Mathematics Revision Notes
S1 Chapters
Chapter List
Ch 1.
Directed Numbers and the Number Line
P.3
Ch 2.
Introduction to Algebra
P.5
Ch 3.
Algebraic Equations in One Unknown
P.10
Ch 4.
Percentages (I)
P.13
Ch 5.
Estimation in Numbers and Measurement
P.19
Ch 6.
Introduction to Geometry
P.21
Ch 7.
Symmetry and Transformation
P.23
Ch 8.
Areas and Volumes (I)
P.27
Ch 9.
Congruence and Similarity
P.30
Ch 10.
Introduction to Coordinates
P.32
Ch 11.
Angles related to Lines
P.38
Ch 12.
Manipulation of Simple Polynomials
P.40
Ch 13.
Introduction of Statistics and Statistical Diagrams
P.43
Mark Distribution from Chapter 1 to Chapter 13
50
47
30
26
19
9
13
Ch
12
11
Ch
10
Ch
9
8
Ch
7
3
2
0
Ch
6
0
Ch
5
Ch
4
Ch
3
Ch
Ch
1
0
0
2
0
9
7
Ch
9
10
Ch
20
Ch
Weight
40
S1 Chapters
P. 2
4
13
Ch
Ch
13
12
11
0
Ch
2
12
Ch
10
0
Ch
9
11
Ch
9
18
Ch
0
10
Ch
8
1
Ch
9
Ch
7
10
Ch
8
Ch
6
0
Ch
0
7
Ch
5
0
Ch
0
6
Ch
4
0
Ch
8
5
Ch
3
40
Ch
2
4
Ch
2
1
P1 Marks
20
Ch
3
Ch
Ch
0
Ch
0
2
1
0
Ch
Ch
P2 Marks
Overview of S1 Chapters
40
30
17
5
6
0
9
8
7
6
3
3
2
1
0
Mathematics Revision Notes
S1-Ch1
Directed Numbers and the Number Line
(prepared by Chris Wong)
Section List
Sec 1.1 Concept of Directed Numbers
Sec 1.2 Addition and Subtraction of Directed Numbers
Sec 1.3 Multiplication and Division of Directed Numbers
Sec 1.4 Mixed Operations of Directed Numbers
DSE Core Paper 1
Sa Pr
12 13 14
15 16 17
18 19 20
Sec 1.1
Sec 1.2
Sec 1.3
Sec 1.4
DSE Core Paper 2
Sa Pr
Sec 1.1
Sec 1.2
Sec 1.3
Sec 1.4
12 13 14
15 16 17
18 19 20
Directed Numbers and the Number Line
DSE Question Table
1.1
1.2 1.3 1.4
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
Section 1.1 - Concept of Directed Numbers
Section 1.2 - Addition and Subtraction of Directed Numbers
Section 1.3 - Multiplication and Division of Directed Numbers
Section 1.4 - Mixed Operations of Directed Numbers
S1-Ch1
P. 4
Mathematics Revision Notes
S1-Ch2
Introduction to Algebra
(prepared by Chris Wong)
Section List
Sec 2.1 Algebra Language
Sec 2.2 Formulas and Method of Substitution
Sec 2.3 Number Patterns
DSE Core Paper 1
Sa Pr
12 13 14
15 16 17
18 19 20
Sec 2.1
Sec 2.2
Sec 2.3
DSE Core Paper 2
Sec 2.1
Sec 2.2
Sec 2.3
Sa Pr
12 13 14
15 16 17
18 19 20
1
1
1
1
1
1
1
1
1
Introduction to Algebra
DSE Question Table
2.1 2.2
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
2.3
Samp-P2-Q11
2012-P2-Q12
2014-P2-Q14
2015-P2-Q13
2016-P2-Q14
2017-P2-Q13
2018-P2-Q12
2019-P2-Q14
2020-P2-Q12
Paper 2 - B
Cross Topic
Section 2.1 - Algebra Language
Section 2.2 - Formulas and Method of Substitution
Section 2.3 - Number Patterns
Paper 2 - A
1.
Let an be the n th term of a sequence. If a1 = 4, a2 = 5 and an+2 = an + an+1 for any positive integer n ,
then a10 =
A. 13 .
B. 157 .
C. 254 .
D. 411 .
[
S1-Ch2
P. 6
DSE2012S-CoreP2-Q11]
Introduction to Algebra
2.
In the figure, the 1st pattern consists of 1 dot. For any positive integer n , the (n + 1)th pattern is formed by
adding n dots to the n th pattern. Find the number of dots in the 8th pattern.
A. 22
B. 29
C. 36
D. 37
[
3.
DSE2012-CoreP2-Q12]
Let an be the n th term of a sequence. If a2 = 7 , a4 = 63 and an+2 = an+1 + an for any positive integer n ,
then a5 =
A. 56 .
B. 70 .
C. 91 .
D. 119 .
[
4.
DSE2014-CoreP2-Q14]
In the figure, the 1st pattern consists of 5 dots. For any positive integer n , the (n + 1)th pattern is formed
by adding 4 dots to the nth pattern. Find the number of dots in the 6th pattern.
A. 21
B. 25
C. 29
D. 33
[
S1-Ch2
P. 7
DSE2015-CoreP2-Q13]
Introduction to Algebra
5.
