Mathematics

Section A

  1. Algebra

  2. Calculus

  3. Integration and its Applications

    • Indefinite Integrals of Simple Functions
    • Evaluation of Indefinite Integrals
    • Definite Integrals
  4. Differential Equations

    • Order and Degree of Differential Equations
    • Formulating and Solving Differential Equations
    • Variable Separable
  5. Probability Distributions

    • Random Variables and its Probability Distribution
    • Expected Value of a Random Variable
    • Variance and Standard Deviation of a Random Variable
    • Binomial Distribution
  6. Linear Programming

    • Mathematical Formulation of Linear Programming Problem
    • Graphical Method of Solution for Problems in Two Variables
    • Feasible and Infeasible Regions
    • Optimal Feasible Solution

Section B1: Mathematics

UNIT I: RELATIONS AND FUNCTIONS
  1. Relations and Functions

  2. Inverse Trigonometric Functions

UNIT II: ALGEBRA
  1. Matrices

    • Concept, notation, order, equality
    • types of matrices
    • zero matrix
    • transpose of a matrix, symmetric and skew symmetric matrices.
    • Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication.
    • Non-commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2).
    • Concept of elementary row and column operations.
    • Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
  2. Determinants

UNIT III: CALCULUS
  1. Continuity and Differentiability

    • Continuity and differentiability
    • derivative of composite functions, chain rule
    • derivatives of inverse trigonometric functions
    • derivative of implicit function.
    • Concepts of exponential, logarithmic functions.
    • Derivatives of log x and e^x.
    • Logarithmic differentiation.
    • Derivative of functions expressed in parametric forms.
    • Second-order derivatives.
    • Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations.
  2. Applications of Derivatives

  3. Integrals

    • Integration as inverseprocess of differentiation.
    • Integration of avarietyof functions bysubstitution, bypartial fractions andbyparts, only simple integrals ofthetype –
    • $$\int \frac{dx}{x^2 \pm a^2}, \quad \int \frac{dx}{\sqrt{x^2 \pm a^2}}, \quad \int \frac{dx}{\sqrt{a^2 - x^2}}, \quad \int \frac{dx}{ax^2 + bx + c}, \quad \int \frac{dx}{\sqrt{ax^2 + bx + c}},$$
    • $$\int \frac{(px + q)}{ax^2 + bx + c} , dx, \quad \int \frac{(px + q)}{\sqrt{ax^2 + bx + c}} , dx, \quad \int \sqrt{a^2 \pm x^2} , dx \quad \text{and} \quad \int \sqrt{x^2 - a^2} , dx,$$
    • $$\int \sqrt{ax^2 + bx + c} , dx \quad \text{and} \quad \int (px + q) \sqrt{ax^2 + bx + c} , dx $$ to be evaluated.
    • Definite integrals as a limit of a sum.
    • Fundamental Theorem of Calculus (without proof)
    • Basic properties of definite integrals and evaluation of definite integrals.
  4. Applications of the Integrals

    • Applications in finding the area under simple curves,
    • especially lines,
    • arcs of circles/parabolas/ellipses (in standard form only),
    • area between the two above said curves (the region should be clearly identifiable).
  5. Differential Equations

UNIT IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY
  1. Vectors

  2. Three-dimensional Geometry

Unit V: Linear Programming
Unit VI: Probability
  • Multiplication theorem on probability.
  • Conditional probability, independent events, total probability, Baye’s theorem.
  • Random variable and its probability distribution, mean and variance of a random variable.
  • Repeated independent (Bernoulli) trials and Binomial distribution.

Section B2: Applied Mathematics

Unit I: Numbers, Quantification and Numerical Applications
  1. Modulo Arithmetic

    • Define modulus of an integer
    • Apply arithmetic operations using modular arithmetic rules
  2. Congruence Modulo

    • Define congruence modulo
    • Apply the definition in various problems
  3. Allegation and Mixture

    • Understand the rule of allegation to produce a mixture at a given price
    • Determine the mean price of a mixture
    • Apply rule of allegation
  4. Numerical Problems

    • Solve real life problems mathematically
  5. Boats and Streams

    • Distinguish between upstream and downstream
    • Express the problem in the form of an equation
  6. Pipes and Cisterns

    • Determine the time taken by two or more pipes to fill or drain
  7. Races and Games

    • Compare the performance of two players w.r.t. time, distance taken/distance covered/ work done from the given data
  8. Partnership

    • Differentiate between active partner and sleeping partner
    • Determine the gain or loss to be divided among the partners in the ratio of their investment with due consideration of the time
  9. Numerical Inequalities

    • Describe the basic concepts of numerical inequalities
    • Understand and write numerical inequalities
UNIT II: ALGEBRA
  1. Matrices and Types of Matrices

    • Define matrix
    • Identify different kinds of matrices
  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
UNIT III: CALCULUS
  1. Higher Order Derivatives

    • Determine second and higher order derivatives
    • Understand differentiation of parametric functions and implicit functions
    • Identify dependent and independent variables
  2. Marginal Cost and Marginal Revenue Using Derivatives

    • Define marginal cost and marginal revenue
    • Find marginal cost and marginal revenue
  3. 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
UNIT IV: PROBABILITY DISTRIBUTIONS
  1. Probability Distribution

    • Understand the concept of Random Variables and its Probability Distributions
    • Find probability distribution of discrete random variable
  2. Mathematical Expectation

    • Apply arithmetic mean of frequency distribution to find the expected value of a random variable
  3. Variance

    • Calculate the Variance and S.D. of a random variable
UNIT V: INDEX NUMBERS AND TIME BASED DATA
  1. Index Numbers

    • Define Index numbers as a special type of average
  2. Construction of Index Numbers

    • Construct different types of index numbers
  3. Test of Adequacy of Index Numbers

    • Apply time reversal test
  4. Time Series

    • Identify time series as chronological data
  5. Components of Time Series

    • Distinguish between different components of time series
  6. Time Series Analysis for Univariate Data

    • Solve practical problems based on statistical data and interpret
UNIT VI: INFERENTIAL STATISTICS
  1. Population and Sample

    • Define Population and Sample
    • Differentiate between population and sample
    • Define a representative sample from a population
  2. Parameter and Statistics and Statistical Inferences

    • 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
UNIT VII: FINANCIAL MATHEMATICS
  1. Perpetuity, Sinking Funds

    • Explain the concept of perpetuity and sinking fund
    • Calculate perpetuity
    • Differentiate between sinking fund and saving account
  2. Valuation of Bonds

    • Define the concept of valuation of bond and related terms
    • Calculation of Bond Using Present Value Approach
  3. Calculation of EMI

    • Explain the concept of EMI
    • Calculate EMI using various methods
  4. 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
UNIT VIII: LINEAR PROGRAMMING
  1. Introduction and Related Terminology

    • Familiarize with terms related to Linear Programming Problem
  2. Mathematical Formulation of Linear Programming Problem

    • Formulate Linear Programming Problem
  3. Different Types of Linear Programming Problems

    • Identify and formulate different types of LPP
  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
  5. Feasible and Infeasible Regions

    • Identify feasible, infeasible and bounded regions
  6. Feasible and Infeasible Solutions, Optimal Feasible Solution

    • Understand feasible and infeasible solutions
    • Find optimal feasible solution

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