No.

Units

Marks

I

Numbers, Quantification and Numerical

Applications

11

II

Algebra

10

III

Calculus

15

IV

Probability Distributions

10

V

Inferential Statistics

5

VI

Time-based data

6

VII

Financial Mathematics

15

VIII

Linear Programming

8

Total

80

Internal Assessment

20

CLASS- XII

Sl.

No.

Contents

Learning Outcomes: Students will be able to

Notes / Explanation

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

· 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

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,

i) (AB)⁻¹ = B⁻¹A⁻¹

ii) (A⁻¹)⁻¹ = A

iii) (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

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 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 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.

· 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

● 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 of differential equation

● Solve simple differential equation using variable separable method only

● Formation of differential equation 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:

𝑃(𝑟) = 𝑛_𝐶 𝑝^𝑟𝑞^𝑛−𝑟

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: 𝑃(𝑋) =

𝜆𝑥𝑒−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 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 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 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 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

● 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

● 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 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

● Define the concept of valuation of bond and related terms.

● Calculate value of bond using present value approach

● 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

· 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

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

7.5

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

· 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 constraints

CBSE Class 12th Applied Maths Syllabus 2025-26 FREE PDF Download