| 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
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| 1.2 | Congruence Modulo | - Define congruence modulo
- Apply the definition in various problems
| - Definition and meaning
- Solution using congruence modulo
- Equivalence class
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| 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
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| 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
|
| Unit-2 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
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| 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 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 -1A –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 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 |
| 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
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| 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 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 |
| 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) 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 |
| 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
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