# Reduction formulae

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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:

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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