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F.1002/CBSE-Acad/Maths/Circular/20 1st April,2020 Circular No Acad-22/ 2020 All Heads of Institutions affiliated to CBSE

Subject: Introduction of ‘Applied Mathematics (241)’ as an Academic Elective at Senior Secondary Level

As part of Systemic Reforms, National Curriculum Framework 2005 recommends that Subjects such as Mathematics and English could be examined at two levels; standard and higher level. In the long term, all subjects could be offered at two levels with students doing at least three/two of the six at standard level and the remaining three/ four at higher level. As per this recommendation, Board has already offered two levels of examinations in Mathematics at secondary level.

As Secondary School Education prepares students to explore future career options after graduating from the school, Mathematics is an important subject which helps students to choose various fields of their choices. Mathematics is widely used in higher studies in the field of Economics, Commerce, Social Sciences and many others. It has been observed that the existing syllabus of Mathematics aligns well with Science subjects, but does not align well with Commerce or Social Science-based subjects in university education. By keeping this in mind, one more elective course in Mathematics syllabus will be offered for Sr. Secondary classes with an aim to provide students relevant experience in Mathematics which can be used in the fields other than Physical Sciences.

A course by this name was earlier designed as a skill subject. The academic course that has been designed now by the same name has several value additions to develop substantial mathematical skills and methods needed in other subject areas. Topics covered in two years shall aim to enable students to use mathematical knowledge in the field of business, economics and social sciences. It aims to promote appreciation of mathematical power and simplicity for its countless applications in diverse field. It is expected that the new subject of Applied Mathematics is taught by connecting concepts to the application in various fields, thereby enabling students to develop 21st century competencies such as critical thinking, problem solving, logical reasoning, mathematical thinking, etc.

Objectives:

a) To develop an understanding of basic mathematical and statistical tools and their applications in the field of commerce (business/finance/economics) and social sciences;

b) To model real world experiences/problems into mathematical expressions using numerical/algebraic/graphical representation;

c) To make sense from the data by organizing, representing, interpreting, analysing, and to make meaningful inferences from the real-world situations;

d) To develop logical reasoning skills and apply the same in problem solving;

e) To reinforce mathematical communication by formulating conjectures, validating logical arguments and testing hypothesis;

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f) To make connections between Mathematics and other disciplines.

Therefore, as per the recommendations of the Curriculum Committee of the Board, and duly approved by the Governing body, CBSE is introducing a new Academic Elective at Senior. Secondary level named as ‘Applied Mathematics (Code No. 241)’ starting from the current Academic Year, that is from 2020-21 session for Class XI. The details of this scheme are as under:

The course can be opted for the class XI students from this Academic year (2020-21) as an Academic elective.

Students who want to opt for higher studies in Mathematics at the University level as an Elective or want to take admission in Mathematics Honours course or Engineering course may take Mathematics (041). The Applied Mathematics (241) course is designed to enhance the knowledge and skills of Mathematics that are required to be successful in different fields of their future career. Therefore this course may be selected by students keeping this aspect in mind.

Students, Parents and Schools may take a conscious choice in selection of this subject by taking into consideration the scope of this subject in getting admission to different courses at University level.

Approach of Applied Mathematics subject will be practical in nature and the students are expected to learn through practical applications of Mathematics in different disciplines.

In view of introduction of Applied Mathematics as an Academic elective, the students who have passed Basic Mathematics (241) in class X are now allowed to offer the new academic elective Applied Mathematics (241) at Senior Secondary Level.

Accordingly, the students who have passed Basic Mathematics (241) as well as Standard Mathematics (041) in Class X of CBSE exam are eligible for this course. However, it is once again clarified that students who have passed Mathematics-Basic (241) at Secondary level are not eligible to opt for Mathematics (041) at Sr. Secondary Level.

The affiliated schools who have permission for running Mathematics course in their school at Sr. Secondary Level shall be automatically eligible to run this course with the availability of necessary infrastructure and human resources.

The detailed modalities for opting this subject shall be made available at the time of registration for this course at class XI.

It may be noted that those students who have earlier offered Applied Mathematics as a Skill Elective will offer Applied Mathematics as an Academic Elective in the current year 2020-21 for class XII. They have to follow the syllabus of Applied Mathematics offered as an Academic subject. Applied Mathematics Course will not be available from the current Academic Session 2020-21 as Skill subject.

