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
(i) The statement is true for n = 1, i.e., P(1) is true, and
(ii) If the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1, i.e., truth of P(k) implies the truth of P (k + 1).
Then, P(n) is true for all natural numbers n.
Property (i) is simply a statement of fact. There may be situations when a statement is true for all n ≥ 4. In this case, step 1 will start from n = 4 and we shall verify the result for n = 4, i.e., P(4). Suppose there is a given statement P(n) involving the natural number n such that
Property (ii) is a conditional property. It does not assert that the given statement is true for n = k, but only that if it is true for n = k, then it is also true for n = k +1. So, to prove that the property holds , only prove that conditional proposition:
If the statement is true for n = k, then it is also true for n = k + 1.
This is sometimes referred to as the inductive step. The assumption that the given statement is true for n = k in this inductive step is called the inductive hypothesis.
For example, frequently in mathematics, a formula will be discovered that appears to fit a pattern like
`1 = 1^2 =1`
`4 = 2^2 = 1 + 3`
`9 = 3^2 = 1 + 3 + 5`
`16 = 4^2 = 1 + 3 + 5 + 7`, etc.
It is worth to be noted that the sum of the first two odd natural numbers is the square of second natural number, sum of the first three odd natural numbers is the square of third natural number and so on.Thus, from this pattern it appears that
`1 + 3 + 5 + 7 + ... + (2"n" – 1) = "n"^2`, i.e,
the sum of the first n odd natural numbers is the square of n.
Let us write
`"P"("n"): 1 + 3 + 5 + 7 + ... + (2"n" – 1) = "n"^2`. We wish to prove that P(n) is true for all n.
The first step in a proof that uses mathematical induction is to prove that P (1) is true. This step is called the basic step. Obviously
`1 = 1^2`, i.e., P(1) is true.
The next step is called the inductive step. Here, we suppose that P (k) is true for some positive integer k and we need to prove that P (k + 1) is true. Since P (k) is true, we have
`1 + 3 + 5 + 7 + ... + (2k – 1) = k^2` ... (1)
Consider
`1 + 3 + 5 + 7 + ... + (2k – 1) + {2(k +1) – 1}` ... (2)
=` k^2 + (2k + 1) = (k + 1)^2` [Using (1)]
Therefore, P (k + 1) is true and the inductive proof is now completed. Hence P(n) is true for all natural numbers n.
