Civil Service Main Examination Syllabus Mathematics - Optional
PAPER
- I
(1)
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.
(2)
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.
(3)
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.
(4)
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.
(5)
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.
(6)
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.
(2)
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.
(3)
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.
(4)
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.
(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 equation,
Laplace equation and their solutions.
(6)
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.
(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-dimensional and axisymmetric motion; Sources and
sinks, vortex motion; Navier-Stokes equation for a viscous fluid.