CBSE Class 12 Term Wise Syllabus 2021-22 Applied Maths

Applied Mathematics (XI-XII) Term-wise
(Code-241)
Session- 2021-22

Secondary School Education prepares students to explore future career options after
graduating from schools. Mathematics is an important subject that helps students to
choose various fields of their choices. Mathematics is widely used in higher studies as an
allied subject in the field of Economics, Commerce, Social Sciences and many others. It
has been observed that the syllabus of Mathematics in senior secondary grades meant
for Science subjects may not be appropriate for the students who wish to pursue
Commerce or Social Science-based subjects in university education. By keeping this in
mind, one more elective course in the Mathematics syllabus is developed for Senior
Secondary classes with an aim to provide students relevant experience in Mathematics
that can be used in fields other than Physical Sciences.
This course is designed to develop substantial mathematical skills and methods needed
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 counterexamples. 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 a coordinate system. The course
content will help students to develop a sound understanding of descriptive and inferential
statistics which they can use to describe and analyze a given 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 applications 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 of the data by organizing, representing, interpreting, analyzing, and
making meaningful inferences from real-world situations.
d) To develop logical reasoning skills and apply the same in simple problem-solving.
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 (2021-22)

Term 1

One Paper
Time: 90 minutes Max. Marks: 40

No. Units Marks
I Numbers, Quantification and Numerical 09
Applications
II Algebra 09
III Mathematical Reasoning 06
IV Calculus 04
VI Descriptive Statistics 12
Total 40

CLASS- XI
Term - 1
Sl. Contents Learning Outcomes: Notes / Explanation
No. Students will be able to
UNIT – 1 NUMBERS, QUANTIFICATION AND NUMERICAL APPLICATIONS
Numbers & Quantification
1.2 Binary Numbers ● Express decimal ● Definition of number system
numbers in binary (decimal and binary)
system ● Conversion from decimal to binary
● Express binary numbers system and vice - versa
in decimal system
1.4 Indices, ● Relate indices and ● Applications of rules of indices
Logarithm and logarithm /antilogarithm ● Introduction of logarithm and
Antilogarithm ● Find logarithm and antilogarithm
antilogarithms of given ● Common and Natural logarithm
number
1.5 Laws and ● Enlist the laws and ● Fundamental laws of logarithm
properties of properties of logarithms
logarithms ● Apply laws of logarithm
1.6 Simple ● Use logarithm in ● Express the problem in the form of
applications of different applications an equation and apply logarithm/
logarithm and antilogarithm
antilogarithm
Numerical Applications
1.7 Averages ● Determine average for a ● Definition and meaning
given data ● Problems on average, weighted
average
1.8 Clock ● Evaluate the angular ● Number of rotations of minute
value of a minute hand / hour hand of a clock in a
● Calculate the angle day
formed between two ● Number of times minute hand and
hands of clock at given hour hand coincides in a day
time
● Calculate the time for
which hands of clock
meet
1.9 Calendar ● Determine Odd days in ● Definition of odd days
a month/ year/ century ● Odd days in a year/ century.
● Decode the day for the ● Day corresponding to a given date
given date
1.10 Time, Work and ● Establish the ● Basic concept of time and work
Distance relationship between ● Problems on time taken / distance
work and time covered / work done
● Compare the work done
by the individual / group
w.r.t. time
● Calculate the time taken/
distance covered/ Work
done from the given
data
1.11 Mensuration ● Solve problems based ● Comparison between 2D and 3D
on surface area and shapes
● Combination of solids

volume of 2D and 3D ● Transforming one solid shape to
shapes another
● Calculate the volume/
surface area for solid
formed using two or
more shapes
1.12 Seating ● Create suitable seating ● Linear and circular seating
arrangement plan/ draft as per given arrangement
conditions ● Position of a person in a seating
(Linear/circular) arrangement
● Locate the position of a
person in a seating
arrangement
UNIT – 2 ALGEBRA
Sets
2.1 Introduction to ● Define set as well-defined ● Definition of a Set
sets – definition collection of objects ● Examples and Non-examples of
Set
2.2 Representation ● Represent a set in Roster ● Write elements of a set in Set
of sets form and Set builder form Builder form and Roster Form
● Convert a set given in Roster form
into Set builder form and vice-
versa
2.3 Types of sets  Identify different types of  Types of Sets: Finite Set, Infinite
and their sets on the basis of Set, Empty Set, Singleton Set
notations number of elements in
the set
 Differentiate between
equal set and
equivalence set
2.4 Subsets  Enlist all subsets of a set  Subset of a given set
 Find number of subsets  Familiarity with terms like
of a given set Superset, Improper subset,
 Find number of elements Universal set, Power set
of a power set
2.5 Intervals  Express subset of real  Open interval, closed interval,
numbers as intervals semi open interval and semi
closed interval
2.6 Venn diagrams  Apply the concept of  Venn diagrams as the pictorial
Venn diagram to representation of relationship
understand the between sets
relationship between  Practical Problems based on Venn
sets Diagrams
 Solve problems using
Venn diagram
2.7 Operations on  Perform operations on  Operations on sets include
sets sets to solve practical i) Union of sets
problems ii) Intersection of sets
iii) Difference of sets
iv) Complement of a set
v) De Morgan’s Laws

