Rajasthan University M.Sc Maths entrance exam syllabus

[Kiran Chandar]

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Rajasthan University M.Sc Maths entrance exam syllabus

Topics for Subject Specific Part B:

Algebra- Definitions and simple properties of Groups and Subgroups, Permutation group, Cyclic group, Cosets, Lagrange’s theorem on the order of subgroups of a finite order group, Morphism of groups, Cayley’s theorem, Normal subgroups and quotient groups; Definitions and simple properties of Rings and Sub rings, Integral domain and Field.


Real Analysis- Review of differentiation and integration, Real numbers a complete ordered field, Limit Point, Closed and Open sets, Union and Intersection of such sets; Concept of compactness, Connected sets;

Real sequence - Limit and Convergence of a sequence and Monotonic sequences, Cauchy’s sequences and sub sequences;

Series - Infinite series and convergent series, Tests for convergences of a series, Alternating series, Absolute convergence, Properties of Functions on closed intervals, Properties of derivable functions, Darbuox’s and Rolle’s theorem, Cauchy’s and Lagrange’s mean value theorems and their applications.

Complex Analysis- Review of complex number system, Complex trigonometry & exponential functions and their simple properties, Complex Valued Functions – Limits, Continuity and Differentiability. Analytic functions, Cauchy – Riemann equations.

Dynamics- Velocity and Acceleration – along radial and Transverse directions, along normal and tangential directions; SHM, Hook’s law, Motion in resisting medium – Resistance varies as velocity and square of velocity, Motion on a smooth curve in a vertical plane, Motion on the inside and outside of a smooth vertical circle.

Differential Equations- Degree and order of a differential equation, Equations of first order and first degree, Equations in which the variables are separable, Homogeneous equations and equations reducible to homogeneous form, Linear equations and equations reducible to linear form, Exact differential equations and equations reducible to exact form, First order but higher degree differential equations, linear differential equations with constant coefficients, Complementary function and particular integral, Homogenous linear differentials equations.

Partial differential equations of the first order. Lagrange's linear equation. Charpit's general method of solution. Homogeneous and non-homogeneous linear partial differential equations with constant coefficients.

Co-ordinate Geometry for Three Dimensions- Sphere, Cone and Cylinder.

Calculus- Curvature, Partial differentiation, Maxima and Minima of functions of two variables, Asymptotes, Multiple Points, Double and Triple Integral, Gamma and Beta functions.

Vector Calculus- Scalar point functions, Vector point functions, Differentiation and Integration of vector point functions, Directional derivative, Differential operators, Gradient, Divergence and Curl.

Linear programming problem- Problem Formulation. Graphical solution of linear programming problems. Basic solution. Some basic properties of convex sets, Simplex method for solution of a L.P.P. to simple problems.






Contact Details-

Rajasthan University
Jln Marg, Rajasthan 302005 ‎
0141 271 1070

Map Location-

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[sumit]

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Rajasthan University M.Sc Maths entrance exam syllabus
MSc. Mathematics Entrance Syllabus
Analysis
Riemann integral. Integrability of continuous and monotonic functions, The fundamental theorem of
integral calculus, Mean value theorems of integral calculus. Partial derivation and differentiability of
real-valued functions of two variables. Schwarz and t Youngs theorem. Implicit function theorem.
Improper integrals and their convergence, Comparison tests, Abels and Dirichlets tests, Frullanis
integral. Integral as a function of a parameter Continuity, derivability and integrability of an integral of a
function of a parameter. Fourier series of half and full intervals.
Complex numbers as ordered pair. Geometric representation of Complex numbers. Stereographic
projection. Continuity and differentiability of Complex functions. Analytic functions. Cauchy-Riemann
equations. Harmonic functions. Mobius transformations. Fixed point. Cross ratio. Inverse points and
critical mappings. Conformal mappings.
Definition and examples of metric spaces. Neighborhood. Limit points. Interior points. Open and closed
sets. Closure and interior. Boundary points. Sub-space of a metric space. Cauchy sequences.
Completeness. Cantors intersection theorem. Contraction principle. Real numbers as a complete ordered
field. Dense subsets. Baire Category theorem. Separable, second countable and first countable spaces.
Continuous functions. Extension theorem. Uniform continuity. Compactness. Sequential compactness.
Totally bounded spaces. Finite intersection property. Continuous functions and compact sets.
Connectedness.
Algebra & Linear Algebra
Group Automorphism, inner automorphisms, automorphism groups, Congjugacy relation and
centraliser, Normaliser, Counting principle and the class equation of a finite group, Cauchys theorem
and Sylows theorems for finite abelian groups and non abelian groups.
Ring theory Ring homonorphism, Ideals and Quotient Rings, Field of Quotients of an Integral Domain.
Euclidean Rings, polynomial Rings, Polynomials over the Rational Field, Polynomial Rings over
Commutaive Rings, Unique factorization domain.
Definition and examples of vector spaces. Subspaces. Sum and direct sum of subspaces. Linear span.
Linear dependence, independence and their basic properties. Finite dimensional vector spaces. Existence
theorem for bases. Invariance of the number of elements of a basis set. Dimension. Existence of
complementary subspace of a subs pace of a finite dimensional vector space. Dimension of sums of
subspaces. Quotient space and its dimension.
Linear transformations and their representation as matrices. The algebra of linear transformations. The
rank nullity theorem. Change of basis. Dual space. Bidual space and natural isomorphism. Adjoint of a
linear transformation. Eigenvalues and eigenvector of a linear transformation. Diagonalisation. Bilinear,
Quadratic and Hermitian forms.
Inner Product Spaces, Cauchy-Schwarz inequality Orthogonal vectors. Orthogonal Complements.
Orthonormal sets and bases. Bessels inequality for finite dimensional spaces. Gram Schmidt
Orthogonalization process.