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Paper-I

**Linear Algebra: **Vector
spaces over R and C, linear dependence and independence, subspaces,
bases, dimension; Linear transformations, rank and nullity, matrix of a
linear transformation.

Algebra of Matrices; Row and column reduction,
Echelon form, congruence’s and similarity; Rank of a matrix; Inverse of
a matrix; Solution of system of linear equations; Eigenvalues and
eigenvectors, characteristic polynomial, Cayley-Hamilton theorem,
Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and
unitary matrices and their eigenvalues.**Calculus: **Real
numbers, functions of a real variable, limits, continuity,
differentiability, mean-value theorem, Taylor's theorem with remainders,
indeterminate forms, maxima and minima, asymptotes; Curve tracing;
Functions of two or three variables: limits, continuity, partial
derivatives, maxima and minima, Lagrange's method of multipliers,
Jacobian.

Riemann's definition of definite integrals; Indefinite
integrals; Infinite and improper integrals; Double and triple integrals
(evaluation techniques only); Areas, surface and volumes.**Analytic Geometry: **Cartesian
and polar coordinates in three dimensions, second degree equations in
three variables, reduction to canonical forms, straight lines, shortest
distance between two skew lines; Plane, sphere, cone, cylinder,
paraboloid, ellipsoid, hyperboloid of one and two sheets and their
properties.**Ordinary Differential Equations: **Formulation
of differential equations; Equations of first order and first degree,
integrating factor; Orthogonal trajectory; Equations of first order but
not of first degree, Clairaut's equation, singular solution. Second and
higher order linear equations with constant coefficients, complementary
function, particular integral and general solution.

Second order
linear equations with variable coefficients, Euler-Cauchy equation;
Determination of complete solution when one solution is known using
method of variation of parameters.

Laplace and Inverse Laplace
transforms and their properties; Laplace transforms of elementary
functions. Application to initial value problems for 2nd order linear
equations with constant coefficients.**Dynamics & Statics: **Rectilinear
motion, simple harmonic motion, motion in a plane, projectiles;
constrained motion; Work and energy, conservation of energy; Kepler's
laws, orbits under central forces.

Equilibrium of a system of
particles; Work and potential energy, friction; common catenary;
Principle of virtual work; Stability of equilibrium, equilibrium of
forces in three dimensions.**Vector Analysis: **Scalar
and vector fields, differentiation of vector field of a scalar
variable; Gradient, divergence and curl in cartesian and cylindrical
coordinates; Higher order derivatives; Vector identities and vector
equations. Application to geometry: Curves in space, Curvature and
torsion; Serret-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s
identities.

Paper-II

**(1) Algebra: **Groups,
subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups,
quotient groups, homomorphism of groups, basic isomorphism theorems,
permutation groups, Cayley’s theorem.

Rings, subrings and ideals,
homomorphisms of rings; Integral domains, principal ideal domains,
Euclidean domains and unique factorization domains; Fields, quotient
fields.**Real Analysis: **Real number system as an
ordered field with least upper bound property; Sequences, limit of a
sequence, Cauchy sequence, completeness of real line; Series and its
convergence, absolute and conditional convergence of series of real and
complex terms, rearrangement of series.

Continuity and uniform continuity of functions, properties of continuous functions on compact sets.

Riemann integral, improper integrals; Fundamental theorems of integral calculus.

Uniform
convergence, continuity, differentiability and integrability for
sequences and series of functions; Partial derivatives of functions of
several (two or three) variables, maxima and minima.**Complex Analysis: **Analytic
functions, Cauchy-Riemann equations, Cauchy's theorem, Cauchy's
integral formula, power series representation of an analytic function,
Taylor’s series; Singularities; Laurent's series; Cauchy's residue
theorem; Contour integration.**Linear Programming: **Linear
programming problems, basic solution, basic feasible solution and
optimal solution; Graphical method and simplex method of solutions;
Duality.

Transportation and assignment problems.**Partial differential equations: **Family
of surfaces in three dimensions and formulation of partial differential
equations; Solution of quasilinear partial differential equations of
the first order, Cauchy's method of characteristics; Linear partial
differential equations of the second order with constant coefficients,
canonical form; Equation of a vibrating string, heat equation, Laplace
equation and their solutions.**Numerical Analysis and Computer programming: **Numerical
methods: Solution of algebraic and transcendental equations of one
variable by bisection, Regula-Falsi and Newton-Raphson methods; solution
of system of linear equations by Gaussian elimination and Gauss-Jordan
(direct), Gauss-Seidel(iterative) methods. Newton's (forward and
backward) interpolation, Lagrange's interpolation.

Numerical integration: Trapezoidal rule, Simpson's rules, Gaussian quadrature formula.

Numerical solution of ordinary differential equations: Euler and Runga Kutta-methods.

Computer
Programming: Binary system; Arithmetic and logical operations on
numbers; Octal and Hexadecimal systems; Conversion to and from decimal
systems; Algebra of binary numbers.

Elements of computer systems and concept of memory; Basic logic gates and truth tables, Boolean algebra, normal forms.

Representation of unsigned integers, signed integers and reals, double precision reals and long integers.

Algorithms and flow charts for solving numerical analysis problems.**Mechanics and Fluid Dynamics: **Generalized
coordinates; D' Alembert's principle and Lagrange's equations; Hamilton
equations; Moment of inertia; Motion of rigid bodies in two dimensions.

Equation
of continuity; Euler's equation of motion for inviscid flow;
Stream-lines, path of a particle; Potential flow; Two-dimensional and
axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes
equation for a viscous fluid.