PURE MATHEMATICS
Total Marks  200
PAPER
I (Marks  100)
Candidates will he asked to attempt three questions from Section A and
two questions from section B.
SECTION A
Modern Algebra
Groups, subgroups, Languages Theorem, cyclic groups, normal sub groups,
quotient groups, Fundamental theorem of homomorphism, Isomorphism theorems
of groups, Inner automorphisms, Conjugate elements, conjugate subgroups,
Commutator subgroups.
Rings, Subrings, Integral domains, Quotient fields, Isomorphism theorems,
Field extension and finite fields.
Vector spaces, Linear independence, Bases, Dimension 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.
Solution of a system of homogeneous and nonhomogeneous linear equations,
Properties of determinants, CayleyHamilton theorem, Eigenvalues and eigenvectors,
Reduction to canonical forms, specially diagonalisation.
SECTION B
Geometry
Conic sections in Cartesian coordinates, Plane polar coordinates and their
use to represent the straight line and conic sections, (artesian and spherical
polar coordinates in three dimensions, The plane, the sphere, the ellipsoid,
the paraboloid and the hyperbiloid in Cartesian and spherical polar coordinates.
Vector equations for Plane and for
spacecurves. The arc length. The osculating plane. The tangent, normal
and binormal, Curvature and torsion, SerreFrenet’s
formulae, Vector equations for surfaces, The first and second fundamental
forms, Normal, principal, Gaussian and mean curvatures,
PAPERII (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
Real Numbers, Limits, Continuity,
Differentiabiliry, Indefinite integration, Mean value theorems, Taylor’s theorem, Indeterminate forms, Asymptotes.
Curve tracing, Definite integrals, Functions of several variables, Partial
derivatives. Maxima and minima Jacobians, Double and triple integration (techniques
only). Applications of Beta and Gamma func tions. Areas and Volumes. RiemannStieltje’s
integral, Improper integrals and their conditions of existences, Implicit
function theorem, Absolute and conditional convergence of series of real
terms, Rearrangement of series, Uniform convergence of series,
Metric spaces, Open and closed spheres, Closure, Interior
and Exterior of a set. Sequences 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.
SUGGESTED READINGS

Title 
Author 
1 
Advance Calculus 
Kaplan, W. 
2 
Analytical Function Theory Vol. I 
Hille, E. 
3 
An Introduction to Differential Geometry 
Wilmore, T.S. 
4 
Complex Analysis 
Goodstein, G.R.G. 
5 
Calculus with Analytical Geometry 
Yusuf, S.M. 
6 
Differential Geometry of Three Dimensions 
Weatherburn, C.E. 
7 
Elements of Complex Analysis 
Pennisi, L.L. 
8 
Theory of Groups 
Majeed, A. 
9 
Mathematical Methods 
Yusuf, S.M. 
10 
Mathematical Analysis 
Apostal, T.M. 
11 
Principles of Mathematical Analysis 
Rudin, W. 
12 
The Theory of Groups 
Macdonald, I.N. 
13 
Topics in Algebra 
Herstein, I.N. 