Civil Service Mains Optional MATHEMATICS

MATHEMATICS


PAPER - I
(1) Linear Algebra:
Vector spaces over R and C, linear depen-dence and independence, subspaces,
bases, dimension; Linear transformations,
rank and nullity, matrix of a linear transfor-mation.
Algebra of Matrices; Row and column re-duction, Echelon form, congruence’s and
similarity; Rank of a matrix; Inverse of a
matrix; Solution of system of linear equa-tions; Eigenvalues and eigenvectors, char-acteristic polynomial, Cayley-Hamilton
theorem, Symmetric, skew-symmetric, Her-mitian, skew-Hermitian, orthogonal and
unitary matrices and their eigenvalues.

(2) Calculus:
Real numbers, functions of a real variable,
limits, continuity, differentiability, mean-value theorem, Taylor’s theorem with re-mainders, 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 (evalu-ation techniques only); Areas, surface and
volumes.

(3) Analytic Geometry:
Cartesian and polar coordinates in three
dimensions, second degree equations in
three variables, reduction to canonical
forms, straight lines, shortest distance be-tween two skew lines; Plane, sphere, cone,
cylinder, paraboloid, ellipsoid, hyperboloid
of one and two sheets and their properties.

(4) Ordinary Differential Equations:
Formulation of differential equations; Equa-tions of first order and first degree, inte-grating factor; Orthogonal trajectory; Equa-tions 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 vari-able 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 2ndorder linear equa-tions with constant coefficients.

(5) Dynamics & Statics:
Rectilinear motion, simple harmonic mo-tion, motion in a plane, projectiles; con-strained motion; Work and energy, conser-vation of energy; Kepler’s laws, orbits un-der central forces.
Equilibrium of a system of particles; Work
and potential energy, friction; common cat-enary; Principle of virtual work; Stability of
equilibrium, equilibrium of forces in three
dimensions.

(6) Vector Analysis:
Scalar and vector fields, differentiation of
vector field of a scalar variable; Gradient,
divergence and curl in cartesian and cylin-drical coordinates; Higher order deriva-tives; Vector identities and vector equa-tions.
Application to geometry: Curves in space,
Curvature and torsion; Serret-Frenet’s for-mulae.
Gauss and Stokes’ theorems, Green’s iden-tities.


PAPER - II

(1) Algebra:
Groups, subgroups, cyclic groups, cosets,
Lagrange’s Theorem, normal subgroups,
quotient groups, homomorphism of
groups, basic isomorphism theorems, per-mutation groups, Cayley’s theorem.
Rings, subrings and ideals, homomor-phisms of rings; Integral domains, princi-pal ideal domains, Euclidean domains and
unique factorization domains; Fields, quo-tient fields.

(2) Real Analysis:
Real number system as an ordered field
with least upper bound property; Se-quences, limit of a sequence, Cauchy se-quence, completeness of real line; Series
and its convergence, absolute and condi-tional convergence of series of real and
complex terms, rearrangement of series.
Continuity and uniform continuity of func-tions, properties of continuous functions on
compact sets.
Riemann integral, improper integrals; Fun-damental theorems of integral calculus.
Uniform convergence, continuity, differen-tiability and integrability for sequences and
series of functions; Partial derivatives of
functions of several (two or three) variables,
maxima and minima.

(3) Complex Analysis:
Analytic functions, Cauchy-Riemann equa-tions, 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.

(4) Linear Programming:

Linear programming problems, basic so-lution, basic feasible solution and optimal
solution; Graphical method and simplex
method of solutions; Duality.
Transportation and assignment problems.

(5) 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 equa-tion, Laplace equation and their solutions.

(6) Numerical Analysis and Computer
programming:
Numerical methods: Solution of algebraic
and transcendental equations of one vari-able by bisection, Regula-Falsi and New-ton-Raphson methods; solution of system
of linear equations by Gaussian elimina-tion and Gauss-Jordan (direct), Gauss-Seidel(iterative) methods. Newton’s (for-ward and backward) interpolation,
Lagrange’s interpolation.
Numerical integration: Trapezoidal rule,
Simpson’s rules, Gaussian quadrature for-mula.
Numerical solution of ordinary differential
equations: Euler and Runga Kutta-methods.
Computer Programming: Binary system;
Arithmetic and logical operations on num-bers; Octal and Hexadecimal systems;
Conversion to and from decimal systems;
Algebra of binary numbers.
Elements of computer systems and con-cept of memory; Basic logic gates and truth
tables, Boolean algebra, normal forms.
Representation of unsigned integers,
signed integers and reals, double preci-sion reals and long integers.
Algorithms and flow charts for solving nu-merical analysis problems.

(7) 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-dimen-sional and axisymmetric motion; Sources
and sinks, vortex motion; Navier-Stokes
equation for a viscous fluid.