Relations
2.8 Ordered pairs  Explain the significance  Ordered pair, order of elements in
of specific arrangement an ordered pair and equality of
Cartesian of elements in a pair ordered pairs
product of two  Write Cartesian product  Cartesian product of two non-
sets of two sets empty sets
 Find the number of
elements in a Cartesian
product of two sets
2.9 Relations  Express relation as a  Definition of Relation, examples
subset of Cartesian pertaining to relations in the real
product number system
 Find domain and range
of a relation
2.10 Types of  Define and illustrate  Types of relations:
relations different types of Empty relation, universal relation,
relations: Empty relation reflexive relation, symmetric
and universal relation relation, transitive relation,
 Examine whether the equivalence relation
relation is equivalence or  Introducing a function as a special
not type of relation
 Define function as a  Examples and non-examples of
special type of relation functions
 Categorize relations that
are functions and non-
functions
Sequences and Series
2.11 Sequence and ● Differentiate between ● Sequence:𝑎1 , 𝑎2 , 𝑎3, … , 𝑎𝑛
Series sequence and series ● Series: 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛

2.12 Arithmetic ● Identify Arithmetic ● General term of AP:
Progression Progression (AP) 𝑡 𝑛 = 𝑎 + (𝑛 − 1)𝑑
● Establish the formulae of ● Sum of n terms of AP :
𝑛
finding 𝑛𝑡ℎ term and sum Sn = 2 [2𝑎 + (𝑛 − 1)𝑑]
of n terms
● Solve application 𝑎+𝑏
AM of 𝑎 𝑎𝑛𝑑 𝑏= 2
problems based on AP
● Find arithmetic mean
(AM) of two positive
numbers

2.13 Geometric  Identify Geometric ● General term of GP:
Progression Progression (GP) 𝑡𝑛 = 𝑎𝑟 𝑛−1
● Sum of n terms of a GP:
● Derive the 𝑛𝑡ℎ term and 𝑎(𝑟 𝑛 −1)
Sn = 𝑟−1
sum of n terms of a
given GP
 Sum of infinite term of GP =
𝑎
● Solve problems based 1−𝑟
, where −1 < 𝑟 < 1
on applications of GP
● Geometric mean of a and b = √𝑎𝑏
● Find geometric mean
(GM) of two positive
numbers
● Solve problems based
on relation between AM
and GM
2.14 Applications of ● Apply appropriate Applications based on
AP and GP formulas of AP and GP ● Economy Stimulation
to solve application ● The Virus spread etc.
problems
UNIT -3 MATHEMATICAL REASONING
3.1 Mathematical ● Identify mathematically ● Meaning of mathematical
reasoning acceptable statements statements
● Express the implications ● Negation
of the compound ● Compound statements
statement ● Quantifiers
● Validate mathematical ● Converse and Contrapositive of
statements the statement
● Implications
● Validating statements
3.2 Logical ● Solve logical problems ● Odd man out
reasoning involving odd man out, ● Syllogism
syllogism, blood relation ● Blood relations
and coding decoding ● Coding Decoding
UNIT – 4 CALCULUS
4.1 Functions  Identify dependent and ● Dependent variable and
independent variables independent variable
 Define a function using ● Function as a rule or law that
dependent and defines a relationship between
independent variable one variable (the independent
variable) and another variable
(the dependent variable)
4.2 Domain and  Define domain, range ● Domain as a set of all values of
Range of a and co-domain of a independent variable
function given function ● Co-domain as a set of all values
of dependent variable
● Range of a function as set of all
possible resulting values of
dependent variable
4.3 Types of  Define various types of  Following types of functions with
functions functions definitions and characteristics

 Identify domain, co- Constant function, Identity
domain and range of the function, Polynomial function,
function Rational function, Logarithm
function, Exponential function,
Modulus function, Greatest
integer function, Signum function,
Algebraic function
4.4 Graphical ● Representation of ● Graph of some polynomial
representation function graphically functions, Logarithm function,
of functions Exponential Function, Modulus
function, Greatest integer function,
Signum function
UNIT- 6 DESCRIPTIVE STATISTICS
6.1 Types of data  Identify real life situations  Examples of raw data from
for collecting data different surveys, sports
 Categorize data based  Multi-variate data from not more
on nature of data than three variables
(Primary and Secondary  Collection of data up to three
Data, Raw and variables from real life examples,
Organized Data) such as, data of students (age,
 Identify and differentiate weight, height)
univariate, bivariate and
multi-variate data
 Identify and differentiate
discrete data and
continuous data
 Collect raw data from
practical examples
6.2 Data on various ● Describe nominal, ordinal, ● Examples and non-examples of
scales interval and ratio scale of data on different scales
data collection ● Benefit and limitations of collecting
● Collect and classify data data on various scales
on different scales of
measurement
6.3 Data ● Organize raw data in ● Data organization in
representation discrete and continuous increasing/decreasing order, using
and data form frequency table and in class
visualization ● Represent data on intervals of various length
nominal and ordinal ● Graphical representation of data
scales of measurement using pie-chart/bar
using pie chart and bar graphs/histogram using class
graphs interval of equal and unequal length
● Represent data on ● Visualization of data using Excel
interval and ratio scale Spreadsheet or any other computer
using histogram and assisted tool
frequency polygon
● Represent bivariate
continuous data using line
graph
● Choose appropriate graph
to represent data of
various kinds

