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

Unit2 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, Nonsingular matrix
 AB=AB
 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 ^{1}A ^{–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 antiderivative
  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 (nontrigonometric 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 surplusproducer 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 nonrepresentative 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 nonrepresentative 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 PopulationSampling DistributionSample
  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  tTest (one sample ttest and two independent groups ttest)   Define a hypothesis
 Differentiate between Null and Alternate hypothesis
 Define and calculate degree of freedom
 Test Null hypothesis and make inferences using ttest statistic for one group / two independent groups
  Examples and nonexamples of Null and Alternate hypothesis (only nondirectional 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 ttable

Unit – 6 Index Numbers And Time Based Data 
6.4  Time Series   Identify time series as chronological data
 
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) FlatRate Method ii) ReducingBalance 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
 Isocost/ Isoprofit 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