Frequently Asked Questions (FAQs)on introduction of Applied Mathematics at Senior Secondary level are enclosed for ready reference. Support material, Design of question papers

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and Sample Question Papers for this subject will be made available on www.cbseacademic.nic.in in due course of time.

(Dr. Joseph Emmanuel)

Director (Academics) Encl : Frequently Asked Questionson introduction of Applied Mathematics

Copy to the respective Heads of Directorates, Organizations and Institutions as indicated below

with a request to disseminate the information to all the schools under their jurisdiction:

1. The Commissioner, Kendriya Vidyalaya Sangathan, 18-Institutional Area, Shaheed Jeet Singh Marg, New Delhi-16

2. The Commissioner, Navodaya Vidyalaya Samiti, B-15, Sector-62, Institutional Area, Noida-201309

3. Additional Chief Secretary/Principal Secretary/Secretary School Education, Governments of Delhi/Sikkim/Chandigarh/Arunachal Pradesh/Andaman And Nicobar Islands/Odisha/Chhattisgarh/Tripura

4. The Director of Education, Directorate of Education, Govt. of NCT of Delhi, Old Secretariat, Delhi-110 054

5. The Director of Public Instructions (Schools), Union Territory Secretariat, Sector 9, Chandigarh-160 017

6. The Director of Education, Govt. of Sikkim, Gangtok, Sikkim –737101 7. The Director of School Education, Govt. of Arunachal Pradesh, Itanagar –791 111 8. The Director of Education, Govt. of A&N Islands, Port Blair – 744101 9. The Director of Education, S.I.E., CBSE Cell, VIP Road, Junglee Ghat, P.O. 744103,

A&N Island 10. The Director, Central Tibetan School Administration, ESSESS Plaza, Community

Centre, Sector 3, Rohini 11. The Additional Director General of Army Education, A – Wing, Sena Bhawan, DHQ, PO,

New Delhi-110001 12. The Secretary AWES, Integrated Headquarters of MoD (Army), FDRC Building No.

202, Shankar Vihar (Near APS), Delhi Cantt-110010 13. The Under Secretary (EE-1), MHRD, Govt. of India, Department of SE&L, Shastri

Bhawan, New Delhi-01 14. All Regional Directors/Regional Officers of CBSE with the request to send this circular

to all the Heads of the affiliated schools of the Board in their respective regions 15. All Joint Secretaries/ Deputy Secretaries/ Assistant Secretaries/ Analysts (Academic,

Training and Skill Unit), CBSE 16. In charge IT Unit with the request to put this circular on the CBSE Academic website 17. The Head Public Relations Unit, CBSE 18. PPS to Chairperson, CBSE 19. SPS to Secretary, CBSE 20. SPS to Director (EDUSAT, Research and Development), CBSE 21. SPS to Director (Information Technology), CBSE 22. SPS to Director (Academics), CBSE 23. SPS to Controller of Examinations, CBSE 24. SPS to Director (SE & T), CBSE 25. SPS to Director (Professional Exams), CBSE

1

Syllabus

Course Title: Applied Mathematics

(Code-241)

Grade XI-XII

2

Applied Mathematics

Grade XI-XII

Secondary School Education prepares students to explore future career options after graduating from the school. Mathematics is an important subject helps students to choose various fields of their choices. Mathematics is widely used in higher studies in the field of Economics, Commerce, Social Sciences and many other. It has been observed that the syllabus of Mathematics meant for Science subjects may not be appropriate for the students pursuing Commerce or Social Science-based subjects in university education. By keeping this in mind, one more elective course in Mathematics syllabus is developed for Sr. Secondary classes with an aim to provide students relevant experience in Mathematics which can be used in the fields other than Physical Sciences.

This course is designed to develop substantial mathematical skills and methods needing in other subject areas. Topics covered in two years aim to enable students to use mathematical knowledge in the field of business, economic and social sciences. It aims to promote appreciation of mathematical power and simplicity for its countless applications in diverse fields. The course continues to develop mathematical language and symbolism to communicate and relate everyday experiences mathematically. In addition, it reinforces the logical reasoning skills of formulating and validating mathematical arguments, framing examples, finding counter examples. It encourages students to engage in mathematical investigations and to build connections within mathematical topics and with other disciplines. The course prepares students to use algebraic methods as a means of representation and as a problem-solving tool. It also enables students to interpret two dimensional geometrical figures using algebra and to further deduce properties of geometrical figures in coordinate system. The course content will help students to develop sound understanding of descriptive and inferential statistics which they can use to describe and analysis a give set of data and to further make meaningful inferences out of it. Data based case studies from the field of business, economics, psychology, education, biology and census data will be used to appreciate the power of data in contemporary society.