6.4 Data Interpretation
Measure of ● Understand meaning of ● Mean deviation around mean and
Dispersion dispersion in a data set median
● Differentiate between ● Standard deviation and variance
range, quartile deviation, ● Examples of different kinds of data
mean deviation and helping students to choose and
standard deviation compare different measures of
● Calculate range, quartile dispersion
deviation, mean deviation
and standard deviation for
ungrouped and grouped
data set
● Choose appropriate
measure of dispersion to
calculate spread of data
Skewness and ● Define Skewness and ● Examples of symmetrical and
Kurtosis Kurtosis using graphical asymmetrical data
representation of a data ● Visualization of graphical
set representation of data using Excel
● Interpret Skewness and Spreadsheet or any other computer
Kurtosis of a frequency assisted tool
distribution by plotting the
graph
● Calculate coefficient of
Skewness and interpret
the results
6.5 Percentile rank ● Define Percentile rank ● Emphasis on visualizing, analysing
and Quartile and Quartile rank and interpreting percentile and
rank ● Calculate and interpret quartile rank scores
Percentile and Quartile
rank of scores in a given
data set
6.6 Correlation ● Define correlation in ● Emphasis on application, analysis
values of two data sets and interpreting the results of
● Calculate Product coefficient of correlation using
moment correlation for practical examples
ungrouped and grouped
data
● Calculate Karl Pearson’s
coefficient of correlation
● Calculate Spearman’s
rank correlation
● Interpret the coefficient of
correlation

Internal Assessment:
The weightage of internal assessment may be as under:

Term Area and Assessment Area Marks
Weightage allocated
Term 1 Project Project work and record 5
Term-end Presentation + Viva of the Project 5
Total 10

Note: The list of suggested projects is given in the end

Assessment Plan (First term)

1. First term assessment of the course is out of 50 marks.
2. The assessment plan consists of an External Exam and Internal Assessment.
3. There will be end term external exam of 40 marks.
4. The weightage of the Internal Assessment is 10 marks. It consists of a project work.
Teachers may choose a project from the suggested list

Grade XI
Term 2

One Paper
Time: 2 Hours Max. Marks: 40

No. Units Marks
II Algebra (Continued) 06
IV Calculus (Continued) 06
V Probability 08
VII Basics of Financial Mathematics 15
VIII Coordinate Geometry 05
Total 40

CLASS- XI
Term - 2
Sl. Contents Learning Outcomes: Notes / Explanation
No. Students will be able to
Permutations and Combinations
2.15 Factorial ● Define factorial of a ● Definition of factorial:
number n! = n(n-1)(n-2)…3.2.1
● Calculate factorial of a Usage of factorial in counting
number principles
2.16 Fundamental ● Appreciate how to count ● Fundamental Principle of Addition
Principle of without counting ● Fundamental Principle of
Counting Multiplication
2.17 Permutations ● Define permutation ● Permutation as arrangement of
objects in a definite order taken
● Apply the concept of some or all at a time
permutation to solve ● Theorems under different conditions
simple problems 𝑛!
resulting in nPr=(𝑛−𝑟)! or 𝑛𝑟 or
𝑛!
𝑛1 !𝑛2 !…𝑛𝑘 !
arrangements

2.20 Combinations ● Define combination -The number of combinations of
n different objects taken r at a
● Differentiate between 𝑛!
time is given by nCr= 𝑟!.(𝑛−𝑟)!
permutation and
combination Some results on combinations:
n
● C0 = 1 = nCn
● Apply the formula of ●
n
Ca = nCb ⇒ a=b or a+ b=n
combination to solve the ●
n
Cr = nCn-r
related problems ●
n
Cr + nCr-1 = n+1Cr
UNIT – 4 CALCULUS
4.5 Concepts of ● Define limit of a function ● Left hand limit, Right hand limit,
limits and ● Solve problems based Limit of a function, Continuity of a
continuity of a on the algebra of limits function
function ● Define continuity of a
function
4.6 Instantaneous ● Define instantaneous ∆𝑦 𝑓(𝑥+∆𝑥)−𝑓(𝑥)
● The ratio ∆𝑥 = ∆𝑥
as
rate of change rate of change
instantaneous rate of change,
where ∆𝑦 is change in 𝑦 and ∆𝑥 is
change in 𝑥 at any instant
4.7 Differentiation ● Find the derivative of ● Derivatives of functions (non-
as a process of the functions trigonometric only)
finding
derivative
4.8 Derivatives of ● Find the derivative of ● If 𝑦 = 𝑓(𝑢) where 𝑢 = 𝑔(𝑥) then
algebraic function of a function differential coefficient of 𝑦 w.r.t x is
functions using 𝑑𝑦 𝑑𝑦 𝑑𝑢
= .
Chain Rule 𝑑𝑥 𝑑𝑢 𝑑𝑥

UNIT – 5 PROBABILITY
5.1 Introduction ● Appreciate the use of ● Probability as quantitative
probability in daily life measure of uncertainty
situations ● Use of probability in determining
the insurance premium, weather
forecasts etc.

