Mathematics — Learning Path
Part 1: FOUNDATIONS
1.1 Arithmetic & Number Sense
- 1 Number Systems
- 2 Operations and Properties
- 3 Fractions and Decimals
- 4 Percentages and Ratios
- 5 Exponents and Powers
- 6 Roots and Radicals
1.2 Mathematical Thinking
- 7 Sets - Introduction
- 8 Set Operations
- 9 Relations
- 10 Logic and Propositions
- 11 Mathematical Proof - Direct Proof
- 12 Proof by Contradiction
- 13 Proof by Contrapositive
- 14 Mathematical Induction
- 15 Number Theory Basics
- 16 Modular Arithmetic
- 17 Real Numbers - Properties
- 18 Intervals and Inequalities Preview
Part 2: ALGEBRA
2.1 Fundamentals of Algebra
- 19 Algebraic Expressions
- 20 Laws of Exponents - Algebra
- 21 Radicals and Rational Exponents
- 22 Polynomials - Introduction
- 23 Polynomial Operations
- 24 Polynomial Factorization
- 25 Algebraic Identities
- 26 Linear Equations in One Variable
- 27 Linear Equations in Two Variables
- 28 Systems of Linear Equations
- 29 Absolute Value - Equations
- 30 Absolute Value - Inequalities
2.2 Quadratic Equations & Polynomials
- 31 Quadratic Equations - Introduction
- 32 Quadratic Formula
- 33 Discriminant and Nature of Roots
- 34 Sum and Product of Roots
- 35 Quadratic Equations - Advanced
- 36 Common Roots
- 37 Quadratic Expression and Function
- 38 Range of Quadratic Expression
- 39 Location of Roots
- 40 Quadratic Inequalities
- 41 Solving Inequalities Using Location of Roots
- 42 Quadratic in Two Variables
- 43 Polynomial Equations
- 44 Remainder and Factor Theorems
- 45 Cubic Equations
- 46 Higher Degree Equations
- 47 Repeated Roots
- 48 Extraneous and Lost Roots
2.3 Complex Numbers
2.3 Complex Numbers
- 49 Complex Numbers - Introduction
- 50 Algebra of Complex Numbers
- 51 Division of Complex Numbers
- 52 Square Root of Complex Number
- 53 Modulus of Complex Number
- 54 Argument of Complex Number
- 55 Polar Form of Complex Numbers
- 56 De Moivre's Theorem
- 57 Roots of Unity
- 58 nth Roots of Complex Numbers
- 59 Geometry with Complex Numbers - Basics
- 60 Geometry with Complex Numbers - Lines
- 61 Geometry with Complex Numbers - Circles
- 62 Loci in Argand Plane
2.4 Sequences and Series
- 63 Sequences - Introduction
- 64 Arithmetic Progression
- 65 Sum of Arithmetic Progression
- 66 Arithmetic Mean
- 67 Geometric Progression
- 68 Sum of Geometric Progression
- 69 Geometric Mean
- 70 Harmonic Progression
- 71 Harmonic Mean
- 72 AM-GM-HM Inequality
- 73 Special Series
- 74 Sigma and Pi Notation
- 75 Arithmetico-Geometric Series
- 76 Method of Differences
- 77 Miscellaneous Series
2.5 Permutations and Combinations
- 78 Fundamental Principle of Counting
- 79 Factorial Notation
- 80 Permutations - Basics
- 81 Permutations - Special Cases
- 82 Permutations with Restrictions
- 83 Combinations - Basics
- 84 Properties of Combinations
- 85 Combinations with Restrictions
- 86 Division and Distribution
- 87 Multinomial Theorem
- 88 Inclusion-Exclusion Principle
- 89 Derangements
2.6 Binomial Theorem
- 90 Binomial Theorem for Positive Integer
- 91 Binomial Coefficients
- 92 General and Middle Terms
- 93 Greatest Term
- 94 Binomial Theorem - Applications
- 95 Special Expansions
- 96 Binomial with Complex Numbers
