Bsc Part - I Mathmatics |
PAPER I -Algebra and Trigonometry |
ALGEBRA
Symmetric. Skew symmetric. Hermitian and skew hermitian matrices. Elementary operations on matrices. Inverse of a matrix. linear independence of row and column matrices. Row rank, column rank and rank of a matrix, Equivalence of column and row ranks. EigenValues eigenvectors and the characteristic equation of a matrix. Cayley Hamilton Marks theorem and its use in finding inverse of a matrix. Applications of matrices to a system of linear (both homogenous and non-homogeneous) equations. Theorems on consistency of a system of linear equations.
Relations between the roots and coefficients of general polynomial equation in one variable. Transformation of equations. Descarte's rule of signs. Solution of cubic equations (Cardon method). Biquadratic equations.
Mappings, Equivalence relations and partitions. Congruence modulo n.
Definition of a group with examples and simple properties. Subgroups. Generation of groups. Cyclic groups. Coset decomposition. Lagrange's theorem and its consequences. Fermat's and Euler's theorems. Homomorphism and Isomorphism. Normal subgroups. Quotient groups. The fundamental theorem of homomorphism, Permutation groups. Even and odd permutations. The alternating groups and Cayley's theorem. Automorphisms, Inner automorphism, Automorphism groups and their computations. Conjugacy relation. Normaliser, Counting principle and the class equation of afinite group. Center for Group of prime-order, Abelianizing of a group and its universal property. Sylow's theorems. pSylowsubgroup. Structure theorem forfiniteAbelian groups.
I ntroductionto rings, subrings, intergral domains and fields. Characteristic of a ring.
Trigonometry
De Moivre's theorem and its applications, Direct and inverse circular and hyperbolic functions. Logarithm of a complex quantity. Expansion of Marks trigonometrical functions. Gregory's series. Summation of series.
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PAPER II -Calculus |
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DIFFERENTIAL CALCULUS
Definition of the limit of a function. Basic properties of limits. Continuous functions and classification of discontinuities. Differentiability. Successive differentiation. Leibnitz theorem. Maclaurin and Taylor series expansions. Asymptotes. Curvature. Tests for concavity and convexity. Points of inflexion. Multiple points. Tracing of curves in Cartesian and polarcoordinates.
INTEGRAL CALCULUS
Integration of irrational algebraic functions and transcendental functions. Reduction formulae. Definite integrals. Quadrature. Rectification. Volumes and surfaces of solids of revolution.
ORDINARY DIFFERENTIAL EQUATIONS
Degree and order of a differential equation, Equations of first order and first degree. Equations in which the variables are separable. Homogeneous equations. Linear equations and equations reducible to the linear form. Exact differential equations. First order higher degree equations solvable for x, y, p. Clairaut's form and singular solutions. Geometrical meaning of a differential equation. Orthogonal trajectories. Linear differential equations with constant coefficients. Homogeneous linear ordinary differential equations: •
Linear differential equations of second order. Transformation of the equation by changing the dependent variable/the independent variable. Method of variation of parameters.
Ordinary simultaneous differential equations.
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PAPER III -(Vector Analysis and Geometry) |
VECTOR ANALYSIS
Scalar and vector product of three vectors. Product of four vectors. Reciprocal Vectors. Vector differentiation, Gradient, divergence and curl. Vector integration. Theorems of Gauss, Green, Stokes and problems based on these.
GEOMETRY 2-D
General equation of second degree. Tracing of conics_ System of conics. Confocal conics. Polar equation of a conic.
3-0
Plane, The Straight line and the plane, Sphere, Cone, Cylinder.
Central conicoids, Paraboloids. Plane Sections of Conicoids Generating lines. confocal Conicoids. Reduction of Second degree equations.
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