1
Chapter 7 Exponential Functions
7A p.2
7B p.16
Chapter 8 Logarithmic Functions
8A p.27
8B p.39
8C p.48
8D p.56
Chapter 9 Rational Functions
9A p.64
9B p.75
9C p.86
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2
F4B: Chapter 7A
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 1
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 2
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 3
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 4
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 5
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 6
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise 7A Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise ○ Complete and Checked Teacher’s
3
7A Level 2 ○ Problems encountered ○ Skipped
Signature ___________ ( )
Maths Corner Exercise 7A Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature
___________ ( )
E-Class Multiple Choice Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
4
4B Lesson Worksheet 7.0 (Refer to Book 4B P.7.3)
Objective: To review positive integral indices, zero index, negative integral indices and laws of integral
indices.
Positive Integral Indices
Without using a calculator, find the value of each of the following expressions. [Nos. 1–6]
1. 15 = 2. 33 = �Review Ex: 1, 2
3. (−4)3 = 4. (−9)2 =
5.
4
2
1
=
) () (
1=
6.
3
10
1
− =
) () (
1=
Zero Index and Negative Integral Indices
Without using a calculator, find the value of each of the following expressions. [Nos. 7–10]
7. (−12)0 = 8.
0
4
3
= �Review Ex: 3–6
9. 2−3 =) () (
1 10. (−4)−2 =
) () (
1
=
=
Laws of Integral Indices
Without using a calculator, find the value of each of the following expressions. [Nos. 11–18]
11. 37 ⋅ 3–4 12. 4
7
5
5
= ( )( ) + ( ) = ( )( ) − ( )
= =
= =
13. (23)2 14. 43
3
2
1
−
= 2( ) × ( )
=
=
n
m
a
a= am – n am ⋅ an = am + n
(ab)n = anbn (am) n = amn
�Review Ex: 7–18
=
5
15.
2
8
7
16. 2–3 × 2–1
=) (
) (
) (
) (
=
=
17. (90 × 3–1)–2 18.
2
25
3−
×
31
5
2
−
=
=
Simplify the following expressions and express the answers with positive indices. [Nos. 19–22]
19. a9 ⋅ a–6 20. 3
5
a
a �Review Ex: 19–27
= =
21. 4
32 )(−
b
ba 22. (ab)–3
3
b
a
= ) (
) () (
b
ba =
=
����Level Up Question����
23. Is the result of 52 014 × (–0.2)2 015 less than 0? Explain your answer.
n
b
a
=
n
n
b
a
Is the value of (–0.2)2 015 equal to 0?
6
4B Lesson Worksheet 7.1A & B (Refer to Book 4B P.7.4)
Objective: To understand the root of a number and rational indices.
Review: Positive Integral Indices, Zero Index and Negative Integral Indices
Without using a calculator, find the value of each of the following expressions. [Nos. 1–3]
1. (−5)2 = 2. 110 = 3. 4−3 =) () (
1
=
The Root of a Number
If xn = a, where n is a positive integer, then x is an nth root of a.
Without using a calculator, find the value of each of the following expressions. [Nos. 4–7]
4. 49 5. 4 16 � Ex 7A: 1
= =
6. 3 27− 7. 3 64−
= =
Rational Indices
If a > 0, m and n are integers, and n > 0, then
(i) na
1
= n a (ii) n
m
a = mn a )( =n m
a
Express each of the following expressions in the form ap, where p is a rational number and a > 0. [Nos. 8–11]
8. 6 a 9. 3)( a � Ex 7A: 2
= ) (
1
a = ) (
) (
a
10. 5 7
a 11. 23 )( −a
= =
Instant Example 1 Instant Practice 1
Without using a calculator, find the values of 2
1
16
and 3
4
8 .
2
1
16 = 16 3
4
8 = 43 )8(
= 4 = 24
= 16
Without using a calculator, find the values of 3
1
27
and 3
2
64 .
3
1
27 = 3 3
2
64 = 23 ) (
= = ( )2
=
�
49 denotes the positive
square root of 49 only.
( )3 = −27
( )4 = 16
7
Without using a calculator, find the value of each of the following expressions. [Nos. 12–15]
12. 2
1
49 = 13. 3
1
125 = � Ex 7A: 3, 4
=
14. 2
3
9 = 3) ( 15. 4
5
16 =
= ( )3
=
Instant Example 2 Instant Practice 2
Without using a calculator, evaluate
3
2
27
1−
.
3
2
27
1−
=
2
3
27
1−
=
2
3
1−
=2
3
1
1
= 9
Without using a calculator, evaluate
2
3
25
1−
.
2
3
25
1−
=
3
) (
1−
=
3
) (
1−
=3
) (
1
1
=
Without using a calculator, find the value of each of the following expressions. [Nos. 16–17]
16. 2
3
16
1−
=
3
) (
1−
17.
5
2
32
1−
= � Ex 7A: 5, 6
=
3
) (
1−
=
����Level Up Question����
18. Without using a calculator, find the value of each of the following expressions.
(a) 3
125
8 (b)
4
1
81
16−
8
4B Lesson Worksheet 7.1C (Refer to Book 4B P.7.9)
Objective: To understand the laws of rational indices.
Review: Laws of Integral Indices
Simplify the following expressions and express the answers with positive indices. [Nos. 1–2]
1. 4
35
a
aa−⋅
= a( ) + ( ) − ( ) 2.
214
−
c
ba=
2
2) (
) (
)(a
= a( ) =) () (
) () (
cb
a ×
=
=
Laws of Rational Indices
Let a > 0, b > 0, p and q be any rational numbers.
(i) ap ⋅ aq = ap + q (ii) q
p
a
a= ap − q (iii) (ap)q = apq
(iv) (ab)p = apbp (v) p
b
a
=
p
p
b
a Note: a–q =
qa
1
Without using a calculator, find the value of each of the following expressions. [Nos. 3–8]
3. 3
1
2 × 3
5
2 = 2( ) + ( ) 4. 2
7
3 ÷ 2
5
3 = 3( ) − ( )
= 2( ) = 3( )
= =
5.
2
1
4
3
2
3
1616
16
×
= 16( ) − ( ) − ( ) 6. 3
2
4
3
)9( = 9( ) × ( )
= 16( ) = 9( )
= =
7. 3
136 )72( × = 2( ) × ( ) × 7( ) × ( ) 8.
2
1
2
4
5
2
=
) () (
) () (
5
2×
×
= =)(
2 ) (
=
�
am ⋅ bn = am + n
n
m
a
a= am – n
a−n =n
a
1
(am) n = amn
(ab)n = an bn
n
b
a
=
n
n
b
a
9
Simplify the following expressions and express the answers with positive indices. [Nos. 9–14]
9. a × 4
1
a = a( ) + ( ) 10. 2a × 4
1−
a = a( ) + ( ) 11. 2
1
a ÷ 3
1
a = a( ) − ( ) � Ex 7A: 11–16
= = =
12.
5
1
b
b= b( ) − ( ) 13. 3
1
4
9
)(a = a( ) × ( ) 14. (a4b)2 = a( ) × ( )b( )
= = =
Instant Example 1 Instant Practice 1
Simplify 5
1
a ÷ 4
1
5
2
)(a and express the answer with
positive index.
5
1
a ÷ 4
1
5
2
)(a = 5
1
a ÷ 4
1
5
2×
a
= 5
1
a ÷ 10
1
a
= 10
1
5
1−
a
= 10
1
a
Simplify 3
1
4 )(a ÷3 2
a and express the answer with
positive index.
3
1
4 )(a ÷3 2
a = a( ) × ( ) ÷ a( )
= a( ) ÷ a( )
= a( ) − ( )
= Simplify the following expressions and express the answers with positive indices. [Nos. 15–18]
15. 3
2
9 )(ab = a( )b( ) × ( ) 16. 6
1
3
2
b
a=
) () (
) () (
×
×
b
a � Ex 7A: 17–25
= =
17. 2
1
4
186 )( aba ÷ = 18. 5
a ×2
1
a
b=
����Level Up Question����
19. Let n be a rational number.
(a) Express 92n as a power of 3. (b) Hence, simplify n
nn
2
1
9
33 +⋅.
10
New Century Mathematics (Second Edition) 4B
7 Exponential Functions
� Consolidation Exercise 7A
Level 1
1. Without using a calculator, find the value of each of the following expressions.
(a) 3 27 (b) 4 256 (c) 6 64
(d) 3 8− (e) 3 125− (f) 5 243−
2. Express each of the following expressions in the form xk, where k is a rational number and x > 0.
(a) 3 x (b) 4 x (c) 5 3x
(d) 4 5−x (e) 73 )( x (f) 56 )( −
x
Without using a calculator, find the value of each of the following expressions. [Nos. 3–11]
3. 3
1
64 4. 2
3
49 5. 5
3
32
6. 3
7
8−
7. 3
1
125−
8. 3
4
27
1
9. 2
3
9
1−
10.
2
1
25
36
11.
4
3
81
16−
Use a calculator to find the value of each of the following expressions. [Nos. 12–17]
(Give the answers correct to 3 significant figures.)
12. 5 16 13. 54 )32( 14. 29 )121( −
15. 6
5
80 16. 5
8
25
12
17.
3
1
5
6−
Simplify each of the following expressions and express the answers with positive indices. [Nos. 18–31]
18. (a) 3
1
2aa × (b) 3
2
2
3−
×bb
(c) 3
4
5
2
cc ×−
(d) 6
1
2
1−−
× dd
11
19. (a)
2
1
4
1
x
x (b)
1
3
2
−y
y
(c)
3
1
2
1
p
p−
(d)
4
3
3
2
−
−
q
q
20. (a) 2
1
4 )(h (b) 24
3
)( −k
(c) 6
5
10
3
)(−
m (d) 3
4
8
3
)(−−
n
21. (a) 3
4
3)( −m (b)
63
2
)(
1
−n
(c) 7
3
7
1
c (d)
9
9
4
1
−
−
k
22. (a) a × 32
1
)(a (b) 23
4
3
2
)( −× xx (c) 4
1
22
5
)( −−
× yy
23. (a) a2 ÷ 3
1
4 )(a (b) 2
1
32
3
)( mm ÷− (c) 6
5
43
2
)(−
÷ nn
24. (a) 2
1
a × 3 a (b) 5 x × 34 )( x (c) 3 y × 65 )( −y
25. (a) 4 a ÷ a (b) 3 4
m ÷4 3
m (c) 53 )( −n ÷ 3−
n
26. (a) 5
2
4
55 )( ba (b) 4
3
3
8
)(−
xy (c) 6
1
2
3
)(−−
sr
27. (a) 6
1
2 )( ba × 3
1
a (b) 3
1
34 )( yx × y−2 (c) 2
3
24 )(−
−kh × 04
kh
28. (a) 3
4
23 )( dc− ÷ 3
2
d (b) 63
4
3
1
)(−
ba ÷ (ab−2) (c) 24
1
2
3
)( −−
sr ÷ )( 2
3
1sr
−
29. (a) 3
2
3
2
b
a×
b
1 (b)
3
1
y×
4
3
4
−
y
x
30. (a) 2
1
4
b
a÷
b
a 4
3
(b)
5
2
2
1
−
m
n÷
5
1
5
3−
n
m
31. (a) 2
1
3
2
)4( ba (b) 2
1
2
1
2 )9(−
−ba
12
Simplify each of the following expressions, where n is a rational number and n ≠ 0. [Nos. 32–35]
32. nnaa
2
)( × 33.
n
n
a
a
3
1
34. na )4( 2
1−
× 3
1
)( na
− 35.
3
4
32
)27(
)(
n
n
a
a
−
Level 2
Without using a calculator, find the value of each of the following expressions. [Nos. 36–39]
36. 4
81
256 37. 3
343
512−
38. 2
1
25
121−
39.
3
2
250
432−
Simplify each of the following expressions and express the answers with positive indices. [Nos. 40–53]
40. (a) 2
1
3)25(−
a × 3
1
a (b) 3 28x × 4
3−
x (c) 4 316 −y × 3 4
y
41. (a) 53
1
4
)(
36
c
c
−
(b)
3
1
2
35
)125(
)(
−
m
m (c)
3
3 27
−
−
p
p
42. (a) 2
1
a × 3 a ×4 1−
a (b) 3 2
k ÷ 3)( −k × k
(c) 3
1
n ×3 2
n ÷ 32
1
)3( −−
n (d) 5h ÷ 2
1
2 )4( h ÷ 43 )( h
43. (a) 2
3
3 2
a
a× a (b)
3
3
1
−
−
b
b× 2
3
b
44. (a)
24 3
x
x÷ 3 x (b)
3
2
1
2
−y
y÷ 2
1−
y
45. (a) 43
3 22
ba
ba ⋅ (b)
yx
yx22
3
4
1
)(−
46. (a) 2
b
a÷
3 5
3
1
b
a−
(b)
4
3
4 2
y
x÷
3 1
4
1
−x
y
13
47. (a) 23 123
1
)( −baa (b) 3232
3
2 )( yxyx−
−
48. (a) 323
12
kh
kh−
−
(b)
2
3 2
3
2
ab
ba
49. (a) 33
1
2 )( ba− × 2
ab (b) 3 42 −yx × 3
2
2 )(−
yx
50. (a) 3
1
2
3
134 )(4 nmnm−× (b) 13 −
qp × 3 28 qp−
51. (a) 3
1
2
3
b
a× 3
2a
b (b)
5
39
n
m−
×2
1
2
3−
m
n
52. (a) 4
3
2
8
−b
a÷
3 2
2
1
a
b (b)
2
1
3
2
m
mn÷ 3
2
8n
m−
53. (a) xx (b) aaa
(c) ba 814 (d) 53 1625 st
Simplify each of the following expressions, where all the unknowns are rational numbers. [Nos. 54–67]
54. k
k
4
2 13 +
55. 22
1
3
27+
−
x
x
56. 146
366+
⋅a
aa
57. nn
n
525
55 2
⋅
⋅−
58. 46
12
2
16+
−
x
x
59. 32
12
27
9n
n
−
−
60. 3a + 2⋅3a + 3a + 1 61. 42k + 42k + 1 + 42k + 2
62. 1
1
22
2−
+
+ nn
n
63. k
kk
2
1212
3
33 −+ −
64. 1313
1323
44
44−+
−+
−
+aa
aa
65. 21
11
66
66−+
+−
−
−nn
nn
66. 2
2
)3(2
9)3(3x
xx + 67.
131
3
28
)2(3++ − nn
n
14
Answers
Consolidation Exercise 7A
1. (a) 3 (b) 4
(c) 2 (d) −2
(e) −5 (f) −3
2. (a) 3
1
x (b) 4
1
x
(c) 5
3
x (d) 4
5−
x
(e) 3
7
x (f) 6
5−
x
3. 4 4. 343
5. 8 6. 128
1
7. 5
1 8.
81
1
9. 27 10. 5
6
11. 8
27 12. 1.74
13. 76.1 14. 0.344
15. 38.5 16. 0.309
17. 0.941
18. (a) 3
7
a (b) 6
5
b
(c) 15
14
c (d)
3
2
1
d
19. (a)
4
1
1
x
(b) 3
5
y
(c)
6
5
1
p
(d) 12
1
q
20. (a) h2 (b)
2
3
1
k
(c)
4
1
1
m
(d) 2
1
n
21. (a) 4
1
m (b) n4
(c) 3
1
c (d)
4
1
k
22. (a) 2
5
a (b) 2
1
x
(c) 3
1
y
23. (a) 3
2
a (b) 5
1
m (c) n4
24. (a) 6
5
a (b) 20
19
x (c)
15
13
1
y
25. (a)
4
3
1
a
(b) 12
7
m (c)
6
1
1
n
26. (a) 2
1
2ba (b)
2
4
3
y
x (c)
6
1
4
1
s
r
27. (a) 6
1
3
2
ba (b) y
x 3
4
(c) 2
3
h
k
28. (a) 4
2
c
d (b)
6b
a (c)
2
4
s
r
29. (a) 3
3
4
b
a (b)
4
3
1
x
30. (a)
ba 4
1
1 (b)
2
1
n
m
31. (a) 2
1
3
1
2 ba (b)
4
1
3b
a
32. 2
1+
na 33. 31
n
a−
34.
