TABLE OF LEARNING MATERIALS I-1
MATHEMATICS (6A-2A) / PENCIL SKILLS PROGRAMME (ZI-ZII)
The black mark (
) indicates Starting Points.
SCT: Standard Completion Time (Min./Sheet)
This is the time in which the student should complete
each worksheet, including time taken for corrections.
(SCT is not used in Levels 6A-5A and ZI-ZII.)
December 2012
Pencil Skills Programme
ZI
ZII
6A
5A
4A
SCT
3A
SCT
2A
SCT
1-010
Colouring 1
001-010
Counting (Up to 5) 1
Number Reading Exercises (Up to 30) 1
Number Tracing Exercises 1
0.5–2
Numbers up to 100 Part 1
0.5–2
Review up to 3A
1–2
11-020
Colouring 2
011-020
Counting (Up to 5) 2
Number Reading Exercises (Up to 30) 2
Number Tracing Exercises 2
0.5–2
Numbers up to 100 Part 2
0.5–2
Adding 4 Part 1 (Up to 12+4)
1–2
21-030
Straight Lines 1
021-030
Counting (Up to 5) 3
Number Reading Exercises (Up to 30) 3
Number Tracing Exercises 3
0.5–2
Numbers up to 100 Part 3
0.5–2
Adding 4 Part 2 (Up to 16+4)
1–2
31-040
Straight Lines 2
031-040
Counting (Up to 10) 1
Number Reading Exercises (Up to 30) 4
Number Tracing Exercises 4
0.5–2
Numbers up to 100 Part 4
0.5–2
Adding 5 Part 1 (Up to 12+5)
1–2
41-050
Straight Lines 3
041-050
Counting (Up to 10) 2
Number Reading Exercises (Up to 30) 5
Number Writing Exercises up to 10 Part 1 0.5–2
Numbers up to 100 Part 5
0.5–2
Adding 5 Part 2 (Up to 15+5)
1–2
51-060
Curved Lines 1
051-060
Counting (Up to 10) 3
Number Reading Exercises (Up to 30) 6
Number Writing Exercises up to 10 Part 2 0.5–2
Numbers up to 100 Part 6
0.5–2
Adding up to 5 Part 1
1–2
61-070
Curved Lines 2
061-070
Counting (Up to 10) 4
Number Reading Exercises (Up to 30) 7
Number Writing Exercises up to 10 Part 3 0.5–2
Numbers up to 120
1–2
Adding up to 5 Part 2
1–2
71-080
Curved Lines 3
071-080
Counting (Up to 10) 5
Number Reading Exercises (Up to 30) 8
Number Writing Exercises up to 10 Part 4 0.5–2
Adding 1 Part 1 (Up to 12+1)
1–2
Adding 6 Part 1 (Up to 12+6)
1–2
81-090
Curved Lines 4
081-090
Counting (Up to 10) 6
Number Reading Exercises (Up to 30) 9
Number Writing Exercises up to 10 Part 5 0.5–2
Adding 1 Part 2 (Up to 18+1)
1–2
Adding 6 Part 2 (Up to 14+6)
1–2
91-100
Curved Lines 5
091-100
Counting (Up to 10) 7
Number Reading Exercises (Up to 30) 10
Number Writing Exercises up to 10 Part 6 0.5–2
Adding 1 Part 3 (Up to 24+1)
1–2
Adding 7 Part 1 (Up to 11+7)
1–2
1-010
Shapes and Pictures 1
101-110
Number Reading Exercises (Up to 10) 1
Sequence of Numbers (Up to 30) 1
Number Writing Exercises up to 20 Part 1 0.5–2
Adding 1 Part 4 (Up to 30+1)
1–2
Adding 7 Part 2 (Up to 13+7)
1–2
11-020
Shapes and Pictures 2
111-120
Number Reading Exercises (Up to 10) 2
Sequence of Numbers (Up to 30) 2
Number Writing Exercises up to 20 Part 2 0.5–2
Adding 1 Part 5 (Up to 60+1)
1–2
Adding up to 7 Part 1
1–2
21-030
Shapes and Pictures 3
121-130
Number Reading Exercises (Up to 10) 3
Sequence of Numbers (Up to 30) 3
Number Writing Exercises up to 30 Part 1 0.5–2
Adding 1 Part 6 (Up to 1000+1)
1–2
Adding up to 7 Part 2
1–2
31-040
Shapes and Pictures
(Stories) 1
131-140
Number Reading Exercises (Up to 10) 4
Sequence of Numbers (Up to 40) 1
Number Writing Exercises up to 30 Part 2 0.5–2
Adding 2 Part 1 (Up to 14+2)
1–2
Adding 8 Part 1 (Up to 11+8)
1–2
41-050
Shapes and Pictures
(Stories) 2
141-150
Number Reading Exercises (Up to 10) 5
Sequence of Numbers (Up to 40) 2
Numbers up to 50 Part 1
0.5–2
Adding 2 Part 2 (Up to 18+2)
1–2
Adding 8 Part 2 (Up to 12+8)
1–2
51-060
Shapes and Pictures
(Stories) 3
151-160
Number of Dots (Up to 10) 1
Sequence of Numbers (Up to 40) 3
Numbers up to 50 Part 2
0.5–2
Adding 2 Part 3 (Up to 32+2)
1–2
Adding 9 (Up to 12+9)
1–2
61-070
Back and Forth 1
161-170
Number of Dots (Up to 10) 2
Sequence of Numbers (Up to 50) 1
Numbers up to 50 Part 3
0.5–2
Adding 3 Part 1 (Up to 14+3)
1–2
Adding 9 and 10 (Up to 12+9 and 15+10)
1–2
71-080
Back and Forth 2
171-180
Number of Dots (Up to 10) 3
Sequence of Numbers (Up to 50) 2
Numbers up to 50 Part 4
0.5–2
Adding 3 Part 2 (Up to 21+3)
1–2
Adding up to 10 Part 1
1–2
81-090
Corners and Curves 1
181-190
Number of Dots (Up to 10) 4
Sequence of Numbers (Up to 50) 3
Numbers up to 50 Part 5
0.5–2
Adding up to 3 Part 1
1–2
Adding up to 10 Part 2
1–2
91-100
Corners and Curves 2
191-200
Number of Dots (Up to 10) 5
Large Numbers
Numbers up to 50 Part 6
0.5–2
Adding up to 3 Part 2
1–2
Adding up to 10 Part 3
1–2
Levels ZI & ZII (Shapes and Pictures)
Level 6A (Counting)
Z II 11a
6 A 1a
Draw a line from the circle to the star.
Count the pictures aloud, “1, 2,” while pointing to
each one.
Level 5A (Reading Numbers)
5 A 24b
Level 3A (Adding 1)
Level 4A (Writing Numbers)
` Write the number that comes next.
Write the numbers.
13
3 + 1 =
Three plus
(1)
tree
To the parent(s):
First demonstrate how to point to the pictures and count them. After
finishing the exercise on each side of the worksheet, tick the box at
the bottom.
13
14
15
(1)
4 + 5 =
(2)
5 + 5 =
(3)
3 + 5 =
(4)
3 + 6 =
(5)
6 + 6 =
(6)
8 + 6 =
(7)
5 + 7 =
(8)
7 + 7 =
(9)
9 + 7 =
( 10 )
8 + 7 =
one equals
4 + 1 = 5
Four
12
` Add.
3
11 12 13 14 15
11
Level 2A (Adding up to 10)
2 A 200a
3 A 74b
4 A 104a
Read the numbers.
11 12
jp.yagi.ka 2021/03/19
sg.wendy
2019/11/27 09:28:23
13:46:22 DD
DR
#
plus
one equals
(2)
5 + 1 =
(3)
6 + 1 =
(4)
8 + 1 =
© 2016 Kumon Institute of Education KIE KIE 2016 GB
M_TLM_1.indd 1
2016/10/26 10:11
TABLE OF LEARNING MATERIALS I-2
MATHEMATICS (A-F)
The black mark (
) indicates Starting Points.
