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 automorphisms. 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 homogeneous and nonhomogeneous linear equations. Properties of determinants. Cayley-Hamilton theorem, Eigenvalues 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. Serret-Frenet formulae. Vector equations for surfaces. The first and second fundamental forms. Normal, principal, Gaussian and mean curvatures