Mathematics

Examination Syllabus -Mathematics

1. Algebra:
Elements of Set Theory; Algebra of Real and
Complex numbers including Demovire's theorem;
Polynomials and Polynomial equations, relation
between Coefficients and Roots, symmetric
functions of roots; Elements of Group Theory;
Sub-Group, Cyclic groups, Permutation, Groups
and their elementary properties. Rings,
Integral Domains and Fields and their
elementary properties.

2. Vector Spaces and Matrices:
Vector Space, Linear Dependence and Independence.
Sub-spaces. Basis and Dimensions, Finite
Dimensional Vector Spaces. Linear Transformation
of a Finite Dimensional Vector Space, Matrix
Representation. Singular and Nonsingular
Transformations. Rank and Nullity.
Matrices : Addition, Multiplication, Determinants
of a Matrix, Properties of Determinants of
order, Inverse of a Matrix, Cramer's rule.

3. Geometry and Vectors:
Analytic Geometry of straight lines and conics
in Cartesian and Polar coordinates; Three
Dimensional geometry for planes, straight lines,
sphere, cone and cylinder. Addition, Subtraction
and Products of Vectors and Simple applications
to Geometry.

4. Calculus:
Functions, Sequences, Series, Limits, Continuity,
Derivatives. Application of Derivatives : Rates
of change, Tangents, Normals, Maxima, Minima,
Rolle's Theorem, Mean Value Theorems of Lagrange
and Cauchy, Asymptotes, Curvature. Methods of
finding indefinite integrals, Definite Integrals,
Fundamental Theorem of integrals Calculus.
Application of definite integrals to area,
Length of a plane curve, Volume and Surfaces
of revolution.

5. Ordinary Differential Equations:
Order and Degree of a Differential Equation, First
order differential Equations, Singular solution,
Geometrical interpretation, Second order equations
with constant coefficients.

6. Mechanics:
Concepts of particles-Lamina; Rigid Body; Displacements;
force; Mass; weight; Motion; Velocity; Speed;
Acceleration; Parallelogram of forces; Parallelogram
of velocity, acceleration; resultant; equilibrium
of coplanar forces; Moments; Couples; Friction;
Centre of mass, Gravity; Laws of motion; Motion of
a particle in a straight line; simple Harmonic
Motion; Motion under conservative forces; Motion
under gravity; Projectile; Escape velocity;
Motion of artificial satellites.

7. Elements of Computer Programming:
Binary system, Octal and Hexadecimal systems.
Conversion to and from Decimal systems. Codes,
Bits, Bytes and Words. Memory of a computer,
Arithmetic and Logical operations on numbers.
Precisions. AND, OR, XOR, NOT and Shit/Rotate
operators, Algorithms and Flow Charts.


Main Examination Syllabus - Mathematics

Paper-I

Section-A

* Linear Algebra

Vector, space, linear dependance and independance,
subspaces, bases, dimensions. Finite dimensional
vector spaces.

Matrices, Cayley-Hamiliton theorem, eigenvalues and
eigenvectors, matrix of linear transformation, row
and column reduction, Echelon form, eqivalence,
congruences and similarity, reduction to cannonical
form, rank, orthogonal, symmetrical, skew symmetrical,
unitary, hermitian, skew-hermitian forms–their
eigenvalues. Orthogonal and unitary reduction of
quadratic and hermitian forms, positive definite
quardratic forms.

* Calculus

Real numbers, limits, continuity, differerentiability,
mean-value theorems, Taylor's theorem with remainders,
indeterminate forms, maximas and minima, asymptotes.
Functions of several variables: continuity,
differentiability, partial derivatives, maxima and
minima, Lagrange's method of multipliers, Jacobian.
Riemann's definition of definite integrals, indefinite
integrals, infinite and improper intergrals, beta and
gamma functions. Double and triple integrals (evaluation
techniques only). Areas, surface and volumes, centre
of gravity.

* Analytic Geometry

Cartesian and polar coordinates in two and three dimesnions,
second degree equations in two and three dimensions,
reduction to cannonical forms, straight lines, shortest
distance between two skew lines, plane, sphere, cone,
cylinder., paraboloid, ellipsoid, hyperboloid of one
and two sheets and their properties.

