JEE Advanced Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in JEE Advanced, not a summary of it.

1. 1

   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
2. 2

   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
3. 3

   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
4. 4

   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
5. 5

   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
6. 6

   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
7. 7

   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
8. 8

   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
9. 9

   Vectors

   5 topics

   • Addition of vectors
   • Scalar multiplication
   • Dot and cross products
   • Scalar and vector triple products
   • Geometrical interpretations

Mathematics flashcards for JEE Advanced

21 of 50 cards from the Mathematics deck — real questions with worked answers.

1. 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. $\{x : x \text{ is a natural number} \leq 3\}$.

2. 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. $\mathbb{N}$.

3. 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 $B \subseteq A$.

4. 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 $A$).

5. 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 $2^{n}$.

6. 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.

7. 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 $A \cap B = \varnothing$.

8. 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 $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.

9. State the idempotent and identity laws for sets (with universal set $U$).

   Idempotent: $A \cup A = A$, $A \cap A = A$. Identity: $A \cup \varnothing = A$, $A \cap U = A$, $A \cup U = U$, $A \cap \varnothing = \varnothing$.

10. 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 $A' = U - A = \{x \in U : x \notin A\}$.

11. 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 $B$. $A \,\triangle\, B = (A - B) \cup (B - A) = (A \cup B) - (A \cap B)$.

12. State De Morgan's laws for the complement of a union and an intersection.

    $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.

13. State De Morgan's laws expressed in terms of set difference.

    $A - (B \cup C) = (A - B) \cap (A - C)$ and $A - (B \cap C) = (A - B) \cup (A - C)$.

14. 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: $n(A \cup B \cup C) = n(A)+n(B)+n(C) - n(A\cap B) - n(B\cap C) - n(A\cap C) + n(A\cap B\cap C)$.

15. 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 $b = d$.

16. 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 $B$.

17. 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.

18. 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 $B$).

19. 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 $2^{pq}$.

20. 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 $(b,c) \in R \Rightarrow (a,c) \in R$.

21. 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 '$=$'.

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