SCT: Standard Completion Time (Min./Sheet)
This is the time in which the student should complete
each worksheet, including time taken for corrections.
#
October 2016
A
SCT
B
SCT
C
SCT
D
SCT
E
SCT
F
SCT
001-010
Addition 1 (Review up to 2A)
1–2
Addition (Review up to A)
1–2
Review up to B
2–3
Review up to C
2–3
Review up to D
2–3
Review up to E 1
3–5
011-020
Addition 2 (Up to sum of 12)
1–2
Addition to 100 Part 1
1–2
Multiplication up to 3
2–3
Multiplication: 2 Digits * 2 Digits 1
2–4
Addition of Fractions 1
2–3
Review up to E 2
3–5
021-030
Addition 3 (Up to sum of 15)
1–2
Addition to 100 Part 2
2–3
Multiplication up to 5
2–3
Multiplication: 2 Digits * 2 Digits 2
2–4
Addition of Fractions 2
3–4
Multiplication and Division of 3 Fractions
4–6
031-040
Addition 4 (Up to sum of 18)
1–2
Addition to 100 Part 3
2–3
Multiplication up to 7
2–3
Multiplication: 2 Digits * 2 Digits 3
3–4
Addition of Fractions 3
3–4
Addition of 3 Fractions 1
3–5
041-050
Addition 5 (Up to sum of 20)
1–2
Addition of 2-Digit Numbers 1
2–3
Multiplication up to 9
2–3
Multiplication: 3 Digits * 2 Digits
3–5
Addition of Fractions 4
3–4
Addition of 3 Fractions 2
4–6
051-060
Addition 6 (Up to sum of 24)
1–2
Addition of 2-Digit Numbers 2
2–3
Multiplication: 2 Digits * 1 Digit 1
2–4
Addition and Subtraction
2–4
Addition of Fractions 5
3–5
Addition and Subtraction of 3 Fractions
4–6
061-070
Addition 7 (Up to sum of 28)
1–2
Addition of 2-Digit Numbers 3
2–3
Multiplication: 2 Digits * 1 Digit 2
2–4
Multiplication and Division 1
3–4
Addition of Fractions 6
3–5
Order of Operations (3 Fractions) 1
3–5
071-080
Addition 8 (Summary of addition)
2–3
Addition of 3-Digit Numbers 1
2–4
Multiplication: 2 Digits * 1 Digit 3
2–4
Multiplication and Division 2
3–4
Addition of Fractions 7
3–5
Order of Operations (3 Fractions) 2
3–5
081-090
Subtraction 1 (Subtracting 1)
1–2
Addition of 3-Digit Numbers 2
2–4
Multiplication: 2 Digits * 1 Digit 4
2–4
Division by 2-Digit Numbers 1
3–4
Addition of Fractions 8
3–5
Order of Operations (3 Fractions) 3
3–5
091-100
Subtraction 2 (Subtracting 2)
1–2
Addition of 3-Digit Numbers 3
3–5
Multiplication: 2 Digits * 1 Digit 5
2–4
Division by 2-Digit Numbers 2
3–5
Addition of Fractions 9
4–6
Order of Operations
(3-or-More Fractions) 1
4–6
101-110
Subtraction 3 (Subtracting 3)
1–2
Subtraction 1 (Review up to A)
1–2
Multiplication: 3 or 4 Digits * 1 Digit
3–5
Division by 2-Digit Numbers 3
3–5
Subtraction of Fractions 1
3–4
Order of Operations
(3-or-More Fractions) 2
4–6
111-120
Subtraction 4 (Subtracting up to 3)
1–2
Subtraction 2 (Review up to A)
2–3
Introduction to Division
2–3
Division by 2-Digit Numbers 4
3–5
Subtraction of Fractions 2
3–5
Order of Operations
(3-or-More Fractions) 3
4–6
121-130
Subtraction 5 (Subtracting up to 5)
1–2
Subtraction of 2-Digit Numbers 1
2–3
Division with Remainders 1
2–3
Division by 2-Digit Numbers 5
3–5
Subtraction of Fractions 3
3–5
Order of Operations
(3-or-More Fractions) 4
4–6
131-140
Subtraction 6 (From numbers up to 10)
1–2
Subtraction of 2-Digit Numbers 2
2–3
Division with Remainders 2
2–3
Division: Quotients of 2-or-More Digits 1
3–5
Addition and Subtraction of Fractions
4–6
Fractions and Decimals 1
3–5
141-150
Subtraction 7 (From numbers up to 11)
1–2
Subtraction of 2-Digit Numbers 3
2–3
Division with Remainders 3
2–3
Division: Quotients of 2-or-More Digits 2
4–6
Multiplication of Fractions 1
3–4
Fractions and Decimals 2
4–6
151-160
Subtraction 8 (From numbers up to 12)
1–2
Addition and Subtraction of
2-Digit Numbers
2–4
Division with Remainders 4
2–3
Fractions
3–5
Multiplication of Fractions 2
3–5
Fractions and Decimals 3
4–6
161-170
Subtraction 9 (From numbers up to 14)
1–2
Subtraction of 3-Digit Numbers 1
2–3
Division: 2 Digits / 1 Digit 1
2–3
Reduction 1
2–3
Division of Fractions
3–5
Solving Word Problems 1
4–6
171-180
Subtraction 10 (From numbers up to 16)
1–2
Subtraction of 3-Digit Numbers 2
2–4
Division: 2 Digits / 1 Digit 2
2–3
Reduction 2
2–3
Multiplication and Division of Fractions
3–5
Solving Word Problems 2
5–7
181-190
Subtraction 11 (From numbers up to 20)
1–2
Subtraction of 3-Digit Numbers 3
2–4
Division: 3 Digits / 1 Digit 1
3–4
Reduction 3
2–4
Four Operations of 2 Fractions 1
3–5
Decimals 1
3–5
191-200
Subtraction 12 (Summary of subtraction)
2–3
Subtraction of 3-Digit Numbers 4
3–5
Division: 3 Digits / 1 Digit 2
3–5
Reduction 4
2–4
Four Operations of 2 Fractions 2
3–5
Decimals 2
3–5
Level A (Subtraction)
B 2B
1 a(Addition #
Level
to 100)
Addition up to sum of 2-Digit Numbers 2
B 21
#
C 5C
2 a(Multiplication)
Level
Multiplication: 2 Digits × 1 Digit 1
Name
A 200a
1$2
(1)
(4)
jp.yagi.ka 2021/03/19
sg.wendy
2019/11/27 09:28:23
13:46:22 DD
DR
(5)
12 - 7 =
15 - 4 =
16 - 8 =
(6)
13 - 7 =
(7)
17 - 9 =
(8)
14 - 5 =
(9)
18 - 6 =
( 10 )
19 - 2 =
( 11 )
20 - 3 =
© 2012 Kumon Institute of Education KIE KIE 2012 GB
© 2012 Kumon Institute of Education KIE KIE 2012 GB
(3)
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(1)
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(4)
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＊： B 2 1 ( 6 )
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7 $ 10
11 $ 20
( 6Date
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(7)
(7)
(8)
(8)
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#
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Order of Operations (3-or-More Fractions)
4
` Calculate.
Date
Time
3
9
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2
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4
5
(8)
=
38
39
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4
6
2 1
2
9
` Add.
(2)
/
Date
#
7 $ 12
F 121
Name
Name
(4)
51 0
(5)
+
(8)
2 1
4$6
` Add.
( 5Date
)
Time
R
(3)
9 $ 16
2$3
Addition of Fractions 6
Name
2 1
E 70
Name
#
Division by 2-Digit Numbers 1
` Divide.
*
*
2$4
D 81a
C 52
Name
` Multiply.
(1)
/
Date
#
Multiplication: 2 Digits × 1 Digit 1
` Multiply.
Name
` Add.
