PNG  IHDR;IDATxܻn0K )(pA 7LeG{ §㻢|ذaÆ 6lذaÆ 6lذaÆ 6lom$^yذag5bÆ 6lذaÆ 6lذa{ 6lذaÆ `}HFkm,mӪôô! x|'ܢ˟;E:9&ᶒ}{v]n&6 h_tڠ͵-ҫZ;Z$.Pkž)!o>}leQfJTu іچ\X=8Rن4`Vwl>nG^is"ms$ui?wbs[m6K4O.4%/bC%t Mז -lG6mrz2s%9s@-k9=)kB5\+͂Zsٲ Rn~GRC wIcIn7jJhۛNCS|j08yiHKֶۛkɈ+;SzL/F*\Ԕ#"5m2[S=gnaPeғL lذaÆ 6l^ḵaÆ 6lذaÆ 6lذa; _ذaÆ 6lذaÆ 6lذaÆ RIENDB` ࡱ> B[ bjbj >"ΐΐ,1118iD1vfQL$܂~`Z2Z2Z20fQfQfQZ2\ fQZ2fQfQt_y1"M1=< v*F0vvޅJޅT_y_yޅ}ydv$TfQX)D-xMvZ2Z2Z2Z2ޅ : @9NEW SCHEME Scheme of Examination of B.A. 1st Semester Mathematics (w.e.f. 2018-2019) Paper CodeTitle of the PaperAllocation of Periods  Maximum MarksTheory Internal AssessmentTotalBM 111 Algebra 6 periods/ 4 hours per week 276 100BM 112Calculus 6 periods/ 4 hours per week277BM 113Solid Geometry 6 periods/ 4 hours per week267 Note:- The other conditions will remain the same as per relevant ordinance and rules and regulations of the University. (w.e.f. 2018-19) Algebra Paper: BM 111 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions (each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Symmetric, Skew symmetric, Hermitian and skew Hermitian matrices. Elementary Operations on matrices. Rank of a matrices. Inverse of a matrix. Linear dependence and independence of rows and columns of matrices. Row rank and column rank of a matrix. Eigenvalues, eigenvectors and the characteristic equation of a matrix. Minimal polynomial of a matrix. Cayley Hamilton theorem and its use in finding the inverse of a matrix. Section II Applications of matrices to a system of linear (both homogeneous and nonhomogeneous) equations. Theorems on consistency of a system of linear equations. Unitary and Orthogonal Matrices, Bilinear and Quadratic forms. Section III Relations between the roots and coefficients of general polynomial equation in one variable. Solutions of polynomial equations having conditions on roots. Common roots and multiple roots. Transformation of equations. Section IV :EMBED Equation.3 Nature of the roots of an equation Descartes rule of signs. Solutions of cubic equations (Cardons method). Biquadratic equations and their solutions. Books Recommended : H.S. Hall and S.R. Knight : Higher Algebra, H.M. Publications 1994. Shanti Narayan : A Text Books of Matrices. Chandrika Prasad : Text Book on Algebra and Theory of Equations. Pothishala Private Ltd., Allahabad. (w.e.f. 2018-19) Calculus Paper: BM 112 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections (I-IV) will contain two questions (each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions (each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Definition of the limit of a function. Basic properties of limits, Continuous functions and classification of discontinuities. Differentiability. Successive differentiation. Leibnitz theorem. Maclaurin and Taylor series expansions. Section II Asymptotes in Cartesian coordinates, intersection of curve and its asymptotes, asymptotes in polar coordinates. Curvature, radius of curvature for Cartesian curves, parametric curves, polar curves. Newtons method. Radius of curvature for pedal curves. Tangential polar equations. Centre of curvature. Circle of curvature. Chord of curvature, evolutes. Tests for concavity and convexity. Points of inflexion. Multiple points. Cusps, nodes & conjugate points. Type of cusps. Section III : Tracing of curves in Cartesian, parametric and polar co-ordinates. Reduction formulae. Rectification, intrinsic equations of curve. Section IV : Quardrature (area)Sectorial area. Area bounded by closed curves. Volumes and surfaces of solids of revolution. Theorems of Pappus and Guilden. Books Recommended : Differential and Integral Calculus : Shanti Narayan. Murray R. Spiegel : Theory and Problems of Advanced Calculus. Schauns Outline series. Schaum Publishing Co., New York. N. Piskunov : Differential and integral Calculus. Peace Publishers, Moscow. Gorakh Prasad : Differential Calculus. Pothishasla Pvt. Ltd., Allahabad. Gorakh Prasad : Integral Calculus. Pothishala Pvt. Ltd., Allahabad. (w.e.f. 2018-19) Solid Geometry Paper: BM 113 Max. Marks: 5 x 4 = 20 1 x 6 = 6Total = 26 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections (I-IV) will contain two questions (each carrying 5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions (each carrying 1 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I : General equation of second degree. Tracing of conics. Tangent