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JEE Advanced Mathematics Syllabus

The Mathematics syllabus for JEE Advanced in full: 9 chapters, 82 topics and 46 sub-topics, every one listed below, plus 50 Mathematics flashcards with answers.

9Chapters

82Topics

46Sub-topics

50Flashcards

Mathematics syllabus: all 9 chapters

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

Sets, Relations and Functions
16 topics
- Sets and their representations
- Algebra of sets
- Intersection, complement, difference and symmetric difference of sets
- De-Morgan’s laws on union, intersection, difference
- Cartesian product of finite sets
- Ordered pair
- Relations, domain and codomain of relations
- Equivalence relation
- Function as a special case of relation
- Functions as mappings
- Domain, codomain, range of functions
- Invertible functions
- Even and odd functions
- Into, onto and one-to-one functions
- Special functions (polynomial, trigonometric, exponential, logarithmic, power, absolute value, greatest integer etc.)
- Sum, difference, product and composition of functions
Algebra
8 topics
- Algebra of complex numbers
  - Addition
  - Multiplication
  - Conjugation
  - Polar representation
  - Properties of modulus and principal argument
  - Triangle inequality
  - Cube roots of unity
  - Geometric interpretations
- Statement of fundamental theorem of algebra
- Quadratic equations with real coefficients
  - Relations between roots and coefficients
  - Formation of quadratic equations with given roots
  - Symmetric functions of roots
- Arithmetic and geometric progressions
  - Arithmetic and geometric means
  - Sums of finite arithmetic and geometric progressions
  - Infinite geometric series
  - Sum of the first n natural numbers
  - Sums of squares and cubes of the first n natural numbers
- Logarithms and their properties
- Permutations and combinations
- Binomial theorem for a positive integral index
- Properties of binomial coefficients
Matrices
11 topics
- Matrices as a rectangular array of real numbers
- Equality of matrices
- Addition, multiplication by a scalar and product of matrices
- Transpose of a matrix
- Elementary row and column transformations
- Determinant of a square matrix of order up to three
- Adjoint of a matrix
- Inverse of a square matrix of order up to three
- Properties of these matrix operations
- Diagonal, symmetric and skew-symmetric matrices and their properties
- Solutions of simultaneous linear equations in two or three variables
Probability and Statistics
11 topics
- Random experiment, sample space, different types of events (impossible, simple, compound)
- Addition and multiplication rules of probability
- Conditional probability
- Independence of events
- Total probability
- Bayes Theorem
- Computation of probability of events using permutations and combinations
- Measure of central tendency and dispersion
  - Mean
  - Median
  - Mode
  - Mean deviation
  - Standard deviation
  - Variance of grouped and ungrouped data
- Analysis of the frequency distribution with same mean but different variance
- Random variable
- Mean and variance of the random variable
Trigonometry
5 topics
- Trigonometric functions, their periodicity and graphs
- Addition and subtraction formulae
- Formulae involving multiple and sub-multiple angles
- General solution of trigonometric equations
- Inverse trigonometric functions (principal value only) and their elementary properties
Analytical Geometry
6 topics
- Two dimensions
  - Cartesian coordinates
  - Distance between two points
  - Section formulae
  - Shift of origin
- Equation of a straight line
  - Various forms
  - Angle between two lines
  - Distance of a point from a line
  - Lines through the point of intersection of two given lines
  - Equation of the bisector of the angle between two lines
  - Concurrency of lines
- Centroid, Orthocentre, Incentre and Circumcentre of a triangle
- Equation of a circle
  - Various forms
  - Equations of tangent, normal and chord
  - Parametric equations of a circle
  - Intersection of a circle with a straight line or a circle
  - Equation of a circle through the points of intersection of two circles and those of a circle and a straight line
- Equations of a parabola, ellipse and hyperbola
  - Standard form
  - Foci, directrices and eccentricity
  - Parametric equations
  - Equations of tangent and normal
- Locus problems
Differential Calculus
8 topics
- Limit of a function at a real number
- Continuity of a function
- Derivative of a function
  - Derivative of the sum, difference, product and quotient of two functions
  - Chain rule
  - Derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions
- Tangents and normals
- Increasing and decreasing functions
- Maximum and minimum values of a function
- Rolle’s theorem and Lagrange’s mean value theorem
  - Geometric interpretation of the two theorems
- Derivatives up to order two of implicit functions
  - Geometric interpretation of derivatives
Integral Calculus
12 topics
- Integration as the inverse process of differentiation
- Indefinite integrals of standard functions
- Definite integrals as the limit of sums
- Definite integral and their properties
- Fundamental theorem of integral calculus
- Integration by parts
- Integration by the methods of substitution and partial fractions
- Application of definite integrals to the determination of areas bounded by simple curves
- Formation of ordinary differential equations
- Solution of homogeneous differential equations of first order and first degree
- Separation of variables method
- Linear first order differential equations
Vectors
5 topics
- Addition of vectors
- Scalar multiplication
- Dot and cross products
- Scalar and vector triple products
- Geometrical interpretations

