UNIT – I:

       Cayley-Hamilton theorem – statement only – Characteristic equation – Characteristic roots and Characteristic vectors – properties – problems

       Polynomial equations – theorems – every n^th  degree equation has got exactly n roots – An equation with rational coefficients has irrational roots in conjugate pairs – an equation with real coefficients has imaginary roots in conjugate pairs – an conjugate pairs – (statement only) – problems – Relation between roots and coefficients

UNIT – II:

       Transformation of equations – roots with sign changed – roots multiplied by a given number – squares of the roots – increasing or decreasing the roots of a given equation by a given number – removing the second term of a given equation – transformations in general – problems

UNIT – III:

       Expansions of sin nθ, cos nθ, tan nθ – Expansions of sinⁿ θ, cosⁿθ, - Expansions of sinθ, cosθ, tanθ in terms of θ – problems – Hyperbolic and inverse hyperbolic functions – properties – problems – Logarithm of a complex number

UNIT – IV:

       Derivatives – Definition of a derivative, differentiation techniques – Standard formulae – differentiation of Implicit functions – successive differentiation – The n^th derivative – Standard results – Leibnitz formula for the nth derivative and applications – Meaning of the derivative – Simple problems for all the above sections.

UNIT – V:

       Partial derivatives – Definition – successive partial derivatives – Function of a function rule – Total differential co-efficient – Implicit functions – Homogeneous functions and Euler’s Theorem – problems – Curvature, radius of curvature in Cartesian and polar co-ordinates only – simple problems.

Reference Books:

1.   T.K. Manickavasagam pillai and S. Narayanan: ALGEBRA (Vol I) Year of Publication 2004. Vijay Nicole Imprints Pvt Ltd, C-7, Nelson Chambers, 115, Nelson Manickam Road, Chennai - 29

2. P.R. Vittal.: ALGEBRA, ANALYTICAL GEOMETRY AND TRINOMETRY – Year of Publication 

    2000 Margham Publications, 24, Rmaeswaram Road, T.Nagar, Chennai – 17

3. T.K. Manickavasagam pillai and S. Narayanan: TRIGONOMETRY Year of Publication 2004.

     Vijay Nicole Imprints P Ltd, C-7, Nelson Chambers, 115, Nelson Manickam Road,

    Chennai – 29

4. N.P. Bali – TRIGONOMETRY – Year of Publication 1994. Krishna Prakasn Mandhir, 9,  

    Shivaji Road, Meerut (UP)

5. T.K. Manickavasagam pillai and S. Narayanan.: CALCULUS (Vol I) Year of Publicaton  

    2004. Vijay Nicole Imprints P Ltd, C-7, Nelson Chambers, 115, Nelson Manickam Road,   

    Chennai – 29

6. P.R. Vittal.: CALCULUS – Year of Publication 2000. Margham Publications, 24,

    Rameswaram Road, T.Nagar, Chennai - 17

UNIT – I:

       Definition of integration – List of standard formulae – Different types of integration – Integration by substitution – Integration of rational functions – Integration of irrational functions – Integration by partial fractions – Integrals of the type -  and  Integration by parts – simple problems for all the above sections.

UNIT – II:

       Definition – Properties of definite integral with problems – Reduction formulae – Bernoulli’s Formula – Reduction formula for       simple problems for all the above sections

UNIT – III:

       Differential Equations – Equations of first order and higher degree – Equations solvable for p- solvable for x- solvable for y- Clairaut’s form – Exact differential equations and method of obtaining solution to an exact differential equation – problems.

       Second order differential equations with constant coefficients – Particular integrals of e^ax V where is of the form x, x², sin ax and cos ax – problems – Second order differential equations with variable coefficients.