In the figure, the 1st pattern consists of 9 dots. For any positive integer n , the (n + 1) th pattern is formed
by adding 5 dots to the n th pattern. Find the number of dots in the 7th pattern.
A. 29
B. 34
C. 39
D. 44
[
6.
DSE2016-CoreP2-Q14]
In the figure, the 1st pattern consists of 1 dot. For any positive integer n , the (n + 1)th pattern is formed
by adding (2n + 2) dots to the n th pattern. Find the number of dots in the 7th pattern.
A. 41
B. 55
C. 71
D. 161
[
7.
DSE2017-CoreP2-Q13]
Let an be the n th term of a sequence. If a3 = 21 , a6 = 89 and an+2 = an + an+1 for any positive integer n ,
then a1 =
A. 8 .
B. 13 .
C. 34 .
D. 55 .
[
S1-Ch2
P. 8
DSE2018-CoreP2-Q12]
Introduction to Algebra
8.
In the figure, the 1st pattern consists of 6 dots. For any positive integer n , the (n + 1)th pattern is formed
by adding 4 dots to the n th pattern. Find the number of dots in the 9th pattern.
A. 30
B. 34
C. 38
D. 42
[
9.
DSE2019-CoreP2-Q14]
In the figure, the 1st pattern consists of 3 dots. For any positive integer n , the (n + 1)th pattern is formed
by adding (2n + 1) dots to the n th pattern. Find the number of dots in the 7th pattern.
A. 15
B. 27
C. 38
D. 51
[
Answers
1.
C
2.
B
3.
D
4.
B
5.
C
6.
B
7.
A
8.
C
9.
D
S1-Ch2
P. 9
DSE2020-CoreP2-Q12]
Mathematics Revision Notes
S1-Ch3
Algebraic Equations in One Unknown
(prepared by Chris Wong)
Section List
Sec 3.1 Algebraic Equations in One Unknown
Sec 3.2 More about Solving Equations
Sec 3.3 Applications of Algebraic Equations in One Unknown
DSE Core Paper 1
Sec 3.1
Sec 3.2
Sec 3.3
Sa Pr
12 13 14
15 16 17
4
4
4
4
18 19 20
2
DSE Core Paper 2
Sa Pr
Sec 3.1
Sec 3.2
Sec 3.3
12 13 14
1
15 16 17
18 19 20
Algebraic Equations in One Unknown
DSE Question Table
3.1 3.2
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
3.3
Samp-P1-Q5
2012-P1-Q5
2015-P1-Q7
2017-P1-Q4
2014-P2-Q8
2019-P1-Q7
Section 3.1 - Algebraic Equations in One Unknown
Section 3.2 - More about Solving Equations
Section 3.3 - Applications of Algebraic Equations in One Unknown
Paper 1 - A(1)
1.
In a football league, each team gains 3 points for a win, 1 point for a draw and 0 point for a loss. The
champion of the league plays 36 games and gains a total of 84 points. Given that the champion does not
lose any games, find the number of games that the champion wins.
(4 marks)
DSE2012S-CoreP1-Q05]
[
2.
There are 132 guards in an exhibition centre consisting of 6 zones. Each zone has the same number of
guards. In each zone, there are 4 more female guards than male guards. Find the number of male guards
in the exhibition centre.
(4 marks)
[
3.
The number of apples owned by Ada is 4 times that owned by Billy. If Ada gives 12 of her apples to
Billy, they will have the same number of apples. Find the total number of apples owned by Ada and Billy.
(4 marks)
[
4.
DSE2012-CoreP1-Q05]
DSE2015-CoreP1-Q07]
There are only two kinds of admission tickets for a theatre: regular tickets and concessionary tickets. The
prices of a regular ticket and a concessionary ticket are $126 and $78 respectively. On a certain day, the
number of regular tickets sold is 5 times the number of concessionary tickets sold and the sum of money for
the admission tickets sold is $50 976 . Find the total number of admission tickets sold that day. (4 marks)
[
Paper 2 - A
S1-Ch3
P. 11
DSE2017-CoreP1-Q04]
Algebraic Equations in One Unknown
5.
The price of 2 bowls and 3 cups is $ 506 . If the price of 5 bowls and the price of 4 cups are the same, then
the price of a bowl is
A. $ 88 .
B. $ 92 .
C. $ 110 .
D. $ 115 .
[
DSE2014-CoreP2-Q08]
Cross Topic
6.
In a playground, the ratio of the number of adults to the number of children is 13 : 6 . If 9 adults and 24
children enter the playground, then the ratio of the number of adults to the number of children is 8 : 7 . Find
the original number of adults in the playground.
(4 marks)
[
Answers
1.
24
2.
The number of male guards in the exhibition centre is 54.
3.
40
4.
432
5.
A
6.