5.2 Random ● Define random ● Sample space as set of all
experiment and experiment and sample possible outcomes
sample space space with suitable
examples
5.3 Event  Define an event  Types of Event:
 Recognize and Impossible and sure event,
differentiate different Independent and dependent
types of events and find event, mutually exclusive and
their probabilities exhaustive event
5.4 Conditional  Define the concept of  Conditional Probability of event E
Probability conditional probability given that F has occurred is:
 Apply reasoning skills to 𝑃(𝐸|𝐹) =
𝑃(𝐸∩𝐹)
, 𝑃(𝐹) ≠ 0
solve problems based 𝑃(𝐹)

on conditional
probability
5.5 Total Probability  Interpret mathematical  Total Probability:
information and identify Let 𝐸1, 𝐸2 , …,𝐸𝑛 be a partition of the
situations when to apply sample space S, then probability of
total probability an event A associated with S is:
 Solve problems based 𝑃(𝐴) = ∑𝑛𝑗=1 𝑃(𝐸𝑗 )𝑃(𝐴|𝐸𝐽 )
on application of total
probability
5.6 Bayes’  State Bayes’ theorem ● Bayes’ Theorem:
Theorem  Solve practical problems If 𝐸1 , 𝐸2 , … , 𝐸𝑛 be n non empty
based on Bayes’ events which constitute a partition
Theorem of a sample space 𝑆 and 𝐴 be any
event with non zero probability,
then:
𝑃(𝐸𝑖 )𝑃(𝐴|𝐸𝑖 )
𝑃(𝐸𝑖 |𝐴) = ∑𝑛
𝑗=1 𝑃(𝐸𝑗 )𝑃(𝐴|𝐸𝑗 )
UNIT – 7 FINANCIAL MATHEMATICS
7.1 Interest and ● Define the concept of ● Impact of high interest rates and
Interest Rates Interest Rates low interest rates on the business
● Compare the difference
between Nominal
Interest Rate, Effective
Rate and Real Interest
Rate
● Solve Practical
applications of interest
rate
7.2 Accumulation ● Interpret the concept of ● Meaning and significance of
with simple and simple and compound simple and compound interest
compound interest ● Compound interest rates
interest ● Calculate Simple applications on various financial
Interest and Compound products
Interest
7.3 Simple and ● Explain the meaning, ● Concept of Equivalency
compound nature and concept of ● Annual Equivalency Rate
interest rates equivalency
with ● Analyze various
equivalency examples for

understanding annual
equivalency rate
7.4 Effective rate of ● Define with examples ● Effective Annual Interest Rate
interest the concept of effective = (1 + i/n)n – 1
rate of interest where:
i = Nominal Interest Rate
n = No. of Periods
7.5 Present value,  Interpret the concept of  Formula for Present Value:
net present compounding and PV = CF/(1 + r)n
value and future discounting along with Where:
value practical applications CF = Cash Flow in Future Period
 Compute net present r = Periodic Rate of return or Interest
value (also called the discount rate or the
 Apply net present value required rate of return)
in capital budgeting n = no. of periods
decisions  Use of PVAF, FVAF tables for
practical purposes
 Solve problems based on
Application of net present value
7.6 Annuities, ● Explain the concept of ● Definition, Formulae and
Calculating Immediate Annuity, Examples
value of Annuity due and
Regular Annuity Deferred Annuity
● Calculate General
Annuity
7.7 Simple ● Calculate the future ● Examples of regular annuity:
applications of value of regular annuity, Mortgage Payment, Car Loan
regular annuity due Payments, Leases, Rent Payment,
annuities (upto ● Apply the concept of Insurance payouts etc.
3 period) Annuity in real life
situations
7.8 Tax, calculation  Explain fundamentals of  Computation of income tax
of tax, simple taxation Add Income from
applications of  Differentiate between Salary, house property,
tax calculation Direct and indirect tax business or profession, capital
in Goods and  Define and explain GST gain, other sources, etc.
service tax, Less deductions
Income Tax
 Calculate GST
PF, PPF, LIC, Housing loan, FD,
 Explain rules under NSC etc.
State Goods and
Services Tax (SGST)  Assess the Individuals under
Central Goods and Income Tax Act
Services Tax (CGST)  Formula for GST
and Union Territory Different Tax heads under GST
Goods and Services Tax
(UTGST)
7.9 Bills, tariff rates,  Describe the meaning of  Tariff rates- its basis of
fixed charge, bills and its various types determination
surcharge,  Analyze the meaning  Concept of fixed charge service
service charge and rules determining charge and their applications in
tariff rates various sectors of Indian economy
 Explain the concept of
fixed charge

