1.1 |
Modulo Arithmetic |
- Define modulus of an integer
- Apply arithmetic operations using modular arithmetic rules
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- 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
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- 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
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- 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
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- 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
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- 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
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- 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
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- 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
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Unit-2 Algebra |
2.1 |
Define matrix |
- Identify different kinds of matrices
- Find the size / order of matrices
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- The entries, rows and columns of matrices
- Present a set of data in a matrix form
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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
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- 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
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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
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- 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
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- Singular matrix, Non-singular matrix
- |AB|=|A||B|
- Simple problems to find determinant value
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2.5 |
Inverse of a matrix |
- Define the inverse o
- f a square matrix
- Apply properties of inverse of matrices
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- 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
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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
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- Solution of system of simultaneous equations upto three variables only (non- homogeneous equations)
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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
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- Simple problems based on higher order derivatives
- Differentiation of parametric functions and implicit functions (upto 2 nd order)
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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
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- 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)
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3.3 |
Marginal Cost and Marginal Revenue using derivatives |
- Define marginal cost and marginal revenue
- Find marginal cost and marginal revenue
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- Examples related to marginal cost, marginal revenue, etc.
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3.4 |
Increasing /Decreasing Functions |
- Determine whether a function is increasing or decreasing
- Determine the conditions for a function to be increasing or decreasing
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- Simple problems related to increasing and decreasing behaviour of a function in the given interval
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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
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- 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
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Integration and its Applications |
3.6 |
Integration |
- Understand and determine indefinite integrals of simple functions as anti-derivative
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- Integration as a reverse process of differentiation
- Vocabulary and Notations related to Integration
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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
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- Simple integrals based on each method (non-trigonometric function)
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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
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- Evaluation of definite integrals using properties
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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
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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
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Differential Equations and Modeling |
3.10 |
Differential Equations |
- Recognize a differential equation
- Find the order and degree of a differential equation
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- Definition, order, degree and examples
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3.11 |
Formulating and Solving Differential Equations |
- Formulate differential equation
- Verify the solution of differential equation
- Solve simple differential equation
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- Formation of differential equation by eliminating arbitrary constants
- Solution of simple differential equations (direct integration only)
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3.12 |
Application of Differential Equations |
- Define Growth and Decay Model
- Apply the differential equations to solve Growth and Decay Models
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- Growth and Decay Model in Biological sciences, Economics and business, etc.
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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
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- Definition and example of discrete and continuous random variable and their distribution
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4.2 |
Mathematical Expectation |
- Apply arithmetic mean of frequency distribution to find the expected value of a random variable
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- The expected value of discrete random variable as summation of product of discrete random variable by the probability of its occurrence.
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4.3 |
Variance |
- Calculate the Variance and S.D. of a random variable
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- 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
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- 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
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4.5 |
Normal Distribution |
- Understand normal distribution is a Continuous distribution
- Evaluate the Mean and Variance of Poisson distribution
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- Characteristics of Poisson Probability distribution Poisson formula:
P(x) = (λ x. e -λ)/𝑥!
- Mean = Variance = 𝜆
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4.6 |
Normal Distribution |
- Understand normal distribution is a Continuous distribution
- Evaluate value of Standard normal variate
- Area relationship between Mean and Standard Deviation
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- 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.
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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
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- 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)
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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
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- 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
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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
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- 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
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Unit – 6 Index Numbers And Time Based Data |
6.4 |
Time Series |
- Identify time series as chronological data
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6.5 |
Components of Time Series |
- Distinguish between different components of time series
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- Secular trend
- Seasonal variation
- Cyclical variation
- Irregular variation
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6.6 |
Time Series analysis for univariate data |
- Solve practical problems based on statistical data and Interpret the result
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- Fitting a straight line trend and estimating the value
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6.7 |
Secular Trend |
- Understand the long term tendency
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- The tendency of the variable to increase or decrease over a long period of time
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6.8 |
Methods of Measuring trend |
- Demonstrate the techniques of finding trend by different methods
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- Moving Average method
- Method of Least Squares
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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
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- Meaning of Perpetuity and Sinking Fund
- Real life examples of sinking fund
- Advantages of Sinking Fund
- Sinking Fund vs. Savings account
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7.3 |
Calculation of EMI |
- Explain the concept of EMI
- Calculate EMI using various methods
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- 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.
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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
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- Formula for calculation of Rate of Return, Nominal Rate of Return
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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
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- Meaning and use of Compound Annual Growth Rate
- Formula for Compound Annual Growth Rate
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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
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- Meaning and formula for Linear Method of Depreciation
- Advantages and disadvantages of Linear Method
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Unit - 8 Linear Programming |
8.1 |
Introduction and related terminology |
- Familiarize with terms related to Linear Programming Problem
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- Need for framing linear programming problem
- Definition of Decision Variable, Constraints, Objective function, Optimization and Non Negative conditions
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8.2 |
Mathematical formulation of Linear Programming Problem |
- Formulate Linear Programming Problem
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- 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
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8.3 |
Different types of Linear Programming Problems |
- Identify and formulate different types of LPP
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- Formulate various types of LPP’s like Manufacturing Problem, Diet Problem, Transportation Problem, etc.
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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
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- Corner Point Method for the Optimal solution of LPP
- Iso-cost/ Iso-profit Method
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8.5 |
Feasible and Infeasible Regions |
- Identify feasible, infeasible, bounded and unbounded regions
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- Definition and Examples to explain the terms
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8.6 |
Feasible and infeasible solutions, optimal feasible solution |
- Understand feasible and infeasible solutions
- Find optimal feasible solution
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- Problems based on optimization
- Examples of finding the solutions by graphical method
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