6
5
4n
n
a
35. 81
2na
36. 3
4 37.
7
8−
38. 11
5 39.
36
25
40. (a)
6
7
5
1
a
(b)
12
1
2
x
(c) 12
7
2y
15
41. (a) 3
1
6c− (b) 15
19
5m (c) 6
11
3p−
42. (a) 12
7
a (b) 6
19
k
(c)
2
1
27
n
(d) 6
1
2
1h
43. (a) a (b) b
1
44. (a)
6
5
1
x
(b) 2
3
y
45. (a)
3
4
2
1
b
a (b)
2
7
2
1
1
yx
46. (a)
3
1
6
5
b
a (b)
y
x 6
1
47. (a)
3
2
3
5
b
a (b)
2
5
2
3
x
y
48. (a) k
h3
5
(b) 3
2
3
2
ba
49. (a)
2
11
2
a
b (b)
23
2
1
yx
50. (a) 23
5
2 nm (b)
6
1
6
5
2
q
p
51. (a)
3
1
3
1
b
a (b)
42
1
3
nm
52. (a) ba 3
11
(b) 3
4
2mn
53. (a) 4
3
x (b) 4
7
a
(c) 4
1
2
1
6 ba (d) 4
5
2
3
10 st
54. 2k + 1 55. 31 − 5x
56. 6−a − 1 57. 53n + 1
58. 2x − 4 59. 3
87
3
−n
60. 6⋅3a 61. 21⋅42k
62. 3
4 63.
3
8
64. 3
13 65.
43
42−
66. 2 67. 2
1
16
F4B: Chapter 7B
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 7
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 8
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 9
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 10
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise 7B Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 7B Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 7B Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature
___________ ( )
E-Class Multiple Choice Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
17
4B Lesson Worksheet 7.2A (Refer to Book 4B P.7.15)
Objective: To understand the properties of exponential functions and their graphs.
Exponential Functions
Let x and y be variables and a be a constant, where a > 0 and a ≠ 1.
Then y = ax is called an exponential function with base a.
Properties of Graphs of the Exponential Functions y = ax
Range of a a > 1 0 < a < 1
Graph of y = ax
Common
properties
1. The graph cuts the y-axis at the point (0 , 1). 2. The graph does not cut the x-axis. It lies above the x-axis. For each value
of x, the value of y is greater than 0.
3. The graphs of
x
ay
=
1and y = ax are the images of each other when
reflected in the y-axis.
Different
properties
1. The graph slopes upward from left to right.
2. When the value of x increases, the value of y increases and so does the rate of increase of y.
1. The graph slopes downward from left to right.
2. When the value of x increases, the value of y decreases and so does the rate of decrease of y.
1. 2.
Coordinates of P = ( , ) Coordinates of P = ( , )
When the value of x increases, When the value of x increases,
the value of y ( increases / decreases ). the value of y ( increases / decreases ).
3. 4.
Coordinates of P = ( , ) Coordinates of P = ( , )
When the value of x increases, When the value of x increases,
the value of y ( increases / decreases ). the value of y ( increases / decreases ).
x
y
O
y = 8x
P
x
y
O
P
y = 0.3x
x
y
O
P
y =
x
5
1
x
y
O
y = 4x
P
x
y
0
1
y = ax
x
y
1
0
y = ax
�
4 ( < / > ) 1 0 <
5
1< 1
8 ( < / > ) 1
18
5. The figure shows the graph of y = 2x. 6. The figure shows the graph of
x
y
=
6
1.
Sketch the graph of
x
y
=
2
1in the same figure. Sketch the graph of y = 6x in the same figure.
7. The two curves in the figure represent the graphs of the exponential
functions y = 5x and y = 0.2x. Write down the corresponding
exponential function for each of C1 and C2. �Ex 7B: 3–5
C1:
C2:
8. The two curves in the figure represent the graphs of the exponential
functions y = 3x and y = 9x. Write down the corresponding exponential
function for each of C1 and C2.
C1:
C2:
9. The two curves in the figure represent the graphs of the exponential
functions y = 0.4x and y = 0.7x. Write down the corresponding
exponential function for each of C1 and C2.
C1:
C2:
����Level Up Question����
10. The figure shows the graphs of the exponential functions y = 8x
and y = ax, where a is an integer. Paul claims that the value of a
can be 10. Do you agree? Explain your answer.
x
y
O
y =
x
6
1
x
y
O
y = 2x
x
y
O
C1 C2
x
y
O
C1
C2
x
y
O
C1 C2
x
y
O
y = 8x
y = ax
x
y
O
?
?
19
y
x
C2 C1
O
y
x
C3 C4
O
New Century Mathematics (Second Edition) 4B
7 Exponential Functions
� Consolidation Exercise 7B
Level 1
1. In each of the following, when the value of x increases, does the value of the exponential function
increase or decrease?
(a) y = 6x (b) y = 0.4x (c) y =
x
6
1
(d) y = 1.2x (e) y = 3−x (f) y =x5
1
2. In each of the following, the graph of the exponential function is reflected in the y-axis. Write down
the corresponding function of each graph obtained.
(a) y =
x
4
3 (b) y = 4x (c) y = 8−x
(d) y =x9
1 (e) y = 0.3x (f) y = 2.5x
3. The two curves in the figure represent the graphs of the exponential
functions y = 3x and y = 7x. Write down the corresponding exponential
function for each of C1 and C2.
4. The two curves in the figure represent the graphs of the exponential
functions y =
x
2
1and y = 8x. Write down the corresponding
exponential function for each of C3 and C4.
20
y
x
C5
O
C6
y
x
C8
O
C9
C7
5. The two curves in the figure represent the graphs of the exponential
functions y = 0.3x and y =
x
5
3. Write down the corresponding
exponential function for each of C5 and C6.
6. The three curves in the figure represent the graphs of the exponential
functions y = 3.5x, y = 3−x and y = 0.7x. Write down the corresponding
exponential function for each of C7, C8 and C9.
7. (a) Sketch the graph of y = 2.5x in the figure.
(b) Write down the y-intercept of the graph.
(c) According to each of the following conditions, write down the
range of values of y of the function y = 2.5x.
(i) x ≥ 0
(ii) x < 0
8. (a) Sketch the graph of y = 0.6x in the figure.
(b) The graph cuts the y-axis at the point P. Write down the
coordinates of the point P.
(c) According to each of the following conditions, write down the
range of values of y of the function y = 0.6x.
(i) x ≤ 0
(ii) x > 0
7
6
5
4
3
2
1
x −2 −1 0 1 2
y
3
2
1
x −2 −1
y
0 1 2
21
9. Let y = 2−x.
(a) Complete the table below.
x −2 −1 0 1 2
y
(b) Sketch the graph of y = 2−x in the figure.
(c) Write down the coordinates of the point where the graph cuts the
y-axis.
(d) According to each of the following conditions, write down the
range of values of x of the function y = 2−x.
(i) 0 < y < 1
(ii) y ≥ 1
10. The value $P of a machine after t years can be represented by the formula P = 320 000(0.85)t. Find
the value of the machine after 4 years.
11. The selling price $V of a tablet computer after x months is given by the formula V = 2 500(0.95)x.
(a) Find the present selling price of the tablet computer.
(b) Find the selling price of the tablet computer after 6 months.
(Give the answer correct to 3 significant figures.)
12. The diameter d cm of the trunk of a tree t years after it was planted for 10 years can be represented by
the formula d = 20(1.02)t.
(a) Find the diameter of the trunk of the tree after it was planted for 10 years.
(b) Find the diameter of the trunk of the tree after it was planted for 13 years.
(Give the answer correct to 3 significant figures.)
13. The number R of birds in a forest after t weeks can be estimated by the formula
R = 6 000 + 1 250(0.84)t. Find the number of birds in the forest after 2 weeks.
14. An experiment is conducted to study the effect of a certain drug on the growth of bacteria. t hours
after the drug is applied, the number N of bacteria remained can be represented by the following
formula:
N = k(0.9)t, where k is a constant.
It is known that the number of bacteria after 2 hours is 4 050.
(a) Find the value of k.
(b) Find the number of bacteria at the beginning of the experiment.
(c) Find the number of bacteria after 5 hours.
(Give the answer correct to the nearest integer.)
4
3
2
1
x −2 −1
y
0 1 2
22
y
x O
y = 6x
y
x O
C2
C1
15. The number of animals of a certain species was 100 000 at the end of 2010. The number N of the
animals t years after the end of 2010 can be represented by the following formula:
N = a(0.95)t, where a is a constant.
(a) Find the value of a.
(b) Find the number of the animals at the end of 2015.
(Give the answer correct to the nearest integer.)
16. The course fee $V of a training course t years after 2014 can be represented by the following formula:
V = 20 000kt, where k is a positive constant.
It is known that the course fee in 2016 is $22 050.
(a) Find the value of k.
(b) Find the course fee of the training course in 2018.
(Give the answer correct to 3 significant figures.)
17. Mary carries out a diet plan. Her weight W kg t months after the beginning of the diet plan can be
represented by the following formula:
W = 80pt, where p is a constant.
It is known that the weight of Mary after 1 month is 78.4 kg.
(a) Find the value of p.
(b) Mary claims that she can lose more than 20 kg within 12 months. Do you agree? Explain your
answer.
Level 2
18. The figure shows the graph of y = 6x. Sketch the graph of y = 6−x in
the same figure.
19. The figure shows the graphs of y = 4.5−x and y = 1.5−x.
(a) Which curve, C1 or C2, represents the graph of y = 4.5−x?
(b) Sketch the graph of y = 4.5x in the same figure.
23
y
x O
C3 C4 C5
P
y
x O
y = ax y = bx
y
x O
y = bx y = 4x
y = ax
20. The three curves in the figure represent the graphs of the exponential
functions y = 0.35x, y = 2.5x and y = 5−x.
(a) Complete the table below.
Curve Corresponding exponential function
C3
C4
C5
(b) Write down the coordinates of the point P in the figure.
(c) Sketch the graph of y =
x
5
2in the figure.
21. The figure shows the graphs of the exponential functions y = ax and
y = bx, where a and b are constants. The graph of y = bx is the image
of the graph of y = ax when reflected in the y-axis. If the difference
between a and b is greater than 4, suggest a pair of possible values of
a and b. Explain your answer.
22. The figure shows the graphs of the exponential functions y = ax, y = bx
and y = 4x, where a and b are constants. The graph of y = ax is the
image of the graph of y = 4x when reflected in the y-axis.
(a) Find the value of a.
(b) Is the value of b greater than4
1? Explain your answer.
23. A researcher finds that the number N of a kind of fish in a pond under some controlled conditions after
t months can be represented by the following formula:
N = a(1.06)t, where a is a constant.
At the beginning of the research, the number of that kind of fish in the pond is 50 000.
(a) Find the value of a.
(b) Find the number of that kind of fish in the pond 3 months from the beginning of the research.
(c) Find the number of that kind of fish increased from t = 2 to t = 5.
(Give the answers correct to the nearest integer if necessary.)
24
24. The value of a smart phone is $4 645 three months after it is on sale. The value $P of the smart phone
t months after it is on sale can be represented by the following formula:
P = 1 000 + k(0.9)t, where k is a constant.
(a) Find the value of k.
(b) Find the depreciation of the smart phone after 1 year.
(Give the answer correct to 3 significant figures.)
25. The value of a car in 2014 is $200 000. Its value $V after t years can be represented by the following
formula:
V = 200 000at, where a is a constant.
It is known that the value of the car after 1 year is $186 000.
(a) Find the value of a.
(b) Find the decrease in the value of the car from 2014 to 2018.
(Give the answer correct to the nearest integer.)
26. In a factory, when x thousand of a kind of product are produced in a day, the amount Q units of a
particular pollutant in the air can be estimated by the formula:
Q = 100kx, where k is a positive constant.
It is known that the amount of the pollutant in the air is 121 units when 2 thousand of that kind of
product are produced in a certain day.
(a) Find the value of k.
(b) Can the amount of the pollutant in the air be less than 100 units? Explain your answer.
27. After a new ride has been in operation for k months, the monthly number N of visitors in a theme park
can be represented by the following formula:
N = p + 80 000(1.15)k, where p is a constant.
It is known that the monthly number of visitors is 120 000 when k = 2.
(a) Find the value of p.
(b) Is it possible that the monthly number of visitors in the theme park is less than 93 000? Explain
your answer.
28. The temperature T °C inside a balloon after t hours can be represented by the following formula:
T = 30 + 70at, where a is a positive constant.
It is known that the temperature of the air inside the balloon after 2 hours is 74.8°C.
(a) Find the temperature of the air inside the balloon after 3 hours.
(b) Can the temperature of the air inside the balloon be higher than 100°C? Explain your answer.
25
29. The value $V of a flat t years after 2010 can be represented by the following formula:
V = 1 500 000kt, where k is a constant.
It is known that the value of the flat in 2013 was $1 889 568.
(a) Find the value of k.
(b) If the value of the flat increases by more than 40% from 2012 to 2016, then the owner will sell
the flat. Will the owner sell the flat? Explain your answer.
30. A dose of 80 mg of medicine is given to a patient. The amount A mg of the medicine in the body of
the patient after t hours can be represented by the following formula:
A = kat, where a and k are constants.
(a) Find the value of k.
(b) It is given that the amount of the medicine in the body of the patient after 1 hour is 56 mg.
(i) Find the value of a.
(ii) Is the amount of the medicine in the body of the patient less than 10 mg after 6 hours?
Explain your answer.
31. Ms Tang invested $500 000 in 2010. The amount $A obtained by her n (0 ≤ n ≤ 4) years after 2010 can
be represented by the following formula:
A = 500 000rn, where r is a positive constant.
It is known that the amount obtained by Ms Tang in 2012 was $605 000.
(a) Find the value of r.
(b) Find the amount obtained by Ms Tang in 2014.
(c) In 2015, Ms Tang will invest the amount $B obtained in (b) with a new offer. The amount $P
obtained by her t years after 2015 can be represented by the following formula:
P = B(1.2)t
Ms Tang claims that she will obtain an amount more than $1 800 000 in 2020. Do you agree?
Explain your answer.
32. The carbon dioxide level N (in suitable units) in cinema A t hours from the beginning of a movie can
be represented by the formula:
N = P(1 − 0.4t) + 200, where P is a constant.
It is known that the carbon dioxide level in cinema A is 440 units 1 hour from the beginning of the
movie.
(a) Find the value of P.
(b) The carbon dioxide level N1 (in suitable units) in cinema B t hours from the beginning of a movie
can be represented by N1 = 300 − 100(0.16)t. Is it possible that the carbon dioxide level in cinema
B is greater than that in cinema A? Explain your answer.
