## CBSE Class 12 Mathematics Syllabus 2024-25

### Course  Structure

One Paper – Max Marks: 80

 No. Units No. of Periods Marks I. Relations and Functions 30 08 II. Algebra 50 10 III. Calculus 80 35 IV. Vectors and Three – Dimensional Geometry 30 14 V. Linear Programming 20 05 VI. Probability 30 08 Total 240 80 Internal Assessment 20

#### Unit-I: Relations and Functions

1. Relations and Functions – 15 Periods

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

1. Inverse Trigonometric Functions 15 Periods Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.

#### Unit-II: Algebra

• Matrices – 25 Periods

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

• Determinants – 25 Periods

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

• Continuity and Differentiability – 20 Periods

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. 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  – 10 Periods

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

• Integrals – 20 Periods

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

• Applications of the Integrals – 15 Periods

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

• Differential Equations – 15 Periods

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. Solutions of linear differential equation of the type:

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

• Vectors – 15 Periods

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 – 15 Periods

Direction cosines and direction ratios of a line joining two Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.

#### Unit-V: Linear Programming

• Linear Programming – 20 Periods

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

#### Unit-VI: Probability

• Probability – 30 Periods

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

MATHEMATICS (Code No. – 041)

QUESTION PAPER DESIGN CLASS – XII

(2024-25)

Time: 3 hours                                                                                                                Max. Marks: 80

 S. No. Typology of Questions Total Marks % Weightage 1 Remembering: Exhibit memory of previously learned material by recalling facts, terms, basic concepts, and answers. Understanding: Demonstrate understanding of facts and ideas by organizing, comparing, translating, interpreting, giving descriptions, and stating main ideas 44 55 2 Applying: Solve problems to new situations by applying acquired knowledge, facts, techniques and rules in a different way. 20 25 3 Analysing : Examine and break information into parts by identifying motives or causes. Make inferences and find evidence to support generalizations Evaluating: Present and defend opinions by making judgments about information, validity of ideas, or quality of work based on a set of criteria. Creating: Compile information together in a different way by combining elements in a new pattern or proposing alternative solutions 16 20 Total 80 100
1. No chapter wise Care to be taken to cover all the chapters
2. Suitable internal variations may be made for generating various templates keeping the overall weightage to different form of questions and typology of questions same.
##### Choice(s):

There will be no overall choice in the question paper. However, 33% internal choices will be given in all the sections

 INTERNAL ASSESSMENT                                                                                       20 MARKS Periodic Tests ( Best 2 out of 3 test conducted)     10 Marks Mathematics Activities                                                  10 Marks

Note: For activities NCERT Lab Manual may be referred.

## Conduct of Periodic Tests:

Periodic Test is a Pen and Paper assessment which is to be conducted by the respective subject teacher. The format of periodic test must have questions items with a balance mix, such as, very short answer (VSA), short answer (SA) and long answer (LA) to effectively assess the knowledge, understanding, application, skills, analysis, evaluation and synthesis. Depending on the nature of subject, the subject teacher will have the liberty of incorporating any other types of questions too. The modalities of the PT are as follows:

a) Mode: The periodic test is to be taken in the form of pen-paper test.

b) Schedule: In the entire Academic Year, three Periodic Tests in each subject may be conducted as follows:

This is only a suggestive schedule and schools may conduct periodic tests as per their convenience. The winter bound schools would develop their own schedule with similar time gaps between two consecutive tests.

c) Average of Marks: Once schools complete the conduct of all the three periodic tests, they will convert the weightage of each of the three tests into ten marks each for identifying best two tests. The best two will be taken into consideration and the average of the two shall be taken as the final marks for PT.

d) The school will ensure simple documentation to keep a record of performance as suggested in detail circular no.Acad-05/2017.

e) Sharing of Feedback/Performance: The students’ achievement in each test must be shared with the students and their parents to give them an overview of the level of learning that has taken place during different periods. Feedback will help parents formulate interventions (conducive ambience, support materials, motivation and morale-boosting) to further enhance learning. A teacher, while sharing the feedback with student or parent, should be empathetic, non- judgmental and motivating. It is recommended that the teacher share best examples/performances of IA with the class to motivate all learners.

## Assessment of Activity Work:

Throughout the year any 10 activities shall be performed by the student from the activities given in the NCERT Laboratory Manual for the respective class (XI or XII) which is available on the link: http://www.ncert.nic.in/exemplar/labmanuals.htmla record of the same may be kept by the student. An year end test on the activity may be conducted

The weightage are as under:

• The activities performed by the student throughout the year and record keeping : 5 marks
• Assessment of the activity performed during the year end test: 3 marks
• Viva-voce: 2 marks

## Prescribed Books:

1. Mathematics Textbook for Class XI, NCERT Publications
2. Mathematics Part I – Textbook for Class XII, NCERT Publication
3. Mathematics Part II – Textbook for Class XII, NCERT Publication