CBSE Class 12 Applied Mathematics Syllabus for academic session 2026-27
This page contains the CBSE Class 12 Applied Mathematics syllabus for the academic session 2026-27, as prescribed by CBSE curriculum.
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:
Course Objectives:
•
To develop an understanding of essential mathematical and statistical concepts that are relevant to areas such as business, economic and social sciences.
•
To enable students to interpret real-life situations into structured numerical, algebraic and graphical representations for analysis and decision making.
•
To develop ability to organise, analyse and interpret data, and to draw meaningful conclusions in practical contexts.
•
To strengthen logical thinking and reasoning by engaging students in problem-solving situations that require nuance understanding of qualification and relative change.
•
To develop clarity in mathematical communication, including the ability to justify solutions, examine assumptions and validate results.
•
To help students recognise connections between mathematics and other disciplines, and to use these connections meaningfully.
Number of Paper:
1
Time:
3 Hours
Max Marks:
80
No.
Units
Marks
I
Numbers, Quantification and 10 Numerical Applications
11
II
Algebra
10
III
Calculus
15
IV
Combinatorics Distributions
10
V
Inferential Statistics
05
VI
Time-based data
06
VII
Financial Mathematics
15
VIII
Linear Programming
08
Total
80
Internal Assessment
20
Syllabus – Learning Outcomes
UNIT – 1 NUMBERS, QUANTIFICATION AND NUMERICAL APPLICATIONS
Numbers & Quantification
1.1
Modulo Arithmetic
•
Define modulus of an integer
•
Apply arithmetic operations using modular arithmetic rules
1.2
Congruence Modulo
•
Define congruence modulo
•
Apply the definition in various problems
1.3
Alligation and Mixture
•
Understand the rule of alligation to produce a mixture at a given price
•
Determine the mean price of a mixture
•
Apply rule of allegation
1.4
Boats and Streams (upstream and downstream) (Numerical Problems Solve real life problems mathematically)
•
Distinguish between upstream and downstream
•
Express the problem in the form of an equation
Pipes and Cisterns (Numerical Problems Solve real life problems mathematically)
•
Determine the time taken by two or more pipes to fill or empty the tank
Races and Games (Numerical Problems Solve real life problems mathematically)
•
Compare the performance of two players w.r.t. time, distance
1.5
Numerical Inequalities
•
Describe the basic concepts of numerical inequalities
•
Understand and write numerical inequalities
UNIT-2 ALGEBRA
2.1
Matrices and types of matrices
•
Define matrix
•
Identify different kinds of matrices. Find the size / order of matrices
2.2
Equality of matrices, Transpose of a matrix, Symmetric and Skew symmetric matrix
•
Determine equality of two matrices
•
Write transpose of given matrix
•
Define symmetric and skew symmetric matrix
2.3
Algebra of Matrices
•
Perform operations like addition & subtraction on matrices of same order
•
Perform multiplication of two matrices of appropriate order
•
Perform multiplication of a scalar with matrix
2.4
Determinants
•
Find determinant of a square matrix
2.5
Inverse of a matrix
•
Define the inverse of a square matrix
•
Apply properties of inverse of matrices
2.6
Solving a system of simultaneous equations using matrix method and Cramer’s rule
•
Solve the system of simultaneous equations using
i)
Cramer’s Rule
ii)
Inverse of coefficient matrix
•
Formulate real life problems into a system of simultaneous linear equations and solve it using these methods
UNIT – 3 CALCULUS
Differentiation and its Applications
3.1
Derivatives up to second order
•
Determine derivatives up to second order
•
Understand differentiation of parametric functions and implicit functions
3.2
Application of Derivatives
•
Determine the rate of change of various quantities
3.3
Marginal Cost and Marginal Revenue using derivatives
•
Define marginal cost and marginal revenue
•
Find marginal cost and marginal revenue
3.4
Increasing /Decreasing Functions
•
Determine whether a function is increasing or decreasing
•
Determine the conditions for a function to be increasing or decreasing
3.5
Maxima and Minima
•
Determine critical points of the function
•
Find the point(s) of local maxima and local minima and corresponding local maximum and local minimum values
•
Find the absolute maximum and absolute minimum value of a function
•
Solve applied problems related to optimization of cost, revenue and profit only.
