SYLLABUS for JEE (Main)

 Syllabus for Paper-1 (B.E./B.Tech.)- Mathematics, 

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1. UNIT 1: SETS, RELATIONS AND FUNCTIONS: Sets and their representation: Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Type of relations, equivalence relations, functions; one-one, into and onto functions, the composition of functions.        

Sure. Here is a definition of sets, relations, and functions with examples and formulas:

Sets

A set is a collection of distinct objects. Objects in a set are called elements or members of the set. Sets can be represented in a variety of ways, including:

• Roster form: This is the most common way to represent a set. In roster form, the elements of the set are listed between braces, separated by commas. For example, the set of all even numbers between 1 and 10 can be represented as follows:

{2, 4, 6, 8, 10}

• Set-builder form: This form is used to define sets by specifying a property that all of the elements of the set must satisfy. For example, the set of all even numbers between 1 and 10 can also be represented as follows:

{x ∈ ℕ | 2 ≤ x ≤ 10 and x is even}

• Venn diagrams: Venn diagrams are graphical representations of sets. In a Venn diagram, each set is represented by a circle. The intersection of two sets is represented by the overlap of their circles, and the union of two sets is represented by the area covered by both circles.

Union and Intersection of Sets

The union of two sets A and B is the collection of all elements that are in A or in B or in both A and B. The intersection of two sets A and B is the collection of all elements that are in both A and B.

Formally, the union and intersection of two sets A and B are defined as follows:

A ∪ B = {x | x ∈ A or x ∈ B}
A ∩ B = {x | x ∈ A and x ∈ B}

Complement of a Set

The complement of a set A relative to a universal set U is the collection of all elements of U that are not in A.

Formally, the complement of a set A is defined as follows:

A' = {x | x ∈ U and x ∉ A}

Algebraic Properties of Sets

The union, intersection, and complement of sets have a number of algebraic properties. Some of the most important of these properties are as follows:

• Commutativity: The union and intersection of sets are commutative, meaning that the order in which the sets are listed does not matter. For example, the following two sets are equal:

A ∪ B = B ∪ A
A ∩ B = B ∩ A

• Associativity: The union and intersection of sets are associative, meaning that the way in which the parentheses are grouped does not matter. For example, the following two sets are equal:

(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)

• Distributivity: The union of sets distributes over the intersection of sets, meaning that the following two sets are equal:

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Power Set

The power set of a set A is the collection of all subsets of A. The power set of a set A is denoted by P(A).

Formally, the power set of a set A is defined as follows:

P(A) = {X | X ⊆ A}

Relation

A relation R on a set A is a subset of the Cartesian product A × A. The Cartesian product of A × A is the set of all ordered pairs of elements from A.

In other words, a relation R on a set A is a collection of ordered pairs of elements from A.

For example, the following relation R on the set A = {1, 2, 3} defines the relationship of "is less than":

R = {(1, 2), (1, 3), (2, 3)}

Types of Relations

There are many different types of relations, but some of the most common include:

• Reflexive relations: A reflexive relation is a relation where every element of the set is related to itself. For example, the relation of equality is a reflexive relation.
• Symmetric relations: A symmetric relation is a relation where if two elements are related, then the second element is also related to the first element. For example, the relation of friendship is a symmetric relation.
• Transitive relations: A transitive relation is a relation where if two elements are related and the second element is related to a third element, then the first element is also related 
• UNIT 2: COMPLEX NUMBERS AND QUADRATIC EQUATIONS: Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a + ib and their representation in a plane, Argand diagram, algebra of complex number, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions Relations between roots and coefficient, nature of roots, the formation of quadratic equations with given roots. 

    Complex Numbers A complex number is a number of the form z=a+bi, where a and b are real numbers and i is the imaginary unit, defined as i 2 =−1. Complex numbers can be represented in a plane called the Argand plane. In the Argand plane, the real part of the complex number z is plotted on the horizontal axis, and the imaginary part of z is plotted on the vertical axis. The modulus of a complex number z=a+bi is denoted by ∣z∣ and is defined as follows: |z| = \sqrt{a^2 + b^2} The argument of a complex number z=a+bi is denoted by argz and is defined as the angle that the line connecting the origin to the point (a,b) in the Argand plane makes with the positive real axis. Algebra of Complex Numbers Complex numbers can be added, subtracted, multiplied, and divided in a similar way to real numbers. However, there are a few special rules that need to be kept in mind when performing these operations. For example, the product of two complex numbers z 1 ​ =a+bi and z 2 ​ =c+di is defined as follows: (a + bi)(c + di) = (ac - bd) + (ad + bc)i Quadratic Equations A quadratic equation is an equation of the form ax 2 +bx+c=0, where a, b, and c are real numbers and a  =0. Quadratic equations can be solved using the quadratic formula, which is given as follows: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} The discriminant of a quadratic equation is denoted by D and is defined as follows: D = b^2 - 4ac The nature of the roots of a quadratic equation depends on the value of the discriminant. If D>0, then the quadratic equation has two real roots. If D=0, then the quadratic equation has two equal real roots. If D<0, then the quadratic equation has two complex roots. Relations Between Roots and Coefficients The sum of the roots of a quadratic equation ax 2 +bx+c=0 is given by − a b ​ . The product of the roots of a quadratic equation ax 2 +bx+c=0 is given by a c ​ . Examples Here are some examples of complex numbers and quadratic equations: The complex number z=2+3i can be represented as the point (2,3) in the Argand plane. The modulus of the complex number z=2+3i is given by ∣z∣= 2 2 +3 2 ​ = 13 ​ . The argument of the complex number z=2+3i is given by argz=tan −1 ( 2 3 ​ )≈56.31 ∘ . The quadratic equation x 2 −4x+3=0 has two real roots, which are given by the quadratic formula: x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = 2 \pm \sqrt{5} The quadratic equation x 2 +2x+5=0 has two complex roots, which are given by the quadratic formula: x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = -1 \pm 2i I hope this helps!