= 2x
5.

9.

(a) f (x + h) = 2 ] x + h g2 - 7 (x + h) + 3
= 2 (x 2 + 2xh + h 2) - 7x - 7h + 3
= 2x 2 + 4xh + 2h 2 - 7x - 7h + 3
2
2
(b) f (x + h) - f (x) = (2x + 4xh + 2h - 7x - 7h + 3)
- (2x 2 - 7x + 3)
= 2x 2 + 4xh + 2h 2 - 7x - 7h + 3
- 2x 2 + 7x - 3
= 4xh + 2h 2 - 7h

811

812

Maths In Focus Mathematics Extension 1 Preliminary Course

(c) f ] x + h g - f ] x g h (d) f l] x g = 4x - 7
6.

2.

(a)

4.

(c) f ] 2 + h g - f ] 2 g = h + 5h
2

(d) f ] 2 + h g - f ] 2 g h (e) f l] 2 g = 5
7.

x
-1
3

(b) 2x 3 - x 2

(d) 16x 3 - 24x

(c) 2x

(c)

8x 7
- 6x 5
3

(d) 4x

(e)

(f) 2x 2 - 2x + 2

(b) f ] 2 + h g = h 2 + 5h + 11

(a) f ] 2 g = 11

(a) 4x + 1 (b) 8x - 12
(e) 6x 2 + 6x - 3

3.

4xh + 2h 2 - 7h h h ] 4x + 2h - 7 g
=
h
= 4x + 2h - 7
=

6.

h 2 + 5h
=
h h ]h + 5 g
=
h
=h+5

f l] x g = 16x - 7 dy dx

= 60x 9 - 40x 7 + 35x 4 - 3

8.

gl] x g = - 20x - 5

11.

(a) f ] -1 g = -7

dV
= 4rr 2 dr 14. (a) 12

(b) f ] -1 + h g - f ] -1 g = 4h 3 - 12h 2 + 12h
(a) f ] 3 g = 8

(c) f l] 3 g = 6

9.

(a) f l] 1 g = - 13

ds
= 10t - 20 dt 7.

12. 3

(b) x = ! 2

dv
= 30t dt 10.

13. (a) 5

9.

dh
= 40 - 4t dt (b) - 5

(c) x = 4

15. 18

(c) 12

8.

5. - 56

(b) f ] 3 + h g - f ] 3 g = 6h + h 2

Exercises 8.6
1.

10. (a) y = x 2 + 2x
Substitute _ x + dx, y + dy i:

2.

y + dy = ] x + dx g2 + 2 (x + dx)
= x 2 + 2xdx + dx 2 + 2x + 2dx
Since y = x 2 + 2x dy = 2xdx + dx 2 + 2dx dy 2 x d x + d x 2 + 2d x
(b)
= dx dx d x ] 2x + d x + 2 g
=
dx
= 2x + dx + 2

(c)

dy dx 12. (a) f l] x g = 2x

(g)

dy dx dy dx = 3x 2

(c) -12
(b)

dy dx (d) 15

3.

(d) -18

(i) - 4

(j) 149

(a) 1
7

1
26

(b)

(g) -

1
25

1
71

(ii) -

(a) (i) 6

1
6

1
20

1
20

(d) (i) -

(b) (i) 8

1
24
1
(ii) 11

(e) (i) 11

(d) (i) - 8

1
8

(e) 18

1
43

(e)

(j) -

(f) 27

1
10

1
5

1
8
1
(ii)
8

(ii) -

dy dx (a) 27x - y - 47 = 0 (b) 7x - y - 1 = 0
(c) 4x + y + 17 = 0 (d) 36x - y - 47 = 0
(e) 44t - v - 82 = 0

5.

(a) x + 24y - 555 = 0
(c) x - 17y - 516 = 0
(e) x + 2y - 9 = 0

6.

(a) (i) 7x - y + 4 = 0 (ii) x + 7y - 78 = 0
(b) (i) 10x - y + 36 = 0 (ii) x + 10y - 57 = 0
(c) (i) 10x + y - 6 = 0 (ii) x - 10y - 41 = 0
(d) (i) 2x + y + 2 = 0 (ii) x - 2y - 19 = 0
(e) (i) 2x - y + 2 = 0 (ii) x + 2y - 9 = 0

7.

x = !3

(e) - 9

= 10x - 1

(f) f l] x g = 6x 2 + 5
(h) f l] x g = - 6x 2

13. (a) 0.252 (b) 0.25 (c) 0.2498
(b) - 0.03992

(c) - 0.04

11. (1, 2)

Exercises 8.5
(a) 1 (b) 5 (c) 2x + 3 (d) 10x - 1 (e) 3x 2 + 4x - 7
(f) 6x 2 - 14x + 7 (g) 12x 3 - 4x + 5
(h) 6x 5 - 25x 4 - 8x 3 (i) 10x 4 - 12x 2 + 2x - 2
(j) 40x 9 - 63x 8

(b) x - 8y + 58 = 0
(d) x - 45y + 3153 = 0

8. (1, 2) and (-1, 0)

15. -1
10. (0, 1)

1.

(c)
(h)

(ii) -

(c) (i) 24

= 2x + 5

(d)

= 3x 2 - 4 x + 3

14. (a) - 0.04008

(c) 11

(h) 136

4.

(c) f l] x g = 8x - 4
(e)

(b) -13

(f)

= 2x + 2

11. (a) 2 (b) 5

(a) 72
(g) 11

(b) 17

13. (a) (1, -1)

9. (- 5, -7)

3
15
n
12. d - 1 , - 4
4
16

(b) 6x - y - 7 = 0

14. 10t - h - 7 = 0

15. 4x - 2y - 19 = 0

1
4

ANSWERS

Exercises 8.7

(k) 1

1.

(a) - 3x - 4
(e) x

-

1
2

(c) 1.2x - 0.8

(b) 1.4x 0.4

+ 3x - 2

(f) x

-

2
3

(g) 6x

-

1
4

1 -2 x 2

(d)
(h) x

-

(l)

x

2 2-x
- 2 ] 5x + 3 g

+ 2-x =

] 2x - 1 g2

4.

2.
2.

(a) (f) (j) -

3.
8.

1
27
1
8

1

(b)

x2
1

2x

3

3

1

(h)

x7

(i) -

6.

x

x4

2

1

1
32

6. −3 7. 2x + 3 x + 1
1.

(b) -

15. d 5,

1
16

10. x - y + 9 = 0

12. x + 16y - 16 = 0

13. (9, 3)

2
2
n, d - 5, - n
5
5

(w) 2.

9

5

2 ] 4 + x g3

(u) -

^ 4x 3 - 9x 2 + 3 h

^ x 4 - 3x 3 + 3x h

3. 40

2

3

4 ] 3x - 1 g3
(x)

(v) -

16 3 4x + 1
3

4. (4, 1) 5. x = 2, -1

- 2x 2

(g)

^ 2x 2 - x h2

9. 34x - y + 29 = 0

1
2

4

] 7 - x g9

6. 8x + y + 7 = 0

(a) 8x 3 + 9x 2 (b) 12x - 1 (c) 30x + 21
(d) 72x 5 - 16x 3 (e) 30x 4 - 4x
(f) x ] 5x + 2 g ] x + 1 g2 (g) 8 ] 9x - 1 g ] 3x - 2 g4
(h) 3x 3 ] 16 - 7x g ] 4 - x g 2 (i) ] 10x + 13 g ] 2x + 5 g3
4
5
(j) 10x ^ x 3 + 5x 2 - 3 h ^ x 2 + 1 h + ^ 3x 2 + 10x h ^ x 2 + 1 h
^ 13x 3 + 60x 2 + 3x - 20 h ^ x 2 + 1 h4
=x

(c)

- x + 14x
2

(e)
(h)

x4
-6
] x - 2 g2

=
(i)

x 4 - 12x 2
^x - 4h

2

2

- x + 14 x3 - 34
] 4x - 3 g2

=

x 2 ^ x 2 - 12 h
^ x 2 - 4 h2

(f)

11
] x + 3 g2

(j)

-14
] 3x + 1 g2

4x ] x - 3 g
4x 2 - 12x
=
] 2x - 3 g2
] 2x - 3 g2
^ 3x 2 - 7 h
2x 2 ] x + 6 g
- 18x
2x 3 + 12x 2
=
(m)
(n)
2
2
] x + 4 g2
] x + 4 g2
^x - 5h
2x 3 + 9x 2 + 7
] x + 3 g2

(o)

(q)

(p)

3x 2 + 8x - 5
] 3x + 4 g2

^ x 2 - x - 1 h2

(s)
(t)

(l)

x 4 - 2x 3 - 4x 2 - 1

] 7x + 2 g4 - 28 ] x - 1 g ] 7x + 2 g3

(u)

1
2

-

2]x + 5g - x ]x + 5g
(r)
x+5

1
2

] 2x - 9 g2 ] 20x + 51 g
6 ] 5x + 1 g ] 2x - 9 g2 - 5 ] 2x - 9 g3
=
2
] 5x + 1 g
] 5x + 1 g2
] 7x + 2 g8

=

- 21x + 30
] 7x + 2 g5

15 ] 2x - 5 g3 ] 3x + 4 g4 - 6 ] 3x + 4 g5 ] 2x - 5 g2
] 2x - 5 g6
] 3x + 4 g4 ] 4x - 33 g
3
=
] 2x - 5 g4

5

4

15
] x + 5 g2

2

27

(y)

(b)

- 3x 2 - 6x - 7

2 ] 2x + 7 g10

Exercises 8.9
1.

16
] 5x + 1 g2

(k)

6
(a) 4 ] x + 3 g3 (b) 6 ] 2x - 1 g2 (c) 70x ^ 5x 2 - 4 h
5
4
(d) 48 ] 8x + 3 g (e) - 5 ] 1 - x g (f) 135 ] 5x + 9 g8
3
(g) 4 ] x - 4 g (h) 4 ^ 6x 2 + 3 h ^ 2x 3 + 3x h
^ x 2 + 5x - 1 h 7
(i) 8 ] 2x + 5 g
1
3
5
(j) 6 ^ 6x 5 - 4x h ^ x 6 - 2x 2 + 3 h (k) ] 3x - 1 g 2
2
2
5
-4
(l) 2 ] 4 - x g- 3 (m) - 6x ^ x 2 - 9 h
(n) ] 5x + 4 g 3
3
1
3^ 2
3
4
3x - 14x + 1 h ^ x 3 - 7x 2 + x h
(o)
(p)
4
2 3x + 4
8x
5
2
(q) (r) (s) 3
] 5x - 2 g2
^ x 2 + 1 h5
7 - 3x

(t) -

5. 176

7

7. 69x - y - 129 = 0

- 6 ! 30
3

-2
] 2x - 1 g2

(a)
(d)

Exercises 8.8
1.

x=

3x 2

8

=

Exercises 8.10
5.

2 x3

10x - y - 9 = 0

7

5

9. 3x + 16y - 8 = 0

14. x = 4

3 x
2

(e)

x6

1

7+

15

12

-

4. −3

11. (a) -

(d) -

6 6 x5

10

26

8.

(c)

2 x
(g) -

2 x3
1

5

5
11
=2x - 1
] 2x - 1 g2

+

3. 1264

3
2

4 - 3x
2 2-x

3x + 1
3 x+1 -2 x+1
3x + 5
(v)
= x+1 2 ] x + 1 g3
2x - 3
(w)

2 x-1 -2 x-1
- 2x + 1
=
] 2x - 3 g2
2 x - 1 ] 2x - 3 g2

x ] x - 9 g2
(x)

x2 + 1

- 2 ] x - 9 g x2 + 1
] x - 9 g4

=

- x 2 - 9x - 2

x2 + 1 ] x - 9 g3

813

814

Maths In Focus Mathematics Extension 1 Preliminary Course

5
9

2.

1
8

6.

x - 18y + 8 = 0

3. - 1

4. x = 0, 1

(b)

5. x = - 9, 3

7. 17x - 25y - 19 = 0

Exercises 8.11
1.

(a)

2.

dy dx (b)

(b) Substitute Q into both equations.
(c) y = x 2 - 4 has m 1 = 4 y = x 2 - 8x + 12 has m 2 = - 4
(d) 28c 4l

(d)
(e)

dy dx dy dx dy dx =

dy

3. (a)

= 10x - 3

11
] 2x + 1 g2

dx

=

5 x3
2

(f)

dy dx dv
= 4t - 3 dt 5. (a) 1

(a) x = - 2

(a) f l] x g = 32 ] 4x + 9 g3
(c)

dy dx 4. 71c 34l

5. 162c 54l

(d) 0c

3. 8c 8l

6. (a) X = ^ 4, 16 h, Y = ^ -1, 6 h

10 x3 (b) 20

6. 10

7. 42

(c) x = 2
(b)

dy dx = ] 9x - 1 g ] 3x - 1 g (d)

(e) f l] x g =
(c) m = 6

= 9 (2x + 4) (x 2 + 4x - 2)8

=-

(b) x = 1

9.

(b) P = ^ 3, 9 h

dx

= 40x ] 2x - 1 g3 + 5 ] 2x - 1 g4 = 5 ] 2x - 1 g3 (10x - 1)

4.

(a)

dy

(c)

8.

2.

= 42x 5 - 9x 2 + 2x - 8

=dy dx 5
] x - 3 g2

=-

4 x2 1
5 5 x4

10.

y

(b) At X: m 1 = 12, m 2 = 7
At Y : m 1 = - 8, m 2 = - 3
(c) At X: 3c 22l At Y : 11c19l
7.

71c 34l, 8c 58l 8. (a) (0, 0), (2, 8), (-1, -1)
(b) 63c 26l at (0, 0), 4c 42l at (2, 8), 71c 34l at (-1, -1)

9.

At (0, 0), m 1 = 0 and m 2 = 4
At (2, 4), m 1 = 4 and m 2 = 0 Angle at both is 75c 58l

l
10. 164c 45 at (0, 0), 178c 37l at (- 3, - 33), 146c 19l at (1, 3)

Test yourself 8
1.

(a)

11. 9x - y - 7 = 0

12. (2, 3)

14. (- 2, 71), (5, - 272)
17. 9
20.

7
10

13.

dS
= 8rr dr 15. 4x - y - 6 = 0

18. 12x + y - 4 = 0

19.

16. 3525

ds
1
= u + at, t =
5
dt

21. 17c6l at (3, 9), 53c8l at (-1, 1)

22. 175c 26l at (2, 4), 177c 40l at (4, 16)

ANSWERS

Challenge exercise 8

y

(b)

1.

f ] 1 g = - 3, f l] 1 g = - 36

3.

dx
= 8t 3 + 300t 2; t = 0, - 37.5 dt 4.

2x + y = 0, 3x - y - 3 = 0, 6x - y + 12 = 0

5.

^ 2, 2 h, ^ - 2, -14 h, x + 12y - 26 = 0, x + 12y + 170 = 0

6.

3
4

7.

5 ] 5x + 1 g3 ] x - 9 g4 + 15 ] x - 9 g5 ] 5x + 1 g2
= 10 ] 5x + 1 g2 ] x - 9 g4 (4x - 13)

8.

9.

2. -

13
18

2 ] 4x - 9 g4 - 16 ] 2x + 1 g ] 4x - 9 g3
] 4x - 9 g8
- 2 ] 12x + 17 g
=
] 4x - 9 g5 x= - 6 ! 204
- 3 ! 51
=
6
12

11. a = 1
14.
2

1
27

1
1
1 1 ! 13 n 13. x = ,
12. P = d - 2 , 6
4 16
3
3
10

16. (a) Substitute (1, 1) into both curves: y = ] 3x - 2 g5:
LHS = 1
RHS = ] 3 # 1 - 2 g5
= 15
=1
= LHS
So (1, 1) lies on the curve y = ] 3x - 2 g5
5x - 3
:
x+1
LHS = 1
5#1-3
RHS =
1+1
2
=
2
=1
= LHS

90c

21. ^ - 4, -73 h

10. 2x + y - 25 = 0

15. 3x - y + 5 = 0, Q = ^ 0, 5 h, PQ =

1

25. x = 0, 2, 6
28. p = 1

1
2

3 ] 4 - 5x g

23.

4x 4 3x - 2

29.

8r 3 dV =
3
dr
33. -

27.