at any point to the conic, chord of contact, pole of line to the conic, director circle of conic. System of conics. Confocal conics. Polar equation of a conic, tangent and normal to the conic. Section II : Sphere: Plane section of a sphere. Sphere through a given circle. Intersection of two spheres, radical plane of two spheres. Co-oxal system of spheres Cones. Right circular cone, enveloping cone and reciprocal cone. Cylinder: Right circular cylinder and enveloping cylinder. Section III : Central Conicoids: Equation of tangent plane. Director sphere. Normal to the conicoids. Polar plane of a point. Enveloping cone of a coincoid. Enveloping cylinder of a coincoid. Section IV : Paraboloids: Circular section, Plane sections of conicoids. Generating lines. Confocal conicoid. Reduction of second degree equations. Books Recommended 1. R.J.T. Bill, Elementary Treatise on Coordinary Geometry of Three Dimensions, MacMillan India Ltd. 1994. 2. P.K. Jain and Khalil Ahmad : A Textbook of Analytical Geometry of Three Dimensions, Wiley Eastern Ltd. 1999. NEW SCHEME Scheme of Examination of B.A. 2nd Semester Mathematics (w.e.f. 2018-2019) Paper CodeTitle of the PaperAllocation of Periods  Maximum MarksTheory Internal AssessmentTotalBM 121 Number Theory and Trigonometry 6 periods/ 4 hours per week 276 100BM 122Ordinary Differential Equations 6 periods/ 4 hours per week277BM 123 Vector Calculus 6 periods/ 4 hours per week267 Note:- The other conditions will remain the same as per relevant ordinance and rules and regulations of the University. (w.e.f. 2018-19) Number Theory and Trigonometry Paper: BM 121 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions(each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I : Divisibility, G.C.D.(greatest common divisors), L.C.M.(least common multiple) Primes, Fundamental Theorem of Arithemetic. Linear Congruences, Fermats theorem. Wilsons theorem and its converse. Linear Diophanatine equations in two variables Section II : Complete residue system and reduced residue system modulo m. Eulers function Eulers generalization of Fermats theorem. Chinese Remainder Theorem. Quadratic residues. Legendre symbols. Lemma of Gauss; Gauss reciprocity law. Greatest integer function [x]. The number of divisors and the sum of divisors of a natural number n (The functions d(n) and ((n)). Moebius function and Moebius inversion formula. Section - III : De Moivres Theorem and its Applications. Expansion of trigonometrical functions. Direct circular and hyperbolic functions and their properties. Section IV : Inverse circular and hyperbolic functions and their properties. Logarithm of a complex quantity. Gregorys series. Summation of Trigonometry series. Books Recommended : S.L. Loney : Plane Trigonometry Part II, Macmillan and Company, London. R.S. Verma and K.S. Sukla : Text Book on Trigonometry, Pothishala Pvt. Ltd. Allahabad. Ivan Ninen and H.S. Zuckerman. An Introduction to the Theory of Numbers. (w.e.f. 2018-19) Ordinary Differential Equations Paper: BM 122 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions(each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I : Geometrical meaning of a differential equation. Exact differential equations, integrating factors. First order higher degree equations solvable for x,y,p Lagranges equations, Clairauts equations. Equation reducible to Clairauts form. Singular solutions. Section II : Orthogonal trajectories: in Cartesian coordinates and polar coordinates. Self orthogonal family of curves.. Linear differential equations with constant coefficients. Homogeneous linear ordinary differential equations. Equations reducible to homogeneous linear ordinary differential equations. Section III : Linear differential equations of second order: Reduction to normal form. Transformation of the equation by changing the dependent variable/ the independent variable. Solution by operators of non-homogeneous linear differential equations. Reduction of order of a differential equation. Method of variations of parameters. Method of undetermined coefficients. Section IV : Ordinary simultaneous differential equations. Solution of simultaneous differential equations involving operators x (d/dx) or t (d/dt) etc. Simultaneous equation of the form dx/P = dy/Q = dz/R. Total differential equations. Condition for Pdx + Qdy +Rdz = 0 to be exact. General method of solving Pdx + Qdy + Rdz = 0 by taking one variable constant. Method of auxiliary equations. Books Recommended : D.A. Murray : Introductory Course in Differential Equations. Orient Longaman (India) . 