39
S1-Ch3
P. 12
DSE2019-CoreP1-Q07]
Mathematics Revision Notes
S1-Ch4
Percentages (I)
(prepared by Chris Wong)
Section List
Sec 4.1 Simple Problems on Percentages
Sec 4.2 Percentage Change
Sec 4.3 Profit and Loss
Sec 4.4 Discount
DSE Core Paper 1
Sa Pr
Sec 4.1
Sec 4.2
Sec 4.3
Sec 4.4
12 13 14
4
15 16 17
2
4
18 19 20
1
4
4
4
4
4
5
4
DSE Core Paper 2
Sa Pr
Sec 4.1
Sec 4.2
Sec 4.3
Sec 4.4
12 13 14
15 16 17
18 19 20
1
1
1
1
1
1
1
Percentages (I)
DSE Question Table
4.1
4.2
2012-P1-Q4
2016-P1-Q5
2020-P1-Q5
4.3
Samp-P1-Q4
2012-P2-Q8
2014-P2-Q9
Samp-P2-Q10
Prac-P2-Q10
2013-P2-Q10
2020-P2-Q9
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
4.4
Prac-P1-Q4
2014-P1-Q6
2015-P1-Q6
2018-P1-Q7
2019-P1-Q5
Prac-P2-Q11
2014-P1-Q5
2017-P1-Q8
Section 4.1 - Simple Problems on Percentages
Paper 2 - A
1.
In a company, 37.5% of the employees are female. If 60% of the male employees and 80% of the female
employees are married, then the percentage of married employees in the company is
A. 32.5% .
B. 45% .
C. 55% .
D. 67.5% .
[
DSE2012-CoreP2-Q08]
Section 4.2 - Percentage Change
Paper 1 - A(1)
2.
The daily wage of Ada is 20% higher than that of Billy while the daily wage of Billy is 20% lower than
that of Christine. It is given that the daily wage of Billy is $480 .
(a) Find the daily wage of Ada.
(b) Who has the highest daily wage? Explain your answer.
(4 marks)
[
S1-Ch4
P. 14
DSE2012-CoreP1-Q04]
Percentages (I)
3.
In a recreation club, there are 180 members and the number of male members is 40% more than the number of female members. Find the difference of the number of male members and the number of female
members.
(4 marks)
[
4.
DSE2016-CoreP1-Q05]
In a recruitment exercise, the number of male applicants is 28% more than the number of female applicants.
The difference of the number of male applicants and the number of female applicants is 91 . Find the number
of male applicants in the recruitment exercise.
(4 marks)
[
DSE2020-CoreP1-Q05]
Paper 2 - A
5.
There are 792 workers in a factory. If the number of male workers is 20% less than that of female workers,
then the number of male workers is
A. 352.
B. 360.
C. 432.
D. 440.
DSE2014-CoreP2-Q09]
[
Cross Topic
6.
If the circumference of a circle is increased by 40% , then the area of the circle is increased by
A. 18% .
B. 20% .
C. 40% .
D. 96% .
DSE2012P-CoreP2-Q11]
[
7.
Consider the formula 2(3m + n) = m + 7 .
(a) Make n the subject of the above formula.
(b) If the value of m is increased by 2, write down the change in the value of n .
(4 marks)
8.
It is given that y varies inversely as
(a) Express y in terms of x .
p
[
DSE2014-CoreP1-Q05]
x . When x = 144 , y = 81 .
(b) If the value of x is increased from 144 to 324, find the change in the value of y .
(5 marks)
[
Section 4.3 - Profit and Loss
Paper 1 - A(1)
S1-Ch4
P. 15
DSE2017-CoreP1-Q08]
Percentages (I)
9.
The marked price of a handbag is $ 560 . It is given that the marked price of the handbag is 40% higher
than the cost.
(a) Find the cost of the handbag.
(b) If the handbag is sold at $ 460 , find the percentage profit.
(4 marks)
[
DSE2012S-CoreP1-Q04]
Paper 2 - A
10. Mary sold two bags for $240 each. She gained 20% on one and lost 20% on the other. After the two
transactions, Mary
A. lost $20 .
B. gained $10 .
C. gained $60 .
D. had no gain and no loss.
[
DSE2012S-CoreP2-Q10]
11. John buys a vase for $ 1 600 . He then sells the vase to Susan at a profit of 20% . At what price should
Susan sell the vase in order to have a profit of 20% ?
A. $ 2 240
B. $ 2 304
C. $ 2 400
D. $ 2 500
[
DSE2012P-CoreP2-Q10]
12. Susan sells two cars for $80 080 each. She gains 30% on one and loses 30% on the other. After the two
transactions, Susan
A. loses $15 840.
B. gains $5 544 .
C. gains $10 296 .
D. has no gain and no loss.
[
DSE2013-CoreP2-Q10]
13. The cost of a toy is x% lower than its selling price. After selling the toy, the percentage profit is 25% . Find
x.
A. 20
B. 25
C. 75
D. 80
[
S1-Ch4
P. 16
DSE2020-CoreP2-Q09]
Percentages (I)
Section 4.4 - Discount
Paper 1 - A(1)
14. The cost of a chair is $ 360 . If the chair is sold at a discount of 20% on its marked price, then the percentage
profit is 30% . Find the marked price of the chair.
(4 marks)
[
DSE2012P-CoreP1-Q04]
15. The marked price of a toy is $255 . The toy is now sold at a discount of 40% on its marked price.
(a) Find the selling price of the toy.
(b) If the percentage profit is 2% , find the cost of the toy.
