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7. (a) x x x x5 5 5 5for and for2 1+ - - - - (b) b b b x3 3 3 3for and for2 1- - (c) a a a a4 4 4 4for and for2 1+ - - - - (d) y y y y2 6 3 6 2 3for and for2 1- - (e) 3 9 3 3 9 3x x x xfor and for2 1+ - - - - (f) 4 4 4 4x x x xfor and for1 2- -
(g) 2 1 2 1k k k k21
21
for and for2 1+ - - - -
(h) 5 2 5 2x x x x52
52
for and for2 1- - +
(i) a b a b a b a bfor and for2 1+ - - - - (j) p q p q q p p qfor and for2 1- -
1. y2 3+^ h 2. x5 2-] g 3. m3 3-] g 4. x2 4 1+] g 5. y6 4 3-^ h 6. xx 2+] g 7. m m 3-] g 8. y y2 2+^ h 9. a a3 5 -] g 10. ab b 1+] g 11. xy x2 2 1-] g 12. mn n3 32 +^ h 13. xy x z2 4 -] g 14. a b a6 3 2+ -] g 15. x x y5 2- +^ h 16. q q3 22 3 -_ i 17. b b5 32 +] g 18. a b ab3 22 2 -] g 19. 5)( 7)(m x+ + 20. 1 2y y- -^ ^h h 21. 7 )(4 3 )( y x+ - 22. 2 6 5a x- +] ]g g 23. 2 1t x y+ -] ^g h 24. 3 2 2 3x a b c- + -] ]g g 25. 3 2 3x x2 +] g 26. 3 2q pq3 2 -_ i 27. ab a b3 5 13 2 +^ h 28. 4 6x x2 -] g 29. 5 7 5m n mn2 3 -^ h 30. 4 6 4ab ab2 3 +^ h 31. r r h2r +] g 32. 3 2x x- +] ]g g 33. ( ) ( )x y4 22+ +
34. 1a- +] g 35. ( ) ( )a ab1 4 32 + -
Exercises 2.8
1. 4 2x b+ +] ]g g 2. 3y a b- +^ ]h g 3. 5 2x x+ +] ]g g 4. 2 3m m- +] ]g g 5. d c a b- +] ]g g 6. 1 3x x2+ +] ^g h 7. 5 3 2a b- +] ]g g 8. 2y x x y- +^ ^h h 9. 1 1y a+ +^ ]h g 10. 5 1x x+ -] ]g g 11. 3)(1 )(y a+ + 12. 2)(1 2 )(m y- -
13. 5 2 3x y x y+ -^ ^h h 14. 4a b ab2+ -^ ]h g 15. 5 3x x- +] ]g g 16. 7)( 4)(x x3+ - 17. 3 7x y- -] ^g h 18. 3 4d e+ -] ]g g 19. 4 3x y- +] ^g h 20. 3 2a b+ -] ]g g 21. 3)( 6)(x x2- + 22. 3q p q- +^ ^h h 23. 2 3 5x x2- -] ^g h 24. 3 4a b c- +] ]g g 25. 7 4y x+ -^ ]h g 26. 4)( 5)(x x3- -
27. (2 3)(2 4) (2 3)( )x x x x2 22 2- + = - +
28. ( ) ( )a b a3 2 3+ + 29. 5( 3)(1 2 )y x- +
30. r r2 3r+ -] ]g g
Exercises 2.9
1. 3 1x x+ +] ]g g 2. 4 3y y+ +^ ^h h 3. 1m 2+] g 4. 4t 2+] g 5. 3 2z z+ -] ]g g 6. 1 6x x+ -] ]g g 7. 3 5v v- -] ]g g 8. 3t 2-] g 9. 10 1x x+ -] ]g g 10. 7 3y y- -^ ^h h 11. 6 3m m- -] ]g g 12. 12 3y y+ -^ ^h h 13. 8 3x x- +] ]g g 14. a 2 2-] g 15. 2 16x x- +] ]g g 16. 4 9y y+ -^ ^h h 17. 6 4n n- -] ]g g 18. x 5 2-] g
19. 9 1p p+ -^ ^h h 20. 2 5k k- -] ]g g 21. 4 3x x+ -] ]g g 22. 7 1m m- +] ]g g 23. 10 2q q+ +^ ^h h 24. 5 1d d- +] ]g g 25. 9 2l l- -] ]g g
Exercises 2.10
1. 2 1)( 5)( a a+ + 2. 5 2 1y y+ +^ ^h h 3. 3 7)( 1)( x x+ + 4. 3 2)( 2)( x x+ + 5. 2 3)( 1)( b b- -
6. 7 2)( 1)( x x- - 7. 3 1 2y y- +^ ^h h 8. 2 3 4x x+ +] ]g g 9. 5 2 3p p- +^ ^h h 10. 3 5 2 1x x+ +] ]g g 11. 2 1)( 6)( y y+ - 12. 5 1 2 1x x- +] ]g g 13. 4 1)(2 3)( t t- - 14. 3 4)(2 3)( x x+ -
15. 6 1 8y y- +^ ^h h 16. 4 3 2n n- -] ]g g 17. 4 1 2 5t t- +] ]g g 18. 3 2 4 5q q+ +^ ^h h 19. r r r r4 1 2 6 4 12 3- + - +=] ] ] ]g g g g 20. 2 5 2 3x x- +] ]g g 21. 6 1 2y y- -^ ^h h 22. 2 3 3 2p p- +^ ^h h 23. 8 7)( 3)( x x+ +
24. 3 4 4 9b b- -] ]g g 25. 6 1)( 9)( x x+ -
26. 3 5x 2+] g 27. 4 3y 2+^ h 28. 5 2k 2-] g 29. 6 1a 2-] g 30. 7 6m 2+] g
Answer S1-S5.indd 543 7/31/09 1:36:02 PM
544 Maths In Focus Mathematics Preliminary Course
Exercises 2.11
1. 1y 2-^ h 2. ( 3)x 2+ 3. ( 5)m 2+ 4. ( 2)t 2-
5. ( 6)x 2- 6. 2 3x 2+] g 7. 4 1b 2-] g 8. 3 2a 2+] g 9. 5 4x 2-] g 10. 7 1y 2+^ h 11. 3 5y 2-^ h 12. 4 3k 2-] g 13. 5 1x 2+] g 14. 9 2a 2-] g 15. 7 6m 2+] g 16.
4. 5 5x x+ -] ]g g 5. 2 7)(2 7)( x x+ - 6. 4 3)(4 3)( y y+ -
7. 1 2 )(1 2 )( z z+ - 8. 5 1 5 1t t+ -] ]g g 9. 3 2 3 2t t+ -] ]g g 10. 3 4 3 4x x+ -] ]g g 11. 2 )( 2 )(x y x y+ -
12. 6 6x y x y+ -^ ^h h 13. 2 3 2 3a b a b+ -] ]g g 14. 10 10x y x y+ -^ ^h h 15. 2 9 2 9a b a b+ -] ]g g 16. 2 2x y x y+ + + -^ ^h h 17. 3)( 1)(a b a b+ - - +
18. 1 1z w z w+ + - -] ]g g 19. 21
21
x x+ -d dn n
20. 3
13
1y y+ -e eo o 21. 2 3 2 1x y x y+ + - +^ ^h h
22. ( )( ) ( )( )( )x x x x x1 1 1 1 12 2 2+ - = + + -
23. 3 2 3 2x y x y3 3+ -_ _i i 24. 4 2 2x y x y x y2 2+ + -_ ^ ^i h h 25. 1)( 1)( 1)( 1)(a a a a4 2+ + + -
Exercises 2.13
1. 2)( 2 4)(b b b2- + + 2. 3 3 9x x x2+ - +] ^g h 3. 1 1t t t2+ - +] ^g h 4. 4)( 4 16)(a a a2- + +
5. 1 )(1 )( x x x2- + + 6. 2 3 4 6 9y y y2+ - +^ _h i 7. ( ) ( )y z y yz z2 2 42 2+ - + 8. 5 )( 5 25 )(x y x xy y2 2- + +
9. 2 3 4 6 9x y x xy y2 2+ - +^ _h i 10. 1 1ab a b ab2 2- + +] ^g h 11. 10 2 )(100 20 4 )( t t t+ - + 2 12.
