CBSE Class 12 Applied Mathematics — Notes, Solutions & Papers

CLASS- XII

Sl. No. Contents Learning Outcomes: Students will be able to Notes / Explanation

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

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)

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 2 nd 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)

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) = 𝑛𝐶𝑟𝑝𝑟𝑞𝑛−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

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)

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

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

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 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, 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 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 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