(4 marks)
[
DSE2014-CoreP1-Q06]
16. The cost of a book is $250 . The book is now sold and the percentage profit is 20% .
(a) Find the selling price of the book.
(b) If the book is sold at a discount of 25% on its marked price, find the marked price of the book.
(4 marks)
[
DSE2015-CoreP1-Q06]
17. The marked price of a vase is 30% above its cost. A loss of $88 is made by selling the vase at a discount
of 40% on its marked price. Find the marked price of the vase.
(5 marks)
[
DSE2018-CoreP1-Q07]
18. A wallet is sold at a discount of 25% on its marked price. The selling price of the wallet is $690 .
(a) Find the marked price of the wallet.
(b) After selling the wallet, the percentage profit is 15% . Find the cost of the wallet.
(4 marks)
[
Answers
1.
D
2.
(a) $ 576
3.
30
(b) Christine has the highest daily wage (= 600)
4.
5.
A
6.
D
7.
(a) n =
8.
(a) y = p (b) −27
9.
(a) $400
7 − 5m
2
(b) The decrease in the value of n = 5
972
x
S1-Ch4
(b) 15%
P. 17
DSE2019-CoreP1-Q05]
Percentages (I)
10. A
11. B
12. A
13. A
14. $585
15. (a) $153
(b) $150
16. (a) $300
(b) $400
17. $520
18. (a) $920 (b) $600
S1-Ch4
P. 18
Mathematics Revision Notes
S1-Ch5
Estimation in Numbers and Measurement
(prepared by Chris Wong)
Section List
Sec 5.1 Introduction to Estimation
Sec 5.2 Estimation in Measurement
DSE Core Paper 1
Sa Pr
12 13 14
15 16 17
18 19 20
Sec 5.1
Sec 5.2
DSE Core Paper 2
Sa Pr
Sec 5.1
Sec 5.2
12 13 14
15 16 17
18 19 20
Estimation in Numbers and Measurement
DSE Question Table
5.1 5.2
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
Section 5.1 - Introduction to Estimation
Section 5.2 - Estimation in Measurement
S1-Ch5
P. 20
Mathematics Revision Notes
S1-Ch6
Introduction to Geometry
(prepared by Chris Wong)
Section List
Sec 6.1 Basic Elements of Geometry
Sec 6.2 Plane Figures
Sec 6.3 Construction of Geometric Figures
Sec 6.4 Three-Dimensional Figures
DSE Core Paper 1
Sa Pr
12 13 14
15 16 17
18 19 20
Sec 6.1
Sec 6.2
Sec 6.3
Sec 6.4
DSE Core Paper 2
Sa Pr
Sec 6.1
Sec 6.2
Sec 6.3
Sec 6.4
12 13 14
15 16 17
18 19 20
Introduction to Geometry
DSE Question Table
6.1
6.2 6.3 6.4
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
Section 6.1 - Basic Elements of Geometry
Section 6.2 - Plane Figures
Section 6.3 - Construction of Geometric Figures
Section 6.4 - Three-Dimensional Figures
S1-Ch6
P. 22
Mathematics Revision Notes
S1-Ch7
Symmetry and Transformation
(prepared by Chris Wong)
Section List
Sec 7.1 Symmetry
Sec 7.2 Transformation
DSE Core Paper 1
Sa Pr
12 13 14
15 16 17
Sec 7.1
Sec 7.2
18 19 20
1
DSE Core Paper 2
Sa Pr
Sec 7.1
Sec 7.2
1
1
1
12 13 14
15 16 17
18 19 20
1
1
1
1
1
DSE Question Table
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
7.1
7.2
Prac-P2-Q24
2013-P2-Q15
2016-P2-Q23
2018-P2-Q23
Samp-P2-Q25
2012-P2-Q22
2015-P2-Q22
2016-P1-Q7
Prac-P2-Q25
Symmetry and Transformation
Section 7.1 - Symmetry
Paper 2 - A
1.
Which of the following parallelograms have rotational symmetry and reflectional symmetry?
I.
II.
6
III.
6
6
6
6
A. I and II only
B. I and III only
C. II and III only
D. I, II and III
[
2.
DSE2012P-CoreP2-Q24]
In the figure, the regular octagon is divided into eight identical isosceles triangles and four of them are
shaded. The number of axes of reflectional symmetry of the octagon is
A. 2 .
B. 4 .
C. 8 .
D. 16 .
[
3.
DSE2013-CoreP2-Q15]
The figure below consists of eight identical regular hexagons. The number of axes of reflectional symmetry
of the figure is
A. 2 .
B. 4 .
C. 6 .
D. 8 .
[
S1-Ch7
P. 24
DSE2016-CoreP2-Q23]
Symmetry and Transformation
4.
The figure below consists of eight identical squares. The number of folds of rotational symmetry of the
figure is
A. 2 .
B. 4 .
C. 6 .
D. 8 .
[
DSE2018-CoreP2-Q23]
Cross Topic
5.
Which of the following statements about a regular 12-sided polygon are true?
I. Each exterior angle is 30◦ .
II. Each interior angle is 150◦ .
III. The number of axes of reflectional symmetry is 6 .
A. I and II only
B. I and III only
C. II and III only
D. I, II and III
[
6.