It is expected that the subject is taught connecting concepts to the application in various fields. The objectives of the course areas are as follows:

Objectives:

a) To develop an understanding of basic mathematical and statistical tools and their applications in the field of commerce (business/ finance/economics) and social sciences;

b) To model real world experiences/problems into mathematical expressions using numerical/algebraic/graphical representation;

c) To make sense from the data by organizing, representing, interpreting, analysing, and to make meaningful inferences from the real-world situations;

d) To develop logical reasoning skills and apply the same in simple problem solving;

3

e) To reinforce mathematical communication by formulating conjectures, validating logical arguments and testing hypothesis;

f) To make connections between Mathematics and other disciplines.

Grade XI

One Paper Total Period–240 (35 Minutes Each)

Three Hours Max Marks: 80 No.

Units No. of

Periods Marks

I. Numbers, Quantification and Numerical Applications

20 09

II. Algebra 35 10

III. Mathematical Reasoning 15 06

IV. Calculus 30 10

V. Probability 30 10

VI. Descriptive Statistics 35 12

VII Basics of Financial Mathematics

55 18

VIII Coordinate Geometry 20 05

Total 240 80

Internal Assessment 20

Unit I Numbers, Quantification and Numerical Applications

Prime Numbers, Encryptions using Prime Numbers Binary Numbers Complex Numbers (Preliminary idea only) Indices, Logarithm and Antilogarithm Laws and properties of logarithms Simple applications of logarithm and antilogarithm Numerical problems on averages, calendar, clock, time, work and distance,

mensuration, seating arrangement

Unit II Algebra

Sets Types of sets Venn diagram De Morgan's laws Problem solving using Venn diagram

4

Relations and types of relations Introduction of Sequences, Series Arithmetic and Geometric progression Relationship between AM and GM Basic concepts of Permutations and Combinations Permutations, Circular Permutations, Permutations with restrictions Combinations with standard results

Unit III Mathematical and Logical Reasoning

Mathematically acceptable statements Connecting words/ phrases in Mathematical statement consolidating the

understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and Mathematics

Problems based on logical reasoning (coding-decoding, odd man out, blood relation, syllogism etc)

Unit IV Calculus

Introducing functions Domain and Range of a function Types of functions (Polynomial function; Rational function; Composite

function; Logarithm function; Exponential function; Modulus function; Greatest Integer function, Signum function)

Graphical representation of functions Concept of limits and continuity of a function Instantaneous rates of change Differentiation as a process of finding derivative Derivatives of algebraic functions using Chain rule Tangent line and equations of tangents

Unit V Probability

Random experiment, sample space, events, mutually exclusive events Independent and Dependent Events Law of Total Probability Bayes’ Theorem

Unit VI Descriptive Statistics

Types of data (raw data, univariate data, bivariate and multi-variate data) Data on various scales (nominal, ordinal, interval and ratio scale) Data representation and visualization Data interpretation (central tendency, dispersion, deviation, variance,

skewness and kurtosis) Percentile rank and quartile rank Correlation (Pearson and Spearman method of correlation) Applications of descriptive statistics using real time data

5

Unit VII Basics of Financial Mathematics

Interest and interest rate Accumulation with simple and compound interest Simple and compound interest rates with equivalency Effective rate of interest Present value, net present value and future value Annuities, calculating value of regular annuity Simple applications of regular annuities (up to 3 period) Tax, calculation of tax and simple applications of tax calculation in Goods and

service tax, Income Tax Bills, tariff rates, fixed charge, surcharge, service charge Calculation and interpretation of electricity bill, water supply bill and other

supply bills

(Comparing interest rates on various types of savings; calculating income tax; electricity bills, water bill; service surcharge using realistic data)

Unit VIII Coordinate Geometry

Straight Line Circles Parabola

(only standard forms and graphical representation on two-dimensional plane)

Practical: Use of spread sheet

Calculating average, interest (simple and compound), creating pictographs, drawing pie chart, bar graphs, calculating central tendency; visualizing graphs (straight line, circles and parabola using real time data)

Suggested practical using spread sheet

1. Plot the graph of functions on excel; study the nature of function at various points, drawing lines of tangents;

2. Create budget of income and spending; 3. Create compare sheet of price, features to buy a product; 4. Prepare best option plan to buy a product by comparing cost, shipping

charges, tax and other hidden cost; 5. Smart purchasing during sale season; 6. Prepare a report card using scores of last four exams and compare the

performance; 7. Collect the data on weather, price, inflation, and pollution. Sketch different

types of graphs.

