CBSE Class 12 Applied Mathematics Syllabus 2022-23
Grade XII (2022-23)
Number of Paper: 1
Total number of Periods: 240 (35 Minutes Each)
Time: 3 Hours
Max Marks: 80
No. |
Units |
No. of Periods |
Marks |
I |
Numbers, Quantification and Numerical Applications |
30 |
11 |
II |
Algebra |
20 |
10 |
III |
Calculus |
50 |
15 |
IV |
Probability Distributions |
35 |
10 |
V |
Inferential Statistics |
10 |
05 |
VI |
Index Numbers and Time-based data |
30 |
06 |
VII |
Financial Mathematics |
50 |
15 |
VIII |
Linear Programming |
15 |
08 |
Total |
240 |
80 |
|
Internal Assessment |
20 |
CLASS XII |
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Sl. No. |
Contents |
Learning Outcomes: Students will be able to |
Notes / Explanation |
UNIT-1 NUMBERS, QUANTIFICATION AND NUMERICAL APPLICATIONS |
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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 |
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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 |
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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 |
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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 |
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1.6 |
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 |
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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 ● 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 of 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) (AT)-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 |
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Differentiation and its Applications |
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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 2nd 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 |
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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 |
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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 |
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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-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: 𝑥 −𝜆 P(x) = 𝜆 . 𝑒 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 where 𝜎 x = value of the random variable 𝜇 = mean 𝜎 = S.D. |
UNIT – 5 INFERENTIAL STATISTICS |
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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 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 |
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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 |
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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 |
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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 |
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
- 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 analyze the data and make meaningful inferences
- Collect data from newspapers on traffic, sports activities and market trends and use excel to study future trends
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