H2 Mathematics Textbook

Front Matter (C)
Cover (C)
Version Date (i)
CC-BY-NC-SA 4.0 License (iii)
Contents (v)
About this Book (xxix)
Tips for the Student (xxxi)
Miscellaneous Tips for the Student (xxxv)
Use of Graphing Calculators (xxxvii)
Apps You Can Use on Your TI Calculator (xxxix)
Preface/Rant (xl)
How and Why JC Maths Fails Our Students (xlii)
Work Work Work Work Work Work (xliv)
Taking Tests Seriously (xlv)
Genuine Understanding (xlvii)
Why Even Type 1 Pragmatists Should Read This Textbook (xlix)
PISA 2018 Maths Results (li)
0. A Few Basics (1)
1. Just To Be Clear (3)
2. PSLE Review: Division (4)
2.1. Long Division (6)
2.2. The Cardinal Sin of Dividing By Zero (8)
2.3. Is One Divided by Zero Infinity? (9)
3. Logic (10)
3.1. True, False, and Indeterminate Statements (11)
3.2. The Conjunction AND and the Disjunction OR (12)
3.3. The Negation NOT (13)
3.4. Equivalence ⟺ (14)
3.5. De Morgan's Laws: Negating the Conjunction and Disjunction (15)
3.6. The Implication ? ⟹ ? (17)
3.7. The Converse ? ⟹ ? (19)
3.8. Affirming the Consequent (or The Fallacy of the Converse) (21)
3.9. The Negation NOT-(? ⟹ ?) (22)
3.10. The Contrapositive NOT-? ⟹ NOT-? (25)
3.11. (? ⟹ ? AND ? ⟹ ?) ⟺ (? ⟺ ?) (27)
3.12. Other Ways to Express ? ⟹ ? (Optional) (28)
3.13. The Four Categorical Propositions and Their Negations (29)
3.14. Chapter Summary (34)
4. Sets (35)
4.1. The Elements of a Set Can Be Pretty Much Anything (36)
4.2. In ∈ and Not In ∉ (38)
4.3. The Order of the Elements Doesn't Matter (39)
4.4. n(?) Is the Number of Elements in the Set ? (40)
4.5. The Ellipsis ``…'' Means Continue in the Obvious Fashion (40)
4.6. Repeated Elements Don't Count (41)
4.7. ℝ Is the Set of Real Numbers (42)
4.8. ℤ Is the Set of Integers (43)
4.9. ℚ Is the Set of Rational Numbers (44)
4.10. A Taxonomy of Numbers (45)
4.11. More Notation: ⁺, ⁻, and ₀ (46)
4.12. The Empty Set ∅ (47)
4.13. Intervals (48)
4.14. Subset Of ⊆ (50)
4.15. Proper Subset Of ⊂ (51)
4.16. Union ∪ (53)
4.17. Intersection ∩ (54)
4.18. Set Minus ∖ (56)
4.19. The Universal Set ℰ (57)
4.20. The Set Complement ?' (58)
4.21. De Morgan's Laws (60)
4.22. Set-Builder Notation (or Set Comprehension) (62)
4.23. Chapter Summary (67)
5. O-Level Review (68)
5.1. Some Mathematical Vocabulary (68)
5.2. The Absolute Value or Modulus Function (69)
5.3. The Factorial ?! (71)
5.4. Exponents (72)
5.5. The Square Root Refers to the Positive Square Root (78)
5.6. Rationalising the Denominator with a Surd (81)
5.7. Logarithms (83)
5.8. Polynomials (85)
I. Functions and Graphs (88)
6. Graphs (90)
6.1. Ordered Pairs (90)
6.2. The Cartesian Plane (94)
6.3. A Graph is Any Set of Points (96)
6.4. The Graph of An Equation (98)
6.5. Graphing with the TI84 (100)
6.6. The Graph of An Equation with Constraints (101)
6.7. Intercepts and Roots (108)
7. Lines (111)
7.1. Horizontal, Vertical, and Oblique Lines (112)
7.2. Another Equation of the Line (113)
7.3. The Gradient of a Line (115)
7.4. Yet Another Equation of the Line (118)
7.5. Perpendicular Lines (120)
7.6. Lines vs Line Segments vs Rays (121)
8. Distance (122)
8.1. The Distance between Two Points on the Real Number Line (122)
8.2. The Distance between Two Points on the Cartesian Plane (123)
8.3. Closeness (or Nearness) (125)
9. Circles (126)
9.1. Graphing Circles on the TI84 (129)
9.2. The Number π (optional) (131)
10. Tangent Lines, Gradients, and Stationary Points (133)
10.1. Tangent Lines (133)
10.2. The Gradient of a Graph at a Point (134)
10.3. Stationary Points (135)
11. Maximum, Minimum, and Turning Points (136)
11.1. Maximum Points (136)
11.2. Minimum Points (140)
11.3. Extrema (144)
11.4. Turning Points (145)
12. Reflection and Symmetry (148)
12.1. The Reflection of a Point in a Point (148)
12.2. The Reflection of a Point in a Line (150)
12.3. Lines of Symmetry (152)
13. Solutions and Solution Sets (155)
14. O-Level Review: The Quadratic Equation ? = ??² + ?? + ? (163)
15. Functions (173)
15.1. What Functions Aren't (184)
15.2. Notation for Functions (187)
15.3. Warning: ? and ?(?) Refer to Different Things (194)
15.4. Nice Functions (195)
15.5. Graphs of Functions (197)
15.6. Piecewise Functions (201)
15.7. Range (206)
16. An Introduction to Continuity (211)
17. When a Function Is Increasing or Decreasing (215)
18. Arithmetic Combinations of Functions (219)
19. Domain Restriction (223)
20. Composite Functions (225)
21. One-to-One Functions (233)
21.1. Two More Characterisations of One-to-One (236)
21.2. A Strictly Increasing/Decreasing Function Is One-to-One (239)
22. Inverse Functions (240)
22.1. Only a One-to-One Function Can Have an Inverse (244)
22.2. Formal Definition of the Inverse (246)
