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JEE Main Paper 2 (B.Arch) Mathematics Syllabus
The Mathematics syllabus for JEE Main Paper 2 (B.Arch) in full: 5 chapters, 18 topics and 47 sub-topics, every one listed below, plus 50 Mathematics flashcards with answers.
Mathematics syllabus: all 5 chapters
Expand any chapter to see its topics and sub-topics.
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Algebra and Number Systems
5 topics- Sets, Relations and Functions
- Types of relations and equivalence relations
- One-one, onto, composite and inverse functions
- Domain, range and graphs of functions
- Complex Numbers and Quadratic Equations
- Algebra of complex numbers, modulus and argument
- Argand plane and polar representation
- Roots and relation between roots and coefficients
- Matrices and Determinants
- Algebra and types of matrices
- Determinants, properties and area of a triangle
- Adjoint, inverse and solution of linear equations
- Permutations, Combinations and Binomial Theorem
- Fundamental principle of counting, nPr and nCr
- Binomial theorem for positive integral index
- General and middle terms
- Sequences and Series
- Arithmetic and geometric progressions
- Sum of special series and infinite GP
- Sets, Relations and Functions
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Calculus
4 topics- Limits, Continuity and Differentiability
- Evaluation of limits and standard limits
- Continuity and differentiability
- Differentiation and Applications
- Derivatives of standard, implicit and parametric functions
- Tangents, normals and rate of change
- Maxima, minima and monotonicity
- Integral Calculus
- Integration by substitution, parts and partial fractions
- Definite integral and its properties
- Area under and between curves
- Differential Equations
- Order, degree and formation
- Variable separable and homogeneous equations
- Linear differential equations of first order
- Limits, Continuity and Differentiability
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Coordinate Geometry and 3D
3 topics- Straight Lines
- Slope and various forms of a line
- Distance, angle and intersection of lines
- Circles and Conic Sections
- Equation of a circle, tangents and normals
- Parabola, ellipse and hyperbola standard forms
- Eccentricity, foci and directrix
- Three Dimensional Geometry
- Direction cosines and distance in space
- Equation of line and plane
- Angle between lines and planes
- Straight Lines
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Trigonometry and Vectors
3 topics- Trigonometric Functions and Identities
- Ratios, identities and compound angles
- Trigonometric equations and general solutions
- Inverse Trigonometric Functions
- Domain, range and principal values
- Properties and simplification
- Vector Algebra
- Addition, components and scalar multiplication
- Dot and cross products
- Scalar and vector triple products
- Trigonometric Functions and Identities
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Statistics, Probability and Reasoning
3 topics- Statistics
- Mean, median and mode of grouped data
- Variance and standard deviation
- Probability
- Addition and multiplication theorems
- Conditional probability and Bayes' theorem
- Binomial distribution
- Mathematical Reasoning
- Statements and logical connectives
- Tautology, contradiction and validity
- Statistics
JEE Main Paper 2 (B.Arch) Mathematics flashcards — 50 questions with answers
What is the definition of a relation from set $A$ to set $B$?
A relation $R$ from $A$ to $B$ is any subset of the Cartesian product $A \times B$. That is, $R \subseteq A \times B$, and $(a,b) \in R$ means $a$ is related to $b$.
State the three properties that define an equivalence relation on a set $A$.
A relation $R$ on $A$ is an equivalence relation if it is: (1) Reflexive: $(a,a) \in R$ for all $a$; (2) Symmetric: $(a,b) \in R \Rightarrow (b,a) \in R$; (3) Transitive: $(a,b) \in R$ and $(b,c) \in R \Rightarrow (a,c) \in R$.
If set $A$ has $m$ elements and set $B$ has $n$ elements, how many relations exist from $A$ to $B$?
There are $2^{mn}$ relations, since $A \times B$ has $mn$ ordered pairs and each relation is a subset of $A \times B$.
Define a reflexive, a symmetric, and a transitive relation with a brief example each.
Reflexive: every element relates to itself, e.g. $'='$ on $\mathbb{R}$. Symmetric: $a R b \Rightarrow b R a$, e.g. 'is parallel to'. Transitive: $a R b, b R c \Rightarrow a R c$, e.g. $'<'$ on $\mathbb{R}$.
What is an equivalence class of an element $a$ under an equivalence relation $R$?
The equivalence class $[a] = \{x \in A : (x,a) \in R\}$, the set of all elements related to $a$. Equivalence classes partition the set $A$ into disjoint subsets.
Define a one-one (injective) function.
A function $f: A \to B$ is one-one if distinct inputs give distinct outputs: $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$ for all $x_1, x_2 \in A$.
Define an onto (surjective) function.
A function $f: A \to B$ is onto if every element of $B$ is the image of at least one element of $A$, i.e. range of $f$ = $B$ (codomain).
What condition must a function satisfy to have an inverse, and what is the inverse?
A function must be bijective (one-one and onto) to be invertible. Its inverse $f^{-1}: B \to A$ satisfies $f^{-1}(y) = x \iff f(x) = y$, so $f^{-1} \circ f = I_A$ and $f \circ f^{-1} = I_B$.
