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B.A.-I MATHEMATICS (Annual) For Examinations 2013, 2014 and 2015 PAPER-I: CALCULUS AND DIFFERENTIAL EQUATIONS Maximum Marks: 100 Time allowed: 3 Hrs. Lectures to be delivered: 5 periods (of 45 minutes duration) per week Pass Marks: 35% Instructions for paper-setters The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two questions from the respective sections of the syllabus. Section E will have only one question, which will consist of 8-10 objective/very short answer type parts covering the whole syllabus. All the questions from sections A, B, C, D and E will carry equal marks. Instructions for candidates Candidates are required to attempt one question each from sections A, B, C and D of the question paper and the entire section E. All the questions carry equal marks. Use of scientific non-programmable calculator is allowed. Section-A Limit and continuity of functions, classification of discontinuities, uniform continuity. Differentiability of a function of one variable. Mean Value theorems, Taylor series and Maclaurin series with Lagrange's form of remainder, Darboux Theorem, Successive differentiation, Leibnitz theorem, L’Hospital rule. Asymptotes, Curvature , Multiple points, Tests for concavity and convexity , points of inflexion, Maxima and minima, Tracing of curves, Evolute and Involute. Section-B Integration of irrational , algebraic and transcendental functions . Reduction formulae. Definite integrals . Quadrature and rectification . Volumes and surfaces of solids of revolution. Double integrals and triple integrals .Change of order of integration,Cylindrical and spherical coodinates, Jacobians. Applications of multiple integrals. Section-C First order differential equations: order and degree of a differential equations, separable differential equations, Homogeneous differential equations, equations reducible to Homogenous differential equations form, Linear differential equations and equations reducible to linear differential equations form, integrating factor and Exact differential equations.Orthogonal trajectories. Higher order differential equations: Solution of Linear homogeneous and non- homogeneous differential equations of higher order with constant coefficients. Differential operator method. Linear non-homogeneous differential equations with variable coefficients, Euler method. Section-D Exact differential equations and equations of particular form.Equations of the second order: the compete solution interms of a known inegral. Change of independent variable, removal of the first derivative and the variation of parameters. Condition of integrability and exactness of total differential equations. Basic concepts of partial differential equations, vibrating string, one dimensional wave equation, separation of variables (Product method). D'Alembet's solution of wave equation.One dimensional heat flow, heat flow in any infinite bar. Laplace equation in spherical coordinates. Recommended books: 1. Differential calculus, Gorakh Prasad. 2. Integral calculus, Gorakh Prasad 3. Mathematical Analysis, Malik and Arora. 4. Piaggio : Elementary treatise of differential equations and their applications, C.B.S. Publishers, Delhi, 1985 5. D.A. Murray: Differential equations, Introductory Course in differential equations. Orient Longman (India).1967 B.A.-I MATHEMATICS (Annual) For Examinations 2013, 2014 and 2015 Paper-II: ALGEBRA Maximum Marks: 100 Time allowed: 3 Hrs. Lectures to be delivered: 5 periods (of 45 minutes duration) per week Pass Marks: 35% Instructions for paper-setters The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two questions from the respective sections of the syllabus. Section E will have only one question, which will consist of 8-10 objective/very short answer type parts covering the whole syllabus. All the questions from sections A, B, C, D and E will carry equal marks. Instructions for candidates Candidates are required to attempt one question each from sections A, B, C and D of the question paper and the entire section E. All the questions carry equal marks. Use of scientific non-programmable calculator is allowed. Section-A Semigroups and groups , Examples,subgroups, Langrange's theorem, Normal subgroups, Quotient groups, Homomorphisms, Fundamental theorem of homomorphism and related theorems. Cyclic Groups. Permutation groups, Cayley's theorem .Automorphisms, Conjugacy and conjugate classes. Section- B Definition and examples of Rings,Elementary properties of Rings. Sub-rings, Integral domains, division rings and fields.Ring of endomorphisms of an Abelian group.Elementary properties of fields. Ideals, quotient rings, principal ideals, examples. Ring Homomorphism, the