Integration and its Applications
3.6
Integration
•
Understand and determine indefinite integrals of simple functions as anti-derivative
3.7
Indefinite Integrals as family of curves
•
Evaluate indefinite integrals of simple algebraic functions by method of:
i)
substitution
ii)
partial fraction
iii)
by parts
3.8
Definite Integrals as area under the curve
•
Define definite integral as area under the curve
•
Understand fundamental theorem of Integral calculus and apply it to evaluate the definite integral
3.9
Application of Integration
Problems based on finding
•
Identify the region representing consumer surplus and producer surplus graphically
•
Apply the definite integral to find consumer surplus-producer surplus
Differential Equations and Modeling
3.10
Differential Equations
•
Recognize a differential equation
•
Find the order and degree of a differential equation
3.11
Formulating and Solving Differential Equations
•
Formulate differential equation
•
Verify the solution of differential equation
•
Solve simple differential equation using variable separable method only
UNIT – 4 PROBABILITY DISTRIBUTIONS
4.1
Probability Distribution
•
Understand the concept of Random Variables and its Probability Distributions
•
Find probability distribution of discrete random variable
4.2
Mathematical Expectation
•
Apply arithmetic mean of frequency distribution to find the expected value of a random variable
4.3
Variance
•
Calculate the Variance and S.D. of a random variable
4.4
Bionomial Distribution
•
Identify the Bernoulli Trials and apply Binomial Distribution
•
Evaluate Mean, Variance and S.D of a binomial distribution
4.5
Poisson Distribution
•
Understand the Conditions of Poisson Distribution
•
Evaluate the Mean and Variance of Poisson distribution
4.6
Normal Distribution
•
Understand normal distribution is a Continuous distribution
•
Evaluate value of Standard normal variate
•
Area relationship between Mean and Standard Deviation
UNIT – 5 INFERENTIAL STATISTICS
5.1
Population and Sample
•
Define Population and Sample
•
Differentiate between population and sample
•
Define a representative sample from a population
•
Differentiate between a representative and non-representative sample
•
Draw a representative sample using simple random sampling
•
Draw a representative sample using systematic random sampling
5.2
Parameter and Statistics and Statistical Interferences
•
Define Parameter with reference to Population
•
Define Statistics with reference to Sample
•
Explain the relation between Parameter and Statistic
•
Explain the limitation of Statistic to generalize the estimation for population
•
Interpret the concept of Statistical Significance and Statistical Inferences
•
State Central Limit Theorem
•
Explain the relation between Population-Sampling Distribution-Sample
5.3
t-Test (one sample t-test and for a small group sample)
•
Define a hypothesis
•
Differentiate between Null and Alternate hypothesis
•
Define and calculate degree of freedom
•
Test Null hypothesis and make inferences using t-test statistic for one group
UNIT – 6 TIME-BASED DATA
6.1
Time Series
•
Identify time series as chronological data
6.2
Components of Time Series
•
Distinguish between different components of time series
6.3
Time Series analysis for univariate data
•
Solve practical problems based on statistical data and interpret the result
6.4
Secular Trend
•
Understand the long-term tendency
6.5
Methods of Measuring trend
•
Demonstrate the techniques of finding trend by different methods
UNIT – 7 FINANCIAL MATHEMATICS
7.1
Perpetuity, Sinking Funds
•
Explain the concept of perpetuity and sinking fund
•
Calculate perpetuity
•
Differentiate between sinking fund and saving account
7.2
Valuation of Bonds
•
Define the concept of valuation of bond and related terms.