7.10 Calculation and  To interpret and analyze  Components of electricity bill/water
interpretation of
electricity bills, water supply and other supply bills:
electricity bill,
bills and other supply i) overcharging of electricity
water supply bill
bills ii) water supply bills
and other  Evaluate how to iii) units consumed in electricity
supply bills calculate units bills
consumed under
electricity bills/water bill
UNIT – 8 COORDINATE GEOMETRY
8.1 Straight line ● Find the slope and ● Gradient of a line
equation of line in ● Equation of line:
various form Parallel to axes, point-slope form,
● Find angle between the two-points form, slope intercept
two lines form, intercept form
● Find the perpendicular ● Application of the straight line in
from a given point on a demand curve related to
line economics problems
● Find the distance
between two parallel
lines
8.2 Circle ● Define a circle ● Circle as a locus of a point in a
● Find different form of plane
equations of a circle ● Equation of a circle in standard
● Solve problems based form, central form, diameter form
on applications of circle and general form

8.3 Parabola ● Define parabola and ● Parabola as a locus of a point in a
related terms plane.
● Define eccentricity of a ● Equation of a parabola in standard
parabola form:
● Derive the equation of ● Focus, Directrix, Axis, Latus
parabola rectum, Eccentricity

Internal Assessment:
The weightage of internal assessment may be as under:

Term Area and Assessment Area Marks
Weightage allocated
Term 2 Practical Performance of practical and record 5
Term-end test of any one practical + Viva 5
Total 10

Practical: Use of spreadsheet
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 spreadsheet

1. Plot the graph of functions on excel study the nature of function at various points,
drawing lines of tangents
2. Create a budget of income and spending
3. Create and compare sheet of price & features to buy a product
4. Prepare the best option plan to buy a product by comparing cost, shipping charges,
tax and other hidden costs
5. Smart purchasing during sale season
6. Prepare a report card using scores of the last four exams and compare the
performance
7. Collect the data on weather, price, inflation, and pollution. Sketch different types of
graphs and analyze the results

Assessment Plan (Second term)

1. Second term assessment of the course is out of 50 marks.
2. The assessment plan consists of an External Exam and Internal Assessment.
3. There will be external exam of 40 marks.
4. The weightage of the Internal Assessment is 10 marks. It consists of practical work.
Teachers can choose activities from the suggested list of practical or they can plan
activities of a similar nature. For data-based practical, teachers are encouraged to use
data from local sources to make it more relevant for students.

Grade XII (2021-22)

Term 1

One Paper
Time: 90 minutes Max. Marks: 40

No. Units Marks
I Numbers, Quantification and Numerical 09
Applications
II Algebra 10
III Calculus 06
IV Probability Distribution 10
VI Index Numbers and Time-based Series 05
Total 40

CLASS XII
Term 1
Sl. No. Contents Learning Outcomes: Notes / Explanation
Students will be able to
UNIT-1 NUMBERS, QUANTIFICATION AND NUMERICAL APPLICATIONS
1.1 Modulo  Define modulus of an integer  Definition and meaning
Arithmetic  Apply arithmetic operations  Introduction to modulo operator
using modular arithmetic rules  Modular addition and
subtraction
1.2 Congruence ● Define congruence modulo ● Definition and meaning
Modulo ● Apply the definition in various ● Solution using congruence
problems modulo
● Equivalence class
1.4 Alligation and ● Understand the rule of alligation ● Meaning and Application of
Mixture to produce a mixture at a given rule of alligation
price ● Mean price of a mixture
● Determine the mean price of a
mixture
● Apply rule of alligation
1.5 Numerical Solve real life problems mathematically
Problems
Boats and ● Distinguish between upstream ● Problems based on speed of
Streams and downstream stream and the speed of boat
(upstream and ● Express the problem in the form in still water
downstream) of an equation
Pipes and ● Determine the time taken by ● Calculation of the portion of
Cisterns two or more pipes to fill or the tank filled or drained by
empty the tank the pipe(s) in unit time

Races and ● Compare the performance of ● Calculation of the time
Games two players w.r.t. time, taken/ distance covered /
distance speed of each player
Partnership ● Differentiate between active ● Definition, Profit division
partner and sleeping partner among the partners
● Determine the gain or loss to
be divided among the
partners in the ratio of their
investment with due
consideration of the time
1.6 Numerical ● Describe the basic concepts of ● Comparison between two
Inequalities numerical inequalities statements/situations which
● Understand and write numerical can be compared numerically
inequalities ● Application of the techniques of
numerical solution of algebraic
inequations
UNIT-2 ALGEBRA
2.1 Matrices and ● Define matrix ● The entries, rows and columns
types of ● Identify different kinds of of matrices
matrices matrices ● Present a set of data in a
● Find the size / order of matrices matrix form
2.2 Equality of  Determine equality of two  Examples of transpose of
matrices, matrices matrix
Transpose of a  Write transpose of given matrix  A square matrix as a sum of
matrix,  Define symmetric and skew symmetric and skew symmetric
Symmetric and symmetric matrix matrix
Skew  Observe that diagonal
symmetric elements of skew symmetric
matrix matrices are always zero
2.3 Algebra of ● Perform operations like addition ● Addition and Subtraction of
Matrices & subtraction on matrices of matrices
same order ● Multiplication of matrices (It
● Perform multiplication of two can be shown to the students
matrices of appropriate order that Matrix multiplication is
● Perform multiplication of a similar to multiplication of two
scalar with matrix polynomials)
● Multiplication of a matrix with a
real number
2.4 Determinants ● Find determinant of a square ● Singular matrix, Non singular
matrix matrix
● Use elementary properties of ● |AB|=|A||B|
determinants ● Simple problems to find
determinant value
2.5 Inverse of a  Define the inverse of a square  Inverse of a matrix using:
matrix matrix a) cofactors
 Apply properties of inverse of  If A and B are invertible square
matrices matrices of same size,
i) (AB)-1=B -1A –1
ii) (A-1)-1 =A
iii) (AT)-1 = (A-1)T