DSE2012-CoreP2-Q22]
If an interior angle of a regular polygon is 5 times an exterior angle of the polygon, which of the following
is/are true?
I. Each interior angle of the polygon is 150◦ .
II. The number of diagonals of the polygon is 6 .
III. The number of folds of rotational symmetry of the polygon is 6 .
A. I only
B. II only
C. I and III only
D. II and III only
[
7.
DSE2015-CoreP2-Q22]
In a polar coordinate system, O is the pole. The polar coordinates of the points A and B are (12, 75◦ ) and
(12, 135◦ ) respectively.
(a) Find ∠ AOB .
(b) Find the perimeter of 4AOB .
(c) Write down the number of folds of rotational symmetry of 4AOB .
(4 marks)
[
S1-Ch7
P. 25
DSE2016-CoreP1-Q07]
Symmetry and Transformation
Section 7.2 - Transformation
Paper 2 - A
8.
In the figure, the two 6-sided polygons show
A. a rotation transformation.
B. a reflection transformation.
C. a translation transformation.
D. a dilation transformation.
[
DSE2012S-CoreP2-Q25]
Cross Topic
9.
If the point (−2, −1) is reflected with respect to the straight line y = −5 , then the coordinates of its image
are
A. (−8, −1)
B. (−2, −9)
C. (−2, 11)
D. (12, −1)
[
Answers
1.
D
2.
B
3.
A
4.
B
5.
A
6.
A
7.
(a) 60◦
8.
A
9.
B
S1-Ch7
(b) 36
(c) 3
P. 26
DSE2012P-CoreP2-Q25]
Mathematics Revision Notes
S1-Ch8
Areas and Volumes (I)
(prepared by Chris Wong)
Section List
Sec 8.1 Areas of Simple Polygons
Sec 8.2 Volumes and Total Surface Areas of Prisms
DSE Core Paper 1
Sa Pr
Sec 8.1
Sec 8.2
12 13 14
15 16 17
18 19 20
5
DSE Core Paper 2
Sa Pr
12 13 14
Sec 8.1
Sec 8.2
15 16 17
18 19 20
1
1
DSE Question Table
8.1
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
8.2
2012-P1-Q9
2016-P2-Q18
2014-P2-Q15
Section 8.1 - Areas of Simple Polygons
Cross Topic
Areas and Volumes (I)
1.
In the figure, AB = AE and ∠B AE = ∠BC D = ∠C DE = 90◦ . If BC = C D = DE = 16 cm , then the area of
the pentagon ABC DE is
B
A. 71 cm2 .
E
B. 128 cm2 .
A
C. 192 cm2 .
2
D. 224 cm .
C
[
D
DSE2014-CoreP2-Q15]
Section 8.2 - Volumes and Total Surface Areas of Prisms
Paper 1 - A(1)
2.
In Figure 2, the volume of the solid right prism ABC DE FG H is 1 020 cm3 . The base ABC D of the prism
is a trapezium, where AD is parallel to BC . It is given that ∠B AD = 90◦ , AB = 12 cm , BC = 6 cm and
DE = 10 cm .
F
G
H
A
B
E
C
D
Figure 2
Find
(a) the length of AD ,
(b) the total surface area of the prism ABC DE FG H .
(5 marks)
[
Paper 2 - A
S1-Ch8
P. 28
DSE2012-CoreP1-Q09]
Areas and Volumes (I)
3.
The figure shows a right prism. Find the volume of the prism.
A. 216 cm3
B. 240 cm
6 cm
3
4 cm
C. 300 cm3
D. 328 cm3
13 cm
5 cm
[
Answers
1.
C
2.
(a) The length of AD is 11 cm.
3.
C
S1-Ch8
(b) C D = 13 cm, total surface area = 624 cm2
P. 29
DSE2016-CoreP2-Q18]
Mathematics Revision Notes
S1-Ch9
Congruence and Similarity
(prepared by Chris Wong)
Section List
Sec 9.1 Concept of Congruence
Sec 9.2 Conditions for Congruent Triangles
Sec 9.3 Concept of Similarity
Sec 9.4 Conditions for Similar Triangles
DSE Core Paper 1
Sa Pr
12 13 14
15 16 17
18 19 20
Sec 9.1
Sec 9.2
Sec 9.3
Sec 9.4
DSE Core Paper 2
Sa Pr
Sec 9.1
Sec 9.2
Sec 9.3
Sec 9.4
12 13 14
15 16 17
18 19 20
Congruence and Similarity
DSE Question Table
9.1
9.2 9.3 9.4
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
Section 9.1 - Concept of Congruence
Section 9.2 - Conditions for Congruent Triangles
Section 9.3 - Concept of Similarity
Section 9.4 - Conditions for Similar Triangles
S1-Ch9
P. 31
Mathematics Revision Notes
S1-Ch10
Introduction to Coordinates
(prepared by Chris Wong)
Section List
Sec 10.1 Introduction to Ordered Pairs
Sec 10.2 Rectangular Coordinate System
Sec 10.3 Distance between Two Points
Sec 10.4 Areas of Plane Figures
Sec 10.5 Transformations of Points on the Coordinate Plane
Sec 10.6 Polar Coordinate System
DSE Core Paper 1
Sa Pr
Sec 10.1
Sec 10.2
Sec 10.3
Sec 10.4
Sec 10.5
Sec 10.6
12
13 14 15
2
16 17 18
3
4
19 20
2
4
2
DSE Core Paper 2
Sa Pr
Sec 10.1
Sec 10.2
Sec 10.3
Sec 10.4
Sec 10.5
Sec 10.6
1
12
13 14 15
16 17 18
1
19 20
1
1
1
1
1
1
1
Introduction to Coordinates
DSE Question Table
10.1
10.2
10.3 10.4
10.5
10.6
Prac-P1-Q6
2013-P1-Q6
Samp-P2-Q26
Prac-P2-Q25
2019-P2-Q25
2020-P2-Q24
Samp-P1-Q8
2014-P1-Q8
2017-P1-Q6
2012-P2-Q23
2014-P2-Q23
2015-P2-Q23
2016-P1-Q7
2017-P2-Q25
2018-P2-Q24
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
Section 10.1 - Introduction to Ordered Pairs
Section 10.2 - Rectangular Coordinate System
Section 10.3 - Distance between Two Points
Section 10.4 - Areas of Plane Figures
Section 10.5 - Transformations of Points on the Coordinate
Plane
Paper 2 - A
1.
