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NCERT 12th Mathematics

Mathematics is the branch of science, which deals with numbers, involves calculations and mainly focuses on the study of quantity, shapes, measurements etc.In more simple words, math is the science, deals with structures, numbers, geometry etc.

Relations

  * Empty and Universal Relation
  * Types of relations − Reflexive , Symmetric , transitive and equivalence relation,One to one and onto functions,composite functions,inverse of a function,Binary operations
  * To prove relation reflexive,trasitive,symmetric and equivalent
  * Finding number of relations

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Functions

  * Definition
  * To prove one - one and onto (injective , surjective , bijective)
  * Composite functions,composite functions and one - one onto
  * finding inverse
  * inverse function - Proof questions
  * Binary operation - defintion
  * whether binary commutative/associative or not
  * binary operation - identity element
  * binary operation - inverse

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Continuity

  * Continuity of function
  * Reasons and Types of Discontinuity
  * Theorems of Continuity
  * Checking continuity at a given point
  * Checking continuity at a any point
  * checking continuity using LHL and RHL
  * Algebra of continous functions
  * continuity of composite function

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Inverse Trigonometric Functions

  * Definition, range, domain, principal value branch
  * Graphs of inverse trigonometric functions
  * Elementary properties of inverse trigonometric functions
  * Simplification & Transformation of Inverse Functions
  * Equations Involving IT
  * Simultaneous Equations & In-equations Involving IT
  * Summation of Series
  * Changing of trignometric variables and then applying formula

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Matrices

  * Definition,formation
  * notation,order of matrices
  * Types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices
  * Operation on matrices: Addition and multiplication and multiplication with a scalar
  * Simple properties of addition, multiplication and scalar multiplication
  * Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2)
  * Transpose of matrices
  * Concept of elementary row and column operations
  * Inverse of matrix using elementary transformation
  * Minors and cofactors,Adjoint of a matrix
  * Rank of a matrix
  * System of equations(Cramer's rule)
  * solutions of simultaneous linear equations in two or three variables
  * Matrix method and Augmented matrix
  * Method of inversion
  * Method of reduction
  * Test of consistency
  * Statement question - Addition and Subtraction of matrices
  * Statement question -multiplication of matrices
  * finding unknown element and matrices
  * proof using property of transpose
  * proof using mathematical induction

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Determinants

  * Introduction
  * Determinant of a square matrix (up to 2 × 2 matrices)
  * Determinant of a square matrix (up to 3 × 3 matrices)
  * Ad joint and inverse of a square matrix
  * minors, co-factors
  * properties of determinants
  * Finding the area of a triangle
  * Equation of line using determinant
  * Consistency, inconsistency and number of solutions of system of linear equations by examples
  * solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix

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Differentiation - 1

  * checking if function is differentiable
  * derivative of composite functions
  * derivative of functions by chain rule
  * derivatives of inverse trigonometric functions, derivative of implicit functions
  * Derivatives of logarithmic and exponential functions
  * Derivative in parametric form
  * "Relationship between Continuity and Differentiability."

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Differentiation - 2

  * Introduction
  * Derivative of composite functions
  * Derivative of inverse functions
  * Derivative of implicit functions
  * Higher order derivatives
  * Geometrical meaning of derivative
  * Logarithmic differentiation
  * Derivatives of Parametric functions

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Applications of Derivatives

  * Monotonicity
  * Maxima and Minima
  * Tangent and Normal
  * rate of change
  * Finding second order derivative - normal form
  * Finding second order derivative - implicit form
  * Rolle's Theorems (without proof) and their geometric interpretation
  * Lagrange's Mean Value Theorems (without proof) and their geometric interpretation
  * To show increasing/decreasing in whole domain
local maxima/minima
  * Absolute minima/maxima
  * To show increasing/decreasing in intervals
  * Find intervals of increasing/decreasing
  * Finding slope of tangent/normal
  * Finding point when tangent is parallel/perpendicular
  * Finding equation of tangent/normal when point and curve is given
  * Finding equation of tangent/normal when slope and curve is given
  * Finding apporximate value of numbers
  * Finding apporximate value of function
  * Finding approximate value - statement questions
  * Finding minimum and maximum values from graph
  * Minima/maxima - Number questions
  * Minima/maxima - geometry questions
  * Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)

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Integrals - Indefinite integration

