# 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

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