Mathematics

Mathematics Syllabus for Standard 12th

Scoring well in mathematics requires a clear understanding of the concepts. Here is the module prepared for students to acquire knowledge about the CBSE mathematics syllabus for standard 12th of the academic year 2022-2023.

Unit-I: Relations and Functions

1. Relations and Functions

• One to one and onto functions
• Types of relations: Reflexive, Symmetric, Transitive and Equivalence Relations

2. Inverse Trigonometric Functions

• Graphs of Inverse Trigonometric functions
• Definition
• Range
• Domain
• Principal Value Branch

Unit-II: Algebra

1. Matrices

• On commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2)
• Operation on Matrices: 
  • Addition and multiplication and multiplication with a scalar
• Simple properties of addition, multiplication and scalar multiplication 
• Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew-symmetric matrices
• Invertible matrices and proof of the uniqueness of inverse, if it exists
  • (here all matrices will have real entries)

2. Determinants

• Adjoint and inverse of a square matrix
• Determinant of a square matrix (up to 3 X 3 matrices)
• Minors
• Cofactors and Applications of determinants in finding the area of a triangle
• 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

Unit-III: Calculus

1. Continuity and Differentiability 

• Second-Order Derivatives

• Concept of Exponential and Logarithmic Functions 

• Derivatives of Logarithmic and Exponential Functions 

• Continuity and Differentiability

• Chain Rule

• Derivative of Inverse Trigonometric Functions, 𝑙𝑖𝑘𝑒 Sin−1 𝑥 , Cos−1 𝑥 And Tan−1 x

• Derivative of Implicit Functions

• Logarithmic Differentiation

• Derivative of functions expressed in Parametric Forms

2. Applications of Derivatives

• Simple problems (That illustrate basic principles and understanding of the subject as well as real-life situations)
• Applications of Derivatives: 
  • Rate of change of bodies
  • Increasing/Decreasing Functions
  • Maxima and Minima (First derivative test motivated geometrically and second derivative test given as a provable tool)

3. Integrals

• Integration as inverse process of differentiation
• Integration of a variety of functions by substitution, by partial fractions and by parts
• Evaluation of simple integrals of the following types and problems based on them

dxx2a2, dxx2a2, dxa2-x2, dxax2+bx+c, dxax2+bx+c, px+qax2+bx+cdx, a2x2dx, x2-a2dx, ax2+bx+cdx

• Basic properties of definite integrals and evaluation of definite integrals
• Fundamental theorem of calculus (Without Proof)

4. Applications of the Integrals 

• Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)

5. Differential Equations

• Solution of differential equations by method of Separation of Variables
• Solutions of homogeneous differential equations of first order and first degree
• Definition
• Order and Degree
• General and particular solutions of a Differential Equation
• Solutions of linear differential equation of the type: 

 dydx + py = q, where p and q are functions of x or constants

 dydx + px = q, where p and q are functions of y or constants 

Unit-IV: Vectors and Three-Dimensional Geometry

1. Vectors

• Definition
• Geometrical Interpretation
• Properties and Application of Scalar (Dot) product of vectors
• Vector (cross) product of vectors
• Direction cosines and direction ratios of a vector
• Types of Vectors (Equal, Unit, Zero, Parallel And Collinear Vectors)
• Position vector of a point
• Negative of a vector
• Components of a Vector
• Addition of Vectors
• Multiplication of a Vector by a scalar
• Position vector of a point dividing a line segment in a given ratio
• Vectors and scalars
• Magnitude and Direction of a vector

2. Three-Dimensional Geometry

• Cartesian Equation and vector equation of a line
• Skew Lines
• Shortest Distance between two lines
• Direction cosines and direction ratios of a line joining two points
• Angle between two lines

Unit-V: Linear Programming

1. Linear Programming

• Introduction
• Optimization
• Feasible and Infeasible Solutions
• Related Terminology such as constraints
• Graphical Method of solution for problems in two variables
• Feasible and infeasible regions (Bounded or Unbounded)
• Objective Function
• Optimal Feasible Solutions (Up To Three Non-Trivial Constraints)

Unit-VI: Probability

1. Probability

• Conditional Probability
• Random Variable and its Probability Distribution
• Bayes’ Theorem
• Independent Events
• Total Probability
• Multiplication Theorem on Probability
• Mean of Random Variable