# CBSE 12th Class Maths Syllabus

## Course Structure

Units Topics Marks
I Relations and Functions 10
II Algebra 13
III Calculus 44
IV Vectors and 3-D Geometry 17
V Linear Programming 6
VI Probability 10
Total 100

## Course Syllabus

### Unit I: Relations and Functions

Chapter 1: Relations and Functions

• Types of relations −
• Reflexive
• Symmetric
• transitive and equivalence relations
• One to one and onto functions
• composite functions
• inverse of a function
• Binary operations

Chapter 2: Inverse Trigonometric Functions

• Definition, range, domain, principal value branch
• Graphs of inverse trigonometric functions
• Elementary properties of inverse trigonometric functions

### Unit II: Algebra

Chapter 1: Matrices

• Concept, notation, order, equality, 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)

• Concept of elementary row and column operations

• Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

Chapter 2: Determinants

• Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle

• Ad joint and inverse of a square matrix

• 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

Chapter 1: Continuity and Differentiability

• Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions

• Concept of exponential and logarithmic functions.

• Derivatives of logarithmic and exponential functions

• Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives

• Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation

Chapter 2: Applications of Derivatives

• Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normal, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool)

• Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)

Chapter 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

$\int \frac{dx}{x^2\pm {a^2}'}$, $\int \frac{dx}{\sqrt{x^2\pm {a^2}'}}$, $\int \frac{dx}{\sqrt{a^2-x^2}}$, $\int \frac{dx}{ax^2+bx+c} \int \frac{dx}{\sqrt{ax^2+bx+c}}$

$\int \frac{px+q}{ax^2+bx+c}dx$, $\int \frac{px+q}{\sqrt{ax^2+bx+c}}dx$, $\int \sqrt{a^2\pm x^2}dx$, $\int \sqrt{x^2-a^2}dx$

$\int \sqrt{ax^2+bx+c}dx$, $\int \left ( px+q \right )\sqrt{ax^2+bx+c}dx$

• Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof)

• Basic properties of definite integrals and evaluation of definite integrals

Chapter 4: Applications of the Integrals

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

• Area between any of the two above said curves (the region should be clearly identifiable)

Chapter 5: Differential Equations

• Definition, order and degree, general and particular solutions of a differential equation

• Formation of differential equation whose general solution is given

• Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree

• Solutions of linear differential equation of the type −

• 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

### Unit IV: Vectors and Three-Dimensional Geometry

Chapter 1: Vectors

• Vectors and scalars, magnitude and direction of a vector

• 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

• Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors

Chapter 2: 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, shortest distance between two lines

• Cartesian and vector equation of a plane

• Angle between −

• Two lines

• Two planes

• A line and a plane

• Distance of a point from a plane

### Unit V: Linear Programming

Chapter 1: Linear Programming

• Introduction
• Related terminology such as −
• 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)

### Unit VI: Probability

Chapter 1: Probability

• Conditional probability
• Multiplication theorem on probability
• Independent events, total probability
• Baye's theorem
• Random variable and its probability distribution
• Mean and variance of random variable
• Repeated independent (Bernoulli) trials and Binomial distribution