22.3. A Method for Showing a Function Is One-to-One and Finding Its Inverse (247)
22.4. Inverse Cancellation Laws (251)
22.5. The Graphs of ? and ?⁻¹ Are Reflections in the Line ? = ? (252)
22.6. When Do the Graphs of ? and ?⁻¹ Intersect? (259)
22.7. Self-Inverses (263)
22.8. If ? Is Strictly Increasing/Decreasing, Then So Is ?⁻¹ (264)
22.9. A Function Is the Inverse of Its Inverse (265)
22.10. Domain Restriction to Create a One-to-One Function (267)
23. Asymptotes and Limit Notation (270)
23.1. Oblique Asymptotes (278)
24. Transformations (280)
24.1. ? = ?(?) + ? (280)
24.2. ? = ?(? + ?) (282)
24.3. ? = ??(?) (285)
24.4. ? = ?(??) (287)
24.5. Combinations of the Above (289)
24.6. ? = |?(?)| (293)
24.7. ? = ?(|?|) (295)
24.8. 1 / ? (297)
24.9. When a Function Is Symmetric in Vertical or Horizontal Lines (301)
25. ln, exp, and e (302)
25.1. The Exponential Function exp (306)
25.2. Euler's Number (308)
26. Trigonometry: Arcs (of a Circle) (310)
27. Trigonometry: Angles (313)
27.1. Angles, Informally Defined Again; Also, the Radian (315)
27.2. Names of Angles (318)
28. Trigonometry: Sine and Cosine (320)
28.1. The Values of Sine and Cosine at 0 and 1 (321)
28.2. Three Values of Sine and Cosine (323)
28.3. Confusing Notation (326)
28.4. Some Formulae Involving Sine and Cosine (327)
28.5. The Area of a Triangle; the Laws of Sines and Cosines (331)
28.6. The Unit-Circle Definitions (333)
28.7. Graphs of Sine and Cosine (343)
28.8. Graphs of Sine and Cosine: Some Observations (348)
29. Trigonometry: Tangent (351)
29.1. Some Formulae Involving Tangent (352)
29.2. Graph of Tangent (355)
29.3. Graph of Tangent: Some Observations (357)
30. Trigonometry: Summary and a Few New Things (358)
30.1. The Signs of Sine, Cosine, and Tangent (359)
30.2. Half-Angle Formulae (361)
31. Trigonometry: Three More Functions (363)
32. Trigonometry: Three Inverse Trigonometric Functions (366)
32.1. Confusing Notation (370)
32.2. Inverse Cancellation Laws, Revisited (371)
32.3. More Compositions (375)
32.4. Addition Formulae for Arcsine and Arccosine (378)
32.5. Addition Formulae for Arctangent (380)
32.6. Harmonic Addition (382)
32.7. Even More Compositions (optional) (384)
33. Elementary Functions (385)
34. Factorising Polynomials (391)
34.1. Long Division of Polynomials (391)
34.2. Factorising Polynomials (395)
34.3. Compare Coefficients (396)
34.4. The Quadratic Formula (399)
34.5. The Remainder Theorem (400)
34.6. The Factor Theorem (401)
34.7. The Intermediate Value Theorem (IVT) (405)
34.8. Factorising a Quartic Polynomial (408)
34.9. Two Warnings (410)
35. Solving Systems of Equations (411)
35.1. Solving Systems of Equations with the TI84 (420)
36. Partial Fractions Decomposition (423)
36.1. Non-Repeated Linear Factors (424)
36.2. Repeated Linear Factors (425)
36.3. Non-Repeated Quadratic Factors (426)
37. Solving Inequalities (427)
37.1. Multiplying an Inequality by an Unknown Constant (427)
37.2. ??² + ?? + ? > 0 (429)
37.3. (?? + ?) / (?? + ?) > 0 (436)
37.4. The Cardinal Sin of Dividing by Zero, Revisited (442)
37.5. Inequalities Involving the Absolute Value Function (443)
37.6. (??² + ?? + ?) / (??² + ?? + ?) > 0 (450)
37.7. Solving Inequalities by Graphical Methods (453)
38. Extraneous Solutions (456)
38.1. Squaring (456)
38.2. Multiplying by Zero (458)
38.3. Removing Logs (459)
39. O-Level Review: The Derivative (461)
39.1. The Gradient (461)
39.2. The Derivative as the Gradient (463)
39.3. The Derivative as Rate of Change (466)
39.4. Differentiability vs Continuity (469)
39.5. Rules of Differentiation (473)
39.6. The Chain Rule (476)
39.7. Increasing and Decreasing (479)
39.8. Stationary and Turning Points (480)
40. Conic Sections (490)
40.1. The Ellipse ?² + ?² = 1 (The Unit Circle) (491)
40.2. The Ellipse ?² / ?² + ?² / ?²  = 1 (492)
40.3. The Hyperbola ? = 1 / ? (494)
40.4. The Hyperbola ?² − ?² = 1 (495)
40.5. The Hyperbola ?² / ?² − ?² / ?² = 1 (496)
40.6. The Hyperbola ?² / ?² − ?² / ?² = 1 (497)
40.7. The Hyperbola ? = (?? + ?) / (?? + ?) (498)
40.8. The Hyperbola ? = (??² + ?? + ?) / (?? + ?) (504)
40.9. Eliminating the Parameter ? (509)
41. Simple Parametric Equations (512)
II. Sequences and Series (517)
42. Sequences (519)
42.1. Sequences Are Functions (521)
42.2. Notation (523)
42.3. Arithmetic Combinations of Sequences (524)
43. Series (525)
43.1. Convergent and Divergent Series (526)
44. Summation Notation ∑ (529)
44.1. Summation Notation for Infinite Series (531)
45. Arithmetic Sequences and Series (534)
45.1. Finite Arithmetic Series (535)
45.2. Infinite Arithmetic Series (536)
46. Geometric Sequences and Series (537)
46.1. Finite Geometric Sequences and Series (538)
46.2. Infinite Geometric Sequences and Series (540)