(2)
/
:
Level
of Fractions)
E 7 0Ea (Addition#
Addition of Fractions 6
© 2016 Kumon Institute of Education KIE KIE 2016 GB
© 2016 Kumon Institute of Education KIE KIE 2016 GB
11 - 6 =
/
Date
#
D 81
Name
© 2016 Kumon Institute of Education KIE KIE 2016 GB
© 2016 Kumon Institute of Education KIE KIE 2016 GB
(2)
11 $ 20
` Add.
D 8D
1 a(Division) #
Level
Division by 2-Digit Numbers 1
© 2012 Kumon Institute of Education KIE KIE 2012 GB
© 2012 Kumon Institute of Education KIE KIE 2012 GB
10 - 3 =
7 $ 10
Addition up to sum of 2-Digit Numbers 2
© 2012 Kumon Institute of Education KIE KIE 2012 GB
© 2012 Kumon Institute of Education KIE KIE 2012 GB
(1)
3$6
B 21a
` Subtract.
C 52
Name
(5)
76 + 72 1 =
(6)
+
=
8
24
D 83 ( 3 )
(6)
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8
+
7
24
=
D 83 ( 3 )
2016/10/26 10:11
TABLE OF LEARNING MATERIALS II-1 (Level G to L)
The black mark (
) indicates most suitable Starting Points.
SCT: Standard Completion Time (Min./Sheet)
This is the set time in which the student should complete
each worksheet, including time taken for correction.
October 2016
G
SCT
H
SCT
I
SCT
J
SCT
K
SCT
L
SCT
001-010
Review up to F 1
3–5
Basics for Level H Mathematics 1
4–6
Basics for Level I Mathematics
4–6
Expansion of Polynomial Products
5–8
Review of Linear Functions
4–6
Logarithmic Functions
6–12
011-020
Review up to F 2
3–5
Basics for Level H Mathematics 2
4–6
Multiplication of Polynomials
4–6
Factorisation I
5–8
Review of Quadratic Functions
5–8
Graphs of Logarithmic Functions
7–14
021-030
Addition and Subtraction of
Positive and Negative Numbers 1
2–4
Literal Equations 1
4–6
Multiplication Using Formulas
4–6
Factorisation II
5–8
Quadratic Functions and Graphs
6–12
Logarithmic Equations and Inequalities
8–16
031-040
Addition and Subtraction of
Positive and Negative Numbers 2
3–5
Literal Equations 2
4–6
Factorisation 1
4–6
Factorisation III
6–10
Determining Equations of Quadratic
Functions
7–14
Modulus Functions
8–16
041-050
Addition and Subtraction of
Positive and Negative Numbers 3
3–5
Simultaneous Equations
in Two Variables 1
4–6
Factorisation 2
4–6
Factorisation IV
6–10
Maxima and Minima of Quadratic
Functions I
7–14
Limits and Derivatives
8–16
051-060
Addition and Subtraction of
Positive and Negative Numbers 4
4–6
Simultaneous Equations
in Two Variables 2
4–6
Factorisation 3
4–6
Factorisation V
7–12
Maxima and Minima of Quadratic
Functions II
7–14
Tangents
12–24
061-070
Multiplication of
Positive and Negative Numbers
3–5
Simultaneous Equations
in Two Variables 3
4–6
Factorisation 4
4–6
Fractional Expressions
6–10
Maxima and Minima of Quadratic
Functions III
8–16
Relative Maxima and Minima I
15–30
071-080
Division of
Positive and Negative Numbers
4–6
Simultaneous Equations
in Two Variables 4
4–6
Factorisation 5
4–6
Irrational Numbers I
5–8
Quadratic Functions and Equations
7–14
Relative Maxima and Minima II
15–30
081-090
Four Operations with
Positive and Negative Numbers 1
4–6
Simultaneous Equations
in Two Variables 5
4–6
Square Roots 1
4–6
Irrational Numbers II
6–10
Quadratic Functions and Inequalities
7–14
Maxima and Minima I
15–30
091-100
Four Operations with
Positive and Negative Numbers 2
4–6
Simultaneous Equations
in Three and Four Variables 1
4–6
Square Roots 2
4–6
Quadratic Equations I
5–8
Quadratic Functions and Solutions of
Quadratic Equations
8–16
Maxima and Minima II
15–30
101-110
Values of Algebraic Expressions 1
4–6
Simultaneous Equations
in Three and Four Variables 2
4–6
Square Roots 3
4–6
Quadratic Equations II
6–10
Higher Degree Functions
6–12
111-120
Values of Algebraic Expressions 2
4–6
Application of Equations
4–6
Quadratic Equations 1
4–6
Quadratic Equations and Complex
Numbers
6–10
121-130
Simplifying Algebraic Expressions 1
3–5
Inequalities 1
4–6
Quadratic Equations 2
4–6
Discriminant
6–10
Graphs of Fractional Functions I
131-140
Simplifying Algebraic Expressions 2
3–5
Inequalities 2
4–6
Quadratic Equations 3
4–6
Root-Coefficient Relationships
6–10
141-150
Simplifying Algebraic Expressions 3
4–6
Functions and Graphs 1
4–6
Graphs of Quadratic Functions 1
4–6
Simultaneous Equations
151-160
Simplifying Algebraic Expressions 4
4–6
Functions and Graphs 2
4–6
Graphs of Quadratic Functions 2
4–6
161-170
Linear Equations 1
3–5
Functions and Graphs 3
4–6
Graphs of Quadratic Functions 3
171-180
Linear Equations 2
4–6
Functions and Graphs 4
4–6
181-190
Linear Equations 3
4–6
Simplifying Monomials and Polynomials 1
191-200
Linear Equations 4
4–6
Simplifying Monomials and Polynomials 2
H-041-050
Level G
#
G 145a
(Simplifying
Algebraic
Expressions)
Simplifying Algebraic
Expressions 3
G 145
Name
G 145a
1
5$8
#
9 16
$
1
Time
:
H-041-050
:
/
Date
2$4
5$8
9 $ 16
Time
:
Ex.
R100%
{
:
5a - 4 ( 3a - 2b ) =
3a - 2 ( 4a - b ) = 3a - 8a + 2b = - 5a + 2b
Date
Time
Name
Grade
Date
Mistakes
/
:
/
to
B1
70%
B1
70%
/
:
[Sol] 1-2x=
16/03/18 18:04
y=
Ans.(x,y)=(
Definite Integrals II
12–24
6–10
Fractional Equations and Inequalities
8–16
Areas I
15–30
Dividing Polynomials
6–10
Graphs of Irrational Functions
7–14
Areas II
15–30
4–6
Remainder Theorem
6–10
Irrational Equations and Inequalities
8–16
Volumes
15–30
The Pythagorean Theorem 1
4–6
Factor Theorem
6–10
Exponential Functions
6–12
Velocity and Distance
15–30
4–6
The Pythagorean Theorem 2
4–6
Proof of Identities
6–10
Graphs of Exponential Functions
7–14
Summary of Differentiation and
Integration I
30–60
4–6
The Pythagorean Theorem 3
4–6
Proof of Inequalities
7–12
Exponential Equations and Inequalities
8–16
Summary of Differentiation and
Integration II
30–60
50%
D
Substituting
x=this into 1 ,
,
I-041-050
)
+3y=30
y=
y=
Ans.(x,y)=(
,
2:08 PM
I 44
Page 8
Name
Date
/
Time
Name
R100%
Grade
0
Factorisation 2
` Factorise using the following formula.
100%
Date
Mistakes
A1~2
:
90%
Time
B3~6
/
:
A1~2 B3~6
Formula
0
x¤+(a+b)x+ab=(x+a)(x+b)
` Factorise using the following formula.