JEE Advanced Mathematics flashcards — 50 questions with answers

What is a set, and what are the two standard ways to represent it?
A set is a well-defined collection of distinct objects (elements). It can be represented in two ways: (1) Roster/tabular form, listing elements e.g. $\{1,2,3\}$, and (2) Set-builder form, stating a defining property e.g. $\{x : x \text{ is a natural number} \leq 3\}$.
Define the empty set, a singleton set, and a finite vs. infinite set.
The empty (null) set $\varnothing$ contains no elements. A singleton has exactly one element, e.g. $\{5\}$. A finite set has a countable (limited) number of elements; an infinite set has unlimited elements, e.g. $\mathbb{N}$.
When are two sets $A$ and $B$ said to be equal?
$A = B$ if and only if every element of $A$ is in $B$ and every element of $B$ is in $A$; equivalently $A \subseteq B$ and $B \subseteq A$.
Define a subset and a proper subset.
$A \subseteq B$ (subset) if every element of $A$ is also in $B$. $A$ is a proper subset, $A \subset B$, if $A \subseteq B$ but $A \neq B$ (i.e. $B$ has at least one element not in $A$).
If a set has $n$ elements, how many subsets and how many proper subsets does it have?
It has $2^{n}$ subsets in total and $2^{n}-1$ proper subsets. The number of elements of the power set is $2^{n}$.
Define the union of two sets $A \cup B$.
$A \cup B = \{x : x \in A \text{ or } x \in B\}$ — the set of all elements belonging to $A$, to $B$, or to both.
Define the intersection of two sets $A \cap B$ and state when they are disjoint.
$A \cap B = \{x : x \in A \text{ and } x \in B\}$. Sets $A$ and $B$ are disjoint when $A \cap B = \varnothing$.
State the commutative, associative and distributive laws of the algebra of sets.
Commutative: $A \cup B = B \cup A$, $A \cap B = B \cap A$. Associative: $A \cup (B \cup C) = (A \cup B) \cup C$, similarly for $\cap$. Distributive: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ and $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.
State the idempotent and identity laws for sets (with universal set $U$).
Idempotent: $A \cup A = A$, $A \cap A = A$. Identity: $A \cup \varnothing = A$, $A \cap U = A$, $A \cup U = U$, $A \cap \varnothing = \varnothing$.
Define the difference of sets $A - B$ and the complement $A'$.
$A - B = \{x : x \in A \text{ and } x \notin B\}$. The complement (relative to universal set $U$) is $A' = U - A = \{x \in U : x \notin A\}$.
Define the symmetric difference $A \,\triangle\, B$ and give two equivalent formulas.
$A \,\triangle\, B$ is the set of elements in exactly one of $A$ or $B$. $A \,\triangle\, B = (A - B) \cup (B - A) = (A \cup B) - (A \cap B)$.
State De Morgan's laws for the complement of a union and an intersection.
$(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.
State De Morgan's laws expressed in terms of set difference.
$A - (B \cup C) = (A - B) \cap (A - C)$ and $A - (B \cap C) = (A - B) \cup (A - C)$.
State the inclusion–exclusion formula for $n(A \cup B)$ and for three sets.
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$. For three sets: $n(A \cup B \cup C) = n(A)+n(B)+n(C) - n(A\cap B) - n(B\cap C) - n(A\cap C) + n(A\cap B\cap C)$.
Define an ordered pair and state the condition for equality of two ordered pairs.
An ordered pair $(a,b)$ is a pair with a fixed first element $a$ and second element $b$, where order matters. $(a,b) = (c,d)$ if and only if $a = c$ and $b = d$.
Define the Cartesian product $A \times B$ of two finite sets.
$A \times B = \{(a,b) : a \in A,\ b \in B\}$ — the set of all ordered pairs whose first element is from $A$ and second from $B$.
If $n(A) = p$ and $n(B) = q$, what is $n(A \times B)$, and is $A \times B = B \times A$ in general?
$n(A \times B) = p \cdot q$. In general $A \times B \neq B \times A$ (they are equal only when $A = B$ or one of them is empty), so the Cartesian product is not commutative.
Define a relation $R$ from set $A$ to set $B$, and its domain, codomain and range.
A relation $R$ from $A$ to $B$ is a subset of $A \times B$. Domain = set of all first components (subset of $A$); Codomain = the whole set $B$; Range = set of all second components actually appearing (subset of $B$).
If $n(A) = p$ and $n(B) = q$, how many distinct relations exist from $A$ to $B$?
Since a relation is any subset of $A \times B$ (which has $pq$ elements), the number of relations is $2^{pq}$.
Define what it means for a relation $R$ on a set to be reflexive, symmetric and transitive.
Reflexive: $(a,a) \in R$ for all $a$. Symmetric: $(a,b) \in R \Rightarrow (b,a) \in R$. Transitive: $(a,b) \in R$ and $(b,c) \in R \Rightarrow (a,c) \in R$.