UNIT – IV:

       Formation of Partial differential Equations by eliminating arbitrary constants and arbitrary functions – Non-linear differential equations of first order – definition – Complete, Particular, singular and general integrals – Solutions of the Partial Differential Equations of standeard types – Clairaut’s form, equations reducible to the Calairaut’s form – simple problems. Lagrange’s Linear Partial differential equations Pp+Qq=R – simple problems only

UNIT – V:

       Laplace transforms – definition – standard formulae – elementary therorems with proof – problems – Inverse Laplace transforms – standard formulae – elementary theorems – problems – applications to solving second order differential equations with constant coefficients.

Reference Books:

1.   T.K. Manikkavasagam & others: CALCULUS (Vol II) year of Publication 2004. Vijay Nicole  Imprints P Ltd, C-7, Nelson Chambers, 115, Nelson Manickam Road, Chennai – 29

2.   P.R. Vittal.: CALCULUS – Year of Publication 2000 Margham Publications, 24, Rameswaram Road, T.Nagar – 17

3.   N.P. Bali: CALCULUS: Year of Publication 1994 Krishna Prakasan Mandhir, 9, Shivaji Road, Meerut (UP)

4.   Dr. P.R. Vittal – DIFFERENTIAL EQUATIONS AND LAPLACE TRANSFORMS Year of Publication 2002. Margham Publications, 24, Rameswaram Road, T.Nagar – 17

5.   S.Narayanan and others – CALCULUS (Vol III) Year of Publication 2004. Vijay Nicole Imprints P Ltd, C-7, Nelson Chambers, 115, Nelson Manickam Road, Chennai – 29

6.   S.Sankarappan and S.Kalavathi – DIFFERENTIAL EQUATIONS AND LAPLACE TRANSFORMATIONS – Year of Publication 2004. Vijay Nicole Imprints P Ltd, C-7, Nelson Chambers, 115, Nelson Manickam Road, Ch-29

UNIT – I:

       Random Variable – Discrete and continuous – Distribution Functions – Marginal and Conditional Distributions – Mathematical expectation, Moment generating function – Characteristic function – Chebechev’s inequality.

UNIT – II:

       Standard distributions – Binomial, Poisson, Rectangular and Normal distributions. Exact Sampling distributions; chi-square distribution, Students ‘t’ distribution – Fisher’s ‘t’ distribution, F distribution – Relationship between them.

UNIT – III:

       Correlation and regression – Correlation coefficient, Rank correlation coefficient, Regression Lines, Regression coefficient – Partial and multiple correlation coefficient (for 3 variables only), Curve Fitting, Fitting of a straight line – Fitting of a second degree parabola – Fitting of power curve – exponential curve.

UNIT – IV:

       Concept of population, sample, statistics – parameter, point estimation – Concept of point estimation – Consistency, Unbiasedness efficiency (Cramer-Rao Inequality) and sufficiency (Rao-Balackwell theorem). Methods of estimation, Maximum Likelihood, Moments and Minimum chi-square methods, properties of these estimators – Interval estimation (concept only).

UNIT – V:

       Tests of significance – Large sample test with regard to proportion, mean, difference between means and proportions – Small sample tests based on ‘t’ F and Chi-square tests.

Text Book:

1.   S.C. Gupta and V.K. Kapoor, Nineth Revised edition (2001), Fundamentals of

     Mathematical Statistics – Sultan chand, New Delhi

Reference Book:

1.   Gupta C.B. and Vijay Gupta, (1998), An introduction to Statistical Methods – Sultan

     Chand, New Delhi

UNIT – I:

       Definition of Gradient of a Scalar point function – Directional derivative of a vector point function – Unit normal vector – divergence and Curl of a vector point function – Definitions – solenoidal and irrotational Vectors – problems.

UNIT – II:

       Fourier series – definition – to find the Fourier coefficients of periodic functions of period 2∏ - even and odd functions – half range series – problems.

UNIT –III:

       Introduction – Fourier integral representation – Fourier integral theorems – statement only – sine and cosine integral representations – transformation of elementary functions – properties of Fourier transforms – linearity property – change of scale – shifting property – simple problems.