49%
Grade
Mistakes
90%
/
to
70%
to
70%
C
7/30/09
2:00 PM
Page 8
J_131-140*.q
7/30/09
2:00 PM
Page 8
J 134å
I 44
:
50%
7~11
/
C
J_131-140*.q
Level J
(Root-Coefficient Relationships)
50%
49%
12~22
:
7~11
D
(3+2) 3'2
( 2 ) x¤+7x+10=
( 1 ) x¤+5x+6=(x+
(3+2) 3'2
( 3 ) x¤+8x+15=
( 2 ) x¤+7x+10=
( 4 ) x¤+3x+2=
( 3 ) x¤+8x+15=
( 5 ) x¤+5x+4=
( 4 ) x¤+3x+2=
( 6 ) x¤-5x+6=(x-3)(x( 5 ) x¤+5x+4=
( 7 ) x¤-6x+8=
( 6 ) x¤-5x+6=(x-3)(x( 8 ) x¤-7x+10=
( 7 ) x¤-6x+8=
( 9 ) x¤-8x+15=
( 8 ) x¤-7x+10=
(10) x¤-7x+12=
( 9 ) x¤-8x+15=
jp.chosa.yu 2016/11/01 14:24:08 R
)(x+2)
(10) x¤-7x+12=
)
)
ME_TLM_2.indd 1
Date
Time
Name
Grade
mistakes
D
49%
12~22
100
0
%
Date
/
/
:
B
70%
/1
50%
2~3
/
D
%
70%
(1)
(å-∫)¤
å¤+∫¤=(å+∫)¤-2å∫=4¤-2*5=6
[Sol] (å-∫)¤=(å+∫)¤-4å∫
=4¤-4*5
=-4
[Sol] (å-∫)¤=(å+∫)¤-4å∫
=4¤-4*5
=-4
(1)
(2)
(å-∫)¤
å¤∫+å∫¤
[Sol] å¤∫+å∫¤=å∫(å+∫)
=5*4
=20
[Sol] å¤∫+å∫¤=å∫(å+∫)
=5*4
=20
( 3 ) å¤+å∫+∫¤=(å+∫)¤-å∫
=16-5
=11
( 3 ) å¤+å∫+∫¤=(å+∫)¤-å∫
=16-5
=11
(2)
å¤∫+å∫¤
jp.chosa.yu 2016/11/01 14:24:55 R
jp.chosa.yu 2016/11/01 14:24:31 R
jp.chosa.yu 2016/11/01 14:23:40 R
jp.chosa.yu 2016/11/01 14:24:55 R
jp.chosa.yu 2016/11/01 14:24:08 R
Date
R
Grade
/
Time
Name
100
0
:
%
Date
/
/1
70%
C
:
2/
1
L_061-070*
50%
D
2
[Sol] y=x¤-6x+4
Ex.
=(x-3)¤-5
When 2≤x≤6.
Since 2≤x≤6, the graph is the
solid curve
shown on the figure
[Sol]
y=x¤-6x+4
below. =(x-3)¤-5
Since 2≤x≤6,
y the graph is the
solid curve shown on the figure
4
below.
O
4
-4
-5
O
y
2 3
2 3
6
6
x
x
From the graph:
• There is-4
a maximum value of 4,
-5
at x=6.
• There
a minimum value of
From
theis graph:
-5, atisx=3.
• There
a maximum value of 4,
Therefore,
at x=6. the range is
•-5≤y≤4.
There is a minimum value of
-5, at x=3.
Therefore, the range is
-5≤y≤4.
O
y
-5
O
-1
the figure below.
3
5
3
5
x
x
From the graph:
• There is a maximum value of
-5
4, at x=0.
• There
a minimum value of
From
theis graph:
-5, atisx=3.
• There
a maximum value of
Therefore,
the range is
4, at x=0.
•-5≤y≤4.
There is a minimum value of
-5, at x=3.
Therefore, the range is
-5≤y≤4.
• The set of x values is called the domain.
The set of y values is called the range.
• The domain is normally written in brackets next to the function,
e.g. y=x¤-6x+4 (2≤x≤6)
• The set of x values is called the domain.
The set of y values is called the range.
• The domain is normally written in brackets next to the function,
e.g. y=x¤-6x+4 (2≤x≤6)
Page 12
Date
L 66
/
Time
Name
R
Grade
100
0
/
:
Grade
100
C
70%
/1
Relative Maxima and Minima I
Time
:
1. Given y=-x‹+x¤+x-1, complete the questions
below.
(1)
B
%
L 66
to
B
%
Date
:
50%
2/
70%
C
49%
D
49%
3~4
to
0
1
Create a variation table to findmistakes
the relative
extreme values.
D
:
50%
2
3~4
1. Given
y=-x‹+x¤+x-1,
complete the questions below.
[Sol] From
y`=-3x¤+2x+1=-(3x+1)(x-1)=0,
1
Createx=a variation
, 1table to find the relative extreme values.
3
[Sol] From y`=-3x¤+2x+1=-(3x+1)(x-1)=0,
… -1 …
…
x
1
1 3
x=- , 1
y` - 3 0
0
+
-
(1)
Since 0 ≤x≤ 5 , draw the
4
-1
4:55 PM
mistakes
3~5
Since 0 ≤x≤ 5 , draw the
[Sol] y=x¤-6x+4
graph on
the figure below.
y=
(x-3)¤-5
y
Page 12
Relative Maxima and Minima I
49%
y= 0≤x≤5.
(x-3)¤-5
When
graph4 on
4/27/09
L 66å
49%
3~5
[Sol] y=x¤-6x+4
(1)
4:55 PM
Name
K 43
to
B
4/27/09
Level L
(Relative
R and Minima)
L 66å Maxima
Ex.
1. Given y=x¤-6x+4, draw the graph corresponding to each given
When
2≤x≤6.
( 1 )set of
When
0≤x≤5.
condition.
Then find the corresponding
y values
(the range).
Ex.
[Sol] å+∫=4, å∫=5
K 43
Page 6
0
49%
expressions.
å¤+∫¤
å¤+∫¤=(å+∫)¤-2å∫=4¤-2*5=6
å¤+∫¤
11:21 AM
Maxima and Minima of Quadratic
Functions
I
Time
to
:
1. Given y=x¤-6x+4,
draw the graph corresponding
to: each given
Grade 100
condition. Then find the corresponding
set of% y values (the
70%
C 50% D
B range).
mistakes
49%
mistakes
[Sol] å+∫=4, å∫=5
L_061-070*
mistakes
4~6
50%
05.3.11
K 43å
:
C
Page 6
Name
J 134
to
11:21 AM
Maxima and Minima of Quadratic
Functions I
1. Ex.
If the roots of x¤-4x+5=0 are å and ∫, evaluate the following
jp.chosa.yu 2016/11/01 14:24:31 R
MNT_M_WS_G_141.indd 9
Root-Coefficient Relationships
J 134å
R
K_041-050
J 134
Name
05.3.11
Level K
R of Quadratic Functions)
(Maxima
K 43åand Minima
Time
to
:
1. If the roots of x¤-4x+5=0 are å and ∫,
evaluate :the following
Grade 100
C
D
B
expressions.
0
1
2~3
4~6
(4+2) 4'2
Ex.
x¤+6x+8=(x+4)(x+2)
( 1 ) x¤+5x+6=(x+
)(x+2)
(4+2) 4'2
R
Root-Coefficient Relationships
Ex.
x¤+6x+8=(x+4)(x+2)
Formula
x¤+(a+b)x+ab=(x+a)(x+b)
)
)
04.12.22
I 44å
Substituting this
into 1 ,
3y=
3y= ,
Ans.(x,y)=(
K_041-050
Page 8
Factorisation 2
49%
x=
+3y=30
)
2:08 PM
Level I
(Factorisation)
R
I 44å
4~6
:
C
D
'''2
[Sol] 1-2 : x=
Substituting
3x= this into 1 ,
+3y=30
y=
jp.chosa.yu 2016/11/01 14:23:40 R
Mistakes
x=
{5x+2y=8
'''2
3y= ,
Ans.(x,y)=(
3x - 2 ( 4x - 3y ) =
Grade
{
Substituting this
into 1 ,
3y=
( 7 ) 3x + 2 (- 4x + 3y ) =
( 8 ) 3x - 2 ( 4x - 3y ) =
50%
) same
8x+3y=30
'''1
7x+2y=12
'1the ones
Note
Two linear equations
with( 2the
variables, such''as
5x+3y=21
'''2
5x+2y=8
'''2
above,
are called simultaneous
linear equations
in two variables.