1967 A.R.Forsyth : A Treatise on Differential Equations, Machmillan and Co. Ltd. London E.A. Codington : Introduction to Differential Equations. S.L.Ross: Differential Equations, John Wiley & Sons B.Rai & D.P. Chaudhary : Ordinary Differential Equations; Narosa, Publishing House Pvt. Ltd. (w.e.f. 2018-19) Vector Calculus Paper: BM 123 Max. Marks: 5 x 4 = 20 1 x 6 = 6Total = 26 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections (I-IV) will contain two questions (each carrying 5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions (each carrying 1 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Scalar and vector product of three vectors, product of four vectors. Reciprocal vectors. Vector differentiation. Scalar Valued point functions, vector valued point functions, derivative along a curve, directional derivatives Section II Gradient of a scalar point function, geometrical interpretation of grad (EMBED Equation.3, character of gradient as a point function. Divergence and curl of vector point function, characters of DivEMBED Equation.3 and Curl EMBED Equation.3 as point function, examples. Gradient, divergence and curl of sums and product and their related vector identities. Laplacian operator. Section III Orthogonal curvilinear coordinates Conditions for orthogonality fundamental triad of mutually orthogonal unit vectors. Gradient, Divergence, Curl and Laplacian operators in terms of orthogonal curvilinear coordinates, Cylindrical co-ordinates and Spherical co-ordinates. Section IV Vector integration; Line integral, Surface integral, Volume integral. Theorems of Gauss, Green & Stokes and problems based on these theorms. Books Recommended: Murrary R. Spiegal : Theory and Problems of Advanced Calculus, Schaum Publishing Company, New York. Murrary R. Spiegal : Vector Analysis, Schaum Publisghing Company, New York. N. Saran and S.N. NIgam. Introduction to Vector Analysis, Pothishala Pvt. Ltd., Allahabad. Shanti Narayna : A Text Book of Vector Calculus. S. Chand & Co., New Delhi. NEW SCHEME Scheme of Examination of B.A. 3rd Semester Mathematics (w.e.f. 2018-2019) Paper CodeTitle of the PaperAllocation of Periods  Maximum MarksTheory Internal AssessmentTotalBM 231 Advanced Calculus6 periods/ 4 hours per week 276 100BM 232Partial Differential Equations 6 periods/ 4 hours per week277BM 233Statics6 periods/ 4 hours per week267 Note:- The other conditions will remain the same as per relevant ordinance and rules and regulations of the University. (w.e.f. 2018-19) Advanced Calculus Paper: BM 231 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions (each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Continuity, Sequential Continuity, properties of continuous functions, Uniform continuity, chain rule of differentiability. Mean value theorems; Rolles Theorem and Lagranges mean value theorem and their geometrical interpretations. Taylors Theorem with various forms of remainders, Darboux intermediate value theorem for derivatives, Indeterminate forms. Section II Limit and continuity of real valued functions of two variables. Partial differentiation. Total Differentials; Composite functions & implicit functions. Change of variables. Homogenous functions & Eulers theorem on homogeneous functions. Taylors theorem for functions of two variables. Section III Differentiability of real valued functions of two variables. Schwarz and Youngs theorem. Implicit function theorem. Maxima, Minima and saddle points of two variables. Lagranges method of multipliers. Section IV Curves: Tangents, Principal normals, Binormals, Serret-Frenet formulae. Locus of the centre of curvature, Spherical curvature, Locus of centre of Spherical curvature, Involutes, evolutes, Bertrand Curves. Surfaces: Tangent planes, one parameter family of surfaces, Envelopes. Books Recommended: C.E. Weatherburn : Differential Geometry of three dimensions, Radhe Publishing House, Calcutta Gabriel Klaumber : Mathematical analysis, Mrcel Dekkar, Inc., New York, 1975 R.R. Goldberg : Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi, 1970 Gorakh Prasad : Differential Calculus, Pothishala Pvt. Ltd., Allahabad S.C. Malik : Mathematical Analysis, Wiley Eastern