fundamental theorem of homomorphism. Section-C Hermitian, Skew-Hermitian, Orthogonal and Unitary matrices, .Elementary operatin on matrices.Inverse of a matrix.Linear independence of row and column vectors, Row rank, Coumn rank and their equivalence. Eigen values, Eigen vectors and the characteristic equation of a matrix, Properties of eigen values for special type of matrices, Cayley-Hamilton theorem. Application of matrices to a system and linear equations, Theorems on consistency of a system of linear equations. Section-D Relations between roots and coefficients of a general polynomial, Tranformation of equation. Descartes's rule of signs, Solution of cubic equations, Biquadratic equations and their solution. De Moivre's theorem and its application, Direct and inverse circular functions, hyperbolic and logarithmic functions.Summation of series. Recommended books: 1. Text book on Algebra and Theory of equations by Chandrika Prasad.Pothishala Pvt. Ltd. 2. Linear Algebra by Schaum Outline series. 3. Trigonometry by S.L. Loney. Mcmillan and Company London. 4. Surjeet Singh and Qazi Zameeruddin: Modern Algebra (Relevant portion) Reference book: Herstein, I.N.: Topics in Algebra B.A.-I MATHEMATICS (Annual) For Examinations 2013, 2014 and 2015 PAPER-III: COORDINATE GEOMETRY AND MATHEMATICAL STATISTICS Maximum Marks: 100 Time allowed: 3 Hrs. Lectures to be delivered: 5 periods (of 45 minutes duration) per week Pass Marks: 35% Instructions for paper-setters The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two questions from the respective sections of the syllabus. Section E will have only one question, which will consist of 8-10 objective/very short answer type parts covering the whole syllabus. All the questions from sections A, B, C, D and E will carry equal marks. Instructions for candidates Candidates are required to attempt one question each from sections A, B, C and D of the question paper and the entire section E. All the questions carry equal marks. Use of scientific non-programmable calculator is allowed. Section-A Transformation of axis, shifting of origin, rotation of axes, reduction of second degree equation into standard form by transformation of coordinates, invariants and identification of curves represented by second degree equation. Pole and polar, pair of tangents from a point, chord of contact, equation of chord in terms of midpoints and diameter of conic, Polar equations of conics and equations of chords, tangents and normals only. Section-B Parabola, tangents and normal, point of contact, length of intercept, parametric representation of parabola, equation of chord joining two points on a parabola. Diameter of a conic. Ellipse, properties of ellipse, parametric representation of ellipse, tangents, normals, equation of chord joining two points on ellipse. Director circle of ellipse, chord of contact, conjugate lines and conjugate diameter. Hyperbola, properties of hyperbola, fundamental rectangle, parametric representation of hyperbola, asymptotes of hyperbola, Conjugate hyperbola, rectangular hyperbola, tangents and normals. Section - C Statistics: Definition, importance and limitations. Measures of Central tendency, measures of dispersion, measures of skewness, measures of Kurtosis; their significance, applications, merits and demerits and mathematical properties. Correlation: Scatter diagram, Karl Pearson’s coefficient and its properties, probable error, Rank correlation coefficient; the merits and limitations. Regression: Principle of least squares, fitting of linear regression, Regression lines, Regression coefficients and their properties, standard error of estimate; the merits and limitations. Section – D Probability: Definition of the probability and its importance; classical and axiomatic approach to probability. Concepts in probability: Random experiment, trial, sample point and sample space. Events: mutually exclusive, exhaustive, independent and equally likely events, Conditional probability, Baye’s theorem. Random variable: Definitions of discrete random variables, probability mass function, continuous random variable, probability density function, probability distribution. Expectation of a random variable and its properties.Variance of a random variable and its properties. Moments, moment generating function and probability generating function. Books Recommended: 1. S.P. Gupta: Statistical Methods 2. S.C. Gupta and V. K. Kapoor: Fundamentals of Mathematical Statistics 3. Goon, Gupta and Das Gupta: Fundamentals of Statistics Vol.-I 4. P.L. Meyer : Introductory Probabilty and Satistics

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posted: | 2/13/2013 |

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