23
4 23
9x x x2
- + +d en o 13.
10 1 100 10 1a b a ab b2 2
+ - +d en o 14. 1 2 1x y x x xy y y2 2+ - + + + + +^ _h i 15. xy z x y xyz z5 25 306 362 2 2+ - +^ _h i 16. a a 19 2 - +- ^ h 17. 1
31
3 9x x x2
- + +d en o 18. 3 3 9 6x y y y xy x x2 2+ + - - + + +^ _h i 19. 1 4 5 7x y x x xy y y2 2+ - + - + - +^ _h i 20. 2 6 )(4 24 2 6 36)( a b a a ab b b2 2+ - + + + + +
Exercises 2.14
1. x x2 3 3+ -] ]g g 2. p p3 3 4+ -^ ^h h 3. y y y5 1 12- + +^ _h i 4. ) (ab a b a2 2 2 1+ -^ h 5. 5 1a 2-] g 6. x x2 3 4- -- ] ]g g 7. z z z3 5 4+ +] ]g g 8. ab ab ab3 2 3 2+ -] ]g g 9. x xx 1 1+ -] ]g g 10. x x2 3 2 2- +] ]g g 11. 5 3m n- +] ]g g 12. x7 2 1- +] g
13. 5 4 4y y y+ + -^ ^ ^h h h 14. 1 2 2 4x x x x2- + - +] ] ^g g h 15. x x x x x x1 1 1 12 2+ - + - + +] ^ ] ^g h g h 16. x x x2 5+ -] ]g g 17. ( )x x3 3 2+ -] g
18. ( ) ( )xy xyy 2 1 2 1+ - 19. b b b3 2 4 2 2- + +] ^g h 20. x x3 3 2 2 5- +] ]g g 21. x3 1 2-] g 22. 2)( 5)( 5)(x x x+ + - 23. 3z z 2+] g 24. 1 1 2 3 2 3x x x x+ - + -] ] ] ]g g g g 25. x x x y x xy y2 2 2 2 2+ - + - +] ] ^ _g g h i 26. ( ) ( )a a a4 3 3+ - 27. x x xx 2 4 25 2- + +] ^g h 28. 2)( 2)( 3)( 3)(a a a a+ - + - 29. 4 ( 5)k k 2+
30. 3( 1) 1) 3)( (x x x+ - +
Exercises 2.15
1. 4 4 2x x x2 2+ + = +] g 2. 6 9 3b b b2 2- + = -] g 3. 10 25 5x x x2 2- + = -] g 4. 8 16 4y y y2 2+ + = +^ h
5. 14 49 7m m m2 2- + = -] g 6. 18 81 9q x q2 2+ + = +^ h
7. 2 1 1x x x2 2+ + = +] g 8. 16 64 8t t t2 2- + = -] g 9. 20 100 10x x x2 2- + = -] g 10. 44 484 22w w w2 2+ + = +] g 11. 32 256 16x x x2 2- + = -] g 12. 3
49
23
y y y22
+ + = +d n
13. 74
4927
x x x22
- + = -d n 14. 41
21
a a a22
+ + = +d n
15. 94
8129
x x x22
+ + = +d n 16. 5
yy
y1625
45
22
2
- + = -d n
17. 11
kk
k16
1214
112
22
- + = -d n
18. 6 9 3x xy y x y2 2 2+ + = +^ h 19. 4 4 2a ab b a b2 2 2- + = -] g 20. 8 16 4p pq q p q2 2 2- + = -^ h Exercises 2.16
1. 2a + 2. 2 1t - 3. 3
4 1y + 4.
2 14
d - 5.
5 2xx-
6. 4
1y -
7. ab a
322
-
-] g 8.
31
ss+
- 9.
11
bb b2
+
+ +
10. 3
5p + 11.
31
aa+
+ 12.
2 4
3
x x
y2 + +
+ 13. 3x -
14. 4 2 1
2
p p
p2 - +
- 15.
2a ba b
-
+
Answer S1-S5.indd 544 7/31/09 1:36:03 PM
545ANSWERS
Exercises 2.17
1. (a) 45x
(b) 15
13 3y + (c)
128a +
(d) 6
4 3p + (e)
613x -
2. (a) 2 1a
b-
(b) 1
2 1
q
p q q2
+
- - +^ _h i (c)
b
x yb
2 1
2
10
2
-
+
]^
gh
(d) ab
x xy y2 2- + (e)
5 2
3 1
x x
x x
- -
- -
] ]] ]
g gg g
3. (a) 5x (b)
xx
x 12
-
- +
] g (c) 3
a ba b
+
+ + (d)
22
xx+
(e) p q
p q p q
p q
p q 11 2 2
+
+ -=
+
- ++^ ^h h (f)
x x
x
1 3
12
+ -
-
] ]]g gg
(g) 2 23 8
x xx
+ -
- +
] ]g g (h) 1
2
a
a2+
+
] g
(i) y y y
y y
2 3 1
3 14 132 2
+ + -
+ +
^ ^ ^_h h h
i (j)
x x x
x
4 4 3
5 22
+ - +
+-
] ] ]]g g g
g
4. (a) y y
xx
3 9
2
8
2
2 - +
+
_]
ig
(b) 15
2 1
y
y y+ +^ ^h h
(c) x x
x x2 3 4
210 42
- -
-+
] ]g g (d) b
b bb 1
3 5 102
2
+
- -
] g (e) x
5. (a) 5 2 3
3 13x x x
x- - +
-
] ] ]g g g (b) 2 2
3 5x x
x+ -
-
] ]g g
(c) p q p q
p pq q
pq
3 5 22 2
+ -
+ -
^ ^h h (d) 2 1
a b a ba ab b2 2
+ -
- - +
] ]g g
(e) x y x y
x yy 1
+ -
+ +
^ ^^h h
h
Exercises 2.18
1. (a) 7.1- (b) 6.9- (c) 48.1 (d) 37.7- (e) 0.6
(f) 2.3 (g) 5.3- 2. 47 3. 7- 4. 375 5. 196-
6. 5.5 7. 377 8. 284 9. 40- 10. 51.935 11. 143
-
12. 22.4 13. 1838.8 14. 43
15. 15 16. 10
17. 2 312 = 18. 23.987 19. 352.47 20. 93 21. 4
Exercises 2.19
1. (a) 2 3 (b) 3 7 (c) 2 6 (d) 5 2 (e) 6 2
(f) 10 2 (g) 4 3 (h) 5 3 (i) 4 2 (j) 3 6
(k) 4 7 (l) 10 3 (m) 8 2 (n) 9 3 (o) 7 5
(p) 6 3 (q) 3 11 (r) 5 5
2. (a) 6 3 (b) 20 5 (c) 28 2 (d) 4 7 (e) 16 5
(f) 8 14 (g) 72 5 (h) 30 2 (i) 14 10 (j) 24 5
3. (a) 18 (b) 20 (c) 176 (d) 128 (e) 75
(f) 160 (g) 117 (h) 98 (i) 363 (j) 1008
4. (a) 45x = (b) 12x = (c) 63x = (d) 50x =
(e) 44x = (f) 147x = (g) 304x = (h) 828x =
(i) 775x = (j) 960x =
Exercises 2.20
1. 3 5 2. 2 3. 6 3 4. 3 3 5. 3 5- 6. 3 6
7. 7 2- 8. 8 5 9. 4 2- 10. 4 5 11. 2 12. 5 3
13. 3- 14. 2 15. 5 7 16. 2 17. 13 6
18. 9 10- 19. 47 3 20. 2 2 35 - 21. 7 5 2-
22. 2 3 4 5- - 23. 7 6 3 5+ 24. 2 2 3- -
25. 17 5 10 2- +
Exercise 2.21
1. 21 2. 15 3. 3 6 4. 10 14 5. 6 6- 6. 30
7. 12 55- 8. 14 9. 60 10. 12 2 3=
11. 2 48 8 3= 12. 15 28 30 7=
13. 2 20 4 5= 14. 84- 15. 2
16. 28 17. 30 18. 2 105- 19. 18
20 . 30 50 150 2= 21. 2 6 22. 4 3 23. 1 24. 6
8
25. 2 3 26. 3 10
1 27.
2 5
1 28.