2.6 Solving system  Solve the system of  Solution of system of
of simultaneous equations using simultaneous equations up to
simultaneous i) Cramer’s Rule three variables only
equations ii) Inverse of coefficient matrix (non- homogeneous
using matrix iii) Row reduction method equations)
method,  Formulate real life problems
Cramer’s rule into a system of simultaneous
and row linear equations and solve it
reduction using these methods
method
2.7 Simple  Apply simple applications of  Real life applications of
applications of matrices and determinants in Matrices and Determinant
matrices and different areas of mathematics,  Leontief Input–output model
determinants physics, coding, encryption etc. that represents the
including  Apply real life applications interdependencies between
Leontief input particularly for Leontief input different sectors of a national
output model output model for two variables economy or different regional
for two in economics economies
variables
UNIT- 3 CALCULUS
Differentiation and its Applications
3.1 Higher Order  Determine second and higher  Simple problems based on
Derivatives order derivatives higher order derivatives
 Understand differentiation of  Differentiation of parametric
parametric functions and functions
implicit functions and implicit functions (upto
2nd order)
3.2 Application of  Understand the gradient of  Gradient / Slope of tangent
Derivatives tangent and normal to a curve and normal to the curve
at a given point  The equation of the tangent
 Write the equation of tangents and normal to the curve
and normal to a curve at a given (simple problems only)
point
3.3 Marginal Cost  Define marginal cost and  Examples related to marginal
and Marginal marginal revenue cost, marginal revenue, etc.
Revenue using  Find marginal cost and marginal
derivatives revenue
3.4 Increasing  Determine whether a function is  Simple problems related to
/Decreasing increasing or decreasing increasing and decreasing
Functions  Determine the conditions for a behaviour of a function in the
function to be increasing or given interval
decreasing
3.5 Maxima and  Determine critical points of the  A point x= c is called the critical
Minima function point of f if
 Find the point(s) of local f is defined at c and
maxima and local minima and f ′ (c) =
corresponding local maximum 0 or f is not differentiable
and local minimum values at c
 Find the absolute maximum and  To find local maxima and local
absolute minimum value of a minima by:
function i) First Derivative Test

 Solve applied problems ii) Second Derivative Test
 Contextualized real life
problems
UNIT- 4 PROBABILITY DISTRIBUTIONS
4.1 Probability ● Understand the concept of  Definition and example of
Distribution Random Variables and its discrete and continuous
Probability Distributions random variable and their
● Find probability distribution of distribution
discrete random variable
4.2 Mathematical ● Apply arithmetic mean of  The expected value of discrete
Expectation frequency distribution to find random variable as summation
the expected value of a random of product of discrete random
variable variable by the probability of its
occurrence.
4.3 Variance ● Calculate the Variance and S.D.  Questions based on variance
of a random variable and standard deviation
4.4 Binomial ● Identify the Bernoulli Trials and  Characteristics of the binomial
Distribution apply Binomial Distribution distribution
● Evaluate Mean, Variance and  Binomial formula:
S.D of a binomial distribution P(r) = nCr pr qn-r
Where n = number of trials
P = probability of
success
q = probability of
failure
Mean =np
Variance = npq
Standard Deviation = √𝑛𝑝𝑞
4.5 Poison ● Understand the Conditions of  Characteristics of Poisson
Distribution Poisson Distribution Probability distribution
● Evaluate the Mean and Poisson formula:
Variance of Poisson distribution 𝜆𝑥 . 𝑒 −𝜆
P(x) = 𝑥!
 Mean = Variance = 𝜆

4.6 Normal ● Understand normal distribution  Characteristics of a normal
Distribution is a Continuous distribution probability distribution
● Evaluate value of Standard  Total area under the curve =
normal variate total probability = 1
● Area relationship between
 Standard Normal Variate:
Mean and Standard Deviation
𝑥− 𝜇
Z= where
𝜎
x = value of the random variable
𝜇 = mean
𝜎 = S.D.
UNIT – 6 INDEX NUMBERS AND TIME BASED DATA
6.1 Index ● Define Index numbers as a ● Meaning and Definition
Numbers special type of average ● Utility of Index Numbers
6.2 Construction of ● Construct different type of index ● Simple Index numbers
Index numbers numbers ● Weighted index numbers

6.3 Test of  Apply time reversal test ● Time reversal test
adequacy of
Index numbers

Internal Assessment:
The weightage of internal assessment may be as under:

Term Area and Assessment Area Marks
Weightage allocated
Term 1 Project Project work and record 5
Term-end Presentation + Viva of the Project 5
Total 10

Note: The list of suggested projects is given in the end

Assessment Plan (First term)

