JEE Main Mathematics (JEE Main) Syllabus: Check the Latest Syllabus

• Roster or Tabular method or List method
• Set-Builder or Rule Method

• De Morgan's Law
• Some Properties of Complement Sets

• Number of Elements in the Cartesian Product of Two Finite Sets
• Cartesian Product of set of the Reals with Itself

• Definition of Relation
• Domain
• Co-domain and Range of a Relation

• Empty Relation
• Universal Relation
• Trivial Relations
• Identity relation
• Symmetric relation
• Transitive relation
• Equivalence Relation
• Antisymmetric relation
• Inverse relation
• One-One Relation (Injective)
• Many-one relation
• Into relation
• Onto relation (Surjective)

• Function, Domain, Co-domain, Range
• Types of function
  1. One-one or One to one or Injective function
  2. Onto or Surjective function
• Representation of Function
• Graph of a function
• Value of funcation
• Some Basic Functions - Constant Function, Identity function, Power Functions, Polynomial Function, Radical Function, Rational Function, Exponential Function, Logarithmic Function, Trigonometric function

• Types of Function based on Elements:
  1) One One Function (or injective)
  2) Many One Function
  3) Onto Function (or surjective)
  4) One One and Onto Function (or bijective)
  5) Into Function
  6) Constant Function
• Types of Function based on Equation:
  1) Identity Function
  2) Linear Function
  3) Quadratic Function
  4) Cubic Function
  5) Polynomial Functions
• Types of Function based on the Range:
  1) Modulus Function
  2) Rational Function
  3) Signum Function
  4) Even and Odd Functions
  5) Periodic Functions
  6) Greatest Integer Function
  7) Inverse Function
  8) Composite Functions
• Types of Function based on the Domain:
  1) Algebraic Functions
  2) Trigonometric Functions
  3) Logarithmic Functions
• Explicit and Implicit Functions
• Value of a Function
• Equal Functions

• Sum, Difference,Product and Quotient of Function
• Addition of two real functions
• Subtraction of a real function from another
• Multiplication by a scalar
• Multiplication of two real functions
• Quotient of two real functions

• Imaginary number
• Complex Number

• Expressing complex number in a + ib form and their representation in a plane

• Equality of two Complex Numbers 
• Conjugate of a Complex Number 
• Properties of `barz`
• Addition of complex numbers - Properties of addition, Scalar Multiplication
• Subtraction of complex numbers - Properties of Subtraction
• Multiplication of complex numbers - Properties of Multiplication
• Powers of i in the complex number
• Division of complex number - Properties of Division
• The square roots of a negative real number
• Identities

• Properties of 1, w, w²

• Quadratic equations in real and complex number system and their solutions Relations between roots and co-efficient

• General form of a matrix
• Types of Matrices
• Equality of Matrices
• Algebraic Operations on Matrices
• Properties of Matrix Addition, Scalar Multiplication and Product of Matrices
• Operation of Transpose of a Matrix and its Properties
• Symmetric and Skew-symmetric Matrices

• Row Matrix
• Column Matrix
• Zero or Null matrix
• Square Matrix
• Diagonal Matrix
• Scalar Matrix
• Unit or Identity Matrix
• Upper Triangular Matrix
• Lower Triangular Matrix
• Triangular Matrix
• Symmetric Matrix
• Skew-Symmetric Matrix
• Determinant of a Matrix
• Singular Matrix
• Transpose of a Matrix

• Determinants of Matrices of different order
• Properties of Determinants
• Application of Factor Theorem to Determinants
• Product of Determinants
• Relation between a Determinant and its Cofactor Determinant
• Area of a Triangle
• Singular and non-singular Matrices

• Property 1 - The value of the determinant remains unchanged if its rows are turned into columns and columns are turned into rows.
• Property 2 -  If any two rows  (or columns)  of a determinant are interchanged then the value of the determinant changes only in sign.
• Property 3 - If any two rows ( or columns) of a  determinant are identical then the value of the determinant is zero.
• Property  4  -  If each element of a row (or column)  of a determinant is multiplied by a  constant k then the value of the new determinant is k times the value of the original determinant.
• Property  5  -  If each element of a row (or column) is expressed as the sum of two numbers then the determinant can be expressed as the sum of two determinants
• Property  6  -  If a constant multiple of all elements of any row  (or column)  is added to the corresponding elements of any other row  (or column  )  then the value of the new determinant so obtained is the same as that of the original determinant. 
• Property 7 -  (Triangle property) - If all the elements of a  determinant above or below the diagonal are zero then the value of the determinant is equal to the product of its diagonal elements.

