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JEE Main Mathematics Syllabus

The Mathematics syllabus for JEE Main in full: 14 chapters, 78 topics and 48 sub-topics, every one listed below, plus 50 Mathematics flashcards with answers.

14Chapters

78Topics

48Sub-topics

50Flashcards

Mathematics syllabus: all 14 chapters

Expand any chapter to see its topics and sub-topics.

Sets, Relations and Functions
3 topics
- 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
  - Composition of functions
Complex numbers and quadratic equation
2 topics
- 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
- Quadratic equations in real and complex number system and their solutions
  - Relations between roots and co-efficient
  - Nature of roots
  - Formation of quadratic equations with given roots
Matrices and Determinants
3 topics
- Matrices
  - Algebra of matrices
  - Type of matrices
- Determinants
  - Evaluation of determinants
  - Area of triangles using determinants
  - Adjoint
  - Evaluation of inverse of a square matrix using determinants
- Test of consistency and solution of simultaneous linear equations in two or three variables using matrices
Permutation and Combination
3 topics
- The fundamental principle of counting
- Permutation as an arrangement and combination as section
- Meaning of P (n,r) and C (n,r)
  - Simple applications
Binomial Theorem
2 topics
- Binomial theorem for a positive integral index
  - General term and middle term
- Simple applications
Sequence and Series
3 topics
- Arithmetic and Geometric progressions
- Insertion of arithmetic, geometric means between two given numbers
- Relation between A.M and G.M
Limit, Continuity, and Differentiability
23 topics
- Real–valued functions
- Algebra of functions
- Polynomials
- Rational functions
- Trigonometric functions
- Logarithmic functions
- Exponential functions
- Inverse function
- Graphs of simple functions
- Limits
- Continuity
- Differentiability
- Differentiation of the sum, difference, product, and quotient of two functions
- Differentiation of trigonometric functions
- Differentiation of inverse trigonometric functions
- Differentiation of logarithmic functions
- Differentiation of exponential functions
- Differentiation of composite functions
- Differentiation of implicit functions
- Derivatives of order up to two
- Applications of derivatives: Rate of change of quantities
- Applications of derivatives: Monotonic-Increasing and decreasing functions
- Applications of derivatives: Maxima and minima of functions of one variable
Integral Calculus
9 topics
- Integral as an anti-derivative
- Fundamental integral involving algebraic, trigonometric, exponential, and logarithmic functions
- Integrations by substitution, by parts, and by partial functions
- Integration using trigonometric identities
- Evaluation of simple integrals
- The fundamental theorem of calculus
- Properties of definite integrals
- Evaluation of definite integrals
- Determining areas of the regions bounded by simple curves in standard form
Differential Equations
2 topics
- Ordinary Differential Equations
  - Order and Degree
- Solution of Differential Equations
  - Method of Separation of Variables
  - Solution of Homogeneous Equations
  - Solution of Linear Equations
Co-ordinate Geometry
9 topics
- Cartesian system of rectangular coordinates in a plane
- Distance formula
- Sections formula
- Locus and its equation
- Slope of a line
- Parallel and perpendicular lines
- Intercepts of a line on the co-ordinate axis
- Straight line
  - Various forms of equations of a line
  - Intersection of lines
  - Angles between two lines
  - Conditions for concurrence of three lines
  - Distance of a point from a line
  - Co-ordinate of the centroid
  - Orthocentre, and circumcentre of a triangle
- Circle, conic sections
  - Standard form of equations of a circle
  - General form of the equation of a circle
  - Radius and central of a circle
  - Equation of a circle when the endpoints of a diameter are given
  - Points of intersection of a line and a circle with the centre at the origin
  - Sections of conics
  - Equations of conic sections (parabola, ellipse, and hyperbola) in standard forms
Three Dimensional Geometry
8 topics
- Coordinates of a Point in Space
- Distance Between Two Points
- Section Formula
- Directions Ratios and Direction Cosines
- Angle Between Two Intersecting Lines
- Skew Lines
- Shortest Distance Between Skew Lines
- Equations of a Line
Vector Algebra
4 topics
- Vectors and Scalars
- Addition of Vectors
- Components of a Vector in Two Dimensions and Three-Dimensional Space
- Scalar and Vector Products
Statistics and Probability
4 topics
- Measures of Discretion
- Calculation of Mean, Median, Mode
  - Of Grouped Data
  - Of Ungrouped Data
- Calculation of Standard Deviation, Variance, Mean Deviation
  - For Grouped Data
  - For Ungrouped Data
- Probability
  - Probability of an Event
  - Addition and Multiplication Theorems of Probability
  - Bayes' Theorem
  - Probability Distribution of a Random Variate
Trigonometry
3 topics
- Trigonometrical identities
- Trigonometrical functions
- Inverse trigonometrical functions
  - Properties

