**PURE MATHEMATICS**

Total Marks – 200

Paper I

Marks – 100

Candidates will be asked to attempt three questions from Section A and two questions from Section B

Section A

Modern Algebra

1. Groups, subgroups, languages, theorem, cyclic groups, normal sub-groups, quotient groups,

fundamental theorem of homomorphism. Isomorphism theorems of groups, inner

automorphosms. Conjugate elements, conjugate sub-groups, commutator sub-groups

2. Rings, sub rings, integral domains, quotient fields, isomorphism theorems, field extension and

finite fields

3. Vector spaces, linear independence, bases, dimensions of a finitely generated space, linear

transformations, matrices and their algebra. Reduction of matrices to their echelon form. Rank

and nullity of a linear transformation

4. Solution of a system of homogenous and non-homogenous linear equations. Properties of

determinants. Cayley-Hamilton theorem, Eigen values and eigenvectors. Reduction to canonical

forms, specially digitalization

Section B

Geometry

1. Conic sections in Cartesian coordinates, Plane polar coordinates and their use to represent the

straight line and conic sections. Cartesian and spherical polar coordinates in three dimension.

The plane, the sphere, the ellipsoid, the paraboloid and the hyperboloid in Cartesian and

spherical polar coordinates

2. Vector equations for plane and for space-curves. The arc length. The osculating plane. The

tangent, normal and bi-normal. Curvature and torsion. Serre-Frenet’s formulae. Vector equations

for surfaces. The first and second fundamental forms. Normal, principal, Gaussian and mean

curvatures

Paper II

Marks – 100

Candidates will be asked to attempt any three questions from Section A and two questions from Section

B

Section A

Calculus and Real Analysis

1. Real numbers, limits, continuity, differentiability, indefinite integration, mean value theorems.

Taylor’s theorems, indeterminate form. Asymptotes, curve tracing, definite integrals, functions of

several variables. Partial derivates. Maxima and minima. Jacobeans, double and triple integration

(Techniques only). Applications of Beta and Gamma functions. Areas and volumes. Riemann-

Stieltje’s integral. Improper integrals and their conditions of existence. Implicit function theorem.

Absolute and conditional convergence of series of real terms. Rearrangement of series, uniform

convergence of series

2. Metric spaces. Open and closed spheres. Closure, interior and exterior of a set

3. Sequence in metric space. Cauchy sequence, convergence of sequences, examples, complete

metric spaces, continuity in metric spaces. Properties of continuous functions

Section B

Complex analysis

Function of a complex variable, Demoiver’s theorem and its applications. Analytic functions, Cauchy’s

theorem. Cauchy’s integral formula, Taylor’s and Laurent’s series. Singularities. Cauchy residue theorem

and contour integration. Fourier series and Fourier transforms. Analytic continuation.