26
Answers
Consolidation Exercise 7B
1. (a) increases (b) decreases
(c) decreases (d) increases
(e) decreases (f) decreases
2. (a) y =
x
3
4 (b) y =
x
4
1
(c) y = 8x (d) y = 9x
(e) y =
x
3
10 (f) y = 0.4x
3. C1: y = 3x, C2: y = 7x
4. C3: y = 8x, C4: y =
x
2
1
5. C5: y = 0.3x, C6: y =
x
5
3
6. C7: y = 0.7x, C8: y = 3−x, C9: y = 3.5x
7. (b) 1
(c) (i) y ≥ 1 (ii) 0 < y < 1
8. (b) (0 , 1)
(c) (i) y ≥ 1 (ii) 0 < y < 1
9. (a) x −2 −1 0 1 2
y 4 2 1 0.5 0.25
(c) (0 , 1)
(d) (i) x > 0 (ii) x ≤ 0
10. $167 042
11. (a) $2 500 (b) $1 840
12. (a) 20 cm (b) 21.2 cm
13. 6 882
14. (a) 5 000 (b) 5 000
(c) 2 952
15. (a) 100 000 (b) 77 378
16. (a) 1.05 (b) $24 300
17. (a) 0.98 (b) no
19. (a) C2
20. (a) Curve
Corresponding
exponential function
C3 y = 0.35x
C4 y = 5−x
C5 y = 2.5x
(b) (0 , 1)
21. a =5
1, b = 5 (or other reasonable answers)
22. (a) 4
1 (b) no
23. (a) 50 000 (b) 59 551
(c) 10 731
24. (a) 5 000 (b) $3 590
25. (a) 0.93 (b) $50 390
26. (a) 1.1 (b) no
27. (a) 14 200 (b) no
28. (a) 65.84°C (b) no
29. (a) 1.08 (b) no
30. (a) 80
(b) (i) 0.7 (ii) yes
31. (a) 1.1 (b) $732 050
(c) yes
32. (a) 400 (b) no
27
F4B: Chapter 8A
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 1
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 2
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 3
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 4
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 5
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 6
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 7
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
28
Maths Corner Exercise 8A Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 8A Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 8A Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature
___________ ( )
E-Class Multiple Choice Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
29
4B Lesson Worksheet 8.1A (Refer to Book 4B P.8.3)
Objective: To understand the definition of logarithms.
Meaning of Logarithms
Let a > 0 and a ≠ 1.
(a) If x = ay, then y = loga x.
(b) If y = loga x, then x = ay.
(c) log10 x (or log x) represents the common logarithm of x.
Instant Example 1 Instant Practice 1
Find the value of log4 64. ∵ 43 = 64 ∴ log4 64 = 3
Find the value of log2 16.
∵ 2( ) = 16 ∴ log2 16 =
Find the values of the following logarithms. [Nos. 1–3] �Ex 8A: 3
1. log6 1 2. log9 3 3. log381
1
∵ 6( ) = 1
∴ log6 1 =
Instant Example 2 Instant Practice 2
If log7 x = 2, find the value of x. ∵ log7 x = 2 ∴ x = 72
= 49
If log5 x = 3, find the value of x.
∵ log5 x = 3 ∴ x = ( )( )
=
In each of the following, find the value of x. [Nos. 4–6] �Ex 8A: 4
4. log4 x = 1 5. x
3
1log = 3 6. log9 x =2
1
∵ log4 x = ( )
∴ x = ( )( )
=
����Level Up Question����
7. It is given that loga2
1=
2
1. Find the value of a.
ma
1= a–m
a = 2
1
a
�
Base = 4 Base = 2
Base = ( )
� loga x is called the logarithm of x with base a.
� loga x is meaningful only when a > 0,
a ≠ 1 and x > 0.
x = ay y = loga x
30
4B Lesson Worksheet 8.1B (Refer to Book 4B P.8.5)
Objective: To use the properties of logarithms to evaluate and simplify expressions.
Properties of Logarithms
Let a, b > 0, a, b ≠ 1 and x, y > 0.
(a) loga ax = x (b) loga 1 = 0 (c) loga a = 1
(d) loga xy = loga x + loga y (e) loga
y
x= loga x – loga y
(f) loga xk = k loga x (k is any real number) (g) loga x =
a
x
b
b
log
log(formula for change of base)
Instant Example 1 Instant Practice 1
Find the value of log5 3 + log5
3
1.
log5 3 + log5
3
1= log5
×
3
13 � loga x + loga y = loga xy
= log5 1
= 0 � loga 1 = 0
Find the value of log2
1+ log 2.
log2
1+ log 2 = log [( ) × ( )]
= log ( )
=
Without using a calculator, find the values of the following. [Nos. 1–10]
1. log10 2 + log10 5 2. log3 81 – log3 9 �Ex 8A: 5, 6
= log10 [( ) × ( )] = log3
)(
)( � loga x – loga y = loga
y
x
= log10 ( ) = log3 ( )
= � loga a = 1 = log3 ( )( )
= � loga ax = x
3. log4 8 + log4 2 4. log10 4 – log10 40
= =
5. 9
2
42
7log
7log 6.
2log
32log �Ex 8A: 7
= 7log)(
7log)(
2
2 � loga xk = k loga x =)(log
)(log
= =
32 = 2 ( )
�
log4 4x = x
31
7. log 3 × log3 10 8. log2 3 × log3 2
= log 3 ×)(
)( � loga x =
a
x
b
b
log
log =
=
9. log4 5 × log5 16 10. log5 9 × log3 25
= =
Simplify the following expressions, where x > 0 and x ≠ 1. [Nos. 11–14]
11. 3
4
log
log
x
x 12. log4 x
3 – 3 log4 x �Ex 8A: 8
= =
13. log24
6x
– 6 log2 x 14. (log x)(logx 100)
= =
����Level Up Question����
15. (a) Express 75 as a product of 3 and 5.
(b) Let log 3 = a and log 5 = b. Express log 75 in terms of a and b.
32
New Century Mathematics (Second Edition) 4B
8 Logarithmic Functions
� Consolidation Exercise 8A
Level 1
1. Convert each of the following expressions into logarithmic form.
(a) 16 = 24 (b) 100 = 102 (c) 1 = 80 (d) 27
1= 3–3
(e) 5 = 3
1
125 (f) 24 = x4 (g) m = 3n (h) 12 = ab
2. Convert each of the following expressions into exponential form.
(a) 3 = log2 8 (b) 3 = log 1 000 (c) 0 = log7 1 (d) –4 = log2 16
1
(e) 6
1= log64 2 (f) 5 = log3 x (g) c = logd 5 (h) 8 = loga b
3. Without using a calculator, find the values of the following logarithms.
(a) log2 25 (b) log3 9 (c) log5 1 (d) log 100
(e) log27 3 (f) log3
9
1 (g) logx x3 (h) y
ylog
4. In each of the following, find the value of x.
(a) log x = 5 (b) log5 x = 4 (c) log1.2 x = 2 (d) log4 x =2
1−
(e) logx 64 = 3 (f) logx
25
1= 2 (g) logx 9 = –2 (h) logx 3 =
4
1−
5. Without using a calculator, find the values of the following.
(a) log6 4 + log6 9 (b) log4 2 + log4 32 (c) log 25 + log 4 (d) log 20 + log 0.5
(e) log5 0.4 + log5 12.5 (f) log3 6 + log3
2
9 (g) log7
2
7 + log7 14 (h) log4
16
3+ log4
12
1
6. Without using a calculator, find the values of the following.
(a) log2 40 – log2 10 (b) log5 100 – log5 4 (c) log 300 – log 30 (d) log6 7 – log6 42
(e) log8 3 – log8 192 (f) log4 56 – log4
2
7 (g) log
5
1– log 20 (h) log3 0.04 – log3 1.08
7. Without using a calculator, find the values of the following.
(a) 3log
27log
2
2 (b) 25log
5log
4
4 (c) 4log
8log (d)
81log
27log
(e) 64log
16log
3
3 (f) 9log
1log
5
5 (g) 3.0log
09.0log
8
8 (h)
25log
2.0log
6
6
33
8. Simplify the following expressions, where x > 0 and x ≠ 1.
(a) 4 log2 x – log2 x4 (b) log3 x + log3
x
3 (c) log (4x) – log (40x) (d)
25
55
log
log
x
x
(e)
24
4
1log
log
x
x (f)
x
x
log
1log
(g) (log5 x)(logx 25) (h) (logx 8)(log4 x)
9. If loga x = 3, find the values of the following expressions.
(a) loga x4 (b) loga x
–6 (c) logx a (d) xa2log
10. If log 5 = m, express the following in terms of m.
(a) log 125 (b) log 0.2 (c) log
5
1 (d) log 500
(e) log 2 (f) log 2.5 (g) log 40 (h) log25 100
11. If log 3 = a and log 5 = b, express the following in terms of a and b.
(a) log 15 (b) log 0.6 (c) log5 3 (d) 5log3
1
12. Use a calculator to find the values of the following, correct to 3 significant figures.
(a) log 18 (b) log 0.37 (c) log
7
32 (d) log
5 36
13. Use a calculator to find the values of the following, correct to 2 decimal places.
(a) log3 8 (b) log6 2.5 (c) log5 18 (d) 7
22log
6
14. Determine whether each of the following is true (T) or false (F).
(a) log3 43 = 4 (b) log 2 + log3 5 = 1
(c) log5 0 = 1 (d) log4 x = –2 has no solutions.
(e) log (–10) is undefined. (f) If x ≠ 0, then log x2 = 2 log x.
(g) log(–2) (–8) = 3 (h) k
x
1log = –k log x for x > 0.
Level 2
15. Without using a calculator, find the values of the following.
(a) log4 3 54 (b) log
5 01.0
1
(c) log3 3 9
27 (d) log5
4 3
2
5
25
−
−
34
16. Without using a calculator, find the values of the following.
(a) log4 4 8 + log4
4 2 (b) log 2–3 + log
25
2
(c) log3 (4 ⋅ 3
5
9 ) – log3 12 (d) log6 3 108 – log6
3 3
(e) 2
1log5 45 + log5 3
27
25 (f)
2
3log 90 – log 0.27
17. Without using a calculator, find the values of the following.
(a) log 12 + log 20 – log 24 (b) log4 3 – log4 75 + log4 100
(c) log2 0.4 + log2
3
8+ log2 60 (d) 4 log5 10 – 2 log5 12 + log5 45
(e) 3 log3 15 + log3 0.6 – 2 log3 5 (f) log6 120 – log6 150 – 2 log6
3
1
18. Without using a calculator, find the values of the following.
(a) 6log
12log18log + (b)
96log6log
16log
33
3
−
(c) 24log4log
54log4log
−
+ (d)
50log20log
2.0log240log
66
66
+
−
(e) (log3 37 )(log7 9) (f)
34
9
1log (log3
52 )
(g) 27log
9log
4
2 (h) log5 3 –25log
45log
3
3
19. Simplify the following expressions, where x > 1.
(a) 3
5
log
log4log
x
xx + (b)
xx
x
log5log
log2
3
−
(c) 34
2
log2
1log2
log3log4
1
xx
xx
−
+ (d)
1log
136log
6
6
+
−
x
x
(e) x
x
3
23
log1
3log1
+
+ (f)
x
x
4
22
log
log
(g) 2 log4 x + 3 log8
x
1 (h)
25
25
25
)5(log
5log)(log
x
xx +
20. Simplify the following expressions, where x, y > 1.
(a) yx
xxy
loglog
log2log 3
+
+ (b)
3
2
log
log2log
y
xyyx −
(c) y
xy
x
x
log21
log 2
+ (d) ))(log(log 52
43 yxxy
(e) 3loglog 3 yy xx− (f) (log x)(logx y
2)(logy 1 000)
(g) )100)(log(log
log
x
x
y
y (h)
))(log(log
log)(log2
322
yxxy
yxy
xx
xx +
35
21. If log 3 = a and log 5 = b, express the following in terms of a and b.
(a) log 75 (b) log 225 (c) log 0.12 (d) 125log
27log
(e)
5
9log
5log (f) log25 15 (g) 50log
3
1 (h) log
9
2
22. If log 2 = c and log 7 = d, express the following in terms of c and d.
(a) log 56 (b) log
8
49 (c) log 1.75 (d)
8log
49log
(e) 14log2 (f) log28 98 (g) log
5
7 (h)
2
35log14
23. If log x = m and log y = n, express the following in terms of m and n, where x, y > 0 and x, y ≠ 1.
(a) log
xy
100 (b) log
3
2
x
y (c) logx xy + logy
y
x (d) 4 23
2log yxx
24. If 10p = 2 and 10q = 7, express log 280 in terms of p and q.
25. If 10x = 2 and 5y = 3, express log 6 in terms of x and y.
26. In each of the following, express y in terms of x.
(a) 2 log x = 3 log y (b) log4 (x2y) = 3
(c) logy x = 5 (d) logx y = 8(logy x)2
(e) log (x + y) = log x + log y (f) log (x – y) = log y – 2 log x
(g) 4 log x + log y = 2 (h) 3 log5 x – log5 y = 2
27. Use a calculator to find the value of x in each of the following, correct to 3 significant figures.
(a) log x = 3.14 (b) log x =70
22−
(c) log4 x = 1.6 (d) 2log3
−=x
28. (a) Show that xaxa =log , where a, x > 0 and a ≠ 1.
(b) Find the value of 13log13log 66 32 ⋅ without using a calculator.
(c) Simplify k
m
k log
)(log log
, where k > 0, k ≠ 1 and m > 1.
29. (a) Show that alog x = xlog a, where a, x > 0.
(b) Using the result of (a), find the values of the following.
(i) (5log 8)(8log 2)
(ii) (3log 25)(2log 9)
36
30. Find the value of (log2 4)(log4 6)(log6 8)…(log14 16) without using a calculator.
31. If x = log 5 + (log 2)i and y = log 4 – (log 25)i, where i = 1− , find the value of 2x – iy.
32. If z = log (x + 2) + [log (x2 + 5x + 5)]i is a real number, where i = 1− , find the values of x and z.
33. α and β are the two roots of the equation x2 – 100x + 10 = 0. Find the values of the following
expressions.
(a) log α2 + 2 log β
(b) logα + β
β
1+ logα + β
α
1
(c) log20 α3 + 3 log20 2β
(d) log (100 – α) + log (100 – β)
34. The minimum value of the quadratic function f(x) = (log4 a)x2 − 4x + 3 is –5. Find the value of a.
35. (a) By taking logarithms on both sides of y = abx, show that
log3 y = log3 a + x log3 b.
(b) The graph in the figure shows the linear relation between log3 y
and x. If y = abx, find the values of a and b.
36. The graph in the figure shows the linear relation between log y and log
x. If y = cxn, find the values of c and n.
(Leave the radical sign ‘√’ in the answers.)
37. The graph in the figure shows the linear relation between log8 y and x.
Express y in terms of x.
38. The figure shows the graph of y = mnx, where m and n are constants.
Write down the relation between x and log2 y.