6

Grade XII

One Paper Total Period–240 (35 Minutes Each)

Three Hours Max Marks: 80

No. Units No. of Periods

Marks

I. Numbers, Quantification and Numerical Applications

20 06

II. Algebra 20 10

III. Inferential Statistics 10 06

IV. Index Numbers and Time-based data

30 10

V. Calculus 60 15

VI. Financial Mathematics 40 15

VII Linear Programming 25 08

VIII Probability 35 10

Total 240 80

Internal Assessment 20

Unit I Numbers, Quantification and Numerical Applications

Modulo Arithmetic Congruence modulo Simple arithmetic functions Allegation or Mixture Numerical problems on boats and streams; partnership; pipes and cistern;

races and games, scheduling Numerical inequalities

Unit II Algebra

Solution of simultaneous linear equations using elimination method (up to 3 variables)

Matrices and types of matrices Algebra of matrices Determinants Inverse of a matrix Cramer’s rule and its application

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Simple applications of matrices and determinants including Leontiff input output model for two variables

Unit III Calculus

Application of derivatives Increasing/Decreasing functions Maxima and Minima Marginal cost and marginal revenue using derivatives Integration Indefinite integral as family of curves Definite integral as area under the curve Integration of simple algebraic functions (primitive, by substitution, by parts) Application of Integration (consumer surplus-producer surplus) Differential equation (definition, order, degree) Formulating and solving linear differential equation Application of differential equation (Growth and Decay Model)

Unit IV Probability

Probability Distribution Mathematical Expectation Variance Binomial Distribution Poisson distribution Normal distribution Basic applications and inferences

Unit V Inferential Statistics

Population and sample Parameter, statistic and statistical inferences t-Test (one sample t-test and two independent groups t-test)

Unit VI Index numbers and Time-based data

Index numbers, uses of index numbers Construction of index numbers (simple and weighted) Tests of adequacy of index numbers (unit test and time reversal test) Time series, Time series analysis for univariant data sets Trend analysis by moving average method Trend analysis by fitting of linear trend line using least squares

Unit VII Financial Mathematics

Perpetuity, Sinking funds Valuation of Bonds (Present value approach and Relative price approach) Calculation of EMI

8

Calculation of returns, nominal rate of return, effective rate of interest Compound annual growth rate Stock, shares and debentures Linear method of depreciation

Unit VIII Linear Programming

Introduction and related terminologies (constraints, objective function, optimization)

Mathematical formulation of linear programming problems Different types of linear programming problems (Transportation and

assignment problem) 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)

Practical: Use of spread sheet

Graphs of exponential function, demand and supply functions on Excel and study the nature of function at various points, maxima/minima

Matrix operations using Excel

Suggested practical using the spreadsheet

1. Plot the graphs of functions on excel and study the graph to find out point of maxima/minima;

2. Probability and dice roll simulation; 3. Matrix multiplication and inverse of a matrix; 4. Stock Market data sheet on excel; 5. Collect the data on weather, price, inflation, and pollution; analyze the data

and make meaningful inferences; 6. Collect data from newspapers on traffic, sports activities and on market trends

and use excel to study future trends.

List of Suggested projects ( class XI /XII)

Use of prime numbers in coding and decoding of messages;

Prime numbers and divisbility rules;

Logrithms for financial calculations such as interest, present value, future vale, profit/loss etc with large values);

9

Cardinality of a set and orders of infinity;

Comparing sets of Natural numbers, rational numbers, real numbers and others;

Use of Venn Diagram in solving practical problems;

Fibonacci Sequence: Its’ history and presence in nature;

Testing the validity of mathematical statements and framing truth tables;

Investigating graphs of functions for their properties;

Visit the census site of India http://www.censusindia.gov.in/Census_Data_2001/Census_Data_Online/Language/State ment3.htm Depict the information given there in a pictorial form;

