Mathematics notes for Class 11, Class 12 and JEE

Roster form and set-builder form for Class 11 Maths and JEE: what a set is, the sets N, Z, Q, R, converting between the forms, with solved examples.

NCERT Section 1.2 15 min read 5 solved examples 12 practice questions

The empty set for Class 11 Maths and JEE: what the null or void set is, its symbols, how to spot one, and why ϕ, {0} and {ϕ} are different.

NCERT Section 1.3 8 min read 3 solved examples 7 practice questions

Finite and infinite sets for Class 11 Maths and JEE: how to tell them apart, n(S) and the cardinal number, roster form with dots, and solved examples.

NCERT Section 1.4 8 min read 3 solved examples 7 practice questions

Equal sets for Class 11 Maths and JEE: when two sets are equal, why order and repetition do not matter, equal vs equivalent sets, and solved examples.

NCERT Section 1.5 9 min read 4 solved examples 7 practice questions

Proper subsets and supersets, ∈ against ⊂, the symbol ⊆, and open, closed and half-open intervals, with solved examples.

NCERT Section 1.6 15 min read 4 solved examples 12 practice questions

Power set for JEE Main: what P(A) is, how to list it, the formula n[P(A)] = 2^ⁿ and why it holds, proper subsets and P(ϕ), with solved examples.

9 min read 4 solved examples 9 practice questions

The universal set for Class 11 Maths and JEE: a set containing every element needed in a problem, how to choose it, and why the choice matters.

NCERT Section 1.7 7 min read 3 solved examples 5 practice questions

Venn diagrams for Class 11 Maths and JEE: the universal set as a rectangle, sets as circles, subsets, disjoint and overlapping sets, with examples.

NCERT Section 1.8 8 min read 3 solved examples 5 practice questions

Union, intersection, difference, disjoint sets, the laws of algebra of sets and n(A ∪ B), with solved examples.

NCERT Section 1.9 16 min read 5 solved examples 11 practice questions

The complement of a set for Class 11 Maths and JEE: A′ as U − A, the complement laws and De Morgan’s laws, with Venn diagrams and examples.

NCERT Section 1.10 12 min read 4 solved examples 9 practice questions

Ordered pairs, n(A × B) = pq, why A × B ≠ B × A, A × ϕ, R × R, laws with ∪ and ∩, and solved examples.

NCERT Section 2.2 17 min read 5 solved examples 9 practice questions

Relations as subsets of A × B, arrow diagrams, domain, codomain and range, and the number of relations, 2 to the power pq.

NCERT Section 2.3 15 min read 5 solved examples 9 practice questions

Function vs relation, domain and range, graphs of the modulus, signum and greatest integer functions, and solved examples.

NCERT Section 2.4 22 min read 7 solved examples 10 practice questions

What a radian is, arc length l = rθ and the radian to degree conversion formula, with solved examples.

NCERT Section 3.2 11 min read 5 solved examples 9 practice questions

Trigonometric functions for Class 11 Maths and JEE: the unit circle, values table, signs in each quadrant (ASTC), domain, range and graphs of all six.

NCERT Section 3.3 17 min read 5 solved examples 9 practice questions

Allied, double, triple and half angles, sum-to-product, with proofs.

NCERT Section 3.4 20 min read 6 solved examples 10 practice questions

a + ib, iota and i^² = −1, real and imaginary parts, purely imaginary numbers and equality, with solved examples.

NCERT Sections 4.1–4.2 10 min read 3 solved examples 8 practice questions

Adding, subtracting, multiplying and dividing a + ib, powers of iota and square roots of negative numbers.

NCERT Section 4.3 14 min read 4 solved examples 10 practice questions

|z| = √(a^² + b^²), the conjugate a − ib, z times its conjugate, the inverse and properties.

NCERT Section 4.4 10 min read 4 solved examples 9 practice questions

x + iy as the point (x, y), |z| as a distance, the conjugate as a reflection, with solved examples.

