CBSE Class XII Applied Mathematics Syllabus 2026 – 27

Applied Mathematics
Subject Code – 241
Classes XI-XII

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.

Grade XII (2026-27)

Number of Paper: 1
Time: 3 Hours
Max Marks: 80

No.	Units	Marks
I	Numbers, Quantification and Numerical Applications	11
II	Algebra	10
III	Calculus	15
IV	Probability Distributions	10
V	Inferential Statistics	05
VI	Time-based data	06
VII	Financial Mathematics	15
VIII	Linear Programming	08
Total		80
Internal Assessment		20

UNIT – 1 NUMBERS, QUANTIFICATION AND NUMERICAL APPLICATIONS

Chapter	Learning Outcomes	Details of content
1.1 Modulo Arithmetic	Apply arithmetic operations using modular arithmetic rules	Definition and meaning Introduction to modulo operator Modular addition and subtraction
1.2 Congruence Modulo	Define congruence modulo Apply the definition in various problems	Definition and meaning Solution using congruence modulo Equivalence class
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	Meaning and Application of rule of alligation Mean price of a mixture
1.4 Numerical Problems	Solve real life problems mathematically	Boats and Streams (upstream and downstream) Distinguish between upstream and downstream Express the problem in the form of an equation Problems based on speed of stream and the speed of boat in still water Pipes and Cisterns Determine the time taken by two or more pipes to fill or empty the tank Calculation of the portion of the tank filled or drained by the pipe(s) in unit time Races and Games Compare the performance of two players w.r.t. time, distance Calculation of the time taken/ distance covered / speed of each player
1.5 Numerical Inequalities	Describe the basic concepts of numerical inequalities Understand and write 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

Chapter	Learning Outcomes	Details of content
2.1 Matrices and types of matrices	Define matrix Identify different kinds of matrices. Find the size / order 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	Determine equality of two matrices Write transpose of given matrix Define 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	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	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	Find determinant of a square matrix	Singular matrix, Non-singular matrix |AB|=|A||B| Simple problems to find determinant value
2.5 Inverse of a matrix	Define the inverse of a square matrix Apply properties of inverse of matrices	Inverse of a matrix using cofactors If A and B are invertible square matrices of same size: (AB)⁻¹ = B⁻¹A⁻¹; (A⁻¹)⁻¹ = A; (A′)⁻¹ = (A⁻¹)′
2.6 Solving system of simultaneous equations using matrix method and Cramer’s rule	Solve the system of simultaneous equations using Cramer’s Rule and Inverse of coefficient matrix Formulate real life problems into a system of simultaneous linear equations and solve it using these methods	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	Determine derivatives up to second order; understand differentiation of parametric functions and implicit functions.	Simple problems based on up to second order derivatives; differentiation of parametric functions and implicit functions (upto 2nd order).
3.2 Application of Derivatives	Determine the rate of change of various quantities.	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	Define marginal cost and marginal revenue; find marginal cost and marginal revenue.	Examples related to marginal cost, marginal revenue, etc.
3.4 Increasing /Decreasing Functions	Determine whether a function is increasing or decreasing; determine the conditions for a function to be increasing or decreasing.	Simple problems related to increasing and decreasing behaviour of a function in the given interval.
3.5 Maxima and Minima	Determine critical points of the function; find local and absolute maxima and minima; solve applied problems related to optimization of cost, revenue and profit only.	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 First Derivative Test and Second Derivative Test. Contextualized real life problems.

Integration and its Applications

3.6 Integration	Understand and determine indefinite integrals of simple functions as anti-derivative.	Integration as a reverse process of differentiation; vocabulary and notations related to Integration.
3.7 Indefinite Integrals as family of curves	Evaluate indefinite integrals of simple algebraic functions by substitution, partial fraction and by parts.	Simple integrals based on each method (non-trigonometric function).
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.	Evaluation of area under simple algebraic curves up to 2nd degree.
3.9 Application of Integration	Identify the region representing consumer surplus and producer surplus graphically; apply the definite integral to find consumer surplus-producer surplus.	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	Recognize a differential equation; find the order and degree of a differential equation.	Definition, order, degree and examples.
3.11 Formulating and Solving Differential Equations	Formulate differential equation; verify the solution; solve simple differential equation using variable separable method only.	Formation by eliminating arbitrary constants; solution of simple differential equations (direct integration 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.	Definition and example of discrete and continuous random variable and their distribution.
4.2 Mathematical Expectation	Apply arithmetic mean of frequency distribution to find the expected value of a random variable.	The expected value of discrete random variable as summation of product of discrete random variable by the probability of its occurrence.
4.3 Variance	Calculate the Variance and S.D. of a random variable.	Questions based on variance and standard deviation.
4.4 Binomial Distribution	Identify the Bernoulli Trials and apply Binomial Distribution; evaluate Mean, Variance and S.D of a binomial distribution.	Characteristics of binomial distribution. Binomial formula: \mathrm{P(r)}=\mathrm{^{n}C_{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 = √npq.
4.5 Poison Distribution	Understand the Conditions of Poisson Distribution; evaluate the Mean and Variance of Poisson distribution.	Characteristics of Poisson Probability distribution. Poisson formula: \mathrm{P(X)}=\frac{\text{λ}^{\mathrm{x}}\mathrm{e}^{-\text{λ}}}{\mathrm{x}!}. Mean = Variance = λ.
4.6 Normal Distribution	Understand normal distribution is a Continuous distribution; evaluate value of Standard normal variate; area relationship between Mean and Standard Deviation.	Characteristics of a normal probability distribution. Total area under the curve = total probability = 1. Standard Normal Variate: \mathrm{Z}=\frac{\mathrm{x}-\text{μ}}{\text{σ}}, where x = value of random variable, μ = mean, σ = S.D.