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Define an equivalence relation and give one example.
A relation that is simultaneously reflexive, symmetric and transitive. Example: 'is congruent modulo $n$' on integers, i.e. $a \sim b \iff n \mid (a-b)$; another is equality '$=$'.
What is an equivalence class, and what is a key property of the equivalence classes of an equivalence relation?
For an equivalence relation $R$ on $A$, the class of $a$ is $[a] = \{x \in A : x R a\}$. The distinct equivalence classes partition $A$ into mutually disjoint non-empty subsets whose union is $A$.
How is a function defined as a special case of a relation?
A relation $f$ from $A$ to $B$ is a function if every element of $A$ is associated with exactly one element of $B$ — i.e. each $a \in A$ has a unique image $f(a) \in B$.
For a function $f: A \to B$, define domain, codomain and range.
Domain = $A$ (set of inputs). Codomain = $B$ (set in which outputs lie). Range = $f(A) = \{f(a) : a \in A\}$, the set of actual outputs, which satisfies range $\subseteq B$.
Define a one-to-one (injective) function.
$f: A \to B$ is one-to-one if distinct inputs have distinct images: $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$ (equivalently $x_1 \neq x_2 \Rightarrow f(x_1) \neq f(x_2)$).
Define an onto (surjective) function and an into function.
$f: A \to B$ is onto if range $= B$, i.e. every $b \in B$ has at least one preimage. It is into if range $\subset B$ (some element of $B$ is not an image).
Define a bijective function and state the condition for invertibility.
A function is bijective if it is both one-to-one and onto. A function $f$ is invertible if and only if it is bijective; then a unique inverse $f^{-1}: B \to A$ exists with $f^{-1}(f(x)) = x$.
For finite sets with $n(A) = m$ and $n(B) = n$, give the count of total functions and of one-to-one functions from $A$ to $B$.
Total functions: $n^{m}$. One-to-one functions (require $n \geq m$): $n(n-1)(n-2)\cdots(n-m+1) = \dfrac{n!}{(n-m)!}$.
Define even and odd functions and give one example of each.
Even: $f(-x) = f(x)$ for all $x$ (graph symmetric about the $y$-axis), e.g. $f(x) = x^{2}$ or $\cos x$. Odd: $f(-x) = -f(x)$ (symmetric about origin), e.g. $f(x) = x^{3}$ or $\sin x$.
How do you express any function as the sum of an even and an odd function?
$f(x) = \underbrace{\frac{f(x)+f(-x)}{2}}_{\text{even part}} + \underbrace{\frac{f(x)-f(-x)}{2}}_{\text{odd part}}$.
Define the greatest integer (floor) function and state its range.
$f(x) = \lfloor x \rfloor$ is the greatest integer not exceeding $x$, e.g. $\lfloor 2.7 \rfloor = 2$, $\lfloor -1.3 \rfloor = -2$. Domain $\mathbb{R}$, range $\mathbb{Z}$ (it is a step function).
Define the absolute value (modulus) function and give its domain and range.
$f(x) = |x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$. Domain $\mathbb{R}$, range $[0, \infty)$. It is an even function.
State the domain and range of the exponential function $f(x) = a^{x}$ ($a>0,\ a \neq 1$) and the logarithmic function $g(x) = \log_a x$.
Exponential $a^{x}$: domain $\mathbb{R}$, range $(0, \infty)$. Logarithm $\log_a x$: domain $(0, \infty)$, range $\mathbb{R}$. They are inverses of each other.
What is a polynomial function, and what is the domain of a power function $f(x)=x^{n}$?
A polynomial function is $f(x) = a_n x^{n} + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$ with constant coefficients and non-negative integer powers. A power function $x^{n}$ with $n \in \mathbb{N}$ has domain $\mathbb{R}$.
State the domains and ranges of the basic trigonometric functions $\sin x$ and $\tan x$.
$\sin x$: domain $\mathbb{R}$, range $[-1,1]$. $\tan x$: domain $\mathbb{R} \setminus \{(2n+1)\frac{\pi}{2}\}$, range $\mathbb{R}$.
Define the sum, difference and product of two functions $f$ and $g$, and state their domain.
$(f+g)(x)=f(x)+g(x)$, $(f-g)(x)=f(x)-g(x)$, $(fg)(x)=f(x)\,g(x)$. Each has domain $\text{dom}(f) \cap \text{dom}(g)$.
