Exam Prep

JEE Main Mathematics: study sheet

14 units, with a key formula and what each covers, from this hub's syllabus notes. Not yet checked by a teacher: check it against your own notes.

The exam

Mathematics is one of three subjects in Paper 1 (B.E./B.Tech): 25 questions, all compulsory. Section A has 20 multiple-choice questions and Section B has 5 numerical-value questions. The whole paper is 75 questions, 300 marks, in 3 hours.

+4 for a correct answer and −1 for a wrong one, in both sections. Unanswered questions score 0. NTA does not publish a chapter-wise weightage.

1. Sets, Relations and Functions
No official chapter weightage
$n(A \cup B) = n(A) + n(B) - n(A \cap B), \quad (g \circ f)(x) = g(f(x))$
- Sets and their representation
- Union, intersection and complement of sets and their algebraic properties
- Power set
- Relations
- Functions
- Types of relations, equivalence relations
- One-one, into and onto functions
- The composition of functions
2. Complex Numbers and Quadratic Equations
No official chapter weightage
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}, \quad |a + ib| = \sqrt{a^2 + b^2}$
- 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 numbers
- Modulus and argument (or amplitude) of a complex number
- Quadratic equations in real and complex number systems and their solutions
- Relations between roots and coefficients, nature of roots
- The formation of quadratic equations with given roots
3. Matrices and Determinants
No official chapter weightage
$A^{-1} = \frac{\operatorname{adj} A}{|A|}, \quad |AB| = |A|\,|B|$
- Matrices, algebra of matrices, types of matrices
- Determinants and matrices of order two and three, evaluation of determinants
- Area of triangles using determinants
- Adjoint and inverse of a square matrix
- Test of consistency and solution of simultaneous linear equations in two or three variables using matrices
4. Permutations and Combinations
No official chapter weightage
${}^{n}P_{r} = \frac{n!}{(n-r)!}, \quad {}^{n}C_{r} = \frac{n!}{r!\,(n-r)!}$
- The fundamental principle of counting
- Permutations and combinations
- Meaning of P(n, r) and C(n, r)
- Simple applications
5. Binomial Theorem
No official chapter weightage
$(x + a)^n = \sum_{r=0}^{n} {}^{n}C_{r}\, x^{n-r} a^{r}, \quad T_{r+1} = {}^{n}C_{r}\, x^{n-r} a^{r}$
- Binomial theorem for a positive integral index
- General term and middle term
- Simple applications
6. Sequences and Series
No official chapter weightage
$a_n = a + (n-1)d, \quad S_n = \frac{n}{2}\,[2a + (n-1)d], \quad \frac{a+b}{2} \ge \sqrt{ab} \ (a, b > 0)$
- Arithmetic and geometric progressions
- Insertion of arithmetic and geometric means between two given numbers
- Relation between A.M. and G.M.
7. Limits, Continuity and Differentiability
No official chapter weightage
$\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{e^x - 1}{x} = 1, \quad \frac{d}{dx}(uv) = u'v + uv'$
- Real-valued functions, algebra of functions
- Polynomial, rational, trigonometric, logarithmic and exponential functions; inverse functions
- Graphs of simple functions
- Limits
- Differentiation of the sum, difference, product and quotient of two functions
- Continuity and differentiability
- Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions
- Derivatives of order up to two
- Rate of change of quantities
- Monotonic (increasing and decreasing) functions
- Maxima and minima of functions of one variable
8. Integral Calculus
No official chapter weightage
$\int_{a}^{b} f(x)\,dx = \int_{a}^{b} f(a+b-x)\,dx, \quad \int u\,dv = uv - \int v\,du$
- Integral as an anti-derivative
- Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions
- Integration by substitution, by parts and by partial fractions
- Integration using trigonometric identities
- Simple integrals of the type ∫dx/(x² + a²), ∫dx/√(x² ± a²), ∫dx/(a² − x²), ∫dx/√(a² − x²), ∫dx/(ax² + bx + c), ∫dx/√(ax² + bx + c), ∫(px + q)dx/(ax² + bx + c), ∫(px + q)dx/√(ax² + bx + c), ∫√(a² ± x²)dx and ∫√(x² − a²)dx
- The fundamental theorem of calculus
- Properties of definite integrals, evaluation of definite integrals
- Areas of the regions bounded by simple curves in standard forms
9. Differential Equations
No official chapter weightage
$\frac{dy}{dx} + p(x)\,y = q(x) \;\Rightarrow\; y\,e^{\int p\,dx} = \int q\,e^{\int p\,dx}\,dx + C$
- Ordinary differential equations, their order and degree
- Solution by the method of separation of variables
- Solution of a homogeneous differential equation
- Linear differential equations of the type dy/dx + p(x)y = q(x)
10. Coordinate Geometry
No official chapter weightage
$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}, \quad x^2 + y^2 + 2gx + 2fy + c = 0, \quad y^2 = 4ax, \quad \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
- Cartesian system of rectangular coordinates in a plane, distance formula, section formula
- Locus and its equation
- The slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes
- Various forms of equations of a line, intersection of lines, angles between two lines
- Conditions for concurrence of three lines, the distance of a point from a line
- Coordinates of the centroid, orthocentre and circumcentre of a triangle
- Standard and general forms of the equation of a circle, its radius and centre
- 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 the parabola, ellipse and hyperbola in standard forms
11. Three-Dimensional Geometry
No official chapter weightage
$\cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}}, \quad d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$
- Coordinates of a point in space, the distance between two points, section formula
- Direction ratios and direction cosines
- The angle between two intersecting lines
- Equation of a line
- Skew lines, the shortest distance between them and its equation
12. Vector Algebra
No official chapter weightage
$\vec{a} \cdot \vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta, \quad |\vec{a} \times \vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta$
- Vectors and scalars, the addition of vectors
- Components of a vector in two dimensions and three-dimensional space
- Scalar and vector products
13. Statistics and Probability
No official chapter weightage
$\sigma^2 = \frac{1}{n}\sum (x_i - \bar{x})^2, \quad P(A_i \mid B) = \frac{P(A_i)\,P(B \mid A_i)}{\sum_j P(A_j)\,P(B \mid A_j)}$
- Measures of dispersion
- Mean, median and mode of grouped and ungrouped data
- Standard deviation, variance and mean deviation for grouped and ungrouped data
- Probability of an event
- Addition theorem of probability
- Multiplication theorem of probability
- Bayes' theorem
- Probability distribution of a random variable
14. Trigonometry
No official chapter weightage
$\sin^2\theta + \cos^2\theta = 1, \quad \sin(A + B) = \sin A\cos B + \cos A\sin B, \quad \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$
- Trigonometric identities and trigonometric functions
- Inverse trigonometric functions and their properties

The exam

1. Sets, Relations and Functions

2. Complex Numbers and Quadratic Equations

3. Matrices and Determinants

4. Permutations and Combinations

5. Binomial Theorem

6. Sequences and Series

7. Limits, Continuity and Differentiability

8. Integral Calculus

9. Differential Equations

10. Coordinate Geometry

11. Three-Dimensional Geometry

12. Vector Algebra

13. Statistics and Probability

14. Trigonometry