UNIT – IV:                 

       Probability – Probabilities and Events – Conditional probability – Random Variables and Expected values – Convergence and Correlation – Continuous Random Variables – Normal Random Variables – Properties of Normal Random Variables – The Central Limit  Theorem – Simple problems – Geometric Brownian Motion – G.B.M. as a limit of simple models – Brownian Motion – Simple problems – Interest rates – present value analysis – Rate of return – continuation of varying interest rates – An example of option pricing – other examples of pricing via arbitrage.

UNIT – V:

       The Arbitrage theorem – The multiperiod Binomial model – proof of the Arbitrage Theorem – Black Scholes formula – properties of the Black – Scholes option cost – Derivation of Black Scholes formula – simple problems Additional results on options – Call options on Dividend paying Securities – Pricing American put options – Adding jumps to Geometric Brownian motion – Estimating the Volatility Parameter – Simple problems

TEXT BOOKS:

1.   P.R. Vittal and V. Malini: VECTOR CALCULUS – Year of Publication 1997

Margham Publications, 24, Rameswaram Road, T.Nagar, Chennai – 17

2.   P.R. Vittal: CALCULUS – Year of Publication 2000,  Margham Publications, 24,

     Rameswaram Road, T.Nagar, Chennai – 17

3. Sheldon M.Ross – AN ELEMENTARY INTRODUCTION TO MATHEMATICAL FINACNE 2^ND  

    Edition – Cambridge University Press – 2005

4. P. Duraipandian: VECTOR CALCULUS – Year of Publication 1984. S. Viswanathan & CO,

   38, Mcnicals Road, Chetput, Chennai - 31

5. K. Viswanathan & S. Selvaraj: VECTOR CALCULUS – Year of Publication 1984. Emerald Publishers, 135, Annasalai, Chennai – 2

1.   T.K. Manickavasagam pillai and S. Narayanan: CALCULUS (Vol III) – Year of Publication 2004. Vijay Nicole Imprints P Ltd, C-7, Nelson Chambers, 115, Nelson Manickam Road, Ch-29

UNIT – I:

              Introduction – Force – definition – Parallelogram Law of forces, Triangular Law of forces and converse of Triangular Law of forces. – Lami’s theorem – problems – Like and unlide parallel forces – Problems – Moments – definition – Varignon’s theorem – problems. Couples – Definition of Couple – moment of a Couple – Theorems – Problems three forces acting on a rigid body – Coplanar forces – General conditions of equilibrium of coplanar forces acting on a rigid body – problems.

UNIT – II:

       Introduction – Friction – definition – coefficient of friction – Limiting friction – Angle of friction and Cone of friction – Laws of friction – Equilibrium of a particle on a rough inclined plane under any force – problems

       Common catenary – definition – sag and span – Intrinsic, Parametric and Cartesian equations of a catenary – properties – Suspension bridge – approximation to the shape of a catenary – problems

UNIT - III

       Introduction – Velocity – definition – resultant velocity – parallelogram law – acceleration – Motion of a particle along a straight line under uniform acceleration – problems – Simple Harmonic Motion – Definition – Equations of S.H.M. – Properties of S.H.M. – problems – Impulse and impulsive force – definitions only – Principle of Conservation of linear momentum – Newton’s experimental law – Change in K.E. is equal to Work Done – Direct and oblique impact of two smooth spheres – Problems – Impact of a smooth sphere on a fixed plane – problems

UNIT – IV:

       Introduction – Projectile, Trajectory, horizontal range, velocity of projection and angle of projection – definitions – The path of a projectile in a parabola – Range and time of flight on a horizontal plane – Problems – Range and time of flight on an inclined plane – problems.