•
Calculate value of bond using present value approach
7.3
Calculation of EMI
•
Explain the concept of EMI
•
Calculate EMI using various methods
7.4
Compound Annual Growth Rate
•
Understand the concept of Compound Annual Growth Rate
•
Differentiate between Compound Annual Growth Rate and Annual Growth Rate
•
Calculate Compound Annual Growth Rate
7.5
Linear Method of Depreciation
•
Define the concept of linear method of Depreciation
•
Interpret cost, residual value and useful life of an asset from the given information
•
Calculate depreciation
UNIT – 8 LINEAR PROGRAMMING
8.1
Introduction and related terminology
•
Familiarize with terms related to Linear Programming Problem
8.2
Mathematical formulation of Linear Programming Problem
•
Formulate Linear Programming Problem upto 3 non-trivial constraints
8.3
Different types of Linear Programming Problems
•
Identify and formulate different types of LPP
8.4
Graphical method of solution for problems in two variables
•
Draw the Graph for a system of linear inequalities involving two variables and to find its solution graphically
8.5
Feasible and Infeasible Regions
•
Identify feasible, infeasible, bounded and unbounded regions
8.6
Feasible and infeasible solutions, optimal feasible solution
•
Understand feasible and infeasible solutions
•
Find optimal 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/Language/Statement3.html 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
Notes / Explanation of Syllabus
UNIT – 1 NUMBERS, QUANTIFICATION AND NUMERICAL APPLICATIONS
Numbers & Quantification
1.1
Modulo Arithmetic
•
Definition and meaning
•
Introduction to modulo operator
•
Modular addition and subtraction
1.2
Congruence Modulo
•
Definition and meaning
•
Solution using congruence modulo
•
Equivalence class
1.3
Alligation and Mixture
•
Meaning and Application of rule of alligation
•
Mean price of a mixture
1.4
Boats and Streams (upstream and downstream) (Numerical Problems Solve real life problems mathematically)
•
Problems based on speed of stream and the speed of boat in still water
Pipes and Cisterns (Numerical Problems Solve real life problems mathematically)
•
Calculation of the portion of the tank filled or drained by the pipe(s) in unit time
Races and Games (Numerical Problems Solve real life problems mathematically)
•
Calculation of the time taken/ distance covered / speed of each player
1.5
Numerical Inequalities
•
Comparison between two statements/situations which can be compared numerically
•
Application of the techniques of numerical solution of algebraic inequations
UNIT-2 ALGEBRA
2.1
Matrices and types of matrices
•
The entries, rows and columns of matrices
•
Present a set of data in a matrix form
2.2
Equality of matrices, Transpose of a matrix, Symmetric and Skew symmetric matrix
•
Examples of transpose of matrix
•
A square matrix as a sum of symmetric and skew symmetric matrix
•
Observe that diagonal elements of skew symmetric matrices are always zero
2.3
Algebra of Matrices
•
Addition and Subtraction of matrices
•
Multiplication of matrices (It can be shown to the students that Matrix multiplication is similar to multiplication of two polynomials)
•
Multiplication of a matrix with a real number
2.4
Determinants
•
Singular matrix, Non-singular matrix
•
|AB|=|A||B|
•
Simple problems to find determinant value
2.5
Inverse of a matrix
•
Inverse of a matrix using cofactors
•
If A and B are invertible square matrices of same size,
i)
{(AB)^{-1} = B^{-1}A^{-1}}
ii)
{(A^{-1})^{-1} = A}
iii)
{(A')^{-1} = (A^{-1})^{'}}
2.6
Solving a system of simultaneous equations using matrix method and Cramer’s rule
•
Solution of system of simultaneous equations up to three variables only (non-homogeneous equations)
UNIT – 3 CALCULUS
Differentiation and its Applications
3.1
Derivatives up to second order
•
Simple problems based on up to second order derivatives
•
Differentiation of parametric functions and implicit functions (upto 2nd order)
3.2
Application of Derivatives
•
To find the rate of change of quantities such as area and volume with respect to time or its dimension
3.3
Marginal Cost and Marginal Revenue using derivatives
•
Examples related to marginal cost, marginal revenue, etc.
3.4
Increasing /Decreasing Functions
•
Simple problems related to increasing and decreasing behaviour of a function in the given interval
3.5
Maxima and Minima
•
A point {𝑥 = 𝑐} is called the critical point of f if f is defined at 𝑐 and {𝑓′(𝑐) = 0} or f is not differentiable at 𝑐
•
To find local maxima and local minima by:
i)
First Derivative Test
ii)
Second Derivative Test
•
Contextualized real life problems
Integration and its Applications
3.6
Integration
•
Integration as a reverse process of differentiation
•
Vocabulary and Notations related to Integration
3.7
Indefinite Integrals as family of curves
•
Simple integrals based on each method (non-trigonometric function)
3.8
Definite Integrals as area under the curve
•
Evaluation of area under simple algebraic curves up to 2nd degree.
3.9
Application of Integration
Problems based on finding
•
Total cost when Marginal Cost is given
•
Total Revenue when Marginal Revenue is given
•
Equilibrium price and equilibrium quantity and hence consumer and producer surplus
Differential Equations and Modeling
3.10
Differential Equations
•
Definition, order, degree and examples of differential equation
3.11
Formulating and Solving Differential Equations
•
Formation of differential equation by eliminating arbitrary constants
•
Solution of simple differential equations (direct integration only)
UNIT – 4 PROBABILITY DISTRIBUTIONS
4.1
Probability Distribution
•
Definition and example of discrete and continuous random variable and their distribution
4.2
Mathematical Expectation
•
The expected value of discrete random variable as summation of product of discrete random variable by the probability of its occurrence.