30. k = 4
1
48

5 22
22
31. x - y - 4 = 0

34. a = -1, b = 2, c = 4

35. S = 8rr - 8r + 2rrh
36. (a) 6x 2 - 5 ] 3x - 1 g ] 3x - 5 g3

(b)

- ] 5x + 6 g

] x - 3 g4 2x + 1

4 ! 13
6

1
1
n
(b) Q = d - 4 , 12
7
49

5x - 3 x+1 Practice assessment task set 2
- 0.77

l
(b) 22c 45

5.

- 0.309

3 3 o, 12x - 12 3 y + 31 = 0
2

7.

m1 m2 =

20. (a) x = 90c, 270c

9.

7
12

1
1 3
, -1 , 1
2
2 5

26. a = - 14, b = 7

32. 4x - y - 13 = 0

1.

19. x =

x

So perpendicular

` (1, 1) is a point of intersection

11
18. e 1 ,
12

22. 3x - 9y - 14 = 0

360c

38. (a) x + 7y - 80 = 0

So (1, 1) lies on the curve y =

17. n = 8

270c

24. (a) 16x + 32y + 1 = 0, 4x - 2y - 1 = 0
1
(b) m 1 $ m 2 = - # 2
2
= -1

37. x =

y=

180c

2. 1

3. 5x + 2y - 1 = 0

6. (a)

3 cm 2

(b) AC =

4. ^ 2, - 2 h
13 cm, BD = 1 cm

3
8
1
# - = -1; A = d -1, 1 n
4
6
2

8. x = 15c

815

816

Maths In Focus Mathematics Extension 1 Preliminary Course

10.

19. i = 120c, 240c
23. y = 16.5
26. 7

20. - 1

2
3

21. 2

24. 3x + y - 5 = 0

27. x = 3

22. a = 115c 56l

25. 1

2
1x 13
3

28. - 3

29. Show perpendicular distance from ^ 0, 0 h to the line is
2 units, or solving simultaneous equations gives only one solution.
30. (a) g ] 2 g = 1, g ] - 3 g = - 6
(b)
11.

31. 3x 2 - 4x

32. -

34. x = - 2, y = -17
12. 45c 49’ 13. Domain: all real x ! real y ! 0

1
; range: all
2

14.

1

33. 17.5 m

2

35. (a) AB = 7.0 m

36. 3 cos i 37. (a) 2x - y + 4 = 0

(b) 27.8 m2

(b) P ^ - 2, 0 h, Q ^ 0, 4 h

2

(c) 4 units

39. 15 units2 40. f (- x) = ] - x g6 - ] - x g2 - 3
= x6 - x2 - 3
= f (x)

38. 127 m

41. 16x 2 ^ 2x 2 + 1 h + ^ 2x 2 + 1 h = ^ 18x 2 + 1 h ^ 2x 2 + 1 h
3

42. - 4

1
# y #9
3

4

43. -

44. (a) 3x - y - 4 = 0

15.

47.
50.

8 units 13
1
2x - 7
5
] x + 1 g2

3 x2 (b) x - y - 2 = 0

(c) x + 3y + 10 = 0
45.

3

(d) R = ^ -10, 0 h

46. Domain: all x ! - 4; range: all y ! 0
48. 4.9 km 49. 8x - 7 - 10x - 3
51. 2x - 3

53. x + 6y - 56 = 0
55. a = 2, b = - 9

52.

-17 - 2x x 2 + 5x

=

- ] 17 + 2x g x 2 + 5x

54. f ] - 2 g = - 45, f l] - 2 g = 48
56. 7x - 5y + 9 = 0

57. 47x - y + 109 = 0 58. x = - 0.25 59. ^ 33, -17 h
16. sin 4 i 17. 2 units 18. x - 8y + 15 = 0

60.

3+1
2 2

=

6+ 2
4

62. x = 63c 26l, 243c 26l

61. 67c 37l

ANSWERS

63.

4.

(a) x = 112c, y = 56c, z = 34c
(c) x = 55c, y = 43c

(b) x = 49c

(d) x = 166c, y = 7c

(e) x = 62c, b = 31c
(f) x = y = 32c, z = 58c, v = 32c, w = 17c
(h) y = 102c

(g) x = 5c

(i) x = 57c 30l, y = 32c 30l

(j) x = 75c, y = 77c, z = 13c
5.
64. (a) cos i (b) cos ^ i + b h

(c) tan 14a

65. 3

D ABC < D DEC
;

67. 12c 32l at both points

66. x 1 4, x 2 4.6

1 range: y $ 0
2
(b) domain: all real x ! -7 range: all real y ! 0

(b) x = 5.5 cm

68. (a) domain: x $

(c) domain: - 2 # x # 2
69. a = -15, b = -1

6.

x = 30° (angle at centre is double the + at the circumference) y = (180° - 30°) ' 2 (+ sum of isosceles D)
= 75c

7.

360° - x = 2 # 110° (+ at the centre is double the
+ at the circumference)
` x = 140° y = 70° (similarly)

8.

+ABC = 90c (+ in semicircle)
)
` +BAC = 90c - 29c (+ sum of D
= 61c
` x = 61c (+ in same segment)

9.

+STV = +WUV (+ in same segment)
+TSV = +UWV (similarly)
+TVS = +UVW (vertically opposite +s)

range: - 2 # y # 0

70. cos 2i

71. (a) (0, 0), (1, 3), (-1, -1), (2, 20)
(b) 63c 26l at (0, 0), 2c 20l at (1, 3), 40c 36l at (-1, -1),
0c 22l at (2, 20)
72. (a) x = 360n ! 45c

(b) x = 180n + 30c

(c) x = 180n + ] -1 gn # 60c
73. (a) (1, 1) (b) 2 13 units

(c) - 1

1
2

(d) 3x + 2y - 5 = 0
74. (a)

75. (b), (d) 76. (a)

79. (b), (d)

77. (c)

(a) +DCE = +ACB (vertically opposite +s)
+EDC = +BAC (+s in the same segment)
+DEC = +ABC (similarly)
` Since all pairs of +s are equal,

` Since all pairs of angles are equal,

78. (c)

D STV ||| DWUV x = 2.4 cm

80. (c)
10.

+B = 90c (+ in semicircle)
AC 2 = AB 2 + BC 2
= 62 + 32
= 36 + 9
= 45
AC = 45
=3 5
1
Radius = AC
2
3 5
=
cm
2

11.

+OAC = 30c (Base +s of isosceles D)
+BAO = 25c (similarly)
` +CAB = 30c + 25c
= 55c x = 2 +CAB
(+ at the centre is double the
+ at the circumference)
= 2 # 55c
= 110c

Chapter 9: Properties of the circle
The proofs given as answers to this chapter are informal. Also, they may not be the only way to answer the question.

Exercises 9.1
1.

(a) i = 32c (b) x = 8 cm (c) i = a = 68c 30l (d) i = 31c
(e) x = 9 mm

2.

16r cm 9

(f) i = 22c30l

3. (a) i = 29c

(c) a = 83c , b = 42c
(f) y = 97c

(b) x = 18c

(d) x = 68c

(e) x = 10 cm

(g) x = 15c, y = 150c, z = 75c

(h) x = 47c, y = 43c, z = 94c
(j) x = y = 39c

(i) b = 40c

817

818

Maths In Focus Mathematics Extension 1 Preliminary Course

12. (a) x = 52c, y = 76c

6.

(b) AC = BD
(equal diameters)
Diagonals are equal so ABCD is a rectangle.
(opposite sides of a rectangle)
` AD = BC

OB = 8.3 cm

8.

x Z 4.4 m, a = 78c, b = 38c, i = 64c

9.

OA = r

13. +ECB = 33c
(angles in same segment)
+EBC = 180 - ] 114 + 33 g (angle sum of triangle)
= 33°
`+ECB = +ADE
These are equal alternate angles.
` AD < BC

AC =
OC =
=
=

14. (a) +AOB = 90c (given)
+ABC = 90c (angle in semi-circle)
` AOB = +ABC
+
` A is common
+
`D AOB ||| D ABC (AAA)
(Note 2 pairs of angles equal means 3 pairs will be equal by angle sum of triangle.)
(b) AO = BO
AB =

r +r
2

=
=

(equal radii)

=

=

2 r2

=
=

2 # r2
2r

x
2

(perpendicular from the centre bisects a chord)

x 2 r -d n
2
2

x2
4
4r 2 x 2
4
4
2
4r - x 2
4
4r 2 - x 2
2
4r 2 - x 2
2

2r + 4r 2 - x 2
2

10. (a) +ECD = +ACB

(AAA)

AC
BC
=
CE
CD

So BC =

AC. CD = BC.CE

2r

15. Obtuse +BOD = 2i

(angle at centre double angle at circumference)

Reflex +BOD = 360 - 2i

(angle of revolution)

1
+BCD = +BOD
2

Exercises 9.3
1.

(a) x = 107c, y = 94c (b) i = 134c, c = 90c
(c) x = 112c, y = 112c, z = 68c (d) x = 92c, y = 114c
(e) b = 73c, a = 107c, c = 107c (f) x = 141c, y = 63c
(g) x = 65c, y = 43c
(h) w = 89c, x = 86c, y = 54c, z = 35c
(i) w = 69c, x = 111c, y = 82c, z = 98c (j) x = 118c

2.

(a) x = 62c, y = 31c (b) x = 75c, y = 105c
(c) x = 88c, y = 65c (d) x = 62c, y = 82c, z = 36c
(e) x = 90c, y = 113c (f) x = 38c, y = 71c
(g) x = 85c, y = 95c (h) x = 48c, y = 78c
(i) x = 107c, y = 73c
(j) a = 81c, b = 55c, c = 83c, d = 16c, e = 28c

3.

(a) +A = 180c - 58c

(+A and +B cointerior angles,
AD ; BC)

+D = 180c - 58c

(+C and +D cointerior angles,
AD ; BC)

(angle at centre double angle at circumference)

1
= (360 - 2i)
2
= 180 - i
So +BCD and +DAB are supplementary (add to180c)

Exercises 9.2

5.

(angles in same segment)

`DABC ||| DCDE
(b) By similar triangles

But AO = BO so AB = BC

2.

(vertically opposite angles)

+A = +E

By similar triangles
AO
BO
=
AB
BC

1.

(Pythagoras’ theorem)

r2 -

CD = r +

2

7. x = 4.7 m, y = 1.8 m

(a) x = 5 cm (b) y = 15 cm (c) x = 2.4 m (d) x = 42c
(e) z = 90c (f) x Z 10.3 m (g) x = 6 m, y = 3 m
(h) m Z 13.4 cm (i) y Z 5 cm (j) x = 5 mm
41 cm

3. 144 mm

4. 25.6 cm

CE = 11.5 2 - 6.9 2
= 9.2
CD = 2 # 9.2 (perpendicular from O bisects chord)
= 18.4
= AB

So +A = 180c - +C and +D = 180c - +B
Since opposite angles are supplementary, ABCD is a cyclic quadrilateral.
(b) +B = +D = 90c

(given)

` +B = 180c - +D
Let +A = x
+C = 360 - ] 90 + 90 + x g

(angle sum of quadrilateral)

ANSWERS

= 360 - 180 - x
= 180 - x
= 180 - +A
Since opposite angles are supplementary, ABCD is a cyclic quadrilateral.
(c)

+CDA = 180 - i

(straight angle)

` +B = 180c - +CDA

6.

(a) x = 67c (b) y Z 7.5 cm (c) x = 72c, y = 121c
(d) x = 63c, y = 126c (e) x = 8.9 m, y Z 5.1 m
(f) x = 63c, y = 63c (g) x = 98c, y = 65c, z = 17c
(h) x = 57c, y = 57c (i) x = 72c, y = 15c
(j) x = 61c, y = 70c, z = 52c

7.

(a) x = 26c, y = 74c, z = 48c (b) x = 68c, y = 44c, z = 68c
(c) x = y = z = 45c (d) x = 70c, y = 31c
(e) x = 20c, y = 57c, z = 103c (f) x Z 5.4 cm
(g) x Z 7.7 cm (h) x = 77c, y = 13c
(i) x Z 1.2 cm, y Z 2.1 cm (j) x = 55c, y = 112c, z = 57c

8.

AB Z 13 m

Let +A = x
+C = 360 - ] 90 + 90 + x g (angle sum of quadrilateral)
= 360 - 180 - x
= 180 - x
= 180 - +A
Since opposite angles are supplementary, ABCD is a cyclic quadrilateral.

1.

Exercises 9.4
1.

2.

3.

Test yourself 9

(a) i = 47c (b) x = 5 m (c) y = 11.3 cm
(d) x = y = 26c (e) a = 64c, b = 32c (f) i = 57c
(g) p = 145 Z 12 cm (h) y = 10 mm (i) x Z 5.79 cm
(j) x = 33c, y = 33c
(a) x = 10 cm (b) x = 64c, y = 26c (c) x = 13 cm
(d) x = 27c, y = 54c (e) y = 5 cm (f) x = 32c, y = 7c
(g) x = 72c, y = 42c (h) x = 35c, y = 90c
(i) m = 23c, n = 67c, p = 67c, q = 23c
(j) x = 71c, y = 62c
+OAB = 90c (tangent = to radius) z = 90c - 48c (+ sum of D AOB)
= 42c
(equal radii)
OA = OC
` +OAC = +OCA = y

i = 56c

4.

x = y = 12 cm

5.

z = 19c (+s in same segment) y = 180c - (131c + 19c ) (+ sum of D )
= 30c x = 30c (+s in same segment)

6.

x = 10 cm

7.

a = 3c , b = 44c , c = 136c

(+ at centre twice + at circumference)

= 50c
+OCA = 90c (tangent perpendicular to radius) b = 90c - 83c
`
= 7c
OC = OE (equal radii)
` D OCE is isosceles

(base +s of isosceles D )

y = (180c - 48c) ' 2
(+ sum of D OAC)

= 66c
+ACD = 180c - +AED

`

+OCE = + OEC = c
2c + 100c = 180c (+ sum of D)
2c = 80c c = 40c
Reflex +COE = 360c - 100c (+ of revolution)
= 260c d = 360c - (260c + 50c + 7c)

(opposite +s of cyclic quad.)

= 180c - 62c
= 118c
= 52c
= +OAB - +OAC
= 90c - 66c
= 24c
1
v = +AOC
2

y+u
66c + u u +BAC
`
x

(+ at centre twice + at circumference)

1
# 48c
2
= 24c
=

4.

21 cm

5.

AC 2 + BC 2 = 3.9 2 + 5.2 2
= 42.25
AB 2 = 6.5 2
= 42.25
`
AB 2 = AC 2 + BC 2
` +ACB = 90c (by Pythagoras’ theorem)
` A lies on a diameter of the circle (tangent ⊥ radius)

3. x = 7.2 m

1
# 100c
2

a=

8.

`

`

2. y = 2.3 mm

(+ sum of quadrilateral)

= 43c

9.

17 cm

10. 5.3 m

12. a = 61c, b = 29c
15. 18 cm

11. a = 101c, b = 98c
13. 14.9 cm

14. x = 4.9 m

16. a = 127c, b = 53c

17. +D = 180c - (80c + 53c) (+ sum of T)
= 47c
` y = 47c (+s in same segment) x = 47c (+s in alternate segment)
18. x = 55c, y = 56c, z = 54c

819

820

Maths In Focus Mathematics Extension 1 Preliminary Course

6.

19. +C is common
+A = +CBD (+s in alternate segment)

Let +ODC = x and +OAB = y.
Then you can find all these angles

(giving reasons).

` D BCD ||| D ABC ] AAA g
20. (a) +OCB = +OCA = 90c (given)
OA = OB (equal radii)
OC is common
` DOAC / DOBC ] RHS g
(b) AC = BC (corresponding sides in / D s)
∴ OC bisects AB

Challenge exercise 9
1.

6 cm

2.

Let +DOB
Then +EDO
EO
`
+OED

= +DCB = x (base +s of isosceles D ODC)
= 2x (ext. + of D)
= DO (equal radii)
= +EDO = 2x (base of+s of isosceles D EOD)

+AOC + +COB + +BOD + +AOD = 360c
(+ of revolution)

90c - y + x + +COB + y + 90c - x +
+AOD = 360c
+COB ++AOD + 180c = 360c
`
+COB ++AOD = 180c

+EOD = 180c - (+OED + +EDO) (+ sum of D EOD)
= 180c - 4x
+AOE = 180c - (+EOD + +DOB)
(+AOC straight +)

= 180c - (180c - 4x + x)
= 3x
` +AOE = 3+DCB
3.

