#### Topics

##### Angle and Its Measurement

##### 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
- Cramer’s Rule
- Consistency of Three Equations in Two Variables
- Area of Triangle and Collinearity of Three Points
- Introduction to Matrices
- Types of Matrices
- Algebra of Matrices
- Properties of Matrix Multiplication
- Properties of Transpose of a Matrix

##### Straight Line

##### Circle

##### Conic Sections

##### Measures of Dispersion

##### Probability

##### Complex Numbers

##### Sequences and Series

##### Permutations and Combination

- Fundamental Principles of Counting
- Invariance Principle
- Factorial Notation
- Concept of Permutations
- Permutations When All Objects Are Distinct
- Permutations When Repetitions Are Allowed
- Permutations When Some Objects Are Identical
- Circular Permutations
- Properties of Permutations
- Concept of Combinations
- Properties of Combinations

##### Methods of Induction and Binomial Theorem

##### Sets and Relations

##### Functions

##### Limits

##### Continuity

##### Differentiation

#### 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.