Ltd., Allahabad. Shanti Narayan : A Course in Mathemtical Analysis, S.Chand and company, New Delhi Murray, R. Spiegel : Theory and Problems of Advanced Calculus, Schaum Publishing co., New York (w.e.f. 2018-19) Partial Differential Equations Paper: BM 232 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions(each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Partial differential equations: Formation, order and degree, Linear and Non-Linear Partial differential equations of the first order: Complete solution, singular solution, General solution, Solution of Lagranges linear equations, Charpits general method of solution. Compatible systems of first order equations, Jacobis method. Section II Linear partial differential equations of second and higher orders, Linear and non-linear homogenious and non-homogenious equations with constant co-efficients, Partial differential eqution with variable co-efficients reducible to equations with constant coefficients, their complimentary functions and particular Integrals, Equations reducible to linear equations with constant co-efficients. Section III Classification of linear partial differential equations of second order, Hyperbolic, parabolic and elliptic types, Reduction of second order linear partial differential equations to Canonical (Normal) forms and their solutions, Solution of linear hyperbolic equations, Monges method for partial differential equations of second order. Section IV Cauchys problem for second order partial differential equations, Characteristic equations and characteristic curves of second order partial differential equation, Method of separation of variables: Solution of Laplaces equation, Wave equation (one and two dimensions), Diffusion (Heat) equation (one and two dimension) in Cartesian Co-ordinate system. Books Recommended: D.A.Murray: Introductory Course on Differential Equations, Orient Longman, (India), 1967 Erwin Kreyszing : Advanced Engineering Mathematics, John Wiley & Sons, Inc., New York, 1999 A.R. Forsyth : A Treatise on Differential Equations, Macmillan and Co. Ltd. Ian N.Sneddon : Elements of Partial Differential Equations, McGraw Hill Book Company, 1988 Frank Ayres : Theory and Problems of Differential Equations, McGraw Hill Book Company, 1972 J.N. Sharma & Kehar Singh : Partial Differential Equations (w.e.f. 2018-19) Statics Paper: BM 233 Max. Marks: 5 x 4 = 20 1 x 6 = 6Total = 26 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions (each carrying 5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions (each carrying 1 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Composition and resolution of forces. Parallel forces. Moments and Couples. Section II Analytical conditions of equilibrium of coplanar forces. Friction. Centre of Gravity. Section III Virtual work. Forces in three dimensions. Poinsots central axis. Section IV Wrenches. Null lines and planes. Stable and unstable equilibrium. Books Recommended: S.L. Loney : Statics, Macmillan Company, London R.S. Verma : A Text Book on Statics, Pothishala Pvt. Ltd., Allahabad NEW SCHEME Scheme of Examination of B.A. 4th Semester Mathematics (w.e.f. 2018-2019) Paper CodeTitle of the PaperAllocation of Periods  Maximum Marks Theory Intternal Assess-mentPracticalTotalBM 241 Sequences and Series6 periods/ 4 hours per week 276-- 100BM 242Special Functions and Integral transforms 6 periods/ 4 hours per week277--BM 243Programming in C and Numerical Methods6 periods/ 4 hours per week20--13 Note:- The other conditions will remain the same as per relevant ordinance and rules and regulations of the University. (w.e.f. 2018-19) Sequences and Series Paper: BM 241 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions(each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Boundedness of the set of real numbers; least upper bound, greatest lower bound of a set, neighborhoods, interior points, isolated points, limit points, open sets, closed set, interior of a set, closure of a set in real numbers and their properties. Bolzano-Weiestrass theorem, Open covers, Compact sets and Heine-Borel Theorem. Section II Sequence: Real Sequences and their convergence, Theorem on limits of sequence, Bounded and monotonic sequences, Cauchys sequence, Cauchy general principle of convergence, Subsequences, Subsequential limits. Infinite series: Convergence and divergence of Infinite Series, Comparison Tests