47. Rules of Summation Notation (542)
48. The Method of Differences (544)
III. Vectors (549)
49. Introduction to Vectors (551)
49.1. The Magnitude or Length of a Vector (553)
49.2. When Are Two Vectors Identical? (554)
49.3. A Vector and a Point Are Different Things (555)
49.4. Two More Ways to Denote a Vector (556)
49.5. Position Vectors (557)
49.6. The Zero Vector (557)
49.7. Displacement Vectors (558)
49.8. Sum and Difference of Points and Vectors (559)
49.9. Sum, Additive Inverse, and Difference of Vectors (561)
49.10. Scalar Multiplication of a Vector (564)
49.11. When Do Two Vectors Point in the Same Direction? (565)
49.12. When Are Two Vectors Parallel? (566)
49.13. Unit Vectors (567)
49.14. The Standard Basis Vectors (569)
49.15. Any Vector Is A Linear Combination of Two Other Vectors (570)
49.16. The Ratio Theorem (572)
50. Lines (574)
50.1. Direction Vector (574)
50.2. Cartesian to Vector Equations (578)
50.3. Pedantic Points to Test/Reinforce Your Understanding (583)
50.4. Vector to Cartesian Equations (584)
51. The Scalar Product (587)
51.1. A Vector's Scalar Product with Itself (589)
52. The Angle Between Two Vectors (590)
52.1. Pythagoras' Theorem and Triangle Inequality (597)
52.2. Direction Cosines (598)
53. The Angle Between Two Lines (601)
54. Vectors vs Scalars (607)
55. The Projection and Rejection Vectors (610)
56. Collinearity (614)
57. The Vector Product (616)
57.1. The Angle between Two Vectors Using the Vector Product (619)
57.2. The Length of the Rejection Vector (620)
58. The Foot of the Perpendicular From a Point to a Line (621)
58.1. The Distance Between a Point and a Line (623)
59. Three-Dimensional (3D) Space (628)
59.1. Graphs (in 3D) (630)
60. Vectors (in 3D) (632)
60.1. The Magnitude or Length of a Vector (634)
60.2. Sums and Differences of Points and Vectors (635)
60.3. Sum, Additive Inverse, and Difference of Vectors (638)
60.4. Scalar Multiplication and When Two Vectors Are Parallel (642)
60.5. Unit Vectors (644)
60.6. The Standard Basis Vectors (645)
60.7. The Ratio Theorem (646)
61. The Scalar Product (in 3D) (647)
61.1. The Angle between Two Vectors (649)
61.2. Direction Cosines (652)
62. The Projection and Rejection Vectors (in 3D) (653)
63. Lines (in 3D) (657)
63.1. Vector to Cartesian Equations (660)
63.2. Cartesian to Vector Equations (665)
63.3. Parallel and Perpendicular Lines (668)
63.4. Intersecting Lines (670)
63.5. Skew Lines (672)
63.6. The Angle Between Two Lines (674)
63.7. Collinearity (in 3D) (678)
64. The Vector Product (in 3D) (681)
64.1. The Right-Hand Rule (685)
64.2. The Length of the Vector Product (686)
64.3. The Length of the Rejection Vector (687)
65. The Distance Between a Point and a Line (in 3D) (688)
66. Planes: Introduction (693)
66.1. The Analogy Between a Plane, a Line, and a Point (697)
67. Planes: Formally Defined in Vector Form (699)
67.1. The Normal Vector (703)
68. Planes in Cartesian Form (709)
68.1. Finding Points on a Plane (711)
68.2. Finding Vectors on a Plane (712)
69. Planes in Parametric Form (714)
69.1. Planes in Parametric Form (717)
69.2. Parametric to Vector or Cartesian Form (721)
70. Four Ways to Uniquely Determine a Plane (723)
71. The Angle between a Line and a Plane (727)
71.1. When a Line and a Plane Are Parallel, Perp., or Intersect (729)
72. The Angle between Two Planes (733)
72.1. When Two Planes Are Parallel, Perp., or Intersect (734)
73. Point-Plane Foot of the Perpendicular and Distance (739)
73.1. Formula Method (742)
74. Coplanarity (746)
74.1. Coplanarity of Lines (749)
IV. Complex Numbers (752)
75. Complex Numbers: Introduction (754)
75.1. The Real and Imaginary Parts of Complex Numbers (757)
75.2. Complex Numbers in Ordered Pair Notation (758)
76. Some Arithmetic of Complex Numbers (759)
76.1. Addition and Subtraction (759)
76.2. Multiplication (760)
76.3. Conjugation (762)
76.4. Division (764)
77. Solving Polynomial Equations (765)
77.1. The Fundamental Theorem of Algebra (766)
77.2. The Complex Conjugate Root Theorem (768)
78. The Argand Diagram (770)
79. Complex Numbers in Polar Form (771)
79.1. The Argument: An Informal Introduction (772)
79.2. The Argument: Formally Defined (773)
79.3. Complex Numbers in Polar Form (775)
80. Complex Numbers in Exponential Form (776)
80.1. Complex Numbers in Exponential Form (777)
80.2. Some Useful Formulae for Sine and Cosine (778)
81. More Arithmetic of Complex Numbers (779)
81.1. The Reciprocal (783)
81.2. Division (784)
V. Calculus (787)
82. Limits (789)
82.1. Limits, Informally Defined (790)
82.2. Examples Where The Limit Does Not Exist (795)
82.3. Rules for Limits (804)
83. Continuity, Revisited (806)