3 5
1 29.
21
30. 2 2
3 31.
2
3 32.
2 5
9 33.
2 2
5 34.
32
35. 75
Exercises 2.22
1. (a) 10 6+ (b) 2 6 15- (c) 12 8 15+
(d) 5 14 2 21- (e) 6 4 18 6 12 2- + = - +
(f) 5 33 3 21+ (g) 6 12 6- - (h) 5 5 15-
(i) 6 30+ (j) 2 54 6 6 6 6+ = +
(k) 8 12 12 8 24 3- + = - + (l) 210 14 15-
(m) 10 6 120- (n) 10 2 2- - (o) 4 3 12-
2. (a) 10 3 6 3 5 9 3+ + +
(b) 10 35 2 14- - +
(c) 2 10 6 10 15 15 6- + -
(d) 12 18 8 1224 36 8 12
20 60 10 305 15 10 30
+ - - =
+ - -
(e) 52 13 10- (f) 15 15 18 10 6 6- + -
(g) 4 (h) 1- (i) 12- (j) 43 (k) 3 (l) 241-
(m) 6- (n) 7 2 10+ (o) 11 4 6- (p) 25 6 14+
(q) 57 12 15+ (r) 27 4 35-
(s) 77 12 40 77 24 10- = - (t) 53 12 10+
3. (a) 18 (b) 108 2 (c) 432 2 (d) 19 6 2+ (e) 9
4. (a) 21, 80a b= = (b) 19, 7a b= = -
5. (a) 1a - (b) p pp2 1 2 1- - -^ h 6. 25k = 7. 2 3 5x y xy- - 8. 17, 240a b= =
9. 107, 42a b= = - 10. 9 5 units2+
Answer S1-S5.indd 545 8/7/09 12:28:47 PM
546 Maths In Focus Mathematics Preliminary Course
Exercises 2.23
1. (a) 77
(b) 46
(c) 5
2 15 (d)
106 14
53 14
=
(e) 3
3 6+ (f)
212 5 2-
(g) 5
5 2 10+
(h) 14
3 14 4 7- (i)
208 5 3 10+
(j) 35
4 15 2 10-
2. (a) 4 4 3 243 2- -= ^ h (b) 47
6 7 3+- ^ h
(c) 19
2 15 4 1819
15 6 22-=
-- -^ ^h h
(d) 13
19 8 313
8 3 19-=
-- ^ h (e) 6 2 5 3 5 2+ + +
(f) 2
6 15 9 6 2 10 6- + -
3. (a) 2 2
(b) 2 3 32 6 3 2 3 3 6 2 3- + - + = - - + -^ h
(c) 39
22 5 14 2+
(d)
106 6 16 3 84 8 14
6 21 145
3 8 3 4
- - - +
=- + + -
^ h
(e) 4- (f) 4 2
(g) 15
20 12 19 6 25 3 615
19 6 65 3 6+ + -=
+ -
(h) 6
6 9 2 2 3+ + (i)
214 6 9 3+
(j) 415
30330 30 5- -
(k) 13
28 2 6 7 3- -
(l) 2
2 15 2 10 2 6 3 5+ - - -
4. (a) 45, 10a b= = (b) 1, 8a b= = (c) 21
,21
a b= - =
(d) 195
,98
a b= - = - (e) 5, 32a b= =
5.
2
2
3 2 23
2 1
2 1
2
4
2 1
2 1
2 1
2 1
2
4
2
2
2 1
2 1 2 12
4 2
2 12 2 2 1
2
13 2 2
2
2 2
2 2
# #
+
-+
=+
-
-
-+
=-
- -+
=-
- - ++
=-
+
= - +
=
^^ ^
hh h
So rational
6. (a) 4 (b) 14 (c) 16
7. 3
3 5 2 15 3- - -
8. 3 2 2
2
2
8
3 2 2
2
3 2 2
3 2 2
2
8
2
2
3 2 2
2 3 2 22
8 2
9 4 26 4 2
4 2
16 4 2
4 2
6 4 2 4 26
2 2
# #
#
++
=+ -
-+
=-
-+
=-
-+
=-
+
= - +
=
^^
hh
So rational
9. x 3 2= - +^ h 10. 4
4 4b
b b-
+ +
Test yourself 2
1. (a) 2y- (b) 4a + (c) 6k5- (d) 15
5 3x y+ (e) 3 8a b-
(f) 6 2 (g) 4 5
2. (a) 6 6x x+ -] ]g g (b) 3 1a a+ -] ]g g (c) ab b4 2-] g (d) 3)(5 )(y x- + (e) n p2 32 - +^ h (f) 2 )(4 2 )( x x x2- + +
3. (a) 4 6b - (b) 2 5 3x x2 + - (c) 4 17m +
(d) 16 24 9x x2 - + (e) 25p2 - (f) 1 7a- -
(g) 2 6 5 3- (h) 3 3 6 21 2 7- + -
4. (a) a ab 3 9
822 + +^ h (b)
2
15
m 2-] g
5. 157.464V = 6. (a) 17 (b) 17
6 15 9-
7. 3 2
4 5x x
x+ -
+
] ]g g 8. (a) 36 (b) 2- (c) 2 (d) 216 (e) 2
9. (a) 5
1 (b) 8 10. 11.25d =
11. (a) 15
2 3 (b)
22 6+
12. (a) 3 6 6 4 3 4 2- - + (b) 11 4 7+
13. (a) 3( 3)( 3)x x- + (b) x x6 3 1- +] ]g g (c) y y y5 2 2 42+ - +^ _h i
14. (a) 3y
x4
3
(b) 3 1
1x -
15. (a) 99 (b) 24 3
16. (a) a b2 2- (b) 2a ab b2 2+ + (c) 2a ab b2 2- +
17. (a) a b 2-] g (b) a b a ab b2 2- + +] ^g h 18.
23 3 1+
19. (a) 4 3
abb a+
(b) 10
3 11x -
20. 7
21 5 46 2- -
Answer S1-S5.indd 546 7/31/09 1:36:05 PM
547ANSWERS
21. (a) 6 2 (b) 8 6- (c) 2 3 (d) 3
4 (e) 30a b2
(f) 3n
m4 (g) 2 3x y-
22. (a) 2 6 4+ (b) 10 14 5 21 6 10 3 15- - +
(c) 7 (d) 43 (e) 65 6 14-
23. (a) 7
3 7 (b)
156
(c) 5 1
2+
(d) 15
12 2 6- (e)
5320 3 15 4 10 3 6+ + +
24. (a) 10
10x + (b)
2117 15a -
(c) ( 1)( 1)x x
x3 2+
-
-
(d) 1
1k -
(e) 3
15 6 15 3 15 2- - -
25. (a) 48n = (b) 175n = (c) 392n =
(d) n 5547= (e) 1445n =
26. 312171
27. (b), (c) 28. (d) 29. (a), (d) 30. (c)
31. (c) 32. (b) 33. (a) 34. (d) 35. (b)
Challenge exercise 2
1. (a) 2 8 6a b ab a2 2 3- + (b) 4y4 -
(c) 8 60 150 125x x x3 2- + -
2. 17
17 3 2 5 20+ + 3.