of positive terms Infinite series, Cauchys general principle of Convergence of series, Convergence and divergence of geometric series, Hyper Harmonic series or p-series. Section III Infinite series: D-Alemberts ratio test, Raabes test, Logarithmic test, de Morgan and Bertrands test, Cauchys Nth root test, Gauss Test, Cauchys integral test, Cauchys condensation test. Section IV Alternating series, Leibnitzs test, absolute and conditional convergence, Arbitrary series: abels lemma, Abels test, Dirichlets test, Insertion and removal of parenthesis, re-arrangement of terms in a series, Dirichlets theorem, Riemanns Re-arrangement theorem, Pringsheims theorem (statement only), Multiplication of series, Cauchy product of series, (definitions and examples only) Convergence and absolute convergence of infinite products. Books Recommended: R.R. Goldberg : Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi, 1970 S.C. Malik : Mathematical Analysis, Wiley Eastern Ltd., Allahabad. Shanti Narayan : A Course in Mathematical Analysis, S.Chand and company, New Delhi Murray, R. Spiegel : Theory and Problems of Advanced Calculus, Schaum Publishing co., New York T.M. Apostol: Mathematical Analysis, Narosa Publishing House, New Delhi, 1985 Earl D. Rainville, Infinite Series, The Macmillan Co., New York (w.e.f. 2018-19) Special Functions and Integral Transforms Paper: BM 242 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions(each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Series solution of differential equations Power series method, Definitions of Beta and Gamma functions. Bessel equation and its solution: Bessel functions and their properties-Convergence, recurrence, Relations and generating functions, Orthogonality of Bessel functions. Section II Legendre and Hermite differentials equations and their solutions: Legendre and Hermite functions and their properties-Recurrence Relations and generating functions. Orhogonality of Legendre and Hermite polynomials. Rodrigues Formula for Legendre & Hermite Polynomials, Laplace Integral Representation of Legendre polynomial. Section III Laplace Transforms Existence theorem for Laplace transforms, Linearity of the Laplace transforms, Shifting theorems, Laplace transforms of derivatives and integrals, Differentiation and integration of Laplace transforms, Convolution theorem, Inverse Laplace transforms, convolution theorem, Inverse Laplace transforms of derivatives and integrals, solution of ordinary differential equations using Laplace transform. Section IV Fourier transforms: Linearity property, Shifting, Modulation, Convolution Theorem, Fourier Transform of Derivatives, Relations between Fourier transform and Laplace transform, Parsevals identity for Fourier transforms, solution of differential Equations using Fourier Transforms. Books Recommended: Erwin Kreyszing : Advanced Engineering Mathematics, John Wiley & Sons, Inc., New York, 1999 A.R. Forsyth : A Treatise on Differential Equations, Macmillan and Co. Ltd. I.N. Sneddon : Special Functions on mathematics, Physics & Chemistry. W.W. Bell : Special Functions for Scientists & Engineers. I.N. Sneddon: the use of integral transform, McGraw Hill, 1972 Murray R. Spiegel: Laplace transform, Schaums Series. (w.e.f. 2018-19) Programming in C and Numerical Methods Part-A (Theory) Paper: BM 243 Max. Marks: 3.5 x 4 = 14 1 x 6 = 6Total = 20Time: 3 Hours Note:- The question paper will consist of five sections. Each of the first four sections (I-IV) will contain two questions (each carrying 3.5 marks), and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions ( each carrying 1 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Programmers model of a computer, Algorithms, Flow charts, Data types, Operators and expressions, Input / outputs functions. Section II Decisions control structure: Decision statements, Logical and conditional statements, Implementation of Loops, Switch Statement & Case control structures. Functions, Preprocessors and Arrays. Section III Strings: Character Data Type, Standard String handling Functions, Arithmetic Operations on Characters. Structures: Definition, using Structures, use of Structures in Arrays and Arrays in Structures. Pointers: Pointers Data type, Pointers and Arrays, Pointers and Functions. Solution of Algebraic and Transcendental equations: Bisection method, Regula-Falsi method, Secant