4
2 0
log3 y
x
–3 0
log y
log x 30°
y
4
0x
y = mnx
(–2 , 16)
–2
3 0
log8 y
x
37
Answers
Consolidation Exercise 8A
1. (a) 4 = log2 16 (b) 2 = log 100
(c) 0 = log8 1 (d) –3 = log327
1
(e) 3
1= log125 5 (f) 4 = logx 24
(g) n = log3 m (h) b = loga 12
2. (a) 8 = 23 (b) 1 000 = 103
(c) 1 = 70 (d) 16
1= 2–4
(e) 2 = 6
1
64 (f) x = 35
(g) 5 = d c (h) b = a8
3. (a) 5 (b) 2 (c) 0 (d) 2
(e) 3
1 (f) –2 (g) 3 (h) 2
4. (a) 100 000 (b) 625
(c) 1.44 (d) 2
1
(e) 4 (f) 5
1
(g) 3
1 (h)
81
1
5. (a) 2 (b) 3 (c) 2 (d) 1
(e) 1 (f) 3 (g) 2 (h) –3
6. (a) 2 (b) 2 (c) 1 (d) –1
(e) –2 (f) 2 (g) –2 (h) –3
7. (a) 3 (b) 2
1 (c)
2
3 (d)
4
3
(e) 3
2 (f) 0 (g) 2 (h)
2
1−
8. (a) 0 (b) 1 (c) –1 (d) 2
5
(e) 2
1− (f) –2 (g) 2 (h)
2
3
9. (a) 12 (b) –18 (c) 3
1 (d)
2
3
10. (a) 3m (b) –m
(c) 2
m− (d) m + 2
(e) 1 – m (f) 2m – 1
(g) 3 – 2m (h) m
1
11. (a) a + b (b) a – b (c) b
a (d)
a
b−
12. (a) 1.26 (b) –0.432 (c) 0.660 (d)
0.311
13. (a) 1.89 (b) 0.51 (c) 0.90 (d) 1.28
14. (a) F (b) F
(c) F (d) F
(e) T (f) F
(g) F (h) T
15. (a) 3
5 (b)
5
2 (c)
6
5 (d)
4
13−
16. (a) 2
1 (b) –2 (c)
3
7
(d) 3
2 (e)
6
7 (f)
2
7
17. (a) 1 (b) 1 (c) 6
(d) 5 (e) 4 (f) 2
5
18. (a) 3 (b) –1 (c) –3 (d) 1
(e) 3 (f) 6
5− (g)
3
8 (h)
2
1−
19. (a) 3 (b) 9
1− (c) –2 (d) 1
(e) 2 (f) 4 (g) 0 (h) 1
20. (a) 3 (b) –3 (c) 1 (d) 6
5
(e) 0 (f) 6 (g) 4
1 (h) 1
21. (a) a + 2b (b) 2a + 2b
(c) a – 2b (d) b
a
(e) ba
b
24 − (f)
b
ba
2
+
(g) a
b 1+− (h)
2
1(1 – 2a – b)
22. (a) 3c + d (b) 2d – 3c
(c) d – 2c (d) c
d
3
2
(e) c
dc
2
+ (f)
dc
dc
+
+
2
2
(g) c +2
d– 1 (h)
)(2
21
dc
dc
+
+−
23. (a) 2 – m – n (b) 2n –3
m
(c) m
n
n
m+ (d)
m
nm
8
23 +
24. 2p + q + 1 25. x + y – xy
38
26. (a) y = 3
2
x (b) y =2
64
x
(c) y = 5
1
x (d) y = x2
(e) y =1−x
x (f) y =
12
3
+x
x
(g) y =4
100
x (h) y =
25
3x
27. (a) 1 380 (b) 0.485 (c) 9.19 (d)
0.460
28. (b) 13 (c) log m
29. (b) (i) 8 (ii) 9
30. 4 31. 0
32. x = –1, z = 0
33. (a) 2 (b) 2
1− (c) 3 (d) 1
34. 2 35. (b) a = 81, b
=9
1
36. c = 310 , n =3
1 37. y = 22x – 6
38. log2 y = 2 – x
39
F4B: Chapter 8B
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 8
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise 8B Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 8B Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 8B Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature
___________ ( )
E-Class Multiple Choice Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
40
4B Lesson Worksheet 8.2 (Refer to Book 4B P.8.16)
Objective: To understand the properties of logarithmic functions and their graphs.
Logarithmic Functions
Let x and y be variables, a be a constant, where a > 0 and a ≠ 1. For x > 0, y = loga x is called a logarithmic
function with base a.
Properties of the Graphs of Logarithmic Functions
Range of a a > 1 0 < a < 1
Graph of
y = loga x
Common
properties
(a) The graph does not cut the y-axis. It lies on the right of the y-axis.
(b) The graph cuts the x-axis at the point (1 , 0).
(c) The graphs of y = x
a
1log and y = loga x are the images of each other when
reflected in the x-axis.
(d) The graphs of y = ax and y = loga x are the images of each other when reflected in
the line y = x.
Different
properties
When the value of x increases, the value
of y increases and the rate of increase of y
decreases.
When the value of x increases, the value
of y decreases and so does the rate of
decrease of y.
1. 2.
Coordinates of P = ( , ) Coordinates of P = ( , )
When the value of x increases, When the value of x increases,
the value of y ( increases / decreases ). the value of y ( increases / decreases ).
3. The two curves in the figure represent the graphs of the
logarithmic functions y = log5 x and y = x
5
1log . Write down
the corresponding logarithmic function for each of C1 and C2.
C1:
C2:
`
y = log3 x
x O P
y
x O P
y
y = x
6
1log
�
C1
y
x O
C2
� Ex 8B: 4, 5
0 <6
1< 1 3 ( < / > ) 1
41
4. The two curves in the figure represent the graphs of the
logarithmic functions y = log x and y = log6 x. Write down
the corresponding logarithmic function for each of C1 and C2.
C1:
C2:
5. The two curves in the figure represent the graphs of the
functions y = 8x and y = log8 x. Write down the corresponding
function for each of C1 and C2.
C1:
C2:
6. The figure shows the graph of y = log4 x. 7. The figure shows the graph of xy
3
1log= .
Sketch the graph of y = log4
1 x in the same figure. Sketch the graph of y = log3 x in the same figure.
8. The figure shows the graph of y = 6x. 9. The figure shows the graph of y = log4 x.
Sketch the graph of y = log6 x in the same figure. Sketch the graph of y = 4x in the same figure.
����Level Up Question����
10. The two curves in the figure represent the graphs of the logarithmic
functions y = log0.2 x and y = log0.5 x.
(a) Write down the corresponding function for each of C1 and C2.
(b) The straight line x = k passes through the point of intersection of
C1 and C2. Find the value of k.
x O
y y = x
y = log4 x
x O
y y = 6x y = x
x
C1
O
y
C2
� Ex 8B: 6
� Ex 8B: 7, 8
C2 y
O
C1
x
y = log4 x
y
x O
y
x O
y = x
3
1log
x
y
O
C1
C2
42
New Century Mathematics (Second Edition) 4B
8 Logarithmic Functions
� Consolidation Exercise 8B
Level 1
1. Each of the following graphs of functions is reflected in the x-axis. What is the function represented
by each graph obtained?
(a) y = log x (b) y = log0.5 x
(c) y = x
6
1log (d) y = x
2
7log
Each of the following graphs of functions is reflected in the line y = x. What is the function represented by
each graph obtained? [Nos. 2–3]
2. (a) y = 5x (b) y = 1.5x
(c) y =
x
3
5 (d) y =
x)24(
3. (a) y = log4 x (b) y = log0.7 x
(c) y = x
4
3log (d) y = logπ x
4. The two curves in the figure represent the graphs of the logarithmic
functions y = log4 x and y = log0.2 x. Write down the corresponding
logarithmic functions for C1 and C2.
5. In the figure, the two curves represent the graphs of the logarithmic
functions y = log3 x and y = log6 x.
(a) Write down the corresponding logarithmic functions for C3 and
C4.
(b) C3 and C4 intersect at a point P. Find the coordinates of P.
6. The two curves in the figure represent the graphs of the logarithmic
functions y = log0.1 x and y = log0.4 x.
(a) Write down the corresponding logarithmic functions for C5 and
C6.
(b) The straight line x = k passes through the point of intersection of
C5 and C6. Find the value of k.
x
y
O C5
C6
x
y
O
C1
C2
x
y
O
C3
C4
P
43
According to the graph given in each of the following, sketch the graph of the required function in the
same given graph. [Nos. 7–9]
7. (a) y = x
4
1log (b) y = log6 x
8. (a) y = log3 x (b) y = log0.3 x
9. (a) y = 2x (b) y = 0.7x
3
2
1
x
–1
–2
–2 –1
y
0 1 2 3
y = x
y = 0.3x
3
2
1
x
–1
–2
–2 –1
y
0 1 2 3
y = x
y = 3x
3
2
1
x
–1
–2
–2 –1
y
0 1 2 3
y = x
y = log0.7 x
3
2
1
x
–1
–2
–2 –1
y
0 1 2 3
y = x
y = log2 x
2
1
x
–1
–2
0 1 2 3 4
y
y = x
6
1log
2
1
x
–1
–2
0 1 2 3 4
y
y = log4 x
44
Level 2
10. In each of the following, the graph of the function is reflected in the x-axis, and the image is then
reflected in the line y = x. What is the function represented by each of the final graphs?
(a) y = log2.5 x
(b) y = log0.8 x
11. In each of the following, the graph of the function is reflected in the line y = x, and the image is then
reflected in the y-axis. What is the function represented by each of the final graphs?
(a) y = x
7
2log
(b) y = log1.6 x
12. The four curves in the figure represent the graphs of the logarithmic functions y = log0.4 x, y = log0.6 x,
y = log3 x and y = log5 x.
(a) Write down the corresponding logarithmic functions for C1, C2, C3 and C4.
(b) The straight line y = 2x + k passes through the point of intersection of the four curves. Find the
value of k.
13. The three curves in the figure represent the graphs of the functions y = log0.2 x, y = 0.5x and y
= 12
1 2 +x .
(a) Write down the corresponding functions for C5, C6 and C7.
(b) C5 cuts the y-axis at the point P(0 , k). Angela claims that C7 cuts the x-axis at the point Q(k , 0).
Do you agree? Explain your answer.
O
C5
x
y
C6
C7
x
y
C1
C2
C3
C4
O
45
14. The graph of y = logp x is the image of the graph of y = logq x when reflected in the x-axis, where p
and q are constants with p > q. If the sum of p and q is less than 3, suggest a pair of possible values of
p and q. Explain your answer.
15. The graph of y = log(k – 1) x is the image of the graph of y = (k – 3)2x when reflected in the line y = x,
where k is a constant. Find the value of k.
16. The figure shows the graphs of y = logm x, y = logn x and y = log3 x, where m and n are constants. The
graph of y = logm x is the image of the graph of y = log3 x when reflected in the x-axis.
(a) Find the value of m.
(b) Write down two possible values of n. Explain your answer.
17. (a) According to the graph of y = 5x, sketch the graph of y = log5 x in the same given graph.
(b) From the graph of y = log5 x obtained,
(i) write down the range of values of y when x ≥ 1,
(ii) write down the range of values of x when y < 0.
(c) Using the result of (a), sketch the graph of y = log0.2 x in the same given graph.
3
2
1
x
–1
–2
–2 –1
y
0 1 2 3
y = x
y = 5x
x
y
O
y = log3 x
y = logm x
y = logn x
46
18. In the figure, C is the image of the graph of y = 3x when reflected in the line L: y = x.
(a) Write down the corresponding function for C.
(b) The y-coordinate of the point G is 2. The point H is the image of G when reflected in the line y =
x.
(i) Find the equation of the straight line passing through G and H.
(ii) Is L the perpendicular bisector of GH? Explain your answer.
19. The two curves C1 and C2 in the figure represent the graphs of the logarithmic functions y = log4 x and
y = log2 x.
(a) Write down the corresponding functions for C1 and C2.
(b) It is given that P(8 , k) is a point on C1.
(i) Find the value of k.
(ii) Q and R are two points on C2. Q has the same y-coordinate as P and PQ ⊥ PR. Find the
coordinates of Q and R.
y
x
C1
C2
O
L: y = x
x
y
O
C
y = 3x
G
H
47
Answers
Consolidation Exercise 8B 1. (a) y =
10
1log x (b) y = log2 x
(c) y = log6 x (d) y =7
2log x
2. (a) y = log5 x (b) y = log1.5 x
(c) y =3
5log x (d) y =24
log x
3. (a) y = 4x (b) y = 0.7x
(c) y =
x
4
3 (d) y = πx
4. C1: y = log4 x, C2: y = log0.2 x
5. (a) C3: y = log3 x, C4: y = log6 x
(b) (1 , 0)
6. (a) C5: y = log0.1 x, C6: y = log0.4 x
(b) 1
10. (a) y = 0.4x (b) y = 1.25x
11. (a) y =
x
2
7 (b) y = 0.625x
12. (a) C1: y = log3 x, C2: y = log5 x,
C3: y = log0.4 x, C4: y = log0.6 x
(b) –2
13. (a) C5: y = 12
1 2 +x , C6: y = 0.5x,
C7: y = log0.2 x
(b) yes
14. p = 2, q =2
1 (or other reasonable answers)
15. 5
16. (a) 3
1
(b) 4
1,
5
1 (or other reasonable answers)
17. (b) (i) y ≥ 0 (ii) 0 < x < 1
18. (a) y = log3 x
(b) (i) x + y – 11 = 0
(ii) yes
19. (a) C1: y = log2 x, C2: y = log4 x
(b) (i) 3
(ii) Q(64 , 3), R
2
3 , 8
48
F4B: Chapter 8C
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 9
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 10
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 11
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 12
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 13
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 14
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 15
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 16
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
49
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise 8C Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 8C Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 8C Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature
___________ ( )
E-Class Multiple Choice Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
50
4B Lesson Worksheet 8.3A & B (Refer to Book 4B P.8.26)
Objective: To solve exponential equations and logarithmic equations.
Review: Properties of Logarithms
Let a > 0, a ≠ 1 and x, y > 0. In each of the following, fill in the blanks. [Nos. 1–4]
1. If x = ay, then y = log( ) ( ). 2. If y = loga x, then x = ( )( ).
3. loga x4 = ( ) log( ) ( ) 4. yalog = ( ) log( ) ( )
Exponential Equations
An equation involving unknown index (indices) is called an exponential equation.
Let a > 0 and a ≠ 1.
(a) If ax = ay, then x = y. (b) If ax = y, then x = loga y =a
y
log
log.
Instant Example 1 Instant Practice 1
Solve 43x + 1 = 45 – x.
43x + 1 = 45 – x ∴ 3x + 1 = 5 – x
4x = 4
x = 1
Solve 54x – 2 = 54 + 2x.
54x – 2 = 54 + 2x ∴ =
=
Solve the following exponential equations. [Nos. 5–6]
5. 2x – 2 = 2 6. 32 – 3x =9
1 �Ex 8C: 1–6
2x – 2 = 2( ) =
∴ =
=
Instant Example 2 Instant Practice 2
Solve 3x = 11 and give the answer correct
to 3 significant figures.
3x = 11
log 3x = log 11 �
x log 3 = log 11 � loga xk = k loga x
x =3log
11log
= 2.18, cor. to 3 sig. fig.
Solve 8x = 16 and give the answer correct
to 3 significant figures.
8x = 16
log 8x = ( )
( ) log 8 = ( )
x =)(
)(
=
�
Rewrite9
1as a
power of 3.
� When a > 0 and a ≠ 1, we can obtain x = y from ax = ay.
y = y( )
� Change both sides of the equation to powers of the same base.
Take common logarithms on both sides of the equation.
Take common logarithms on both sides of the equation.
51
Solve the following exponential equations and give the answers correct to 3 significant figures. [Nos. 7–8]
7. 19x = 1.9 8. 52x = 15 �Ex 8C: 7–12
Logarithmic Equations
An equation involving unknowns in logarithms is called a logarithmic equation.
Let a > 0 and a ≠ 1.
(a) If loga x = y, then x = ay. (b) If loga x = loga y, then x = y.
Instant Example 3 Instant Practice 3
Solve log5 x = 3 and log2 (y + 4) = log2 3. Solve log2 x = 4 and log3 (y + 7) = log3 (3 – y).
log5 x = 3 ∴ x = 53
= 125
log2 (y + 4) = log2 3 ∴ y + 4 = 3
y = –1
log2 x = 4 ∴ ( ) = ( )( )
=
log3 (y + 7) = log3 (3 – y) ∴ ( ) = ( )
=
Solve the following logarithmic equations. [Nos. 9–12]
9. log3 x = –2 10. log4 x = –3 �Ex 8C: 13–18
11. log (x + 3) = 4 log 3 12. log4 (2x + 1) = 2 log4 5
����Level Up Question����
13. If log5 8x = 2 log25 3y, find x : y.
Change both sides of the equation to logarithms of the same base.