83.1. Functions with a Single Discontinuity (809)
83.2. An Example of a Function That Is Discontinuous Everywhere (817)
83.3. Functions That Seem Discontinuous But Aren't (818)
83.4. Every Elementary Function Is Continuous (821)
83.5. Continuity Allows Us to ``Move'' Limits (822)
83.6. Every Elementary Function is Continuous: Proof (Optional) (825)
83.7. Continuity at Isolated Points (optional) (830)
84. The Derivative, Revisited (832)
84.1. Differentiable ⟺ Approximately Linear (838)
84.2. Continuity vs Differentiability (840)
84.3. The Derivative as a Function (841)
84.4. ``Most'' Elementary Functions Are Differentiable (845)
85. Differentiation Notation (847)
85.1. Using d /d ? as the Differentiation Operator (851)
85.2. Using d /d ? as Shorthand (854)
85.3. d ?/d ? Is Not a Fraction; Nonetheless ... (856)
86. Rules of Differentiation, Revisited (860)
86.1. Proving Some Basic Rules of Differentiation (optional) (861)
86.2. Proving the Product Rule (optional) (866)
86.3. Proving the Quotient Rule (optional) (868)
86.4. Proving the Chain Rule (optional) (870)
87. Some Techniques of Differentiation (871)
87.1. Implicit Differentiation (871)
87.2. The Inverse Function Theorem (IFT) (880)
87.3. Term-by-Term Differentiation (optional) (882)
88. Parametric Equations (884)
88.1. Parametric Differentiation (888)
89. The Second and Higher Derivatives (890)
89.1. Higher Derivatives (896)
89.2. Smooth (or ``Infinitely Differentiable'') Functions (900)
89.3. Confusing Notation (902)
90. The Increasing/Decreasing Test (906)
91. Determining the Nature of a Stationary Point (910)
91.1. The First Derivative Test for Extrema (FDTE) (914)
91.2. A Situation Where a Stationary Point Is Also a Strict Global Extremum (920)
91.3. Fermat's Theorem on Extrema (921)
91.4. The Second Derivative Test for Extrema (SDTE) (923)
92. Concavity (930)
92.1. The First Derivative Test for Concavity (FDTC) (934)
92.2. The Second Derivative Test for Concavity (SDTC) (936)
92.3. The Graphical Test for Concavity (GTC) (940)
92.4. A Linear Function Is One That's Both Concave and Convex (944)
93. Inflexion Points (945)
93.1. The First Derivative Test for Inflexion Points (FDTI) (947)
93.2. The Second Derivative Test for Inflexion Points (SDTI) (949)
93.3. The Tangent Line Test (TLT) (950)
93.4. Non-Stationary Points of Inflexion (optional) (952)
94. A Summary of Chapters 90, 91, 92, and 93 (953)
95. More Techniques of Differentiation (954)
95.1. Relating the Graph of ?′ to That of ? (954)
95.2. Equations of Tangents and Normals (955)
95.3. Connected Rates of Change Problems (957)
96. More Fun With Your TI84 (960)
96.1. Locating Maximum and Minimum Points Using Your TI84 (961)
96.2. Find the Derivative at a Point Using Your TI84 (963)
97. Power Series (967)
97.1. A Power Series Is Simply an ``Infinite Polynomial'' (967)
97.2. A Power Series Can Converge or Diverge (970)
97.3. A Power Series and Its Interval of Convergence (972)
97.4. (Some) Functions Can be Represented by a Power Series (974)
97.5. Term-by-Term Differentiation of a Power Series (975)
97.6. Chapter Summary (978)
98. The Maclaurin Series (979)
98.1. (Some) Functions Can Be Rep'd by Their Maclaurin Series (984)
98.2. The Five Standard Series and Their Intervals of Convergence (989)
98.3. Sine and Cosine, Formally Defined (991)
98.4. Maclaurin Polynomials as Approximations (993)
98.5. Creating New Series Using Substitution (996)
98.6. Creating New Series Using Multiplication (1000)
98.7. Repeated Differentiation to Find a Maclaurin Series (1006)
98.8. Repeated Implicit Differentiation to Find a Maclaurin Series (1007)
99. Antidifferentiation (1010)
99.1. Antiderivatives Are Not Unique ... (1011)
99.2. ... But An Antiderivative Is Unique Up to a COI (1011)
99.3. Shorthand: The Antidifferentiation Symbol ∫ (1013)
99.4. Rules of Antidifferentiation (1017)
99.5. Every Continuous Function Has an Antiderivative (1020)
99.6. Be Careful with This Common Practice (1021)
99.7. The ``Punctuation Mark'' dx (optional) (1022)
100. The Definite Integral (1025)
100.1. Notation for Integration (1028)
100.2. The Definite Integral, Informally Defined (1029)
100.3. An Important Warning (1031)
100.4. A Sketch of How We Can Find the Area under a Curve (1032)
100.5. If a Function Is Continuous, Then Its Definite Integral Exists (1036)
100.6. Rules of Integration (1037)
100.7. Term-by-Term Integration (optional) (1040)
101. The Fundamental Theorems of Calculus (1042)
101.1. Definite Integral Functions (1042)
101.2. The First Fundamental Theorem of Calculus (FTC1) (1045)
101.3. The Second Fundamental Theorem of Calculus (FTC2) (1049)
101.4. Why Integration and Antidifferentiation Use the Same Notation (1051)