Define the composite of two functions $f: A \to B$ and $g: B \to C$.
The composite $g \circ f: A \to C$ is defined by $(g \circ f)(x) = g(f(x))$. Composition is associative but generally not commutative: $g \circ f \neq f \circ g$.
How many one-one functions can be defined from a set with $m$ elements to a set with $n$ elements ($m \leq n$)?
The number of one-one functions is $\frac{n!}{(n-m)!} = {}^{n}P_{m}$.
What is the domain and range of $f(x) = \sqrt{x}$?
Domain: $[0, \infty)$ (since the radicand must be non-negative). Range: $[0, \infty)$ (the principal square root is non-negative).
What is the domain and range of $f(x) = \frac{1}{x}$?
Domain: $\mathbb{R} \setminus \{0\}$ (all reals except $0$). Range: $\mathbb{R} \setminus \{0\}$ (all reals except $0$).
What are the domain and range of the modulus function $f(x) = |x|$?
Domain: $\mathbb{R}$ (all real numbers). Range: $[0, \infty)$ (non-negative reals). Its graph is a V-shape with vertex at the origin.
Define the imaginary unit $i$ and give the cyclic values of its powers.
$i = \sqrt{-1}$, so $i^{2} = -1$. The powers cycle with period 4: $i^{1} = i$, $i^{2} = -1$, $i^{3} = -i$, $i^{4} = 1$.
For a complex number $z = a + ib$, define its modulus and conjugate.
Modulus: $|z| = \sqrt{a^{2} + b^{2}}$. Conjugate: $\bar{z} = a - ib$. They satisfy $z \bar{z} = |z|^{2} = a^{2} + b^{2}$.
How do you find the multiplicative inverse of a complex number $z = a + ib$?
$z^{-1} = \frac{\bar{z}}{|z|^{2}} = \frac{a - ib}{a^{2} + b^{2}}$, valid for $z \neq 0$.
State the key properties of the modulus of complex numbers $z_1$ and $z_2$.
$|z_1 z_2| = |z_1||z_2|$, $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$, and the triangle inequality $|z_1 + z_2| \leq |z_1| + |z_2|$.
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, where $\tan\theta = \frac{b}{a}$. The principal argument lies in $(-\pi, \pi]$.
What is the polar (trigonometric) form of a complex number?
$z = r(\cos\theta + i\sin\theta)$, where $r = |z| = \sqrt{a^{2}+b^{2}}$ is the modulus and $\theta = \arg(z)$ is the argument. Often abbreviated as $r\,\text{cis}\,\theta$.
What is the Argand plane and how is a complex number represented on it?
The Argand plane is the coordinate plane used to represent complex numbers, with the horizontal axis as the real axis and the vertical axis as the imaginary axis. The number $z = a + ib$ is plotted as the point $(a, b)$.
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State the quadratic formula for the roots of $ax^{2} + bx + c = 0$.
$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$, where $a \neq 0$. The expression $b^{2} - 4ac$ is the discriminant.
How does the discriminant $D = b^{2} - 4ac$ determine the nature of roots of a quadratic equation?
If $D > 0$: two distinct real roots. If $D = 0$: two equal (repeated) real roots. If $D < 0$: two complex conjugate roots (no real roots).
State the relation between the roots $\alpha, \beta$ and coefficients of $ax^{2}+bx+c=0$.
Sum of roots: $\alpha + \beta = -\frac{b}{a}$. Product of roots: $\alpha\beta = \frac{c}{a}$.
How do you form a quadratic equation given its roots $\alpha$ and $\beta$?
$x^{2} - (\alpha + \beta)x + \alpha\beta = 0$, i.e. $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$.
Define the order of a matrix and what it means for a matrix to be square.
A matrix with $m$ rows and $n$ columns has order $m \times n$. It is a square matrix when $m = n$ (equal number of rows and columns).
Define a diagonal matrix, a scalar matrix, and an identity matrix.
Diagonal matrix: a square matrix where all non-diagonal entries are $0$. Scalar matrix: a diagonal matrix with all diagonal entries equal. Identity matrix $I$: a scalar matrix with all diagonal entries equal to $1$.
What is a symmetric matrix and a skew-symmetric matrix?
Symmetric: $A^{T} = A$ (i.e. $a_{ij} = a_{ji}$). Skew-symmetric: $A^{T} = -A$ (i.e. $a_{ij} = -a_{ji}$), which forces all diagonal entries to be $0$.
When is the product $AB$ of two matrices defined, and what is the order of the result?
$AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ has order $m \times p$.
Give the formula for the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$.
$\det = ad - bc$.
State three key properties of determinants.
(1) Interchanging two rows (or columns) changes the sign of the determinant. (2) If two rows/columns are identical, the determinant is $0$. (3) Multiplying a row/column by $k$ multiplies the determinant by $k$.
What is the formula for the area of a triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$ using a determinant?
$\text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|$. The points are collinear when this determinant equals $0$.
Define the adjoint (adjugate) of a square matrix $A$.
The adjoint $\text{adj}(A)$ is the transpose of the cofactor matrix of $A$. It satisfies $A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| I$.
State the formula for the inverse of a square matrix $A$.
