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Monday, February 21, 2011

OJEE Syllabus Lateral Entry BSc Mathematics Paper-I

OJEE Syllabus Lateral Entry BSc Mathematics Paper-I Odisha 2011 Orissa Joint Entrance Examination 2011 

B.Sc. Paper-I (B. Sc Mathematics)

Algebra : Mappings. Equivalence relations and partition. Congruence modulo n relation.

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 theorem and its use in finding inverse of a matrix. Applications of matrices to a system of linear (both homogenous and nonhomogenous) equations. Theorems on consistency of a system of linear equations.

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 An. Cayley’s theorem. Introduction to rings, subrings, integral domains and fields. Characteristic of a ring.

Differential Calculus : Definition of the limit of a function. Basic properties of limits. Continuous functions and classification of discontinuities. Differentiability. Successive differentiation. Leibnritz theorem. Maclaurin and Taylor series expansions. Asymptotes. Curvature. Tests for concavity and convexity. Points of inflexion. Multiple points. Tracing of curves in Cartesian and polar coordinates.

Integral Calculus : Integration of irrational algebraic functions and trancscendental 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 coefficient. 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.

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 : General equation of second degree. Tracing of conics. System of conics. Confocal conics. Polar equation of a conic.

The straight line and the plane, sphere, cone, cylinder.

Advanced calculus : Continuity. Sequential continuity. Properties of continuous functions. Uniform continuity. Chain rule of differentiability. Mean value theorems and their geometrical interpretations. Darboux’s intermediate value theorem for derivatives. Taylor’s theorem with various forms of remainders.

Limit and continuity of functions of two variables. Partial differentiation. Change of variables. Euler’s theorem of homogeneous functions. Taylor’s theorem for functions of two variables. Jacobians.

Envelopes. Evolutes. Maxima, minima and saddle points of functions of two variables. Lagrange’s multiplier method. Indeterminate forms.

Beta and Gamma functions. Double and tripe integrals. Dirichlet’s integrals. Change of order of integration in double integrals.

Definition of a sequence. Theorems of limits of sequences. Bounded and monotonic sequences. Cauchy’s convergence criterion. Series of non-negative terms. Comparison tests. Cauchy’s integral test. Ratio tests. Raabe’s, logarithmic, De Morgan and Bertrand’s tests. Alternating series. Leibnitz’s theorem. Absolute and conditional convergence.

Series solutions of differential equations-Power series method, Bessel, Legendre and Hypergeometric equations. Bessel, Legendre and Hypergeometric functions and their properties-convergence, recurrence and generating relations. Orthogonality of functions. Sturm-Liouville problem. Orthogonality of eigen-functions. Reality of eigenvalues. Orthogonality of Bessel functions and Legendere polynomials.

Laplace Transformation - Linearity of the Laplace transformation. Existence theorem for Laplace transforms. Lapalce transforms of derivatives and integrals. Shifting theorems. Differentiation and integration of transforms. Convolution theorem. Solution of integral equation and systems of differential equation using the Laplace transformation.

Statics : Analytical conditions of equilibrium of Coplanar forces. Virtual work, Catenary.

Dynamics : Velocities and accelerations along radial and transverse directions, and along tangential and normal directions. Simple harmonic motion. Elastic strings.

Motion on smooth and rough plane curves. Motion in a resisting medium. Motion of particles of varying mass. Central Orbits. Kepler’s laws of motion.

Numerical Analysis : Solution of equations: Bisection, Secant, Regula falsi, Newton’s Method, Roots of Polynomials

Interpolation: Lagrange and Hermite Interpolation, Divided Difference, Different schemes, Interpolation formula using Differences.

Numerical Differentiation

Numerical Quadrature : Newton-Cotes formula, Gauss quadrature formula, Chebychev’s Formulas.

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