101.5. Some New Notation and More Examples and Exercises (1053)
102. More Techniques of Antidifferentiation (1054)
102.1. Factorisation (1054)
102.2. Partial Fractions: Finding ∫1 / (??² + ?? + ?) d? where ?² − 4?? > 0 (1055)
102.3. Building a Divisor of the Denominator (1057)
102.4. More Rules of Antidifferentiation (1059)
102.5. Completing the Square: ∫1 / (??² + ?? + ?) d? where ?² − 4?? < 0 (1061)
102.6. ∫1 / √(??² + ?? + ?)  d? in the Special Case where ? < 0 (1064)
102.7. Using Trigonometric Identities (1067)
102.8. Integration by Parts (IBP) (1069)
103. The Substitution Rule (1073)
103.1. The Substitution Rule Is the Inverse of the Chain Rule (1079)
103.2. Skipping Steps (1080)
103.3. ∫?′exp?   d?= exp? + ? (1081)
103.4. ∫?′/?  d?= ln |?| + ? (1083)
103.5. ∫(?)ⁿ??′   d?= (?)ⁿ⁺¹/ (? + 1) + ? (1086)
103.6. Building a Derivative (1088)
103.7. Antidifferentiating the Inverse Trigonometric Functions (1089)
103.8. Integration with the Substitution Rule (1090)
103.9. More Challenging Applications of the Substitution Rule (1091)
104. Term-by-Term Antidifferentiation and Integration of a Power Series (1095)
104.1. Formal Results (optional) (1099)
105. More Definite Integrals (1100)
105.1. Area between a Curve and Lines Parallel to Axes (1101)
105.2. Area between a Curve and a Line (1102)
105.3. Area between Two Curves (1103)
105.4. Area below the ?-Axis (1104)
105.5. Area under a Curve Defined Parametrically (1105)
105.6. Volume of Rotation about the ?-axis (1107)
105.7. Volume of Rotation about the ?-axis (1113)
105.8. Finding Definite Integrals Using Your TI84 (1115)
106. Introduction to Differential Equations (1116)
106.1. d? / d? = ?(?) (1116)
106.2. d? / d? = ?(?) (1119)
106.3. d²? / d?² = ?(?) (1123)
106.4. Word Problems (1125)
106.5. Yet Another Way to Define Sine and Cosine (optional) (1127)
107. Revisiting ln, logb, and exp (optional) (1128)
107.1. Revisiting Logarithms (1131)
107.2. Revisiting the Exponential Function (1133)
VI. Probability and Statistics (1135)
108. How to Count: Four Principles (1137)
108.1. How to Count: The Addition Principle (1138)
108.2. How to Count: The Multiplication Principle (1141)
108.3. How to Count: The Inclusion-Exclusion Principle (1144)
108.4. How to Count: The Complements Principle (1146)
109. How to Count: Permutations (1147)
109.1. Permutations with Repeated Elements (1150)
109.2. Circular Permutations (1155)
109.3. Partial Permutations (1158)
109.4. Permutations with Restrictions (1159)
110. How to Count: Combinations (1161)
110.1. Pascal's Triangle (1164)
110.2. The Combination as Binomial Coefficient (1165)
110.3. The Number of Subsets of a Set is 2ⁿ (1167)
111. Probability: Introduction (1168)
111.1. Mathematical Modelling (1168)
111.2. The Experiment as a Model of Scenarios Involving Chance (1170)
111.3. The Kolmogorov Axioms (1175)
111.4. Implications of the Kolmogorov Axioms (1176)
112. Probability: Conditional Probability (1178)
112.1. The Conditional Probability Fallacy (CPF) (1180)
112.2. Two-Boys Problem (Fun, Optional) (1184)
113. Probability: Independence (1186)
113.1. Warning: Not Everything is Independent (1189)
113.2. Probability: Independence of Multiple Events (1191)
114. Fun Probability Puzzles (1192)
114.1. The Monty Hall Problem (1192)
114.2. The Birthday Problem (1195)
115. Random Variables: Introduction (1196)
115.1. A Random Variable vs. Its Observed Values (1197)
115.2. ? = ? Denotes the Event {? ∈ ?: ?(?) = ?} (1198)
115.3. The Probability Distribution of a Random Variable (1199)
115.4. Random Variables Are Simply Functions (1202)
116. Random Variables: Independence (1204)
117. Random Variables: Expectation (1207)
117.1. The Expected Value of a Constant R.V. is Constant (1209)
117.2. The Expectation Operator is Linear (1211)
118. Random Variables: Variance (1213)
118.1. The Variance of a Constant R.V. is 0 (1218)
118.2. Standard Deviation (1219)
118.3. The Variance Operator is Not Linear (1220)
118.4. The Definition of the Variance (Optional) (1222)
119. The Coin-Flips Problem (Fun, Optional) (1223)
120. The Bernoulli Trial and the Bernoulli Distribution (1224)
120.1. Mean and Variance of the Bernoulli Random Variable (1226)
121. The Binomial Distribution (1227)
121.1. Probability Distribution of the Binomial R.V. (1229)
121.2. The Mean and Variance of the Binomial Random Variable (1230)
122. The Continuous Uniform Distribution (1232)
122.1. The Continuous Uniform Distribution (1233)
122.2. The Cumulative Distribution Function (CDF) (1235)
122.3. Important Digression: P(? ≤ ?) = P(? < ?) (1236)
122.4. The Probability Density Function (PDF) (1237)
123. The Normal Distribution (1238)