2 2
142
or
4. ab
xa
bx
ab
x4 2
2
2
2 2
+ + += d n
5. (a) x x4 9+ +] ]g g (b) ( ) ( )x y x y x y x y x y3 2 3 3 22 2 2- + = + - +_ _ _i i i (c) 5 7 25 35 49x x x2+ - +] ^g h (d) 2 2 2b a a- + -] ] ]g g g
ACD` D has a right angle at ACD+ AC` is perpendicular to DC
Answer S1-S5.indd 553 7/31/09 1:36:10 PM
554 Maths In Focus Mathematics Preliminary Course
9. AB b3= 10. xx y2 2+
11. d t tt t t t
t t
20 3 15 2400 120 9 225 60 413 180 625
2 2 2
2 2
2
= - + -
= - + + - +
= - +
] ]g g
12. 1471 mm 13. 683 m 14. 12.6 m 15. 134.6 cm
16. 4.3 m 17. 42.7 cm
18. 1.3 1.1 2.9 1.5 2.25and2 2 2+ = =
. . .1 3 1 1 1 52 2 2!+ so the triangle is not right angled the property is not a rectangle
19. No. The diagonal of the boot is the longest available space and it is only 1.4 m.
20. (a) 6 4BC2 2 2= - 20= 20BC = 6AO cm= (equal radii) So 6 4AC2 2 2= - 20= 20AC = Since ,BC AC= OC bisects AB
(b) OCA OCB 90c+ += = (given) OA OB= (equal radii) OC is common OAC OBC` /D D ( RHS ) So AC BC= (corresponding sides in congruent triangles) OC bisects AB
Exercises 4.7
1. (a) x 94c= (b) y 104c= (c) x 111c= (d) x 60c= (e) y 72c= (f) °, °x y102 51= = (g) °, °x y43 47= =
2. ABED is isosceles.
( s )
( )
B ECBE DEB
76180 76104
base equal
straight s
` cc cc
+ ++ +
+
+
= =
= = -
=
D
DD
62 104 104 360270 360
90
(angle sum of quadrilateral)c c c cc c
c
++
+
+ + + =
+ =
=
CD is perpendicular to AD
3. (a)
( )( , )
( , )
( , )
D x
C x
xx
A C xB xB D x
A D AB DC
C D AD BC
B C AB DC
180
180 180
180 180
180180
and cointerior angles
and cointerior angles
and cointerior angles`
`
c
c c
c c
cc
+
+
+ +++ +
+ +
+ +
+ +
<
<
<
= -
= - -
= - +
=
= =
= -
= = -
(b) x x x x180 180360
Angle sum c cc
= + + - + -
=
4. ,a b150 74c c= =
5. (a) 5 , 3 , 108 , 72a b x z ym m c c= = = = = (b) , ,x y z53 56 71c c c= = = (c) 5 , 68x y cm ca b= = = =
(d) , ,121 52 77c c ca b i= = = (e) 60x c= (f) ,x y3 7= =
6. ( ), ),
ADB CDBCDB ABDADB DBCABD DBC
BD ABC
BD ADCAB DCAD BC
bisects
bisects(alternate angles(alternate angles )
`
`
+ ++ ++ ++ +
+
+
<
<
=
=
=
=
7. (a) ..
AD BCAB DC
3 85 3
cmcm
(given)(given)
= =
= =
Since two pairs of opposite sides are equal, ABCD is a parallelogram.
(b) AB DCAB DC
7cm (given)
(given)<
= =
Since one pair of opposite sides is both equal and parallel, ABCD is a parallelogram.
(c) 54 126180
X M c cc
+ ++ = +
=
These are supplementary cointerior angles. XY MN` <
XM YNAlso, (given)<
XMNY is a parallelogram
(d) AE ECDE EB
56
cmcm
(given)(given)
= =
= =
Since the diagonals bisect each other, ABCD is a parallelogram.
8. (a) ,x 5 66cm ci= = (b) , ,90 25 65c c ca b c= = = (c) ,x y3 5cm cm= = (d) ,x y58 39c c= = (e) x 12 cm=
9. 6.4 cm 10. 59 , 31 , 59ECB EDC ADEc c c+ + += = =
So AG AC= (corresponding sides in congruent triangles)
( )S n
81080
2 1802 180#
#
cc
c
= -
= -
=
] g
AHG
81080
135
`c
c
+ =
=
HGA HAG+ += (base angles in isosceles triangle)
HAG2
180 135
22 30
`c
c
+ =-
= l
(angle sum of triangle)
GAC 135 2 22 30
90# c
c+ = -
=
l
We can similarly prove all interior angles are 90c and adjacent sides equal . So ACEG is a square .
14. EDC5
5
108
2 180# c
c
+ =-
=
] g
ED CD= (equal sides in regular pentagon)
So EDC is an isosceles triangle. DEC ECD`+ += (base angles in isosceles triangle)
36
DEC2
180 108c
c
+ =-
=
(angle sum of triangle)
108 3672
AEC cc
+ = -
=
Similarly, using triangle ABC , we can prove that 72EAC c+ = So EAC is an isosceles triangle. (Alternatively you could prove EDC and ABC congruent triangles and then AC EC= are corresponding sides in congruent triangles.)
23. 4, 11 1, 4x y x yor= = = - = - 24. ,x y2 1= = -
25. 7 26. 7.02 cm 27. 2 1 4 2 1x x x2- + +] ^g h
28. 43
6 15 2 6+ 29. 7 30. $643.08 31. 1.1
32. 2 10 3 5 2 2 3- + - + 33. $83.57
34. , ,x y w z22 29 90c c c= = = = 35. 56.7 cm2
36. a ba
b21 10
21
10
=- 37. ,x x6 252
2 1 - 38. 81
39. x 7- - 40. 41
x = 41. ,x x3 3# $- 42. 61
43. Given diagonal AC in rhombus ABCD :
)
)
AB BCDAC ACBBAC ACBDAC BAC
AD BCABC
(adjacent sides in rhombus)(alternate s,(base s of isosceles
`
+ ++ ++ +
+
+
<
D
=
=
=
=
` diagonal AC bisects the angle it meets. Similarly, diagonal BD bisects the angle it meets.
44. x 3 1+ -] g 45. 6 12 8x x x3 2+ + + 46. 2
517
4
47. 53x c=
48. ,x y98 41c c= = 49. 3 2
1
x +
50. (a) 12 8x y- (b) 2 31 (c) 3 9
3
x x
x2 - +
- (d) 3 2 1+
(e) 1 1
5
x x
x
+ -
- +
] ]]g g
g (f)
611 3
(g) x y zx z
y14 7 11
14 11
7
=- -
(h) 5 1 2
3a a b b+ +] ]g g (i) 8 5 (j) 13
21
51. . , .x y2 7 3 1= = 52. 25x = 53. r2
cm3 r
=
54. 17.3 cm
55. DEA xEAD xCD x x
xABC xABC DEA
A222
LetThen (base s of isosceles )
(exterior of )
(opposite s of gram are equal)
EAD
`
`
+++
++ +
+
+
+ <
D
D
=
=
= +
=
=
=
56. 52
57. 5% 58. 2.2 10 kmh8 1#
- 59. 20k =
60. 9xy y 61. 147 16c l 62. 5.57 m2
63. (a) a b a a ab b b5 2 2 4 2 4 4 42 2+ - - - + + +] ^g h (b) 3 4 6 2a b a b c+ - +] ]g g
Answer S1-S5.indd 557 7/31/09 1:36:13 PM
558 Maths In Focus Mathematics Preliminary Course
64. x181
543
1#-
65. (BCEF is a gram)<
(BC AD ABCDBC FEAD FE
is a gram)
`
< <
<
<
BC ADBC FEAD FE
Also opposite sides of gram
similarly`
<=
=
=
^^
hh
Since AD and FE are both parallel and equal, AFED is a parallelogram.