UNIT – 5 INFERENTIAL STATISTICS

5.1 Population and Sample	Define Population and Sample; differentiate between population and sample; define and draw a representative sample using simple random sampling and systematic random sampling.	Population data from census, economic surveys and other contexts from practical life; representative and non-representative samples; unbiased and biased sampling; problems based on random sampling (sample size less than 100).
5.2 Parameter and Statistics and Statistical Interferences	Define Parameter with reference to Population and Statistics with reference to Sample; explain their relation and limitations; interpret Statistical Significance and Statistical Inferences; state Central Limit Theorem; explain Population-Sampling Distribution-Sample.	Conceptual understanding limited to Mean and Standard deviation; examples highlighting limitations of generalizing results; conceptual understanding of Statistical Significance/Statistical Inferences and Sampling Distribution through simulation and graphs.
5.3 t-Test (one sample t-test and for a small group sample)	Define a hypothesis; differentiate Null and Alternate hypothesis; define and calculate degree of freedom; test Null hypothesis and make inferences using t-test statistic for one group.	Examples and non-examples of Null and Alternate hypothesis (only non-directional alternate hypothesis); framing hypotheses; testing a Null Hypothesis for small sample size (t-test for one group).

UNIT – 6 TIME-BASED DATA

6.1 Time Series	Identify time series as chronological data.	Meaning and Definition.
6.2 Components of Time Series	Distinguish between different components of time series.	Secular trend; Seasonal variation; Cyclical variation; Irregular variation.
6.3 Time Series analysis for univariate data	Solve practical problems based on statistical data and interpret the result.	Fitting a straight-line trend and estimating the value.
6.4 Secular Trend	Understand the long-term tendency.	The tendency of the variable to increase or decrease over a long period of time.
6.5 Methods of Measuring trend	Demonstrate the techniques of finding trend by different methods.	Moving Average method; Method of Least Squares.

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.	Meaning, real life examples and advantages of Sinking Fund; Sinking Fund vs. Savings account.
7.2 Valuation of Bonds	Define the concept of valuation of bond and related terms; calculate value of bond using present value approach.	Coupon rate, Maturity rate and Current price; Present Value Approach.
7.3 Calculation of EMI	Explain the concept of EMI; calculate EMI using various methods.	Flat-Rate Method and Reducing-Balance Method; real life examples for loans and purchase of assets.
7.4 Compound Annual Growth Rate	Understand CAGR; differentiate CAGR and Annual Growth Rate; calculate CAGR.	Meaning, use and formula for Compound Annual Growth Rate.
7.5 Linear method of Depreciation	Define linear depreciation; interpret cost, residual value and useful life; calculate depreciation.	Meaning and formula; advantages and disadvantages of Linear Method.

UNIT – 8 LINEAR PROGRAMMING

8.1 Introduction and related terminology	Familiarize with terms related to Linear Programming Problem; need for framing linear programming problem.	Decision Variable, Constraints, Objective function, Optimization and Non negative constraints.
8.2 Mathematical formulation of Linear Programming Problem	Formulate Linear Programming Problem upto 3 non-trivial constraints.	Set the problem in terms of decision variables, identify the objective function and constraints, and express the problem in inequations.
8.3 Different types of Linear Programming Problems	Identify and formulate different types of LPP.	Manufacturing Problem, Diet Problem etc.
8.4 Graphical method of solution for problems in two variables	Draw the Graph for a system of linear inequalities involving two variables and find its solution graphically.	Corner Point Method for the Optimal solution of LPP.
8.5 Feasible and Infeasible Regions	Identify feasible, infeasible, bounded and unbounded regions.	Definition and Examples to explain the terms.
8.6 Feasible and infeasible solutions, optimal feasible solution	Understand feasible and infeasible solutions; find optimal feasible solution.	Problems based on optimization; examples of finding solutions by graphical method.