•  Inverse of a nonsingular matrix by elementary transformation
•  Inverse of a square matrix by adjoint method

• Consistent System
• Inconsistent System
• Solution of a system of linear equations using the inverse of a matrix

• Write transpose of given matrix

• Define symmetric and skew symmetric matrix

• Concept
• Elementary Transformations and Equivalent matrices
• Echelon form and finding the rank of the matrix (upto the order of 3×4)

1. A system of equations with exactly one solution
2. A system of equations with infinitely many solutions
3. A system of equations that has no solution

•  Testing the consistency of non homogeneous linear equations (two and three variables) by rank method

• Tree Diagram 
• Addition Principle 
• Multiplication principle

• Permutation
• Permutation of repeated things
• Permutations when all the objects are not distinct
• Number of Permutations Under Certain Restricted Conditions
• Circular Permutations

• ⁿC_r , ⁿC_n =1, ⁿC₀ = 1, ⁿC_r = ⁿC_n–r, ⁿC_x = ⁿC_y, then x + y = n or x = y, ⁿ⁺¹C_r = ⁿC_r-1 + ⁿC_r
• When all things are different
• When all things are not different.
• Mixed problems on permutation and combinations.

• Motivating the Application of the Method by Looking at Natural Numbers as the Least Inductive Subset of Real Numbers

• Problems Based on Sum of Series
• Problems Based on Inequality and Divisibility

• History of Binomial Theorem

• Statement and Proof of the Binomial Theorem for Positive Integral Indices
• Proof of Binomial Therom by Induction
• Special Case in Binomial Therom
• Pascal's Triangle
• Binomial theorem for any positive integer n
• Some special cases-(In the expansion of (a + b)ⁿ)

• N^th 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.)

• Definition of Limit
• One-Sided Limit
• Left-hand Limit
• Right-hand Limit
• Existence of a limit of a function at a point x = a
• Algebra of limits:
  Let f(x) and g(x) be two functions such that
  `lim_(x→a) f(x) = l and lim_(x → a) g(x) = m, then`
  1. `lim_(x → a) [f(x) ± g(x)] = lim_(x → a) f(x) ± lim_(x → a) g(x) = l ± m`
  2. `lim_(x → a) [f(x) xx g(x)] = lim_(x→ a) f(x) xx lim_(x→ a) g(x) = l xx m`
  3. `lim_(x → a) [kf(x)] = k xx lim_(x→ a) f(x) = kl, "where" ‘k’ "is a constant"`
  4. `lim_(x → a) f(x)/g(x) = (lim_(x → a) f(x))/(lim_(x → a) g(x)) = l/m "where" m≠ 0`.

• Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations

Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)

• First and Second Derivative test
• Determine critical points of the function
• Find the point(s) of local maxima and local minima and corresponding local maximum and local minimum values
• Find the absolute maximum and absolute minimum value of a function

• Evaluation of Limits when X → ∞
• Evaluation of Limits of the form 1^∞

• ∫ tan x dx = log | sec x |  + C
• ∫ cot x dx = log | sin x | + C
• ∫ sec x dx = log | sec x + tan x | + C
• ∫ cosec x dx = log | cosec x – cot x | + C

• `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`
• Integral of the type ∫ e^x [ f(x) + f'(x)] dx = e^xf(x) + C
• Integrals of some more types

1. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
2. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
3. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`