4.3
Variance
•
Questions based on variance and standard deviation
4.4
Bionomial Distribution
•
Characteristics of binomial distribution
•
Binomial formula:
{P(r) = ^nC_r p^r q^{n-r}}
Where n = number of trials
p =probability of success
q = probability of failure
Mean = np
Variance = npq
Standard deviation = \sqrt{npq}
4.5
Poisson Distribution
•
Characteristics of Poisson Probability distribution
Poisson formula {P(X) = \dfrac{λ^xe^{-y}}{xL!}}
•
Mean = Variance = λ
4.6
Normal Distribution
•
Characteristics of a normal probability distribution
•
Total area under the curve = total probability = 1
•
Standard Normal Variate:
{Z = \dfrac{x - μ}{σ}}
where x = value of random variable
where μ = mean
where σ = S.D
UNIT – 5 INFERENTIAL STATISTICS
5.1
Population and Sample
•
Population data from census, economic surveys and other contexts from practical life
•
Examples of drawing more than one sample set from the same population
•
Examples of representative and non-representative sample
•
Unbiased and biased sampling
•
Problems based on random sampling using simple random sampling and systematic random sampling (sample size less than 100)
5.2
Parameter and Statistics and Statistical Interferences
•
Conceptual understanding of Parameter and Statistics
•
Examples of Parameter and Statistic limited to Mean and Standard deviation only
•
Examples to highlight limitations of generalizing results from sample to population
•
Only conceptual understanding of Statistical Significance/Statistical Inferences
•
Only conceptual understanding of Sampling Distribution through simulation and graphs
5.3
t-Test (one sample t-test and for a small group sample)
•
Examples and non-examples of Null and Alternate hypothesis (only non- directional alternate hypothesis)
•
Framing of Null and Alternate hypothesis
•
Testing a Null Hypothesis to make Statistical Inferences for small sample size (for small sample size: t- test for one group)
UNIT – 6 TIME-BASED DATA
6.1
Time Series
•
Meaning and Definition
6.2
Components of Time Series
•
Secular trend
•
Seasonal variations
•
Cyclical variations
•
Irregular variations
6.3
Time Series analysis for univariate data
•
Fitting a straight-line trend and estimating the value
•
Seasonal variation analysis
6.4
Secular Trend
•
The tendency of the variable to increase or decrease over a long period of time
6.5
Methods of Measuring trend
•
Moving Average method
•
Method of Least Squares
UNIT – 7 FINANCIAL MATHEMATICS
7.1
Perpetuity, Sinking Funds
•
Meaning of Perpetuity and Sinking Fund
•
Real life examples of sinking fund
•
Advantages of Sinking Fund
•
Sinking Fund vs. Savings account
7.2
Valuation of Bonds
•
Meaning of Bond Valuation
•
Terms related to valuation of bond: Coupon rate, Maturity rate and Current price.
•
Bond Valuation Method: Present Value Approach
7.3
Calculation of EMI
•
Methods to calculate EMI:
i)
Flat-Rate Method
ii)
Reducing-Balance Method
•
Real life examples to calculate EMI of various types of loans, purchase of assets, etc.
7.4
Compound Annual Growth Rate
•
Meaning and use of Compound Annual Growth Rate
•
Formula for Compound Annual Growth Rate
7.5
Linear Method of Depreciation
•
Meaning and formula for Linear Method of Depreciation
•
Advantages and disadvantages of Linear Method
UNIT – 8 LINEAR PROGRAMMING
8.1
Introduction and related terminology
•
Need for framing linear programming problem
•
Definition of Decision Variable, Constraints, Objective function, Optimization and Non negative constraints
8.2
Mathematical formulation of Linear Programming Problem
•
Set the problem in terms of decision variables, identify the objective function, identify the set of problem constraints
•
Express the problem in terms of inequations
8.3
Different types of Linear Programming Problems
•
Formulate various types of LPP’s like Manufacturing Problem, Diet Problem etc.
8.4
Graphical method of solution for problems in two variables
•
Corner Point Method for the Optimal solution of LPP
8.5
Feasible and Infeasible Regions
•
Definition and Examples to explain the terms
8.6
Feasible and infeasible solutions, optimal feasible solution
•
Problems based on optimization
•
Examples of finding the solutions by graphical method