Let+DAB
Then+DAC
+ACB
+ADB
+DBA
+CBA
+DBA + +CBA

7.

= x and +CAB = y
=x+y
= +DAB = x (+s in alternate segment)
= +CAB = y (similarly)
= 180c - (x + y) (+ sum of D ADB)
= 180c - (x + y) (+ sum of D ACB)
= 180c (DBC is straight +)

D
Let ABCD be a kite with AB = AD and BC = DC, and
+ADC = +ABC = 90°.
AC is common.

(a) AD = DB = BE = EC = CF = FA (equal radii)

∴ by SSS (or RHS) D ABC / D ADC

` AB = BC = CA
`D ABC is equilateral

`+BAC = +DAC and +BCA = +DCA
(corresponding 1s in congruent D s)

(b) rr units
(c)
5.

3 r2 -

Let +BAC = +DAC = a
Then +BAD = 2a
+BCA = +DCA = 90c - a (+sum of D )
+BCD = 180c - 2a
`
Opposite angles are supplementary.

2 3-r
1 2 o units 2 rr = r 2 e
2
2

+BDE = +ABD + +BAD (ext. + of D BAD)
` 2a = +ABD + a a = +ABD
` D BAD is isosceles with AD = BD
+CDE = +ACD + +CAD (ext. +of D CAD )
` 2b = +ACD + b b = +ACD
` DCAD is isosceles with AD = CD
` AD = BD = CD
So a circle can be drawn through A, B and C with centre D.

C

A

` 180c - (x + y) + 180c - (x + y) = 180c
180c = 2 (x + y)
`
90c = x + y
`
`
+DAC = 90c
4.

B

∴ ABCD is a cyclic quadrilateral, and A, B, C and D are concyclic points
Since +ABC = 90c, AC is a diameter. (+ in semicircle)
8.

25rr 2 units 2
28

ANSWERS

9.

Now ABCE is a cyclic quadrilateral, so

Let interval AB subtend angles of x at +ADB and +ACB.

+AEC + +B = 180c (opposite +s supplementary)
Also, +D + +B = 180c (given)
+D = +AEC
These are equal corresponding angles, so
DA < EA (this is impossible!)
∴ A, B, C and D must be concyclic
∴ ABCD is a cyclic quadrilateral.

Chapter 10: The quadratic function
Assume A, B, C and D are not concyclic. Draw a circle through A, B and C that cuts AD at E.

Exercises 10.1

But +AEB and +EDB are equal corresponding angles.
` EB || DB (this is impossible!)
∴ A, B, C, D must be concyclic
10. Let ABCD be a quadrilateral with opposite angles supplementary. i.e. +A + +C = 180c and +B + +D = 180c
Assume the points are not concyclic. Draw a circle through A, B and C, cutting CD at E.

Axis of symmetry x = - 1, minimum value - 1

2.
Then +AEB = +BCA = x (+s in same segment)

1.

Axis of symmetry x = - 1.5, minimum value - 7.5

3.

Axis of symmetry x = - 1.5, minimum value - 0.25

4.

Axis of symmetry x = 0, minimum value - 4

5.

Axis of symmetry x =

6.

Axis of symmetry x = 1, maximum value -6

7.

Axis of symmetry x = - 1, maximum point ^ - 1, 7 h

8.

Minimum value -1, 2 solutions

9.

Minimum value 3.75, no solutions

3 7
3
n
, minimum point d ,
8 16
8

10. Minimum value 0, 1 solution
11. (a) x = -3; (-3, -12)
1
1 1
(c) x = 1 ; d 1 , 3 n
4
4 8

(b) x = -4; (-4, 17)
1
1
1
(d) x = -1 ; d -1 , -13 n
4
4
4

(e) x = -3; ^ -3, -23 h
12. (a) (i) x = -1
(b) (i) x = 1

(ii) -3

(iii) (-1, -3)

(ii) 1

(iii) (1, 1)

821

822

Maths In Focus Mathematics Extension 1 Preliminary Course

13. (a) Minimum (-1, 0)
(b) Minimum (4, -23)
(c) Minimum (-2, -7)
(d) Minimum (1, -1)
(e) Minimum (2, -11)
1
1
(f) Minimum d - , -3 n
4
8

(c) (i) 5.83, 0.17

(ii) Minimum -8 y (iii)
10
8
6

(g) Maximum (-1, 6)

4

(h) Maximum (2, 11)
1 3
(i) Maximum d , 7 n
2 4

2
1

-4 -3 -2 -1
-2

(j) Maximum (1, -3)

2

3

4

5

x

6

-4
14. (a) (i) -2
(iii)

-6

(ii) Minimum 0 y -8
-10

5
4
3

(d) (i) -2, 0

2

y

(iii)

1
-4 -3 -2 -1
-1

5

x

2

1

4

-2

3

-3

2
1

(b) (i) -1, 3 (ii) Minimum -4

-2

5

-3

4

(e) (i) ! 3

3
2

-2
-3

(ii) Minimum -18 y (iii)

1

-4 -3 -2 -1
-1

x

2

1

-4 -3 -2 -1
-1

y

(iii)

(ii) Minimum -1

1

2

3

4

2

x

1
-4 -3 -2 -1
-2

1

-4

-4

-6

-5

-8
-10
-12
-14
-16
-18
(f) (i) -1,

2
3

(ii) Minimum - 2

1
12

2

3

4

5

x

ANSWERS

y

(iii)

(i) (i) 0.56, -3.56

5

(ii) Minimum 4

1
4

y

(iii)

4

41
4

5

3

4

2

3

1
1

-4 -3 -2 -1
-1

3

2

4

2

x

5

1

2
3

-2

-4 -3 -2 -1
-1

-2 1 -3
12
-4

1

2

3

x

5

4

-2
-3

-5
-6
(j) (i) 2.87, -0.87
(g) (i) 1.65, -3.65 (ii) Maximum 7

y

(iii)

y

(iii)

(ii) Maximum 7

7
6

7

5

6

4

5

3

4

2

3

1

2

1

-4 -3 -2 -1
-1

1
2

1

-4 -3 -2 -1
-1

3

4

5

x

2

3

4

5

x

-2
-3

-2
-3
15. (a) 4

(h) (i) 1.3, -2.3 (ii) Maximum 3

(c)

1
4

y
7

y

(iii)

(b) None

6

5

5

4

3

1
4

4

3

3

2

2
1

1

-4 -3 -2 -1
-1
-2
-3

1

2

3

4

5

x

-4 -3 -2 -1
-1
-2
-3

1

2

3

4

5

x

823

824

Maths In Focus Mathematics Extension 1 Preliminary Course

16. (a) None

(b) 6

y

19.

3
4

8

y

(c)

6

14

4

12

2

10
8

-4

6

1

-3 -2 -1
-2

3

2

4

x

5

-4

4
2

-6
2

1

-4 -3 -2 -1
-1

3

4

x

5

Graph is always above the x-axis so y 2 0 for all x
` 3x 2 - 2x + 4 2 0 for all x

-2

y

20.

-3

8

7
17. (a) - 3
8

6

(b) None

4

y

(c)

2

2

1

-4 -3 -2 -1
-2

1
1

-4 -3 -2 -1
-2

2

3

4

x

5

2

3

4

5

x

-4
-6

-4
-6

Graph is always above the x-axis so y 2 0 for all x
` x 2 + x + 2 2 0 for all x

-8
-10

4

-14

2

-16
-18

-4 -3 -2 -1
-2

y

18. (a)

y

21.

-12

1

2

3

4

5

-4

8

-6

6

-8

4

-10
-12

2

-4 -3 -2 -1
-1

1

-2

2

3

4

5

x

-14
-16
-18

-3
(b) x 1 2, x 2 3

(c) 2 # x # 3

Graph is always below the x-axis so y 1 0 for all x
` - x 2 + 2x - 7 1 0 for all x

x

ANSWERS

22.

y

8.

2
1

-4 -3 -2 -1
-1

3

2

1

4

5

9.

-6

1.

x 1 -3, x 2 3

2. - 1 # n # 0

4.

x 1 - 2, x 2 2

5. 0 # y # 6

7.

x 1 - 4, x 2 2

8. p # - 3, p $ - 1

3. a # 0, a $ 2
6. 0 1 t 1 2

1
11. 1 1 h 1 2
2
14. q 1 3, q 2 6

15. All real x

17. - 3 1 x 1 5

1
19. y 1 - , y 2 5
3

20. x # - 2, x $ 4

1
3

30. - 2

24. -

18. - 6 # t # 2

29. x #

21. -

1
1x 10
2

1
#x 10
2

26. x $ - 1, x 1 - 2

28. x 1 - 6, x 2 - 3

9. m 1 2, m 2 4

12. - 4 # x # 5

16. n # - 4, n $ 3

25. 1 1 x 1 1

1
2

(1)
(2)

x 2 + 3 = 2x + 6 x 2 - 2x - 3 = 0 b 2 - 4ac = ] - 2 g2 - 4 ] 1 g ] - 3 g
= 16
20
So there are 2 points of intersection

Exercises 10.2

23. 0 1 x # 1

14. 0 # b # 2

Substitute (2) in (1):

Graph is always below the x-axis so y 1 0 for all x
` - 5x 2 + 4x -1 1 0 for all x

1
3

3

11. m 1 - 3, m 2 3

y = x2 + 3

-7

22. 0 1 x 1

13. p 1 -

1

16. Solving simultaneously: y = 2x + 6

-5

1
#k #7
2

10. 0 1 k 1 4

15. p # - 2, p $ 6

-4

13. - 2

k # - 5, k $ 3

12. k # - 1, k $ 1

-3

10. x # - 3, x $ 2

b 2 - 4ac = ] - 1 g2 - 4 ] 3 g ] 7 g
= - 83
10
So 3x 2 - x + 7 2 0 for all x

x

-2

a =320

27. 2 1 x # 2

2
5

2
,x 21
3

2
# x 1 -2
2

Exercises 10.3
1.

(a) 20 (b) -47 (c) -12 (d) 49 (e) 9
(h) 64 (i) 17 (j) 0

(f) -16 (g) 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.

1
7
6. p 2 2 7. k 2 - 2 p = 1 4. k = ! 2 5. b # 12
8

17. 3x + y - 4 = 0 y = x 2 + 5x + 3
From (1): y = - 3x + 4
Substitute (2) in (3): x 2 + 5x + 3 = - 3x + 4 x 2 + 8x - 1 = 0 b 2 - 4ac = 8 2 - 4 ] 1 g ] - 1 g
= 68
20
So there are 2 points of intersection

(1)
(2)

18. y = - x - 4 y = x2
Substitute (2) in (1): x2 = - x - 4
2
x +x+4=0 b 2 - 4ac = 1 2 - 4 ] 1 g ] 4 g
= - 15
10
So there are no points of intersection

(1)
(2)

19. y = 5x - 2 y = x 2 + 3x - 1
Substitute (2) in (1): x 2 + 3x - 1 = 5 x - 2 x 2 - 2x + 1 = 0 b 2 - 4ac = ] - 2 g2 - 4 ] 1 g ] 1 g
=0
So there is 1 point of intersection
` the line is a tangent to the parabola

(1)
(2)

20. p = 3

1
4

21. (c) and (d)

(3)

825

826

Maths In Focus Mathematics Extension 1 Preliminary Course

Exercises 10.4
1.

2.

19. (a) k = 2
20. (a) m = 1

(a) a = 1, b = 2, c = -6
(b) a = 2, b = -11, c = 15
(c) a = 1, b = 1, c = - 2
(d) a = 1, b = 7, c = 18
(e) a = 3, b = -11, c = -16
(f) a = 4, b = 17, c = 11
(g) a = 2, b = -12, c = -9
(h) a = 3, b = - 8, c = 2
(i) a = - 1, b = 10, c = - 24
(j) a = - 2, b = 0, c = - 1

(b) k = - 3
(b) m 1

Exercises 10.6
1.

(a) x = - 2, 3

RHS = a ] x - 2 g ] x + 3 g + b ] x - 2 g + c
= 1 ] x - 2 g ] x + 3 g + 1 ] x - 2 g + 17
= x 2 + 3x - 2x - 6 + x - 2 + 17
= x 2 + 2x + 9
= RHS
` true

3.

9.

(b) y = x 2 - 3x
(d) y = x 2 + 4x - 9

x = ! 1, !2

7.

x = ! 2.19, !0.46, !1.93, !0.52

8.

(a) x = 0c , 90c , 180c , 360c (b) x = 90c , 180c , 270c
(c) x = 90c , 210c , 330c (d) x = 60c , 90c , 270c , 300c
(e) x = 0c , 180c , 270c , 360c
(a) x = 0c , 45c , 180c , 225c , 360c
(b) x = 0c , 180c , 360c
(c) x = 0c , 30c , 150c , 180c , 360c
(d) x = 45c , 60c ,135c , 120c , 225c , 240c , 315c , 300c
(e) x = 30c , 60c , 120c , 150c , 210c , 240c , 300c , 330c x+3+ (e) y = - x 2 - 2x + 1

Exercises 10.5
(a) a + b = - 2, ab = 1
(b) a + b = 1.5, ab = - 3
(c) a + b = 0.2, ab = - 1.8
(d) a + b = - 7, ab = 1
2
(e) a + b = 2 , ab = 1
3

Let u = x + 3

4.

m = 0.5

5. k = - 32

9.

k = -5

10. m = ! 3

u 2 - 5u + 2 = 0 b 2 - 4ac = ] - 5 g2 - 4 ] 1 g ] 2 g
= 17
20

(a) x 2 + 3x - 10 = 0 (b) x 2 - 4x - 21 = 0
(c) x 2 + 5x + 4 = 0 (d) x 2 - 8x + 11 = 0
(e) x 2 - 2x - 27 = 0
6. b = 4

7. k = 1

11. k = - 1

14. b = - 6, c = 8

12. n = - 1, 3
15. a = 0, b = - 1

2

Test yourself 10

(c) k = - 1.8

(b) p # - 2 3 , p $ 2 3

(d) k = 3

1.

(a) 0 # x # 3

2.

(e) k # - 1, k $ 0

3 3

So u has 2 real irrational roots.
` x + 3 and so x has 2 real irrational roots

8. p = 13

1
16. ab = 1 ` b = a (c) p = !

2
=5
x+3

2
# (x + 3) = 5 # (x + 3)
]x + 3g
] x + 3 g2 + 2 = 5 ] x + 3 g
] x + 3 g2 - 5 ] x + 3 g + 2 = 0

3.

18. (a) p = ! 2 3

6. x = - 1

(x + 3) # (x + 3) +

(d) 21

(b) k = - 1, 0

(e) a = - 2, - 2 ! 6

(a) x = 0, 3 (b) p = 1 (c) x = 1 (d) x = 1 (e) x = 1, 3

a = 0, b = - 4, c = - 21

17. (a) k = - 1

1! 5
2

10.

K = 1, L = 6, M = 7.5 8. 12 ] x + 5 g + ] 2x - 3 g - 65 - 2

13. p = 2, r = - 7

(c) x =

9.

7.

(c) - 0.5

(b) y = ! 2, ! 2

5.

6. a = 2, b = 1, c = - 1
2

(a) 3 (b) - 6

(a) x = ! 3

(d) x = 3, 5

4.

A = 1, B = 5 , C = - 6

2.

(c) x = 4, 5

(d) x = 1.37, - 4.37, 0.79, - 3.79

5.

1.

(b) x = 2, 3

1
(e) x = 1 , 4
2

x 2 - 4x + 5 = x ] x - 2 g - 2 ] x + 1 g + 3 + 4

4.

(a) x = -1, - 4 (b) y = 2, 5 (c) x = - 4, 2
(d) n = - 1, 4 (e) a = - 3, 5 (f) p = 3, 4 (g) x = 2, - 4
(h) k = 5, 12 (i) t = 6, - 4 (j) b = -12, - 4

2.

3.