[Sol] 1-2 :
[Sol] 1-2 :
( 1 ) 8x+3y=30 '''1
( 2 ) 7x+2y=12 '''1
x=+3y=30
( 6 ) 5x - 2 ( x - 3y ) =
( 7 ) 3x + 2 (- 4x + 3y ) =
to
2~3
/
{
3x=
{5x+3y=21
7–14
© 2002 Kumon
Institute
of Education
2005 GB
in Malaysia
© 2002
Kumon
Institute of KIE
Education
KIEPrinted
2005 GB
Printed in Malaysia
( 5 ) 3x - 4 ( x - 2y ) =
( 6 ) 5x - 2 ( x - 3y ) =
100%
{
Graphs of Fractional Functions II
H41
:
C
2~3
4~6
<
>
Ex.
5x+2y=11 '''1 0 <Method
1. Number
equation.
` Solve the
following equations
in theeach
example.
3x+2y=5
'''2 as shown
2. Subtract one equation from the
[Sol]
1-2 :
<Method
< other to>remove one variable.
Ex.
5x+2y=11
'''1
1. Number5x+2y=11
each equation.
2x=6
3x+2y=5
'''2
2. Subtract
one equation from the
- 3x+2y=05
2x=3 :
[Sol] 1-2
other ˘
to2x-2y=06
remove one variable.
Substituting
x.
5x+2y=11
2x=6 this into 1 , 3. Solve for
- 3x+2y=05
5*3+2y=11
4. Substitute
the value of x in one
2x=3
˘ 2x-2y=06
of the given
equations and solve
2y=-4
Substituting this
into 1 ,
3. for
Solve
y. for x.
y=-2
5*3+2y=11
Substitute
the value of x in one
5. Write
the answer.
Ans.(x,y)=( 3 ,-2) 4.
of the given equations and solve
2y=-4
for same
y. variables, such as the ones
Note Two linear y=-2
equations with the
Write
the answer.
Ans.are
(x,y)=(
3 ,-2) 5.
above,
called simultaneous
linear
equations
in two variables.
(1)
12–24
© 2002 Kumon
Institute
of Education
2005 GB
© 2002
Kumon
Institute of KIE
Education
KIE 2005 GB
( 4 ) 4a - 2 (- 3a - 5b ) =
( 5 ) 3x - 4 ( x - 2y ) =
Definite Integrals I
© 2002 Kumon
Institute
of Education
2005 GB
in Malaysia
© 2002
Kumon
Institute of KIE
Education
KIEPrinted
2005 GB
Printed in Malaysia
+22((-=) =
3a4a3a
b )5b
((43)) 4a
7–14
© 1998 Kumon
© 1998
Institute
Kumon
of Education
Institute of KIE
Education
2005 GB
KIE 2005 GB
-23((4a
=
2a+
a+
((32)) 3a
-4b
b ))=
MNT_M_WS_G_141.indd 9
H41
Page 2
Simultaneous Equations in Two
` Solve the following equations as shown in
the example.
Time
:
Variables 1
-34((a
5a3a
)=
((21)) 2a
+4b2b
)=
(8)
6:47 PM
8–16
04.12.22
Higher Degree Equations and Inequalities 7–14
Applications to Equations and Inequalities 15–30
Indefinite and Definite Integrals
I-041-050
0
` Simplify.
(1)
Page 2
Simultaneous Equations in Two
Variables 1
/
to
04.12.22
H41å
G 145
© 1998 Kumon
© 1998
Institute
Kumon
of Education
Institute of KIE
Education
2005 GB
KIE 2005 GB
© 2016 Kumon Institute of Education KIE KIE 2016 GB
© 2016 Kumon Institute of Education KIE KIE 2016 GB
3a - 2 ( 4a - b ) = 3a - 8a + 2b = - 5a + 2b
6:47 PM
Name
/
to
04.12.22
Level H
R
(Simultaneous
Equations)
H41å
Name
` Simplify.
Ex.
/
Date
2$4
Simplifying Algebraic Expressions 3
sg.wendy 2021/03/19 09:28:23 DD
R
yx
(2)
1
(2)
1
2
2
1
… - 32
3
27
…
10
1
32
Find
x-intercept(s)
and
y-intercept,
- 3graph.
Thethe
relative
minimum
value
is - 27then
, atdraw
x= the
.
x-intercept(s)
Find
the -x‹+x¤+x-1=0,
x-intercept(s) and y-intercept, then draw the graph.
From
-x¤(x-1)+(x-1)=0
x-intercept(s)
-(x-1)¤(x+1)=0
From -x‹+x¤+x-1=0,
x=1, -1
-x¤(x-1)+(x-1)=0
Ans. (1, 0), (-1, 0)
-(x-1)¤(x+1)=0
x=1, -1
y-intercept
Ans. (1, 0), (-1, 0)
Substituting x=0, y=-1
Ans. (0, -1)
y-intercept
Substituting x=0, y=-1
jp.chosa.yu 2016/11/01 14:25:17 R
…
0
0
y` +
The relative maximum value is 0 , at x= 1 .
y
- 32
0
27
32
1
The relative minimum value is , at x= .
The relative maximum value is 027 , at x= 1 3.
Ans. (0, -1)
y
-1
-1
y1
1�
3
O
1
1�
3
O
�
�
-1
32
27
-1
1
x
1
x
32
27
© 2016 Kumon Institute of Education KIE KIE 2018 GB
jp.chosa.yu 2016/11/01 14:25:17 R
16/03/18 18:04
18/03/23 14:36
TABLE OF LEARNING MATERIALS II-2 (Level M to X)
R
SCT: Standard Completion Time (Min./Sheet)
This is the set time in which the student should complete
each worksheet, including time taken for correction.