NCERT Section 4.5 10 min read 3 solved examples 8 practice questions

Argument of a complex number for JEE Main: the principal argument in each quadrant, polar form r(cos θ + i sin θ), products and quotients, with examples.

12 min read 4 solved examples 9 practice questions

Quadratic equations for JEE Main (Class 11): complex roots, discriminant and nature of roots, sum and product of roots, forming equations, with examples.

13 min read 6 solved examples 9 practice questions

Inequalities for Class 11 Maths: the signs <, >, ≤ and ≥, strict and slack inequalities, numerical, literal, double and linear ones, with solved examples.

NCERT Sections 5.1–5.2 9 min read 3 solved examples 8 practice questions

Solving linear inequalities for Class 11 Maths: the two rules, why a negative multiplier reverses the sign, number-line graphs, systems and word problems.

NCERT Section 5.3 12 min read 6 solved examples 10 practice questions

Multiplication and addition principles, filling places with or without repetition, solved examples.

NCERT Sections 6.1–6.2 9 min read 4 solved examples 8 practice questions

The nPr formula, factorials, repeated letters, vowels together or apart, dictionary order, and solved examples.

NCERT Section 6.3 14 min read 6 solved examples 11 practice questions

The nCr formula, its link with nPr, its properties and Pascal’s rule, at least and at most selections, solved examples.

NCERT Section 6.4 12 min read 5 solved examples 11 practice questions

The (a + b)^ⁿ formula, Pascal’s triangle, binomial coefficients, special cases and their sums, with solved examples.

NCERT Sections 7.1–7.2 18 min read 4 solved examples 16 practice questions

General term of a binomial expansion for JEE Main: T_ᵣ₊₁ = ^ⁿC_ᵣ a^ⁿ⁻ʳ b^ʳ, middle terms for even and odd n, the term independent of x, with solved examples.

12 min read 5 solved examples 11 practice questions

Sequences for Class 11 Maths and JEE: the general term a_ₙ, finite and infinite sequences, recurrences and the Fibonacci sequence, with solved examples.

NCERT Section 8.2 11 min read 3 solved examples 8 practice questions

Series for Class 11 Maths and JEE: series vs sequence, finite and infinite series, the sum of a series and sigma notation, with solved examples.

NCERT Section 8.3 10 min read 3 solved examples 7 practice questions

Arithmetic progression formulas for JEE Main: the nth term, the sum of n terms, 1 + 2 + … + n and arithmetic means, with solved examples and practice.

14 min read 4 solved examples 8 practice questions

The nth term ar^ⁿ⁻¹, the sum of n terms, the sum to infinity and geometric means, with solved examples.

NCERT Section 8.4 14 min read 6 solved examples 27 practice questions

The A.M.–G.M. inequality A ≥ G, its proof, when A = G, and least values, with solved examples.

NCERT Section 8.5 12 min read 4 solved examples 8 practice questions

Proofs, external division, mid-point, triangle area, collinear points and solved examples.

NCERT Section 9.1 20 min read 6 solved examples 13 practice questions

Locus of a point for JEE Main: the P(h, k) method, perpendicular bisectors, circles, angle bisectors and loci with a parameter, with solved examples.

14 min read 5 solved examples 10 practice questions

Inclination, m = tan θ, slope from two points, parallel and perpendicular lines and the angle between two lines.

NCERT Section 9.2 17 min read 6 solved examples 12 practice questions

Point-slope, two-point, slope-intercept, intercept and general forms, a formula sheet and solved NCERT questions.

NCERT Section 9.3 17 min read 6 solved examples 26 practice questions

The formula and its proof, parallel lines, the foot of the perpendicular and the image of a point.

NCERT Section 9.4 16 min read 7 solved examples 10 practice questions

Intersection of two lines for JEE Main: the formula, concurrent lines, and the centroid, circumcentre and orthocentre of a triangle, with solved examples.

17 min read 6 solved examples 13 practice questions

How a plane cutting a double cone gives a circle, ellipse, parabola or hyperbola, degenerate cases and eccentricity.

