Mathematics

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Mathematics
Note: There will be one Question Paper which will contain Two Sections i.e. Section A and Section B [B1 and B2].
Section A will have 15 questions covering both i.e. Mathematics/Applied Mathematics which will be compulsory for all candidates
Section B1 will have 35 questions from Mathematics out of which 25 questions need to be attempted
Section B2 will have 35 questions purely from Applied Mathematics out of which 25 question will be attempted

SECTION A

  1. Algebra
    1. Matrices and types of Matrices
    2. Equality of Matrices, transpose of a Matrix, Symmetric and Skew Symmetric Matrix
    3. Algebra of Matrices
    4. Determinants
    5. Inverse of a Matrix
    6. Solving of simultaneous equations using Matrix Method
  2. Calculus
    1. Higher order derivatives
    2. Tangents and Normals
    3. Increasing and Decreasing Functions
    4. Maxima and Minima
  3. Integration and its Applications
    1. Advertising
      1. Indefinite integrals of simple functions
      2. Evaluation of indefinite integrals
      3. Definite Integrals
      4. Application of Integration as area under the curve
  4. Differential Equations
    1. Order and degree of differential equations
    2. Formulating and solving of differential equations with variable separable
  5. Probability Distributions
    1. Random variables and its probability distribution
    2. Expected value of a random variable
    3. Variance and Standard Deviation of a random variable
    4. Binomial Distribution
  6. Linear Programming
    1. Mathematical formulation of Linear Programming Problem
    2. Graphical method of solution for problems in two variables
    3. Feasible and infeasible regions
    4. Optimal feasible solution

Section B1: Mathematics

UNIT1: RELATIONSAND FUNCTIONS

  1. Relations and Functions
    Typesofrelations:Reflexive,symmetric,transitive and equivalence relations.One toone and onto functions,compositefunctions,inverseofa function.Binaryoperations
  2. InverseTrigonometricFunctions
    Definition,range, domain, principal value branches. Graphs ofinverse trigonometric functions. Elementarypropertiesofinverse trigonometric functions.

UNIT2: ALGEBRA

  1. Matrices
    Concept,notation,order, equality,typesofmatrices, zeromatrix,transpose of amatrix,symmetric andskewsymmetricmatrices.Addition,multiplicationandscalarmultiplicationofmatrices,simple propertiesof addition,multiplicationandscalarmultiplication.Non-commutativityofmultiplication ofmatrices and existence of non-zeromatriceswhose productisthe zeromatrix (restrictto square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proofoftheuniquenessofinverse,ifit exists;(Here allmatriceswillhave real entries).
  2. Determinants
    Determinantof asquarematrix (upto3×3matrices),propertiesofdeterminants,minors, cofactors and applications of determinantsin finding the area of a triangle. Adjoint and inverse of a square matrix.Consistency,inconsistencyandnumberofsolutionsofsystemoflinearequationsbyexamples, solvingsystemoflinear equationsin twoorthreevariables(havingunique solution)usinginverse of amatrix.

    3. UNIT 3: CALCULUS

    1. Continuity and Differentiability
      Continuityand differentiability,derivative of composite functions, chainrule, derivativesofinverse trigonometricfunctions,derivativeofimplicitfunction.Conceptsofexponential,logarithmicfunctions. Derivativesoflog x ande x .Logarithmicdifferentiation.Derivativeoffunctions expressedinparametric forms. Second-order derivatives.Rolle’s and Lagrange’s Mean ValueTheorems(without proof) and their geometric interpretations.
    2. Applications of Derivatives
      Applicationsofderivatives:Rateof change,increasing/decreasingfunctions,tangents andnormals, approximation,maximaandminima(firstderivativetestmotivatedgeometricallyandsecondderivative test given as a provable tool).Simple problems(thatillustrate basic principles and understandingof the subject aswell asreal-life situations). Tangent and Normal.
    3. Integrals
      Integrationasinverseprocessofdifferentiation.Integration of avarietyoffunctionsbysubstitution, bypartialfractions and byparts, onlysimple integrals ofthe type –

      to be evaluated.

      Definite integrals as a limit of a sum. Fundamental Theorem of Calculus(without proof). Basic propertiesofdefinite integrals andevaluationofdefinite integrals.
    4. Applications of the Integrals
      Applicationsin finding the area undersimple curves, especially lines, arcs of circles/parabolas/ellipses(in standard formonly), area between the two above said curves(the region should be cleraly identifiable).
    5. DifferentialEquations
      Definition,order and degree, general andparticularsolutions of adifferential equation.Formationof differential equationwhose generalsolution is given.Solution of differential equations bymethod of separation of variables, homogeneous differential equations offirst order and first degree. Solutions oflinear differential equation ofthe type –

UNIT 4: VECTORSAND THREE-DIMENSIONALGEOMETRY

  1. Vectors
    Vectors and scalars,magnitude and direction of a vector. Direction cosines/ratios of vectors.Types of vectors(equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Scalar(dot) product of vectors, projection of a vector on a line.Vector(cross) product of vectors,scalartriple product.
  2. Three-dimensional Geometry
    Direction cosines/ratios of a line joiningtwo points.Cartesian andvector equation of a line, coplanar and skewlines,shortest distance between two lines.Cartesian and vector equation of a plane.Angle between (i)two lines,(ii)two planes,(iii) a line and a plane.Distance of a pointfroma plane.