JEE Main Mathematics flashcards — 50 questions with answers

What is a set, and what are the two standard ways of representing a set?
A set is a well-defined collection of distinct objects (elements). It can be represented in (1) Roster (tabular) form, listing all elements, e.g. $A=\{1,2,3\}$; and (2) Set-builder form, stating a defining property, e.g. $A=\{x : x \in \mathbb{N},\ x<4\}$.
Define the empty set, a singleton set, and a finite vs. infinite set.
The empty (null) set $\varnothing$ contains no elements. A singleton set contains exactly one element. A finite set has a countable, definite number of elements; an infinite set has unlimited elements (e.g. $\mathbb{N}$).
When are two sets said to be equal, and when is $A$ a subset of $B$?
Two sets are equal if they have exactly the same elements: $A=B \iff (A\subseteq B \text{ and } B\subseteq A)$. $A$ is a subset of $B$, $A\subseteq B$, if every element of $A$ is also in $B$.
What is the power set of a set $A$, and how many elements does it have if $|A|=n$?
The power set $P(A)$ is the set of all subsets of $A$ (including $\varnothing$ and $A$ itself). If $|A|=n$, then $|P(A)|=2^{n}$, and the number of proper subsets is $2^{n}-1$.
Define the union and intersection of two sets $A$ and $B$ using set-builder notation.
$A\cup B=\{x : x\in A \text{ or } x\in B\}$ (elements in at least one set). $A\cap B=\{x : x\in A \text{ and } x\in B\}$ (elements common to both).
Define the complement $A'$ of a set $A$ and the difference $A-B$.
With respect to a universal set $U$, the complement is $A'=U-A=\{x : x\in U,\ x\notin A\}$. The difference $A-B=\{x : x\in A \text{ and } x\notin B\}$.
State the commutative and associative laws for union and intersection of sets.
Commutative: $A\cup B=B\cup A$ and $A\cap B=B\cap A$. Associative: $(A\cup B)\cup C=A\cup(B\cup C)$ and $(A\cap B)\cap C=A\cap(B\cap C)$.
State the distributive laws of sets.
$A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$ and $A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$.
State De Morgan's laws for sets.
$(A\cup B)'=A'\cap B'$ and $(A\cap B)'=A'\cup B'$.
State the identity, complement, and idempotent laws for sets.
Identity: $A\cup\varnothing=A$, $A\cap U=A$. Complement: $A\cup A'=U$, $A\cap A'=\varnothing$. Idempotent: $A\cup A=A$, $A\cap A=A$.
State the formula for $|A\cup B|$ and for $|A\cup B\cup C|$ (inclusion-exclusion).
$|A\cup B|=|A|+|B|-|A\cap B|$. For three sets: $|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|B\cap C|-|A\cap C|+|A\cap B\cap C|$.
What is the Cartesian product $A\times B$, and how many elements does it contain?
$A\times B=\{(a,b) : a\in A,\ b\in B\}$, the set of all ordered pairs. If $|A|=m$ and $|B|=n$, then $|A\times B|=mn$.
Define a relation $R$ from set $A$ to set $B$. How many relations are possible from $A$ to $B$?
A relation $R$ from $A$ to $B$ is any subset of $A\times B$. If $|A|=m$ and $|B|=n$, the total number of relations from $A$ to $B$ is $2^{mn}$.
Define the domain and range of a relation $R$.
The domain is the set of all first elements of the ordered pairs in $R$; the range is the set of all second elements. For $R\subseteq A\times B$, the codomain is $B$, and the range is a subset of the codomain.
Define a reflexive relation and a symmetric relation on a set $A$.
$R$ is reflexive if $(a,a)\in R$ for all $a\in A$. $R$ is symmetric if $(a,b)\in R \implies (b,a)\in R$ for all $a,b\in A$.
Define a transitive relation and an antisymmetric relation on a set $A$.
$R$ is transitive if $(a,b)\in R$ and $(b,c)\in R \implies (a,c)\in R$. $R$ is antisymmetric if $(a,b)\in R$ and $(b,a)\in R \implies a=b$.
What is an equivalence relation?
A relation $R$ on a set $A$ is an equivalence relation if it is simultaneously reflexive, symmetric, and transitive.
What is an equivalence class, and what key property do the equivalence classes of a relation have?
For $a\in A$, the equivalence class is $[a]=\{x\in A : (x,a)\in R\}$. The distinct equivalence classes are mutually disjoint and their union is $A$; thus they form a partition of $A$.
Define the identity relation and the universal relation on a set $A$.
Identity relation: $I_A=\{(a,a) : a\in A\}$. Universal relation: $A\times A$ (relates every element to every element). Both, along with the empty relation, are special cases.
Define a function (mapping) $f:A\to B$.
A function $f:A\to B$ is a relation that assigns to each element of $A$ exactly one element of $B$. Every element of the domain $A$ must have one and only one image in $B$.