If the point (−4, 3) is rotated anti-clockwise about the origin through 180◦ , then the coordinates of its image
are
A. (−3, 4) .
B. (3, 4) .
C. (−4, −3) .
D. (4, −3) .
[
S1-Ch10
P. 33
DSE2012S-CoreP2-Q26]
Introduction to Coordinates
2.
If the point (−2, −1) is reflected with respect to the straight line y = −5 , then the coordinates of its image
are
A. (−8, −1)
B. (−2, −9)
C. (−2, 11)
D. (12, −1)
[
3.
DSE2012P-CoreP2-Q25]
The coordinates of the point A are (−5, −2) . A is translated rightwards by 9 units to the point B . B is then
rotated anticlockwise about the origin through 90◦ to the point C . Find the y -coordinate of C .
A. −4
B. −2
C. 2
D. 4
[
DSE2019-CoreP2-Q25]
Cross Topic
4.
In Figure 3, the coordinates of the point A are (−2, 5) . A is rotated clockwise about the origin O through
90◦ to A 0 . A 00 is the reflection image of A with respect to the y -axis.
y
A(−2, 5)
O
x
Figure 3
(a) Write down the coordinates of A 0 and A 00 .
(b) Is O A 00 perpendicular to A A 0 ? Explain your answer.
(5 marks)
[
5.
DSE2012S-CoreP1-Q08]
The coordinates of the points P and Q are (−3, 5) and (2, −7) respectively. P is rotated anticlockwise about
the origin O through 270◦ to P 0 . Q is translated leftwards by 21 units to Q 0 .
(a) Write down the coordinates of P 0 and Q 0 .
(b) Prove that PQ is perpendicular to P 0Q 0 .
(5 marks)
[
S1-Ch10
P. 34
DSE2014-CoreP1-Q08]
Introduction to Coordinates
6.
The coordinates of the points A and B are (−3, 4) and (9, −9) respectively. A is rotated anticlockwise about
the origin through 90◦ to A 0 . B 0 is the reflection image of B with respect to the x -axis.
(a) Write down the coordinates of A 0 and B 0 .
(b) Prove that AB is perpendicular to A 0 B 0 .
(4 marks)
[
DSE2017-CoreP1-Q06]
Section 10.6 - Polar Coordinate System
Paper 1 - A(1)
7.
In a polar coordinate system, the polar coordinates of the points A , B and C are (13, 157◦ ) , (14 , 247◦ ) and
(15, 337◦ ) respectively.
(a) Let O be the pole. Are A , O and C collinear? Explain your answer.
(b) Find the area of 4ABC .
(4 marks)
[
8.
DSE2012P-CoreP1-Q06]
In a polar coordinate system, O is the pole. The polar coordinates of the points A and B are (26, 10◦ ) and
(26, 130◦ ) respectively. Let L be the axis of reflectional symmetry of 4O AB .
(a) Describe the geometric relationship between L and ∠ AOB .
(b) Find the polar coordinates of the point of intersection of L and AB .
(4 marks)
[
DSE2013-CoreP1-Q06]
Paper 2 - A
9.
The point P is translated leftwards by 4 units to the point Q . If the coordinates of the reflection image of
Q with respect to the y -axis are (5, −1) , then the polar coordinates of P are
A. (1, 45◦ ) .
B. (1, 225◦ ) .
p
C. ( 2, 45◦ ) .
p
D. ( 2, 225◦ ) .
[
DSE2020-CoreP2-Q24]
Cross Topic
p
10. The rectangular coordinates of the point P are (−3, −3 3) . If P is rotated anticlockwise about the origin
through 90◦ , then the polar coordinates of its image are
A. (3, 150◦ ) .
B. (3, 330◦ ) .
C. (6, 150◦ ) .
D. (6, 330◦ ) .
[
S1-Ch10
P. 35
DSE2012-CoreP2-Q23]
Introduction to Coordinates
p
11. The rectangular coordinates of the point P are (−1, 3) . If P is reflected with respect to the x -axis, then
the polar coordinates of its image are
A. (2, 210◦ ) .
B. (2, 240◦ ) .
C. (4, 210◦ ) .
D. (4, 240◦ ) .
[
p
DSE2014-CoreP2-Q23]
12. The rectangular coordinates of the point A are ( 3 , −1) . If A is reflected with respect to the y -axis, then
the polar coordinates of its image are
A. (1, 210◦ ) .
B. (1, 240◦ ) .
C. (2, 210◦ ) .
D. (2, 240◦ ) .
[
DSE2015-CoreP2-Q23]
13. In a polar coordinate system, O is the pole. The polar coordinates of the points A and B are (12, 75◦ ) and
(12, 135◦ ) respectively.