123.1. The Normal Distribution, in General (1245)
123.2. Sum of Independent Normal Random Variables (1254)
123.3. The Central Limit Theorem and The Normal Approximation (1257)
124. The CLT is Amazing (Optional) (1259)
124.1. The Normal Distribution in Nature (1259)
124.2. Illustrating the Central Limit Theorem (CLT) (1263)
124.3. Why Are So Many Things Normally Distributed? (1269)
124.4. Don't Assume That Everything is Normal (1270)
125. Statistics: Introduction (Optional) (1276)
125.1. Probability vs. Statistics (1276)
125.2. Objectivists vs Subjectivists (1277)
126. Sampling (1279)
126.1. Population (1279)
126.2. Population Mean and Population Variance (1280)
126.3. Parameter (1281)
126.4. Distribution of a Population (1282)
126.5. A Random Sample (1283)
126.6. Sample Mean and Sample Variance (1285)
126.7. Sample Mean and Sample Variance are Unbiased Estimators (1291)
126.8. The Sample Mean is a Random Variable (1294)
126.9. The Distribution of the Sample Mean (1295)
126.10. Non-Random Samples (1296)
127. Null Hypothesis Significance Testing (NHST) (1297)
127.1. One-Tailed vs Two-Tailed Tests (1301)
127.2. The Abuse of NHST (Optional) (1304)
127.3. Common Misinterpretations of the Margin of Error (Optional) (1305)
127.4. Critical Region and Critical Value (1308)
127.5. Testing of a Pop. Mean (Small Sample, Normal Distribution, σ² Known) (1310)
127.6. Testing of a Pop. Mean (Large Sample, Any Distribution, σ² Known) (1312)
127.7. Testing of a Pop. Mean (Large Sample, Any Distribution, σ² Unknown) (1314)
127.8. Formulation of Hypotheses (1316)
128. Correlation and Linear Regression (1317)
128.1. Bivariate Data and Scatter Diagrams (1317)
128.2. Product Moment Correlation Coefficient (PMCC) (1319)
128.3. Correlation Does Not Imply Causation (Optional) (1324)
128.4. Linear Regression (1325)
128.5. Ordinary Least Squares (OLS) (1327)
128.6. TI84 to Calculate the PMCC and the OLS Estimates (1332)
128.7. Interpolation and Extrapolation (1334)
128.8. Transformations to Achieve Linearity (1342)
128.9. The Higher the PMCC, the Better the Model? (1346)
VII. Ten-Year Series (1348)
129. Past-Year Questions for Part I. Functions and Graphs (1352)
130. Past-Year Questions for Part II. Sequences and Series (1367)
131. Past-Year Questions for Part III. Vectors (1379)
132. Past-Year Questions for Part IV. Complex Numbers (1389)
133. Past-Year Questions for Part V. Calculus (1396)
134. Past-Year Questions for Part VI. Prob. and Stats. (1430)
135. All Past-Year Questions, Listed and Categorised (1475)
135.1. 2019 (9758) (1475)
135.2. 2018 (9758) (1476)
135.3. 2017 (9758) (1477)
135.4. 2016 (9740) (1479)
135.5. 2015 (9740) (1480)
135.6. 2014 (9740) (1481)
135.7. 2013 (9740) (1482)
135.8. 2012 (9740) (1483)
135.9. 2011 (9740) (1484)
135.10. 2010 (9740) (1485)
135.11. 2009 (9740) (1486)
135.12. 2008 (9740) (1487)
135.13. 2007 (9740) (1488)
135.14. 2008 (9233) (1489)
135.15. 2007 (9233) (1490)
135.16. 2006 (9233) (1491)
136. H1 Maths Questions (2016--19) (1492)
136.1. H1 Maths 2019 Questions (1492)
136.2. H1 Maths 2019 Answers (1497)
136.3. H1 Maths 2018 Questions (1500)
136.4. H1 Maths 2018 Answers (1505)
136.5. H1 Maths 2017 Questions (1508)
136.6. H1 Maths 2017 Answers (1513)
136.7. H1 Maths 2016 Questions (1516)
136.8. H1 Maths 2016 Answers (1520)
VIII. Appendices (1523)
137. Appendices for Part 0. A Few Basics (1524)
137.1. Division (1524)
137.2. Logic (1524)
137.3. Sets (1526)
138. Appendices for Part I. Functions and Graphs (1530)
138.1. Ordered Pairs and Ordered ?-tuples (1530)
138.2. The Cartesian Plane (1532)
138.3. Lines (1533)
138.4. Distance (1536)
138.5. Maximum and Minimum Points (1538)
138.6. Distance between a Line and a Point (1539)
138.7. Reflections (1541)
138.8. The Quadratic Equation (1543)
138.9. Functions (1547)
138.10. When a Function Is Increasing or Decreasing at a Point (1547)
138.11. One-to-One Functions (1549)
138.12. Inverse Functions (1551)
138.13. Standard Definitions of the Identity and Inclusion Functions (1554)
138.14. Left, Right, and ``Full'' Inverses (1555)
138.15. When Do ? and ?⁻¹ Intersect? (1558)
138.16. Transformations (1559)
138.17. Trigonometry (1563)
138.18. Factorising Polynomials (1575)
138.19. Conic Sections (1576)
138.20. Inequalities (1581)
139. Appendices for Part II. Sequences and Series (1582)
139.1. Convergence and Divergence (1583)
140. Appendices for Part III. Vectors (1588)
140.1. Some General Definitions (1588)
140.2. Some Basic Results (1590)
140.3. Scalar Product (1592)
140.4. Angles (1595)
140.5. The Relationship Between Two Lines (1596)
140.6. Lines in 3D Space (1597)
140.7. Projection Vectors (1599)
140.8. The Vector Product (1601)