(c) y = 2x 2 - 3x + 7

3 - 10
3 + 10
,m2
2
2

(c) m = - 3

m = 2, p = - 5, q = 2

10. (a) y = x 2 - x - 5

(c) k = 2

(b) n 1 - 3, n 2 3

a = 1, b = - 9, c = 14

4.

a =120
D = b 2 - 4ac
= ] -2 g 2 -4 # 1 # 7
= - 24
10
` positive definite

3. (a) x = 2

(c) - 2 # y # 2
(b) - 3

ANSWERS

2 1
(d) 18 (e) 30 6. x = 1 ,
3 3

5.

(a) 6 (b) 3

7.

(a) iv (b) ii (c) iii

8.

a = -1 1 0
D = b 2 - 4ac
= 3 2 - 4 # (-1) # (- 4)
= -7
10

(c) 2

(d) ii

(e) i

(a) x = -

1
4

(b) 6

1
4

1
13. x = - , 3
2

(c) k = 3

14. m 1 -

16. (a) i (b) i (c) iii

(d) i

17. (a) iii (b) i (c) i

9
16

(d) k = 3

(e) k = 2

ab
1
a
LHS
a

19. (a) x + 3x - 28 = 0

(e) ii

1
1
(c) 1 y 1
5
3

y = x 2 - 5x + 4

5.

11

9.

12. Circle, centre ^ 1, -2 h, radius 4

13. y = -5
15. x = -7

16. x = 3

18. x = !4

x 2 - 10x + y 2 + 4y + 25 = 0

5.

12x - 26y - 1 = 0

7.

3x 2 - 32x + 3y 2 - 50y + 251 = 0

8.

5x 2 - 102x + 5y 2 + 58y - 154 = 0

9.

(b) x - 10x + 18 = 0

x2 + y2 = 1

3.

(b) n # - 3, n 2 3

2. x 2 + 2x + y 2 + 2y - 79 = 0

x 2 - 4x + 20y - 36 = 0

11. y 2 + 8x - 32 = 0

1
(d) x # - 10, x 2 - 2
2

4. 8x - 6y + 13 = 0

6. y = ! x

(e) 4 1 x # 7

13. x 2 + 12y = 0

10. x 2 - 20y = 0

12. x 2 - 2x + 8y - 7 = 0

14. x 2 - 5x + y 2 - 2y - 11 = 0

15. x 2 + 3x + y 2 - y - 4 = 0

3. a = 4, b = - 3, c = 7
7. p 2 0.75

4. x = ! 2

8. Show D = 0

16. x 2 + x + y 2 - 2y - 17 = 0
17. 2x 2 + 4x + 2y 2 - 6y + 47 = 0
18. 2x 2 + 2x + 2y 2 + 4y + 27 = 0
19. 3x + 4y + 25 = 0, 3x + 4y - 15 = 0
20. 12x - 5y - 14 = 0, 12x - 5y + 12 = 0

10. A = 2, B = - 19, C = 67 or A = - 2, B = 13, C = - 61
4x + 1

3
1
=
+
11. 2 x-2 x+1 x -x-2
1 - 21
1 + 21
,k$
2
2

13. x = 30c , 90c , 150c

11. Circle x 2 + y 2 = 1 (centre origin, radius 1)

1.

x = !1

12. k #

10. line y = 2

2

4
7

6. n = - 2.3375

9. lines x = !5

Exercises 11.2

D = ] k - 4 g2 $ 0 and a perfect square ∴ real, rational roots

2.

5. A spiral

20. Circle, centre ^ -4, 5 h, radius 1

1
=
a c = a k
=
k
= RHS = 1

Challenge exercise 10
1.

4. A (parabolic) arc

19. Circle, centre ^ -2, 4 h, radius 6

20. x = 1, 3
21. (a) x 1 - 1, x 2 -

lines y = !
1

17. y = !8

∴ roots are reciprocals for all x.
2

2. A straight line parallel to the ladder.

14. Circle, centre (1, 1), radius 3

15. x = 0, 2

(d) ii

18. For reciprocal roots b

A circle, centre the origin, radius 2 (equation x2 + y2 = 4 i

8.

(b) k = 1

The straight line - 2 1 x 1 2 or | x | 1 2

7.

11. x = 30c , 150c , 270c

An arc

6.

1
8

A circle

3.

10. 3 ] x - 2 g2 + 12 ] x + 3 g - 41
12. (a) k = 3

Exercises 11.1
1.

` - 4 + 3x - x 2 1 0 for all x
9.

Chapter 11: Locus and the parabola

3! 5
14. x = 1,
2

15. x = 60c , 90c , 270c , 300c

16. - 23

21. x - 2y - 3 ! 5 5 = 0
22. x - 7y + 9 = 0, 7x + y - 5 = 0
23. 7x - 4y - 30 = 0, 32x + 56y - 35 = 0
24. xy - 16x - 7y + 40 = 0
25. x 2 - 6x - 3y 2 - 12y + 9 = 0

827

828

Maths In Focus Mathematics Extension 1 Preliminary Course

Problem

23.

12x + 5y - 40 = 0, 12x + 5y + 38 = 0

Exercises 11.3
1.

(a) Radius 10, centre (0, 0) (b) Radius 5 , centre (0, 0)
(c) Radius 4, centre (4, 5) (d) Radius 7, centre (5, −6)
(e) Radius 9, centre (0, 3)

2.

(a) x 2 + y 2 = 16 (b) x 2 - 6x + y 2 - 4y - 12 = 0
(c) x 2 + 2x + y 2 - 10y + 17 = 0
(d) x 2 - 4x + y 2 - 6y - 23 = 0
(e) x 2 + 8x + y 2 - 4y - 5 = 0
(f) x 2 + y 2 + 4y + 3 = 0 (g) x 2 - 8x + y 2 - 4y - 29 = 0
(h) x 2 + 6x + y 2 + 8y - 56 = 0 (i) x 2 + 4x + y 2 - 1 = 0
(j) x 2 + 8x + y 2 + 14y + 62 = 0

3.

x 2 + 4x + y 2 + 4y - 8 = 0

6.

x 2 + 6x + y 2 - 16y + 69 = 0

7.

x 2 - 10x + y 2 + 4y + 27 = 0

9.

x 2 - 2x + y 2 - 10y + 25 = 0

5. x 2 - 2x + y 2 - 48 = 0

26. 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.
27. (a) x 2 + 2x + y 2 - 6y - 15 = 0
(b) ^ 2, 7 h, ^ -1, -2 h (c) Z = ^ -1, 8 h
1
(d) m zx # m yx = - # 3
3
= -1
` +ZXY = 90c

8. x 2 + y 2 - 9 = 0

(b) x 2 = 36y

(c) x 2 = 4y

(f) x 2 = 12y

(g) x 2 = 24y

(i) x = 8y
2

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)

(a) x 2 = -4y

(b) x 2 = -12y
(e) x = -24y

(g) x 2 = -32y

(h) x 2 = -8y

4.

(a) (i) (0, −1) (ii) y = 1 (b) (i) (0, −6) (ii) y = 6
(c) (i) (0, −2) (ii) y = 2 (d) (i) (0, −12) (ii) y = 12
(e) (i) (0, −5) (ii) y = 5 (f) (i) (0, −4) (ii) y = 4
(g) (i) (0, −8) (ii) y = 8 (h) (i) (0, −10) (ii) y = 10

6.

22. x 2 + 2x + y 2 + 2y - 23 = 0

(i) x 2 = -60y

(a) (i) (0, 1) (ii) y = -1 (b) (i) (0, 7) (ii) y = -7
(c) (i) (0, 4) (ii) y = -4 (d) (i) (0, 9) (ii) y = -9
(e) (i) (0, 10) (ii) y = -10 (f) (i) (0, 11) (ii) y = -11
1
1
(g) (i) (0, 3) (ii) y = -3 (h) (i) c (0, 1 m (ii) y = -1
2
2
3
1
1
(i) (i) c 0, 2 m (ii) y = -2
(j) (i) c 0, 3 m
4
2
2
3
(ii) y = -3
4

5.

21. (a) Both circles have centre ^ 1, -2 h
(b) 1 unit

(f) x 2 = -36y

3.

1
1
(i) (i) c 0, - m (ii) y =
2
2

20. Show perpendicular distance from the line to ^ 4, -2 h is
5 units, or solve simultaneous equations.

(c) x 2 = -16y

(j) x = -52y

17. Centre (4, 7), radius 8

19.

2

2

15. Centre ^ 2, 5 h , radius 5

1
1
18. Centre d - 1 , 1 n , radius 2
2
2

(h) x 2 = 44y

2

(d) x = -28y
12. x 2 + y 2 + 6y + 1 = 0

(d) x 2 = 16y

(j) x = 48y

2

11. x 2 - 8x + y 2 - 6y + 22 = 0

16. Centre ^ - 1, -6 h , radius 7

(a) x 2 = 20y
(e) x 2 = 40y

1.

2

14. Centre ^ 3, -1 h , radius 4

(b) x 2 - 4x + y 2 + 10y + 13 = 0

Exercises 11.4

10. x + 12x + y - 2y + 1 = 0
2

34 units

25. (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 .

28. (a) 4 units

x 2 - 18x + y 2 + 8y + 96 = 0

4.

24.

56 units

1
1
(j) (i) c 0, -5 m (ii) y = 5
2
2

(a) x 2 = 28y (b) x 2 = 44y (c) x 2 = -24y
(d) x 2 = 8y (e) x 2 = !12y (f) x 2 = !32y
1
(g) x 2 = 32y (h) x 2 = y
7
(a) Focus ^ 0, 2 h, directrix y = -2, focal length 2
(b) Focus ^ 0, 6 h, directrix y = -6, focal length 6

(c) Focus ^ 0, -3 h, directrix y = 3, focal length 3
1
1
1
(d) Focus d 0, n, directrix y = - , focal length
2
2
2
3
3
3
(e) Focus d 0, -1 n, directrix y = 1 , focal length 1
4
4
4
1
1
1
(f) Focus d 0, n, directrix y = - , focal length
8
8
8

ANSWERS

7.

y =2

8. ^ 4, 4 h

9.

7.

10. ^ 4, -2 h and ^ -4, -2 h ; 8 units
11. (a) x 2 = - 12y

(b) y = 3

(c) 33

x = 4 (latus rectum)

9.

3
1
X = d -1 , - n
2
8

^ 9, - 6 h, ^ 81, 18 h

(d) 4

2 units 13

(e) 11.7 units2

Exercises 11.6

13. (a) x - 4y + 2 = 0 (b) ^ 0, 1 h does not lie on the line
(c) x 2 - 4x + y 2 - 2y + 1 = 0
(d) Substitute ^ 0, 1 h into the equation of the circle.
14. (a) Substitute Q into the equation of the parabola.
(b) _ q 2 - 1 i x - 2qy + 2aq = 0
(c) Equation of latus rectum is y = a. Solving with x 2 = 4ay gives two endpoints A ^ -2a, a h, B ^ 2a, a h .
Length of AB = 4a.

1.

(a) ] x - 3 g2 = 8 ^ y + 3 h

(b) ] x - 5 g2 = 4 ^ y + 6 h

(c) ] x - 1 g = 4 ^ y + 3 h (d) ] x - 4 g2 = -12 ^ y - 3 h
(e) ] x - 6 g2 = 8 ^ y + 7 h (f) ] x + 7 g2 = -16 ^ y - 3 h
(g) ] x - 2 g2 = -4 ^ y - 5 h (h) ] x + 9 g2 = 12 ^ y + 6 h
2

(i) ] x + 1 g2 = - 4 ^ y - 2 h

2.

(j) ] x - 3 g2 = 8 ^ y + 1 h

(a) ^ y - 4 h2 = 4 ] x + 4 g (b) ^ y - 1 h2 = 8 ] x + 2 g
(c) ^ y + 2 h2 = 12 ] x + 1 g (d) ^ y - 10 h2 = - 4 ] x - 29 g
(e) ^ y + 3 h2 = - 16 ] x - 1 g

(f) ^ y - 6 h2 = 8 ] x + 4 g

(g) ^ y + 5 h2 = - 24 ] x - 2 g (h) ^ y + 12 h2 = 4 ] x + 36 g

Exercises 11.5

2.

5
1
units2
(b) d - 5, - 4 n (c) 10
6
12

10. (a) 5x - 12y - 25 = 0

1 units 3

12. (a) Substitute the point into the equation.
3
(b) 3x + 4y - 3 = 0 (c) d 2, - n
4

1.

8. 12, ^ 3, 6 h, ^ 3, -6 h

(a) y 2 = 8x (b) y 2 = 20x (c) y 2 = 56x (d) y 2 = 36x
(e) y 2 = 32x (f) y 2 = 24x (g) y 2 = 28x (h) y 2 = 12x
(i) y 2 = 16x (j) y 2 = 4x

(i) ^ y - 2 h2 = - 20 ] x - 1 g (j) ^ y + 4 h2 = - 8 ] x - 2 g
3.

2

(b) x 2 + 8x - 4y + 16 = 0

(c) x - 4x - 8y - 12 = 0

(d) x 2 - 6x - 8y + 41 = 0

2

(a) y = -36x (b) y = - 16x (c) y = -40x
(d) y 2 = -24x (e) y 2 = - 8x (f) y 2 = -48x
(g) y 2 = - 44x (h) y 2 = -20x (i) y 2 = -12x
(j) y 2 = -28x
2

(a) x 2 + 2x - 8y + 9 = 0

2

(e) x 2 + 4x - 16y + 20 = 0

(f) x 2 + 2x + 16y + 1 = 0

(g) x - 8x + 20y - 24 = 0

(h) x 2 + 10x + 8y + 1 = 0

(i) x 2 + 6x + 12y + 45 = 0

(j) x 2 + 4y + 24 = 0

2

(k) y - 6y - 12x - 3 = 0

(l) y 2 - 8y - 4x + 8 = 0

2

3.

4.

5.

(a) (i) (2, 0)
(c) (i) (4, 0)
(e) (i) (7, 0)
(g) (i) (6, 0)
1
(i) (i) c , 0 m
4

= -2
= -4
= -7
= -6
1
(ii) x = 4

(ii) x
(ii) x
(ii) x
(ii) x

(a) (i) (−2, 0) (ii) x = 2 (b) (i) (−3, 0) (ii) x = 3
(c) (i) (−7, 0) (ii) x = 7 (d) (i) (−1, 0) (ii) x = 1
(e) (i) (−6, 0) (ii) x = 6 (f) (i) (−13, 0) (ii) x = 13
1
1
(g) (i) (−15, 0) (ii) x = 15 (h) (i) c - , 0 m (ii) x =
2
2
1
1
1
1
(i) (i) c - 6 , 0 m (ii) x = 6
(j) (i) c - 1 , 0 m (ii) x = 1
4
4
2
2
(a) y 2 = 20x

(b) y 2 = 4x

(e) y 2 = !36x
6.

(b) (i) (3, 0) (ii) x = -3
(d) (i) (1, 0) (ii) x = -1
(f) (i) (8, 0) (ii) x = -8
(h) (i) (9, 0) (ii) x = -9
1
1
(j) (i) c 4 , 0 m (ii) x = -4
2
2

(c) y 2 = -16x

(f) y 2 = !8x

(m) y 2 - 8x + 32 = 0

(o) y + 2y - 8x - 7 = 0

(h) y 2 =

(s) y - 4y + 2x + 5 = 0
4.

(a) (i) (3, −2)

(ii) y = -4

(b) (i) (1, 1)

(ii) y = - 3

(c) (i) (−2, 0)

(ii) y = -2

(d) (i) (4, 2)

(ii) y = - 4

(e) (i) (−5, −1)

(ii) y = -5

(f) (i) (3, 1)

(g) (i) (−1, 0)

(ii) y = 4

(h) (i) (2, 0)

(i) (i) (4, −2)

(ii) y = 4

(j) (i) (−2, −3)

(ii) y = 3

(ii) y = 2
(ii) y = 5

5.