UNIT – V:

       Standard results of Moments of Inertia of standard bodies – Parallel axes theorem and perpendicular axes theorem – Statement only – motion of a rigid body about a fixed horizontal axis – K.E. – Moment of Momentum – Equation of Motion

TEXT BOOK:

1. P. Duraipandian: MECHANICS Year of Publication 1984, Emerald Publishers, 135,
    Anna Salai, Ch – 2

Reference Books:

1.   S. Narayanan – STATICS: Year of Publication 1986, S. Chand & Co. Chennai

2.   Dr. M. K. Venkataraman – STATICS Year of Publication 1994, Agastjoar Publications, 9A, Clives Building, 33, Nandhi Koil St, Theppakulam (Post), Trichy – 620002

3.   Dr. M.K. Venkataraman  -DYNAMICS  Year of Publication 1994, Agastjoar Publications, 9A, Clives Building, 33, Nandhi Koil St, Theppakulam (Post), Trichy – 620002

UNIT – I:

       Introduction – accounting concepts and conventions – journal ledger – subsidiary books – trial balance – final accounts of a sole trader with adjustments.

UNIT – II:

       Bills of exchange – accommodation bills – average due date – account current

UNIT – III:

       Final accounts of non-trading concerns – receipts and payments account – income and expenditure account – balance sheet.

UNIT – IV:

       Single entry system – statement of affairs method – conversion method – self balancing ledger.

UNIT – V:

       Branch accounts – dependent branch stock and debtors system – independent including foreign branch.

Text Book:

1.   R.L. Gupta and V.K. Gupta – Financial Accounting – Sultan Chand, New Delhi.

 

UNIT – I:

 

       Subgroups – Definition – Examples – Properties – problems – Cosets – Lagrange’s Theorem- Corollary – Normal Subgroups – Quotient groups – Definition – Properties – Examples – Problems.

 

       Homomorphism – Definition – Examples – properties – Kernel of a Homomorphism – properties. Isomorphism – Definition – properties – Fundamental theorem on Homomorphism.

 

UNIT – II:

 

       Rings – Definition – Examples – Ring of real Quaternion – Properties – Special Classes of Rings – Zero divisor – Integral domain – definition – Properties – Unit – Division ring – Field – Defintion – Examples – Properties – Ring of Gaussian Integers – Boolean Ring – Sub rings and Sub-field – Definition – Examples – Properties – Ideals – Definition – Examples – Properties – Quotient rings – Principal Ideal – Maximal Ideal, Prime Ideal – Definition – Properties – The characteristic of an integral domain – Definition – Properties.

 

UNIT – III:

 

       Vector Space – definition and simple properties – example – subspaces – Quotient spaces definition – Sums & direct sums – definition – Linear dependence and Linear independence of vectors – Definition – problems – linear span L(S) – Basis and Dimension – definition – properties – theorems – Homomorphism – definition – Isomorphism – theorems – Kernel of a homomorphism – simple theorems.

 

UNIT – IV:

 

       Definition – properties – examples – norm of a vector – Schwarz inequality – triangle inequality – parallelogram law – orthogonal vectors – definition – orthogonal complement – properties – orthonormal set – definition – properties – simple theorems – problems.

 

UNIT - V:

 

       The algebra of Linear transformations – definition – theorems – minimal polynomial – Invertible and Singular transformations – examples – Rank of a linear transformation – theorems – problems – Eigen values and Eigen vectors – definition – theorems – problems.

 

 

Text Books:

 

1.   M.L. Santiago – MODERN ALGEBRA – Year of Publication 1994, Tata Mc Graw – Hill, New  Delhi

 

Reference Books:

 

1.   Dr. R. Bala Krishnan and Dr. N. Ramabadran – A TEXT BOOK OF MODERN ALGEBRA –

     Year of Publication 1994, Vikas Publishing House, NewDelhi

 

2.   A.R. Vasistha – A FIRST COURSE IN MODERN ALGEBRA – Year of Publication 1983,

     Krishna Prakasan Mandhir, 9, Shivaji Road, Meerut (UP)

 

3.   I.N. Herstein – TOPICS IN ALGEBRA – 2nd edition, Year of Publication 1975, John Wiley,

     New York

 

4. K. Viswanatha Naik – MODERN ALGEBRA – Year of Publication 1988, Emerald Publishers,

    135, Anna Salai, Chennai – 2

UNIT – I:

       Field of Real numbers – Axioms – Bounded sets – lub & glb – Completeness axioms – Archimedean property – Definition of Rational numbers – Monotone sequences – Theorem on nested intervals – Dedikind’s cut property – square roots – absolute value.