NCERT Sections 10.1–10.2 12 min read 4 solved examples 9 practice questions

(x − h)^² + (y − k)^² = r^², the general form, centre and radius, and all of NCERT Exercise 10.1 solved.

NCERT Section 10.3 14 min read 6 solved examples 12 practice questions

Diameter form of circle for JEE Main: the circle on given ends of a diameter, where a line meets a circle, the condition of tangency and chord length.

12 min read 5 solved examples 10 practice questions

Focus and directrix, the four standard equations, latus rectum 4a and focal distance, with solved NCERT examples.

NCERT Section 10.4 14 min read 6 solved examples 19 practice questions

Foci, axes, a^² = b^² + c^², eccentricity c/a, standard equations and latus rectum 2b^²/a, with solved NCERT examples.

NCERT Section 10.5 16 min read 6 solved examples 23 practice questions

Foci, axes, c^² = a^² + b^², eccentricity c/a, latus rectum 2b^²/a, ellipse vs hyperbola, with solved NCERT examples.

NCERT Section 10.6 15 min read 6 solved examples 14 practice questions

The x, y and z axes, the XY, YZ and ZX planes with their equations, and the signs of all eight octants.

NCERT Sections 11.1–11.2 11 min read 6 solved examples 12 practice questions

The triplet (x, y, z), points on the x-axis or in the XY-plane, and distances from the planes.

NCERT Section 11.3 12 min read 6 solved examples 12 practice questions

The distance between two points in space, its proof by Pythagoras, distance from the origin, solved examples.

NCERT Section 11.4 15 min read 6 solved examples 12 practice questions

Section formula in 3D for JEE Main: internal and external division, the mid-point and centroid, and the ratio in which a coordinate plane cuts a segment.

15 min read 6 solved examples 12 practice questions

The intuitive idea of derivatives via average velocity, secants and the tangent, with solved examples.

NCERT Sections 12.1–12.2 13 min read 6 solved examples 12 practice questions

Limit of a function, left- and right-hand limits, algebra of limits, 0/0 forms and a formula sheet, with solved examples.

NCERT Section 12.3 18 min read 8 solved examples 16 practice questions

The sandwich theorem, the proof that sin x/x tends to 1, the standard limits and solved examples.

NCERT Section 12.4 14 min read 6 solved examples 12 practice questions

First principle, slope of the tangent, product and quotient rules, formulas for x^ⁿ, sin x and cos x, and solved examples.

NCERT Section 12.5 16 min read 8 solved examples 22 practice questions

Formulas for lists and frequency tables, the grouped median, mean vs median and why an average is not enough.

NCERT Section 13.1 12 min read 6 solved examples 13 practice questions

The mode formula for grouped data (JEE Main): the modal class, what l, h, f_₁, f_₀ and f_₂ mean, why it works and bimodal data, with solved examples.

12 min read 6 solved examples 12 practice questions

Range, quartile deviation, mean deviation and SD compared, why deviations cancel, with solved examples.

NCERT Section 13.2 10 min read 6 solved examples 12 practice questions

The formula for lists, frequency tables and grouped data, coefficient of range, its limits and solved examples.

NCERT Section 13.3 8 min read 6 solved examples 12 practice questions

Formulas about the mean and the median for lists, frequency tables and grouped data, with every NCERT example worked.

NCERT Section 13.4 19 min read 7 solved examples 12 practice questions

Formulas for lists, frequency tables and grouped data, the short-cut method and solved examples.

NCERT Section 13.5 19 min read 7 solved examples 12 practice questions

Sample space, types of events, A or B, not A, mutually exclusive and exhaustive events, with solved examples.

NCERT Section 14.1 15 min read 6 solved examples 12 practice questions

The three axioms, equally likely outcomes, the addition theorem, P(not A), formulas and solved examples.

NCERT Section 14.2 17 min read 7 solved examples 13 practice questions

Types of relations for Class 12 Maths and JEE: empty and universal relations, reflexive, symmetric and transitive tests, equivalence relations and classes.

NCERT Sections 1.1–1.2 21 min read 6 solved examples 15 practice questions

Types of functions for Class 12 Maths and JEE: one-one, many-one, onto, into and bijective functions, how to prove each, and how to count them.