1. First term assessment of the course is out of 50 marks.
2. The assessment plan consists of an External Exam and Internal Assessment.
3. There will be end term external exam of 40 marks.
4. The weightage of the Internal Assessment is 10 marks. It consists of a project
work. Teachers may choose a project from the suggested list

Grade XII
Term 2

One Paper
Time: 2 Hours Max. Marks: 40

No. Units Marks
III Calculus (Continued) 09
V Inferential Statistics 05
VI Index Numbers and Time series 05
(Continued)
VII Financial Mathematics 15
VIII Linear Programming 06
Total 40

CLASS XII
Term 2
Sl. No. Contents Learning Outcomes: Notes / Explanation
Students will be able to
UNIT- 3 CALCULUS
Integration and its Applications
3.5 Integration  Understand and determine  Integration as a reverse
indefinite integrals of simple process of differentiation
functions as anti-derivative  Vocabulary and Notations
related to Integration
3.6 Indefinite  Evaluate indefinite integrals of  Simple integrals based on
Integrals as simple algebraic functions by each method (non-
family of method of: trigonometric function)
curves i) substitution
ii) partial fraction
iii) by parts
3.7 Definite ● Define definite integral as area ● Evaluation of definite integrals
Integrals as under the curve using properties
area under the ● Understand fundamental
curve theorem of Integral calculus and
apply it to evaluate the definite
integral
● Apply properties of definite
integrals to solve the problems
3.9 Application of ● Identify the region representing Problems based on finding
Integration C.S. and P.S. graphically ● Total cost when Marginal Cost
● Apply the definite integral to find is given
consumer surplus-producer ● Total Revenue when Marginal
surplus Revenue is given
● Equilibrium price and
equilibrium quantity and hence
consumer and producer
surplus
Differential Equations and Modeling
3.10 Differential ● Recognize a differential ● Definition, order, degree and
Equations equation examples
● Find the order and degree of a
differential equation
3.12 Application of ● Define Growth and Decay ● Growth and Decay Model in
Differential Model Biological sciences,
Equations ● Apply the differential equations Economics and business, etc.
to solve Growth and Decay
Models
UNIT - 5 INFERENTIAL STATISTICS
5.1 Population and  Define Population and Sample  Population data from census,
Sample  Differentiate between economic surveys and other
population and sample contexts from practical life
 Define a representative sample
from a population

 Differentiate between a  Examples of drawing more
representative and non- than one sample set from the
representative sample same population
 Draw a representative sample  Examples of representative
using simple random sampling and non-representative sample
 Draw a representative sample  Unbiased and biased sampling
using and systematic random  Problems based on random
sampling sampling using simple random
sampling and systematic
random sampling (sample size
less than 100)
5.2 Parameter and  Define Parameter with  Conceptual understanding of
Statistics and reference to Population Parameter and Statistics
Statistical  Define Statistics with reference  Examples of Parameter and
Interferences to Sample Statistic limited to Mean and
 Explain the relation between Standard deviation only
Parameter and Statistic  Examples to highlight
 Explain the limitation of Statistic limitations of generalizing
to generalize the estimation for results from sample to
population population
 Interpret the concept of  Only conceptual understanding
Statistical Significance and of Statistical
Statistical Inferences Significance/Statistical
 State Central Limit Theorem Inferences
 Explain the relation between  Only conceptual understanding
Population-Sampling of Sampling Distribution
Distribution-Sample through simulation and graphs

5.3 t-Test (one ● Define a hypothesis ● Examples and non-examples
sample t-test ● Differentiate between Null and of Null and Alternate
and two Alternate hypothesis hypothesis (only non-
independent ● Define and calculate degree of directional alternate
groups t-test) freedom hypothesis)
● Test Null hypothesis and make ● Framing of Null and Alternate
inferences using t-test statistic hypothesis
for one group / two independent ● Testing a Null Hypothesis to
groups make Statistical Inferences for
small sample size
● (for small sample size: t- test
for one group and two
independent groups
● Use of t-table
UNIT – 6 INDEX NUMBERS AND TIME-BASED DATA
6.4 Time Series ● Identify time series as ● Meaning and Definition
chronological data
6.5 Components of ● Distinguish between different ● Secular trend
Time Series components of time series ● Seasonal variation
● Cyclical variation
● Irregular variation
6.6 Time Series ● Solve practical problems based ● Fitting a straight line trend and
analysis for on statistical data and Interpret estimating the value
univariate data the result