66. 11.95b m= 67. (a) 34 cm (b) 30 cm 2
68. 75
18 3 31 2 25 5+ - 69. 20 70. 32 m
71. BD bisects AC So AD DC= 90BDC BDA c+ += = (given) BD is common BAD BCD` /D D ( SAS ) AB CB` = (corresponding sides in congruent
triangles) So triangle ABC is isosceles
72. 2
x y2 2+ 73. (b) 74. (c) 75. (a) 76. (b) 77. (b)
78. (d) 79. (d)
Chapter 5 : Functions and graphs
Exercises 5.1
1. Yes 2. No 3. No 4. Yes 5. Yes 6. Yes 7. No
8. Yes 9. Yes 10. No 11. Yes 12. No 13. Yes
14. No 15. Yes
Exercises 5.2
1. 4, 0f f1 3= - =] ]g g 2. , ,h h h0 2 2 2 4 14= - = - =] ] ]g g g
3. 25, 1, 9, 4f f f f5 1 3 2= - - = - = - - = -] ] ] ]g g g g 4. 14
5. 35- 6. 9x = 7. x 5!= 8. x 3= - 9. ,z 1 4= -
10. 2 9, 2 2 9f p p f x h x h= - + = + -^ ]h g
11. 1 2g x x2- = +] g 12. f k k k k1 12= - + +] ] ^g g h 13. ; ,t t1 2 4= - = - 14. 0
15. 125, 1, 1f f f5 1 1= = - = -] ] ]g g g
16. 0 4 1 3f f f2 2 1- - + - = - + = -] ] ]g g g
17. 10 18. 7 19. 28-
20. (a) 3 (b) 3 3 3 0x - = - = Denominator cannot be 0 so the function doesn’t exist for .x 3= (c) 4
21. 2 5f x h f x xh h h2+ - = + -] ]g g 22. 4 2 1x h+ +
23. x c5 -] g 24. 3 5k2 + 25. (a) 2 (b) 0 (c) 2n n4 2+ +
Exercises 5.3
1. (a) x -intercept 32
, y -intercept -2
(b) x -intercept -10, y -intercept 4 (c) x -intercept 12, y -intercept 4 (d) x -intercepts 0, -3, y -intercept 0 (e) x -intercepts 2! , y -intercept -4 (f) x -intercepts -2, -3, y -intercept 6 (g) x -intercepts 3, 5, y -intercept 15
(h) x -intercept 53- , y -intercept 5 (i) x -intercept -3, no y -intercept (j) x -intercept ,3! y -intercept 9
2. 2
( )
f x xxf x
2
even function
2
`
- = - -
= -
=
2] ]g g
3. (a) 1f x x2 6= +^ h (b) f x x x2 12 6 3= + +] g7 A
11. (a) No value of n (b) Yes, when n is odd (1, 3, 5, …)
12. (a) (i) x 02 (ii) x 01 (iii) Even
(b) (i) x 21 (ii) x 22 (iii) Neither
(c) (i) x2 21 1- (ii) ,x x2 21 2- (iii) Neither
(d) (i) All real x 0! (ii) None (iii) Odd
(e) (i) None (ii) All real x (iii) Neither
Exercises 5.4
1. (a) x -intercept 2, y -intercept -2
(b) x -intercept 121
- , y -intercept 3
(c) x -intercept 21
, y -intercept 1
(d) x -intercept -3, y -intercept 3
(e) x -intercept 32
, y -intercept 31
-
Answer S1-S5.indd 558 7/31/09 1:36:13 PM
559ANSWERS
2. (a)
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(b) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(c) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(d) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2-1
1
(e) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2-1
112
(f) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2-1
1
(g) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2-1
1
23
-
(h) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
Answer S1-S5.indd 559 7/31/09 1:36:14 PM
560 Maths In Focus Mathematics Preliminary Course
(i) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(j) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-1111
2
3. (a) ,x yall real all real" ", , (b) :,x y y 2all real =" ", , (c) : ,x x y4 all real= -! "+ , (d) : ,x x y2 all real=! "+ , (e) , :x y y 3all real =! "+ ,
4. (a) Odd (b) Even (c) Neither (d) Odd (e) Odd
5. y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-1111
2
(3, -1)
Exercises 5.5
1. (a) x -intercepts 0, -2, y -intercept 0 (b) x -intercepts 0, 3, y -intercept 0 (c) x -intercepts ! 1, y -intercept -1 (d) x -intercepts -1, 2, y -intercept -2 (e) x -intercepts 1, 8, y -intercept 8
2. (a) y
x-4
-5
-3 -2-1 2 3 4 5
2
1
3
4
5
6
-3-4
-2-1
1
(b) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2
-11
(c) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2-1
1
Answer S1-S5.indd 560 7/31/09 1:36:15 PM
561ANSWERS
(d) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
6
-3
-4
-2
-1 1
(e) y
x-4
-5
-3 -2 -1 2 3 4 5
21
3
4
5
6
-3
-4
-2
-11
(f) y
x-4
-10
-3 -2 -1 2 3 4 5
4
6
8
2
10
12
-6
-8
-4-2
1
(g) y
x-4
-5
-3 -2 -1 2 3 4 5
21
3
4
5
-3
-4
-6
-2
-11
(h) y
x
-5
-3-4 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-6
-2
-1 1
(i) y
x-4
-5
-3 -2 -1 3 4 5
2
1
3
4
5
-3
-4
-6
-2-1 2111
2
(j) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3-4
-6
-2-1
1
3. (a) (i) x -intercepts 3, 4, y -intercept 12 (ii) {all real x },
:y y41
$ -( 2 (b) (i) x -intercepts 0, -4, y -intercept 0 (ii) {all real x }, :y y 4$ -" , (c) (i) x -intercepts -2, 4, y -intercept -8 (ii) {all real x }, : 9y y $ -" , (d) (i) x -intercept 3, y -intercept 9 (ii) {all real x }, :y y 0$" , (e) (i) x -intercepts ,2! y -intercept 4 (ii) {all real x }, :y y 4#" ,
4. (a) {all real x }, :y y 5$ -" , (b) {all real x }, :y y 9$ -" ,
Answer S1-S5.indd 561 7/31/09 1:36:16 PM
562 Maths In Focus Mathematics Preliminary Course
(c) {all real x }, :y y 241
$ -( 2 (d) {all real x }, :y y 0#" , (e) {all real x }, : 0y y $" ,
5. (a) y0 9# # (b) y0 4# # (c) y1 24# #-
(d) y4 21# #- (e) y18 241
# #-
6. (a) (i) x 02 (ii) x 01 (b) (i) x 01 (ii) x 02
(c) (i) x 02 (ii) x 01 (d) (i) x 21 (ii) x 22 (e) (i) x 52 - (ii) x 51 -
7.
( )
f x xx
f xeven
2
2
`
- = - -
= -
=
] ]g g
8. (a) Even (b) Even (c) Even (d) Neither (e) Neither (f) Even (g) Neither (h) Neither (i) Neither (j) Neither
Exercises 5.6
1. (a) x -intercept 0, y -intercept 0 (b) No x -intercepts, y -intercept 7 (c) x -intercepts ,2! y -intercept -2 (d) x -intercept 0, y -intercept 0 (e) x -intercepts ,3! y -intercept 3 (f) x -intercept -6, y -intercept 6
(g) x -intercept 32
, y -intercept 2
(h) x -intercept 54
- , y -intercept 4
(i) x -intercept 71
, y -intercept 1
(j) No x -intercepts, y -intercept 9
2. (a) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(b) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(c) y
-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(d) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(e) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(f) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
Answer S1-S5.indd 562 7/31/09 1:36:16 PM
563ANSWERS
(g) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(h) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(i) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(j) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
3. (a) {all real x }, :y y 0$" , (b) {all real x }, :y y 8$ -" , (c) {all real x }, :y y 0$" , (d) {all real x }, :y y 3$ -" , (e) {all real x }, :y y 0#" ,
4. (a) (i) x 22 (ii) x 21 (b) (i) x 02 (ii) x 01
(c) (i) x 121
2 (ii) x 121
1 (d) (i) x 02 (ii) x 01
(e) (i) x 01 (ii) x 02
5. (a) 0 2y# # (b) y8 4# #- - (c) 0 6y# #
(d) 0 11y# # (e) y1 0# #-
6. (a) x 32 - (b) x 01 (c) x 92 (d) x 22 (e) x 21 -
7. (a) x 3!= (b) ,x x1 12 1 - (c) x2 2# #-
(d) ,x 1 3= - - (e) 3x = (f) ,x 1 2= (g) x3 51 1-
(h) x4 2# #- (i) ,x x4 02 1 (j) ,x x2 4# $
(k) x4 1# #- (l) ,x x0 1# $ (m) ,x 221
= -
(n) No solutions (o) 0x = (p) 1x = (q) ,x 0 2= -
(r) No solutions (s) 31
x = ( t) 0, 6x =
Exercises 5.7
1. (a) (i) {all real x : x ! 0}, {all real y : y ! 0} (ii) no y -intercept
(iii) y
x-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
(b) (i) {all real : },x x 0! {all real :y y 0! } (ii) no y -intercept
(iii) y
x-2 -1 2
2
1
-2
-1
1
Answer S1-S5.indd 563 7/31/09 1:36:17 PM
564 Maths In Focus Mathematics Preliminary Course
(c) (i) {all real :x x 1! - }, {all real : 0y y ! } (ii) 1
(iii) y
x-2 -1 2
2
1
-2
-1
1
(d) (i) {all real :x x 2! }, {all real : 0y y ! } (ii) 121
-
(iii) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(e) (i) {all real :x x 2! - }, {all real : 0y y ! } (ii) 61
(iii) y
x-2 -1 2
2
1
-2
-1
1
(f) (i) {all real :x x 3! }, {all real :y y 0! } (ii) 32
(iii) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(g) (i) {all real : 1x x ! }, {all real : 0y y ! } (ii) -4
(iii) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
(h) (i) {all real : 1x x ! - }, {all real : 0y y ! } (ii) -2
(iii) y
x-4
-5
-3 -2-1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
Answer S1-S5.indd 564 7/31/09 1:36:18 PM
565ANSWERS
(i) (i) :x x21
all real !' 1 , {all real : 0y y ! } (ii) 32
-
(iii) y
x-2 -1 2
2
1
-2
-1
1
23
-
12
(j) (i) {all real :x x 2! - }, {all real :y y 0! } (ii) -3
(iii) y
x-4
-5
-3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-4
-2
-11
2.
