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University of Rajasthan - M.A./M.Sc. (Previous) Mathematics- 2009
SYLLABUS FOR ENTRANCE TEST
For Admission to M.A./M.Sc. (Previous) Mathematics- 2009
I. Algebra – 20 Questions
Definition 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.
Definition and simple properties of Rings and Subrings. Integral domain and Field.
Characteristics of a Ring and Field. Ideals and Quotient Ring. Maximal Ideal and Prime
II. Real Analysis – 15 Questions
Real numbers as complete ordered field, Limit point, Bolzano-Weierstrass theorem,
Closed and Open sets, Union and Intersection of such sets. Concept of compactness.
Heine-Borel theorem. Connected sets. Real sequences- Limit and Convergence of a
sequence, Monotonic sequences.Cauchy’s sequences, Subsequences, Series – Infinite
series and convergent series. Tests for convergence of a series. Alternating series,
Absolute convergence. Properties of continuous functions on closed intervals. Properties
of derivable functions, Darboux’s and Rolle’s theorem. Riemann integration – Lower and
Upper Riemann integrals, Riemann integrability.
III. Complex Analysis – 20 Questions
Complex Valued Functions – Limits, Continuity and Differentiability. Analytic
functions, Cauchy-Riemann equations. Harmonic functions. Complex integration,
Complex line integrals, Cauchy integral theorem, Indefinite integral, Fundamental
theorem of integral calculus for complex functions. Cauchy integral formula, Analyticity
of the derivative of analytic functions, Morera’s theorem, Poisson integral formula,
Liouville’ theorem.Taylor’s theorem. Laurent’s theorem. Maximum modulus theorem.
Singularities of an analytic function, Branch point, Meromorphic and Entire functions.
Residue at a singularity, Cauchy’s residue theorem.
IV. Dynamics – 10 Questions
Velocity and Acceleration – along radial and transverse directions, along tangential and
normal 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. Moment of inertia – M.I. of rods,
Circular rings, Circular disks, Solid and Hollow spheres, Rectangular lamina, Ellipse and
V. Differential Equations – 10 Questions
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 which can be made exact. First order but
higher degree differential equations solvable for x,y and p. Clairaut’s form. Linear
equations with constant coefficients, Complimentary function and Particular integral.
Homogeneous linear differential equations.
VI. Co-ordinate Geometry for Three Dimensions – 10 Questions
Sphere, Cone and Cylinder.
VII. Calculus - 10 Questions
Curvature, Partial differentiation, Maxima and Minima of functions of two variables.
Asymptotes. Multiple points, Double and Triple integrals. Gamma and Beta Functions.
VIII. Vector Calculus – 05 Questions
Scalar point function. Vector point function. Differentiation and integration of vector
point functions. Directional derivative. Differential operators. Gradient, Divergence and
Curl. Theorems of Gauss, Green, Stokes (without proof) and problems based on these
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