52
New Century Mathematics (Second Edition) 4B
8 Logarithmic Functions
� Consolidation Exercise 8C
Level 1
Without using logarithms, solve the following exponential equations. [Nos. 1–6]
1. 42x = 43 + x 2. 62x + 3 = 64 – x
3. 34x – 1 = 27 4. 52x + 1 =3 25
5. 103 – x = 3 100 6. 23 + 2x =5 4
1
Solve the following exponential equations and give the answers correct to 3 significant figures. [Nos. 7–
12]
7. 5x = 8 8. 8x = 1.6
9. 1.2x = 6 10. 3x = 7
11. 6x + 2 = 18 12. 41 – 2x =7
1
Solve the following logarithmic equations. [Nos. 13–20]
13. log3 (x – 2) = 2 14. log7 (4 – x) = 0
15. 3 log4 x – 2 = 1 16. 3 log8 x + 1 = 0
17. log5 (x + 2) = 3 log5 2 18. log (8x) – 2 log 4 = 0
19. log8 (1 + 2x) = –2 log8 3 20. log6 (2 – 3x) + 4 log6 2 = 0
21. The value $P of an antique vase after n years is given by the formula:
P = 3 000 × 1.06n
If the value of the antique vase will be $6 030 after k years, find the value of k.
(Give the answer correct to the nearest integer.)
22. One end of a metal rod is heated to 400°C. The temperature T°C at a point d m from that end point on
the metal rod is given by the formula:
log T = A – 0.8d, where A is a constant.
(a) Find the value of A.
(b) Find the temperature at the point 0.3 m from that end point on the metal rod.
(Give the answers correct to 3 significant figures.)
53
Level 2
Without using logarithms, solve the following exponential equations. [Nos. 23–34]
23. 92x + 1 = 31 – x 24. 83 + x =124
1−x
25.
x−
2
5
1= 25x + 2 26. 36 =
x
x
−
+
3
2
36
6
27. 24 – x ⋅ 4x + 3 = 32 28. 27x + 1 ⋅ 92 – 2x = 3
29. 6x + 1 – 6x =36
5 30. 2x + 2 + 2x = 210
31. 5x – 52 + x + 120 = 0 32. 33 + 2x – 9x + 1 – 54 = 0 33. 3x ⋅ 6x + 2 = 9x + 1 34. 24x + 3 + 42x + 28 = 16x + 1
Solve the following exponential equations and give the answers correct to 3 significant figures.
[Nos. 35–42]
35. 5(3x + 1) = 12 36. 2(10x + 1) – 0.5 = 0
37. 122
7−x
= 11 38. 4x = 7x – 2
39. 6x ⋅ 4x + 2 = 24 40. 53x = 8(33 – x)
41. 8x + 2 – 3(23x + 1) = 9 42. 2x + 2 ⋅ 3x + 1 = 4x
Solve the following logarithmic equations. [Nos. 43–52]
43. log100 x + log x2 = 5 44. log4 x – 2 log8 x = –1
45. x
x
3
9
log
27log= 2 46. log7 (5 – x) – log7 (3 + x) = 1
47. log (2x + 3) + 1 = log (4x – 2) 48. log3 (2x + 1) = log3 (7x – 1) – 1 49. 2 log25 (x – 3) + 1 = log5 (3x + 1) 50. log2 (x + 3) + log4 (x + 3) = 3 51. log (log x) = 0 52. log3 [log2 (log4 x)] = 1
53. Solve log4 (22x + 1 + 8) = x + 1.
54. (a) Solve log u = log (2u + 4) – 1.
(b) Using the result of (a), solve log 2x = log (2x + 1 + 4) – 1.
54
55. (a) It is given that 9 ≤ log k < 10.
(i) Find the range of values of k.
(ii) Hence, determine the number of digits of k.
(b) It is given that log 2 = 0.301, correct to 3 decimal places. Find the number of digits of 230 without
using a calculator.
56. The depreciation rate of a mobile phone is 20% every year. Find the minimum number of years
required for the mobile phone to be less than half of its original value.
57. Billy deposits $100 000 into a bank at an interest rate of 3% p.a. compounded monthly. At least how
many months later can Billy receive an amount more than $115 000?
58. Alfred deposits $30 000 into a bank at an interest rate of 12% p.a. compounded yearly. At the same
time, Cathy deposits $50 000 into a bank at an interest rate of 5% p.a. compounded yearly. At least
how many years later will the total amount received by Alfred be more than that received by Cathy?
59. A reservoir holds 3 000 000 m3 of water initially. The volume of water in the reservoir decreases by
k% every month, where k is a constant. The volume of water in the reservoir after 6 months is 2 205 000
m3.
(a) Find the value of k, correct to the nearest integer.
(b) The reservoir is said to be emergent if the volume of water in the reservoir is less than 1 000 000 m3.
At least how many months later will the reservoir be emergent?
60. The total cost C (in thousand dollars) of producing n electronic devices in a factory can be estimated
by the following formula: C = a + log3 (n + 1), where a is a constant.
The total cost of producing 80 electronic devices in the factory is $9 000.
(a) Find the value of a.
(b) Find the number of electronic devices produced in the factory with a total cost of $12 000.
(c) 100 electronic devices are produced in the factory on a day and packed in box A. 50 electronic
devices are produced on another day and packed in box B. The production manager claims that
the cost of each electronic device in box A is less than half of that in box B. Do you agree?
Explain your answer.
61. An astronomer defined Scale A and Scale B to represent the
magnitudes of brightness of stars as shown in the table. S and T are
the magnitudes of brightness on Scale A and Scale B respectively,
while L is the relative brightness of the star. The magnitude of
brightness of star X is 2 on Scale A.
(a) Find the magnitude of brightness of star X on Scale B, correct to 3 significant figures.
(b) The relative brightness of another star Y is 10 times that of star X. Is the magnitude of brightness
of star Y half of that of star X on Scale B? Explain your answer.
Scale Formula
A S = 3 – log10 L
B T = 5 – log5 L
55
Answers
Consolidation Exercise 8C
1. 3 2. 3
1 3. 1 4.
6
1−
5. 3
7 6.
10
17− 7. 1.29 8. 0.226
9. 9.83 10. 0.886 11. –0.387 12. 1.20
13. 11 14. 3 15. 4 16. 2
1
17. 6 18. 2 19. 9
4− 20.
48
31
21. 12
22. (a) 2.60 (b) 230°C
23. 5
1− 24. –1 25. –6 26. 2
27. –5 28. 6 29. –2 30. 2
3
31. 1 32. 2
1 33. –2 34.
2
1
35. –0.203 36. –1.60 37. 0.174 38. 6.95
39. 0.128 40. 0.907 41. –0.896 42. –6.13
43. 100 44. 64 45. 3 46. –2
47. no real solutions 48. 4 49. 8
50. 1 51. 10 52. 65 536 53. 1
54. (a) 2
1 (b) –1
55. (a) (i) 109 ≤ k < 1010
(ii) 10
(b) 10
56. 4
57. 56 months
58. 8 years
59. (a) 5 (b) 22 months
60. (a) 5 (b) 2 186
(c) no
61. (a) 3.57 (b) no
56
F4B: Chapter 8D
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 17
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 18
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 19
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 20
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise 8D Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 8D Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 8D Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature
___________ ( )
E-Class Multiple Choice Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
57
4B Lesson Worksheet 8.4A (Refer to Book 4B P.8.35)
Objective: To solve problems related to the Richter scale.
Richter Scale
Magnitude M on the Richter scale = log A + K,
where A units represents the amplitude of the seismic wave measured and K is a constant.
1. An earthquake occurred in city X and the 2. An earthquake of magnitude 8 occurred in city B.
amplitude recorded was 40 units. Using the Using the formula M = log A + 4.24, find the
formula M = log A + 4.24, find the magnitude amplitude recorded, correct to 1 decimal place.
of the earthquake, correct to 1 decimal place. ( ) = log A + ( )
M = ( ) + 4.24 log A =
= ( ), cor. to 1 d.p.
∴ The magnitude of the earthquake was
( ).
3. An earthquake occurred in both city P and 4. Two earthquakes occurred in city X. The
city Q. The amplitudes recorded were 100 units amplitudes recorded in the first and the second
and 400 units respectively. Find the difference earthquakes were 810 units and 540 units
in the magnitudes of the earthquakes in these respectively. Find the difference in the
two cities. magnitudes of the two earthquakes. �Ex 8D: 3, 4
(Give the answer correct to 1 decimal place.) (Give the answer correct to 1 decimal place.)
Magnitude of the earthquake in city P
=
Magnitude of the earthquake in city Q
=
Difference in the magnitudes of the earthquakes
in these two cities
= ( ) – ( )
=
∴ The magnitudes of the earthquakes in these
two cities differ by ( ).
�
If loga x = y, then x = ay.
�Ex 8D: 1, 2
log M – log N =N
Mlog
58
5. The magnitude of an earthquake that occurred 6. The magnitudes of two earthquakes that occurred
in a city was 7 on the Richter scale. Later, an in city X and city Y were 7.2 and 4.8 respectively.
aftershock of magnitude 5 was recorded in the How many times was the amplitude of the
same city. How many times was the amplitude earthquake in city X as large as that in city Y?
of the first earthquake as large as that of the (Give the answer correct to the nearest integer.)
aftershock? �Ex 8D: 12, 13
Let the amplitude of the aftershock be A units. Let the amplitude of
( ) = log A + K be A units.
Let the required times be x.
The magnitude of the first earthquake can be
expressed as:
( ) = log xA + K
( ) = log ( ) + log ( ) + K
( ) = log ( ) + ( )
=
∴ The amplitude of the first earthquake was
( ) times as large as that of the aftershock.
����Level Up Question����
7. An earthquake occurred in both city P and city Q. The amplitude recorded in city P was 360 units
more than that recorded in city Q. If the magnitudes of the earthquakes in the two cities differ by
1.5, find the amplitude recorded in city Q. (Give the answer correct to the nearest integer.)
log MN = log M + log N
59
4B Lesson Worksheet 8.4B (Refer to Book 4B P.8.37)
Objective: To solve problems related to the intensity level of sound.
Intensity Level of Sound
For a sound of intensity I (W/m2), its intensity level D (dB) is:
D = 10 log0I
I, where I0 = 10–12 W/m2.
1. If the intensity of a sound is 10–7 W/m2, then 2. If the intensity of a sound is 10–2 W/m2, then
the intensity level of the sound the intensity level of the sound
= 10 log)(
)(dB =
= 10 log 10( ) – ( ) dB
= 10 log 10( ) dB
= 10 ⋅ ( ) dB
= dB
3. The intensity of the sound measured in a contruction site is 10–9 W/m2. Find the intensity level of the sound.
Instant Example 1 Instant Practice 1
The intensity level of a sound produced by a bus
is 74 dB. Find the intensity of the sound, correct
to 3 significant figures.
Let the intensity of the sound be I W/m2.
74 = 10 log1210−
I
7.4 = log1210−
I ∴
1210−
I= 107.4
I = 10–12 × 107.4
= 10–4.6
= 2.51 × 10–5, cor. to 3 sig. fig. ∴ The intensity of the sound is 2.51 × 10–5 W/m2.
The intensity level of a sound produced by a
machine is 105 dB. Find the intensity of the sound,
correct to 3 significant figures.
Let the intensity of the sound be .
( ) = 10 log)(
)(
( ) = log)(
)( ∴
)(
)(= ( )( )
I =
�
�Ex 8D: 6
n
m
a
a= am – n
am ⋅ an = am + n
60
4. The intensity level of a sound produced by a 5. The intensities of the sound measured in a bank
vacuum cleaner is 57 dB. Find the intensity of and a karaoke lounge are 10–5 W/m2 and
the sound, correct to 3 significant figures. 10–3 W/m2 respectively. Find the difference in
�Ex 8D: 7 the intensity levels of the two places. �Ex 8D: 8
Difference in the intensity levels of the two places
=
−
)()(
log10)()(
log10 dB
= ( )( ) dB
6. The intensity of a sound produced by an instrument is 10–5.7 W/m2. If the intensity level of the sound is
increased by 8 dB, find the new intensity of the sound.
(Give the answer correct to 3 significant figures.)
Let the new intensity of the sound be .
Original intensity level = 10 log)(
)(dB
10 log)(
)(– 10 log
)(
)(= ( )
=
����Level Up Question����
7. The intensity of the sound in a taxi is 10–6 W/m2.
(a) Find the intensity level of the sound.
(b) If the intensity level of the sound is halved now, find the new intensity of the sound.
Take out the common factor.
61
New Century Mathematics (Second Edition) 4B
8 Logarithmic Functions
� Consolidation Exercise 8D
[In this exercise, unless otherwise stated, give the answers correct to 1 decimal place if necessary.]
Level 1
1. An earthquake of magnitude 9 occurred in an area. By using the formula M = log A + 4.24, find the
amplitude of the earthquake, correct to 3 significant figures.
2. An earthquake occurred in a region. The amplitude of the earthquake was measured to be 42 units. By
using the formula M = log A + 4.24, find the magnitude of the earthquake.
3. An earthquake occurred in two cities P and Q. The amplitudes recorded were 80 units and 320 units
respectively. Find the difference in the magnitudes of the earthquake in the two cities.
4. Two earthquakes occurred in a city. The amplitude recorded in the first earthquake was 5 times that
recorded in the second earthquake. Find the difference in the magnitudes of the two earthquakes.
5. The magnitude of an earthquake is 8. Using the formula M =0
log3
2
E
E, where M represents
the
magnitude of the earthquake, E units represents the energy released and E0 = 6.3 × 104, find the energy
released in the earthquake.
6. The intensity level of the sound recorded in a library is 30 dB. Find the intensity of the sound.
7. The intensity of the sound produced by an engine is 1 W/m2. Find the intensity level of the sound.
8. The intensity levels of sound measured in two offices are 35 dB and 36 dB respectively. Find the
difference in the intensities of sound of the two offices, correct to 3 significant figures.
9. People feel ear pain when hearing a sound at intensity level of 130 dB, which is called the threshold of
pain. The intensity of the sound recorded in a concert is 11 W/m2. Is the intensity level of the sound
recorded in the concert above the threshold of pain? Explain your answer.
10. For the two numbers 5234 and 2543, which one is larger? Explain your answer.
11. For the two numbers –0.4–321 and
123
8
1−
− , which one is larger? Explain your answer.
62
Level 2
12. The magnitude of an earthquake that occurred in area P was 8. Another earthquake occurred in area Q,
where the amplitude recorded was
50
1 of that recorded in area P. Find the magnitude of the
earthquake recorded in area Q.
13. An earthquake of magnitude 7.4 occurred in a town. Later, there was an aftershock of magnitude 7 in
the same town. How many times was the amplitude of the first earthquake as large as that of the
aftershock?
14. The magnitudes of two earthquakes that occurred in a region differ by 2. Find the ratio of the
amplitude of the stronger earthquake to that of the weaker earthquake.
15. An earthquake occurred in both city X and city Y. The amplitude recorded in city X was 300 units less
than that recorded in city Y. If the magnitudes of the earthquake in the two cities differ by 0.3, find the
amplitude recorded in city Y.
16. Three earthquakes occurred in a region. The magnitudes of the first and second earthquakes are 8.2
and 7.5 respectively. The amplitude of the third earthquake is
8
1 times that of the first earthquake.
(a) Which earthquake, the second or the third earthquake, has a larger magnitude? Explain your
answer.
(b) Find the difference between the magnitudes of the second and the third earthquakes.