  * Integration as the inverse process of differentiation
  * indefinite integrals of standard functions
  * definite integrals and their properties
  * By using formula ( Indirect, Derived)
  * By using trignometric formulae
  * Integration by substitution(algebric,exponential,log function,trignometric identities)
  * Integration using trigo identities(sine and cos formula,a-b formula,2x formula,3x formula,CD and CD inv formulae,Inv trigo formula)
  * Integration by parts(exponential function)
  * Integration by partial function
  * Integration by Specific Formula
  * application of definite integrals to the determination of areas involving simple curves

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Integrals - Definite Integration

  * Introduction
  * Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof)
  * Basic properties of definite integrals and evaluation of definite integrals
  * Definite Integrals - By formulae
  * Definite Integrals - By Partial Fraction
  * Definite Integrals - By e Fraction
  * Definite Integrals - By substitution
  * Definite Integrals - By propeties

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Applications of the Integrals

  * Introduction to integrals
  * Area under curve
  * Area bounded by curve and horizontal or vertical line
  * Area between curve and line
  * Area between curve and curve
  * Some Standard Areas Numericals
  * Curve Tracing
  * Variable Area, Greatest & Least value, Average Value of a Function

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Differential Equations

  * Definition, order, degree and formation of differential equations
  * general and particular solutions of a differential equation
  * Formation of differential equation whose general solution is given
  * Solution of differential equations and variable separable form
  * Reducible to variable separable type
  * Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree
  * Homogeneous and reducible to homogeneous differential equations
  * Linear differential equations(dy/dx + py = q, where p and q are functions of x or constants,dx/dy + px = q, where p and q are functions of y or constants)
  * Solving differential equation when equation is given or statement is given
  * Solving Linerar differential equation when equation is given or statement is given

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Vectors

  * Meaning & types of vectors - Vectors and scalars, magnitude and direction of a vector
  * "Vectors and its Types (Zero Vector,Unit Vector Position Vector,Co-initial Vector,Like and UnlikeVectors,Co-planar Vector,Collinear Vector,Equal Vector,Displacement Vector,Negative of a Vector)"
  * Triangle and Parallelogram Laws of Vector Addition
  * Direction cosines and direction ratios of a vector
  * Section Formula
  * Orthogonal system of unit vectors, scalar product and properties
  * Vector product of two vectors & related properties
  * Scalar Triple Product & Properties
  * Vector triple product and reciprocal system of vectors
  * Linear Combination of vectors
  * Applications of Vectors in Geometry
  * Collinearity of two products and 3 points or 3 position vectors

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Three - dimensional Geometry

  * Direction cosines and direction ratios of a line joining two points
Cartesian equation and vector equation of a line, coplanar and skew lines
  * Equation of line - given point and vector
  * Equation of line - given two points
  * Two lines
  * Two planes
  * A line and a plane
  * shortest distance between two lines- two parallel lines, two skew lines
  * Distance of a point from a plane
  * Equation of Plane - in normal form
  * Equation of Plane - perpendicular to vector and passing through point
  * Equation of Plane - pasing through 3 non collinear point
  * Equation of Plane - intercept form
  * Equation of Plane - passing through intersection of planes
  * Coplanarity of two lines
  * Angle between two planes
  * Distance of point from plane
  * Angle between line and plane
  * Equation of line under planes condition
  * Point with lines and planes

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Linear Programming

  * Introduction and Mathematical Description of LP
  * Mathematical Formulation of Optimal Product Line, Diet and Transportation Problems
  * Constraints
  * Objective function
  * Optimization
  * Different types of linear programming (L.P.) Problems
  * Mathematical formulation of L.P. Problems
  * Graphical method of solution for problems in two variables
  * Feasible and infeasible regions (bounded and unbounded)
  * Feasible and infeasible solutions
  * Optimal feasible solutions (up to three non-trivial constraints)
  * Corner Point Method
  * Iso- Profit/ Iso-Cost Method
  * Different Types of LPP
  * Linear equation given - bounded
  * Linear equation given - not feasible
  * Linear equation given - unbounded
  * Diet problems
  * Manufacturing problems
  * Transport problems

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Probability - 2

  * Introduction to Probability
  * Different methods to find probability
  * Conditional probability
  * Multiplication theorem on probability
  * Independent events, total probability
  * Baye's theorem
  * Definition
  * Probability distribution
  * Probability mass function
  * cumulative distribution function
  * Probability distribution function
  * Mean or Expectation of random variable
  * Variance and Standerd deviation of random variable
  * Bernoulli trials
  * PYQ for JEE Mains

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