(a) (i) (0, −1) (ii) x = -2 (b) (i) (2, 4) (ii) x = - 4
(c) (i) (0, 3) (ii) x = -4 (d) (i) (3, −2) (ii) x = -5
(e) (i) (7, 1) (ii) x = -5 (f) (i) (1, −5) (ii) x = 5
(g) (i) (11, −7) (ii) x = 13 (h) (i) (−3, 6) (ii) x = 7
1
1
(i) (i) (−7, 2) (ii) x = 9 (j) (i) c -10 , -3 m (ii) x = 9
2
2

6.

x 2 - 12y + 36 = 0

(a) Focus ^ 2, 0 h, directrix x = - 2, focal length 2

(c) Focus ^ -3, 0 h, directrix x = 3, focal length 3

(r) y 2 - 6y + 16x + 25 = 0
(t) y 2 - 2y + 2x - 6 = 0

2

1 x 2

(b) Focus ^ 1, 0 h, directrix x = - 1, focal length 1

(p) y 2 + 8y + 12x + 4 = 0

(q) y 2 - 2y + 4x - 11 = 0

(d) y 2 = 12x

(g) y 2 = 12x

(n) y 2 + 4y - 16x - 12 = 0

2

1
1
1
(d) Focus d 1 , 0 n, directrix x = - 1 , focal length 1
2
2
2

7.

x 2 + 4x - 8y - 4 = 0, x 2 + 4x + 8y + 12 = 0

8.

x 2 - 2x - 4y - 19 = 0

9. y 2 - 12y + 12x + 12 = 0

1
1
1
(e) Focus d -1 , 0 n, directrix x = 1 , focal length 1
4
4
4

10. x 2 - 2x - 12y + 1 = 0

11. x 2 - 2x - 28y + 29 = 0

(f) Focus d

12. y 2 + 4y + 24x - 44 = 0

13. y 2 - 6y - 32x + 9 = 0

14. x 2 - 6x + 8y - 15 = 0

15. y 2 + 2y - 16x + 49 = 0

1
1
1
, 0 n, directrix x = - , focal length
12
12
12

16. x 2 + 6x + 4y - 7 = 0

17. x 2 - 4x - 12y - 8 = 0

829

830

Maths In Focus Mathematics Extension 1 Preliminary Course

18. y 2 + 2y + 16x - 95 = 0

Exercises 11.8

19. (a) Vertex ^ - 2, 1 h, focus ^ - 2, 3 h, directrix y = -1

1.

(a)

(b) Vertex ^ 3, 2 h, focus ^ 3, 5 h, directrix y = -1

(c) Vertex ^ 1, -1 h, focus ^ 1, - 2 h, directrix y = 0
(d) Vertex ^ 3, 4 h, focus ^ 7, 4 h, directrix x = -1

(e) Vertex ^ 0, - 2 h, focus ^ 6, - 2 h, directrix x = -6

(f) Vertex ^ - 5, 0 h, focus ^ - 7, 0 h, directrix x = -3
20. Vertex ^ - 1, 4 h, focus ^ -1, -3 h , directrix y = 11, axis x = - 1, maximum value 4
21. x 2 - 4x - 8y + 12 = 0 or x 2 - 4x + 8y - 36 = 0
(b) d 0, 7

22. (a) 8x 2 + 9y - 72 = 0

23
9
n, y = 8
32
32

(b)

23. (a)

(c)
3
1
(b) d -1, -8 n, y = -9
4
4
24. x 2 + 4x + 8y - 20 = 0

25. 0.3 m

Exercises 11.7
1.
5.

m= dy dx

1
3

=x

2. m = -4

3. m = -1

6. x - y - 2 = 0

4. m =

1
2

7. x - 2y + 12 = 0

8.

x + y - 6 = 0, x - y - 18 = 0

9.

(d)

x - 2y - 2 = 0, 2x + y - 9 = 0

7 1
10. 4x + y - 8 = 0, M = d 1 , n
8 2
11. x + y - 9 = 0, P = ^ - 18, 27 h
12. Q = ^ 33, 60.5 h
13. x + 4y + 144 = 0, 4x + 2y + 9 = 0, ^ 18, -40.5 h ; show the point lies on the parabola by substituting it into the equation of the parabola
14. x - y - 4 = 0, R = ^ 4, 0 h
15. (a) Substitute P into the equation of the parabola
(b) x + py - 2p - p 3 = 0
(c) Substitute ^ 0, 1 h into the equation of the normal.
0 + p - 2p - p 3 = 0
0 = p3 + p
= p (p 2 + 1)
2
Since p ! 0, p + 1 = 0

(e)

ANSWERS

(f)

2.

(a) (i) p

(ii) -

1 p (iii) y - px + p 2 = 0

(iv) x + py = p 3 + 2p
1
(iii) y - qx + 3p 2 = 0
(b) (i) q (ii) q
(iv) x + qy = 3q 3 + 6q
1
(iii) y - tx + 2t 2 = 0
(c) (i) t (ii) t

2.

(a) x - 2y - 2 = 0 (b) 2x - y - 11 = 0
(c) y = x 2 + 3x + 2 (d) y = 16x 2 - 1 (e) xy = 2

3.

(a) x = 2t, y = t 2

(iv) x + ty = 2t 3 + 4t
1
(iii) y - nx + 5n 2 = 0
(d) (i) n (ii) n

(b) x = 6t, y = 3t 2

(c) x = - 4t, y = -2t 2

(d) x = 8t, y = 4t 2

(e) x = - 18t, y = - 9t

2

(f) x = 10t, y = 5t 2

t t2 ,y=
4
2
5t 2
(j) x = - 5t, y = 2

(g) x = -3t, y = (i) x =

(iv) x + ny = 5n 3 + 10n
1
(iii) y - px + 6p 2 = 0
(e) (i) p (ii) p
(iv) x + py = 6p 3 + 12p
1
(iii) y + kx - 4k 2 = 0
(f) (i) −k (ii) k t t2 ,y=
4
8

3t
2

2

(h) x =

4.

(iv) x + qy = -q 3 - 2q
1
(iii) y + tx - 2t 2 = 0
(h) (i) −t (ii) t (a) x 2 = 16y (b) x 2 = 20y (c) x 2 = 4y (d) x 2 = -28y
(e) x 2 = - 8y (f) x 2 = 4ay (g) x 2 = -4y (h) x 2 = 24y
(i) x 2 = -2y (j) x 2 = 4ay

5.

(iv) x - ky = 4k 3 + 8k
1
(iii) y - qx - q 2 = 0
(g) (i) q (ii) q

(a) Substitute _ 6t, - 3t i into the equation
(b) P = ^ -12, -12 h
(c) 2x - y + 12 = 0

(iv) x - ty = 2t 3 + 4t
1
(iii) y - mx - 3m 2 = 0
(i) (i) m (ii) m

2

6.

(a) Q = ^ - 8, 4 h

7.

^ 4, 0 h, x = - 4

9.

(a) x 2 = 24y

(iv) x + my = -3m 3 - 6m
1
(iii) y + ax - 8a 2 = 0
(j) (i) −a (ii) a (b) x - y + 12 = 0

(iv) x - ay = 8a 3 + 16a

8. P = ^ 4, -4 h; 4x + 3y - 4 = 0

(b)

1
4

10. 3x - y - 18 = 0

3.

(ii) 8 -4pq ^ p + q h, 4 _ p 2 + pq + q 2 + 2 i B

Exercises 11.9
1.

t+n
(a) (i)
2
(b) (i)
(c) (i)
(d) (i)
(e) (i)

p+q
2
m+n
2
p+q
2
a+b
2

(f) (i) -

p+q
2

a+b
(g) (i) 2
(h) (i)

p+q
2

(i) (i) (j) (i)

s+t
2

p+q
2

(a) (i) ^ p + q, pq h (ii) 7 - pq ^ p + q h, p 2 + pq + q 2 + 2 A
(b) (i) 7 4 ^ p + q h, 4pq A
(c) (i) 6 2 ] a + b g, 2ab @

(ii) 7 -2ab ] a + b g, 2 ^ a 2 + ab + b 2 + 2 h A

1
(ii) y - ] t + n g x + 4tn = 0
2

(d) (i) 6 3 ] s + t g, 3st @

(ii) y -

1
^ p + q h x + 2pq = 0
2

(ii) y -

1
] m + n g x + 3mn = 0
2

(ii) 7 -5tw ] t + w g, 5 ^ t 2 + tw + w 2 + 2 h A

(ii) y -

1
^ p + q h x + 5pq = 0
2

(ii) 8 -6pq ^ p + q h, -6 _ p 2 + pq + q 2 + 2 i B

(ii) y -

1
] a + b g x + ab = 0
2

(ii) y +

(ii) 7 -3st ] s + t g, 3 ^ s 2 + st + t 2 + 2 h A

(e) (i) 6 5 ] t + w g, 5tw @

(f) (i) 7 6 ^ p + q h, -6pq A

(g) (i) 6 4 ] m + n g, -4mn @

(ii) 7 -4mn ] m + n g, - 4 ^ m 2 + mn + n 2 + 2 h A

(h) (i) 7 10 ^ p + q h, -10pq A

1
^ p + q h x - 2pq = 0
2

(ii) 8 -10pq ^ p + q h, -10 _ p 2 + pq + q 2 + 2 i B
(i) (i) 6 5 ] h + k g, - 5hk @

1
(ii) y + ] a + b g x - 6ab = 0
2

(ii) 7 -5hk ] h + k g, - 5 ^ h 2 + hk + k 2 + 2 h A

1
(ii) y - ^ p + q h x - 4pq = 0
2
(ii) y +

1
] s + t g x - st = 0
2

1
(ii) y - ^ p + q h x - 7pq = 0
2

(j) (i) 7 -3 ^ p + q h, - 3pq A

(ii) 8 3pq ^ p + q h, - 3 _ p 2 + pq + q 2 + 2 i B

4.

(a) (i) xx 1 = 4 _ y + y 1 i (ii) y - y 1 = -

4 x - x1 i x1 _

(b) (i) xx 1 = 6 _ y + y 1 i (ii) y - y 1 = -

6 x - x1 i x1 _

831

832

Maths In Focus Mathematics Extension 1 Preliminary Course

(c) (i) xx 1 = 8 _ y + y 1 i (ii) y - y 1 = -

8 x - x1 i x1 _

(d) (i) xx 1 = 2 _ y + y 1 i (ii) y - y 1 = -

2 x - x1 i x1 _

10
(e) (i) xx 1 = 10 _ y + y 1 i (ii) y - y 1 = - _ x - x 1 i x1 (f) (i) xx 1 = -2 _ y + y 1 i

(ii) y - y 1 =
(ii) y - y 1 =

(ii) y - y 1 =
(ii) y - y 1 =

22 x - x1 i x1 _

(j) (i) xx 1 = -14 _ y + y 1 i

(ii) y - y 1 =

14 x - x1 i x1 _

(a) xx 1 = 8 _ y + y 1 i (b) xx 1 = 2 _ y + y 1 i
(c) xx 1 = 4 _ y + y 1 i (d) xx 1 = 6 _ y + y 1 i

(e) xx 1 = 10 _ y + y 1 i (f) xx 1 = -2 _ y + y 1 i

(g) xx 1 = -12 _ y + y 1 i

(h) xx 1 = -4 _ y + y 1 i

Substitute in (1): y - pa (q + p) + ap 2 = 0 y - apq - ap 2 + ap 2 = 0 y - apq = 0 y = apq
(b) Substitute ^ 0, 2 h into equation.

16. (a) 3x + 4y - 8 = 0

17. (a) For proof, see no. 9 above

(i) xx 1 = -8 _ y + y 1 i
6.
7.
9.

(j) xx 1 = -18 _ y + y 1 i

18. (a) 15x + 8y + 4 = 0

(a) y - px + ap 2 = 0

(b) xx 0 = 2a _ y + y 0 i

19. (a) x + 3y - 3 = 0

1
] t + r g x + 2tr = 0
2
x2 y=18 dy x =9 dx 9t 2 o At e -9t, 2 dy -9t n = -d
9
dx
=t
y-

8. x + 2y - 36 = 0

For normal, m 1 m 2 = -1
1
t
The equation is given by y - y 1 = m (x - x 1)
` m2 = -

9t 2
1
= - (x + 9t) t 2
2ty + 9t 3 = -2 (x + 9t)
= -2x - 18t
2x + 2ty + 9t 3 + 18t = 0
`

y+

10. x + ty = at 3 + 2at

11. 3x - 4y + 4 = 0

12. Substitute focus ^ 0, -1 h into equation
3x + 4y + 4 = 0.
13. Equation of chord
1
y - ^ p + q h x + apq = 0
2
Substitute ^ 0, a h into equation
1
a - ( p + q) 0 + apq = 0
2
a + apq = 0 apq = - a pq = - 1

(2)

- px + qx + ap 2 - aq 2 = 0 x (q - p) - a (q 2 - p 2) = 0 x (q - p) - a (q + p) (q - p) = 0 x - a (q + p) = 0 x = a (q + p)

12 x - x1 i x1 _

(i) (i) xx 1 = -22 _ y + y 1 i

(1)

] 1 g - ] 2 g:

4 x - x1 i x1 _

(h) (i) xx 1 = -12 _ y + y 1 i

5.

15. Equation of tangent at P: y - px + ap 2 = 0
Equation of tangent at Q: y - qx + aq 2 = 0

2 x - x1 i x1 _

(g) (i) xx 1 = -4 _ y + y 1 i

14. ^ - 2, -1 h

(b) N = _ 0, ap 2 + 2a i

1
1
(b) N = c - , - m
4
32

(b) ^ - 6, 3 h c 2,

1 m 3

20. (a) F = ^ 0, 6 h
(b) 3x + 4y - 24 = 0
(c) Q = ^ - 24, 24 h

(d) P : x - 2y - 3 = 0; Q: 2x + y + 24 = 0
1
(e) m 1 m 2 = # -2 = -1, ` tangents at P, Q are
2
perpendicular
(f) R = ^ -9, -6 h
(g) directrix: y = - a = - 6, ` R lies on directrix
21. P = ^ -2, -1.5 h
22. x - y + 9 = 0
23. m 1 m 2 = pq
= - 1 (since pq = - 1 for focal chord)
` tangents are perpendicular
24. Tangents intersect at 6 a ^ p + q h, apq @
i.e.
y = apq
= - a (since pq = - 1 for focal chord)
Directrix: y = - a
` tangents meet on the directrix
25.

y=

x2
4a

dy x =
2a
dx
At P _ x 0, y 0 i, dy dx

=

x0
2a

ANSWERS

(c) Directrix y = -3
Point of intersection = ^ - 8, -3 h
So the point lies on the directrix.

The equation is given by y - y 1 = m(x - x 1) x0 ` y - y0 =
(x - x 0)
2a
2ay - 2ay 0 = x 0 (x - x 0)

6.
7.

= xx 0 - x 02
= xx 0 - 4ay 0 (since x = 4ay 0)
2ay + 2ay 0 = xx 0
2a(y + y 0) = xx 0
2
0

(a) 3x + 4y - 8 = 0
(b) Q = d 2,

y - px + 2p 2 = 0; y - qx + 2q 2 = 0; y = - 2

8.

x 2 = 16 ^ y - 6 h

dy dx =

12. (a) PO has gradient

1 n 2

dy

x
4

=

1
2

FR = p + 1
= PF

5.

#

(b) y = - 6a

Test yourself 11
2. x 2 - 4x - 8y - 4 = 0

(a) y - tx + 3t 2 = 0
(b) Y = _ 0, - 3t 2 i
(c) F = ^ 0, 3 h
TF = FY = 3 ^ t 2 + 1 h
(a) y + qx - 5q = 0
(b) R = _ 0, 5q 2 i
(c) F = ^ 0, - 5 h
FR = FQ = 5 _ q 2 + 1 i
So triangle FQR is isosceles.
` +FQR = +FRQ (base angles of isosceles triangle)
2

(a) 4x + 3y - 9 = 0
(b) Focus (0, 3)
Substitute into equation:
LHS = 4 ] 0 g + 3 ] 3 g -9
=0
= RHS
So it is a focal chord.

1.

8x + 6y - 29 = 0

3.

Centre ^ 3, 1 h, radius 4

5.

(a) ^ 8, 8 h

6.

x 2 + y 2 = 25

8.

(a) y - px + p 2 = 0
(b) p 2 + 1
(c) R = _ 0, - p 2 i and F = ^ 0, 1 h
2

4.

q
2

15. (a) x 2 = 9a ^ y - 5a h

= -1
So the tangents are perpendicular.

3.

p

2

14. (a) T = 6 a ^ p + q h, apq @

2
4
dx
1
` m2 =
2

2.