17. The intensity of the sound from a speaker is 10–5 W/m2.
(a) Find the intensity level of the sound produced.
(b) If the intensity level of the sound is increased by 5 dB, find the new intensity of the sound,
correct to 3 significant figures.
18. The intensity level of the sound recorded near a residential area is 60 dB. After a construction site is
set up near the area, the intensity of the sound becomes 30 times of that before. Find the intensity
level of the sound after the construction site is set up.
19. The intensity level of a sound is 50 dB. If the intensity level of the sound is decreased by 20%, is the
new intensity of the sound
10
1 of the original? Explain your answer.
20. The intensity level of the sound recorded near a highway is 120 dB. If noise barrier A is set up, the
intensity of the sound will become
40
1 of that before. If noise barrier B is set up, the intensity level of
the sound will be decreased by 15 dB. Which noise barrier is more effective? Explain your answer.
21. If the intensity I W/m2 of a sound is increased to 100 times the original, the new intensity level of the
sound will be doubled.
(a) Find the original intensity of the sound.
(b) If the intensity I W/m2 of the sound is increased to 10 times the original, will the new intensity
level of the sound be increased by 50%? Explain your answer.
63
Answers
Consolidation Exercise 8D
1. 57 500 units 2. 5.9
3. 0.6 4. 0.7
5. 6.3 × 1016 units 6. 10–9 W/m2
7. 120 dB 8. 8.19 × 10–10
W/m2
9. yes 10. 5234
11. 123
8
1
− 12. 6.3
13. 2.5 times 14. 100 : 1
15. 601.4 units
16. (a) second earthquake
(b) 0.2
17. (a) 70 dB
(b) 3.16 × 10–5 W/m2
18. 74.8 dB
19. yes
20. noise barrier A
21. (a) 10–10 W/m2
(b) yes
64
F4B: Chapter 9A
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 1
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 2
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 3
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 4
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise 9A Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 9A Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 9A Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature
___________ ( )
E-Class Multiple Choice Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
65
4B Lesson Worksheet 9.0 (Refer to Book 4B P.9.3)
Objective: To review the highest common factor (H.C.F.) and the least common multiple (L.C.M.) of
integers, factorization of polynomials, domain of a function and the factor theorem.
Highest Common Factor (H.C.F.) and Least Common Multiple (L.C.M.)
Find the H.C.F. and L.C.M. of each of the following groups of numbers. [Nos. 1–2]
1. 12, 16 2. 15, 21 �Review Ex: 1, 2
12 = ( )2 × ( ) 15 = ( ) × ( )
16 = ( )4 21 = ( ) × ( )
∴ H.C.F. = ∴ H.C.F. =
L.C.M. = L.C.M. =
Factorization
Factorize each of the following expressions. [Nos. 3–11] �Review Ex: 3–11
3. x2 – 3x + 2 4. x2 + 10x + 25 5. 3x2 + 2xy – y2
= = =
6. x2 – 4xy + 4y2 7. 4x2 – 9y2 8. 2x + 2 + xy + y
= = =
9. 4x2 – 9(x + 1)2 10. x3 + 64 11. 8x3 – 27y3
= = =
12. (a) Factorize a2 – 16. 13. (a) Factorize 9a2 + 24ab + 16b2.
(b) Hence, factorize a4 – 16. (b) Hence, factorize 9a2 + 24ab + 16b2 – 3a – 4b.
� � 64 = ( )3
�Review Ex: 12–14
a2 + 2ab + b2 ≡ (a + b)2
a2 – 2ab + b2 ≡ (a – b)2 a2 – b2 ≡ (a + b)(a – b)
a3 + b3 ≡ (a + b)(a2 – ab + b2) a3 – b3 ≡ (a – b)(a2 + ab + b2)
66
Domain of a Function
Find the domain of each of the following functions. [Nos. 14–15]
14. f(x) = x2 – 3x + 2 15. f(x) =7
2
−x �Review Ex: 15, 16
Domain: ∵ The value of the denominator
cannot be 0.
∴ ( ) ≠ 0
∴ The domain is
.
Factor Theorem
16. Let f(x) = x3 + 5x2 + 3x – 9. 17. Let f(x) = x3 + 4x2 + x – 6. �Review Ex: 17
(a) Show that x – 1 is a factor of f(x). (a) Show that x + 2 is a factor of f(x).
(b) Factorize f(x). (b) Factorize f(x).
����Level Up Question����
18. It is given that x + 1 is a factor of f(x) = 2x3 – 7x2 + k.
(a) Find the value of k.
(b) Factorize f(x).
67
4B Lesson Worksheet 9.1 (Refer to Book 4B P.9.5)
Objective: To understand the concepts of H.C.F. and L.C.M. of polynomials.
Highest Common Factor (H.C.F.) and Least Common Multiple (L.C.M.)
(a) Among the common factors of two or more polynomials, if the polynomial P is the one having the
highest degree, then P is called the highest common factor of the polynomials.
(b) Among the common multiples of two or more polynomials, if the polynomial Q is the one having the
lowest degree, then Q is called the least common multiple of the polynomials.
Find the H.C.F. and L.C.M. of each of the following groups of polynomials. [Nos. 1–2]
1. x4, xy2 2. 6x5y, 9x3y2 � Ex 9A: 1, 2
x4 = ( ) 6x5y = ( ) × ( ) × ( ) × ( )
xy2 = ( ) × ( ) 9x3y2 = ( ) × ( ) × ( )
∴ H.C.F. =
L.C.M. =
Instant Example 1 Instant Practice 1
Find the H.C.F. and L.C.M. of x3(x + 1) and
x(x + 1)2.
x3(x + 1) = x3 × (x + 1)
x(x + 1)2 = x × (x + 1)2 ∴ H.C.F. = x(x + 1)
L.C.M. = x3(x + 1)2
Find the H.C.F. and L.C.M. of x2(x − 3) and
x(x − 3)5.
x2(x − 3) =
x(x − 3)5 = ∴ H.C.F. =
L.C.M. =
Find the H.C.F. and L.C.M. of each of the following groups of polynomials. [Nos. 3–8]
3. x2(x + 3)3, x3(x + 3) 4. x5(3x – 1)3, x3(3x – 1)2 � Ex 9A: 3–10
x2(x + 3)3 =
x3(x + 3) =
∴ H.C.F. =
L.C.M. =
5. (x + 2)(x − 4)3, (x + 2)2(x − 4)2 6. (x − 5)6(2x + 1), (x − 5)3(2x + 1)4
(x + 2)(x − 4)3 =
(x + 2)2(x − 4)2 =
∴ H.C.F. =
L.C.M. =
�
�Take the highest power in each group with the same base.
Take the lowest power in each group with the same base.
x and x + 1 are two bases.
( ) and ( ) are two bases.
68
7. 2(x − 1)4(x − 8), 6(x − 1)2(x − 8)3 8. (x + 2)(3x − 4), (x − 7)(3x − 4)2
Instant Example 2 Instant Practice 2
Find the H.C.F. and L.C.M. of x2 + 2x + 1 and
x2(x + 1).
x2 + 2x + 1 = (x + 1)2
x2(x + 1) = x2(x + 1) ∴ H.C.F. = x + 1
L.C.M. = x2(x + 1)2
Find the H.C.F. and L.C.M. of x2 − 3x + 2 and
x(x − 2)5.
x2 − 3x + 2 =
x(x − 2)5 = ∴ H.C.F. =
L.C.M. =
Find the H.C.F. and L.C.M. of each of the following groups of polynomials. [Nos. 9–12]
9. (x + 2)(x + 4)2, x2 + 5x + 4 10. x2 − 6x + 9, x2 + x − 12 � Ex 9A: 11–14
11. 2x2 − 3x + 1, 4x2 − 1 12. 2x2 + 6x + 4, x2 − 6x − 7
����Level Up Question����
13. Let f(x) = 3xy3 − 6x2y2 and g(x) = xy2 – 5x2y + 6x3.
(a) Factorize f(x) and g(x).
(b) Find the H.C.F. and L.C.M. of f(x) and g(x).
69
New Century Mathematics (Second Edition) 4B
9 Rational Functions
� Consolidation Exercise 9A
Level 1
Find the H.C.F. and L.C.M. of each of the following groups of polynomials. [Nos. 1–4]
1. (a) x4, xy4 (b) ab, a3b2
2. (a) uv2w4, u2v5w3 (b) p4q6r2, p7q3r9
3. (a) x2y3z, x2yz2, xy4z (b) abc2, a4bc, a3c
4. (a) 6p2q3, 4p3q2 (b) 8u3vw5, 9u7v8w
(c) 4y4z2, 10xy6z5 (d) 12a5b4, 15a3b2c, 5a4c3
Find the H.C.F. of each of the following groups of polynomials. [Nos. 5–7]
5. (a) x4(x – 3)2, x(x – 3)5
(b) x3(x + 5)(x – 2), x2(x + 5)4
6. (a) (x – 2)(x + 1)2(x + 4), (x + 1)(x + 4)(x – 2)3
(b) (x – 1)(x – 2)(2x + 3)3, (x – 1)(x – 2)2(2x + 3)2
7. (a) 9x2(x + 2)3, 3x3(x – 1)(x + 2)
(b) 8x4(x + 7)2(2x – 1)4, 20x(x + 7)3(2x – 1)2
Find the L.C.M. of each of the following groups of polynomials. [Nos. 8–10]
8. (a) x2(x – 3)2, x(x – 3)4
(b) (x – 9)3(x + 6)2, (x + 6)(x – 9)2
9. (a) x2(x – 1)(x – 4)2, x3(x – 1)2(x – 4)4
(b) (x + 2)(x – 4)2(3x – 2)3, (x – 4)(x + 2)2(3x – 2)
10. (a) 2(3x + 1)2(2x – 7)3, 6(3x + 1)(2x – 7)4
(b) 16x4(x – 5)3(x + 8)2, 12x3(x – 5)(x + 8)3
Find the H.C.F. and L.C.M. of each of the following groups of polynomials. [Nos. 11–14]
11. (x – 6)5(4x + 9)2, (x – 6)2(4x + 9)3, (x – 6)4(4x + 9)
12. x(x + 3)4(2x + 5)2, x2(x + 3)5, (x + 3)3(2x + 5)4
70
13. 5x2(x – 8)3(3x – 2), 10x(x – 8)(3x – 2)4, 15x4(x – 8)4(3x – 2)3
14. 9x(x + 1)(2x + 3)3, 15x2(x + 1)3(x + 2)5, 6x3(x + 2)2(2x + 3)4
15. (a) Factorize x2 + x – 2 and 2x2 + 4x.
(b) Find the H.C.F. of the two polynomials in (a).
16. (a) Factorize 6x2 – x – 1 and 8x2 – 8x + 2.
(b) Find the H.C.F. of the two polynomials in (a).
17. (a) Factorize x2 – 6x + 5 and 2x2 – 50.
(b) Find the L.C.M. of x2 – 6x + 5 and 2x2 – 50.
18. (a) Factorize 2x2 – 3x – 5 and 3x3 + 6x2 + 3x.
(b) Find the L.C.M. of 2x2 – 3x – 5 and 3x3 + 6x2 + 3x.
19. Let f(x) = x4 – 2x3 – 15x2 and g(x) = 27x – 3x3. Find the H.C.F. of f(x) and g(x).
20. Let h(x) = 98x3 – 28x2 + 2x and k(x) = 7x4 – 15x3 + 2x2. Find the H.C.F. of h(x) and k(x).
21. Let p(x) = 4x2 – 5x – 6 and q(x) = 12x3 + 9x2. Find the L.C.M. of p(x) and q(x).
22. Let u(x) = 3x2 + 14x – 5 and v(x) = 18x3 – 12x2 + 2x. Find the L.C.M. of u(x) and v(x).
23. Find the H.C.F. and L.C.M. of x2 – 8x + 16 and 5x2 – 10x – 40.
24. Find the H.C.F. and L.C.M. of x3 – 64x and x2 + 2x – 48.
25. Find the H.C.F. and L.C.M. of 4x2 – 4, 6x2 – 14x + 8 and 8x2 + 4x – 12.
26. Find the H.C.F. and L.C.M. of 3x2 + 7x – 20, 12x2 – 2x – 30 and 18x2 – 60x + 50.
Level 2
Find the H.C.F. and L.C.M. of each of the following groups of polynomials. [Nos. 27–32]
27. 2p2 – 18q2, 5p2 – 14pq – 3q2
28. 6a2 – 10ab – 4b2, 4a2 – 16b2
29. 6u2 – 21uv – 12v2, 12u2 + 12uv + 3v2
30. 2x3 + 3x2y – 2xy2, 2x3y + 2x3 – x2y2 – x2y
71
31. 45h2 – 5k2, 15h2 + 10hk – 5k2, 6h2 – 5hk + k2
32. 2m2 + 4mn + 2n2, 12m2 + 9mn – 3n2, m3 + n3
33. Let f(x) = x2 – 4x – 21 and g(x) = x3 + 2x2 – 5x – 6.
(a) Factorize f(x).
(b) Show that g(2) = 0. Hence, factorize g(x).
(c) Find the H.C.F. of f(x) and g(x).
34. Let p(x) = 3x2 + 20x + 12 and q(x) = 2x3 + 13x2 + 5x – 6.
(a) Factorize p(x).
(b) Show that 2x – 1 is a factor of q(x). Hence, factorize q(x).
(c) Find the L.C.M. of p(x) and q(x).
35. Let u(x) = x3 + 6x2 – 27x and v(x) = x3 – 2x2 – 15x + 36.
(a) Factorize u(x).
(b) Find the value of v(3). Hence, factorize v(x).
(c) Find the L.C.M. of u(x) and v(x).
36. Let h(x) = 9x3 + 30x2 + 24x and k(x) = 3x4 + 4x3 – 9x2 – 10x.
(a) Factorize h(x).
(b) Find the value of k(–2). Hence, factorize k(x).
(c) Find the H.C.F. of h(x) and k(x).
37. Let f(x) = x3 + kx2 + 8x – 4 and g(x) = x2 + 3x – 4, where k is a constant. It is given that x – 1 is a factor
of f(x).
(a) Find the value of k.
(b) Find the H.C.F. and L.C.M. of f(x) and g(x).
38. Let u(x) = 4x3 + 2ax2 + 13x + b and v(x) = ax2 – 2x – b, where a and b are constants. It is given that the
H.C.F. of u(x) and v(x) is 2x + 1.
(a) Find the values of a and b.
(b) Find the L.C.M. of u(x) and v(x).
39. Let R = 2x3 + 3x2 – 1 and S = 2x3 + 7x2 + 2x – 3. Find the H.C.F. of R and S.
40. Let P = x3 + 2x2 – 15x and Q = x3 + 6x2 – 15x – 100. Find the H.C.F. of P and Q.
41. Let H = 9x3 + 15x2 – 6x and K = 3x3 + 11x2 + 8x – 4. Find the L.C.M. of H and K.
42. Let M = 4x4 – 24x3 + 36x2 and N = 8x3 – 30x2 + 16x + 6. Find the L.C.M. of M and N.
72
43. Let r(x) = 4x2 – 18x + 20 and s(x) = r(x) + 5x3 – 19x2 + 18x.
(a) Find s(x).
(b) Find the L.C.M. of r(x) and s(x).
44. Let f(x) = 15x2 + 5x – 20 and g(x) = 18x3 – 26x + 8 – x[f(x)].
(a) Find g(x).
(b) Find the H.C.F. of f(x) and g(x).
45. Let f(x) = x3 + 2x2 – 24x and g(x) = x3 – 27x + 54.
(a) Find the value of g(3). Hence, factorize g(x).
