Topics
Angle and Its Measurement
- Directed Angle
- Angles of Different Measurements
- Angles in Standard Position
- Measures of Angles
- Area of a Sector of a Circle
- Length of an Arc of a Circle
Trigonometry - 1
- Introduction of Trigonometry
- Trigonometric Functions with the Help of a Circle
- Signs of Trigonometric Functions in Different Quadrants
- Range of Cosθ and Sinθ
- Trigonometric Functions of Specific Angles
- Trigonometric Functions of Negative Angles
- Fundamental Identities
- Periodicity of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Graphs of Trigonometric Functions
- Polar Co-ordinate System
Trigonometry - 2
- Trigonometric Functions of Sum and Difference of Angles
- Trigonometric Functions of Allied Angels
- Trigonometric Functions of Multiple Angles
- Trigonometric Functions of Double Angles
- Trigonometric Functions of Triple Angle
- Factorization Formulae
- Formulae for Conversion of Sum Or Difference into Product
- Formulae for Conversion of Product in to Sum Or Difference
- Trigonometric Functions of Angles of a Triangle
Determinants and Matrices
- Definition and Expansion of Determinants
- Minors and Cofactors of Elements of Determinants
- Properties of Determinants
- Application of Determinants
- Determinant method
- Consistency of Three Equations in Two Variables
- Area of Triangle and Collinearity of Three Points
- Introduction of Matrices
- Types of Matrices
- Algebra of Matrices
- Properties of Matrix Multiplication
- Properties of Transpose of a Matrix
Straight Line
- Locus of a Points in a Co-ordinate Plane
- Straight Lines
- Equations of Line in Different Forms
- General Form of Equation of a Line
- Family of Lines
Circle
- Different Forms of Equation of a Circle
- General Equation of a Circle
- Parametric Form of a Circle
- Tangent
- Condition of tangency
- Tangents from a Point to the Circle
- Director circle
Conic Sections
- Double Cone
- Conic Sections
- Parabola
- Ellipse
- Hyperbola
Measures of Dispersion
- Meaning and Definition of Dispersion
- Measures of Dispersion
- Range of Data
- Variance
- Standard Deviation
- Change of Origin and Scale of Variance and Standard Deviation
- Standard Deviation for Combined Data
- Coefficient of Variation
Probability
- Basic Terminologies
- Event and Its Types
- Concept of Probability
- Addition Theorem for Two Events
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ Theorem
- Odds (Ratio of Two Complementary Probabilities)
Complex Numbers
- Introduction of Complex Number
- Concept of Complex Numbers
- Algebraic Operations of Complex Numbers
- Square Root of a Complex Number
- Fundamental Theorem of Algebra
- Argand Diagram Or Complex Plane
- De Moivres Theorem
- Cube Root of Unity
- Set of Points in Complex Plane
Sequences and Series
- Concept of Sequences
- Arithmetic Progression (A.P.)
- Geometric Progression (G. P.)
- Harmonic Progression (H. P.)
- Arithmetico Geometric Series
- Power Series
Permutations and Combination
- Fundamental Principles of Counting
- Invariance Principle
- Factorial Notation
- Permutations
- Permutations When All Objects Are Distinct
- Permutations When Repetitions Are Allowed
- Permutations When Some Objects Are Identical
- Circular Permutations
- Properties of Permutations
- Combination
- Properties of Combinations
Methods of Induction and Binomial Theorem
- Principle of Mathematical Induction
- Binomial Theorem for Positive Integral Index
- General Term in Expansion of (a + b)n
- Middle term(s) in the expansion of (a + b)n
- Binomial Theorem for Negative Index Or Fraction
- Binomial Coefficients
Sets and Relations
- Sets and Their Representations
- Types of Sets
- Operations on Sets
- Intervals
- Concept of Relation
Functions
- Concept of Functions
- Algebra of Functions
Limits
- Concept of Limits
- Factorization Method
- Rationalization Method
- Limits of Trigonometric Functions
- Substitution Method
- Limits of Exponential and Logarithmic Functions
- Limit at Infinity
Continuity
- Continuous and Discontinuous Functions
Differentiation
- Definition of Derivative and Differentiability
- Rules of Differentiation (Without Proof)
- Derivative of Algebraic Functions
- Derivatives of Trigonometric Functions
- Derivative of Logarithmic Functions
- Derivatives of Exponential Functions
- L' Hospital'S Theorem
Notes
A sequence `a_1, a_2, a_3,…, an,…` is called arithmetic sequence or arithmetic progression if `a_(n + 1) = a_n + d, n ∈ N`, where `a_1` is called the first term and the constant term d is called the common difference of the A.P.
The `n^(th)` term (general term) of the A.P. is` a^n = a + (n – 1) d`.
The sum to n term of A.P is `S_n= n/2[2a+(n-1)d]`
We can also write, `S_n = n/2[a+l]`
We can verify the following simple properties of an A.P. :
(1) If a constant is added to each term of an A.P., the resulting sequence is also an A.P.
(2) If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.
(4) If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.
(5) If each term of an A.P. is divided by a non-zero constant then the resulting sequence is also an A.P.
Arithmetic mean:
Given two numbers a and b. We can insert a number A between them so that a, A, b is an A.P. Such a number A is called the arithmetic mean (A.M.) of the numbers a and b. Note that, in this case, we have
A – a = b – A, i.e., A =`(a+b)/2`
We may also interpret the A.M. between two numbers a and b as their average `(a+b)/2.`
For example, the A.M. of two numbers 4 and 16 is 10. We have, thus constructed an A.P. 4, 10, 16 by inserting a number 10 between 4 and 16.
The Arithmetic mean is `d = (b - a)/(n + 1)`
