Group A
Calculus of functions
of one variable: Successive
differentiation, Leibnitz theorem, Roll’s and Mean value theorems. Taylor's and
Maclaurin’s expansion theorems. Fundamental theorem of integral calculus.
Elementary reduction formulae for integrals. Applications to length, area,
volume, surface area of revolution, moments of centre of gravity. Infinite
series-convergence, divergence ratio tests, etc.
Calculus of functions
of several variables: Partial derivatives, gradient and directional
derivatives.
Differentiation of implicit functions,
exact differentials, tangents, normals, maxima; minima, saddle points. Method
of Lagrange’s multiplier. Multiple integrals.
Vector Calculus: Scalar and vector
fields. Line and surface integrals. Gradient and divergence. Green's and
Stoke's theorems and their applications
Linear Algebra: Vector spaces-linear
independence and dependence of vectors, inner products, linear transformations.
Matrices and determinants. Systems of linear equations consistency and
inconsistency. Gauss elimination, rank of a matrix, inverse of a matrix. .
Eigen values and eigenvectors of a matrix, diagonalization of a matrix. .
Group B
Ordinary Differential
Equations (ODEs): Formation
of ODEs, definition of order, degree and solutions. ODEs of first. order;
separable variables, homogeneous. And non-homogeneous .equations, exactness and
integrating factors, linear equations and Bernoulli's equations {general linear
ODEs of nth order, solutions of homogeneous and non-homogeneous equations,
operator method, methods of undetermined coefficients and variation of
parameters). Solutions of simple simultaneous ODEs. Partial differential
equations and its applications. Transforms theory-Laplace, Fourier, etc.
Numerical Methods: Difference operators
forward, backward, central, shift and average. operators, and relations.
between them. Newton's forward and backward interpolations. Lagrange’s
interpolation and the error formula for interpolation.
Numerical differentiation and
integration. Trapezoidal rule and Simpson's one-third rule, including error
formulae.
Introduction to
Probability and Statistics: Basic concepts, including introduction to
probability theory, Venn diagrams., central limit theorem, mean, mode and median.
Properties of Beta, Poisson, Exponential and Normal distributions. Correlation
and regression, Students t-distribution test, Chi-square and F tests of
significance.