( )
f x x
xf x
2
2
odd function`
- =-
= -
= -
] g
3. (a) 1y91
## (b) 1y31# # (c) y2
21
21
# #- -
(d) 3y73
## (e) 2 y81
# #- -
4. (a) 1 3x# # (b) 1 4x# # (c) 6 0x# #-
(d) 1 4x# # (e) 1 2x# #
Exercises 5.8
1. (a) (i) y
x-3
3
3
-3
(ii) : , :x x y y3 3 3 3# # # #- -! "+ , (b) (i) y
x-4
4
4
-4
(ii) : , :x x y y4 4 4 4# # # #- -! "+ , (c) (i)
(2, 1)
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
y
x
Answer S1-S5.indd 565 7/31/09 1:36:18 PM
566 Maths In Focus Mathematics Preliminary Course
(ii) : 0 4 , : 1 3x x y y# # ## -! "+ , (d) (i)
-4
-5
-3 -2 -1 2 3 4
2
1
3
4
5
-3
-4
-2
-11
y
x
(ii) : , :x x y y4 2 3 3# # # #- -! "+ , (e) (i)
-4 -3 -2 -1 2 3 4
2
1
3
4
5
-2
-1
(-2, 1)
1
y
x
(ii) : , :x x y y3 1 0 2# # # #- -! "+ , 2. (a) (i) Below x -axis
(ii) y
x-5 5
-5
(iii) : , :x x y y5 5 5 0# # # #- -! "+ , (b) (i) Above x -axis
(ii) y
x-1
1
1
(iii) : , :x x y y1 1 0 1# # # #-! "+ , (c) (i) Above x -axis
(ii) y
x-6
6
6
(iii) : , :x x y y6 6 0 6# # # #-! "+ , (d) (i) Below x -axis
(ii) y
x-8 8
-8
(iii) : , :x x y y8 8 8 0# # # #- -! "+ ,
Answer S1-S5.indd 566 7/31/09 1:36:19 PM
567ANSWERS
(e) (i) Below x -axis
(ii) y
x- 7
- 7
7
(iii) : , :x x y y7 7 7 0# # # #- -" #, - 3. (a) Radius 10, centre (0, 0) (b) Radius 5 , centre (0, 0)
(c) Radius 4, centre (4, 5) (d) Radius 7, centre (5, -6) (e) Radius 9, centre (0, 3)
4. (a) 16x y2 2+ =
(b) 6 4 12 0x x y y2 2- + - - =
(c) 2 10 17 0x x y y2 2+ + - + =
(d) 4 6 23 0x x y y2 2- + - - =
(e) 8 4 5 0x x y y2 2+ + - - =
(f) 4 3 0x y y2 2+ + + =
(g) 8 4 29 0x x y y2 2- + - - =
(h) 6 8 56 0x x y y2 2+ + + - =
(i) 4 1 0x x y2 2+ + - =
(j) 8 14 62 0x x y y2 2+ + + + =
Exercises 5.9
1. (a) {all real x }, {all real y } (b) {all real x }, {y: y = -4} (c) {x: x = 3}, {all real y } (d) {all real x }, { y : y $ -1 }
(e) {all real x }, {all real y } (f) {all real x }, : 1241
y y #' 1 (g) { : 8 8}, { : 8 8}x x y y# # # #- -
(h) {all real :t t 4! }, {all real ( ): ( )f t f t 0! }
(i) {all real : 0!z z }, {all real :g g 5!zz^ ^h h }
(j) {all real x }, { :y y 0$ }
2. (a) { x : 0x $ }, { y : y 0$ } (b) { x : x 2$ }, { y : y 0$ } (c) {all real x }, { y : y 0$ } (d) {all real x }, { y : y 2$ - }
(e) : 221
, { : }x x y y 0$ #-' 1
(f) {all real x }, { :y y 5# } (g) {all real x }, { : }y y 02
(h) {all real x }, { : }y y 01
(i) {all real :x x 0! }, {all real :y y 1! } (j) {all real :x x 0! }, {all real :y y 2! }
3. (a) ,x 0 5= (b) , ,x 3 1 2= - (c) , ,x 0 2 4=
(d) ,x 0 4!= (e) x 7!= 4. (a) x1 1# #-
(b) { : }x x1 1# #-
5. (a) { : , }x x x1 2# $- (b) { : , }t t t6 0# $-
6. (a) { y : y9 3# #- }
(b) { y : y0 9# # } (c) { y : y8 1# #- }
(d) :51
1y y# #' 1 (e) { y : 0 4y# # }
(f) { y : y1 15# #- } (g) { y : y1 0# #- }
(h) :y y1 8# #-" , (i) { y : 4 21y# #- }
(j) :y y61
64
# #-' 1 7. (a) {all real :x x 1! - }
(b) x -intercept: 0y =
01
3x
=+
0 3= This is impossible so there is no x -intercept (c) {all real :y y 0! }
8. (a) {all real :x x 0! } (b) {all real :y y 1!! }
9. (a) y
x-4 -3 -2 -1 2 3 4 5
10
5
15
20
25
-15
-10
-51
(b) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-8
-4
-21
(c) y
x-4 -3 -2 -1 2 3 4 5
10
5
15
20
25
-15
-10
-51
Answer S1-S5.indd 567 7/31/09 1:36:20 PM
568 Maths In Focus Mathematics Preliminary Course
(d) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-8
-4
-21
(e) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-8
-4
-21
(f) y
x-10 10
10
-10
(g) y
x-1
1
2
3
-1
1
10. (a) : : 0x x y y1$ $" ", , (b) y
x2 3
2
1
-11
11. y
x-1
4
3
2
1
5
6
-1 1
12. (a) (i) {all real x }, {all real y } (ii) All x (iii) None (b) (i) {all real x }, :y y 22 -" , (ii) x 02 (iii) x 01 (c) (i) {all real :x x 0! }, {all real : 0y y ! } (ii) None (iii) All 0x ! (d) (i) {all real x }, {all real y } (ii) All x (iii) None (e) (i) {all real x }, :y y 02" , (ii) All x (iii) None
2. (a) Continuous (b) Discontinuous at 1x = - (c) Continuous (d) Continuous (e) Discontinuous at x 2!=
3. (a)
(b)
(c)
Exercises 5.11
1. (a) y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-4
-2
-11
(b) y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-4
-2
-11
(c) y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-4
-2
-11
(d) y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-4
-2
-11
Answer S1-S5.indd 569 7/31/09 1:36:21 PM
570 Maths In Focus Mathematics Preliminary Course
(e)
y = x +1
y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-4
-2
-11
(f)
y = 2x-3
y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-2
-11
(g)
x + y = 1
y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-2
-11
-4
(h)
3x - y - 6 = 0
y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-2
-11
-4
-5
-6
(i)
x + 2y - 2 = 0
y
x-4 -3 -2 -1 2 3 4
2
1
3
4
5
6
-3
-2
-11
-4
-5
-6
(j)
x-4 -3 -2 -1 2 3 41
y
2
1
3
4
5
6
-3
-2
-1
-4
-5
-6
x =12
Answer S1-S5.indd 570 7/31/09 1:36:22 PM
571ANSWERS
2. (a) x 32 - (b) y 2$ - (c) y x 1$ + (d) y x 422 -
(e) y 2x$
3. (a) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-2
-11
-4
-5
y = x2 - 1
(b)
-3 3
3
-3
y
x
(c) y
x-1 1
1
-1
(d)
x-3-4 -2 -1 2 3 4 51
y = x 2
y
1
2
3
4
5
-3
-2
-1
-4
-5
(e) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-8
-4
-21
y = x3
4. (a) y x3 21 - (b) y x 222 +
(c) x y 492 21+
(d) x y 812 22+
(e) ,x y5 21 2
5. (a) y
x-4 -3 -2 -1 2 3 4
3
1
2
4
5
-2
-11
Answer S1-S5.indd 571 7/31/09 1:36:23 PM
572 Maths In Focus Mathematics Preliminary Course
(b) y
x-4 -3 -2 -1 2 3 4
3
1
2
4
5
-2
-11
(c) y
