**CBSE Class 12th Maths syllabus 2019. The Central Board of Secondary Education (CBSE) will conduct board examinations from February for the coming academic year 2019. The board has already released a list of vocational subjects for the examinations that will be conducted from February to March 2019, the schedule and date of the examinations will be released later.**

Apart from the 40 different vocational subjects, the board will conduct exams for typography and Computer Applications (English), web applications, graphics, office communication, et al, in February as these subjects have larger practical component, and shorter theory papers.

**Class 12 – The Maths syllabus for class 12 is divided into six units, namely:**

1. Relations and Functions

2. Algebra

3. Calculus

4. Vectors and Three – Dimensional Geometry

5. Linear Programming

6. Probability

**2. Mathematical Modelling**

1. Chapters with Time Allocation

2. Relations and Functions Periods

3. Inverse Trigonometric Functions Periods

4. Matrices Periods

5. Determinants Periods

6. Continuity and Differentiability Periods

7. Applications of Derivatives Periods

8. Integrals Periods

9. Applications of the Integrals Periods

10. Differential Equations Periods

11. Vectors Periods

12. Three-dimensional Geometry Periods

13. Linear Programming Periods

14. Probability Periods

The unit I: Relations and Functions

1. Relations and Functions Types of relations: Reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, the inverse of a function. Binary operations.

2. Inverse Trigonometric Functions Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

Unit II: Albegra

1. Matrices Concept, notation, order, equality, types of matrices, zero matrices, the transpose of a matrix, symmetric and skew-symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Non-commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order

2). The 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). 2. Determinants Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency, and a number of solutions of the system of linear equations by examples, solving system of linear equations in two or three variables (having a unique solution) using the inverse of a matrix.

Unit III: Calculus:

1. Continuity and Differentiability Continuity and differentiability, a derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. The concept of exponential and logarithmic functions and their derivatives. Logarithmic differentiation. A derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations.

2. Applications of Derivatives Applications of derivatives: Rate of change, increasing/decreasing functions, tangents, and normals, 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).

3. Integrals Integration as an inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type ± + + ± – + + ? ? ? ? ? 2 2 2 2 2 2 2 2 , , , , , dx dx dx dx dx x a ax bx c x a a x ax bx c ++ ±- ++ ++ ? ? ? ? 2 2 2 2 2 2 ( ) ( ) ,, px q px q dx dx a x dx and x a dx ax bx c ax bx c to be evaluated. Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ ellipses (in standard form only), area between the two above said curves (the region should be clearly identifiable).

5. Differential Equations Definition, order, and degree, general and particular solutions of a differential equation. Formation of the differential equation whose general solution is given. A solution of differential equations by the method of separation of variables, homogeneous differential equations of first order and first degree. Solutions of the linear differential equation of the type: P Q, dy y dx += where P and Q are functions of x.

Unit IV: Vectors and Three-Dimensional Geometry:

1. Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, the addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Scalar (dot) product of vectors, the projection of a vector on a line. Vector (cross) product of vectors.

2. Three-dimensional Geometry Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, the shortest distance between two lines. Cartesian and vector equation of a plane. The angle between (i) two lines, (ii) two planes, (iii) a line and a plane. The distance of a point from a plane.

Unit V: 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, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit VI: Probability

Multiplication theorem on probability. Conditional probability, independent events, total probability, Baye’s theorem. Random variable and its probability distribution, mean and variance of the haphazard variable. Repeated independent (Bernoulli) trials and Binomial distribution.

**Appendix 1. Proofs in Mathematics**

Through a variety of examples related to mathematics and already familiar to the learner, bring out different kinds of proofs: direct, contrapositive, by contradiction, by counter-example.

**2. Mathematical Modelling**

Modeling real-life problems where many constraints may really need to be ignored (continuing from Class XI). However, now the models concerned would use techniques/results of matrices, calculus, and linear programming.

Also Read: List of Architecture entrance exams, courses, colleges and eligibility