Pure Mathematics

Section-A (40- marks)
I. Modern Algebra
 Group, subgroups, Lagranges theorem, Cyclic groups, Normal subgroups, Quotient
groups. Fundamental theorem of homomorphism. Isomorphism theorems of groups,
Inner automorphisms. Conjugate elements, conjugate subgroups. Commutator
subgroups.
 Ring, 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 non-homogeneous linear equations.
Properties of determinants.
Section-B (40- marks)
II. Calculus & Analytic Geometry
 Real Numbers. Limits. Continuity. Differentiability. Indefinite integration. Mean value
theorems. Taylor’s theorem, Indeterminate forms. Asymptotes. Curve tracing.
Definite integrals. Functions of several variables. Partial derivatives. Maxima and
minima. Jacobnians, 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 existences. Implicit function theorem.
 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 dimensions. The plane, the sphere, the ellipsoid, the paraboloid
and the hyperboloid in Cartesian and spherical polar coordinates.
Section-C (20-marks)
III. Complex Variables
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.

SUGGESTED READINGS

S.No.TitleAuthor
1.Advanced CalculusKaplan, W.
2.Analytic Function Theory Vol.1Hille, E.
3.CalculusAnton H.,Biven I and Davis, S.
4.Complex AnalysisGoodstein G.R.G.
5.Complex VariablesMurray R. Spiegel
6.Calculus with Analytic GeometryYusuf, S.M.
7.Calculus and Analytic GeometryZia ul Haq
8.Elements of Complex AnalysisPennisi, L.L.
9.Theory of GroupsMajeed, A.
10.Mathematical MethodsYusuf, S.M.
11.Mathematical TechniquesKaramat H.Dar
12.Mathematical AnalysisApostal, T.M.
13.The Theory of GroupsMacdonald, I.N.
14.Topics in AlgebraHerstein, I.N.

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