March 2018
XV
SCT
001-010
Surface Vectors I
6–12
25–50
011-020
Surface Vectors II
Concavity of Curves
25–50
021-030
15–30
Maxima and Minima
25–50
Recurrence Relations
20–40
Various Applications of
Differentiation
15–30
Mathematical Induction
20–40
Loci II
15–30
Infinite Sequences
071-080
Regions
15–30
081-090
Trigonometric Ratios I
091-100
M
SCT
N
SCT
O
SCT
XM
SCT
XP
SCT
001-010
Points and Lines I
10–20
Arithmetic Sequences
10–20
Tangents and Normals
20–40
01-010
Matrix Definitions, Addition and
Subtraction
6–12
01-010
Introduction to Permutations
6–12
011-020
Points and Lines II
10–20
Geometric Sequences
10–20
Increasing/Decreasing Functions
and Relative Extreme Values
8–16
11-020
Matrix Multiplication
8–16
11-020
Permutations
6–12
021-030
Points and Lines III
15–30
Various Sequences I
15–30
Surface Vectors III
8–16
21-030
Inverse Matrices
10–20
21-030
Permutations and Combinations
8–16
031-040
Circles I
15–30
Various Sequences II
031-040
Surface Vectors IV
10–20
31-040
Equations and Matrices
10–20
31-040
Combinations
10–20
041-050
Circles II
15–30
25–50
041-050
Coordinates in Space
6–12
41-050
Mapping I
8–16
41-050
Binomial Theorem
12–24
051-060
Loci I
Indefinite Integrals I
15–30
051-060
Vectors in Space
8–16
51-060
Mapping II
10–20
51-060
Probability
8–16
061-070
10–20
Indefinite Integrals II
15–30
061-070
Basics of Inner Products of
Vectors
8–16
61-070
Transformations I
8–16
61-070
Conditional Probability
10–20
Infinite Geometric Sequences
15–30
Integration by Substitution
15–30
071-080
Inner Products of Vectors
8–16
71-080
Transformations II
10–20
71-080
Independent Trials
12–24
10–20
Infinite Geometric Series
15–30
Integration by Parts
20–40
081-090
Applications and Summary of
Vectors
10–20
81-090
Transformations III
10–20
81-090
Expected Value
12–24
Trigonometric Ratios II
10–20
Infinite Series
15–30
Definite Integrals
15–30
091-100
Vectors and Figures
10–20
101-110
Properties of
Trigonometric Functions I
10–20
Limits of Functions I
15–30
Integration by Substitution for
Definite Integrals
15–30
101-110
Equations of Lines and Planes
in Space I
10–20
111-120
Properties of
Trigonometric Functions II
XS
SCT
10–20
Limits of Functions II
Integration by Parts for Definite Integrals and
15–30
20–40
Functions Represented by Definite Integrals
111-120
Equations of Lines and Planes
in Space II
10–20
01-010
Introduction to Statistics
8–16
121-130
Trigonometric Equations
10–20
Limits of Trigonometric Functions
15–30
Integration by Quadrature and
Proof of Inequalities
25–50
121-130
Equations of Lines and Planes
in Space III
12–24
11-020
Binomial Distribution
8–16
131-140
Graphs of
Trigonometric Functions
15–30
Continuous and
Discontinuous Functions
20–40
Areas
25–50
131-140
Equations of Lines, Planes
and Spheres
12–24
21-030
Probability Density Function
10–20
141-150
Trigonometric Inequalities
15–30
Differentiation I
15–30
Volumes
25–50
31-040
Standard Normal Distribution I
12–24
151-160
Addition Formulas I
15–30
Differentiation II
15–30
Length of a Curve,
Velocity and Distance
25–50
41-050
Standard Normal Distribution II
12–24
161-170
Addition Formulas II
15–30
Differentiation of
Trigonometric Functions
15–30
Differential Equations
25–50
51-060
Sampling
15–30
171-180
Addition Formulas III
15–30
Differentiation of Logarithmic and
Exponential Functions
15–30
Natural/Social Science and
Differential Equations
30–60
61-070
Hypothesis Testing
15–30
181-190
Sine and Cosine Rules
15–30
Applications of Calculus I
30–60
191-200
Area of Triangles
15–30
Applications of Calculus II
30–60
Differentiation of Various Functions
20–40
and Higher Order Derivatives
Various Properties of Derivatives
25–50
XV 091-100*
#
1
2
:
to
H
b
c
2
2
a =C H a+B
H2+c 2-2bc cos A is true.
Therefore,
=b
B
a
2
B
B
=( b sin A ) 2+( c-b cos A ) 2
Answers: in order b sin A, b cos A, c-b cos A, b sin A, c-b cos A
2
2
sin A+ cos A=1
=b 2+c 2-2bc cos A
2
A is an obtuse
WhenTherefore,
true. B is an obtuse angle,
a 2=bangle,
+c 2-2bc cos A isWhen
C H=b sin ( 180^-A )=b sin A
C H=b sin A
Answers: in order b sin A, b cos A, c-b cos A, b sin A, c-b cos A
B H=A B+A H
B H=A H-A B
A is ancos
When=c+b
obtuse
angle,)
( 180^-A
When B is an obtuse angle,
=b cos A-c
C H=b sin A
a 2=C H 2+B H 2
B H=A H-A2B
=( b sin A ) +( b cos A-c ) 2
=b cos A-c
=b 2+c 2-2bc cos A
a 2=C H 2+B H 2
=c-b
A
C H=b
180^-A
)=b sin A
sin (cos
2
2
=C HB+A
+B H
a 2H=A
B
H
=(
b sin A
) +(
c-b cos
=c+b
( 180^-A
) A)
cos
2
2
=b
+c 2cos
-2bc
=c-b
A cos A
Ca
2
=C H 2+B H 2
c
A
B
a
B
A
A
C
H
B
b
b
Therefore,
when
either A or B is an obtuse angle, a 2=b 2+c 2-2bc cosaA.
A
B
B
A or B Bis a right A
angle. A
It His also true
when either
c
c
A
B
Similarly, b 2=c 2+a 2-2ca cos B and c 2=a 2+b 2-2ab cos C are true.
2
2
2
ARule.
Therefore,
when
or B is an obtuse angle, a =b +c -2bc cos A.
This
is called
the either
Cosine
It is also true when either A or B is a right angle.
Similarly, b 2=c 2+a 2-2ca cos B and c 2=a 2+b 2-2ab cos C are true.
H
This is called the Cosine Rule.
ME_WS_M_181.indd 7
ME_WS_M_181.indd 7
ME_TLM_2.indd 2
AA
BB
A
B
a
a
a
a
a
a
a
a
a
a
a
a
a
a
[ f ( x ) g ( x ) ]s=fs( x ) g ( x )+f ( x ) gs( x )
1
n
n
b
bb
a
b
a
a
Using this constant value e,
log e e
1
1
xA
log a e=
ys=
=
1
When a=e, log
therefore,
from
,
x logA
x e a= log e e=1
x ; log
ea
e ays= 1
When a=e, log e a= log e e=1; therefore, from A, ys= x
x
log a M=
When a=e, log e a= log e e=1; therefore, from A, ys=
1
When examining the value of ( 1+k ) k1 by
When examining
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When
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it approaches a constant value as shown on
the right. e is an irrational number and
e=2.7182818x.
ME_WS_N_171.indd 1
18/03/02 11:56
18/03/02 11:56
ME_WS_N_171.indd 1
ME_WS_N_171.indd 1
k
k
1
x
log b M
log b a
( 1+k )
( 1+k )
0.1
2.59374x
0.1
2.59374x
0.01
2.70481
x
0.01
2.70481x
0.001
2.71692
x
0.001
2.71692x
1
0.0001 2.71814
k
k
( 1+k ) x
0.0001
2.71814
0.00001 2.71826x
x
0.00001 2.71826
0.1
2.59374x
0.01
2.70481x
0.001
2.71692x
0.0001 2.71814x
0.00001 2.71826x
k
k
-0.1
-0.1
-0.01
-0.01
-0.001
-0.001
-0.0001
k
-0.0001
-0.00001
-0.00001
-0.1
-0.01
-0.001
-0.0001
-0.00001
( 1+k )
( 1+k )
2.86797x
2.86797x
2.73199
x
2.73199x
2.71964
x
2.71964x
1
2.71841
k
( 1+k ) x
2.71841
2.71829x
x
2.71829
2.86797x
2.73199x
2.71964x
2.71841x
18/03/02
2.71829
x
18/03/02
:
Time
3$5
/
:
to
/
N 144
:
/
to
:
Integration by Parts
Y
This method
called
Integration
Parts.
f ( xis
) gs(
x ) dx=f
( x ) g ( by
x )fs( x ) g ( x ) dx
Y
Y
Find the following indef inite integrals using Integration by Parts.
This method is called Integration by Parts.
Ex.
Yx cos x dx=Yx ( sin x )s dx
Y f ( x ) gs( x ) dx
= x sin x -Y( x )s sin x dx
sin x )s dx
Yx cos x dx==xYx (xY f ( x ) gs( x ) dx
sin
f ( x ) g ( x ) -Y fs( x ) g ( x ) dx
Y sin x dx
= x sin x -Y( x )s sin x dx
x+C
=x sin x+ cos
=x sin x-Y sin x dx
f ( x ) g ( x ) -Y fs( x ) g ( x ) dx
x ( x+
- cos
xx+C
)s dx
Yx sin x dx==xYsin
cos
=-x cos x-Y( x )s( - cos x ) dx
Yx sin x dx=
Yx ( - cos x )s dx
=-x cos x+Y cos x dx
=-x cos x- ( x )s( - cos x ) dx
=-x cos x+Y
sin x+C
=-x cos x+Y cos x dx
Find the following indef inite integrals using Integration by Parts.
Ex.