Show the other 30 Mathematics flashcards

Define a one-one (injective) function.
A function $f:A\to B$ is one-one (injective) if distinct elements have distinct images: $f(x_1)=f(x_2) \implies x_1=x_2$, equivalently $x_1\neq x_2 \implies f(x_1)\neq f(x_2)$.
Define an onto (surjective) function and an into function.
$f:A\to B$ is onto (surjective) if every element of $B$ is the image of at least one element of $A$, i.e. range $=B$. If the range is a proper subset of $B$, the function is called an into function.
Define a bijective function and state when $f:A\to B$ has an inverse.
A function is bijective if it is both one-one and onto. A function $f:A\to B$ is invertible (its inverse $f^{-1}$ exists) if and only if it is bijective.
If $|A|=m$ and $|B|=n$, how many functions and how many one-one functions exist from $A$ to $B$?
Total functions from $A$ to $B$: $n^{m}$. Number of one-one functions (requires $n\geq m$): $\dfrac{n!}{(n-m)!}={}^{n}P_{m}$.
Define the composition of functions $g\circ f$, including its domain condition.
For $f:A\to B$ and $g:B\to C$, the composition $(g\circ f):A\to C$ is defined by $(g\circ f)(x)=g(f(x))$. The range of $f$ must lie within the domain of $g$.
Is composition of functions commutative or associative? State the relationship with inverses.
Composition is associative: $(f\circ g)\circ h=f\circ(g\circ h)$, but generally not commutative: $f\circ g\neq g\circ f$. Also $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$, and $f\circ f^{-1}=I$.
How is a complex number defined as an ordered pair of real numbers, and how are equality, addition, and multiplication defined?
A complex number is an ordered pair $(a,b)$ of reals. Equality: $(a,b)=(c,d)\iff a=c,\ b=d$. Addition: $(a,b)+(c,d)=(a+c,\ b+d)$. Multiplication: $(a,b)\cdot(c,d)=(ac-bd,\ ad+bc)$.
Write a complex number in the form $a+ib$, and identify the real and imaginary parts. What is $i$?
$z=a+ib$ where $a,b\in\mathbb{R}$, $i=\sqrt{-1}$ with $i^{2}=-1$. Here $\operatorname{Re}(z)=a$ is the real part and $\operatorname{Im}(z)=b$ is the imaginary part.
What are the values of $i^{2}$, $i^{3}$, $i^{4}$, and the general value of $i^{n}$?
$i^{2}=-1$, $i^{3}=-i$, $i^{4}=1$. The powers cycle with period 4, so $i^{n}=i^{n\bmod 4}$ (e.g. $i^{4k}=1,\ i^{4k+1}=i,\ i^{4k+2}=-1,\ i^{4k+3}=-i$).
What is the Argand diagram, and how is a complex number $z=a+ib$ represented in it?
The Argand diagram (complex plane) represents $z=a+ib$ as the point $(a,b)$, with the real part on the horizontal (real) axis and the imaginary part on the vertical (imaginary) axis.
Define the conjugate $\bar{z}$ of $z=a+ib$ and state $z\bar{z}$.
