Topics
Sets
 Sets and Their Representations
 Empty Set (Null or Void Set)
 Finite and Infinite Sets
 Equal Sets
 Subsets
 Power Set
 Universal Set
 Venn Diagrams
 Intrdouction of Operations on Sets
 Union of Sets
 Intersection of Sets
 Difference of Sets
 Complement of a Set
 Practical Problems on Union and Intersection of Two Sets
 Proper and Improper Subset
 Open and Close Intervals
 Disjoint Sets
 Element Count Set
Mathematical Reasoning
 Mathematically Acceptable Statements
 New Statements from Old
 Special Words Or Phrases
 Contrapositive and Converse
 Introduction of Validating Statements
 Validation by Contradiction
 Difference Between Contradiction, Converse and Contrapositive
 Consolidating the Understanding
Sets and Functions
Relations and Functions
 Cartesian Product of Sets
 Concept of Relation
 Concept of Functions
 Some Functions and Their Graphs
 Algebra of Real Functions
 Ordered Pairs
 Equality of Ordered Pairs
 Pictorial Diagrams
 Graph of Function
 Pictorial Representation of a Function
 Exponential Function
 Logarithmic Functions
 Brief Review of Cartesian System of Rectanglar Coordinates
Algebra
Trigonometric Functions
 Concept of Angle
 Introduction of Trigonometric Functions
 Signs of Trigonometric Functions
 Domain and Range of Trigonometric Functions
 Trigonometric Functions of Sum and Difference of Two Angles
 Trigonometric Equations
 Trigonometric Functions
 Truth of the Identity
 Negative Function Or Trigonometric Functions of Negative Angles
 90 Degree Plusminus X Function
 Conversion from One Measure to Another
 180 Degree Plusminus X Function
 2X Function
 3X Function
 Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications
 Graphs of Trigonometric Functions
 Transformation Formulae
 Values of Trigonometric Functions at Multiples and Submultiples of an Angle
 Sine and Cosine Formulae and Their Applications
Coordinate Geometry
Calculus
Complex Numbers and Quadratic Equations
 Concept of Complex Numbers
 Algebraic Operations of Complex Numbers
 The Modulus and the Conjugate of a Complex Number
 Argand Plane and Polar Representation
 Quadratic Equations
 Algebra of Complex Numbers  Equality
 Algebraic Properties of Complex Numbers
 Need for Complex Numbers
 Square Root of a Complex Number
Mathematical Reasoning
Linear Inequalities
Statistics and Probability
Permutations and Combinations
 Fundamental Principles of Counting
 Permutations
 Combination
 Introduction of Permutations and Combinations
 Permutation Formula to Rescue and Type of Permutation
 Smaller Set from Bigger Set
 Derivation of Formulae and Their Connections
 Simple Applications of Permutations and Combinations
 Factorial N (N!) Permutations and Combinations
Principle of Mathematical Induction
Binomial Theorem
 Introduction of Binomial Theorem
 Binomial Theorem for Positive Integral Indices
 General and Middle Terms
 Proof of Binomial Therom by Pattern
 Proof of Binomial Therom by Combination
 Rth Term from End
 Simple Applications of Binomial Theorem
Sequence and Series
Straight Lines
 Slope of a Line
 Various Forms of the Equation of a Line
 General Equation of a Line
 Distance of a Point from a Line
 Brief Recall of Two Dimensional Geometry from Earlier Classes
 Shifting of Origin
 Equation of Family of Lines Passing Through the Point of Intersection of Two Lines
Conic Sections
 Sections of a Cone
 Concept of Circle
 Introduction of Parabola
 Standard Equations of Parabola
 Latus Rectum
 Introduction of Ellipse
 Relationship Between Semimajor Axis, Semiminor Axis and the Distance of the Focus from the Centre of the Ellipse
 Special Cases of an Ellipse
 Eccentricity
 Standard Equations of an Ellipse
 Latus Rectum
 Introduction of Hyperbola
 Eccentricity
 Standard Equation of Hyperbola
 Latus Rectum
 Standard Equation of a Circle
Introduction to Threedimensional Geometry
Limits and Derivatives
 Intuitive Idea of Derivatives
 Introduction of Limits
 Introduction to Calculus
 Algebra of Limits
 Limits of Polynomials and Rational Functions
 Limits of Trigonometric Functions
 Introduction of Derivatives
 Algebra of Derivative of Functions
 Derivative of Polynomials and Trigonometric Functions
 Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically
 Limits of Logarithmic Functions
 Limits of Exponential Functions
 Derivative of Slope of Tangent of the Curve
 Theorem for Any Positive Integer n
 Graphical Interpretation of Derivative
 Derive Derivation of x^n
Statistics
 Measures of Dispersion
 Concept of Range
 Mean Deviation
 Introduction of Variance and Standard Deviation
 Standard Deviation
 Standard Deviation of a Discrete Frequency Distribution
 Standard Deviation of a Continuous Frequency Distribution
 Shortcut Method to Find Variance and Standard Deviation
 Introduction of Analysis of Frequency Distributions
 Comparison of Two Frequency Distributions with Same Mean
 Statistics Concept
 Central Tendency  Mean
 Central Tendency  Median
 Concept of Mode
 Measures of Dispersion  Quartile Deviation
 Standard Deviation  by Short Cut Method
Probability
 Random Experiments
 Introduction of Event
 Occurrence of an Event
 Types of Events
 Algebra of Events
 Exhaustive Events
 Mutually Exclusive Events
 Axiomatic Approach to Probability
 Probability of 'Not', 'And' and 'Or' Events
Definition
Mode: The mode is the value of the observation which occurs most frequently, i.e., an observation with the maximum frequency is called the mode.
Notes
Mode:

The mode is the value of the observation which occurs most frequently, i.e., an observation with the maximum frequency is called the mode.
 The mode of a set of observations is the observation that occurs most often.

It is possible that more than one value may have the same maximum frequency. In such situations, the data is said to be multimodal.
Mode of Large Data:

Putting the same observations together and counting them is not easy if the number of observations is large. In such cases, we tabulate the data.

Tabulation can begin by putting tally marks and finding the frequency.
Example
Find the mode of the given set of numbers: 1, 1, 2, 4, 3, 2, 1, 2, 2, 4
Arranging the numbers with same values together, we get 1, 1, 1, 2, 2, 2, 2, 3, 4, 4
Mode of this data is 2 because it occurs more frequently than other observations.
Example
The following are the margins of victory in the football matches of a league.
1, 3, 2, 5, 1, 4, 6, 2, 5, 2, 2, 2, 4, 1, 2, 3, 1, 1, 2, 3, 2, 6, 4, 3, 2, 1, 1, 4, 2, 1, 5, 3, 3, 2, 3, 2, 4, 2, 1, 2.
Find the mode of this data.
Let us put the data in a tabular form:
Margins of Victory  Tally Bars  Number of Matches 
1  `cancel()`   9 
2  `cancel()cancel()`  14 
3  `cancel()`   7 
4  `cancel()`  5 
5    3 
6    2 
Total  40 
Looking at the table, we can quickly say that 2 is the ‘mode’ since 2 has occurred the highest number of times. Thus, most of the matches have been won with a victory margin of 2 goals.