Class XII Mathematics

Types of relations: reflexive, symmetric, transitive, and equivalence relations, equivalence classes, types of functions: one-one (injective), onto (surjective), and bijective, composition of functions, invertible functions, and binary operations.

Definition, range, domain, principal value branches, graphs of inverse trigonometric functions, elementary properties and identities, and simplification of complex inverse trigonometric expressions.

Concept of matrix as rectangular array of numbers, order, types of matrices (row, column, zero, square, diagonal, scalar, identity), equality of matrices, addition, scalar multiplication, matrix multiplication, transpose, symmetric and skew-symmetric matrices, and elementary transformations.

Determinant of a square matrix up to 3x3, properties of determinants, minors, cofactors, evaluation by row/column expansion, application to area of triangle, adjoint of a matrix, inverse of a matrix via adjoint, and solving system of linear equations by Matrix method and Cramer's rule.

Continuity of a function at a point and on an interval, algebra of continuous functions, differentiability at a point, relation between continuity and differentiability, derivative of composite functions (Chain rule), derivative of implicit functions, and derivatives of inverse trigonometric functions.

Exponential and logarithmic functions, derivatives of e^x and ln x, logarithmic differentiation, derivatives of functions expressed in parametric forms, second-order derivatives, Rolle's Theorem, and Lagrange's Mean Value Theorem.

Rate of change of quantities, increasing and decreasing functions, tangents and normals to curves, approximation using differentials, maxima and minima: local extrema, first derivative test, second derivative test, absolute extrema on closed intervals, and real-life optimization problems.

Integration as reverse process of differentiation, standard algebraic and trigonometric integrals, integration by method of substitution, integration by parts (ILATE rule), integration of rational functions by partial fractions, and integration of special irrational and quadratic forms.

Definite integral as limit of a Riemann sum, Fundamental Theorem of Calculus (statement only), evaluation of definite integrals by substitution, and essential properties of definite integrals used for symmetry and periodic simplifications.

Applications in finding the area under simple curves, especially lines, circles, parabolas, ellipses in standard form, area bounded between two curves, and geometric symmetry considerations.

Definition, order and degree of a differential equation, general and particular solutions, formation of differential equations, solution of differential equations by variable separable method, homogeneous differential equations of first order and first degree, and linear differential equations.

Vectors and scalars, position vector, direction cosines, unit vector, addition of vectors, scalar multiplication, components of a vector in 2D and 3D, scalar (dot) product and projection, vector (cross) product and area of parallelogram/triangle, and scalar triple product.

Direction cosines and direction ratios of a line, equation of a line in vector and Cartesian forms, angle between two lines, skew lines and shortest distance, equation of a plane in vector and Cartesian forms, angle between planes, line and plane, and distance of a point from a plane.

Mathematical formulation of real-life LPPs (manufacturing, diet, transportation, allocation), linear objective function Z = ax + by, linear constraints and non-negativity restrictions, graphical representation of feasible region, Corner Point Method, bounded and unbounded feasible regions, and determination of optimal solutions.

Conditional probability P(A|B), properties, multiplication rule of probability, independent events, theorem of total probability, Bayes' theorem for posterior probabilities, random variables, probability distributions, mean and variance of discrete random variables, Bernoulli trials, and binomial probability distribution.