Gauhati University Question Papers for Mathematics 5th Semester
Gauhati University Question Papers for Mathematics 5th Semester
Question Paper from 2010 available
More than 50 question papers every semester
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Year
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Paper 101
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Paper 102
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2010
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2011
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2012
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2013
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2014
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2015
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2016
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2017
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5th Semester
Revised
Syllabus of Mathematics
For
Three
year Degree Course ( Major Course) Paper-M501
Real and
Complex Analysis
Unit1: Limit
and continuity of a function of several variables, partial derivatives,
differentiability, Young’s and Schwarz’s theorems, differentials of higher
orders, differentiation of composite functions, change of variables , Taylor’s
theorem for two variables, implicit functions, only statement of implicit
theorem on two variables with its applications, jacobians, maxima
and minima, LaGrange’s method of multiplier
Unit2: Riemann
integral, integrability conditions, Riemann integral as a limit, some classes
of integrable functions , the fundamental theorem of
integral calculus, statement and application of M.V. theorems of integral
calculus.
Unit3: Improper
integrals and their convergence, various forms of comparison
tests, absolute and conditional convergence, Abel’s and Dirichlet’s tests,
beta and gamma functions, Frullani’s integral, integral as a function
of parameter( excluding improper integrals), continuity, derivability and
integrability of an integral as a function of a parameter.
Unit4: Theorems
on limit and continuity of a function of complex variable, uniform
continuity, differentiability of a function of complex variable, analytic
functions, Cauchy- Riemann equations, harmonic functions, differentials,
derivatives of elementary
functions, L’Hospital’s
rule , stereographic
projection.
Unit5: Rectifiable
curves, integral along an oriented curve, fundamental Cauchy theorem, proof applying green’s theorem, Cauchy
integral formula, mobius transformation, fixed points, inverse points and
critical mappings, conformal mappings
5th Semester
Revised
Syllabus of Mathematics
For
Three
year Degree Course ( Major Course) Paper-M502
Unit1:Definition
and examples of metric spaces, neighbourhoods, limit points, interior
points, open and closed sets, closure and interior, equivalent metrics,
subspace of a metric space, Cauchy sequences, completeness, Cantor’s intersection
theorem.
Unit2: Dense
subsets, Baire’s category theorem, separable, second countable and first
countable spaces, continuous functions, extension theorem, uniform continuity,isometry
and homeomorphism.
Unit3: Compactness,
sequential compactness, totally bounded spaces, finite intersection property, continuous functions and
compact sets, connectedness, components, continuous functions and connected
sets.
Unit4: Definition
and examples of topological spaces, metric topology, closed sets,
closure, Kuratoski closure operator and neighbourhood systems, dense
subsets, neighbourhoods, interior, exterior and boundary, accumulation points
and derived sets, bases and sub bases, subspaces and relative topology, continuous
functions and homeomorphism.
Unit5: Definition
and examples of normed linear spaces, Banach spaces, inner product spaces
and Hilbert space, some elementary
properties.
5th Semester
Revised
Syllabus of Mathematics
For
Three
year Degree Course ( Major Course) Paper-M503
Unit1:Section
of a sphere by a plane, spherical triangles, properties of spherical and
polar triangles, fundamental formulae of spherical triangles, sine formula,
cosine formula, sine- cosine formula, cot formula, Napier’s rule of
circular
parts.
Unit2: The standard( or geometric) celestial
sphere, system of coordinates, conversion of one coordinate system to
the another system, diurnal motion of heavenly bodies, sidereal time,
solar time(mean), rising and setting of stars, circumpolar star, dip of the
horizon,
rate of
change of zenith distance and azimuth, examples.
Unit3: Planetary motion:
annual motion of the sun, planetary motion, synodic period, orbital
period,Keplar’s law of planetary motion, deduction of Keplar’s law
from Newton’s law of gravitation, the equation of the orbit, velocity
of a planet in its orbit, components of linear velocity perpendicular
to the radius vector and to the major axis, direct and retrograde motion
in a plane, laws of refraction: refraction for small zenith distance,
general formula for refraction,Cassini’s hypothesis, differential
equation for
refraction,
effect of refraction on sunrise, sunset, right ascension and declination, shape
of the disc of the
sun.
Unit4: Geocentric
parallax, parallax of the moon, right ascension and declination,
parallax
on zenith distance and azimuth, stellar or annual parallax, effect of parallax
on the star longitude, and latitude, effect of stellar parallax on right
ascension and declination.
Lunar
eclipses section of the shadow cone at moon’s geocentric distance, condition of
lunar eclipse in terms of it, solar eclipses, the angle
subtended at the earth’s center by the
centers
of the sun and the moon at the beginning or end of a solar eclipse,
condition of solar eclipse in terms of this angle, idea of ecliptic
limits, frequency of eclipses.
.
5th Semester
Revised
Syllabus of Mathematics
For
Three
year Degree Course ( Major Course) Paper-M504
Unit1: Moments
and products of inertia, parallel axes theorem, theorem of six
constants, the momental ellipsoid, equimomental systems, principle axes.
Unit2: D’Alembert’s
principle, the general equation of motion of a rigid body, motion of
the centre of inertia and motion relative to the centre of inertia.
Unit3: Motion
about a fixed axis, the compound pendulum, centre of percussion.
Unit4: Motion
of a body in two dimension under finite and impulsive forces. 10 marks
Unit5: Conservation
of momentum and energy, generalized coordinates, LaGrange’s
equations, initial motions.
5th Semester
Revised
Syllabus of Mathematics
For
Three
year Degree Course ( Major Course) Paper-M505
Unit1: Random experiment,
sample space , events, classical definition of probability and the
theorems of total and compound probability based on this definition, axiomatic
approach to the notion of probability, important theorems based on
this approach, conditional probability and independent events,
Bay’s theorem.
Unit2:Random variables,
discrete and continuous probability distributions, probability function
and distribution function, probability mass function and probability
density function, joint distributions, marginal distribution, independent
random variables, change of variables, conditional distribution.
Unit3: Mathematical
expectation, basic theorems on expectation(proofs required only in case of
discrete random variables), variance and standard deviation, moments
and moment generating functions, covariance conditional expectation and
conditional variance, Chebyshev’s inequality, law of large numbers.
Unit4: Some
important probability distributions: Binomial, Poisson and Normal.
5th Semester
Revised
Syllabus of Mathematics
For
Three
year Degree Course ( Major Course) Paper-M506
Optimization
Theory Marks :
Unit1: Partitioning
of matrices, simultaneous equations, basic solution, point sets, lines
and hyper planes, convex sets and their properties, convex functions, convex
cones.
Unit2: General
linear programming problems, mathematical formulation of a linear
programming problem, linear programming problem in matrix
notation, feasible solution, basic solution, degenerate basic solution,
necessary and sufficient condition for the existence of non-degenerate basic
solution, graphical method for the solution of a linear programming problem.
Unit3: simplex method:
fundamental theorem of linear programming problem, basic feasible solution from feasible
solution, determination of improved basic feasible
solution,
optimality conditions, alternative optimal solution, conditions for alternative
optimal solution, theory and application of the simplex method of solution
of a linear programming problem, Charne’s M-technique, two phase method.
Unit4:Principles
of duality in linear programming problem, fundamental duality theorem, simple
problems.
Unit5:The
Transportation and Assignment problem.
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