PUC lecturers recruitment 2012 CET syllabus for Mathematics
Mathematics
(Subject Code - 14)
PAPER -1
Linear Algebra Vector space, bases, dimension of a
finitely generated space, ‘Linear Transformations, Rank and nullity of a linear
transformation, Cayley-Hamilton theorem, Eigenvalues and Eigen vectors.
Matrix of a linear transformation, Row and Column
reduction. Echelon form.
Equivalence, Congruence and similarity. Reduction to
canonical forms. Orthogonal, Symmetrical, skew-symmetrical, unitary, Hermitian
and skew-Hermitian matrices-their eigenvalues, orthogonal and unitary reduction
of quadratic and Hermitian forms. Positive definite quardratic forms, simultaneous
reduction. Calculus:
Real numbers, limits, continuity, differentiability, Mean-value
theorem, Taylor
theorem, indeterminate forms, Maxima and Minima, Curve
Tracing. Asymptotes :
Functions of several variables, partial derivatives
maxima and minima, Jocobian.
Definite and indefinite integrals, Double and triple
integrals (techniques only). Application to Beta and Gamma Functions. Areas, Volumes;
centre of gravity.
Analytic Geometry of Two and Three Dimensions First
and second degree equations in two dimensions in Cartesian and polar coordinates,
Plane, sphere, paraboloid, Ellipsoid, hyperboloid of one and two sheets and
their elementary properties, curves in space, curvature and torsion, Frenet’s formulae.
Differential Equations :
Order and Degree of a differential equaltion; differential
equation of first order and first degree, variables separable. Homogeneous, linear
and exact differential equations. Differential equations with constant
coefficients. The complimentary function and the particular integral of eax, COSax,
sinax, Xm, eax, Cosbx, eax, sinbx. Vector, Tensor, Statics, Dynamics and
Hydrostatics
(i) Vector Analysis - Vector Algebra, Differentation
and Vector function of a scalarvariable, Gradient, divergence and curl in
Cartesian, cylindrical and spherical coordinates and their physical
interpretation. Higher order derivaties. Vector identifies and Vector equations,
Gauss and Stokes Theorems.
(ii) Tensor Analysis: Definition of a Tensor, transformation
of coordinates, contravariant and covariact tensors. Addition and
multiplication of tensors, contraction of tensors, Inner product, fundamental
tensor, christoffel symbols, covariant differentiation. Gradient, Curl and
divergence in tensor notation.
(iii) Statics - Equilibrium of a system of particles,
work and potential energy. Friction, Common category. Principles of Virtual
work. Stability of equilibrium. Equilibrium of forces in three dimensions,
(iv) Dynamics - Degree of freedom and constraints. Rectilinear
motion. Simple
harmonic motion. Motion in a plane. Projectiles. Constrained
motion. Work and
energy. Motion under impulsive forces. Kepler’s laws.
Orbits under central forces.
Motion of varying mass. Motion under resistance.
(v) Hydrostatics - Pressure of heavy fluids, Equilibirum
of fluids under given system of forces. Centre of pressure Thrust on curved
surfaces. Equilibrium of floating bodies. Stability of equilibrium and Pressure
of gases, problems of relating to atmosphere.
PAPER - II
Section - A : Algebra, Real Analysis, Complex
Analysis, Partial differential equations.
Section - B : Mechanics, Hydrodynamics, Numerical
Analysis, Statistics including
probability. Operational research.
Algebra : Group, Sub-groups, normal sub-groups, homomorphism
of groups, quotient groups. Basic isomorphism theorems. Sylow theorems. Permutation
Groups. Cayley’s theorem. Rings and ideals, Principal ideal domains, unique
factorization domains and Euclidean domains. Field Extensions, finite fields. Real
Analysis: Metric spaces, their topology with special reference to Rn,sequence
in
a metric space Cauchy sequence, Completeness, Completion,
Continuous functions, Uniform Continuity. Properties of continuous functions on
Compact sets. Riemann Steiltjes’ Integral, Improper intergrals and their
conditions of existance. Differentiation of functions of several variables. Implicit
function theorem, maxima and minima., Absolute and Conditional Convergence of
series of real and Complex terms, Rearrangement of series. Uniform convergence
infinite products. Continuity, differentiability and integrability for series, Multiple
integrals.Complex Analysis: Analytic functions, cauchy’s theorem, Cauchy’s
integral formula, power series, Taylor’s series, Singularities, Cauchy’s
Residue theorem and Contour integration.
Partial Differential Equations : Formation of partial
differential equations, Types of
Integrals of Partial differential equations of first
order, Charpits methods, Partial
differential equation with constant coefficients.
Mechanics : Generalised Co-ordinates, Constraints, holonomic
and non-holonomic systems, D’Alembert’s principle and Langranges’ equations, Moment
of Inertia, Motion of rigid bodies in two dimension.
Hydrodynamics: Equation of continuity, momentum and
energy, Inviscid Flow Theory:
Two dimensional motion, Streaming motion, sources and
Sinks. Numerical Analysis : Transcedental and Polynomial Equations: Methods of
solution, bisection, regula-false secant and Newton-Raphson and order of its
convergence. Interpolation and Numerical Differentiation: Polynomial
interpolation with equal or unequal step size. Spline interpolation - Gubic
Splines. Numerical differentiation formulae with error terms.
Numerical Intergration : Problems of approximate
quadrative, quadrature formulae with equispaced arguments, Caussian quadrature
Convergence.
Ordinary Differential Equations : Euler’s method, multisytep
Predector-Corrector
Methods - Adam’s and Milne’s method, Convergence and
stability, Runge-Kutta
methods. Probability and Statistics.
1. Statistical Methods : Concept of statistical
population and random sample. Collection and presentation of date. Measure of
location and dispersion. Moment
and Shepard’s corrections. Comulants. Measures of
Skewness of Kurtosis. Curve fitting by least squares Regression, correlation
and correlation ratio. Rank
correlation, partial correlation co-efficient and
Multiple correlation co-efficient.
2. Probability : Discrete sample space, Events, their
union and intersection etc.
Probability - Classical relative frequency and
exiomatic approaches, Probability
in continnum, Probability space conditional
probability and independence, Basic
laws of Probability, Probability of combination of
events, Bayes theorem, Rondom variable Priobability function, Probability
density function, Distribution function, Mathematical expectation, Marginal and
conditional distributions, conditional expectation.
3. Probability distributions : Binomial, Poison, Normal
Gamma, Beta, Cauchy,
Multinmial, Hypergeometric, Negative Bionomial, Chebychev’s
lemma (Weak)
law of large numbers, Central limit theorem for independent
and identical varieties. Standard errors, Sampling distribution of to Fand Chi-square
and their uses interests of significance. Large sample tests for mean and
proportion.
Operational Research : Mathematical Programming : Definition
and some elementary properties of convex sets, simplex methods, degeneracy, duality,
and sensitivity analysis, rectangular games and their solutions, Transportation
and assignment problems. Kuha Tucker condition for non-linear programming. Bellman’s
optimality principle and some elementary applications of dynamic programming. Theory
of Queues: Analysis of steady - State and transient solutions for quenqueing system
with Fission arrivals and exponential service time. Deterministic replacement
models, Sequencing problems with two machines n jobs, 3 machines, n jobs (Special
case) and n machines 2 jobs.
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