NCERT Section 1.3 19 min read 5 solved examples 12 practice questions

Composition of functions for Class 12 Maths and JEE: gof and fog, why order matters, invertible functions, finding the inverse, and NCERT examples solved.

NCERT Section 1.4 16 min read 5 solved examples 10 practice questions

Inverse trigonometric functions for Class 12 Maths and JEE: domain and range of each principal value branch, graphs, principal values and solved examples.

NCERT Sections 2.1–2.2 14 min read 5 solved examples 10 practice questions

Properties of inverse trigonometric functions for Class 12 Maths and JEE: sin^⁻¹(sin x) for any x, substitutions, formulas for JEE and solved examples.

NCERT Section 2.3 17 min read 6 solved examples 16 practice questions

What a matrix is, for Class 12 Maths and JEE: rows, columns, the element a_ᵢⱼ, the order m × n, possible orders, and building a matrix from a rule.

NCERT Sections 3.1–3.2 12 min read 5 solved examples 9 practice questions

Types of matrices for Class 12 Maths and JEE: row, column, square, diagonal, scalar, identity and zero matrices, and when two matrices are equal.

NCERT Section 3.3 12 min read 4 solved examples 8 practice questions

Operations on matrices for Class 12 Maths and JEE: addition, scalar multiples, subtraction, matrix multiplication, its properties and why AB ≠ BA.

NCERT Section 3.4 17 min read 6 solved examples 17 practice questions

Transpose of a matrix for Class 12 Maths and JEE: how rows become columns, its properties, why (AB)′ = B′A′, and solved 2 × 2 and 3 × 3 examples.

NCERT Section 3.5 11 min read 5 solved examples 9 practice questions

Symmetric and skew symmetric matrices for Class 12 Maths and JEE: definitions, properties, the zero diagonal, any square matrix as their sum, examples.

NCERT Section 3.6 10 min read 5 solved examples 10 practice questions

Invertible matrices for Class 12 Maths and JEE: what the inverse of a matrix is, why it is unique, (AB)^⁻¹ = B^⁻¹A^⁻¹, how to find it, with solved examples.

NCERT Section 3.7 13 min read 4 solved examples 9 practice questions

Determinant of a matrix for Class 12 Maths and JEE: 2 × 2 and 3 × 3, expanding along any row or column, |kA| = k^ⁿ|A|, the properties and solved examples.

NCERT Sections 4.1–4.2 12 min read 5 solved examples 8 practice questions

Area of a triangle using determinants for Class 12 Maths and JEE: the formula, collinear points, the line through two points, with solved examples.

NCERT Section 4.3 11 min read 4 solved examples 8 practice questions

Minors and cofactors for Class 12 Maths and JEE: M_ᵢⱼ and A_ᵢⱼ = (−1)^ⁱ⁺ʲ M_ᵢⱼ, expansion by cofactors, the zero-sum rule and NCERT examples solved.

NCERT Section 4.4 13 min read 5 solved examples 8 practice questions

Adjoint and inverse of a matrix for Class 12 Maths and JEE: adj A, A(adj A) = |A| I, singular matrices, A^⁻¹ = (1/|A|) adj A and NCERT examples solved.

NCERT Section 4.5 14 min read 5 solved examples 12 practice questions

System of linear equations for Class 12 Maths and JEE: matrix method X = A^⁻¹B, consistent and inconsistent systems, the |A| = 0 test and solved examples.

NCERT Section 4.6 14 min read 5 solved examples 10 practice questions

Continuity of a function for Class 12 Maths and JEE: continuity at a point, one-sided limits, types of discontinuity, sums and composites, solved examples.

NCERT Section 5.2 14 min read 5 solved examples 11 practice questions

Differentiability for Class 12 Maths and JEE: why |x| fails at 0, continuity vs differentiability, the chain rule, implicit and inverse trig derivatives.