       Bounded sequences – Null Sequence – Convergent sequence – Subsequences – Bolzano – Wirestrass Theorem – Cauchy’s Criterion for convergence – Limit superior and limit inferior of a bounded sequence.

UNIT – II:

       Intervals – Closed sets – open sets – Neighbourhoods – finite and infinite sets – Heine – Borel covering theorem – Limit of a function at a point – Deleted neighbourhoods – Limits and continuity – Characterization of limits – Algebra of limits.

       Continuity of a function at a point – Algebra of continuity – examples – one sided continuity – composition – continuous function on an interval – Intermediate value theorem – Continuous function on a closed interval – Monotonic Continuous Functions – Inverse function theorems – Uniform continuity.

UNIT – III:

       Functions of a complex variable – limit of a function at a point – theorems on limits – continuity – derivatives – Cauchy – Riemann equations – necessary and sufficient conditions – analytic function – examples – harmonic function – properties – to find an analytic function whose real of imaginary part is given.

UNIT – IV:

       Simply – connected domain – Cauchy’s fundamental theorem – proof using Goursat’s lemma – Cauchy’s theorem for multiply connected domains – Cauchy’s integral formula & Cauchy’s formula for the first derivative – Morera’s theorem.

       Cauchy’s inequality – Liouville’s theorem – Fundamental theorem of Algebra – Maximum modulus theorem – Taylor’s series – Laurent’s series – problems.

UNIT – V:

       Singularities – types of singularities – isolated singularity – removable singularity – pole – essential singularity – determination of the nature of singularity – residue – definition – calculation of residues – Cauchy’s Residue theorem – Contour integration – integration around a unit circle – integration along the real axis – Jordan’s lemma (stalemate only) – integration of functions with poles on the real axis.

Text Book:

       For Units I and II

1.   Sterling K. Barberian – A FIRST COURSE IN REAL ANALYSIS – Year of Publication 2004, Springer (India P Ltd, New Delhi).

For Units III & IV

1.   Ruel V Churchill & James ward Brown – COMPLEX VARIABLES AND APPLICATIONS (IV – edition ) Year of Publication 1986

McGraw Hill International Book Company, New York.

Reference Books:

1.   Richard R. Goldberg – METHODS OF REAL ANALYSIS – Year of Publication 1970 IBM Publishing, New Delhi

2.   P. Duraipandian  Laxmi Duraipandian, & D. Muhilan – COMPLEX ANALYSIS – Year of Publication 1988, Emerald Publishers, 135, Anna Salai, Chennai – 2

UNIT – I:

              Introduction – Definition of O.R. – Scope, phases and Limitations of O.R. – Linear Programming Problem – Graphical Method – Definitions of bounded, unbounded and optimal solutions – procedure of solving LPP by graphical method – problems – Simplex technique Definitions of Basic, nonbasic variables – basic solutions – slack variables and optimal solution, simplex procedure of solving LPP – problems. Two – Phase Simplex method – Procedure of solving an LPP by two-phase simplex method – problems.

      UNIT – II:

              Introduction – Balanced and unbalanced T.P, Feasible solution – Basic feasible solution – Optimum solution – degeneracy in a T.P, - Mathematical formulation – North – West Corner rule – Vogell’s approximation method (unit penalty method) Method of Matrix minima (Least cost Method) – problems – algorithm of Optimality test (Modi Method) – Problems.

              Introduction – Definition of Assignment problem, balanced and unbalanced assignment problem – restrictions on assignment problem – Mathematical formulation – formulation and solution of an assignment problem (Hungarian method) – degeneracy in an assignment problem – problems

UNIT – III:

       Introduction – Definition – Basic assumptions – n jobs to be operated on two machines – problems – n-jobs to be operated on three machines – problems – n-jobs to be operated on m machines – problems

       Definition of Inventory models – Type of Inventory models:

(i)                   Uniform rate of demand, infinite rate of production and no shortage

(ii)                  Uniform rate of demand, finite rate of replacement and no shortage

(iii)                 Uniform rate of demand instantaneous production with shortage – Books works – Problems.