(a) Find ∠ AOB .
(b) Find the perimeter of 4AOB .
(c) Write down the number of folds of rotational symmetry of 4AOB .
(4 marks)
[
DSE2016-CoreP1-Q07]
14. The polar coordinates of the points P , Q and R are (3, 160◦ ) , (4, 280◦ ) and (6, 340◦ ) respectively. The
perpendicular distance from Q to P R is
A. 2 .
B. 3 .
p
C. 2 3 .
p
D. 3 3 .
[
◦
DSE2017-CoreP2-Q25]
◦
15. The polar coordinates of the points C , D and E are (16, 127 ), (12, 217 ) and (5, 307◦ ) respectively. Find
the perimeter of 4C DE .
A. 54
B. 78
C. 126
D. 130
[
S1-Ch10
P. 36
DSE2018-CoreP2-Q24]
Introduction to Coordinates
Answers
1.
D
2.
B
3.
D
4.
(a) A 0 = (5, 2), A 00 = (2, 5)
5.
(a) P 0 = (5, 3) and Q 0 = (−19, −7)
6.
(a) A 0 = (−4, −3)
7.
(a) A , O and C are collinear.
8.
(a) L is the angle bisector of ∠ AOB .
9.
D
(b) A A 0 is not perpendicular to O A 00
(b) PQ and P 0Q 0 are perpendicular
(b)B 0 = (9, 9)
(b) Area of 4ABC = 196
(b) the required polar coordinates are (13, 70◦ ) .
10. D
11. B
12. C
13. (a) 60◦
(b) 36
(c) 3
14. C
15. A
S1-Ch10
P. 37
Mathematics Revision Notes
S1-Ch11
Angles related to Lines
(prepared by Chris Wong)
Section List
Sec 11.1 Angles related to Intersecting Lines
Sec 11.2 Angles related to Parallel Lines
Sec 11.3 Identifying Parallel Lines
DSE Core Paper 1
Sa Pr
12 13 14
15 16 17
18 19 20
Sec 11.1
Sec 11.2
Sec 11.3
DSE Core Paper 2
Sa Pr
12 13 14
Sec 11.1
Sec 11.2
Sec 11.3
15 16 17
18 19 20
1
1
DSE Question Table
11.1
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
11.2
2016-P2-Q15
2020-P2-Q19
11.3
Angles related to Lines
Section 11.1 - Angles related to Intersecting Lines
Section 11.2 - Angles related to Parallel Lines
Paper 2 - A
1.
According to the figure, which of the following must be true?
I.
a + c = 180◦
II.
a + b − c = 180◦
c
III. b + c = 360◦
A. I only
b
B. II only
a
C. I and III only
D. II and III only
[
2.
DSE2016-CoreP2-Q15]
According to the figure, which of the following must be true?
I.
u − v + w = 0◦
u
II. u + v − w = 180◦
v
III. u + v + w = 450◦
w
A. I only
B. II only
C. I and III only
D. II and III only
[
Section 11.3 - Identifying Parallel Lines
Answers
1.
B
2.
B
S1-Ch11
P. 39
DSE2020-CoreP2-Q19]
Mathematics Revision Notes
S1-Ch12
Manipulation of Simple Polynomials
(prepared by Chris Wong)
Section List
Sec 12.1 Laws of Positive Integral Indices
Sec 12.2 Polynomials
Sec 12.3 Addition and Subtraction of Polynomials
Sec 12.4 Multiplication of Polynomials
DSE Core Paper 1
Sa Pr
12
13 14 15
16 17 18
19 20
Sec 12.1
Sec 12.2
Sec 12.3
Sec 12.4
DSE Core Paper 2
Sa Pr
Sec 12.1
Sec 12.2
Sec 12.3
Sec 12.4
1
12
13 14 15
1
16 17 18
19 20
1
Manipulation of Simple Polynomials
DSE Question Table
12.1
12.2 12.3
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
12.4
Prac-P2-Q1
2015-P2-Q1
2019-P2-Q1
Paper 2 - B
Cross Topic
Section 12.1 - Laws of Positive Integral Indices
Section 12.2 - Polynomials
Section 12.3 - Addition and Subtraction of Polynomials
Section 12.4 - Multiplication of Polynomials
Paper 2 - A
1.
x 3 (2x + x) =
A. 3x 4 .
B. 2x 5 .
C. 3x 5 .
D. 2x 6 .
[
2.