1) `int (dx)/(x^2 - a^2) = 1/(2a) log |(x - a)/(x + a)| + C`

2) `int (dx)/(a^2 - x^2) = 1/(2a) log |(a + x)/(a - x)| + C`

3) `int (dx)/(x^2 - a^2) = 1/a  tan^(-1) (x/a) + C`

4) `int (dx)/sqrt (x^2 - a^2) = log |x + sqrt (x^2-a^2)| + C`

5) `int (dx)/sqrt (a^2 - x^2) = sin ^(-1) (x/a) +C`

6)  `int (dx)/sqrt (x^2 + a^2) = log |x + sqrt (x^2 + a^2)| + C`

7) To find the integral `int (dx)/(ax^2 + bx + c)`

8) To find the integral of the type `int (dx)/sqrt(ax^2 + bx + c)`

9) To find the integral of the type `int (px + q)/(ax^2 + bx + c) dx`

10) For the evaluation of the integral of the type `int (px + q)/sqrt(ax^2 + bx + c) dx`

Area function, First fundamental theorem of integral calculus and Second fundamental theorem of integral calculus

1. `int_a^a f(x) dx = 0`
2. `int_a^b f(x) dx = - int_b^a f(x) dx`
3. `int_a^b f(x) dx = int_a^b f(t) dt`
4. `int_a^b f(x) dx = int_a^c f(x) dx + int_c^b f(x) dx`
   where a < c < b, i.e., c ∈ [a, b]
5. `int_a^b f(x) dx = int_a^b f(a + b - x) dx`
6. `int_0^a f(x) dx = int_0^a f(a - x) dx`
7. `int_0^(2a) f(x) dx = int_0^a f(x) dx + int_0^a f(2a - x) dx`
8. `int_(-a)^a f(x) dx = 2. int_0^a f(x) dx`, if f(x) even function
   = 0,  if f(x) is odd function

• Area of the Region Bounded by a Curve & X-axis Between two Ordinates
• Area of the Region Bounded by a Curve & Y-axis Between two Abscissa 
• Circle-line, elipse-line, parabola-line

• Simple curves: lines, parabolas, polynomial functions

• Definition of ordinary differential equation
• Order and degree of a differential equation
• Formation of ordinary differential equation

• Solution of a Differential Equation

• Formation of Differential equations from Physical Situations
• Formation of Differential Equations from Geometrical Problems

• Solutions of linear differential equation of the type:

1. `dy/dx` + py = q, where p and q are functions of x or constants.
2. `dx/dy` + px = q, where p and q are functions of y or constants.

• Differential equations, order and degree.
• Solution of differential equations.
• Variable separable.
• Homogeneous equations
• Linear form `dy/dx` + Py = Q where P and Q are functions of x only. Similarly, for `dx/dy`.

• Section formula for internal division
• Midpoint formula
• Section formula for external division

• Equation of a locus

• Slope of a Line Or Gradient of a Line.
• Parallelism of Line
• Perpendicularity of Line in Term of Slope
• Collinearity of Points
• Slope of a line when coordinates of any two points on the line are given
• Conditions for parallelism and perpendicularity of lines in terms of their slopes
• Angle between two lines
• Collinearity of three points

• Introduction of Distance of a Point from a Line
• Distance between two parallel lines

• Inclination of a line
• Slope of a line
• Perpendicular Lines
• Angle between intersecting lines
• Different Forms of an equation of a straight line
• General form to other forms

• Points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle.

• Geometric description of conic section
• Degenerate Forms
•  Identifying the conics from the general equation of the conic 

• Coplanar
• Skew lines
• Distance between two skew lines
• Distance between parallel lines

3.3.1 Combined equation of the pair of straight lines

3.3.2 Pair of straight lines passing through the origin

3.3.3 Angle between pair of straight lines passing through the origin

3.3.4 The condition for general second degree equation to represent the pair of straight

lines

•  Equation of the bisectors of the angle between the lines
•  General form of Pair of Straight Lines

• Direction cosines of a line passing through two points.

