# CBSE Syllabus Class 12 Applied Mathematics

On April 21st, 2022, CBSE released a reduced syllabus for classes 9 to 12 of the 2022-23 academic session.

After so many changes in the last 2 to 3 years, CBSE has finalised a reduced syllabus for Class 12 teachers and students to follow. However, the term-based examination pattern will NOT BE FOLLOWED in 2022-23.

We were prompt about the syllabus release and have updated the changes in all the subjects for you. Here, you can find:

• simple analysis of all the deleted topics/ chapters to be followed this year.

2022-23 Reduced Syllabus

We have also provided the syllabus for the 2020-21 session that was previously restored so that you can compare the deleted and added topics.

2020-21 Reduced Syllabus

With all this information in hand, both teachers and students will have a defined structure to begin the learning process on time.

<red> Marked in red: <red> Topics/ chapters <red> removed <red> for 2022-23

<green> Marked in red: <green> Topics/ chapters <green> added <green> for 2022-23

Units Unit Name Marks
I Numbers, Quantification and Numerical Applications 11
II Algebra 10
III Calculus 15
IV Probability Distributions 10
V Inferential Statistics 05
VI Index Numbers and Time-based Data (Continued) 06
VII Financial Mathematics 15
VIII Linear Programming 08
INTERNAL ASSESSMENT
Project Work (10m) + Practical Work (10m)
20
TOTAL 100

UNIT I: NUMBERS, QUANTIFICATION AND NUMERICAL APPLICATIONS

1.1 Modulo Arithmetic

• Definition and meaning
• Introduction to modulo operator

1.2 Congruence Modulo

• Definition and meaning
• Solution using congruence modulo
• Equivalence class

<red> 1.3 Simple Arithmetic Functions <red>

<red> Properties and Examples of <red>

• <red> Euler totient function <red>
• <red> Number of divisor function <red>
• <red> Divisor sum function <red>
• <red> Mobius function <red>

1.4 Alligation and Mixture

• Meaning and Application of rule of alligation
• Mean price of a mixture

1.5.1 Numerical Problems

Solve real life problems mathematically.

1.5.2 Boats and Streams (upstream and downstream)

• Problems based on speed of stream and the speed of boat in still water

1.5.3 Pipes and Cisterns

• Calculation of the portion of the tank filled or drained by the pipe(s) in unit time

1.5.4 Races and Games

• Calculation of the time taken/ distance covered/ speed of each player

<red> Partnership (Removed from 2020-21 reduced syllabus) <red>

• <red> Definition, Profit division among the partners <red>

<red> Scheduling <red>

• <red> Definition and meaning <red>
• <red> Use of Gantt chart Simple problems based on FCFS (First come First serve) and SJF and SJF (shortest job first) <red>

1.6 Numerical Inequalities

• Comparison between two statements/situations which can be compared numerically
• Application of the techniques of numerical solution of algebraic inequalities

UNIT II: 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: a) cofactors b) elementary row operations
• If A and B are invertible square matrices of same size,
1. (AB)^-1 = B^-1 A^–1
2. (A^-1)^-1 = A
3. (A^T)^-1 = (A^-1)^T

2.6 Solving system of simultaneous equations using matrix method, Cramer’s rule and row reduction method

• Solution of system of simultaneous equations upto three variables only (non- homogeneous equations)

<red> 2.7 Simple applications of matrices and determinants including Leontiff input output model for two variables (Deleted from 2020-21 reduced syllabus) <red>

• <red> Real life applications of Matrices and Determinant <red>
• <red> Leontiff Input–output model that represents the interdependencies between different sectors of a national economy or different regional economies <red>

UNIT III: CALCULUS

Differentiation and its Applications

3.1 Higher Order Derivatives

• Simple problems based on higher order derivatives
• Differentiation of parametric functions and implicit functions (upto 2nd order)

3.2 Application of Derivatives

• <green> To find the rate of change of quantities such as area and volume with respect to time or its dimension (Added in the 2020-21 reduced syllabus) <green>
• 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

• 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:
1. First Derivative Test
2. Second Derivative Test
• Contextualised 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 definite integrals using properties

<red> Integration of Simple Algebraic Functions (primitive, by substitution, by parts) <red>

3.9 Application of Integration

• Problems based on finding:
1. Total cost when Marginal Cost is given
2. Total Revenue when Marginal Revenue is given
3. Equilibrium price and equilibrium quantity and hence consumer and producer surplus

Differential Equations and Modelling

3.10 Differential Equations

• Definition, order, degree and examples

<green> 3.11 Formulating and Solving Differential Equations (Added in the 2020-21 reduced syllabus) <green>

• <green> Formation of differential equation by eliminating arbitrary constants <green>
• <green> Solution of simple differential equations (direct integration only) <green>

3.12 Application of Differential Equations

• Growth and Decay Model in Biological sciences, Economics and business, etc.