; QO has gradient

(c) x 2 = 2a ^ y - 4a h is a parabola in the form
(x - h) 2 = 4a 0 ^ y - k h where ^ h, k h is the vertex and a 0 is the focal length a ` vertex is ^ 0, 4a h and focal length is
2
a
13. x 2 = 2ay - a 2 or x 2 = 2a d y - n
2

1 m: 2

m 1 m 2 = -2 #

p
2

q
= -1
2
pq = - 4
`
(b) x 2 = 2a ^ y - 4a h m1 m2 =

At P (−8, 8): dy -8
=
4 dx ` m 1 = -2
At q c 2,

(b) x 2 = 2a ^ y - a h

10. (a) y = -a

(c) (−3, −2)
(d)

9. x 2 = 2a ^ y - a h

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

Exercises 11.10
1.

x 2 = 2 a ^ y - 2a h

x 2 + x + y 2 - 3y - 10 = 0

4. (a) ^ 1, - 3 h

(b) ^ 4, - 3 h

(b) 2x - y - 8 = 0

10. (a) (i) ^ 1, 1 h

(b) ^ 0, - 2 h

7. (a) y = 2

(ii) ^ 1, 2 h

9. x 2 - 8x + 16y - 16 = 0

(b) y = 0

11. 2x + 3y + 6 = 0

12. 14 units

13. y = - 24x

2

2

14. x - 8y + 16 = 0

15. 4x - 3y - 16 = 0, 4x - 3y + 14 = 0
16. y = x, y = - x
19. (a) x 2 = 12y

17. y 2 = 20x

18. (a) -

1
2

(b) 2

(b) y 2 = - 32x

20. (a) x - 4y + 72 = 0

1
(b) d 9, 20 n
4

21. Sub ^ 0, 4 h : LHS = 7 # 0 - 3 # 4 + 12 = 0 = RHS
2
22. d , -7 n
9

23. 3x - 2y + 40 = 0

24. x 2 - 10y + 100 = 0

25. y - 3x + 9a = 0

26. (a) x - y - 3 = 0 (b) R = ^ 0, -3 h
(c) F = ^ 0, 3 h FP = FR = 6

833

834

Maths In Focus Mathematics Extension 1 Preliminary Course

27. (a) y a-

1
^ p + q h x + apq = 0
2

1
(p + q) # 0 + apq
2
a + apq apq pq

(b) Sub ^ 0, a h:

=0

6ap + 4aq 3ap 2 + 2aq 2 p ,
5
5

(b) x 2 = 2a ^ y + 2a h

=0
= -a
= -1

19. y = 0
20. (a) T = ^ 6, - 20 h

28. x - 2y + 48 = 0
(b) ^ 6, 9 h and ^ - 2, 1 h

29. (a) x - y + 3 = 0

18. (a) N = f

m x = t and m y = s m1 - m2 tan i =
1 + m1 m2 t-s tan 45c =
1 + ts t-s 1=
1 + ts t-s 1=
1 + ts
1 + ts = t - s
1 + s = t - ts
= t (1 - s)
1+s
=t
1-s
t-s
-1 =
1 + ts
-1 - ts = t - s s - 1 = t + ts
= t (1 + s) s-1 =t
1+s

(c)

30. y = - a

Challenge exercise 11
1.

(a) 8x + 6y - 29 = 0
(b) Midpoint of AB lies on line; m 1 m 2 = -1

2.

(a) x 2 - 2x + y 2 - 6y - 15 = 0
(b) Put y = 0 into equation

3.

y = 1 - 2x 2

5.

(a) 4x - 2y + 9 = 0; x + 2y - 24 = 0
(b) m 1 m 2 = - 1 (c) X = ^ 3, 10.5 h
(d) 3x - 4y + 8 = 0; focus ^ 0, 2 h lies on the line

6.

^ 0, 0 h

7.

(a) 2x - 4y - 1 = 0; 2x + y + 4 = 0
(b) Point lies on line y = - 1

8.

y = - 2 x 2 + 4x - 2

1
4. d 2 , - 3 n
2

or

9. 3x + y + 2 = 0

10.

(b) P = 6 a ] t + s g, ats @

Practice assessment task set 3
1.

m ≤ 2, m ≥ 3

4.

24 cm

2
6. (a)
3

2. 4x + 3y - 16 = 0

3. x 2 = 8y

Centre ^ - 3, 5 h, radius 7

5.

1
(b) 3

(c) 1

1
9

7.

Focus ^ 0, -2 h, directrix y = 2

8.

x = - 5 or - 6

10.

9. k = - 1

+AFE = +CBE (ext.+ equal to opp. interior
+ in cyclic quadrilateral)

+CBE = 180c - +EDC (opp.+s supplementary in cyclic quadrilateral) 11. (a) x 2 + 4x + y 2 - 10y + 21 = 0
(b) ] x + 2 g2 + ^ y - 5 h2 = 8; centre ^ -2, 5 h; radius =
12. -

8 =2 2

11. 3x - 4y - 14 = 0, 3x - 4y + 16 = 0

2 3
3

12. Vertex ^ - 4, -17 h , focus ^ - 4, -16.75 h

13. (a) y + 4y - 16x + 52 = 0
2

14. 4 2 units

` +AFE = 180c - +EDC
These are supplementary cointerior angles.
` AF || CD

(b) 2x - y - 6 = 0

15. x + y - 2y - 2 = 0
2

2

16. 696 mm from the vertex
17. 141x + 127y + 32 = 0; 219x + 23y + 58 = 0

13. x = 0, 3

14. k = 7.2 cm

16. b $ -2 17. i = 16c
^ 0, 0 h and radius 4

15. x + 2y + 2 = 0

18. x 2 + y 2 = 16, circle centre

19. x 2 + 4x + y 2 + 6y - 12 = 0

ANSWERS

Obtuse +AOC = 2+ADC (+at centre double + at circumference)
Reflex +AOC = 2+ABC (similarly)
Obtuse +AOC + reflex +AOC = 360c (+ of revolution)
`
2+ADC + 2+ABC = 360c
+ADC + +ABC = 180c
It can be proved similarly that
+BAD + +BCD = 180c by drawing BO and DO.
` opposite angles in a cyclic quadrilateral are supplementary 20. x 2 - 3x + y 2 - 6y - 17 = 0
21.

+BCD
+DAB
+DBC
+BDC
`+BDC
` +DAE

22. - 0.75

= 90c (+in semicircle)
= 90c (similarly)
= +DAE (+s in same segment)
= 90c - +DBC (+ sum of D BDC)
= 90c - +DAE
= 90c - +BDC

23. 5x 2 - 54x + 5y 2 + 20y - 79 = 0

24. a = 2, b = 1, c = 0
26. -

25. x = 33°, y = 57°

x

37. Centre ^ -5, 3 h, radius 2
38.

9 - x2

27. (a) y - px + ap 2 = 0

(b) R = f

a _ p2 - 1 i p , - ap

(c) 2px + _ p 2 - 1 i y + a - ap 2 = 0
28. x 2 - 4x - 16y + 20 = 0
29.
`

AC = BC and CD = CE (given)
AC
BC
=
CD
CE
+ACB = +ECD (vertically opposite angles)

` since two sides are in proportion and their included angles are equal, ΔABC is similar to ΔCDE y = 5.3 cm
30. x - y - 4 = 0

39. a 2 0
D = b 2 - 4ac
= ] -1 g2 - 4 (1) (3)
= -11
10
Since a 2 0 and D 1 0, x 2 - x + 3 2 0 for all x

31. x 2 + 2x - 16y - 15 = 0
33.

Let +DBA = x and +EBC = y
Then +EDB = x and +DEB = y (alternate +s, DE < AC)
+FDE = 180c - x (+FDB straight +)
+GED = 180c - y (+GEB straight +)
+FGB = +DBA = x (+s in alternate segment)
+GFB = +EBC = y (similarly)
` +FDE = 180c - +FGB and
+GED = 180c - +GFB
Since opposite angles are supplementary, FGED is a cyclic quadrilateral.

32. x = 0, 2

a 10
D = b 2 - 4ac
= 1 2 - 4 (- 1) (- 9)
= - 35
10
Since a 1 0 and D 1 0, - x 2 + x - 9 1 0 for all x

34. 8 (3x - 1) (2x + 5) + 3 (2x + 5) = ] 30x + 7 g (2x + 5) 3
3

4

35. sec x cosec x
36.

40. k = 1

41. 3x + 2y - 9 = 0

42. (a) 217 km

(b) 153c

43. a = 3, b = - 18, c = - 34
45. (a) y = x 2 - 1

44. x 2 4, x 1 3

(b) ^ - 4, 15 h

(c) x - 8y + 124 = 0

46. i = 95c 44’ 47. x = 11c
48. T = 361 ^ 2 0 and a perfect square h
49. x + 2y + 9 = 0

50. k # 3

51.

Let ABCD be a cyclic quadrilateral of circle, centre O.
Join AO and CO.

52. 5x - 4y - 41 = 0

55. - 1

1
# y 1 -1
4

1+ 3
2
2
53. d 3 , -2 n 54.
5
5
3-1
56.

3 6 - 10 + 3 3 - 5
22

835

836

Maths In Focus Mathematics Extension 1 Preliminary Course

57. x = 4.9 cm, y = 11.1 cm
60. 4.5 m

128
2187

61.

59. 8.25 units

1
1
64. y = 1 , 3
2

65. 162c

b 2 - 4ac = -104 1 0
So Pl(x) has no real roots
15. Ql] x g = 3x 2 - 6x + 3 b 2 - 4ac = 0
So Ql] x g has equal roots

+ACB = 90c (+ in semicircle)
`+DCA = 90c (+DCB straight +)
` AD is a diameter of the circle

Exercises 12.2

67. x = 45°, 135°, 225°, 315°

1.

1
1
68. x = - 1, y = 2 or x = - , y = 4
4
4

69. ] a - 2b g ^ a 2 + 2ab + 4b 2 h

70. x = 43

71. -

1
31

73. tan i

72. 1.8 units

14. Pl] x g = 3x 2 - 2x + 9

62. x = 60°, 120°, 240°, 300°

63. 2x + 3y - 3 = 0
66.

58. x = 1

3x 2 + 2x + 5 = ] x + 4 g ] 3x - 10 g + 45

2.

x 2 - 7x + 4 = ] x - 1 g ] x - 6 g - 2

3.

x 3 + x 2 + 2x - 1 = ] x - 3 g ^ x 2 + 4x + 14 h + 41

74. 8x ] 2x + 5 g (x 2 - 1) 3 + 2 (x 2 - 1) 4
= 2 (x 2 - 1) 3 (9x 2 + 20x - 1)

4.

4x 2 + 2x - 3 = ] 2x + 3 g ] 2x - 2 g + 3

5.

x 3 - 5x 2 + x + 2 = ^ x 2 + 3x h ] x - 8 g + ] 25x + 2 g

1
75.
4

6.

x 3 + x 2 - x - 3 = ] x - 2 g ^ x 2 + 3x + 5 h + 7

77. Focus (2, 1), directrix y = 5

7.

5x 3 - 2x 2 + 3x + 1 = ^ x 2 + x h ] 5x - 7 g + ] 10x + 1 g

78. px - y - 9p 2 = 0

8.

76. x + 2x + y - 3y - 25 = 0
2

2

79. x - 2y - 36 = 0

1
^ p + q h x + apq = 0 (b) x 2 = 2a ^ y - 2a h
2
(c) Concave upward parabola, vertex (0, 2a)

80. (a) y -

81. (c)

82. (d)

83. (b)

88. (a)

87. (c)

84. (a)

89. (a), (d)

85. (c)

86. (a)

9.

x 4 - x 3 - 2x 2 + x - 3
= ] x + 4 g ^ x 3 - 5x 2 + 18x - 71 h + 281
2x 4 - 5x 3 + 2x 2 + 2x - 5 = ^ x 2 - 2x h ^ 2x 2 - x h + ] 2x - 5 g

10. 4x 3 - 2x 2 + 6x - 1 = ] 2x + 1 g ^ 2x 2 - 2x + 4 h - 5
11. 6x 2 - 3x + 1 = ] 3x - 2 g d 2x +

90. (c)

1
2
n+ 1
3
3

12. x 4 - 2x 3 - x 2 - 2 = ^ x 2 - x h ^ x 2 - x - 2 h + ] -2x - 2 g
Chapter 12: Polynomials 1

5
4
3
2
13. 3x - 2x - 3x + x - x - 1
4
3
= ] x + 2 g ^ 3x - 8x + 13x 2 - 25x + 49 h - 99

Exercises 12.1

14. x 2 + 5x - 2 = ] x + 1 g ] x + 4 g - 6

1.

(a) 7 (b) 4 (c) 1

2.

(a) -19 (b) -10 (c) -1
3. (a) -6 (b) 5
(c) 2 (d) 1 (e) 2
4. (a) 5 (b) 4 (c) -3 (d) 0

5.

(a) !3

6.

7.

(b) -5

(d) 11 (e) 3 (f) 0 (g) 4

(c) -2, 1

(d) 4 (e) 0

(a) P l ] x g = 12x 3 - 6x 2 - 2x + 4; 3 (b) Pl] x g = 10x; 1
(c) Pl] x g = 108x 11 - 35x 4 + 8; 11
(d) P l ] x g = 7x 6 - 9x 2 + 2x - 7; 6 (e) Pl] x g = 8; 0
(a), (b), (g) 8. (a) a = 0
(d) a = -1
(c) 3

10. (a)

(d) 3

(e) a = 4

(b) b = 10
1
9. (a) -2
2

(c) c = -6
(b) x = 2, -1

(e) x 5

(d) 9

11. (a) 2 (b) 0 (c) 2
12. x = -3, 2

17. x 3 - 3x 2 + 3x - 1 = ^ x 2 + 5 h ] x - 3 g + ] - 2x + 14 g
18. 2x 3 + 4x 2 - x + 8 = ^ x 2 + 3x + 2 h ] 2x - 2 g + ] x + 12 g
4
3
2
19. x - 2x + 4x + 2x + 5
= ^ x 2 + 2x - 1 h ^ x 2 - 4x + 13 h + ] - 28x + 18 g

20. 3x 5 - 2x 3 + x - 1 = ] x + 1 g ^ 3x 4 - 3x 3 + x 2 - x + 2 h - 3

Exercises 12.3

(e) x =

(d) 0

13. x = 0, 1

(e) 2 (f) 4

(g) 3

2
(a) k = 8 (b) k =
7
(e) k = !2

3.

2
, -1
3

(a) 41 (b) -3 (c) -43 (d) 9424 (e) 0
(g) 47 (h) 2321 (i) 31 174 (j) - 3

2.

` f ] x g has no zeros
(c) - 2

16. 2x 4 - x 3 + 5 = ^ x 2 - 2x h ^ 2x 2 + 3x + 6 h + ] 12x + 5 g

1.

D = b 2 - 4ac
= -8
-8 1 0

(b) 9x 3

15. x 4 - 2x 2 + 5x + 4 = ] x - 3 g ^ x 3 + 3x 2 + 7x + 26 h + 82

(a) 0 (b) Yes
(c) x 3 - 4x 2 + x + 6 = ] x - 2 g ^ x 2 - 2x - 3 h
(d) f ] x g = ] x - 2 g ] x - 3 g ] x + 1 g

(c) k = 15 299

(f) 37

(d) k = 6

8
9

ANSWERS

4.

5.
7.

9.

