# NCERT Solutions and Notes for CBSE Class 12 Maths

## CBSE Class 12 Maths Syllabus 2020-21(Revised)

 No. Units No. of Periods Marks I. Relations and Functions 17 08 II. Algebra 35 10 III. Calculus 57 35 IV. Vectors and  Three  – Dimensional Geometry 26 14 V. Linear Programming 13 05 VI. Probability 20 08 Total 168 80 Internal Assessment 20

Unit-I: Relations and Functions

Relations and Functions

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.

Inverse Trigonometric Functions

Definition, range, domain, principal value branch.

Unit-II: Algebra

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. Non- commutativity of multiplication of matrices Invertible matrices

Determinants

Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit-III: Calculus

Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivative 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.

Applications of Derivatives

Applications of derivatives: increasing/decreasing functions, tangents and normals, 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).

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. Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

Applications of the Integrals

Applications in finding the area under simple curves, especially lines, parabolas; area of circles /ellipses  (in standard form only) (the region should be clearly identifiable

Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Solution of  differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree of the type: dy/dx = 𝑓(y/x). Solutions of linear differential equation  of the type:

Solutions of linear differential equation of the type:

(dy/dx) + py = q, where p and q are functions of x or constants.

Unit-IV: Vectors and Three-Dimensional Geometry

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.

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. Distance of a point from a plane.

Unit-V: Linear Programming

Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, different types of  linear programming (L.P.) problems. graphical method of solution for problems in two variables, feasible  and infeasible regions (bounded), feasible and infeasible solutions, optimal feasible solutions (up to  three non-trivial constraints).

Unit-VI: Probability

Probability

Conditional probability, multiplication theorem on probability, independent events, total probability,  Bayes’ theorem, Random variable and its probability distribution.

### Reference Books for Class 12 Math

• CBSE Chapter wise Solutions – Mathematics (Class 12) by Hemant Malhotra
• CBSE Examination Mathematics : 15 Sample Question Papers (Class 12) by Prem Kumar
• Mathematics (Class – XII) by NCERT
• Problems Plus In IIT Mathematics by A Das Gupta
• Senior Secondary School Math 12 by R. S. Aggarwal
• Solutions of Objective Mathematics: Solutions of Exercises and Chapter Tests by Dhanpat Rai