NCERT Section 5.3 16 min read 5 solved examples 13 practice questions

Exponential and logarithmic functions for Class 12 Maths and JEE: graphs, laws of logarithms, change of base, derivatives of e^ˣ and log x, solved examples.

NCERT Section 5.4 12 min read 4 solved examples 9 practice questions

Logarithmic differentiation for Class 12 Maths and JEE: formula for u^v, the derivative of x^x, products and quotients, sums of powers, solved examples.

NCERT Section 5.5 15 min read 5 solved examples 10 practice questions

Parametric differentiation for Class 12 Maths and JEE: derivatives in parametric form, dy/dx = (dy/dt)/(dx/dt), circles, parabolas, solved examples.

NCERT Section 5.6 12 min read 5 solved examples 8 practice questions

Second order derivatives for Class 12 Maths and JEE: d^²y/dx^², y″ and y_₂, relations like y_₂ − 5y_₁ + 6y = 0, implicit and parametric cases, solved examples.

NCERT Section 5.7 14 min read 5 solved examples 8 practice questions

Rate of change of quantities for Class 12 Maths and JEE: dy/dx as a rate, related rates by the chain rule, marginal cost and revenue, with solved examples.

NCERT Section 6.2 12 min read 5 solved examples 12 practice questions

Increasing and decreasing functions for Class 12 Maths and JEE: increasing vs strictly increasing, the sign of f′(x), intervals, with solved examples.

NCERT Section 6.3 14 min read 5 solved examples 11 practice questions

Maxima and minima for Class 12 Maths and JEE: critical points, first and second derivative tests, local and absolute extrema, applied problems solved.

NCERT Section 6.4 20 min read 5 solved examples 12 practice questions

Indefinite integrals for Class 12 Maths and JEE: anti derivatives, the constant C, standard integration formulas, log |x|, properties, solved examples.

NCERT Section 7.2 13 min read 4 solved examples 11 practice questions

Integration by substitution for Class 12 Maths and JEE: choosing t, integrals of tan x, cot x, sec x, cosec x, trigonometric identities, solved examples.

NCERT Section 7.3 15 min read 5 solved examples 12 practice questions

Integrals of particular functions for Class 12 Maths and JEE: 1/(x^² ± a^²), square roots, completing the square, linear numerators, solved examples.

NCERT Section 7.4 15 min read 5 solved examples 12 practice questions

Integration by partial fractions for Class 12 Maths and JEE: improper fractions, linear, repeated and quadratic factors, finding A, B, C, solved examples.

NCERT Section 7.5 14 min read 5 solved examples 12 practice questions

Integration by parts for Class 12 Maths and JEE: the formula, ILATE for the first function, ∫e^ˣ[f + f′] dx, square-root integrals, solved examples.

NCERT Section 7.6 15 min read 5 solved examples 13 practice questions

Definite integral for Class 12 Maths and JEE: lower and upper limits, F(b) − F(a), signed area under a curve, the limit of a sum, solved examples.

NCERT Section 7.7 10 min read 5 solved examples 10 practice questions

Fundamental theorem of calculus for Class 12 Maths and JEE: first and second theorems, area function, F(b) − F(a), d/dx of an integral, solved examples.

NCERT Section 7.8 13 min read 5 solved examples 10 practice questions

Definite integrals by substitution for Class 12 Maths and JEE: changing the limits, limits that swap order, and NCERT Exercise 7.9 integrals solved.

NCERT Section 7.9 10 min read 5 solved examples 10 practice questions

Properties of definite integrals for Class 12 Maths and JEE: P_₀ to P_₇, King’s rule with a + b − x, even and odd functions, [0, 2a], solved examples.

NCERT Section 7.10 13 min read 5 solved examples 14 practice questions

Area under the curve for Class 12 Maths and JEE: vertical and horizontal strips, parts below the x-axis, circle and ellipse areas, solved NCERT questions.

NCERT Section 8.2 12 min read 5 solved examples 10 practice questions

Area between two curves for JEE Main: top minus bottom, limits from intersection points, horizontal strips, solved parabola, line and circle examples.