Prepare a questionnaire to collect information about money spent by your friends in a month on activities like traveling, movies, recharging of the mobiles, etc. and draw interesting conclusions;

Check out the local newspaper and cut out examples of information depicted by graphs. Draw your own conclusions from the graph and compare it with the analysis given in the report;

Analysis of population migration data – positive and negative influence on urbanization;

Each day newspaper tells us about the maximum temperature, minimum temperature, humidity. Collect the data for a period of 30 days and represent it graphically. Compare it with the data available for the same time period for the previous year;

Analysis career graph of a cricketer (batting average for a batsman and bowling average for a bowler). Conclude the best year of his career. It may be extended for other players also – tennis, badminton, athlete;

Vehicle registration data – correlating with pollution and number of accidents;

Visit a village near Delhi and collect data of various crops over past few years from the farmers. Also collect data about temperature variation and rain over the period for a particular crop. Try to find the effect of temperature and rain variations on various crops;

Choose any week of your ongoing semester. Collect data for the past 10 – 15 years for the amount of rainfall received in Delhi during that week. Predict amount of rainfall for the current year;

Weather prediction (prediction of monsoon from past data);

10

Visit Kirana shops near your home and collect the data of sale of certain commodities over a month. Try to figure out the stock of a particular commodity which should be in the store in order to maximize the profit;

Stock price movement ;

Risk assessments by insurance firms from data;

Predicting stock market crash;

Predicting outcome of election – exit polls;

Predicting mortality of infants.

Assessment Plan

1. Overall Assessment of the course is out of 100 marks. 2. Assessment plan consists of External Exam and Internal Assessment. 3. External Exam will be of 03 hours duration Paper/Pencil Test consisting of

80 marks. 4. Weightage of Internal Assessment is of 20 marks. Internal Assessment can

be a combination of activities spread throughout semester/ academic year. Internal Assessment activities include, projects and excel based practical. Teachers can choose activities from the suggested list of practical or they can plan activities of similar nature. For data based practical, teachers are encouraged to use data from local sources to make it more relevant for students.

5. Weightage for each area of internal assessment may be as under:

Sr.No. Area and weightage Assessment Area Marks allocated

1 Project work (10 marks)

Project work and record 5 Year End Presentation/Viva of the Project

5

2 Practical work (10 marks)

Performance of practical and record 5 Yearend test of any one practical 5

Total 20

MATHEMATICS (XI-XII)

(Code No. 041)

Session – 2020-21

The Syllabus in the subject of Mathematics has undergone changes from time to time in

accordance with growth of the subject and emerging needs of the society. Senior Secondary

stage is a launching stage from where the students go either for higher academic education in

Mathematics or for professional courses like Engineering, Physical and Biological science,

Commerce or Computer Applications. The present revised syllabus has been designed in

accordance with National Curriculum Framework 2005 and as per guidelines given in Focus

Group on Teaching of Mathematics 2005 which is to meet the emerging needs of all categories

of students. Motivating the topics from real life situations and other subject areas, greater

emphasis has been laid on application of various concepts.

Objectives

The broad objectives of teaching Mathematics at senior school stage intend to help the students:

to acquire knowledge and critical understanding, particularly by way of motivation and

visualization, of basic concepts, terms, principles, symbols and mastery of underlying

processes and skills.

to feel the flow of reasons while proving a result or solving a problem.

to apply the knowledge and skills acquired to solve problems and wherever possible,

by more than one method.

to develop positive attitude to think, analyze and articulate logically.

to develop interest in the subject by participating in related competitions.

to acquaint students with different aspects of Mathematics used in daily life.

to develop an interest in students to study Mathematics as a discipline.

to develop awareness of the need for national integration, protection of environment,

observance of small family norms, removal of social barriers, elimination of gender

biases.

to develop reverence and respect towards great Mathematicians for their contributions

to the field of Mathematics.

COURSE STRUCTURE

CLASS XI (2020-21)

One Paper Total Period–240 [35 Minutes Each]

Three Hours Max Marks: 80

No. Units No. of Periods Marks

I. Sets and Functions 60 23

II. Algebra 70 30

III. Coordinate Geometry 40 10

IV. Calculus 30 05

V. Mathematical Reasoning 10 02

VI. Statistics and Probability 30 10

Total 240 80

Internal Assessment 20

*No chapter/unit-wise weightage. Care to be taken to cover all the chapters.