• Coplanar
• Skew lines
• Distance between two skew lines
• Distance between parallel lines

• Point-slope Form
• Slope-Intercept form
• Two-points Form
• Double-Intercept form
• Normal Form

• Distance between skew lines
• Distance between parallel lines

• Equation of a plane when a normal to the plane and the distance of the plane from the origin are given
• Equation of a plane perpendicular to a vector and passing through a given point
• Intercept form of the equation of a plane
• Equation of a plane passing through three given non-collinear points
• Equation of a plane passing through a given point and parallel to two given non-parallel vectors
• Equation of a plane passing through two given distinct points and is parallel to a non-zero vector
• Condition for a line to lie in a plane
• Condition for coplanarity of two lines
• Equation of plane containing two non-parallel coplanar lines
• Angle between two planes
• Angle between a line and a plane
• Distance of a point from a plane
• Distance between two parallel planes
• Equation of line of intersection of two planes
• Equation of a plane passing through the line of intersection of two given planes

• Passing through a point and perpendicular to a vector
• Passing through a point and parallel to two vectors
• Passing through three non-collinear points
• In normal form
• Passing through the intersection of two planes

• Vector addition using components
• Components of a vector in two dimensions space
• Components of a vector in three-dimensional space

• Geometrical interpretation
• Application of dot and cross products in plane Trigonometry
• Application of dot and cross products in Geometry
• Application of dot and cross product in Physics

• Addition of Two Vectors
  - Parallelogram Law
  - Triangle Law of addition of two vectors
• Subtraction of two vectors
• Scalar multiplication of a vector

• Section formula for internal division
• Midpoint formula
• Section formula for external division

1. Variance and Standard Deviation for raw data
2. Variance and Standard Deviation for ungrouped frequency distribution
3. Variance and Standard Deviation for grouped frequency distribution

• Mean deviation for grouped data
• Mean deviation for ungrouped data

• Partition of a sample space
• Theorem of total probability

• Probability distribution of a random variable

• Independent Events

• Properties of Δ
• Sine formula: `a/sinA=b/sinB=c/sinC`
• Cosine formula:`cosA=(b^2+c^2-a^2)/(2bc)`, etc
• Area of triangle:Δ = `1/2`bc A etc
• Simple applications of the above

• Introduction of Inverse Trigonometric Functions

Inverse of Sin, Inverse of cosin, Inverse of tan, Inverse of cot, Inverse of Sec, Inverse of Cosec

• Problems involving Angle of Elevation
• Problems involving Angle of Depression
• Problems involving Angle of Elevation and Depression

• Domain and Range of Trignometric Functions and Their Graphs

• Validating the Statements Involving the Connecting Words
• statement with “and”
• Statements with “or”
• “Implies”
• “Implied by”
• Statements with “If then” 
• Statements with “if and only if”

• Idempotent law
• Associative law
• Commutative law
• Distributive law
• Identity law
• Complement law
• Involution law
• DeMorgan’s laws

• Representation of solution of linear inequality in one variable on the number line

• The Law of Sines or Sine Formula

• Law of Sines
• Law of Cosines
• Projection Formula
• Projection Formula
• Half-Angle formula

• Area of a triangle (Heron’s Formula )

• Polar co-ordinates
• Relation between the polar co-ordinates and the Cartesian co-ordinates
• Solving a Triangle
• The Sine rule
• The Projection rule
• Applications of the Sine rule, the Cosine rule and the Projection rule

• If all the sides and all the angles of a polygon are equal, it is called a regular polygon.
• Sum of interior angles of an 'n' sided polygon ( whether it is regular or not) = ( 2n - 4 )rt. angles and sum of its exterior angles = 4 right angles = 360°
• At each vertex of every polygon, Exterior angle + Interior angle = 180°.
• Each interior angle of a regular polygon = `[( 2n - 4 ) "rt. angles"]/[n] = [( 2n - 4 ) xx 90°]/n`
• Each exterior

• Problems involving Angle of Elevation
• Problems involving Angle of Depression
• Problems involving Angle of Elevation and Depression