6.7 Secular Trend ● Understand the long term ● The tendency of the variable to
tendency increase or decrease over a
long period of time
6.8 Methods of ● Demonstrate the techniques of ● Moving Average method
Measuring finding trend by different ● Method of Least Squares
trend methods
UNIT - 7 FINANCIAL MATHEMATICS
7.1 Perpetuity,  Explain the concept of  Meaning of Perpetuity and
Sinking Funds perpetuity and sinking fund Sinking Fund
 Calculate perpetuity  Real life examples of sinking
 Differentiate between sinking fund
fund and saving account  Advantages of Sinking Fund
 Sinking Fund vs. Savings
account
7.2 Valuation of  Define the concept of valuation  Meaning of Bond Valuation
Bonds of bond and related terms  Terms related to valuation of
 Calculate value of bond using bond:
present value approach Coupon rate, Maturity rate and
Current price
 Bond Valuation Methods:
i) Present Value Approach
ii) Relative Price Approach
7.3 Calculation of  Explain the concept of EMI  Methods to calculate EMI:
EMI  Calculate EMI using various i) Flat-Rate Method
methods ii) Reducing-Balance Method
 Real life examples to calculate
EMI of various types of loans,
purchase of assets, etc.
7.4 Calculation of  Explain the concept of rate of  Formula for calculation of Rate
Returns, return and nominal rate of of Return, Nominal Rate of
Nominal Rate return Return
of Return  Calculate rate of return and
nominal rate of return
7.5 Compound  Understand the concept of  Meaning and use of
Annual Growth Compound Annual Growth Rate Compound Annual Growth
Rate  Differentiate between Rate
Compound Annual Growth Rate  Formula for Compound Annual
and Annual Growth Rate Growth Rate
 Calculate Compound Annual
Growth Rate
7.7 Linear method  Define the concept of linear  Meaning and formula for Linear
of Depreciation method of Depreciation Method of Depreciation
 Interpret cost, residual value  Advantages and
and useful life of an asset from disadvantages of Linear
the given information Method
 Calculate depreciation
UNIT - 8 LINEAR PROGRAMMING
8.1 Introduction ● Familiarize with terms related to ● Need for framing linear
and related Linear Programming Problem programming problem
terminology ● Definition of Decision Variable,
Constraints, Objective function,

Optimization and Non Negative
conditions
8.2 Mathematical ● Formulate Linear Programming ● Set the problem in terms of
formulation of Problem decision variables, identify the
Linear objective function, identify the
Programming set of problem constraints,
Problem express the problem in terms
of inequations
8.3 Different types ● Identify and formulate different ● Formulate various types of
of Linear types of LPP LPP’s like Manufacturing
Programming Problem, Diet Problem,
Problems Transportation Problem, etc.
8.4 Graphical ● Draw the Graph for a system of ● Corner Point Method for the
method of linear inequalities involving two Optimal solution of LPP
solution for variables and to find its solution
problems in graphically
two variables
8.5 Feasible and ● Identify feasible, infeasible and ● Definition and Examples to
Infeasible bounded regions explain the terms
Regions
8.6 Feasible and ● Understand feasible and ● Problems based on
infeasible infeasible solutions optimization
solutions, ● Find optimal feasible solution ● Examples of finding the
optimal solutions by graphical method
feasible
solution

Internal Assessment:
The weightage of internal assessment may be as under:

Term Area and Assessment Area Marks
Weightage allocated
Term 2 Practical Performance of practical and record 5
Term-end test of any one practical + Viva 5
Total 10

Practical: Use of spreadsheet

Graphs of an 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

i) Plot the graphs of functions on excel and study the graph to find out the point of
maxima/minima

ii) Probability and dice roll simulation
iii) Matrix multiplication and the inverse of a matrix
iv) Stock Market data sheet on excel
v) Collect the data on weather, price, inflation, and pollution analyze the data and make
meaningful inferences
vi) Collect data from newspapers on traffic, sports activities and market trends and use
excel to study future trends

Assessment Plan (Second term)

1. Second term assessment of the course is out of 50 marks.
2. The assessment plan consists of an External Exam and Internal Assessment.
3. There will be external exam of 40 marks.
4. The weightage of the Internal Assessment is 10 marks. It consists of practical work.
Teachers can choose activities from the suggested list of practical or they can plan
activities of a similar nature. For data-based practical, teachers are encouraged to use
data from local sources to make it more relevant for students.

List of Suggested projects (Class XI /XII)

i) Use of prime numbers in coding and decoding of messages
ii) Prime numbers and divisibility rules
iii) Logarithms for financial calculations such as interest, present value, future value,
profit/loss etc. with large values)
iv) The cardinality of a set and orders of infinity
v) Comparing sets of Natural numbers, rational numbers, real numbers and others
vi) Use of Venn diagram in solving practical problems
vii) Fibonacci sequence: Its' history and presence in nature
viii) Testing the validity of mathematical statements and framing truth tables
ix) Investigating Graphs of functions for their properties
x) Visit the census site of India
http://www.censusindia.gov.in/Census_Data_2001/Census_Data_Online/Languag
e/State ment3.htm Depict the information given there in a pictorial form
xi) Prepare a questionnaire to collect information about money spent by your friends in
a month on activities like travelling, movies, recharging of the mobiles, etc. and draw
interesting conclusions
xii) 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

xiii) Analysis of population migration data – positive and negative influence on
urbanization
xiv) Each day newspaper tells us about the maximum temperature, minimum
temperature, and 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
xv) Analysis of 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
xvi) Vehicle registration data – correlating with pollution and the number of accidents
xvii) Visit a village near Delhi and collect data of various crops over the 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
xviii) 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 the amount of
rainfall for the current year
xix) Weather prediction (prediction of monsoon from past data)
xx) Visit Kirana shops near your home and collect the data regarding the sales 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
xxi) Stock price movement
xxii) Risk assessments by insurance firms from data
xxiii) Predicting stock market crash
xxiv) Predicting the outcome of an election – exit polls
xxv) Predicting mortality of infants