- 97 Binomial Theorem for Rational Index
2.7 Inequalities
Part 3: FUNCTIONS
3.1 Functions - Fundamentals
- 102 Functions - Definition and Notation
- 103 Ways to Define Functions
- 104 Types of Functions
- 105 Domain and Range
- 106 Graphs of Basic Functions
- 107 Special Functions - Part 1
- 108 Special Functions - Part 2
- 109 Even and Odd Functions
- 110 Periodic Functions
- 111 Bounded Functions
3.2 Function Operations and Transformations
- 112 Algebra of Functions
- 113 Composite Functions
- 114 Injective and Surjective Nature of Composites
- 115 Inverse Functions
- 116 Graph Transformations - Translations
- 117 Graph Transformations - Scaling
- 118 Graph Transformations - Reflections
- 119 Functions of Form max/min
3.3 Standard Functions
Part 4: COORDINATE GEOMETRY
4.1 Straight Lines
- 124 Coordinate Geometry - Basics
- 125 Area and Collinearity
- 126 Centres of Triangle
- 127 Locus
- 128 Transformation of Axes
- 129 Straight Line - Forms
- 130 Straight Line - General Equation
- 131 Angle and Conditions
- 132 Distance Formulas
- 133 Angle Bisectors
- 134 Family of Lines
- 135 Pair of Straight Lines
4.2 Circles
- 136 Circle - Standard Forms
- 137 Circle - Special Cases
- 138 Line and Circle
- 139 Tangent and Normal to Circle
- 140 Pair of Tangents and Chord of Contact
- 141 Family of Circles
4.3 Conics - Parabola
- 142 Parabola - Introduction
- 143 Parabola - Position and Parametric Form
- 144 Parabola - Tangent and Normal
- 145 Parabola - Advanced
4.4 Conics - Ellipse
- 146 Ellipse - Introduction
- 147 Ellipse - Auxiliary Circle and Eccentric Angle
- 148 Ellipse - Tangent and Normal
- 149 Ellipse - Advanced
4.5 Conics - Hyperbola
Part 5: TRIGONOMETRY
- 156 Trigonometric Ratios
- 157 Trigonometric Ratios of Any Angle
- 158 Trigonometric Functions and Graphs
- 159 Trigonometric Identities
- 160 Compound Angles
- 161 Multiple Angles
- 162 Transformation Formulas
- 163 Trigonometric Equations
- 164 Trigonometric Equations - Advanced
- 165 Inverse Trigonometric Functions
- 166 Inverse Trigonometric Functions - Properties
- 167 Euler's Formula
Part 6: VECTORS
Part 7: THREE-DIMENSIONAL GEOMETRY
- 178 3D Coordinates
- 179 Direction Cosines and Ratios
- 180 Straight Line in 3D - Equations
- 181 Straight Line in 3D - Angles and Distances
- 182 Straight Line in 3D - Advanced
- 183 Plane - Basic Equations
- 184 Plane - More Forms
- 185 Angle and Intersection
- 186 Distance Formulas for Planes
- 187 Bisector Planes
- 188 Regular Tetrahedron
- 189 Sphere
Part 8: MATRICES AND DETERMINANTS
- 190 Matrices - Introduction
- 191 Matrix Operations
- 192 Transpose of Matrix
- 193 Special Matrices
- 194 Determinants - Introduction
- 195 Properties of Determinants
- 196 Special Determinants
- 197 Determinants in Geometry
- 198 Product of Determinants
- 199 Differentiation of Determinants
- 200 Inverse of Matrix
- 201 Systems of Linear Equations
- 202 Consistency of Systems
- 203 Eigenvalues and Cayley-Hamilton
Part 9: CALCULUS - LIMITS AND CONTINUITY
- 204 Limits - Introduction
- 205 Algebra of Limits
- 206 Indeterminate Forms
- 207 Standard Limits
- 208 Exponential and Logarithmic Limits
- 209 Limits at Infinity
- 210 Sandwich Theorem
- 211 Limits Using Expansion