(a) P ] - 3 g = 81 - 81 - 81 + 81 = 0
` x + 3 is a factor
(b) P ] x g = x ] x + 3 g 2 ] x - 3 g
7
17 a = -1 , b = -1
12
48

Exercises 12.4
1.

y

(a)

6. a = - 6

(a) P ] 3 g = 140 ! 0
` x - 3 is not a factor of P ] x g
(b) k = - 39 8. a = -2, b = -1

6

-1

(a) a = 3, b = 11
(b) f ] x g = ] x + 1 g ^ 3x 3 + 8x 2 + 7x + 2 h
(c) g ] -1 g = 0 (d) f ] x g = ] 3x + 2 g ] x + 1 g3

10. (a) ] x + 2 g ] x - 4 g (b) x ] x + 2 g ] x - 1 g
(c) ] x - 1 g ] x + 4 g ] x - 2 g (d) ] x + 5 g ] x - 3 g ] x + 2 g
(e) ] x - 3 g ] x - 1 g ] x - 7 g (f) ] x + 2 g ] x - 9 g ] x - 5 g
(g) ] x - 3 g ] x - 2 g2 (h) x ] x + 4 g ] x + 1 g 2
(i) ] x - 1 g ] x + 2 g2 (j) ] x + 1 g ] x - 3 g ] x + 2 g
11. (a) P ] x g = ] x - 1 g ] x + 3 g ] x - 2 g (b) -3, 1, 2

2

x

3

y

(b)

(c) Yes

12. (a) Dividing f ] x g by ] x + 5 g ] x - 2 g gives f ] x g = ] x + 5 g ] x - 2 g ^ x 2 + 7x + 12 h
` ] x + 5 g ] x - 2 g is a factor of f ] x g
(b) f ] x g = ] x + 5 g ] x - 2 g ] x + 3 g ] x + 4 g

-4

x

2

13. P ] x g = ] x + 1 g ] x - 4 g ] x + 3 g2
14. (a) P ] -6 g = P ] 5 g = 0

(b) P ] x g = ] x - 4 g ] x + 6 g ] x - 5 g

15. (a) P ] u g = ] u - 2 g ] u - 1 g 2

(b) x = 2, 3

16. (a) f ^ p h = ^ p - 1 h ^ p + 2 h ^ p - 3 h
17. (a) P ] k g = ] 2k - 1 g ] k + 1 g2

y

(c)

1
(b) x = 0, -1 , 1
2

(b) x = 30c , 150c , 270c

18. (a) f ] u g = ] u - 1 g ] u - 3 g ] u - 9 g (b) x = 0, 1, 2
19. x = -5, -4, -2

1

3

x

20. i = 0c, 90c, 120c, 240c, 270c, 360c
21. (a) a = 1, b = 3, c = 4, d = - 2
(b) a = 1, b = -1, c = 8, d = -12
(c) a = 2, b = 0, c = -1, d = 6
(d) a = 1, b = 1, c = 11, d = -12
(e) a = 3, b = 0, c = -1, d = 8
(f) a = 1, b = 1, c = -4, d = -7
(g) a = 5, b = -2, c = -19, d = -43
(h) a = -1, b = 4, c = -1, d = -1
(i) a = -1, b = 3, c = 6, d = -4
(j) a = -1, b = -10, c = -27, d = -20
22. P ] x g = x 3 - x 2 - 12x

23. a = 1, b = -3, c = -6

24. P ] x g = 2x 4 - 4x 3 - 10x 2 + 12x
25. P(x) has degree 3.
Suppose P(x) has 4 zeros, a1, a2, a3 and a4.
Then _ x - a 1 i _ x - a 2 i _ x - a 3 i _ x - a 4 i is a factor of P(x).
So P ] x g = _ x - a 1 i _ x - a 2 i _ x - a 3 i _ x - a 4 i Q ] x g.
` P(x) has at least degree 4
But P(x) only has degree 3.
So it cannot have 4 zeros.

y

(d)

-2

x

837

838

Maths In Focus Mathematics Extension 1 Preliminary Course

(d) (i) A ] x g = x ] 2x - 5 g ] x + 3 g

y

(e)

-5

y

(ii)

50

-2

x

5

2.

(a) (i) P ] x g = x ] x - 4 g ] x + 2 g

(e) (i) P ] x g = - x 2 ] x - 3 g ] x + 1 g

y

(ii)

x

1
22

-3

y

(ii)

-2

x

4

x

3

-1

(b) (i) f ] x g = - x ] x - 1 g ] x + 5 g
(ii)

3.

y

(a) x = 0, 1, -2 y (b)

-5

x

1

-2

(c) (i) P ] x g = x 2 ] x + 1 g ] x + 2 g

4.

y

(ii)

x

1

(a) P ] 2 g = 8 - 12 - 8 + 12
=0
(b) P ] x g = ] x - 2 g ] x - 3 g ] x + 2 g y (c)
12

-2

-1

x
-2

2

3

x

ANSWERS

5.

y

(a)

-4

(e)

-2

y

x

3

-24

-3

x

-18 y (f)

y

(b)

3

2

-3

-1

3

x

2

-2

x

-8
-9
y

(c)

y

(g)

12

1

3

4

1

x

2

x

-4

y

(d)

y

(h)

12
3

-4

1

3

x
-3

1

x

839

840

Maths In Focus Mathematics Extension 1 Preliminary Course

(i)

(c) P ] 4 g = ] 4 + 1 g ] 4 - 4 g2
=0
Pl(x) = 3x 2 - 14x + 8
Pl(4) = 3 ] 4 g2 - 14 (4) + 8
=0

y

6.

4

-2

x

7. y (j)

1

1

-1

x

8.

(a) ] x + 3 g 3 = x 3 + 9x 2 + 27x + 27
Dividing by x 3 + 9x 2 + 27x + 27 gives x 4 + 7x 3 + 9x 2 - 27x - 54 = ^ x 3 + 9x 2 + 27x + 27 h ] x - 2 g so ] x + 3 g 3 is a factor
(b) f ] x g = ] x - 2 g ] x + 3 g 3
(c) f ] -3 g = ] -3 - 2 g ] -3 + 3 g 3
=0
f l(x) = 4x 3 + 21x 2 + 18x - 27 f l(-3) = 4 ] -3 g3 + 21 ] -3 g2 + 18 (-3) - 27
=0
(a) P ] x g = ] x - k g3 Q ] x g where Q(x) has degree n - 3
(b) P ] k g = ] k - k g3 Q ] k g
=0
Pl(x) = ulv + vlu
= Ql(x) ] x - k g3 + 3 ] x - k g2 Q (x)
Pl(k) = Ql(k) ] k - k g3 + 3 ] k - k g2 Q (k)
=0

(a)

y

Exercises 12.5
1.

2.

(a) x = 3, double root (b) x = 0, 2, 7, single roots
(c) x = 0, double root, x = 3, single root
(d) x = - 2, single root, x = 2, double root
(e) x = - 2, triple root (f) x = 0, 2, single roots, x = 1, double root
(g) x = - 1, 3, double roots (h) x = 0, triple root, x = 4, double root (i) x = 1, triple root, x = - 5,
1
single root (j) x = 1 , triple root
2

(b)

y

(a) (i) Positive (ii) Even (b) (i) Negative (ii) Odd
(c) (i) Negative (ii) Even (d) (i) Negative (ii) Odd
(e) (i) Positive (ii) Odd (f) (i) Positive (ii) Even
(g) (i) Positive (ii) Odd (h) (i) Negative (ii) Even
(i) (i) Positive (ii) Odd (j) (i) Positive (ii)Even

3.

P ] x g = ] x + 4 g 2 Yes, unique

4.

(a) P ] x g = k ] x - 1 g Not unique

5.

x

3

x
(b) P ] x g = 5 ] x - 1 g3

(a) ] x - 4 g2 = x 2 - 8x + 16
Dividing by x 2 - 8x + 16 gives x 3 - 7x 2 + 8x + 16 = ^ x 2 - 8x + 16 h ] x + 1 g so ] x - 4 g 2 is a factor
(b) P ] x g = ] x + 1 g ] x - 4 g2

ANSWERS

(c)

10.

y

x

x

-1

11.

y

(d)

y

y

x

2

x

y

12. y (e)

x

-3

x

13.
9.

y

y

1

2

x

x

841

842

Maths In Focus Mathematics Extension 1 Preliminary Course

17. Odd function with positive leading coefficient starts negative and turns around at the double root. It then becomes positive as x becomes very large so it must cross the x-axis again. So there is another root at k 2 -1

y

14.

y

x

15.

x

k

-1

y

x

-2

18. Even function with negative leading coefficient is negative at both ends. The triple root has a point of inflexion so the curve must cross the x-axis to turn negative again. So there is another root at k 2 -2 y 16.

y

4

x

k

-2

x
19. Odd function with positive leading coefficient starts negative and turns around at both the double roots.
It then becomes positive as x becomes very large so it must cross the x-axis again. So there is another root at k 22 y -3

2

k

x

ANSWERS

20. Odd function with negative leading coefficient starts positive and turns around at the double root. It then becomes negative as x becomes very large so it must cross the x-axis again. So there is another root at k 2 1

Test yourself 12 p ]xg = ]x + 3g]x - 3g]x + 5g]x - 1g

2.

(a) 3

3.

P (x) = (x - 6) (x - 1) (x + 2)
= x 3 - 5x 2 - 8x + 12

4.

(a) x 2 + 3x + 2

5.

y

1.

(a) 3 (b) - 3

(b) 9

(c) 1

(d)

1
9

(b) p ] x g = ] x - 5 g ] x + 3 g ] x + 1 g ] x + 2 g
(c) - 3, 0, 1

(d) x 3

6. k 1

x

Exercises 12.6
1.

(b) (i) -2 (ii) -

(a) (i) 2 (ii) 8
(d) (i) 2

1
(ii) -3
4

2
3

(c) (i) -7 (ii) 1

(e) (i) -3 (ii) 0

2. (a) (i) -1 (ii) -2

1
(c) (i) (ii) 3
2

(iii) -8
(iii) -1
3.

(b) (i) 3 (ii) 5 (iii) 2

(d) (i) -3 (ii) 0 (iii) -11 (e) (i) 0 (ii) 7 (iii) 3

(a) (i) -2 (ii) -1 (iii) 1 (iv) 5

(b) (i) 1 (ii) -3

(iii) -2 (iv) -7

(c) (i) 1 (ii) -3 (iii) -2 (iv) -4
1
(d) (i) 1 (ii) -2 (iii) -1 (iv) -1 (e) (i) 6 (ii) 0
2
1
(iii) 0 (iv) 3
4. (a) 5 (b) -5 (c) -1 (d) 35
2
3
5
1
1
(e) 200
5. (a)
(b) (c) (d) 2
2
2
3
1
(e) 2
2
6. (a) - 3 (b) - 5 (c) 1
7. k = - 26
3
1
1
8. a + b = 2, ab = -7 9. a + b = 2 , ab = 2
2
1
10. (a) k = 0 (b) k = 4 (c) k = ±1 (d) k = - , 1
2
3
1
(e) k = 0
11. m = -9
12. a = -1 , b = -9
4
8
13. (a) P ] 1 g = 0

15. (a)

4
2
, q = -17
15
15

1 1
18. x = - , 1
2 2

(a) a = 3

(b) - 5

8.

p (-7) = ] -7 g3 - 7 ] -7 g2 + 5 (-7) - 4
= - 725 ! 0

9.

x = - 1, !3

10. a = 2, b = - 18, c = 40

11. x-intercepts -3, 2, 4; y-intercept 24
12. 3x 5 - 7x 3 + 8x 2 - 5
= (x - 2) (3x 4 + 6x 3 + 5x 2 + 18x + 36) + 67
13. x = 60c, 90c, 180c, 270c, 300c
15. 4, 5
16.

(b) a + b + c = 1, abc = -6

14. a = 1; a + b = -2
(b) p = 8

7.

19. x =

4
15

16. 1
1
1
,!
3
2

17. -5
1 2
20. x = ! 3 , -1 ,
2 3

17. k = - 14

18. 4

14. k = 7.4

843

844

Maths In Focus Mathematics Extension 1 Preliminary Course

19. P (a) = A (a) ] a - a g3
=0
Pl(x) = A (x) 3 ] x - a g2 + Al(x) ] x - a g3
Pl(a) = A (a) 3 ] a - a g2 + Al(a) ] a - a g3
=0

y

24.

20. f ] 5 g = 5 3 - 6 ] 5 g 2 + 12 ] 5 g - 35 = 0 x 21. (a) f ] 5 g = 5 3 - 7 ] 5 g 2 - 5 ] 5 g + 75
=0
(b) f l] x g = 3x 2 - 14x - 5 f l(5) = 3 ] 5 g2 - 14 (5) - 5
=0
(c) Double root at x = 5 (d) f ] x g = ] x + 3 g ] x - 5 g 2 y 22.

25. (a) a = 2, b = - 3, c = 4, d = 5

Challenge exercise 12
1.
3

x

2.

P ] x g = ] x - 1 g ] x + 1 g2 ^ x 2 + x + 1 h
(a) P (b) = ] b - b g7 Q (b)
=0
Pl(x) = ] x - b g7 Ql(x) + Q (x) 7 ] x - b g6
Pl(b) = ] b - b g7 Ql(b) + Q (b) 7 ] b - b g6
=0
(b) a = -7, b = - 1

23. (a) P ] x g = ] x + 6 g 3Q ] x g
3.

y

i = 0c , 45c , 60c , 120c , 180c , 225c , 240c , 300c , 360c

4.

(b)

(a) 3x - y + 2 = 0

(b) ^ 2, 8 h

5. (a)

a - 33
4

(b) a = -14
6.

(a) - 3

(b) 17

7. i = 90c, 210c, 330c

8. a = - 5

If x - a is a factor of P ] x g
Then P (x) = (x - a) Q (x)
P (a) = (a - a) Q (a)
`
=0
10. ^ - 1, -1 h, ^ -3, 5 h 11. P ] x g = - ] x + 1 g 2 ] x - 2 g 3
9.

-6

x

12.

y

a1

a2

x

ANSWERS

Exercises 13.1
16
33

1.

1
10 000

2.

5.

98.5%

6. (a)

8.

1
3

1
9. (a)
6

2
9

3.
3
7

7.

1
(b)
3

4
7

1
20 000

4.

5
(c)
6

(b)

3
20

6.

(a) 39 916 800

8.

Chapter 13: Permutations and combinations

5040

9. 6

12. 1.3 # 10 12
14. (a) 720
16. (a)

1
4

10. 720

13. (a) 39 916 800

(b) 120
(b)

1
62

(b)

3
31

(c)

1
2

11. (a)

1
15

(b)

8
15

(c)

3
5

13. (a)

29
86

(b)

19
43

(c)

67
86

15. (a)

1
6

(b)

4
5

(b) 36

17. (a)

20.

329
982

22. (a)

14
59

1
2

1
3

(c)

(b)

24. 19%

35
59

19
31

(c)

20. (a)

14. 32

(d)

1
2

24
59

(b)

20
31

(b)

(d)

(b)

1
2

(e)

23
44

16. (a)
21
44

(c)

38
59

4
31

23.

19.

(d)

5
18

(b) 91
(c)

6
19

11
31

5
24

(d)

1
24

5.

1
67 600 000

14.

1
17.
5184

18. 6

21. 7 880 400

n ] n - 1 g ] n - 2 g ... ] n - r + 1 g ] n - r g ... 3.2.1 n! =
]n - r g!
] n - r g ] n - r - 1 g ] n - r - 2 g ... 3.2.1
= n ] n - 1 g ] n - 2 g ... ] n - r + 1 g

5!
= 20
]5 - 2 g!

1
9900

7!
= 2520
]7 - 5 g!

11!
= 6 652 800
] 11 - 8 g !

(g)

(e)

8!
= 336
]8 - 3 g!

(c)

9!
= 60 480
]9 - 6g!

8!
= 20 160
]8 - 6g!
(i)

9!
=9
]9 - 1 g!

6!
= 720
]6 - 6 g!

(a) 720 (b) 3 628 800 (c) 1 (d) 35 280 (e) 120
(f) 210 (g) 3 991 680 (h) 715 (i) 56 (j) 330
4. 479 001 600

5. 120

3.

(a) 648 (b) 432 (c) 144 4. (a) 20 (b) 4 (c) 12 (d) 8

5.

(a) 24

7.

(a) 120

(a) 479 001 600 (b) 1320

9.

1
720

(a) 650

8.

20. 360

23. 271 252 800
25.

(b)

2.
16. Yes

Exercises 13.3

3. 720

10!
= 640 800
] 10 - 7 g !

(j)

15. 7

19. 6840

22. 210

24. (a) 9900 (b)

362 880

n ] n - 1 g ] n - 2 g ... ] r + 1 g r ] r - 1 g ... 3.2.1 n! = r! r ] r - 1 g ] r - 2 g ... 3.2.1
= n # ] n - 1 g # ] n - 2 g #...# ] r + 1 g
= n (n - 1) (n - 2) #...# (r + 1)

6!
= 120
]6 - 3 g!

7. 1 000 000

12. (a) 10 000 000 (b) 1000

13. Yes

2.