(b) Let P and Q be the H.C.F. and L.C.M. of f(x) and g(x) respectively.
(i) Find P and Q.
(ii) Are the results of f(x)g(x) and P × Q equal? Explain your answer.
46. Let R = x3 – x2y + xy2 – y3 and S = x3y – x3 + xy3 – xy2.
(a) Find the H.C.F. and L.C.M. of R and S.
(b) Using the results of (a), find R × S.
47. The H.C.F. and L.C.M. of two polynomials are ab4 and 4a3b5c2 respectively. If the first polynomial is
a3b4, find the second polynomial.
48. The H.C.F. and L.C.M. of three polynomials are uv and 8u3v2w6 respectively. If the first and the
second polynomials are u3vw4 and 4u2vw6 respectively, find the third polynomial.
49. Let f(x) = 6x2 + 13x – 28 and g(x) be a polynomial. If the H.C.F. and L.C.M. of f(x) and g(x) are 3x – 4
and 6x3 + 25x2 – 2x – 56 respectively, find g(x).
50. Let f(x) = x3 + kx – 30, where k is a constant. It is given that x + 2 is a factor of f(x).
(a) Find the value of k.
(b) Factorize f(x).
(c) Let g(x) = x2 – 3x – 10 and h(x) be a polynomial. If the H.C.F. and L.C.M. of g(x) and h(x) are
x – 5 and f(x) respectively, find h(x).
73
Answers
Consolidation Exercise 9A
1. (a) H.C.F. = x, L.C.M. = x4y4
(b) H.C.F. = ab, L.C.M. = a3b2
2. (a) H.C.F. = uv2w3, L.C.M. = u2v5w4
(b) H.C.F. = p4q3r2, L.C.M. = p7q6r9
3. (a) H.C.F. = xyz, L.C.M. = x2y4z2
(b) H.C.F. = ac, L.C.M. = a4bc2
4. (a) H.C.F. = 2p2q2, L.C.M. = 12p3q3
(b) H.C.F. = u3vw, L.C.M. = 72u7v8w5
(c) H.C.F. = 2y4z2, L.C.M. = 20xy6z5
(d) H.C.F. = a3, L.C.M. = 60a5b4c3
5. (a) x(x – 3)2 (b) x2(x + 5)
6. (a) (x – 2)(x + 1)(x + 4)
(b) (x – 1)(x – 2)(2x + 3)2
7. (a) 3x2(x + 2)
(b) 4x(x + 7)2(2x – 1)2
8. (a) x2(x – 3)4
(b) (x – 9)3(x + 6)2
9. (a) x3(x – 1)2(x – 4)4
(b) (x + 2)2(x – 4)2(3x – 2)3
10. (a) 6(3x + 1)2(2x – 7)4
(b) 48x4(x – 5)3(x + 8)3
11. H.C.F. = (x – 6)2(4x + 9),
L.C.M. = (x – 6)5(4x + 9)3
12. H.C.F. = (x + 3)3, L.C.M. = x2(x + 3)5(2x +
5)4
13. H.C.F. = 5x(x – 8)(3x – 2),
L.C.M. = 30x4(x – 8)4(3x – 2)4
14. H.C.F. = 3x,
L.C.M. = 90x3(x + 1)3(x + 2)5(2x + 3)4
15. (a) x2 + x – 2 = (x – 1)(x + 2),
2x2 + 4x = 2x(x + 2)
(b) x + 2
16. (a) 6x2 – x – 1 = (2x – 1)(3x + 1),
8x2 – 8x + 2 = 2(2x – 1)2
(b) 2x – 1
17. (a) x2 – 6x + 5 = (x – 1)(x – 5),
2x2 – 50 = 2(x + 5)(x – 5)
(b) 2(x – 1)(x + 5)(x – 5)
18. (a) 2x2 – 3x – 5 = (x + 1)(2x – 5),
3x3 + 6x2 + 3x = 3x(x + 1)2
(b) 3x(x + 1)2(2x – 5)
19. x(x + 3)
20. x(7x – 1)
21. 3x2(x – 2)(4x + 3)
22. 2x(x + 5)(3x – 1)2
23. H.C.F. = x – 4, L.C.M. = 5(x + 2)(x – 4)2
24. H.C.F. = x + 8, L.C.M. = x(x – 6)(x + 8)(x –
8)
25. H.C.F. = 2(x – 1),
L.C.M. = 4(x – 1)(x + 1)(2x + 3)(3x – 4)
26. H.C.F. = 3x – 5,
L.C.M. = 2(x + 4)(2x + 3)(3x – 5)2
27. H.C.F. = p – 3q,
L.C.M. = 2(p – 3q)(p + 3q)(5p + q)
28. H.C.F. = 2(a – 2b),
L.C.M. = 4(a – 2b)(a + 2b)(3a + b)
29. H.C.F. = 3(2u + v),
L.C.M. = 3(2u + v)2(u – 4v)
30. H.C.F. = x(2x – y),
L.C.M. = x2(y + 1)(2x – y)(x + 2y)
31. H.C.F. = 3h – k,
L.C.M. = 5(3h – k)(3h + k)(2h – k)(h + k)
32. H.C.F. = m + n,
L.C.M. = 6(m + n)2(4m – n)(m2 – mn + n2)
33. (a) (x + 3)(x – 7)
(b) (x + 1)(x – 2)(x + 3)
(c) x + 3
34. (a) (x + 6)(3x + 2)
(b) (x + 1)(x + 6)(2x – 1)
(c) (x + 1)(x + 6)(2x – 1)(3x + 2)
35. (a) x(x – 3)(x + 9)
(b) 0, (x – 3)2(x + 4)
(c) x(x – 3)2(x + 4)(x + 9)
36. (a) 3x(x + 2)(3x + 4)
(b) 0, x(x + 1)(x + 2)(3x – 5)
(c) x(x + 2)
74
37. (a) –5
(b) H.C.F. = x – 1,
L.C.M. = (x + 4)(x – 1)(x – 2)2
38. (a) a = 8, b = 3
(b) (x + 3)(2x + 1)2(4x – 3)
39. (x + 1)(2x – 1)
40. x + 5
41. 3x(x + 2)2(3x – 1)
42. 4x2(x – 1)(x – 3)2(4x + 1)
43. (a) 5x3 – 15x2 + 20
(b) 10(x + 1)(x – 2)2(2x – 5)
44. (a) 3x3 – 5x2 – 6x + 8
(b) (x – 1)(3x + 4)
45. (a) 0, (x – 3)2(x + 6)
(b) (i) P = x + 6, Q = x(x – 3)2(x – 4)(x +
6)
(ii) yes
46. (a) H.C.F. = x2 + y2,
L.C.M. = x(y – 1)(x – y)(x2 + y2)
(b) x(y – 1)(x – y)(x2 + y2)2
47. 4ab5c2
48. 8uv2
49. (x + 2)(3x – 4)
50. (a) –19
(b) (x + 2)(x + 3)(x – 5)
(c) (x + 3)(x – 5)
75
F4B: Chapter 9B
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 5
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 6
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 7
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 8
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 9
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 10
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Maths Corner Exercise 9B Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 9B Level 2
○ Complete and Checked ○ Problems encountered
Teacher’s Signature
___________
76
○ Skipped ( )
Maths Corner Exercise 9B Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature
___________ ( )
E-Class Multiple Choice Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
77
4B Lesson Worksheet 9.2A (Refer to Book 4B P.9.12)
Objective: To know the definition of rational functions.
Definition of Rational Functions
If a function can be expressed in the form
Q
P (where P and Q are polynomials, and Q ≠ 0),
then the function is called a rational function.
Instant Example 1 Instant Practice 1
Find the domain of each of the following rational
functions.
(a) f(x) =6
3
+
−
x
x (b) g(x) =
2)1(
2
−x
(a) ∵ The value of the denominator x + 6 cannot
be 0.
∴ x + 6 ≠ 0
i.e. x ≠ −6
∴ The domain is all real numbers except −6.
(b) ∵ The value of the denominator (x − 1)2
cannot be 0.
∴ (x − 1)2 ≠ 0
x − 1 ≠ 0
i.e. x ≠ 1
∴ The domain is all real numbers except 1.
Find the domain of each of the following rational
functions.
(a) f(x) =5
3
−
+
x
x (b) g(x) =
2)2(
9
+x
(a) ∵ The value of the denominator ( )
cannot be ( ).
∴ ( ) ≠ 0
i.e. x ≠ ( )
∴
(b) ∵ The value of the denominator ( )
cannot be ( ).
∴ ( ) ≠ 0
i.e. x ≠ ( )
∴
Find the domain of each of the following rational functions. [Nos. 1–2]
1. f(x) =9
73
+
+
x
x 2. g(x) =
2)5(
14
−
−
x
x � Ex 9B: 1, 2
∵ The value of the denominator ( )
cannot be ( ).
∴
�
x
1is undefined when x = 0.
78
Instant Example 2 Instant Practice 2
Simplify
xx
x
32
962 +
+.
xx
x
32
962 +
+=
)32(
)32(3
+
+
xx
x
=x
3
Simplify
xx
x
88
552 +
+.
xx
x
88
552 +
+=
Simplify each of the following rational functions. [Nos. 3–12]
3. xx
x
2
32
2
+= 4.
xx
xx
66
122
2
+
++= � Ex 9B: 3–8
5. xx
x
123
162
2
−
−= 6.
9
962
2
−
++
x
xx=
7. 169
192
2
+−
−
xx
x= 8.
168
122
2
+−
−−
xx
xx=
9. 572
2542
2
++
−
xx
x= 10.
32
1272
2
−−
+−
xx
xx=
11. 43
232
2
−+
−−
xx
xx= 12.
7103
1342
2
++
−+
xx
xx=
����Level Up Question����
13. (a) Find the domain of the rational function f(x) =2)4(
53
+
+
x
x.
(b) Give another example of a rational function which has the same domain as f(x).
�Take out the common factor 3.
�Take out the common factor x.
79
4B Lesson Worksheet 9.2B (Refer to Book 4B P.9.15)
Objective: To perform the multiplication and division of rational functions.
Multiplication and Division of Rational Functions
Step 1: Factorize the numerator and the denominator of each rational function in an expression.
Step 2: Reduce the fraction.
Step 3: Multiply the rational functions after reducing the fraction.
Instant Example 1 Instant Practice 1
Simplify
13 +x
x×
xx
x
7
132 +
+.
13 +x
x×
xx
x
7
132 +
+
=13 +x
x×
)7(
13
+
+
xx
x
=7
1
+x
Simplify
xx
x
+
−−2
5×
5+x
x.
xx
x
+
−−2
5×
5+x
x
=) (
) (×
) (
) (
=
Simplify each of the following expressions. [Nos. 1–6]
1. 3
2
5
)4(
4
4
x
x
x
x +×
+ 2.
x
x
x
x
9
)2(
2
3 23−
×
− � Ex 9B: 9–15
=) (
) ( =
) (
) (×
) (
) (
=
3. )2)(1(
66
1
)1(
−−
+×
+
−
xx
x
x
xx 4.
52
9
+x×
xx
xx
3
522
2
+
+
=) (
) (×
) (
) ( =
=
5. xx
x
34
42 +
−×
xx
x
287
82
2
− 6.
)3)(2(
93 2
++
+
xx
xx×
2
42
−
−
x
x
=
=
�
� Reduce the fraction after factorizing the numerator and the denominator.
80
Instant Example 2 Instant Practice 2
Simplify
2
23
−
+
x
x÷
x
x 69 +.
2
23
−
+
x
x÷
x
x 69 +=
2
23
−
+
x
x×
69 +x
x
=2
23
−
+
x
x×
)23(3 +x
x
=)2(3 −x
x
Simplify
x
x 88 −÷
3
22
+
−
x
x.
x
x 88 −÷
3
22
+
−
x
x=
) (
) (×
) (
) (
=
Simplify each of the following expressions. [Nos. 7–12]
7. 65
2
+
−
x
x÷
x
xx
7
22 − 8.
2
2
4
5
x
xx −÷
9
153
+
−
x
x � Ex 9B: 16–20
=) (
) (×
) (
) ( =
) (
) (×
) (
) (
=
=
9. xx
x
66
242 +
−÷
x
xx
9
)3)(12( +− 10.
)4)(52(
322
++
−−
xx
xx÷
xx
x
123
32 +
−
=) (
) (×
) (
) ( =
=
11. 155
)5)(2(
+
−+
x
xx÷
9
1022 −
−
x
x 12.
2
34
44
33 2
2 +
−−÷
++
−
x
xx
xx
x
=
=
����Level Up Question����
13. Simplify424
)1)(2( 22
+
−÷
+×
+
−+
x
xx
x
x
x
xx.
� Convert the operation
from ÷ to ×.
81
New Century Mathematics (Second Edition) 4B
9 Rational Functions
� Consolidation Exercise 9B
Level 1
Find the domain of each of the following rational functions. [Nos. 1–4]
1. 5
2
−x 2.
x
x
−
+
4
2
3. 2)3(
1
+x 4.
9
32 +
−
x
x
Simplify each of the following rational functions. [Nos. 5–16]
5. 93
6
+x 6.
xx
x
168
42
3
+
7. )3(2
124
−
−
xx
x 8.
xx
x
2
1052 +
+
9. 2425
52
x
x
−
− 10.
98
12 −−
+
xx
x
11. 12
932 −+
−
xx
x 12.
2
2
)2(
82
−
−+
x
xx
13. 12
)5)(3(2 −−
−+
xx
xx 14.
992
3442
2
++
−+
xx
xx
15. 32162
24102
2
+−
+−
xx
xx 16.
5656
6492
2
−+
−
xx
x
Simplify each of the following expressions. [Nos. 17–32]
17. 32
1
)1(
2
x
x
x
x +×
+ 18.
3
2
)3(
8
4
3
−×
−
h
h
h
h
19. 2
)5)(4(
5
22
xx
xx
x
x
+
++×
+
+ 20.
)6(
9
7
62
23
−
−×
+
−
kk
k
kk
kk
82
21. xx
x
x
xx
23
123
82
482
2
−
+×
+
− 22.
pp
p
pp
pp
+
−×
−
+22
2
3
186
84
26
23. )54)(1(
3
62
12 22
−−
−×
−
+−
xx
xx
x
xx 24.
164
2
187
82 2
2
23
−
−+×
−−
−
t
tt
tt
tt
25. 24
3
)2()2( +÷
+ x
x
x
x 26.
)1)(5(
7
1
)5(
−+÷
−
+
uu
u
u
uu
27. )5(
12
6
22
2
+
+÷
+
+
xx
x
xx
xx 28.
6
)12(3
9
22
23
+
−÷
−
−
v
vv
vv
vv
29. xx
x
x
x
−
+÷
−
+22 3
7
19
26 30.
ss
s
s
ss
62
4
155
1262
22
+
−÷
+
−
31. 96
4
)2)(3(
822
223
++
−÷
−+
−
xx
xx
xx
xx 32.
34
1
68
32 2
2
2
+
−÷
+
−−
z
z
zz
zz
Level 2
Find the domain of each of the following rational functions. [Nos. 33–36]
33. 6)1(
742 ++
−
x
x 34.
xx
x
5
382 −
+
35. 124
72 −−
+
xx
x 36.
2
2
)8(9
53
−−
++
x
xx
Simplify each of the following rational functions. [Nos. 37–44]
37. 2
4
936
464
x
x
−
− 38.
27
1523
2
−
−−
x
xx
39. 22
2
16
520
yx
yxy
−
− 40.
22
2
45
123
nmnm
mnm
+−
−
41. 22
22
4
812
qp
qpqp
−
++ 42.
22
22
2
65
khkh
khkh
+−
−+
43. 22
22
49
2110
ba
baba
−
+− 44.