UNIT – IV:

       Definitions – Newspaper boy problem – Inventory model with one price break – problems. Introduction – definition of steady state, transient state and queue discipline, characteristics of a queuing model – Applications of queuing model – Little’s formula – Classification of queues – Poisson process – properties of Poisson process.

Models

(i)                   (M/M/1) : (µ / FCFS)

(ii)                  (M/M/1) : (N / FCFS)

(iii)                 (M/M/S) : (µ / FCFS) – problems

UNIT – V:

       Introduction – definition of network, event, activity, optimistic time, pessimistic time, the most likely time, critical path, total float and free float – Difference between slack and float – phases of critical path in a PERT network – difference between CPM and PERT – problems.

Text Book:

1.   P.K. Gupta, Manmohan and Kanti swarup – OPERATIONS RESEARCH 9^TH edition, 2001 –

     Sultan Chand & Sons, Chennai

Reference Books:

1.   S. Kalavathy – OPERATIONS RESEARCH 2^ND EDITION – 2002 Publishing House P Ltd,

NewDelhi

2.   P.K. Gupta and D.S. Hira – OPERATIONS RESEARCH 2^nd edition, 1986 – S Chand & Co

Newdelhi

3.   Hamdy Taha – OPERATIONS RESEARCH Year of Publication 1996. Prentice-Hall

     Publications, New Delhi

UNIT – I:

       Structure of C program – character set – constants – variables – operator – Hierarchy of arithmetic operations – writing simple C programs.

UNIT – II:

       Control structure: Logical If–If-Else structure Nested If-Else-GO TO-SWITCH structure – LOOP structures:  WHILE–DO-WHILE-FOR loop – NESTED loops.

UNIT – III:

       Arrays declaration – Array processing – functions: Naming – Declaration – library functions.

UNIT – IV:

       Algebraic equations: Newton – Raphson method, Bisection method, Regula-falsI method. Matrices and linear algebraic equations: Gauss Elimination – matrix inversion – Gauss Siedel method.

UNIT – V:

       Interpolation: Lagrange’s interpolation, Differential Equations: Euler’s method – Runge – Kutta method – Predictor – corrector method. Numerical integration: Simpson’s 1/3 Rule – Trapezoidal rule.

Text Book:

1.   C.Xavier: C LANGUAGE AND NUMERICAL METHODS, New Age international limited, New

     Delhi, 1999

Reference Books:

1.   Kernighan B.W. and Retchie D.M., THE PRORAMMING LANGUAGE, Prentice-Hall India,

     New Delhi, 1977

2.   S.S. Sastry: INTRODUCTORY METHODS OF NUMERICAL ANALYSIS, 3^rd Edn., Prentice-

      Hall of India, New Delhi, 1999

UNIT – I:

       Introduction – probability Axioms – conditional probability – Baye’s theorem – independent events – problems.

UNIT – II:

       Random variable – probability distribution of a random variable – Discrete and continuous variables – problems.

UNIT – III:

       Expected value – Functions of a random variable – Moment generating functions – problems.

UNIT – IV:

       Two point distribution – Binomial distribution – Poisson distribution – Gamma distribution – Normal distribution – Chebychev’s inequality – problems.

UNTI – V:

       Regression model – one way analysis of variance – Two way analysis of variance – problems.

Text Book:

1.   V.K. Rokatgi – AN INTRODUCTION TO PROBABILITY THEORY AND MATHEMATICAL STATISTICS – Year of Publication 1985 Wiely Eastern publications, New Delhi, 1985.

Reference Book:

1.   Marek Fiseh – PROBABILITY THEORY AND MATHEMATICAL STAISTICS John Wiely and

     Sons, New York – 1956

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