CBSE Class 12 Applied Mathematics Syllabus 2024-25

CBSE has released the Latest Updated Syllabus for the New Academic Session 2024-25 on March 23rd, 2024, for class 12.

CBSE Board has released the latest Class 12 Applied Mathematics syllabus which is to be strictly followed. Below please find our detailed analysis of Board Paper pattern, Unit-wise summary for the New Session 2024-25.

We have also updated Oswal Publishers Books as per the Latest Paper Pattern prescribed by Board for Applied Mathematics Curriculum.

Students can directly access the CBSE Applied Mathematics Syllabus for Class 12 of the academic year 2024-25 by clicking on the link below.

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Table Of Contents−

▪ CBSE Class 12 Applied Mathematics Latest Syllabus 2024-25
▪ CBSE Class 12 Applied Mathematics Syllabus 2024-25: Unit-wise Summary
▪ Internal Assessment
▪ The Changes for Class 12 (2024-25) Year-end Board Examinations are as under:
▪ 2023-24 Reduced Syllabus

CBSE Class 12 Applied Mathematics Latest Syllabus 2024-25

No.	Units	No.of Periods	Marks
1	Numbers, Quantification and Numerical Applications	30	11
2	Algebra	20	10
3	Calculus	50	15
4	Probability Distributions	35	10
5	Inferential Statistics	10	05
6	Index Numbers and Time-based data	30	06
7	Financial Mathematics	50	15
8	Linear Programming	15	08
Total		240	80
Internal Assessment		20

CBSE Class 12 Applied Mathematics Syllabus 2024-25: Unit-wise Summary

Sl. No.	Contents	Learning Outcomes: Students will be able to	Notes / Explanation
Unit-1 Numbers, Quantification And Numerical Applications
1.1	Modulo Arithmetic	Define modulus of an integer 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
1.4	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 alligation	Meaning and Application of rule of alligation Mean price of a mixture
1.5	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
1.6	Numerical Inequalities	Describe the basic concepts of numerical inequalities Understand and write numerical inequalities	Meaning and Application of rule of alligationComparison between two statements/situations which can be compared numerically Application of the techniques of numerical solution of algebraic inequations
Unit-2 Algebra
2.1	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 Use elementary properties of determinants	Singular matrix, Non-singular matrix \|AB\|=\|A\|\|B\| Simple problems to find determinant value
2.5	Inverse of a matrix	Define the inverse o f a square matrix Apply properties of inverse of matrices	Inverse of a matrix using: a) cofactors If A and B are invertible square matrices of same size, i) (AB) ^-1=B ^-1A ^–1 ii) (A ^-1) ^-1 =A iii) (A ^T) ^-1 = (A ^-1) ^T
2.6	Solving system of simultaneous equations using matrix method, Cramer’s rule and	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	Solution of system of simultaneous equations upto three variables only (non- homogeneous equations)
Unit- 3 Calculus
Differentiation and its Applications
3.1	Higher Order Derivatives	Determine second and higher order derivatives Understand differentiation of parametric functions and implicit functions	Simple problems based on higher order derivatives Differentiation of parametric functions and implicit functions (upto 2 ^nd order)
3.2	Application of Derivatives	Determine the rate of change of various quantities Understand the gradient of tangent and normal to a curve at a given point Write the equation of tangents and normal to a curve at a given point	To find the rate of change of quantities such as area and volume with respect to time or its dimension Gradient / Slope of tangent and normal to the curve The equation of the tangent and normal to the curve (simple problems only)
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 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	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	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 method of: i) substitution ii) partial fraction iii) 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 Apply properties of definite integrals to solve the problems	Evaluation of definite integrals using properties
3.9	Application of Integration	Identify the region representing C.S. and P.S. 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 of differential equation Solve simple differential equation	Formation of differential equation by eliminating arbitrary constants Solution of simple differential equations (direct integration only)
3.12	Application of Differential Equations	Define Growth and Decay Model Apply the differential equations to solve Growth and Decay Models	Growth and Decay Model in Biological sciences, Economics and business, etc.
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 the binomial distribution Binomial formula: P(r) = nCr pr qn- Where n = number of trials P = probability of success q = probability of failure Mean =np Variance = npq Standard Deviation = √npq
4.5	Normal Distribution	Understand normal distribution is a Continuous distribution Evaluate the Mean and Variance of Poisson distribution	Characteristics of Poisson Probability distribution Poisson formula: P(x) = (λ ^x. e ^-λ)/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: Z =(𝑥− 𝜇)/𝜎 where x = value of the 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 Differentiate between a representative and non-representative sample Draw a representative sample using simple random sampling Draw a representative sample using and systematic random sampling	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	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	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 two independent groups t-test)	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 / two independent groups	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 and two independent groups Use of t-table
Unit – 6 Index Numbers And Time Based Data
6.4	Time Series	Identify time series as chronological data	Meaning and Definition
6.5	Components of Time Series	Distinguish between different components of time series	Secular trend Seasonal variation Cyclical variation Irregular variation
6.6	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
6.7	Secular Trend	Understand the long term tendency	The tendency of the variable to increase or decrease over a long period of time
6.8	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 of Perpetuity and Sinking Fund Real life examples of sinking fund Advantages of Sinking Fund Sinking Fund vs. Savings account
7.3	Calculation of EMI	Explain the concept of EMI Calculate EMI using various methods	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	Calculation of Returns, Nominal Rate of Return	Explain the concept of rate of return and nominal rate of return Calculate rate of return and nominal rate of return	Formula for calculation of Rate of Return, Nominal Rate of Return
7.5	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	Meaning and use of Compound Annual Growth Rate Formula for Compound Annual Growth Rate
7.7	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	Meaning and formula for Linear Method of Depreciation 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 Definition of Decision Variable, Constraints, Objective function, Optimization and Non Negative conditions
8.2	Mathematical formulation of Linear Programming Problem	Formulate 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	Identify and formulate different types of LPP	Formulate various types of LPP’s like Manufacturing Problem, Diet Problem, Transportation 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 to find its solution graphically	Corner Point Method for the Optimal solution of LPP Iso-cost/ Iso-profit Method
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 the solutions by graphical method

Internal Assessment

Overall Assessment of the course is out of 100 marks.
The assessment plan consists of an External Exam and Internal Assessment.
External Exam will be of 03 hours duration Pen/ Paper Test consisting of 80 marks.
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.
Weightage for each area of internal assessment may be as under:

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

The Changes for Class 12 (2024-25) Year-end Board Examinations are as under:

(Class-12)
Periodic Assessment	Academic Session 2023-24	Academic Session 2024-25
Composition of question paper for year-end examination/ Board Examination (Theory)	Competency Focused Questions in the form of MCQs/ Case Based Questions, Source-based Integrated Questions or any other type = 40% Select response type questions (MCQ) = 20% Constructed response questions (Short Answer Questions/Long Answer type Questions, as per existing pattern) = 40%	Competency Focused Questions in the form of MCQs/ Case Based Questions, Source-based Integrated Questions or any other type = 50% Select response type questions (MCQ) = 20% Constructed response questions (Short Answer Questions/Long Answer type Questions, as per existing pattern) = 30%