Topics
Sets and Functions
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
- 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
Relations and Functions
- Fundamental Concepts of Ordered Pairs and Relations
- 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 Co-ordinates
Algebra
Trigonometric Functions
- Angles and Their Measurement in Higher Mathematics
- Trigonometric Ratios
- 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
- Algebra of Complex Numbers - Equality
- Algebraic Properties of Complex Numbers
- Need for Complex Numbers
- Square Root of a Complex Number
Mathematical Reasoning
Linear Inequalities
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
Statistics and Probability
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
- Concept of Slope (or, gradient)
- Various Forms of the Equation of a Line
- Equation of a Straight 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
- Introduction of Parabola
- Standard Equations of Parabola
- Latus Rectum
- Introduction of Ellipse
- Relationship Between Semi-major Axis, Semi-minor 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 Three-dimensional Geometry
Limits and Derivatives
- Intuitive Idea of Derivatives
- Introduction of Limits
- Concept of 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
- 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
- Concepts of Statistics
- Measures of Central Tendency for Different Data Types
- Basic Concept of Mode
- Standard Deviation - by Short Cut Method
- Quartiles and Range in Statistics
Probability
- Concept of Probability
- Occurrence of an Event
- Elementary Types of Events in Probability
- Axiomatic Approach to Probability
- Probability of 'Not', 'And' and 'Or' Events
- Definition: Angle
- Properties of Angle
Angle
An angle consists of two rays that originate from a single initial point. An angle is represented by the symbol ∠.
Arms of an Angle
The two rays forming the angle are called the arms or sides of the angle.
Vertex
The vertex of the angle is the common initial point of two rays.
Diagram of an Angle
Definition
Angle: An angle consists of two rays that originate from a single initial point. An angle is represented by the symbol ∠.
Arms of an Angle: The two rays forming the angle are called the arms or sides of the angle.
Vertex: The vertex of the angle is the common initial point of two rays.
Example:
Maharashtra State Board: Class 5
Properties of Angle
- An angle is formed by the rotation of a ray about a fixed point.
- The initial position of the ray is called the initial side.
- The final position after rotation is the terminal side.
- The fixed point is the vertex.
- Anticlockwise rotation gives a positive angle.
- Clockwise rotation gives a negative angle.
Example:
| Angle |  |
 |
 |
| Name of the angle | ∠PYR OR ∠RYP | ∠LMN OR ∠NML | ∠BOS OR ∠SOB |
| Vertex of the angle | Y | M | O |
| Arms of the angle | YP and YR | ML and MN | OB and OS |
Example
Name the angles in the given figure.
Angles of given polygon: ∠BAD, ∠ADC, ∠DCB, ∠CBA
Example
In the given figure, Identify five pairs of adjacent angles.
Five pairs of adjacent angles are (∠AOE, ∠EOC), (∠EOC, ∠COB), (∠AOC, ∠COB), (∠COB, ∠BOD), (∠EOB, ∠BOD).
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