Unit-I: Sets and Functions

1. Sets (20) Periods

Sets and their representations.Empty set.Finite and Infinite sets.Equal sets.Subsets.Subsets of a

set of real numbers especially intervals (with notations). Power set. Universal set. Venn

diagrams. Union and Intersection of sets.Difference of sets. Complement of a set. Properties of

Complement.

2. Relations & Functions (20) Periods

Ordered pairs.Cartesian product of sets.Number of elements in the Cartesian product of two finite

sets.Cartesian product of the set of reals with itself (upto R x R x R).Definition of relation, pictorial

diagrams, domain, co-domain and range of a relation. Function as a special type of relation.

Pictorial representation of a function, domain, co-domain and range of a function. Real valued

functions, domain and range of these functions, constant, identity, polynomial, rational, modulus,

signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum,

difference, product and quotients of functions.

3. Trigonometric Functions (20) Periods

Positive and negative angles.Measuring angles in radians and in degrees and conversion from

one measure to another.Definition of trigonometric functions with the help of unit circle. Truth of

the identity sin2x + cos2x = 1, for all x. Signs of trigonometric functions. Domain and range of

trigonometric functions and their graphs.Expressing sin (x±y) and cos (x±y) in terms of sinx, siny,

cosx&cosy and their simple applications. Deducing identities like the following:

tan(x ± y) =tan x ± tan y

1 ∓ tan x tan y , cot(x ± y) =

cot x cot y ∓ 1

cot y ± cot x

sinα ± sinβ = 2sin1

2(α ± β)cos

1

2(α ∓ β)

cosα + cosβ = 2cos1

2(α + β)cos

1

2(α − β)

𝑐𝑜𝑠𝛼 − 𝑐𝑜𝑠𝛽 = −2𝑠𝑖𝑛1

2(𝛼 + 𝛽)𝑠𝑖𝑛

1

2(𝛼 − 𝛽)

Identities related to sin2x, cos2x, tan2 x, sin3x, cos3x and tan3x. General solution of trigonometric

equations of the type siny = sina, cosy = cosa and tany = tana.

Unit-II: Algebra

1. Principle of Mathematical Induction (10) Periods

Process of the proof by induction, motivating the application of the method by looking at natural

numbers as the least inductive subset of real numbers.The principle of mathematical induction

and simple applications.

2. Complex Numbers and Quadratic Equations (15) Periods

Need for complex numbers, especially√−1, to be motivated by inability to solve some of the

quardratic equations. Algebraic properties of complex numbers.Argand plane and polar

representation of complex numbers.Statement of Fundamental Theorem of Algebra, solution of

quadratic equations (with real coefficients) in the complex number system.Square root of a

complex number.

3. Linear Inequalities (15) Periods

Linear inequalities.Algebraic solutions of linear inequalities in one variable and their

representation on the number line.Graphical solution of linear inequalities in two

variables.Graphical method of finding a solution of system of linear inequalities in two variables.

4. Permutations and Combinations (10) Periods

Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation

ofFormulae forn and n and their connections, simple applications.

5. Binomial Theorem (10) Periods

Historical perspective, statement and proof of the binomial theorem for positive integral

indices.Pascal's triangle, General and middle term in binomial expansion, simple applications.

6. Sequence and Series (10) Periods

Sequence and Series. Arithmetic Progression (A. P.). Arithmetic Mean (A.M.) Geometric

Progression (G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum,

geometric mean (G.M.), relation between A.M. and G.M. Formulae for the following special

sums.

k , k and k

Unit-III: Coordinate Geometry

1. Straight Lines (10) Periods

Brief recall of two dimensional geometry from earlier classes. Shifting of origin. Slope of a line

and angle between two lines. Various forms of equations of a line: parallel to axis, point -slope

form, slope-intercept form, two-point form, intercept form and normal form. General equation of a

line.Equation of family of lines passing through the point of intersection of two lines.Distance of a

point from a line.

2. Conic Sections (20) Periods

Sections of a cone: circles, ellipse, parabola, hyperbola, a point, a straight line and a pair of

intersecting lines as a degenerated case of a conic section. Standard equations and simple

properties of parabola, ellipse and hyperbola.Standard equation of a circle.

3. Introduction to Three-dimensional Geometry (10) Periods

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance

between two points and section formula.