method, Newton-Raphsons method. Newtons iterative method for finding pth root of a number, Order of convergence of above methods. Section IV Simultaneous linear algebraic equations: Gauss-elimination method, Gauss-Jordan method, Triangularization method (LU decomposition method). Crouts method, Cholesky Decomposition method. Iterative method, Jacobis method, Gauss-Seidals method, Relaxation method. Books Recommended: B.W. Kernighan and D.M. Ritchie : The C Programming Language, 2nd Edition V. Rajaraman : Programming in C, Prentice Hall of India, 1994 Byron S. Gottfried : Theory and Problems of Programming with C, Tata McGraw-Hill Publishing Co. Ltd., 1998 M.K. Jain, S.R.K.Lyengar, R.K. Jain : Numerical Method, Problems and Solutions, New Age International (P) Ltd., 1996 M.K. Jain, S.R.K. Lyengar, R.K. Jain : Numerical Method for Scientific and Engineering Computation, New Age International (P) Ltd., 1999 Computer Oriented Numerical Methods, Prentice Hall of India Pvt. Ltd. Programming in ANSI C, E. Balagurusamy, Tata McGraw-Hill Publishing Co. Ltd. Programming in ANSI C, E. Balagurusamy, Tata McGraw-Hill Publishing Co. Ltd. Babu Ram: Numerical Methods, Pearson Publication. R.S. Gupta, Elements of Numerical Analysis, Macmillans India 2010. Part-B (Practical) Max. Marks: 13 Time: 3 Hours There will be a separate practical paper which will consist simple programs in C and the implementation of Numerical Methods, studied in the paper BM 243 (Part-A). NEW SCHEME Scheme of Examination of B.A. 5th Semester Mathematics (w.e.f. 2018-2019) Paper CodeTitle of the PaperAllocation of Periods  Maximum Marks Theory Internal AssessmentPracticalTotalBM 351 Real Analysis6 periods/ 4 hours per week 276 -- 100BM 352Groups and Rings 6 periods/ 4 hours per week277--BM 363 Numerical Analysis6 periods/ 4 hours per week20--13 Note:- The other conditions will remain the same as per relevant ordinance and rules and regulations of the University. (w.e.f. 2018-19) Real Analysis Paper: BM 351 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks), and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions(each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Riemann integral, Integrabililty of continuous and monotonic functions, The Fundamental theorem of integral calculus. Mean value theorems of integral calculus. Section II Improper integrals and their convergence, Comparison tests, Abels and Dirichlets tests, Frullanis integral, Integral as a function of a parameter. Continuity, Differentiability and integrability of an integral of a function of a parameter. Section III Definition and examples of metric spaces, neighborhoods, limit points, interior points, open and closed sets, closure and interior, boundary points, subspace of a metric space, equivalent metrics, Cauchy sequences, completeness, Cantors intersection theorem, Baires category theorem, contraction Principle Section IV Continuous functions, uniform continuity, compactness for metric spaces, sequential compactness, Bolzano-Weierstrass property, total boundedness, finite intersection property, continuity in relation with compactness, connectedness , components, continuity in relation with connectedness. Book s Recommended: P.K. Jain and Khalil Ahmad: Metric Spaces, 2nd Ed., Narosa, 2004 T.M. Apostol: Mathematical Analysis, Narosa Publishing House, New Delhi, 1985 R.R. Goldberg : Real analysis, Oxford & IBH publishing Co., New Delhi, 1970 D. Somasundaram and B. Choudhary : A First Course in Mathematical Analysis, Narosa Publishing House, New Delhi, 1997 Shanti Narayan : A Course of Mathematical Analysis, S. Chand & Co., New Delhi E.T. Copson, Metric Spaces, Cambridge University Press, 1968. G.F. Simmons : Introduction to Topology and Modern Analysis, McGraw Hill, 1963. (w.e.f. 2018-19) Groups and Rings Paper: BM 352 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions (each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Definition of a group with example and simple properties of groups, Subgroups and Subgroup criteria, Generation of groups, cyclic groups, Cosets, Left and right cosets, Index of a sub-group Coset decomposition, Largrages theorem and its consequences, Normal subgroups, Quotient groups, Section II Homoomorphisms, isomophisms, automorphisms and inner automorphisms of a group. Automorphisms of cyclic groups, Permutations groups. Even and odd permutations. Alternating