$A^{-1} = \frac{1}{|A|}\,\text{adj}(A)$, valid only when $|A| \neq 0$ (i.e. $A$ is non-singular).
What is the condition for a square matrix to be invertible (non-singular)?
A square matrix $A$ is invertible if and only if its determinant is non-zero, $|A| \neq 0$. If $|A| = 0$, the matrix is singular and has no inverse.
Using matrices, how do you solve the linear system $AX = B$ when $A$ is invertible?
Multiply both sides by $A^{-1}$ to get the unique solution $X = A^{-1}B$, valid when $|A| \neq 0$.
In Cramer's rule, what determines whether a system has a unique solution, no solution, or infinitely many?
Let $D = |A|$. If $D \neq 0$: unique solution. If $D = 0$ and all $D_i = 0$: infinitely many solutions (consistent). If $D = 0$ and at least one $D_i \neq 0$: no solution (inconsistent).
State the fundamental principle of counting (multiplication rule).
If one event can occur in $m$ ways and a second independent event in $n$ ways, then both events together can occur in $m \times n$ ways. The addition rule applies to mutually exclusive choices: $m + n$ ways.
What is the formula for permutations ${}^{n}P_{r}$ and what does it count?
${}^{n}P_{r} = \frac{n!}{(n-r)!}$. It counts the number of arrangements (where order matters) of $r$ objects chosen from $n$ distinct objects.
What is the formula for combinations ${}^{n}C_{r}$ and what does it count?
${}^{n}C_{r} = \frac{n!}{r!(n-r)!}$. It counts the number of selections (where order does not matter) of $r$ objects from $n$ distinct objects.
State the relationship between ${}^{n}P_{r}$ and ${}^{n}C_{r}$.
${}^{n}P_{r} = {}^{n}C_{r} \times r!$, because each combination of $r$ objects can be arranged in $r!$ ways.
State two important identities for combinations.
(1) ${}^{n}C_{r} = {}^{n}C_{n-r}$ (symmetry). (2) ${}^{n}C_{r} + {}^{n}C_{r-1} = {}^{n+1}C_{r}$ (Pascal's rule).
State the binomial theorem for a positive integral index $n$.
$(a+b)^{n} = \sum_{r=0}^{n} {}^{n}C_{r}\, a^{n-r} b^{r} = {}^{n}C_0 a^n + {}^{n}C_1 a^{n-1}b + \cdots + {}^{n}C_n b^n$. The expansion has $n+1$ terms.
What is the general term in the binomial expansion of $(a+b)^{n}$?
$T_{r+1} = {}^{n}C_{r}\, a^{n-r} b^{r}$, the $(r+1)$th term, with $r = 0, 1, 2, \ldots, n$.
How do you find the middle term(s) in the expansion of $(a+b)^{n}$?
If $n$ is even: one middle term, the $\left(\frac{n}{2}+1\right)$th term. If $n$ is odd: two middle terms, the $\left(\frac{n+1}{2}\right)$th and $\left(\frac{n+3}{2}\right)$th terms.
What is the sum of the binomial coefficients in the expansion of $(1+x)^{n}$?
Setting $x = 1$ gives $\sum_{r=0}^{n} {}^{n}C_{r} = 2^{n}$. Setting $x = -1$ gives $\sum_{r=0}^{n} (-1)^r {}^{n}C_{r} = 0$.
Give the formula for the $n$th term of an arithmetic progression (AP).
$a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.
Give the formula for the sum of the first $n$ terms of an AP.
$S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)$, where $a$ is the first term, $d$ the common difference, and $l$ the last term.
Give the formula for the $n$th term and the sum of $n$ terms of a geometric progression (GP).
$n$th term: $a_n = a r^{n-1}$. Sum: $S_n = a\dfrac{r^{n}-1}{r-1}$ for $r \neq 1$, where $a$ is the first term and $r$ the common ratio.
What is the sum of an infinite geometric progression and its condition for convergence?
$S_\infty = \frac{a}{1-r}$, valid only when $|r| < 1$. If $|r| \geq 1$, the infinite GP does not converge to a finite sum.
What are the arithmetic mean (AM) and geometric mean (GM) of two positive numbers $a$ and $b$, and how are they related?
$\text{AM} = \frac{a+b}{2}$, $\text{GM} = \sqrt{ab}$. They satisfy the inequality $\text{AM} \geq \text{GM}$, with equality if and only if $a = b$.
JEE Main Paper 2 (B.Arch) Mathematics FAQ
What is in the JEE Main Paper 2 (B.Arch) Mathematics syllabus?
JEE Main Paper 2 (B.Arch) Mathematics is split into 5 chapters: Algebra and Number Systems, Calculus, Coordinate Geometry and 3D, Trigonometry and Vectors and Statistics, Probability and Reasoning. Together they contain 18 topics and 47 sub-topics.
How many chapters are in JEE Main Paper 2 (B.Arch) Mathematics?
5 chapters, containing 18 topics and 47 sub-topics. The largest chapter is Algebra and Number Systems, with 5 topics.
Are there flashcards for JEE Main Paper 2 (B.Arch) 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.