Unit-IV: Calculus

1. Limits and Derivatives (30) Periods

Derivative introduced as rate of change both as that of distance function and

geometrically.Intuitive idea of limit.Limits of polynomials and rational functions trigonometric,

exponential and logarithmic functions.Definition of derivative relate it to scope of tangent of the

curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial

and trigonometric functions.

Unit-V: Mathematical Reasoning

1. Mathematical Reasoning (10) Periods

Mathematically acceptable statements. Connecting words/ phrases - consolidating the

understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied

by", "and", "or", "there exists" and their use through variety of examples related to real life and

Mathema tics. Validating the statements involving the connecting words, difference among

contradiction, converse and contrapositive.

Unit-VI: Statistics and Probability

1. Statistics (15) Periods

Measures of Dispersion: Range, Mean deviation, variance and standard deviation of

ungrouped/grouped data. Analysis of frequency distributions with equal means but different

variances.

2. Probability (15) Periods

Random experiments; outcomes, sample spaces (set representation). Events; occurrence of

events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Axiomatic (set

theoretic) probability, connections with other theories of earlier classes. Probability of an event,

probability of ‘not’, ‘and’ and ‘or’ events.

MATHEMATICS

QUESTION PAPER DESIGN

CLASS – XI (2020-21)

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 weightage. 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

Note: Please refer the guidelines given under XII Mathematics Syllabus:

INTERNAL ASSESSMENT 20 MARKS

Periodic Tests ( Best 2 out of 3 tests conducted) 10 Marks

Mathematics Activities 10 Marks

CLASS-XII (2020-21)

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 andonto

functions, composite functions, inverse of a function.

2. Inverse Trigonometric Functions 15 Periods

Definition, range, domain, principal value branch. Graphs of inversetrigonometric

functionsElementary properties of inverse trigonometricfunctions.

Unit-II: Algebra

1. 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.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).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 25 Periods

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. 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

1. Continuity and Differentiability 20 Periods 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. Rolle’s and Lagrange's Mean Value

Theorems (without proof) and their geometric interpretation.

2. Applications of Derivatives 10 Periods

Applications of derivatives: rate of change of bodies, 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).

3. 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.

dx

x ± a ,

dx

x ± a,

dx

√a − x,

dx

ax + bx + c,

dx

ax

px + q

ax + bx + cdx,

px + q

√ax bx + cdx, a ± x dx, x − a dx

𝑎𝑥 + 𝑏𝑥 + 𝑐𝑑𝑥, (𝑝𝑥 + 𝑞) 𝑎𝑥 + 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.

4. Applications of the Integrals 15 Periods

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).

5. Differential Equations 15 Periods

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

1. 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, scalar triple product of vectors.

2. Three - dimensional Geometry 15 Periods

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 (i) two lines, (ii) two planes, (iii) a line and a plane.Distance of a

point from a plane.

Unit-V: Linear Programming

1. Linear Programming 20 Periods

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 or unbounded),

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

Unit-VI: Probability

1. Probability 30 Periods

Conditional probability, multiplication theorem on probability, independent events, total probability,

Bayes’ theorem, Random variable and its probability distribution, mean and variance of random

variable. Binomial probability distribution.

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

4) Mathematics Exemplar Problem for Class XI, Published by NCERT

5) Mathematics Exemplar Problem for Class XII, Published by NCERT

6) Mathematics Lab Manual class XI, published by NCERT

7) Mathematics Lab Manual class XII, published by NCERT

MATHEMATICS (Code No. - 041)

QUESTION PAPER DESIGN CLASS - XII

(2020 - 21) 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 weightage. 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

Note: For activities NCERT Lab Manual may be referred

INTERNAL ASSESSMENT 20 MARKS

Periodic Tests ( Best 2 out of 3 tests conducted) 10 Marks

Mathematics Activities 10 Marks

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-papertest.

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

conducted as follows:

Test Pre Mid-term (PT-I) Mid-Term (PT-II) Post Mid-Term (PT-III)

Tentative Month July-August November December-January

This is only a suggestive schedule and schools may conduct periodic tests as pertheir

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 levelof

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:

Through out 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.Anyear end test on the activity may be

conducted

The weightage are as under:

The activities performed by the student through out the year and record keeping

: 5 marks

Assessment of the activity performed during the year end test: 3 marks

Viva-voce : 2 marks