The conjugate is $\bar{z}=a-ib$ (reflection of $z$ across the real axis). Then $z\bar{z}=a^{2}+b^{2}=|z|^{2}$, a non-negative real number.
State the properties of conjugates: $\overline{z_1+z_2}$, $\overline{z_1 z_2}$, and $\overline{\left(\frac{z_1}{z_2}\right)}$.
$\overline{z_1+z_2}=\bar{z_1}+\bar{z_2}$, $\overline{z_1 z_2}=\bar{z_1}\,\bar{z_2}$, and $\overline{\left(\dfrac{z_1}{z_2}\right)}=\dfrac{\bar{z_1}}{\bar{z_2}}$. Also $\overline{\bar{z}}=z$ and $z+\bar{z}=2\operatorname{Re}(z)$.
Define the modulus of a complex number $z=a+ib$.
The modulus is $|z|=\sqrt{a^{2}+b^{2}}$, the distance of the point $z$ from the origin in the Argand plane. Note $|z|=|\bar{z}|$ and $|z|^{2}=z\bar{z}$.
State the modulus properties: $|z_1 z_2|$, $\left|\frac{z_1}{z_2}\right|$, and the triangle inequality.
$|z_1 z_2|=|z_1||z_2|$ and $\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|}$. Triangle inequality: $|z_1+z_2|\leq|z_1|+|z_2|$ and $|z_1-z_2|\geq \big||z_1|-|z_2|\big|$.
Define the argument (amplitude) of a complex number $z=a+ib$.
The argument $\theta=\arg(z)$ is the angle the line joining $z$ to the origin makes with the positive real axis, given by $\tan\theta=\dfrac{b}{a}$. The principal value lies in $(-\pi,\pi]$.
Write the polar (trigonometric) form of a complex number and its Euler form.
$z=r(\cos\theta+i\sin\theta)$, where $r=|z|$ and $\theta=\arg(z)$. In Euler form, $z=re^{i\theta}$.
How do you compute the multiplicative inverse $z^{-1}$ of a non-zero complex number $z=a+ib$?
$z^{-1}=\dfrac{1}{z}=\dfrac{\bar{z}}{|z|^{2}}=\dfrac{a-ib}{a^{2}+b^{2}}=\dfrac{a}{a^{2}+b^{2}}-i\dfrac{b}{a^{2}+b^{2}}$.
How is division of complex numbers $\dfrac{z_1}{z_2}$ carried out?
Multiply numerator and denominator by the conjugate of the denominator: $\dfrac{z_1}{z_2}=\dfrac{z_1\,\bar{z_2}}{z_2\,\bar{z_2}}=\dfrac{z_1\,\bar{z_2}}{|z_2|^{2}}$, which removes $i$ from the denominator.
State the general form of a quadratic equation and its standard solution (quadratic formula).
$ax^{2}+bx+c=0$ with $a\neq 0$. Its roots are $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$.
What is the discriminant of a quadratic equation, and what does it determine?
The discriminant is $D=b^{2}-4ac$. Its sign determines the nature (real/complex, equal/distinct) of the roots of $ax^{2}+bx+c=0$.
For a quadratic with real coefficients, what is the nature of roots when $D>0$, $D=0$, and $D<0$?
If $D>0$: roots are real and distinct. If $D=0$: roots are real and equal ($x=-\dfrac{b}{2a}$). If $D<0$: roots are complex (non-real) and occur as conjugate pairs.
When are the roots of $ax^{2}+bx+c=0$ (with rational coefficients) rational, and when irrational?