Define the quotient $\left(\frac{f}{g}\right)(x)$ of two functions and its domain.
$\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}$, with domain $\{x \in \text{dom}(f) \cap \text{dom}(g) : g(x) \neq 0\}$.
Define the composition of functions $(f \circ g)(x)$ and state whether composition is commutative.
$(f \circ g)(x) = f(g(x))$, applying $g$ first then $f$. Composition is generally not commutative: $f \circ g \neq g \circ f$ in general, though it is associative.
Define a complex number and its real and imaginary parts, with the value of $i$.
A complex number is $z = a + bi$ where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$ (so $i^{2} = -1$). $\operatorname{Re}(z) = a$ and $\operatorname{Im}(z) = b$.
State the condition for equality of two complex numbers $a+bi$ and $c+di$.
$a + bi = c + di$ if and only if $a = c$ and $b = d$ (real parts equal and imaginary parts equal).
How do you add and subtract two complex numbers $z_1 = a+bi$ and $z_2 = c+di$?
$z_1 + z_2 = (a+c) + (b+d)i$ and $z_1 - z_2 = (a-c) + (b-d)i$ — add/subtract real and imaginary parts separately.
How do you multiply two complex numbers $(a+bi)(c+di)$?
Expand using $i^{2} = -1$: $(a+bi)(c+di) = (ac - bd) + (ad + bc)i$.
List the algebraic properties (closure, commutativity, associativity, identities) of complex number addition.
Addition of complex numbers is closed, commutative ($z_1+z_2=z_2+z_1$), and associative. The additive identity is $0 = 0+0i$, and the additive inverse of $a+bi$ is $-a-bi$.
What are the powers of $i$ in cyclic form?
$i^{1}=i$, $i^{2}=-1$, $i^{3}=-i$, $i^{4}=1$, and the pattern repeats with period 4: $i^{4k}=1$, $i^{4k+1}=i$, $i^{4k+2}=-1$, $i^{4k+3}=-i$.
Define the conjugate $\bar{z}$ of a complex number and state $z\bar{z}$.
For $z = a+bi$, the conjugate is $\bar{z} = a - bi$. Their product $z\bar{z} = a^{2} + b^{2} = |z|^{2}$, a non-negative real number.
How do you divide complex numbers $\dfrac{a+bi}{c+di}$?
Multiply numerator and denominator by the conjugate of the denominator: $\dfrac{a+bi}{c+di} = \dfrac{(a+bi)(c-di)}{c^{2}+d^{2}} = \dfrac{(ac+bd)+(bc-ad)i}{c^{2}+d^{2}}$.
Compare an into function and an onto function.
Onto (surjective): range $= $ codomain — every codomain element is hit. Into: range $\subset$ codomain — at least one codomain element has no preimage. A function is either onto or into, never both.
Why is the constant function $f(x) = c$ both an example function and not one-to-one (for a domain with $>1$ element)?
It maps every input to the same output $c$, so it is a valid function (each input has one image), but $f(x_1)=f(x_2)=c$ for distinct inputs, violating injectivity; its range is the singleton $\{c\}$.
State the relationship between a relation being a function and the 'vertical line test' / pre-image condition.
A relation is a function iff no input has more than one output. Graphically (vertical line test), every vertical line meets the graph in at most one point; set-theoretically, no two ordered pairs share the same first component.
Distinguish the domain of a relation from the domain of the function it may define.
For a relation $R \subseteq A \times B$, the domain is only the set of first elements appearing in $R$ (a subset of $A$). For a function $f: A \to B$, the domain is the entire set $A$, since every element of $A$ must have an image.

JEE Advanced Mathematics FAQ

What is in the JEE Advanced Mathematics syllabus?

JEE Advanced Mathematics is split into 9 chapters: Sets, Relations and Functions, Algebra, Matrices, Probability and Statistics, Trigonometry, Analytical Geometry, Differential Calculus, Integral Calculus and Vectors. Together they contain 82 topics and 46 sub-topics.

How many chapters are in JEE Advanced Mathematics?

9 chapters, containing 82 topics and 46 sub-topics. The largest chapter is Sets, Relations and Functions, with 16 topics.

Are there flashcards for JEE Advanced 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.