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
Methods of Induction and Binomial Theorem
Sets and Relations
Functions
Limits
Continuity
Differentiation
notes
Set is a collection of well defined objects. By well defined objects, we mean the definition should not vary it should be definite.
Example 1- Collection of vowels of English alphabet. The collection will be a, e, i, o, u. This is a well defined collection.
Example 2- Collection of good maths books available in market. Here, the collection will vary form person to person because of different personal choices and preferences.
There are two methods of representing a set :
(i) Roster or tabular form
(ii) Set-builder form.
Example- Collection of vowels of English alphabet.
A= {a, e, i, o, u} this is the roster form of a set.
Here, A is represented as a set, and the elements of set are enclosed in { }.
Set builder form, A= {x: x is a vowel in English alphabet}
x represents the elements in the enclosed brackets { }.
Note:
1) Order is immaterial. That the sequence in which the elements appear is not important, even if the sequence is changed the meaning remains same.
Take the same example- Collection of vowels of English alphabet.
A= {e, i, o, u, a}
2) Same elements are not repeated.
Example- B= set of all letters of the word SCHOOL
Roster form: B= {S, C, H, O, L}
Set-builder form: B= {x:x is letter used in word SCHOOL}
3) C= Set of all natural numbers.
Set-builder form C = {x :x ∈ N}
Roster form C= {1, 2, 3, 4, ...}
As natural numbers are infinite so we will represent them with three dots.
∈ means belongs to and ∉ means does not belongs to.
Example- A= {1, 2, 3, 4, 5, 6}
Here, 2 ∈ A, 9 ∉ A, 8 ∉ A, 5 ∈ A
We give below a few more examples of sets used particularly in mathematics, viz.
`"N"` : the set of all natural numbers
`"Z"` : the set of all integers
`"Q"` : the set of all rational numbers
`"R"` : the set of real numbers
`"Z"^+` : the set of positive integers
`"Q"^+` : the set of positive rational numbers, and
`"R"^+` : the set of positive real numbers.
description
1) Roster or Tabular method or List method
2) Set-Builder or Rule Method
3) Venn Diagram