DSE2012P-CoreP2-Q01]
(x + 1)(x 2 + x + 1) =
A. x 3 + 1 .
B. (x + 1)3 .
C. x 3 + x 2 + x + 1 .
D. x 3 + 2x 2 + 2x + 1 .
[
S1-Ch12
P. 41
DSE2015-CoreP2-Q01]
Manipulation of Simple Polynomials
3.
(a − b)(a 2 + ab − b 2 ) =
A. (a − b)3 .
B. a 3 − b 3 .
C. a 3 − 2ab 2 + b 3 .
D. a 3 − 2a 2 b + 2ab 2 + b 3 .
[
Answers
1.
A
2.
D
3.
C
S1-Ch12
P. 42
DSE2019-CoreP2-Q01]
Mathematics Revision Notes
S1-Ch13
Introduction of Statistics and Statistical Diagrams
(prepared by Chris Wong)
Section List
Sec 13.1 Introduction to Various Stages of Statistics
Sec 13.2 Construction and Interpretation of Simple Statistical Diagrams
Sec 13.3 Construction and Interpretation of Stem-and-Leaf Diagrams
Sec 13.4 Construction and Interpretation of Scattered Diagrams
Sec 13.5 Constructing Statistical Diagrams with Computer Software
DSE Core Paper 1
Sa Pr
Sec 13.1
Sec 13.2
Sec 13.3
Sec 13.4
Sec 13.5
12
13 14 15
16 17 18
5
19 20
1
DSE Core Paper 2
Sa Pr
Sec 13.1
Sec 13.2
Sec 13.3
Sec 13.4
Sec 13.5
12
13 14 15
1
1
1
16 17 18
19 20
Introduction of Statistics and Statistical Diagrams
DSE Question Table
13.1
Paper 1 - A(1)
Paper 1 - A(2)
Paper 1 - B
Paper 2 - A
Paper 2 - B
Cross Topic
13.2
Samp-P1-Q9
13.3
13.4
13.5
2013-P2-Q30
2014-P2-Q29
2017-P1-Q7
2013-P2-Q28
Section 13.1 - Introduction to Various Stages of Statistics
Section 13.2 - Construction and Interpretation of Simple Statistical Diagrams
Paper 1 - A(1)
1.
In Figure 4, the pie chart shows the distribution of the numbers of traffic accidents occurred in a city in a
year. In that year, the number of traffic accidents occurred in District A is 20% greater than that in District
B.
District B
x°
District A
72°
District C
120°
District E
30°
District D
The distribution of the numbers of
traffic accidents occurred in the city
Figure 4
(a) Find x .
(b) Is the number of traffic accidents occurred in District A greater than that in District C ? Explain your
answer.
(5 marks)
[
S1-Ch13
P. 44
DSE2012S-CoreP1-Q09]
Introduction of Statistics and Statistical Diagrams
Paper 2 - A
2.
The pie charts below show the distributions of the profits of stationery shop X and stationery shop Y from
the sales of stationery in a certain month. Which of the following must be true?
Distribution of the profits of stationery shop X
Pen
60°
Notebook θ
Distribution of the profits of stationery shop Y
Pen
16%
Others
Others
46%
Notebook
k%
162°
68° 36°
Ruler
Ruler
12%
Pencil
Pencil
10%
A. In that month, the profit from the sales of pencils of stationery shop X is the same as that of stationery
shop Y .
B. In that month, the total profit from the sales of pens and notebooks of stationery shop X is less than
the total profit from the sales of rulers and pencils of the shop.
C. k = 14
D. θ = 36◦
[
3.
DSE2013-CoreP2-Q30]
The pie chart below shows the expenditure of John in a certain week. John spends $240 on clothing that
week. Find his expenditure on transportation that week.
A. $40
B. $60
C. $90
Transportation
D. $135
Meals
Clothing
160°
50°
Others
[
Cross Topic
S1-Ch13
P. 45
DSE2014-CoreP2-Q29]
Introduction of Statistics and Statistical Diagrams
4.
The pie chart below shows the distribution of the seasons of birth of the students in a school.
Autumn
Summer
158°
Winter
x
Spring
Distribution of the seasons of birth of the students in the school
If a student is randomly selected from the school, then the probability that the selected student was born in
1
spring is .
9
(a) Find x .
(b) In the school, there are 180 students born in winter. Find the number of students in the school.
(4 marks)
[
DSE2017-CoreP1-Q07]
Section 13.3 - Construction and Interpretation of Stem-andLeaf Diagrams
Section 13.4 - Construction and Interpretation of Scattered
Diagrams
Cross Topic
5.
The scatter diagram below shows the relation between x and y . Which of the following may represent the
relation between x and y ?
y
A. y increases when x increases.
B.
y decreases when x increases.
C.
y varies inversely as x 2 .
D. y varies directly as x −3 .
x
O
[
S1-Ch13
P. 46
DSE2013-CoreP2-Q28]
Introduction of Statistics and Statistical Diagrams
Section 13.5 - Constructing Statistical Diagrams with Computer Software
Answers
2.
(a) x = 60
C.
B
3.
C
4.
(a) 40◦ (b) 900
5.
A
1.
S1-Ch13
(b) the number of traffic accidents occurred in District A is not greater than that in District
P. 47