B.A. (General) in Mathematics

SYLLABUS

ABSTRACT ALGEBRA

Sets and Logic (No question should be asked on this part). The well-ordering principle.
The division algorithm. The fundamental theorem of arithmetic, Congruence modulo.
Equivalance relations and Equivalance classes.
Unit-II
Groups: Definition, Examples and Properties, Permutation and Permutation group,
Subgroups and their properties.
Unit-III
Cosets and Coset decomposition, Lagrange’s theorem and its corollaries, Farmat’s
theorem, Cyclic group.
Unit-IV
Normal subgroup, Centre of a group, Quotient group, Homomorphism and Isomorphism,
Fundamental theorem of homomorphism, Cayley’s theorem.
Unit-V
Ring, Examples and simple properties, Different types of rings, Subring and Ideals,
Divisibility in an integral domain, Polynomial ring, Field and simple properties

3-D Coordinate Geometry & Trigonometry

System of coordinates, Direction Cosine, Angle between two lines, Projections, Distance of a
point from a line.
The plane: General form,Normal form, Intercept form, Reduction of the general form to normal
form , Equation of plane through three points, Angle between two planes, Parallel planes,
Perpendicular distance of a point from the planes, Pair of the planes ,Area of a triangle and
volume of a tetrahedron.
Unit-II
The straight line: Equation of a line in general form, Symmetric form, Two point form,
Reduction of the general equation to the symmetrical form, Straight line and the planes,
Conditions of parallelism and perpendicularity of a line and a plane ,Plane through a given line,
Perpendicular distance formula for the line, Projection of a line on a given plane containing them,
Equation of a straight line intersecting two given lines, Perpendicular distance formula for the line
and coordinates of the foot of the perpendicular, Shortest distance between two lines.
Unit-III
Sphere: General equation of a sphere, Plane section of a sphere, Intersection of two spheres,
Sphere through a given circle, Intersection of a straight line and a sphere, Equation of a tangent
plane to sphere, Condition of tangency. Plane of contact, Polar plane of a given plane, Angle of
intersection of two spheres, Length of tangent, Radical plane, Coaxial system of spheres.

Linear Algebra

Unit-I
Vector Space: Field, Vector space, Subspaces, Base and dimension, Coordinates,
Summary of rows equivalence, Computations concerning subspaces.
Unit-II
Linear Transformations: Linear transformations and their algebra. Isomorphism,
Representation of transformations by matrices.
Linear functionals, Double dual, Transpose of linear transformations.
Unit-III
Polynomials: Algebra of polynomials, Polynomial ideals, Determinant functions and
simple properties.
Unit-IV
Canonical Form: Characteristic values and Characteristic vectors, Annihilating
polynomials, Examples of invariant subspaces.
Diagonalization, Orthogonal diagonalization, Applications to differential equations.
Unit-V
Quadratic forms: Quadratic forms in two and n variables, Cross-product terms of the
quadratic form. Positive definite Quadratic form, Diagonalization of quadratic forms,
Application to conic sections.