140.9. Planes in General (1603)
140.10. Planes in Three-Dimensional Space (1609)
140.11. Four Ways to Uniquely Determine a Plane (1611)
140.12. The Relationship Between a Line and a Plane (1612)
140.13. The Relationship Between Two Planes (1613)
140.14. Distances (1615)
140.15. Point-Plane Distance: Calculus Method (1616)
140.16. The Relationship Between Two Lines in 3D Space (1620)
140.17. A Necessary and Sufficient Condition for Skew Lines (1621)
141. Appendices for Part IV. Complex Numbers (1622)
142. Appendices for Part V. Calculus (1628)
142.1. A Few Useful Results and Terms (1629)
142.2. Limits (1632)
142.3. Infinite Limits and Vertical Asymptotes (1634)
142.4. Limits at Infinity and Horizontal and Oblique Asymptotes (1635)
142.5. Rules for Limits (1636)
142.6. Continuity (1639)
142.7. The Derivative (1646)
142.8. ``Most'' Elementary Functions are Differentiable (1649)
142.9. Leibniz Got the Product Rule Wrong (1652)
142.10. Proving the Chain Rule (1653)
142.11. Fermat's Theorem on Extrema (1657)
142.12. Rolle's Theorem and the Mean Value Theorem (1658)
142.13. The Increasing/Decreasing Test (1660)
142.14. The First and Second Derivative Tests (1661)
142.15. Concavity (1666)
142.16. Inflexion Points (1672)
142.17. Sine and Cosine (1676)
142.18. Power Series and Maclaurin series (1677)
142.19. Antidifferentiation (1678)
142.20. The Definite Integral (1679)
142.21. Proving the First Fundamental Theorem of Calculus (1688)
142.22. ∫1 / (??² + ?? + ?) d? (1690)
142.23. ∫1 / √(??² + ?? + ?) d? (1692)
142.24. The Substitution Rule (1695)
142.25. Revisiting Logarithms and Exponentiation (1697)
143. Appendices for Part VI. Probability and Statistics (1701)
143.1. How to Count (1701)
143.2. Circular Permutations (1703)
143.3. Probability (1704)
143.4. Random Variables (1705)
143.5. The Normal Distribution (1709)
143.6. Sampling (1712)
143.7. Null Hypothesis Significance Testing (1714)
143.8. Calculating the Margin of Error (1715)
143.9. Correlation and Linear Regression (1717)
143.10. Deriving a Linear Model from the Barometric Formula (1719)
IX. Answers to Exercises (1720)
144. Part 0 Answers (A Few Basics) (1721)
144.1. Ch. 1 Answers (Just To Be Clear) (1721)
144.2. Ch. 2 Answers (PSLE Review: Division) (1721)
144.3. Ch. 3 Answers (Logic) (1722)
144.4. Ch. 4 Answers (Sets) (1727)
144.5. Ch. 5 Answers (O-Level Review) (1731)
145. Part I Answers (Functions and Graphs) (1734)
145.1. Ch. 6 Answers (Graphs) (1734)
145.2. Ch. 7 Answers (Lines) (1737)
145.3. Ch. 8 Answers (Distance) (1738)
145.4. Ch. 9 Answers (Circles) (1738)
145.5. Ch. 11 Answers (Maximum and Minimum Points) (1739)
145.6. Ch. 12 Answers (Reflection and Symmetry) (1742)
145.7. Ch. 13 Answers (Solutions and Solution Sets) (1742)
145.8. Ch. 14 Answers (O-Level Review: The Quadratic Equation) (1743)
145.9. Ch. 15 Answers (Functions) (1744)
145.10. Ch. 16 Answers (An Introduction to Continuity) (1748)
145.11. Ch. 17Answers (When A Function Is Increasing or Decreasing) (1748)
145.12. Ch. 18 Answers (Arithmetic Combinations of Functions) (1749)
145.13. Ch. 20 Answers (Composite Functions) (1750)
145.14. Ch. ?? Answers (one-to-one Functions) (1753)
145.15. Ch. 22 Answers (Inverse Functions) (1754)
145.16. Ch. 23 Answers (Asymptotes and Limit Notation) (1759)
145.17. Ch. 24 Answers (Transformations) (1760)
145.18. Ch. 25 Answers (ln, exp, and e) (1765)
145.19. Ch. 39 Answers (O-Level Review: The Derivative) (1766)
145.20. Ch. ?? Answers (O-Level Review: Trigonometry) (1770)
145.21. Ch. 34 Answers (Factorising Polynomials) (1775)
145.22. Ch. 35 Answers (Solving Systems of Equations) (1778)
145.23. Ch. 36 Answers (Partial Fractions Decomposition) (1781)
145.24. Ch. 37 Answers (Solving Inequalities) (1783)
145.25. Ch. 38 Answers (Extraneous Solutions) (1791)
145.26. Ch. 40 Answers (Conic Sections) (1792)
145.27. Ch. 41 Answers (Simple Parametric Equations) (1800)
146. Part II Answers (Sequences and Series) (1806)
146.1. Ch. 42 Answers (Sequences) (1806)
146.2. Ch. 43 Answers (Series) (1806)
146.3. Ch. 44 Answers (Summation Notation Σ) (1807)
146.4. Ch. 45 Answers (Arithmetic Sequences and Series) (1808)
146.5. Ch. 46 Answers (Geometric Sequences and Series) (1808)
146.6. Ch. 47 Answers (Rules of Summation Notation) (1809)
146.7. Ch. 48 Answers (Method of Differences) (1810)
147. Part III Answers (Vectors) (1813)
147.1. Ch. 49 Answers (Introduction to Vectors) (1813)
147.2. Ch. 50 Answers (Lines) (1816)
147.3. Ch. 51 Answers (The Scalar Product) (1817)
147.4. Ch. 52 Answers (The Angle Between Two Vectors) (1818)
147.5. Ch. 53 Answers (The Angle Between Two Lines) (1820)
147.6. Ch. 54 Answers (Vectors vs Scalars) (1820)
147.7. Ch. 55 Answers (Projection Vectors) (1820)