Section-B

* Ordinary Differential Equations

Formulation of differential equations, order and degree,
equations of first order and first degree, integrating
factor, equations of first order but not of first degree,
Clariaut's equation, singular solution.

Higher order linear equations, with constant coefficients,
complementary function and particular integral,
general solution, Euler-Cauchy equation.

Second order linear equations with variable coefficients,
determination of complete solution when one solution is
known, method of variation of parameters.

* Dynamics, Statics and Hydrostatics

Degree of freedom and constraints, rectilinerar motion,
simple harmonic motion, motion in a plane, projectiles,
constrained motion, work and energy, conservation of
energy, motion under impulsive forces, Kepler's laws,
orbits under central forces, motion of varying mass,
motion under resistance.

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.

Pressure of heavy fluids, equilibrium of fluids under
given system of forces Bernoulli's equation, centre of
pressure, thrust on curved surfaces, equilibrium of
floating bodies, stability of equilibrium, metacentre,
pressure of gases.

* Vector Analysis

Scalar and vector fields, triple, products, differentiation
of vector function of a scalar variable, Gradient, divergence
and curl in cartesian, cylindrical and spherical coordinates
and their physical interpretations. Higher order derivatives,
vector identities and vector quations.

Application to Geometry: Curves in space, curvature and
torision. Serret-Frenet's formulae, Gauss and Stokes'
theorems, Green's identities.

Paper-II

Section-A

* Algebra

Groups, subgroups, normal subgroups, homomorphism of groups
quotient groups basic isomorophism theorems, Sylow's group,
permutation groups, Cayley theorem. Rings and ideals, principal
ideal domains, unique factorization domains and Euclidean
domains. Field extensions, finite fields.

* Real Analysis

Real number system, ordered sets, bounds, ordered field, real
number system as an ordered field with least upper bound
property, cauchy sequence, completeness, Continuity and
uniform continuity of functions, properties of continuous
functions on compact sets. Riemann integral, improper
integrals, absolute and conditional convergence of series
of real and complex terms, rearrangement of series.
Uniform convergence, continuity, differentiability and
integrability for sequences and series of functions.
Differentiation of fuctions of several variables, change
in the order of partial derivatives, implict function
theorem, maxima and minima. Multiple integrals.

* Complex Analysis

Analytic function, Cauchy-Riemann equations,
Cauchy's theorem, Cauchy's integral formula,
power series, Taylor's series, Laurent's Series,
Singularities, Cauchy's residue theorem, contour
integration. Conformal mapping, bilinear
transformations.

* 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.
Travelling salesman problmes.

Section-B

* Partial differential equations

Curves and surfaces in three dimesnions, formulation
of partial differential equations, solutions of
equations of type dx/p=dy/q=dz/r; orthogonal
trajectories, pfaffian differential equations;
partial differential equations of the first order,
solution by Cauchy's method of characteristics;
Charpit's method of solutions, linear partial
differential equations of the second order with
constant coefficients, equations of vibrating
string, heat equation, laplace equation.

* 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) methods, Gauss-Seidel(iterative)
method. Newton's (Forward and backward) and Lagrange's
method of interpolation.

* Numerical integration

Simpson's one-third rule, tranpezodial rule,
Gaussian quardrature formula.

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

* Computer Programming

Storage of numbers in Computers, bits, bytes and words,
binary system. arithmetic and logical operations on
numbers. Bitwise operations. AND, OR , XOR, NOT, and
shift/rotate operators. Octal and Hexadecimal Systems.
Conversion to and form decimal Systems.

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

Algorithms and flow charts for solving numerical
analysis problems.

Developing simple programs in Basic for problems
involving techniques covered in the numerical analysis.

* Mechanics and Fluid Dynamics

Generalised coordinates, constraints, holonomic and
non-holonomic , systems. D' Alembert's principle and
Lagrange' equations, Hamilton equations, moment of
intertia, 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 axisymetric motion,
sources and sinks, vortex motion, flow past a cylinder
and a sphere, method of images. Navier-Stokes equation
for a viscous fluid.