11!
11 # 10 # 9 #...# 2 # 1
=
6#5#4#3#2#1
6!
= 11 # 10 # 9 # 8 # 7

(a)

4. 260

3
10.
10 000

9. 64

1
120

8!
8 # 7 # 6 #...# 2 # 1
=
4#3#2#1
4!
=8#7#6#5

(h)

3. 26 5 # 10 4

6. 1 000

1
11. (a) 84 (b)
84

1.

18.

(f)

1.

2. 67 600

26 10 # 10 15

8. 300

1
12

(d)

25. 0.51

456 976

(b)

Exercises 13.4

Exercises 13.2
1.

(b) 3 628 800

19. 6 227 020 800

12. 8

18. (a)

21. (a)

99
124

(d)

11. 5040

15. 5040

(c) 48

17. (a) 479 001 600
10. (a)

7. 40 320

(b) 479 001 600

(a) 56 (b) 336 (c) 1680

(b) 15 600

(c) 358 800

6. (a) 4536

(b) 24
(b) 48

(c) 96

(d) 7 893 600

(b) 2016

(d) 72

(c) 3528

(e) 60

10. (a) 60 480 (b) 2520 (c) 907 200 (d) 151 200
(f) 453 600 (g) 360 (h) 2520 (i) 59 875 200
(j) 90 720
11. (a) 24 (b) 5040
(e) 39 916 800
12. (a) 6

(b) 720

(c) 40 320

(c) 5040

(e) 60

(d) 3 628 800

(d) 362 880

(e) 3 628 800

845

846

Maths In Focus Mathematics Extension 1 Preliminary Course

13. (a) 181 440 (b) 19 958 400 (c) 20 160
(d) 1 814 400 (e) 239 500 800

n!
' (n - r) !
]n - n + r g! n! 1
=
#
]n - r g! r! n!
=
] n - r g !r! n n
Pn - r
Pr
=
]n - r g! r! =

14. (a) 720 (b) 120 15. (a) 362 880 (b) 40 320
16. (a) 3 628 800 (b) 362 880 (c) 181 440
17. (a) 24 (b) 12 (c) 24

`

18. (a) 720 (b) 240 (c) 480 (d) 144

20.

2
9

21. (a) 20! (b) 5!8!7!3! (c)

22. (a) 60 (b) 48 (c) 36 (d)
24.

1
336

P3

=
=
=
=

P5

5!

=
=
=
=

8

`

=

1
6

=
=

3!

8

Pr + r n Pr - 1 =

27. (a) 720 (b) 120 (c) 192

28. (a) x! (b) ] x - 1 g !
(e) ] x - 3 g ] x - 2 g !

29. (a)

23.

n

7
20

Pr =

25. (a) 40 320 (b) 30 240 (c) 21 600

26. (a) 20 (b) 60

8

1
5

n+1

30.

19. (a) 3 628 800 (b) 362 880 (c) 28 800

P3

3!

3!
8!
' 3!
5!
8!
1
#
5!
3!
8!
5!3!
8!
]8 - 5 g!
5!
8!
' 5!
3!
8!
1
#
3!
5!
8!
5!3!
8

=

=
=
=
=
`

=
=
=
=

Pn - r

]n - r g!

=

n

n

Pr = Pr + r Pr - 1

(a)

9!
= 126
] 9 - 5 g !5!

(c)

8!
= 56
] 8 - 3 g !3 !

(e)

1.

P5

Pr

n+1

Exercises 13.5

11!
= 462
] 11 - 5 g !5!

(b)
(d)

12!
] 12 - 7 g !7!

n!
]n - r g! r! n!
' r!
]n - r g! n! 1
#
]n - r g! r! n!
] n - r g !r! n! (n - 5 n - r ? ) !
]n - r g!

= 792

10!
= 210
] 10 - 4 g !4!

2.

(a) (i) 1 (ii) 1 (iii) 1 (iv) 1 n n
(b) (i) C 0 = 1 (ii) C n = 1

3.

(a) 28

4.

(a) Number of arrangements = 15

5!

r!

n

=

(d) 3! ] x - 2 g !

8!
]8 - 3g!

n

(b)

(c) 2! ] x - 2 g !

]n + 1g!
]n + 1 - r g! n! n!
+r
]n - r g!
^n - 5r - 1?h ! n! n!
+r
]n - r g!
^n - 5r - 1?h ! n! rn!
+
]n - r g! ]n - r + 1g!
] n + 1 - r g n! rn! +
]n + 1 - r g]n - r g! ]n + 1 - r g!
] n + 1 - r g n! rn! +
]n + 1 - r g!
]n + 1 - r g! n $ n! + n! - rn! + rn!
]n + 1 - r g! nn! + n!
]n + 1 - r g!
] n + 1 g n!
]n + 1 - r g!
]n + 1g!
]n + 1 - r g!

(b) 84

(c) 462

(v) 1

(d) 5005

(e) 38 760

R1R2

R2R3

R3B1

B1B2

R1R3

R2B1

R3B2

B1B3

R1B1

R2B2

R3B3

R1B2

R2B3

R1B3
(b) 77 520

B2B3

ANSWERS

5.

15 504

6. 210

8.

(a) 720 (b) 120

10. 296 010

7. 2 598 960
9. (a) 2184 (b) 364

11. 4845

13. 23 535 820

27.

12. 2925

14. (a) 792 (b)

5
12

(c)

5
33

15. (a) 100 947 (b) 462 (c) 924 (d) 36 300 (e) 26 334
(f) 74 613 (g) 27 225
16. $105
17. (a) 2 042 975 (b) 55 (c) 462 462 (d) 30 030
18. (a) 3003 (b) (i) 2450 (ii) 588 (iii) 56 (iv) 1176
19. (a) 1.58 # 10 10
(e) 12 271 512

(b) 286 (c) 15 682 524 (d) 5 311 735

20. (a) 395 747 352 (b) 32 332 300 (c) 4 084 080
(d) 145 495 350 (e) 671 571 264
21. (a) 170 544 (b) 36 (c) 20 160 (d) 17 640 (e) 6300

28.

22. (a) 7 (b) 27 132 (c) 13 860 (d) 20 790 (e) 27 720
23. (a) 5 (b) 360 (c) 126
24. (a) 792 (b) 792
(c)

25.

12!
] 12 - 5 g !5!
12!
=
7!5!
12!
12
C7 =
] 12 - 7 g !7!
12!
=
5!7!
12
12
` C5 = C7
12

9
8

C5 =

C 6 = 84

8

C 6 + C 5 = 28 + 56
= 84
9
8
8
`
C6 = C6 + C5
26.

13! b 13 l =
7
] 13 - 7 g !7!
13!
=
6!7!
13! b 13 l =
6
] 13 - 6 g !6!
13!
=
7!6!
13
13
`b 7 l=b 6 l

29.

10! b 10 l =
4
] 10 - 4 g !4!
10!
=
6!4!
9!
9!
b9 l + b9 l =
+
4
3
] 9 - 4 g !4! ] 9 - 3 g !3!
9!
9!
=
+
5!4! 6!3!
6 # 9!
4 # 9!
=
+
6 # 5!4! 4 # 6!3!
6 # 9! 4 # 9!
=
+
6!4!
6!4!
6 # 9! + 4 # 9!
=
6!4!
10 # 9!
=
6!4!
10!
=
6!4!
b 10 l = b 9 l + b 9 l
`
4
3
4 n! bn l = r ] n - r g !r! n! b n l= n-r (n - 5 n - r ? ) ! ] n - r g ! n! =
]n - n + r g! ]n - r g! n! = r! ] n - r g ! n n
` br l = bn - r l n Pr =

n!
]n - r g!

n!
] n - r g !r! n! =
]n - r g!

r! n C r = r! #

` n Pr = r! n C r n! n
30. b l = k ] n - k g !k!
]n - 1 g!
]n - 1 g! bn - 1 l + bn - 1 l =
+
k-1 k ] n - 1 - ] k - 1 g g ! ] k - 1 g ! ] n - 1 - k g !k!
]n - 1 g!
]n - 1 g!
=
+
] n - k g ! ] k - 1 g ! ] n - 1 - k g !k!
]n - k g]n - 1 g! k ]n - 1 g!
=
+ k ] n - k g ! ] k - 1 g ! ] n - k g ] n - 1 - k g !k! k ]n - 1 g! ]n - k g]n - 1 g!
=
+
] n - k g !k!
] n - k g !k!
]k + n - k g]n - 1 g!
=
] n - k g !k! n ]n - 1 g!
=
] n - k g !k! n! =
] n - k g !k! n =b l k 847

848

Maths In Focus Mathematics Extension 1 Preliminary Course

Test yourself 13
1.

4.
13
(b)
22

4
2. (a)
11

(a) 5040 (b) 720

3.

(a) 24 (b) 12

5.

(a) 65 780 (b) 25 200 (c) 252

7.

120

17
(c)
22

(a) ] n - 1 g !

(b)

6.

(a) 792

5
44

8. 2.4 # 10 18

9.

7.

1
9

10. 142 506

11. 990
12. (a) 40 320 (b) 362 880
(c) 80 640 (d) 168
13. (a) 19 958 400 (b) 4 989 600 (c) 181 440
(d) 9 979 200 (e) 181 440 n! n
14. b l = k ] n - k g ! k!

15. (a) 151 200 (b) 881 280

16. 1.08 # 10 17

20. (a) 1

(a) 60 (b) 72 (c) 30

3.

n! bn l = k ] n - k g ! k!

3
40

21
40

(a) 1 860 480

9.

(a) 94 109 400 (b) 7920 (c) 5 527 200
(e) 37 643 760 (f) 23 289 700
4
35

(b)

(b)

(c)

(d)

(d) 93 024

17
35

Practice assessment task set 4
2. P ] x g = ] x - 1 g ] x + 1 g ] x + 4 g 3. y = 3x 4

1
2

(a) 362 880

(b) 4320

5. -1, 2

(c) 282 240
3
(c)
4

6.

^ 19, 10 h

8.

x 2 + y 2 = 2; circle centre ^ 0, 0 h radius

9.

3060c; 161c3l

7. (a) - 4

(b) −2

(d) 10
2

10. Distance from centre ^ 0, 0 h to line is d= | ax 1 + by 1 + c | a2 + b2

40
10
=4
= radius
` line is tangent
=

2. (a) 360 (b) 60

]n - 1 g!
]n - 1 g! bn - 1 l + bn - 1 l =
+
k-1 k (n - 1 - 5 k - 1 ? ) ! ] k - 1 g ! ] n - 1 - k g ! k!
]n - 1 g!
]n - 1 g!
=
+
] n - k g ! ] k - 1 g ! ] n - k - 1 g ! k!
]n - k g]n - 1g! k ]n - 1g!
=
+ k ] n - k g ! ] k - 1 g ! ] n - k g ] n - k - 1 g ! k! k ]n - 1 g! ]n - k g]n - 1 g!
=
+
] n - k g ! k!
] n - k g ! k! k ] n - 1 g !+ ] n - k g ] n - 1 g !
=
] n - k g ! k!
] n - 1 g !5 k + n - k ?
=
] n - k g ! k!
] n - 1 g !n
=
] n - k g ! k! n! =
] n - k g ! k!

n n-1 n-1
` bk l = bk - 1 l + b k l

1
4

8.

4.

Challenge exercise 13
1.

(b) 246

Pr =

1.

n n! c m=
] n - 0 g ! 0!
0
n!
=
n! 0 !
=1
n! n c m= n ] n - n g ! n! n! =
0!n!
=1 n n
` c m=c m
0
n

5. (a) 90 720

` n Pr = r! n C r

10. (a)

18. (a) 15 (b) 181 440 19. 37 015 056

k!

n!
]n - r g! n! r! n C r = r!
] n - r g !r! n! =
]n - r g! n 17. (a) 1 709 316 (b) 203 490 (c) 167 580

(b)

(b)

4. (a) 720 (b) 120
6. 29%

]n - k + 1g!

11. k = -2

1
2

12. x = 74c (+s in alternate segment) y = (180c - 74c) ' 2
= 53c (+ sum in isosceles D)
13. 120

14. x 1 - 2, x 2 2

15. P ] x g = ] x - 2 g2 Q ] x g
Pl(x) = ] x - 2 g2 Ql(x) + 2 (x - 2) Q (x)
P (2) = ] 2 - 2 g 2Q (2)
=0
Pl(2) = ] 2 - 2 g2 Ql(2) + 2 (2 - 2) Q (2)
=0
16. (a) 1
17. 126

(b) 3

(c) ab = -

18. 7

1
,a+b=0
10

19. 7.1 m

20. 131c 38l

ANSWERS

31. Domain: all real x; range: y $ - 3

21. (a) P ] x g = ] x - 1 g ] x - 3 g2
(b)
y

32.

+ACB
+ABC
AC
` by AAS, DABC

33. 46 m2
1

x

3

= +ECD ^ vertically opposite angles h
= +CED (alternate angles AB||ED)
= CD ^ given h
/ DCDE

34. x + y - 3 = 0

35. x 2 - 12x + 36 = ] x - 6 g2

36. 41c 38l

37. y $ 2.5, y # - 6.5
38.

-9

22.
4 1
39. d - 1 , 7 n
7 7
40. (a) 9x - y + 16 = 0
(c) Q = ^ - 20, 0 h

(b) x + 9y + 20 = 0 l (d) 27c 21

41. Domain: all real x ! ! 2; range: all real y
42. (a) sin ^ a - b h
43. (a) 149.1 m
23. P ] -2 g = - 55
P ] x g = ] x + 2 g ^ 2x 2 - 11x + 23 h - 55 on division
` P ] - 2 g is the remainder
24. (a) abc + acd + bcd + abd = -

d
=0
a

(b) 1 (c) -1

25. P ] x g = ] x - 3 g 2 Q ] x g
P ]3 g = ]3 - 3 g 2 Q ]3 g
= 0Q ] 3 g
=0
Pl] x g = 2 ] x - 3 g Q ] x g + ] x - 3 g 2 Ql] x g
Pl(3) = 2 (3 - 3) Q (3) + ] 3 - 3 g 2 Ql(3)
= 2 (0) Q (3) + 0Ql(3)
=0

45.

3

(b) cos 45c =

(b) 46c 48l

2

(c)

^ 3 + 1 h2
8

44. X = ^ 7.5,17.5 h

46. x 11, x 21.6

48. (a) x - 8y + 129 = 0

1

47. x = 150c

1
1
n
(b) R = d 7 , 17
64
8

t 4 + 2t 3 - 6t 2 + 2t + 1
^ 1 + t2 h 2
50. f ] x g = 3x 3 - 7x 2 - 5x - 3 f ]3g = 3]3g3 - 7]3g2 - 5]3g - 3
= 81 - 63 - 15 - 3
=0
So x - 3 is a factor of f ] x g = 3x 3 - 7x 2 - 5x - 3
49.

51. a = 1, b = - 3, c = -1
52. 3x - y - 1 = 0, x + 3y - 7 = 0

26. 1 884 960 27. Radius 3; x 2 + y 2 = 9

53. 2175 cm3; 1045 cm2

28. a = 3, b = -14, c = 9

54. y 1 - 3, - 2 1 y 1 -1

29. (a) 8.1 m (b) 35c 46l

55.

56. (a) x = 60c, 120c, 240c, 300c

30. (a) !1
(b) P (1) = ^ 1 2 - 1 h (1 2 + 5)
=0
2
3
Pl(x) = 6x ^ x 2 - 1 h (x 2 + 5) + ^ x 2 - 1 h $ 2x
3

Pl(1) = 6 (1) ^ 1 2 - 1 h (1 2 + 5) + ^ 1 2 - 1 h $ 2 (1)
=0
2

3

(c) x = 270c

56
65
(b) x = 0c 90c, 360c
,

849

850

Maths In Focus Mathematics Extension 1 Preliminary Course

60. x = - 6.5, y = 2.8

(b)

61. y = - x 4

62. (a) 4

57. (a) 0

(d) 1

(b) −2

(c) −3

63. P ] x g = ] x - 2 g ^ x 2 + 1 h + 5
65. P ] x g = - ] x - 5 g ] x + 1 g 2

1
2

(e) 22
64. 15 504

66. 63

68. (b)

59. 8c 8l

70. (a), (b), (d)

71. (b)
58. i = 90c # n + ] - 1 g n # 135c

69. (c)
72. (b)

73. (a)

67. (a)

75. (d)

74. (d)