B.Sc. (General) in MathematicsGurukula Kangri Vishwavidayalaya
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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.
Groups: Definition, Examples and Properties, Permutation and Permutation group,
Subgroups and their properties.
Cosets and Coset decomposition, Lagrange’s theorem and its corollaries, Farmat’s
theorem, Cyclic group.
Normal subgroup, Centre of a group, Quotient group, Homomorphism and Isomorphism,
Fundamental theorem of homomorphism, Cayley’s theorem.
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.
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.
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.
Vector Space: Field, Vector space, Subspaces, Base and dimension, Coordinates,
Summary of rows equivalence, Computations concerning subspaces.
Linear Transformations: Linear transformations and their algebra. Isomorphism,
Representation of transformations by matrices.
Linear functionals, Double dual, Transpose of linear transformations.
Polynomials: Algebra of polynomials, Polynomial ideals, Determinant functions and
Canonical Form: Characteristic values and Characteristic vectors, Annihilating
polynomials, Examples of invariant subspaces.
Diagonalization, Orthogonal diagonalization, Applications to differential equations.
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.