groups, Cayleys theorem, Center of a group and derived group of a group. Section III Introduction to rings, subrings, integral domains and fields, Characteristics of a ring. Ring homomorphisms, ideals (principle, prime and Maximal) and Quotient rings, Field of quotients of an integral domain. Section IV Euclidean rings, Polynomial rings, Polynomials over the rational field, The Eisensteins criterion, Polynomial rings over commutative rings, Unique factorization domain, R unique factorization domain implies so is R[X1 , X2Xn] Books Recommended: I.N. Herstein : Topics in Algebra, Wiley Eastern Ltd., New Delhi, 1975 2. P.B. Bhattacharya, S.K. Jain and S.R. Nagpal : Basic Abstract Algebra (2nd edition). 3. Vivek Sahai and Vikas Bist : Algebra, NKarosa Publishing House. 4. I.S. Luther and I.B.S. Passi : Algebra, Vol.-II, Norsa Publishing House. 5. J.B. Gallian: Abstract Algebra, Narosa Publishing House. (w.e.f. 2018-19) Numerical Analysis Part-A (Theory) Paper: BM 363 Max. Marks: 3.5 x 4 = 14 1 x 6 = 6Total = 20Time: 3 Hours Note:- The question paper will consist of five sections. Each of the first four sections (I-IV) will contain two questions (each carrying 3.5 marks), and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions (each carrying 1 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Finite Differences operators and their relations. Finding the missing terms and effect of error in a difference tabular values, Interpolation with equal intervals: Newtons forward and Newtons backward interpolation formulae. Interpolation with unequal intervals: Newtons divided difference, Lagranges Interpolation formulae, Hermite Formula. Section II Central Differences: Gauss forward and Gausss backward interpolation formulae, Sterling, Bessel Formula. Probability distribution of random variables, Binomial distribution, Poissons distribution, Normal distribution: Mean, Variance and Fitting. Section III Numerical Differentiation: Derivative of a function using interpolation formulae as studied in Sections I & II. Eigen Value Problems: Power method, Jacobis method, Givens method, House-Holders method, QR method, Lanczos method. Section IV Numerical Integration: Newton-Cotes Quadrature formula, Trapezoidal rule, Simpsons one- third and three-eighth rule, Chebychev formula, Gauss Quadrature formula. Numerical solution of ordinary differential equations: Single step methods-Picards method. Taylors series method, Eulers method, Runge-Kutta Methods. Multiple step methods; Predictor-corrector method, Modified Eulers method, Milne-Simpsons method. Books Recommended: Babu Ram: Numerical Methods, Pearson Publication. R.S. Gupta, Elements of Numerical Analysis, Macmillans India 2010. M.K. Jain, S.R.K.Iyengar, R.K. Jain : Numerical Method, Problems and Solutions, New Age International (P) Ltd., 1996 M.K. Jain, S.R.K. Iyengar, R.K. Jain : Numerical Method for Scientific and Engineering Computation, New Age International (P) Ltd., 1999 C.E. Froberg : Introduction to Numerical Analysis (2nd Edition). Melvin J. Maaron : Numerical Analysis-A Practical Approach, Macmillan Publishing Co., Inc., New York R.Y. Rubnistein : Simulation and the Monte Carlo Methods, John Wiley, 1981 Radhey S. Gupta: Elements of Numerical Analysis, Macmillan Publishing Co. Part-B (Practical) Max. Marks: 13 Time: 3 Hours There will be a separate practical paper which will consist simple programs in C and the implementation of Numerical Methods, studied in the paper BM 363 (Part-A). NEW SCHEME Scheme of Examination of B.A. 6th Semester Mathematics (w.e.f. 2018-2019) Paper CodeTitle of the PaperAllocation of Periods  Maximum MarksTheory Internal AssessmentTotalBM 361 Real and Complex Analysis 6 periods/ 4 hours per week 276 100BM 362Linear Algebra 6 periods/ 4 hours per week277BM 353Dynamics6 periods/ 4 hours per week267 Note:- The other conditions will remain the same as per relevant ordinance and rules and regulations of the University. (w.e.f. 2018-19) Real and Complex Analysis Paper: BM 361 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions (each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Jacobians, Beta and Gama functions, Double and Triple integrals, Dirichlets integrals, change of order of integration in double integrals. Section II Fouriers series: Fourier expansion of piecewise monotonic functions, Properties of Fourier Co-efficients, Dirichlets