12 min read 5 solved examples 11 practice questions

Order and degree of differential equations for Class 12 Maths and JEE: the highest derivative, when the degree is not defined, all of NCERT Exercise 9.1.

NCERT Section 9.2 10 min read 5 solved examples 10 practice questions

General and particular solutions for Class 12 Maths and JEE: checking a solution by substitution, counting arbitrary constants, all of NCERT Exercise 9.2.

NCERT Section 9.3 11 min read 5 solved examples 10 practice questions

Variable separable, homogeneous and linear differential equations for Class 12 Maths and JEE: y = vx, the integrating factor, linear in x, solved examples.

NCERT Section 9.4 19 min read 7 solved examples 13 practice questions

Direction cosines of a vector for Class 12 Maths and JEE: scalars and vectors, magnitude, position vector, l^² + m^² + n^² = 1, solved examples.

NCERT Sections 10.1–10.2 10 min read 4 solved examples 10 practice questions

Types of vectors for Class 12 Maths and JEE: zero and unit vectors, coinitial, collinear, equal and negative vectors, free vectors, with NCERT questions.

NCERT Section 10.3 9 min read 3 solved examples 10 practice questions

Addition of vectors for Class 12 Maths and JEE: triangle law, parallelogram law, subtraction, commutative and associative properties, with NCERT questions.

NCERT Section 10.4 10 min read 5 solved examples 10 practice questions

Section formula in vectors for Class 12 Maths and JEE: internal, external and midpoint forms, scalar multiples, unit vectors, components, collinearity.

NCERT Section 10.5 13 min read 7 solved examples 12 practice questions

Dot and cross products for Class 12 Maths and JEE: angle between vectors, projection, Cauchy–Schwarz, cross product as a determinant, areas.

NCERT Section 10.6 15 min read 7 solved examples 12 practice questions

Direction cosines and direction ratios for Class 12 Maths and JEE: l^² + m^² + n^² = 1, cosines from ratios, two points, collinear points, solved examples.

NCERT Section 11.2 10 min read 5 solved examples 10 practice questions

Equation of a line in 3D for Class 12 Maths and JEE: vector form r = a + λb, parametric and Cartesian forms, the line through two points, solved examples.

NCERT Section 11.3 11 min read 5 solved examples 11 practice questions

Angle between two lines in 3D for Class 12 Maths and JEE: cos θ from direction ratios, perpendicular and parallel lines, vector form, solved examples.

NCERT Section 11.4 11 min read 6 solved examples 11 practice questions

Shortest distance between two lines for Class 12 Maths and JEE: skew lines, the cross-product formula, Cartesian form, parallel lines, NCERT questions.

NCERT Section 11.5 12 min read 5 solved examples 11 practice questions

The common perpendicular, its feet from two dot products, and its equation, with checks.

9 min read 3 solved examples 8 practice questions

Linear programming (LPP) for Class 12 Maths: objective function, constraints, feasible region, corner point method, unbounded regions, solved examples.

NCERT Sections 12.1–12.2 14 min read 6 solved examples 9 practice questions

Conditional probability for Class 12 Maths and JEE: P(E|F) = P(E ∩ F)/P(F), the reduced sample space, properties, coin and dice examples, NCERT questions.

NCERT Section 13.2 9 min read 6 solved examples 14 practice questions

Multiplication theorem of probability for Class 12 Maths and JEE: P(E ∩ F) = P(E) P(F|E), three events, probability trees, draws without replacement.

NCERT Section 13.3 7 min read 4 solved examples 9 practice questions

Independent events for Class 12 Maths and JEE: P(E ∩ F) = P(E) P(F), independent versus mutually exclusive, three events, complements, NCERT questions.

NCERT Section 13.4 10 min read 6 solved examples 12 practice questions

Bayes’ theorem for Class 12 Maths and JEE: partitions, the theorem of total probability, one path over all paths on a tree, NCERT examples and practice.

NCERT Section 13.5 13 min read 6 solved examples 13 practice questions

Building the table, probabilities that add up to 1, mean and variance, with dice and card examples.

10 min read 4 solved examples 9 practice questions