22
33
82
8
vuvu
vu
−−
+
83
Simplify each of the following expressions. [Nos. 45–56]
45. 81
168
127
9102
2
2
2
−
+−×
+−
+−
x
xx
xx
xx
46. 65
82
96
122
2
2
2
++
−−÷
++
−−
xx
xx
xx
xx
47. 34
183
32283
642
2
2
2
+−
−+×
+−
−
xx
xx
xx
x
48. 2510
145
152
1872
2
2
2
+−
−+÷
−−
−+
xx
xx
xx
xx
49. 22
22
36
209
305
5
sr
srsr
sr
sr
−
+−÷
+
−
50. 3322
22
27
424
3612
183
cb
bc
cbcb
cbcb
+
−×
+−
−−
51. )1(3
32223
−×−
−÷
−a
aa
a
aa
a
52. )2)(1(
93
14
32 −+
+÷
+×
−
+
xx
x
x
x
x
x
53. 1)3(
2
2
3 3
2
22
−
−÷
+
−−×
−
+
t
tt
t
tt
t
tt
54. 12
1
44
2
84
2232 +
×++
−÷
+
+
wwww
w
ww
w
55. kh
kh
khkh
khkhkh
+
−÷
++
++×+
33
22
22 8
34
42)3(
56. zy
zy
zy
yzy
zyzy
yzy
5
325
352
25 2
22
23
+
+×
−
−÷
++
−
84
Answers
Consolidation Exercise 9B
1. all real numbers except 5
2. all real numbers except 4
3. all real numbers except –3
4. all real numbers
5. 3
2
+x
6. )2(2
2
+x
x
7. x
2
8. x
5
9. 52
1
+−
x
10. 9
1
−x
11. 4
3
+x
12. 2
4
−
+
x
x
13. 4
5
−
−
x
x
14. 3
12
+
−
x
x
15. )4(2
6
−
−
x
x
16. 72
83
+
+
x
x
17. )1(
22 +xx
18. )3(2
1
−hh
19. x
x 4+
20. 7
9
+
−
k
k
21. 23
)12(6
−
−
x
x
22. )2(
)3(3
−
−
pp
p
23. )54(2
)1(
−
−
x
xx
24. )9(2
)1(2
−
−
t
tt
25. 2
2
)2( +x
x
26. 7
)5( 2+u
27. 6
)5(
+
+
x
xx
28. )9(3
6
−
+
v
v
29. 7
2
+x
x
30. )2(5
12 2
+s
s
31. 2
)3(2
−
+
x
xx
32. )1(2
3
−
−
zz
z
33. all real numbers
34. all real numbers except 0 and 5
35. all real numbers except –2 and 6
36. all real numbers except 5 and 11
37. 9
)4(4 2 +x
38. 93
522 ++
+
xx
x
39. yx
y
+4
5
40. nm
m
−
3
41. qp
qp
−
+
2
6
42. kh
kh
−
+ 6
43. ba
ba
7
3
+
−
44. vu
vuvu
4
42 22
−
+−
45. )9)(3(
)4)(1(
+−
−−
xx
xx
46. 1
47. )43)(1(
)8)(6(
−−
++
xx
xx
48. )7)(3(
)9)(5(
++
+−
xx
xx
85
49. )4(5
6
sr
sr
−
−
50. 22 93
4
cbcb +−−
51. 2
52. )2(3 +x
x
53. 3
1
+t
54. )2(4
12
−
+
w
w
55. kh 2
1
−
56. zy
zy
+
−
86
F4B: Chapter 9C
Date Task Progress
Lesson Worksheet
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
Book Example 11
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 12
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 13
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 14
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 15
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 16
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Book Example 17
○ Complete ○ Problems encountered ○ Skipped
(Video Teaching)
Consolidation Exercise
○ Complete and Checked ○ Problems encountered ○ Skipped
(Full Solution)
87
Maths Corner Exercise 9C Level 1
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 9C Level 2
○ Complete and Checked ○ Problems encountered ○ Skipped
Teacher’s Signature
___________ ( )
Maths Corner Exercise 9C Multiple Choice
○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature
___________ ( )
E-Class Multiple Choice Self-Test
○ Complete and Checked ○ Problems encountered ○ Skipped Mark:
_________
88
4B Lesson Worksheet 9.3A (Refer to Book 4B P.9.21)
Objective: To perform the addition and subtraction of rational functions.
Addition and Subtraction of Rational Functions
Step 1: Simplify each rational function in an expression.
Step 2: Find the L.C.M. of the denominators of the rational functions.
Step 3: Take the L.C.M. as the denominator of each rational function.
Step 4: Perform addition or subtraction of the new numerators.
Step 5: Simplify the results obtained.
Instant Example 1 Instant Practice 1
Simplify1
1
3
2
++
+ xx.
1
1
3
2
++
+ xx
=)1)(3(
)3()1(2
++
+++
xx
xx
=)1)(3(
322
++
+++
xx
xx
=)1)(3(
53
++
+
xx
x
Simplify3
2
2
1
−+
+ xx.
3
2
2
1
−+
+ xx
=) )( (
) (2) ( +
=) )( (
) (
=
Simplify each of the following expressions. [Nos. 1–4]
1. 5
4
2
3
++
− xx 2.
4
1
)4(
4
++
+ xxx �Ex 9C: 3, 4, 7, 9, 11
= =
3. 3
1
62
12 −
+− xxx
4. 5
1
25
152 +
+−
+
xx
x
=3
1
)3)( (
1
−+
− xx =
=
�
⊳ The L.C.M. of the two denominators is (x + 3)(x + 1).
In the final step, take out the common factor
in the numerator.
Remember to simplify the results.
a2 – b2 ≡ (a + b)(a – b)
Add the two polynomials in the numerator and simplify the result.
L.C.M. of denominators = ( )( )
89
Instant Example 2 Instant Practice 2
Simplify5
2
6
3
−−
− xx.
5
2
6
3
−−
− xx=
)5)(6(
)6(2)5(3
−−
−−−
xx
xx
=)5)(6(
122153
−−
+−−
xx
xx
=)5)(6(
3
−−
−
xx
x
Simplifyxx
1
1
4−
−.
xx
1
1
4−
−=
) )( (
) () (4 −
=) )( (
) (
=) )( (
) (
Simplify each of the following expressions. [Nos. 5–8]
5. 4
1
52
2
−−
+ xx 6.
1
3
)1)(1(
4
−−
−+ xxx
x �Ex 9C: 5, 6, 8, 10, 12
= =
7. 3
2
6
32 +
−−+ xxx
x 8.
6
1
65
522 −
−−−
−
xxx
x
=3
2
) )( (
3
+−
x
x =
=
����Level Up Question����
9. Simplify each of the following expressions.
(a) xxxx
3
)2(
2
2
4−
++
+ (b)
xxxxx
x 1
5
3
)5(
72
+−
−−
90
4B Lesson Worksheet 9.3B (Refer to Book 4B P.9.25)
Objective: To perform mixed arithmetic operations of rational functions.
Mixed Arithmetic Operations of Rational Functions
The mixed arithmetic operations of rational functions follow the principle of ‘multiplication and division
first, then addition and subtraction’.
Instant Example 1 Instant Practice 1
Simplify2
11
2
2
−×
−+
− xx
x
x.
2
11
2
2
−×
−+
− xx
x
x
=)2(
1
2
2
−
−+
− xx
x
x
=)2(
)1(2
−
−+
xx
xx
=)2(
13
−
−
xx
x
Simplify54
3
11
1
−×
+−
+ xx
x
x.
54
3
11
1
−×
+−
+ xx
x
x
=) )( (
) (
1
1−
+x
=) )( (
) () ( −
=) )( (
) (
Simplify each of the following expressions. [Nos. 1–4]
1. 12
2
1
1
12
1
+
+×
−+
+ x
x
xx 2.
3
3
3
2
9
22 −
×+
−− xxx
x �Ex 9C: 23
= =) )( (
6
) )( (
2−
x
=
3. 12
1
592
1352 −
−−+
−×
+
xxx
x
x
x 4.
x
x
xxx
x 1
4
1
45
142
−×
−+
+−
+
= =
�
⊳ Do ‘×’ first.
91
Instant Example 2 Instant Practice 2
Simplify11
45
−÷
−−
x
x
xx.
11
45
−÷
−−
x
x
xx=
x
x
xx
1
1
45 −×
−−
=xx
45−
=x
1
Simplifyx
x
xx
23
2
2 +÷+
+.
x
x
xx
23
2
2 +÷+
+=
) (
) (3
2
2×+
+ xx
=) (
) (
2
2+
+x
=) (
) (
Simplify each of the following expressions. [Nos. 5–8]
5. x
x
xx 3
4
1
2
4
6 +÷
+−
+ 6.
x
x
xx −÷
−−
+ 55
1
3
4 �Ex 9C: 24
= = ) (
) (
5
1
3
4 −×
−−
+ xx
=
7.
−+÷
+
−
4
21
23
22
xx
xx 8. 1
1
1−
−÷
+ x
x
x
x
= =
����Level Up Question����
9. Simplify
xx
x
1
1
−
+.
Convert
÷ to ×.
92
New Century Mathematics (Second Edition) 4B
9 Rational Functions
� Consolidation Exercise 9C
Level 1
Simplify each of the following expressions. [Nos. 1–28]
1. 5
7
5
3
−+
− xx 2.
6
7
6
32
+
−+
+
+
x
x
x
x
3. 4
2
4
6
+−
+ xx 4.
9
2
9
45
−
+−
−
−
x
x
x
x
5. 3
4
2
4
++
+ xx 6.
23
1
52
3
++
+ xx
7. 5
3
12
3
+−
+ xx 8.
25
3
13
2
−−
− xx
9. )4(
1
)1(
1
++
− xxxx 10.
)2)(3(
3
)2)(2(
2
+−−
+− xxxx
11. xx
x
−+
− 8
64
8
2
12. 2
2
2 53
37
35
89
xx
xx
xx
x
+−
−−
+−
−
13. )3)(5(
3
5
2
−++
+ xxx 14.
)25)(14(
7
14
5
+−−
− xxx
15. 9
3
)4(
8222 −
++
−
−
x
x
x
x 16.
)9)(7(
213
482
1222 +−
−−
−+
−
xx
x
xx
x
17. 43
4
43
52 +
+−+ xxx
18. 2
5
2115 2 −−
+− xxx
x
19. 4129
7
23
42 +−
+− xxx
20. 4
2
162 +−
− xx
x
21. 86
5
43
222 ++
+−+ xxxx
22. 54
3
152
422 −+
−−+ xxxx
23. xxx
5
1
2
1
3−
++
− 24.
4
2
5
3
3
1
−+
−−
− xxx
25. 4
3
9
8
4
2
−×
+
−+
− xx
x
x 26.
3
5
6
1
5
2
+
+÷
−−
+ x
x
xx
27. 5
34
5
7
−
−×−
− x
x
xx 28.
1
9
2
3
9
4
−
−÷
++
− x
x
xx
93
Level 2
Simplify each of the following expressions. [Nos. 29–42]
29.
4
51
4
−+
x
30. 7
22
5−
+y
31. 222
22
yx
yx
yxy
y
−
−+
− 32.
2233
2
yxyx
yx
yx
x
++
+−
−
33. )92)(4(
73
92
4
4
4
−−
−−
−+
− xx
x
xx 34.
1
1
)1)(6(6
3
+−
+−−
− xxx
x
x
35. 1
1
1
2
1
223 +−
−+
+
−−
+ xx
x
x
x
x 36.
1
2
1
1
1
142 −
++
++
+
xxx
x
37. 22
11
xy
y
yx
x
xy
yx −+
−−
− 38.
22 )(
21
yx
yx
xyx
y
yx −
+−
−+
−
39. 222 )(
12
yx
yx
yxyx
x
+
−+
++
− 40.
33
22
2
2
yx
yxyx
xyx
yx
yx +
+−−
−
+−
−
41. 25
4
2
5
5
122
2
−
+×
−
−+
+ x
x
xx
xx
x 42.
−−
−÷
+−
−−
xxx
x
xx
xx
62
8
1242110
28322
2
43. If x =u
u1
+ and y =u
u1
− , express
xy
x
x
y 22 −− in terms of u. Give the answer in the simplest form.
44. If x =nm
mn
+ and y =
nm
mn
−, express
++
yxyx
11)( in terms of m and n. Give the answer in the
simplest form.
45. (a) Show that x + 1 is a factor of x3 – 9x2 + 14x + 24.
(b) Factorize x3 – 9x2 + 14x + 24.
(c) Simplify
24149
168
4
21
23
2
++−
+−×
−−
xxx
xx
x.
46. (a) Let k be a constant. Find the value of k such that x + 2 is a factor of x3 – 5x2 + kx + 12.
(b) Simplify
30
2416102
5
1
2
32
232
−−
+−−÷
+
−+
+
−
xx
xxx
x
x
x
x.
47. Consider the quadratic function z = −28 + 10x − x2.
(a) Find the maximum value of z and the corresponding value of x.
(b) Simplify
44
42
4
1
2
82
2
22
3
++
++÷
−×
+
−
zz
zz
zzz
z and express the answer in terms of z.
(c) Using the results of (a) and (b), find the minimum value of
44
42
4
1
2
82
2
22
3
++
++÷
−×
+
−
zz
zz
zzz
z.
94
Answers
Consolidation Exercise 9C
1. 5
10
−x
2. 6
43
+
−
x
x
3. 4
4
+x
4. 9
)3(2
−
−
x
x
5. )3)(2(
)52(4
++
+
xx
x
6. )23)(52(
)1(11
++
+
xx
x
7. )5)(12(
)4(3
++
−
xx
x
8. )25)(13(
1
−−
−
xx
x
9. )4)(1(
32
+−
+
xxx
x
10. )3)(2)(2( −+−
−xxx
x
11. x + 8
12. 3
13. )3)(5(
32
−+
−
xx
x
14. )25)(14(
325
+−
+
xx
x
15. )3)(4(
103
−−
−
xx
x
16. )9)(8(
6
++
+−
xx
x
17. )43)(1(
14
+−
+
xx
x
18. )15)(2(
245
−−
−
xx
x
19. 2)23(
112
−
−
x
x
20. )4)(4(
8
−+
−
xx
x
21. )4)(2)(1(
17
++−
−
xxx
x
22. )3)(1(
1
−− xx
23. )1)(1(
5
−+
+
xxx
x
24. )5)(4)(3(
)27(2
−−−
−
xxx
x
25. )9)(4(
65
+−
−
xx
x
26. )6)(5(
15
−+
−
xx
x
27. )5(
)4(3
−
+
xx
x
28. )9)(2(
57
−+
+
xx
x
29. 1
)4(4
+
−
x
x
30. )2(7
21
+
−
y
y
31. ))((
3
yxyx
x
−+
32. ))(( 22
2
yxyxyx
y
++−
33. )92)(4(
)5(9
−−
−
xx
x
34. )1)(6(
9
+−
+
xx
x
35. 1
3
+x
36. )1)(1)(1(
22
3
+−+ xxx
x
37. 22
)1)((
yx
xyyx −−
38. 2
2
)(
2
yxx
y
−−
39. 2
2
))((
4
yxyx
x
+−
40. )( yxx
y
+
41. )5)(2(
6
+−−
xx
42. 4
4
−x
x
43. )1)(1)(1(
22
2
+−+−
uuu
u
95
44. ))((
4 2
nmnm
m
−+
45. (b) (x + 1)(x – 4)(x – 6)
(c) 1
1
+x
46. (a) –8
(b) )2(2
53
+
−
x
x
47. (a) maximum value: –3, x = 5
(b) z
1 (c)
3
1−