conditions, Parsevals identity for Fourier series, Fourier series for even and odd functions, Half range series, Change of Intervals. Section III Extended Complex Plane, Stereographic projection of complex numbers, continuity and differentiability of complex functions, Analytic functions, Cauchy-Riemann equations. Harmonic functions. Section IV Mappings by elementary functions: Translation, rotation, Magnification and Inversion. Conformal Mappings, Mobius transformations. Fixed pints, Cross ratio, Inverse Points and critical mappings. Books Recommended: T.M. Apostol: Mathematical Analysis, Narosa Publishing House, New Delhi, 1985 R.R. Goldberg : Real analysis, Oxford & IBH publishing Co., New Delhi, 1970 D. Somasundaram and B. Choudhary : A First Course in Mathematical, Analysis, Narosa Publishing House, New Delhi, 1997 Shanti Narayan : A Course of Mathematical Analysis, S. Chand & Co., New Delhi R.V. Churchill & J.W. Brown: Complex Variables and Applications, 5th Edition, McGraw-Hill, New York, 1990 Shanti Narayan : Theory of Functions of a Complex Variable, S. Chand & Co., New Delhi. (w.e.f. 2018-19) Linear Algebra Paper: BM 362 Max. Marks: 4.5 x 4 = 18 1.5 x 6 = 9Total = 27 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions(each carrying 4.5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions(each carrying 1.5 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Vector spaces, subspaces, Sum and Direct sum of subspaces, Linear span, Linearly Independent and dependent subsets of a vector space. Finitely generated vector space, Existence theorem for basis of a finitely generated vactor space, Finite dimensional vector spaces, Invariance of the number of elements of bases sets, Dimensions, Quotient space and its dimension. Section II Homomorphism and isomorphism of vector spaces, Linear transformations and linear forms on vactor spaces, Vactor space of all the linear transformations Dual Spaces, Bidual spaces, annihilator of subspaces of finite dimentional vactor spaces, Null Space, Range space of a linear transformation, Rank and Nullity Theorem, Section III Algebra of Liner Transformation, Minimal Polynomial of a linear transformation, Singular and non-singular linear transformations, Matrix of a linear Transformation, Change of basis, Eigen values and Eigen vectors of linear transformations. Section IV Inner product spaces, Cauchy-Schwarz inequality, Orthogonal vectors, Orthogonal complements, Orthogonal sets and Basis, Bessels inequality for finite dimensional vector spaces, Gram-Schmidt, Orthogonalization process, Adjoint of a linear transformation and its properties, Unitary linear transformations. Books Recommended: I.N. Herstein : Topics in Algebra, Wiley Eastern Ltd., New Delhi, 1975 2. P.B. Bhattacharya, S.K. Jain and S.R. Nagpal : Basic Abstract Algebra (2nd edition). 3. Vivek Sahai and Vikas Bist : Algebra, Narosa Publishing House. 4. I.S. Luther and I.B.S. Passi : Algebra, Vol.-II, Narosa Publishing House. (w.e.f. 2018-19) Dynamics Paper: BM 353 Max. Marks: 5 x 4 = 20 1 x 6 = 6Total = 26 Time: 3 Hours Note: The question paper will consist of five sections. Each of the first four sections(I-IV) will contain two questions (each carrying 5 marks) and the students shall be asked to attempt one question from each section. Section-V will contain six short answer type questions (each carrying 1 marks) without any internal choice covering the entire syllabus and shall be compulsory. Section I Velocity and acceleration along radial, transverse, tangential and normal directions. Relative velocity and acceleration. Simple harmonic motion. Elastic strings. Section II Mass, Momentum and Force. Newtons laws of motion. Work, Power and Energy. Definitions of Conservative forces and Impulsive forces. Section III Motion on smooth and rough plane curves. Projectile motion of a particle in a plane. Vector angular velocity. Section IV General motion of a rigid body. Central Orbits, Kepler laws of motion. Motion of a particle in three dimensions. Acceleration in terms of different co-ordinate systems. Books Recommended: S.L.Loney : An Elementary Treatise on the Dynamics of a Particle and a Rigid Bodies, Cambridge University Press, 1956 F. Chorlton : Dynamics, CBS Publishers, New Delhi A.S. Ramsey: Dynamics Part-1&2, CBS Publisher & Distributors.      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