CBSE Class 12 Applied Mathematics Syllabus 2021-22

Grade XII (2021-22)

Number of Paper: 1
Total number of Periods: 240 (35 Minutes Each)
Time: 3 Hours
Max Marks: 80
No. Units No. of Marks
Periods
I Numbers, Quantification and Numerical 30 09
Applications
II Algebra 20 10
III Calculus 50 15
IV Probability Distributions 35 10
V Inferential Statistics 10 05
VI Index Numbers and Time-based data 30 10
VII Financial Mathematics 50 15
VIII Linear Programming 15 06
Total 240 80
Internal Assessment 20

CLASS XII
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
1.3 Simple  Define arithmetic function Properties and Examples of:
Arithmetic  Enlist different arithmetic i) Euler totient function
Functions functions ii) Number of divisor function
 Apply the arithmetic functions iii) Divisor sum function
on given number iv) Mobius function

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
Scheduling ● Define scheduling ● Definition and meaning
● Differentiate between FCFS & ● Use of Gantt chart
SJF Simple problems based on
● Solve problems based on FCFS FCFS (First come First serve)

and SJF and SJF (shortest job first)
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
 Explain elementary row b) elementary row operations
operations and use to it find the  If A and B are invertible square
inverse of a matrix matrices of same size,
 Apply properties of inverse of i) (AB)-1=B -1A –1
matrices 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 upto
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,  Leontiff Input–output model
determinants physics, coding, encryption etc. that represents the
including  Apply real life applications interdependencies between
Leontiff input particularly for Leontiff 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  Determine the rate of change of  To find the rate of change of
Derivatives various quantities quantities such as area and
 Understand the gradient of volume with respect to time or
tangent and normal to a curve its dimension
at a given point  Gradient / Slope of tangent
 Write the equation of tangents and normal to the curve
and normal to a curve at a given  The equation of the tangent
point and normal to the curve
(simple problems only)
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
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
3.11 Formulating ● Formulate differential equation ● Formation of differential
and Solving ● Verify the solution of differential equation by eliminating
Differential equation arbitrary constants
Equations ● Solve simple differential ● Solution of simple differential
equation equations (direct integration
only)
3.12 ● Define Growth and Decay
Application of ● 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- 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 = 𝜎 where
x = value of the random variable
𝜇 = mean
𝜎 = S.D.
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  Examples of drawing more
from a population than one sample set from the
 Differentiate between a same population
representative and non-  Examples of representative
representative sample and non-representative sample
 Draw a representative sample  Unbiased and biased sampling
using simple random sampling  Problems based on random
 Draw a representative sample sampling using simple random
using and systematic random sampling and systematic
sampling 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
6.1 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 unit test and time reversal ● Unit test
adequacy of test ● Time reversal test
Index numbers
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
7.6 Stock, Shares  Explain the concept of stock,  Meaning of Stock, shares and
and shares and debentures debentures
Debentures  Enlist features related to equity  Types of Shares and
shares and debentures Debentures
 Interpret case studies related to  Features and advantages of
shares and debentures (Simple equity shares and debentures
Case studies only)  Real life examples of shares &
debentures
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 ● Iso-cost/ Iso-profit Method
problems in graphically
two variables
8.5 Feasible and ● Identify feasible, infeasible, ● Definition and Examples to
Infeasible bounded and unbounded explain the terms
Regions 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

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

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

Assessment Plan
1. Overall Assessment of the course is out of 100 marks.
2. The assessment plan consists of an External Exam and Internal Assessment.
3. External Exam will be of 03 hours duration Pen/ Paper Test consisting of 80 marks.
4. The weightage of the Internal Assessment is 20 marks. Internal Assessment can be
a combination of activities spread throughout the 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 a
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:

Sl. Area and Assessment Area Marks
No. Weightage allocated
1 Project work Project work and record 5
(10 marks) Year-end Presentation/ Viva of the Project 5
2 Practical work Performance of practical and record 5
(10 marks) Year-end test of any one practical 5
Total 20