Unit 5 :LinearProgramming


Introduction,relatedterminologysuchas constraints,objective function,optimization,differenttypes oflinearprogramming(L.P.)problems,mathematicalformulationofL.P.problems,graphicalmethod ofsolution for problemsin two variables, feasible and infeasible regions, feasible and infeasible solutions,optimalfeasiblesolutions(uptothreenon-trivial constrains).

Unit 6: Probability

Multiplicationstheoremonprobability.Conditionalprobability,independent events,totalprobability, Baye’stheorem.Randomvariable and its probability distribution,mean and variance of haphazard variable.Repeatedindependent(Bernoulli)trials andBinomialdistribution.

Section B2: Applied Mathematics

Unit 1: 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 variousproblems
  3. Allegation andMixture
    • Understand the rule of allegation to produce a mixture at a givenprice
    • Determine the mean price of amixture
    • Apply rule of allegation
  4. Numerical Problems
    • Solve real life problems mathematically
  5. Boats and Streams
    • Distinguish between upstreamand downstream
    • Express the problem in the formof an equation
  6. Pipes and Cisterns
    • Determine the time taken bytwo or more pipes to fill or
  7. Races and Games
    • Compare the performance oftwo players w.r.t. time,
    • Distance taken/distance covered/ Work done from the givendata
  8. Partnership
    • Differentiate between activepartner and sleeping partner
    • Determine the gain or loss tobe divided among the partners in the ratio of their investment with due
    • Consideration of the time volume/surface area for solid formed using two ormore shapes
  9. Numerical Inequalities
    • Describe the basic concepts ofnumerical inequalities
    • Understand and write numericalinequalities

UNIT 2: ALGEBRA

  1. Matrices andtypes of matrices
    • Define matrix
    • Identify different kinds ofmatrices
  2. Equality of matrices, Transpose of amatrix, Symmetric andSkew symmetric matrix
    1. Determine equality of twomatrices
    2. Write transpose of given matrix
    3. Define symmetric and skewsymmetric matrix

UNIT 3: CALCULUS

  1. Higher Order Derivatives
    • Determine second and higherorder derivatives
    • Understand differentiation ofparametric functions and implicit functions Identify dependent andindependent variables
  2. Marginal Cost and Marginal Revenue usingderivatives
    • Define marginal cost andmarginal revenue
    • Find marginal cost and marginalrevenue
  3. Maxima andMinima
    • Determine critical points of thefunction
    • Find the point(s) of local maxima and local minima and corresponding local maximumand local minimum values
    • Find the absolute maximum and absolute minimum value of afunction

UNIT 4: PROBABILITY DISTRIBUTIONS

  1. Probability Distribution
    • Understand the concept of Random Variables and its Probability Distributions
    • Find probability distribution ofdiscrete 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 5: INDEX NUMBERS AND TIME BASED DATA

  1. Index Numbers
    • Define Index numbers as aspecial type of average
  2. Construction of Index numbers
    • Construct different type of indexnumbers
  3. Test of Adequacy of Index Numbers
    • Apply time reversal test

UNIT 6: INDEX NUMBERS AND TIME BASED DATA

  1. Population and Sample
    • Define Population and Sample
    • Differentiate between population and sample
    • Define a representative samplefrom a population
  2. Parameter andStatistics and Statistical Interferences
    • Define Parameter with reference to Population
    • Define Statistics with referenceto Sample
    • Explain the relation betweenParameter and Statistic
    • Explain the limitation of Statisticto generalize the estimation for population
    • Interpret the concept of Statistical Significance andStatistical Inferences
    • State Central Limit Theorem
    • Explain the relation betweenPopulation-Sampling Distribution-Sample
  3. Test of Adequacy of Index Numbers
    • Apply time reversal test

UNIT 7: INDEX NUMBERS AND TIME BASED DATA

  1. Time Series
    • Identify time series aschronological data
  2. Components of Time Series
    • Distinguish between differentcomponents of time series
  3. Time Series analysis for univariate data
    • Solve practical problems basedon statistical data and Interpret

UNIT 8: FINANCIAL MATHEMATICS

  1. Perpetuity, Sinking Funds
    • Explain the concept of perpetuity and sinking fund
    • Calculate perpetuity
    • Differentiate between sinkingfund and saving account
  2. Valuation of Bonds
    • Define the concept of valuationof bond and related terms
    • Calculate value of bond usingpresent 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 linearmethod of Depreciation
    • Interpret cost, residual value and useful life of an asset fromthe given information
    • Calculate depreciation

UNIT 9: LINEAR PROGRAMMING

  1. Introductionand relatedterminology
    • Familiarize with terms related toLinear Programming Problem
  2. Mathematical formulation of Linear Programming Problem
    • Formulate Linear Programming Problem
  3. Different types of Linear Programming Problems
    • Identify and formulate differenttypes 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 andinfeasible solutions, optimal feasible solution
    • Understand feasible andinfeasible solutions
    • Find optimal feasible solution