x-4-5 -3 -2 -1 2 3 4
3
1
2
4
5
-2
-11
6. (a) y
x-4 -3 -1-2 2 3 4
3
1
2
4
5
6
-2
-3
-4
-11
(b) y
x-4 -3 -1-2 2 3 4
3
1
2
4
5
6
-2
-3
-4
-5
-6
-11
y = x - 3
(c) y
x-4 -3 -1-2 2 3 4
3
1
2
4
5
6
-2
-3
-4
-5
-11
y = 3x – 5
-6
(d) y
x-4 -3 -1-2 2 3 4
3
1
2
4
5
6
-2
-3
-4
-5
-6
-11
y = x + 1
y = 3 – x
(e) y
x-3 3
3
-3
y = 1
Answer S1-S5.indd 572 8/1/09 8:13:02 PM
573ANSWERS
(f) y
x-1-2 2
1
2
-2
x = – 1
(g) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-2
-11
-4
-5
y = x2
y = 4
(h) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-4
-21
-8
y = x3
y = 3
x = -2
(i) y
x-1 1
1
1
-1
(j) y
x-4 -3 -1-2 2 3 4
3
1
2
4
5
6
-2
-3
-4
-5
-6
-11
x - y = 2
x - y = -1
7. (a) y
x-4 -3 -2 -1 2 3 4 5
2
1
3
4
5
-3
-2
-11
-4
-5
y = x2
Answer S1-S5.indd 573 7/31/09 1:36:25 PM
574 Maths In Focus Mathematics Preliminary Course
(b) y
x-4 -3 -2 -1 2 3 4
4
2
6
8
-6
-4
-21
-8
y = x3
y = 1
(c) y
x1-2 2
2
-2
x = 1
(d)
1-1 2 3 4
1
2
-2
y
x
y =2x
(e)
-1 2 3-2-3-4 1 4
1
2
-1
-2y =
1x + 2
x
y
8. (a)
x2 3 4 51-1-3-4 -2
y
y = x2
y = 5
x = 2
3
2
1
4
5
-2
-1
-3
-4
-5
(b)
x2 3 41-1-3-4 -2
y
x = 3
y = -1
y = x - 2
3
2
1
4
5
6
-2
-1
-3
-4
-5
-6
Answer S1-S5.indd 574 7/31/09 1:36:26 PM
575ANSWERS
(c)
x2 3 41-1-3-4 -2
y
y = 2x + 1
2x - 3y = 6
3
2
1
4
5
6
-2
-1
-3
-4
-5
-6
(d)
-3 3
3
-3
x
x = -3
y = 2
y
(e)
x2 3 41-1-3-4 -2
y
y = 3
y = |x |
x = 2
3
2
1
4
5
6
-2
-1
-3
Test yourself 5
1. (a) f 2 6- =] g (b) f a a a3 42= - -] g (c) ,x 4 1= -
2. (a)
(b)
(c)
(d)
(e)
(f)
Answer S1-S5.indd 575 7/31/09 1:36:26 PM
576 Maths In Focus Mathematics Preliminary Course
(g)
(h)
3. (a) Domain: all real x ; range: y 641
$ -
(b) Domain: all real x ; range: all real y (c) Domain: 1 1;x# #- range: 1 1y# #- (d) Domain: 1 1;x# #- range: 0 1y# # (e) Domain: 1 1;x# #- range: 1 0y ##- (f) Domain: all real ;x 0! range: all real y 0! (g) Domain: all real x ; range: all real y (h) Domain: all real x ; range: y 0$
4. 15 5. (a) 4 (b) 5 (c) 9 (d) 3 (e) 2
6.
7.
8.
9.
10.
11. (a) y 3# (b) y x 22 + (c) ,y x y 02$ #-
12. (a) Domain: all real ,x 3! range: all real y 0!
7. 47.4 mm 8. 20.3 m 9. (a) 7.4 cm (b) 6.6 cm (c) 9.0 cm
10. (a) 6.8 cm (b) 6.5 cm 11. 38 cm
Exercises 6.4
1. (a) x 39 48c= l (b) 35 06ca = l (c) 37 59ci = l (d) 50 37ca = l (e) 38 54ca = l (f) 50 42cb = l (g) x 44 50c= l (h) 3 10 5ci = l (i) 29 43ca = l (j) 45 37ci = l (k) 57 43ca = l (l) 43 22ci = l (m) 37 38ci = l (n) 64 37ci = l (o) 66 16cb = l (p) 29 56ca = l (q) 54 37ci = l (r) 35 58ca = l (s) °59 2i = l (t) 56 59cc = l
2. 37 57c l 3. 22 14c l 4. 36 52c l 5. 50c
6. (a) 11.4 cm (b) 37 52c l 7. ,31 58 45 44c ca b= =l l
8. (a) 13 m (b) 65 17c l 9. (a) 11 19c l (b) 26 cm
10. 4.96 cm and 17.3 cm 11. (a) 12.9 m (b) 56 34c l
Exercises 6.5
1. (a)
100c
Boat
Beachhouse
North
AnswerS6.indd 579 7/31/09 11:07:53 AM
580 Maths In Focus Mathematics Preliminary Course
(b)
320c
Campsite
Jamie
North
(c)
200c
Seagull
Jetty
North
(d)
50c
Alistair
Bus stop
North
(e)
B Hill285c
Plane
North
(f)
12c
Dam
FarmhouseNorth
(g)
160cHouse
Mohammed
North
(h)
80c
Town
Mine shaft
North
(i)
349cSchool
YvonneNorth
AnswerS6.indd 580 7/31/09 11:07:54 AM
581ANSWERS
(j)
Island
Boat ramp
280c
North
2. (a) 248c (b) 145c (c) 080c (d) 337c (e) 180c
3. 080c 4. 210c 5. 160c 6. 10.4 m
7. 21 m 8. 126.9 m 9. 72 48c l
10. (a) 1056.5 km (b) 2265.8 km (c) 245c
11. 83.1 m 12. 1.8 km 13. 12 m 14. 242c 15. 035c
16. 9.2 m 17. 171 m 18. 9.8 km 19. 51 41c l 20. 2.6 m
21. 9 21c l 22. 1931.9 km 23. 34.6 m 24. 149c
25. 198 m 26. 4.8 km 27. 9.2 m 28. 217c
29. (a) 1.2 km (b) 7.2 km 30. (a) 13.1 m (b) 50 26c l
2. (a) 17 unequal real irrational roots (b) -39 no real roots (c) 1 unequal real rational roots (d) 0 equal real rational roots (e) 33 unequal real irrational roots (f) -16 no real roots (g) 49 unequal real rational roots (h) -116 no real roots (i) 1 unequal real rational roots (j) 48 unequal real irrational roots
3. 1p = 4. k 2!= 5. b87
# - 6. p 22 7. k 2121
2 -
8. a 3 02=
b ac4 1 4 3 7
830
2 2
1
- = - -
= -
] ] ]g g g
So x x3 7 02 2- + for all x
9. ,k k5 3$# - 10. k0 41 1 11. ,m m3 31 2-
12. ,k k1 1# $- 13. 3
p1
1 - 14. b0 221
# #
y
x-4 -3 -2 -1 2 3 4 5
4
2
6
8
-6
-4
-21
Answer S9-S10.indd 596 8/1/09 8:41:30 PM
597ANSWERS
15. ,p p2 6# $-
16. Solving simultaneously: 2 6y x= + (1)
3y x2= + (2)
Substitute (2) in (1):
x xx xb ac
3 2 62 3 04 2 4 1 3
160
2
2
2 2
2
+ = +
- - =
- = - - -
=
] ] ]g g g
So there are 2 points of intersection
17. 3 4 0x y+ - = (1) 5 3y x x2= + + (2) From (1): 3 4y x= - + (3) Substitute (2) in (3):
5 3 3 48 1 0
4 8 4680
x x xx x
b ac 1 1
2
2
2 2
2
+ + = - +
+ - =
- = - -
=
] ]g g
So there are 2 points of intersection
18. 4y x= - - (1) y x2= (2) Substitute (2) in (1):
44 0
4 1 415
0
x xx x
b ac 1 4
2
2
2 2
1
= - -
+ + =
- = -
= -
] ]g g
So there are no points of intersection
19. 5 2y x= - (1) 3 1y x x2= + - (2) Substitute (2) in (1):
x x xx x
b ac
3 1 5 22 1 0
4 2 4 1 10
2
2
2 2
+ - = -
- + =
- = - -
=
] ] ]g g g
So there is 1 point of intersection the line is a tangent to the parabola
8. (a) , , ,x 0 90 180 360c c c c= (b) , ,x 90 180 270c c c= (c) , ,x 90 210 330c c c= (d) , , ,x 60 90 270 300c c c c= (e) , , ,x 0 180 270 360c c c c=
9. (a) , , , ,x 0 45 180 225 360c c c c c= (b) , ,x 0 180 360c c c= (c) , , , ,x 0 30 150 180 360c c c c c= (d) 45 , 60 ,135 , 120 , 225 , 240 , 315 , 300x c c c cc c c c= (e) 30 , 60 , 120 , 150 , 210 , 240 , 300 , 330x c c c c c c c c=
10.