UNIT IV: 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 a discrete random variable is the summation of the product of the discrete random variable by the probability of its occurrence.

4.3 Variance

• Questions based on variance and standard deviation

4.4 Binomial Distribution

• Characteristics of the 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 = √𝑛𝑝𝑞

4.5 Poisson Distribution

• Characteristics of Poisson Probability distribution Poisson formula: P(x) = (𝜆 𝑥 . 𝑒 −𝜆)/ 𝑥!
• 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 = (𝑥−𝜇)/𝜎, where x = value of the random variable, 𝜇 = mean, 𝜎 = S.D.

<red> 4.7 Basic Applications and Inferences <red>

UNIT V: 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 biassed 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 generalising 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)

• Examples and non-examples of Null and Alternate hypothesis (only nondirectional alternative 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 VI: INDEX NUMBERS AND TIME BASED DATA

<red> 6.1 Index Numbers (Deleted from 2020-21 reduced syllabus) <red>

• <red> Meaning and Definition <red>
• <red> Utility of Index Numbers <red>

<red> 6.2 Construction of Index numbers (Deleted from 2020-21 reduced syllabus) <red>

• <red> Simple Index numbers <red>
• <red> Weighted index numbers <red>

<red> 6.3 Test of adequacy of Index numbers (Deleted from 2020-21 reduced syllabus) <red>

• <red> Unit test <red>
• <red> Time reversal test <red>

6.4 Time Series

• Meaning and Definition

6.5 Components of Time Series

• Secular trend
• Seasonal variation
• Cyclical variation
• Irregular variation

6.6 Time Series analysis for univariate data

• Fitting a straight line trend and estimating the value

6.7 Secular Trend

• The tendency of the variable to increase or decrease over a long period of time

6.8 Methods of Measuring trend

• Moving Average method
• Method of Least Squares

UNIT VII: FINANCIAL MATHEMATICS

7.1 Perpetuity, Sinking Funds

• Meaning of Perpetuity and Sinking Fund
• Real life examples 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 Methods:
1. Present Value Approach
2. Relative Price Approach

7.3 Calculation of EMI

• Methods to calculate EMI:
1. Flat-Rate Method
2. 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

• Formula for calculation of Rate of Return, Nominal Rate of Return

7.5 Compound Annual Growth Rate

• Meaning and use of Compound Annual Growth Rate
• Formula for Compound Annual Growth Rate

<red> 7.6 Stock, Shares and Debentures <red>

• <red> Meaning of Stock, shares and debentures <red>
• <red> Types of Shares and Debentures <red>
• <red> Features and advantages of equity shares and debentures <red>
• <red> Real life examples of shares & debentures <red>

7.7 Linear method of Depreciation

• Meaning and formula for Linear Method of Depreciation

UNIT VIII: 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 conditions

8.2 Mathematical formulation of Linear

• 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 LPPs like Manufacturing Problem, Diet Problem, Transportation Problem, etc.

8.4 Graphical Method of Solution for Problems in Two Variables

• Corner Point Method for the Optimal solution of LPP
• <green> Iso-cost/ Iso-profit Method (Added to the 2020-21 reduced syllabus) <green>

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

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.

• Plot the graphs of functions on excel and study the graph to find out the point of maxima/minima
• Probability and dice roll simulation
• Matrix multiplication and the inverse of a matrix
• Stock Market data sheet on excel
• Collect the data on weather, price, inflation, and pollution analyse the data and make meaningful inferences
• 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)

• Use of prime numbers in coding and decoding of messages
• Prime numbers and divisibility rules
• Logarithms for financial calculations such as interest, present value, future value, profit/loss etc. with large values)
• The cardinality of a set and orders of infinity
• Comparing sets of Natural numbers, rational numbers, real numbers and others
• Use of Venn diagram in solving practical problems
• Fibonacci sequence: Its' history and presence in nature
• Testing the validity of mathematical statements and framing truth tables
• Investigating Graphs of functions for their properties
• Visit the census site of India. Depict the information given there in a pictorial form
https://censusindia.gov.in/
• 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
• 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
• Analysis of population migration data – positive and negative influence on urbanisation
• Each day the 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
• 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
• Vehicle registration data – correlating with pollution and the number of accidents
• 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
• 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
• Weather prediction (prediction of monsoon from past data)
• 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 maximise the profit
• Stock price movement
• Risk assessments by insurance firms from data
• Predicting stock market crash
• Predicting the outcome of an election – exit polls
• Predicting mortality of infants