- 212 Special Limit Forms
- 213 L'Hôpital's Rule
- 214 Continuity - Introduction
- 215 Types of Discontinuity
- 216 Properties of Continuous Functions
- 217 Theorems on Continuous Functions
Part 10: CALCULUS - DIFFERENTIATION
- 218 Differentiation - Introduction
- 219 Differentiability
- 220 Reasons for Non-Differentiability
- 221 Derivatives of Basic Functions
- 222 Rules of Differentiation
- 223 Chain Rule
- 224 Derivatives of Trigonometric Functions
- 225 Derivatives of Inverse Trigonometric Functions
- 226 Derivatives of Exponential and Logarithmic Functions
- 227 Logarithmic Differentiation
- 228 Implicit Differentiation
- 229 Parametric Differentiation
- 230 Differentiation of Functions w.r.t. Functions
- 231 Higher Order Derivatives
- 232 Differentiation of Special Functions
- 233 Functional Equations and Differentiation
Part 11: CALCULUS - APPLICATIONS OF DERIVATIVES
- 234 Tangent and Normal
- 235 Tangent and Normal - Advanced
- 236 Rate of Change
- 237 Approximations
- 238 Rolle's Theorem
- 239 Mean Value Theorems
- 240 Monotonicity
- 241 Monotonicity - Applications
- 242 Maxima and Minima - First Derivative Test
- 243 Maxima and Minima - Second Derivative Test
- 244 Concavity and Points of Inflection
- 245 Maxima and Minima - Special Cases
- 246 Optimization Problems
- 247 Curve Sketching
Part 12: CALCULUS - INTEGRATION
- 248 Integration - Introduction
- 249 Basic Integration Formulas
- 250 Integration by Substitution
- 251 Special Integrals - Part 1
- 252 Special Integrals - Part 2
- 253 Special Integrals - Part 3
- 254 Integration by Parts
- 255 Integration by Parts - Special Forms
- 256 Partial Fractions - Review and Integration
- 257 Integration by Cancellation
- 258 Definite Integration - Introduction
- 259 Fundamental Theorem of Calculus
- 260 Properties of Definite Integrals
- 261 Properties - Advanced
- 262 Definite Integration Techniques
- 263 Leibniz Rule
- 264 Definite Integrals - Inequalities
- 265 Sum of Series Using Integration
- 266 Improper Integrals
- 267 Numerical Integration
Part 13: CALCULUS - AREA AND APPLICATIONS
Part 14: DIFFERENTIAL EQUATIONS
- 274 Differential Equations - Introduction
- 275 First Order - Variable Separable
- 276 First Order - Reducible to Separable
- 277 First Order - Linear DE
- 278 First Order - Reducible to Linear
- 279 First Order - Exact Equations
- 280 Second Order - Homogeneous
- 281 Second Order - Non-Homogeneous
- 282 Applications of DE
- 283 Orthogonal and Isogonal Trajectories
Part 15: PROBABILITY
- 284 Probability - Introduction
- 285 Classical Probability
- 286 Axiomatic Approach
- 287 Addition Theorem
- 288 Conditional Probability
- 289 Independent Events
- 290 Bayes' Theorem
- 291 Random Variables - Discrete
- 292 Expectation and Variance - Discrete
- 293 Binomial Distribution
- 294 Other Discrete Distributions
- 295 Continuous Random Variables
- 296 Normal Distribution
- 297 Conditional Probability - Advanced
Part 16: STATISTICS
- 298 Data Organization
- 299 Measures of Central Tendency
- 300 Measures of Dispersion
- 301 Quartiles and Percentiles
- 302 Correlation
- 303 Regression
- 304 Sampling
- 305 Introduction to Inference
Part appendix: ADDITIONAL ADVANCED TOPICS