( ) ( ) ( ) ( )
xx
x xx
x x
x xx x
33
25
3 33
23 5 3
3 2 5 33 5 3 2 0
2
2
# # #
+ ++
=
+ + ++
+ = +
+ + = +
+ - + + =
]] ]
] ]g
g gg g
Let 3u x= +
u ub ac
5 2 04 5 4 1 2
170
2
2 2
2
- + =
- = - -
=
] ] ]g g g
So u has 2 real irrational roots. x 3` + and so x has 2 real irrational roots
Test yourself 9
1. (a) x0 3# # (b) ,n n3 31 2- (c) 2 2y ##-
2. , ,a b c1 9 14= = - = 3. (a) 2x = (b) 3-
4. ab ac1 0
42 4 1 724
0positive definite
2
# #
`
2
1
D
=
= -
= - -
= -
2] g
5. (a) 6 (b) 3 (c) 2 (d) 18 (e) 30 6. ,x 132
31
=
7. (a) iv (b) ii (c) iii (d) ii (e) i
8.
( ) ( )
ab ac
1 04
3 4 1 47
0
2
2# #
1
1
D
= -
= -
= - - -
= -
x x4 3 02` 1- + - for all x
9. (a) 41
x = - (b) 681
10. 3 2 12 3 41x x2- + + -] ]g g 11. , ,x 30 150 270c c c=
12. (a) 341
k = (b) 1k = (c) 3k = (d) 3k = (e) 2k =
13. ,x21
3= - 14. m169
1 - 15. ,x 0 2=
16. (a) i (b) i (c) iii (d) i (e) ii
17. (a) iii (b) i (c) i (d) ii
18.
ac
kk
1
1
1
For reciprocal roots
LHS RHS
ba
ab
aa
=
=
=
= =
∴ roots are reciprocals for all x .
19. (a) 3 28 0x x2 + - = (b) 10 18 0x x2 - + =
20. 1, 3x =
Challenge exercise 9
1. k 4 02$D= -] g and a perfect square ∴ real, rational roots
2. y x x5 42= - + 3. , ,a b c4 3 7= = - = 4. x 2!=
5. 11 6. 2.3375n = - 7. .p 0 752 8. Show 0D =
9. x 1!=
10. 2, 19, 67 2, 13, 61A B C A B Cor= = - = = - = = -
11. 2
4 12
31
1
x x
xx x2 - -
+=
-+
+
12. ,k k2
1 212
1 21# $
- +
13. , ,x 30 90 150c c c= 14. ,x 12
3 5!=
15. , , ,x 60 90 270 300c c c c= 16. 23-
Chapter 10: Locus and the parabola
Exercises 10.1
1. A circle 2. A straight line parallel to the ladder.
3. An arc 4. A (parabolic) arc 5. A spiral
6. The straight line 2 2 | | 2x xor1 1 1-
7. A circle, centre the origin, radius 2 (equation 4x y2 2+ = i
8. lines y 1!= 9. lines x 5!= 10. line 2y =
11. Circle 1x y2 2+ = (centre origin, radius 1)
12. Circle, centre , ,1 2-^ h radius 4 13. 5y = -
Answer S9-S10.indd 598 8/1/09 6:52:45 PM
599ANSWERS
14. Circle, centre (1, 1), radius 3 15. x 7= - 16. 3x =
1. (a) Radius 10, centre (0, 0) (b) Radius 5 , centre (0, 0) (c) Radius 4, centre (4, 5) (d) Radius 7, centre (5, −6) (e) Radius 9, centre (0, 3)
2. (a) 16x y2 2+ = (b) 6 4 12 0x x y y2 2- + - - = (c) 2 10 17 0x x y y2 2+ + - + = (d) 4 6 23 0x x y y2 2- + - - = (e) 8 4 5 0x x y y2 2+ + - - = (f) 4 3 0x y y2 2+ + + = (g) 8 4 29 0x x y y2 2- + - - = (h) 6 8 56 0x x y y2 2+ + + - = (i) 4 1 0x x y2 2+ + - = (j) 8 14 62 0x x y y2 2+ + + + =
3. 18 8 96 0x x y y2 2- + + + =
4. 4 4 8 0x x y y2 2+ + + - = 5. 2 48 0x x y2 2- + - =
6. 6 16 69 0x x y y2 2+ + - + =
7. 10 4 27 0x x y y2 2- + + + = 8. 9 0x y2 2+ - =
9. 2 10 25 0x x y y2 2- + - + =
10. 12 2 1 0x x y y2 2+ + - + =
11. 8 6 22 0x x y y2 2- + - + = 12. 6 1 0x y y2 2+ + + =
13. (a) Radius 3, centre (2, 1) (b) Radius 5, centre (−4, 2) (c) Radius 1, centre (0, 1) (d) Radius 6, centre (5, −3) (e) Radius 1, centre (−1, 1) (f) Radius 6, centre (6, 0) (g) Radius 5, centre (−3, 4) (h) Radius 8, centre (−10, 2) (i) Radius 5, centre (7, −1) (j) Radius 10 , centre (−1, −2)
14. Centre ,3 1-^ h , radius 4 15. Centre ,2 5^ h , radius 5
16. Centre ,1 6- -^ h , radius 7 17. Centre (4, 7), radius 8
18. Centre ,121
1-d n , radius 221
19.
20. Show perpendicular distance from the line to ,4 2-^ h is 5 units, or solve simultaneous equations.
21. (a) Both circles have centre ,1 2-^ h (b) 1 unit
22. 2 2 23 0x x y y2 2+ + + - = 23. 34 units
24. (a) 5 units (b) 3 units and 2 units (c) XY is the sum of the radii. The circles touch each other at a single point, ,0 1^ h .
25. Perpendicular distance from centre ,0 0^ h to the line is equal to the radius 2 units; perpendicular distance from centre ,1 2-^ h to the line is equal to the radius 3 units.
26. (a) 2 6 15 0x x y y2 2+ + - - = (b) , ,,2 7 1 2- -^ ^h h (c) ,Z 1 8= -^ h (d) m m
-d n 13. (a) 4 2 0x y- + = (b) 0, 1^ h does not lie on the line
(c) 4 2 1 0x x y y2 2- + - + = (d) Substitute ,0 1^ h into the equation of the circle.
14. (a) Substitute Q into the equation of the parabola. (b) 1 2 2 0q x qy aq2 - - + =_ i (c) Equation of latus rectum is .y a= Solving with 4x ay2 = gives two endpoints , , ,A a a B a a2 2-^ ^h h . Length of 4AB a= .