(1)
X V 99
Page 18
Name
Date
Time
R
Grade
Mistakes
Mistakes
N 144
f ( x ) gs( x )=[ f ( x ) g ( x ) ]s-fs( x ) g ( x )
Y
4:05 PM
Name
100
0
%
/
/
:
to
B1
70%
:
C2
50%
Level XM
R
X M 62å
)
(Transformations
0
1
2
2^a≈ ·≈^¤=^
a≈ · b≈=0)
Since BE
AD≈=b≈^¤=(‘
0 ,
“ 2AB¤=AC¤
1
1
”-a≈+ b≈’ · (a≈+b≈)=0
2
2
AC=@ 2 AB
A
2^a≈^¤=^b≈^¤=(‘ a≈ · b≈=0)
2. Given ˚ABC, let M be the midpoint of side BC. Show that
“ 2AB¤=AC¤
AB¤+AC¤=2(AM¤+BM¤) using the following two methods.
AC=@ 2 AB
[Sol 1] Let AB≈=a≈ and AC≈=b≈.
2. Given ˚ABC, let M be the midpoint of side BC. Show that
LHS=^a≈^¤+^b≈^¤
AB¤+AC¤=2(AM¤+BM¤) using the following two methods.
¤
¤
1
1
RHS=2ï„ (a≈+b≈)„ +„ (b≈-a≈)„ ö
[Sol 1] Let AB≈=a≈ 2and AC≈=b≈. 2
B
1a≈^¤+^b≈^¤
LHS=^
RHS= (^a≈^¤+2a≈ · b≈+^b≈^¤+^a≈^¤-2a≈ · b≈+^b≈^¤)
2
¤
¤
1
1
RHS=2
a
b
b
a
ï
„
(
+
)„
+„
(
)„
ö
≈
≈
≈
≈
RHS=^a≈^¤+^
2 b≈^¤
2
B
“ LHS=RHS
1
RHS= (^a≈^¤+2a≈ · b≈+^b≈^¤+^a≈^¤-2a≈ · b≈+^b≈^¤)
2 a≈ and BM≈=b≈.
[Sol 2] Let AM≈=
B
A
A
M
C
M
C
jp.chosa.yu 2016/11/01 14:26:58 R
AB¤+AC¤=2(AM¤+BM¤)
%
/
/
:
to
B1
70%
:
C2
50%
0
-y· y
y`=0 · x+ (-1)
˛
Rewriting,
The
matrix equation is:
x`x`
=1 · x+0
1 ·y 0
x
”=”
”” ”
0 -1
y`y`
=0 · x+
(-1) · yy
˛
Í̧ ”
The matrix equation is:
x`
1
0
x
” ”=”
”” ”
y`
0 -1
y
7/23/09
2:48 PM
X P 24
Page 8
1
2
y
O
O
x
Q(x`, y`)
x
Q(x`, y`)
2. Given that P(x, y) is mapped to Q(x`, y`) by the reflection in the origin, find
the matrix equation relating the two points.
[Sol] Since (x, y) is mapped to (x`, y`) by the reflection in the origin,
2. Given that P(x, y) is mapped to Q(x`, y`) by the reflection in the origin, find
x`=-x,=y`=-y
the matrix equation relating the two points.
Rewriting,
[Sol] Since (x,
y) is mapped
x`=(-1)
· x+0 to
· y (x`, y`) by the reflection in the origin,
x`
=-y
y`=-x,=y`
=0 · x+(-1)
·y
˛
Rewriting,
The
matrix equation is:
x`x`
=(-1)-1
· x+0 · y0 x
Í̧ ”
”=”
”” ”
0 -1
y`y`
=0 · x+(-1)
·y y
˛
Í̧
Grade
Mistakes
/
:
Name
100
0
/
to
B1
%
0
Mistakes
70%
X P 24
:
C2~3
D4~6
2~3
4~6
50%
49%
1
[Sol] Examining 5 of the possibilities,
1. Find the number
of ways 5Epeople A, B,DC, D and E can
sit around aB round
A
C
table.
B
E
A
D
E
C
[Sol] Examining 5 of the possibilities,
C
A
D
B
E
C
A
D
D
B
B
E
C
A
C
A
D
B
E
Since
the E
seating
B
A relationship
D E of theCpeople
D in each
B ofC the above
A
possibilities is the same,
D
BshowCthe same
A seating
B arrangement.
E
A
all C
5 possibilities
D
E
Since
the seating
relationship
the people
in each
of the
above
Therefore,
if we were
to use 5! of
to find
the number
of ways,
it would
possibilities
is large.
the same,
be 5 times too
5!
all
5 possibilities
show of
theways
sameis:seating =
arrangement.
Thus,
the total number
24
5 the number of ways, it would
Therefore, if we were to use 5! to find
Number
Circular
Permutations
be 5oftimes
too large.
The ways
of arranging
differentofobjects
in a 5!
ring =
are called
circular
Thus,
the total number
ways is:
24
permutations.
5
The
numberof
ofCircular
circular permutations
of n objects is:
Number
Permutations
The ways
n! of arranging different objects in a ring are called circular
=(n-1)!
n
permutations.
The number of circular permutations of n objects is:
2. Find the number of ways 6 people can sit around a round table.
n!
=(n-1)!
6!
[Sol] n
Using the formula, the number of ways is:
=5!=120
6
2. Find the number of ways 6 people can sit around a round table.
3. Find the number of ways 8 people can sit around a6!
round table.
[Sol] Using the formula, the number of ways is:
=5!=120
6
8!
[Sol] Using the formula, the number of ways is:
=7!=5040
8
3. Find the number of ways 8 people can sit around a round table.
Note: In general, when S is a set, the mapping from S to S is called a transformation of S.
jp.chosa.yu 2016/11/01 14:27:22 R
R
Date
/
/
Permutations and Combinations
Time
1. Find the number of ways 5 people A, B, C,
D and E: can sittoaround a: round
table.
Grade 100 %
B 70% C 50% D 49%
3~4
P(x, y)
Date
Time
X P 24å
49%
[Sol] Since (x, y) is mapped to (x`, y`) by the reflection in the x-axis,
1. Given that P(x, y) is mapped to Q(x`, y`) by the reflection in the x-axis, find
y
x`=x,=y`= -y
the matrix equation relating the two points.
Rewriting,
P(x, y)
[Sol] Since (x,
y) is· x+0
mapped
x`=1
· y to (x`, y`) by the reflection in the x-axis,
Í̧ x`=x,=y`=
Page 8
Permutations and Combinations
X M 62
D3~4
The matrix equation is:
x`
-1
0 x
” ”=”
”” ”
y`
0 -1 y
AB¤+AC¤=^AB≈^¤+^AC≈^¤
AB¤+AC¤=2(AM¤+BM¤)
AB¤+AC¤=^a≈-b≈^¤+^a≈+b≈^¤
AB¤+AC¤=2(^a≈^¤+^b≈^¤)
Name
100
0
Mistakes
RHS=^
a≈^¤+^b≈^¤ AB≈^¤+^AC≈^¤
AB¤+AC¤=^
“ LHS=RHS
AB¤+AC¤=^a≈-b≈^¤+^a≈+b≈^¤
[Sol 2] Let AM
=
a≈ and BM≈=ab≈^¤+^
≈
≈.
AB¤+AC¤=2(^
b≈^¤)
=-x cos x+ sin x+C
Grade
Mistakes
2:48 PM
Level XP
R
X P 24å
and Combinations)
(Permutations
XP 021-030*
Date
/
/
Transformations I
Time
:
: find
1. Given that P(x, y) is mapped to Q(x`, y`) by
the reflection
intothe x-axis,
the matrix equation relating the Grade
two points.
100 %
B 70% C 50% D 49%
3~4
a
R
7/23/09
Name
Date
Time
X M 62å
49%
a
X M 62
Name
Transformations I
X V 99
D3~4
[Sol] Let AB≈=a≈ and AC≈=b≈.
