Concept of Series

Topics

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
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
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
Sets and Functions
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
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
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
Introduction to Three-dimensional Geometry
- Coordinate Axes and Coordinate planes
- Coordinates of a Point in Space
- Distance Between Two Points
- Section Formula
- Three Dimessional Space
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
Algebra
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 Co-ordinates
Sequence and Series
- Concept of Sequences
- Concept of Series
- Arithmetic Progression (A.P.)
- Geometric Progression (G. P.)
- Relationship Between A.M. and G.M.
- Sum to N Terms of Special Series
- Introduction of Sequence and Series
Linear Inequalities
- Inequalities - Introduction
- Algebraic Solutions of Linear Inequalities in One Variable and Their Graphical Representation
- Graphical Solution of Linear Inequalities in Two Variables
- Solution of System of Linear Inequalities in Two Variables
Coordinate Geometry
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
Conic Sections
- Sections of a Cone
- Concept of Circle
- 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
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
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
Mathematical Reasoning
Statistics and Probability
Principle of Mathematical Induction
- Motivation
- Principle of Mathematical Induction

Notes

Let `a_1, a_2, a_3,…,a_n`, be a given sequence. Then, the expression
`a_1 + a_2 + a_3 +,…+ a_n + ...`
is called the series associated with the given sequence.
Series are often represented in compact form, called sigma notation, using the Greek letter ∑(sigma) as means of indicating the summation involved.
Thus the series `a_1 + a_2 + a_3 +...+a_n` is abbreviated as `sum _(k=1)^n a_k.`

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Column I	Column II
(a) 1² + 2² + 3² + ...+ n²	(i) `((n(n + 1))/2)^2`
(b) 1³ + 2³ + 3³ + ... + n³	(ii) n(n + 1)
(c) 2 + 4 + 6 + ... + 2n	(iii) `(n(n + 1)(2n + 1))/6`
(d) 1 + 2 + 3 +...+ n	(iv) `(n(n + 1))/2`

Topics

Trigonometric Functions

Binomial Theorem

Statistics

Sets and Functions

Limits and Derivatives

Mathematical Reasoning

Straight Lines

Introduction to Three-dimensional Geometry

Probability

Algebra

Relations and Functions

Sequence and Series

Linear Inequalities

Coordinate Geometry

Sets

Conic Sections

Calculus

Complex Numbers and Quadratic Equations

Permutations and Combinations

Mathematical Reasoning

Statistics and Probability

Principle of Mathematical Induction

Notes

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