147.8. Ch. 56 Answers (Collinearity) (1821)
147.9. Ch. 57 Answers (The Vector Product) (1822)
147.10. Ch. 58 Answers (The Foot of the Perpendicular) (1823)
147.11. Ch. 59 Answers (Three-Dimensional Space) (1827)
147.12. Ch. 60 Answers (Vectors in 3D) (1827)
147.13. Ch. 61 Answers (The Scalar Product in 3D) (1829)
147.14. Ch. 62 Answers (The Proj. and Rej. Vectors in 3D) (1830)
147.15. Ch. 63 Answers (Lines in 3D) (1831)
147.16. Ch. 64 Answers (The Vector Product in 3D) (1834)
147.17. Ch. 65 Answers: The Distance Between a Point and a Line (1837)
147.18. Ch. 66 Answers (Planes: Introduction) (1840)
147.19. Ch. 67 Answers (Planes: Formally Defined in Vector Form) (1841)
147.20. Ch. 68 Answers (Planes in Cartesian Form) (1842)
147.21. Ch. 69 Answers (Planes in Parametric Form) (1843)
147.22. Ch. 71 Answers (The Angle Between a Line and a Plane) (1846)
147.23. Ch. 72 Answers (The Angle Between Two Planes) (1847)
147.24. Ch. 73 Answers (Point-Plane Foot and Distance) (1849)
147.25. Ch. 74 Answers (Coplanarity) (1852)
148. Part IV Answers (Complex Numbers) (1854)
148.1. Ch. 75 Answers (Complex Numbers: Introduction) (1854)
148.2. Ch. 76 Answers (Some Arithmetic of Complex Numbers) (1854)
148.3. Ch. 77 Answers (Solving Polynomial Equations) (1856)
148.4. Ch. 78 Answers (The Argand Diagram) (1857)
148.5. Ch. 79 Answers (Complex Numbers in Polar Form) (1858)
148.6. Ch. 80 Answers (Complex Numbers in Exponential Form) (1858)
148.7. Ch. 81 Answers (More Arithmetic of Complex Numbers) (1859)
149. Part V Answers (Calculus) (1862)
149.1. Ch. 82 Answers (Limits) (1862)
149.2. Ch. 83 Answers (Continuity, Revisited) (1863)
149.3. Ch. 84 Answers (The Derivative, Revisited) (1864)
149.4. Ch. 85 Answers (Differentiation Notation) (1867)
149.5. Ch. 85 Answers (Rules of Differentiation, Revisited) (1868)
149.6. Ch. 87 Answers (Some Techniques of Differentiation) (1870)
149.7. Ch. 89 Answers (The Second and Higher Derivatives) (1872)
149.8. Ch. 90 Answers (The Increasing/Decreasing Test) (1877)
149.9. Ch. 91 Answers (Determining the Nature of a Stationary Point) (1879)
149.10. Ch. 92 Answers (Concavity) (1883)
149.11. Ch. 93 Answers (Inflexion Points) (1884)
149.12. Ch. 94 Answers (A Summary of Chapters 90, 91, 92, and 93) (1885)
149.13. Ch. 95 Answers (More Techniques of Differentiation) (1886)
149.14. Ch. 96 Answers (More Fun with Your TI84) (1888)
149.15. Ch. 97 Answers (Power Series) (1889)
149.16. Ch. 98 Answers (Maclaurin Series) (1890)
149.17. Ch. 99 Answers (Antidifferentiation) (1899)
149.18. Ch. 100 Answers (Integration) (1901)
149.19. Ch. 101 Answers (The Fundamental Theorems of Calculus) (1903)
149.20. Ch. 102 Answers (More Techniques of Antidifferentiation) (1904)
149.21. Ch. 102 Answers (The Substitution Rule) (1910)
149.22. Ch. 105 Answers (More Definite Integrals) (1914)
149.23. Ch. 106 Answers (Differential Equations) (1918)
149.24. Ch. 107 Answers (Revisiting Logarithms) (1921)
150. Part VI Answers (Probability and Statistics) (1922)
150.1. Ch. 108 Answers (How to Count: Four Principles) (1922)
150.2. Ch. 109 Answers (How to Count: Permutations) (1925)
150.3. Ch. 110 Answers (How to Count: Combinations) (1927)
150.4. Ch. 111 Answers (Probability: Introduction) (1930)
150.5. Ch. 112 Answers (Conditional Probability) (1934)
150.6. Ch. 113 Answers (Probability: Independence) (1935)
150.7. Ch. 115 Answers (Random Variables: Introduction) (1936)
150.8. Ch. 116 Answers (Random Variables: Independence) (1940)
150.9. Ch. 117 Answers (Random Variables: Expectation) (1940)
150.10. Ch. 118 Answers (Random Variables: Variance) (1942)
150.11. Ch. 119 Answers (The Coin-Flips Problem) (1942)
150.12. Ch. 120 Answers (Bernoulli Trial and Distribution) (1942)
150.13. Ch. 121 Answers (Binomial Distribution) (1943)
150.14. Ch. 122 Answers (Continuous Uniform Distribution) (1944)
150.15. Ch. 123 Answers (Normal Distribution) (1945)
150.16. Ch. 126 Answers (Sampling) (1951)
150.17. Ch. 127 Answers (Null Hypothesis Significance Testing) (1954)
150.18. Ch. 128 Answers (Correlation and Linear Regression) (1958)
151. Part VII Answers (2006--19 A-Level Exams) (1962)
151.1. Ch. 129 Answers (Functions and Graphs) (1962)
151.2. Ch. 130 Answers (Sequences and Series) (1996)
151.3. Ch. 131 Answers (Vectors) (2016)
151.4. Ch. 132 Answers (Complex Numbers) (2030)
151.5. Ch. 133 Answers (Calculus) (2053)
151.6. Ch. 134 Answers (Probability and Statistics) (2110)
Back Matter (2141)
Index (2141)
Abbreviations Used in This Textbook (2141)
Singlish Used in This Textbook (2145)
Notation Used in the Main Text (2146)
Notation Used (Appendices) (2147)
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