C
1. Given ˚ABC where A=90°,
1 let D and E be the midpoints of BC and CA
BE≈=-a≈+ b≈
2
respectively. Show that if AD†BE,
AC=@ 2 AB.
b
1
AD≈= (a≈+b≈)
D
2 AC≈=b≈.
[Sol] Let AB≈=a≈ and
E
C
1
Since BE
AD≈=
BE≈≈·=a≈+0 ,b≈
2
b
1
1
”· b≈)(a≈+b≈)=0
≈+ 12 (ba≈’≈+
AD≈a=
D
2
2
B
EA
XP 021-030*
XM 061-070* 08.6.23 0:33 PM Page 4
Date
/
/
Vectors and Figures
Time
:
toof BC and
:
1. Given ˚ABC where A=90°, let D and E
be the midpoints
CA
respectively. Show that if AD†BE,
Grade AC=
100 %@ 2 AB. B 70% C 50% D 49%
If both sides
following
f ( xare
) gs(integrated,
x ) dx=f (then
x ) gthe
( x )fs( x )formula
g ( x ) dxis obtained.
Integration by Parts
X V 99å
O 81
If both sides are integrated, then the following formula is obtained.
can be rearranged into
(1)
1
k
1
k
2
f ( x ) gs( x )=[ f ( x ) g ( x ) ]s-fs( x ) g ( x )
[ f ( x ) g ( x ) ]s=fs( x ) g ( x )+f ( x ) gs( x )
log a a=1
1
k
1
k
Time
Name
3$5
Date
can be rearranged into
The Product Rule
n
a
2
The Product Rule
a
/
Date
#
Integration by Parts
1
08.7.10
Vectors and Figures
Name
O 81a
a
B
c
Integration by Parts
N 171
2
3$4
Differentiation of 1Logarithmic
and
Using
the def inition
of the derivative, differentiate y= log a x. ( x > 0 )
Exponential
Functions
Using the def inition of the derivative, differentiate
y= log
/ a x. ( x >
/ 0)
Date
log a ( x+h )- log a x
[Sol] ys= lim
:
:
log a3x$ 4
Time
to
h c 0 log a1( x+h
h )2
M
[Sol] ys= lim
log M- log N= log
hc0
h
N
M
1
x+h
log M- log N= log
=def
lim
1 log aofx+h
Using the
inition
the derivative, differentiate y= log a x. ( xN> 0 )
hc0
h
log a x
= lim
hc0 h
x
h log a x
1log a ( x+h )= lim
1 log a 1+ x
h
c0 h
[Sol] ys=
h
lim
hc0
log a 1+
= lim
h
M
hc0
x
log M- log N= log
h h
N
1 . As hx+h
Let= h
=k
c 0, k c 0; therefore,
log
lim
a
x
0 h . As h c
Let h c=k
x 0, k c 0; therefore,
x
1
log
h
1
ys=klim
)
a ( 1+k
1 a 1+
c 0 kx
log
=
lim
log a ( 1+k
ys=
)
h c lim
0 h
x
c
n log M= log M
k
0 kx
1
1
n log M= log M
h lim
( 1+k ) ;k1 therefore,
1 hlog
k c 0 . As
Let =
=k
c a0(, 1+k
k c 0) k
x
log
=
a
x klim
c0
x
1
1
k is known to approach a constant value, which is
Asys=
k clim
0, ( 1+k
)
1
log a ( 1+k )
As k ck c0,0 (kx
1+k ) k is known to approach a constant value, which is
n log M= log M
expressed as1e.
1
expressed
as e. log1a ( 1+k ) k
= lim
c0
So, e=k klim
(x
1+k ) k1
c0
So, e=klim
( 1+k )1k
c0
k
As k cthis
is known
Using
e, to approach a constant value, which is
0, (constant
1+k ) value
Using this constant value
e,
log M
log e e
1 e.
1
log M=
expressed
log
ys= 1 aslog
log M
a
a e=
log e e =
1 a xA
log M=
1
x log
x log
ea =
e
xA
ys= x log a e=
log a=1 log a
k
So, e=klim
(
1+k
)
log e a
x log e a
x
x
log a=1
c0
b
=b 2+c 2-2bc cos A
b 2+c 2-2bc cos A
=b
A
C
H
C
=( b sin A ) 2+( b cos A-c ) 2
2
=( b sin A ) 2+(
a c-b cos A )
:
:
XM 061-070* 08.6.23 0:33 PM Page 4
© 2006 Kumon
© 2006
Institute
Kumon
of Education
Institute ofNorth
Education
America
North
KIEAmerica
2008 GBKIE
Printed
2008 GB Printed
in Malaysia in Malaysia
A
2
sin A+ cos A=1
A
A
H
c
/
/
to
to
Page 18
© 2006 Kumon
© 2006
Institute
Kumon
of Education
Institute ofNorth
Education
America
North
KIEAmerica
2008 GBKIE 2008 GB
in Malaysia in Malaysia
B
3$4
/
/
:
:
#
2
4:05 PM
Level XV
X V 99å andR
Figures)
(Vectors
XV 091-100*
O 81
© 2006
Kumon
Institute ofNorth
Education
North
© 2006 Kumon
Institute
of Education
America
KIEAmerica
2008 GBKIE 2008 GB
in Malaysia in Malaysia
2
C 2H=
A H= b cos A
a
=C Hb2sin
+BAH ,
2
C
A
2
b sinB-A
A ) 2H=
+( c-b
Also,=(
B H=A
c-bcos
cosAA )
In qB
, 2-2bc cos A
C 2H+c
=b
2
a
b
[Sol] In
a perpendicular
qABBH=A
C, drop
Also,
B-A
H= c-b cosCAH from
to, A B. In qA C H,
vertex
In qB C
CH
1
© 2016 Kumon
© 2016
Institute
Kumon
of Education
Institute of Education
KIE KIE 2018
KIE GB
KIE 2018 GB
2
© 2016
© 2016
Kumon
Kumon
Institute
© 2016
Institute
Kumon
of Education
of Education
Institute
KIE
of
KIE
Education
KIEKIE
2018
2018
GB
KIE
GBKIE 2018 GB
© 2016 Kumon
© 2016
Institute
Kumon
of Education
Institute of
KIE
Education
KIE 2018
KIE
GBKIE 2018 GB
2
C H=
b sin
a =b +c
-2bc
A.,A H= b cos A
cosA
2
N 171a
M 184
As shown in the diagram below, when A and
prove that
/
DateB are both/ acute angles,
a2=b 2+c 2-2bc cos A.
:
:
Time
to
1
2
C
[Sol] In qA B C, drop a perpendicular C H from
,
vertex
A B. In qA
C Hwhen
A and B are both acute angles, prove that
As shown
in C
thetodiagram
below,
2
sg.wendy 2021/03/19 09:28:23 DD
Exponential Functions
/
:
Time
Name
Name
Name
Date
Date
Time
Time
Name
Answers: all x
/
Date
#
Level O
by Parts#
)
(Integration
O 81a
N 171
N 171
Name
M 184a
Sine and Cosine Rules
Level N
#
N 171a
#
N
1 7 1 a of Logarithmic
Differentiation
and
of Logarithmic
Functions)
(Differentiation
Differentiation
of Logarithmic and
Exponential Functions
M 184
Answers: all x
Answers: all x
Level M
M 1 8 4Rule
a
)
(Cosine
Sine and Cosine Rules
08.7.10
[Sol] Using the formula, the number of ways is:
jp.chosa.yu 2016/11/01 14:27:45 R
Note: In general, when S is a set, the mapping from S to S is called a transformation of S.
8!
=7!=5040
8
11:56
11:56
18/03/02 11:56
ME_WS_O_081.indd 1
18/03/02 11:59
ME_WS_O_081.indd 1
18/03/02 11:59
jp.chosa.yu 2016/11/01 14:26:58 R
jp.chosa.yu 2016/11/01 14:27:22 R
jp.chosa.yu 2016/11/01 14:27:45 R
18/03/23 14:36