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
- Nth Term of Geometric Progression (G.P.)
- General Term of a Geometric Progression (G.P.)
- Sum of First N Terms of a Geometric Progression (G.P.)
- Sum of infinite terms of a G.P.
- Geometric Mean (G.M.)
Notes
Let us consider the following sequences: 2,4,8,16,...,
we have `a_1 = 2 , a_2/a_1 = 2 , a_3/a_2 = 2, a_4/a_3 = 2` and so on.
In above sequence the constant ratio is 2. Such sequences are called geometric sequence or geometric progression abbreviated as G.P.
1) General term of a G .P:
Let us consider a G.P. with first non-zero term ‘a’ and common ratio ‘r’. The second term is obtained by multiplying a by r, thus `a_2` = ar. Similarly, third term is obtained by multiplying `a_2` by r. Thus, `a_3 = a_2r = ar^2`, and so on.
The nth term of a G.P. is given by `a_n =ar^(n-1)`.
The series `a + ar + ar^2 + ... +` `ar^(n–1)`or `a + ar + ar^2 + ...+` `ar^(n–1) +...` are called finite or infinite geometric series, respectively.
2) Sum to n terms of a G .P.:
The first term of a G.P. be a and the common ratio be r.
`s_n =(a(1-r^n))/1-r` or `s_n = (a(r^n -1))/r-1`
3) Geometric Mean (G .M.):
The geometric mean of two positive numbers a
and b is the number `sqrt (ab)` .
`G_1, G_2,…, G_n` be n numbers between positive numbers a and b such that a,`G_1,G_2,G_3,…,G_n`, b is a G.P.
`G_n =ar^n =a (b/a)^(n/(n + 1))`
