JEE Main Paper 2 (B.Arch) 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 Main Paper 2 (B.Arch), not a summary of it.

1. 1

   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
2. 2

   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
3. 3

   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
4. 4

   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
5. 5

   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

Mathematics flashcards for JEE Main Paper 2 (B.Arch)

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

1. 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 $b$.

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

3. 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 $A \times B$.

4. 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 $\mathbb{R}$.

5. 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.

6. 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 $x_1, x_2 \in A$.

7. 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).

8. 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 $f \circ f^{-1} = I_B$.

9. 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: $g \circ f \neq f \circ g$.

10. 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 $\frac{n!}{(n-m)!} = {}^{n}P_{m}$.

11. 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).

12. 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 $0$).

13. 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.

14. 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: $i^{1} = i$, $i^{2} = -1$, $i^{3} = -i$, $i^{4} = 1$.

15. 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 $z \bar{z} = |z|^{2} = a^{2} + b^{2}$.

16. 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 $z \neq 0$.

17. 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 $|z_1 + z_2| \leq |z_1| + |z_2|$.

18. 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 $(-\pi, \pi]$.

19. 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 $r\,\text{cis}\,\theta$.

20. 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 $(a, b)$.

21. 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.

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