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. 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. Im plicit 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.