If $D=b^{2}-4ac$ is a perfect square (and $\geq 0$), the roots are rational; if $D>0$ but not a perfect square, the roots are irrational and occur as conjugate surds $p\pm\sqrt{q}$.
State the relations between the roots $\alpha,\beta$ and the coefficients of $ax^{2}+bx+c=0$.
Sum of roots: $\alpha+\beta=-\dfrac{b}{a}$. Product of roots: $\alpha\beta=\dfrac{c}{a}$.
How do you form a quadratic equation given its roots $\alpha$ and $\beta$?
Use $x^{2}-(\text{sum})x+(\text{product})=0$, i.e. $x^{2}-(\alpha+\beta)x+\alpha\beta=0$.
Express $\alpha^{2}+\beta^{2}$ and $\alpha-\beta$ in terms of the sum and product of roots.
$\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2\alpha\beta$, and $(\alpha-\beta)^{2}=(\alpha+\beta)^{2}-4\alpha\beta$, so $|\alpha-\beta|=\dfrac{\sqrt{D}}{|a|}$.
For a complex (non-real) root of a quadratic with real coefficients, what can be said about the other root?
Complex roots occur in conjugate pairs: if $p+iq$ is a root, then $p-iq$ is also a root (provided the coefficients are real).
What is the condition for the roots of $ax^{2}+bx+c=0$ to be equal in magnitude but opposite in sign, and reciprocal of each other?
Roots are equal and opposite (sum $=0$) when $b=0$. Roots are reciprocals of each other (product $=1$) when $c=a$, i.e. $a=c$.
Solve $x^{2}+1=0$ over the complex number system.
$x^{2}=-1 \implies x=\pm\sqrt{-1}=\pm i$. The roots are $i$ and $-i$.
How do you compute the square root of a complex number, e.g. $\sqrt{a+ib}$ method?
Set $\sqrt{a+ib}=x+iy$, then square to get $x^{2}-y^{2}=a$ and $2xy=b$. Solving these (together with $x^{2}+y^{2}=\sqrt{a^{2}+b^{2}}$) gives the two square roots $\pm(x+iy)$.
What is the geometric meaning of $|z-z_0|=r$ and $|z-z_1|=|z-z_2|$ in the Argand plane?
$|z-z_0|=r$ represents a circle of radius $r$ centered at $z_0$. $|z-z_1|=|z-z_2|$ represents the perpendicular bisector of the segment joining points $z_1$ and $z_2$.

JEE Main Mathematics FAQ

What is in the JEE Main Mathematics syllabus?

JEE Main Mathematics is split into 14 chapters: Sets, Relations and Functions, Complex numbers and quadratic equation, Matrices and Determinants, Permutation and Combination, Binomial Theorem, Sequence and Series, Limit, Continuity, and Differentiability, Integral Calculus, Differential Equations, Co-ordinate Geometry, Three Dimensional Geometry, Vector Algebra, Statistics and Probability and Trigonometry. Together they contain 78 topics and 48 sub-topics.

How many chapters are in JEE Main Mathematics?

14 chapters, containing 78 topics and 48 sub-topics. The largest chapter is Limit, Continuity, and Differentiability, with 23 topics.

Are there flashcards for JEE Main Mathematics?

Yes — a 50-card Mathematics deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.