Public Service Commission Uttar Pradesh Upper Subordinate Services Examination 2007  |  Upper Subordinate Services Examination 2007 Syllabus  |  Upper Subordinate Services Examination Fee  |  Upper Subordinate Services Examination Plan  | 

Public Service Commission Exams in India |  Uttar Pradesh Public Service Commission

Linear Algebra : Vector space, bases, dimensions of a finitely generated space, linear transformation : Rank and nullity of a linear transformation. Cayley Hamiliton theorem, Eigenvalues and Eigen vectors. Matrix of linear transformation, Row and column reduction. Echelon form, Equivalence, Congruence and similarly, Reduction to canonical form. Orthogonal, symmetrical, skew-symmetrical, unitary, Hemitian and skew-Hermitian matrices their eigen values, orthogonal and unitary reduction of quadratic and Hermitian forme. Positive definite quadratic forme. Simultaneous reduction.

Calculus : Real numbers, limits, continuity, differentiability. Mean value theorems, Taylor’s indeterminate forms, Maxima and Minima. Curve Tracing Asymptotes. Functions of several variables, partial derivatives, maxima and minima, Jacobian Definite and indefinite integrals. Double and tripple integrals (techniques only), application to Beta and Gamma Functions, Areas, Volumes, Centre of gravity.

Analytical 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 formulze.

Differential Equations : Order and Degree of a differential equation, differential equation of first order and first degree, variables separable. Homogeneous, linear, and exact differential equations, differential equations with constant coefficients. The complementary function and the particular integral of eax, cosax, sinax, xm, eax, cosbx, eax, sinbx.

Vector Analysis : Vector Algebra, Differentiation of vector function of a scalar variable Gradient, divergence and curl in cartesian, cylindrical and spherical coordinates and their physical interpretation. Higher order derivates. Vector identities and vector, equations, Gauss and stokes Theorems.

Tensor Analysis : Definition of Tensor, Transformation of coordinates, contravariant and contravariant tensors. Addition and multiplication of tensors, contraction of tensors. Inner product, fundamental tensors, Christoffel symbols, contravariant differentiation, Gradiant, curl and divergence in tensor notation.

Statics : 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.

Dynamics : Degree of freedom and constraints. Rectilinear motion Simple harmonic 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 resisting medium.

Hydrostatics : Pressure of heavy fluids. Equilibrium of fluids under given system of forces. Centre of pressure. Thrust on curved surfaces. Equilibrium of floating bodies, stability of equilibrium and pressure and gases, problems relating to atmosphere.

1. Mechanics : Conservation Law. Collisions, impact parameter, scattering cross-section centre of mass and lab systems with transformation of physical quantities, Rutherford Scattering. Motion of a rocket under constant force field. Rotating frames of reference, Coriolis force. Motion of rigid bodies, Dynamics of rotating bodies, Moment of inertia, Theorem of parallel and perpendicular axis. Moment of inertia of sphere, ring cylinder, disc. Angular momentum, Torque and precession of a top. Gyroscope. Central forces, Motion under inverse square law, Kepler’s Laws, Motion of Satellites including geostationary). Galilean Relativity, Special Theory of Relativity, Micheison-Morley Experiment, Lorentz Transformations-addition theorem of velocities. Variation of mass with velocity, Mass-Energy equivalence. Fluid dynamics, streamlines, Reynold number Viscosity, Poiseulle’s formula for the flow of liquid through narrow tubes, turbulence, Bernoulli’s Equation with simple applications.

2. Thermal Physics : Laws of thermodynamics, Entropy, Carnot’s cycle, Isothermal and Adiabatic changes, Thermodynamic Potentials, Helmboltz and Gibbs functions, Maxwell’s relations. The clausius-clapeyron equation, reversible cell, Joul- Kelvin effect, Stefan Boltzmann Law. Kinetic Theory of Gases, Maxwells’ Distribution Law of Velocities, Equipartition of energy, specific heats of gases, mean free patin, Borwnian Motion, Black Body radiation specific heat of solids, Einstein and Debye theories, Wein’s Law, Planck’s Law, solar constant. Shah’s theory of therma ionization and Steliar spectre Production of low temperatures using adiabetic demagnatization and dilution refrigeration. Concept of negative temperature.

3. Waves of Oscillations : Oscillations, simple harmonic motion. Examples of simple harmonic motion mass, spring and LC circuits. Stationary and travelling waves, Damped hormonic motion, forced oscillation and Resonance. Sharpness of resonance. Wave equation, Harmonic solutions, Plane and Spherical waves, Superposition of waves, Two perpendicular simple harmonic motions, Lissajous figures, Fourier analysis of periodic waves-square and triangular waves. Phase and Group Velocities, Beats, Huygen’s principle. Division of amplitude and wave front, Fresnel Biprism, Newton’s rings, Michelson interferometer, Fabry-Perot inter ferometer. Diffraction-Fresnel and Fraunhofe’r. Diffraction as a FourierTransformation. Fresnel and raunhofer diffraction by rectangular and circular apertures. Diffraction by straight edge, Single and multiple slits. Resolving power of granting and Optical Instruments. Rayleigh criterion. Polarization, production and Detection of polarised light (Linear, circular and elliptical) Brewster’s law, Huyghen’s theory of double refraction, optical rotation, polarimeters. Laser sources (helium-Neon, Ruby and semi conductor diode). Concept of spatial and temporal coherence Holography, theory and application.

1. Electricity and Magnetism : Coulomb’s law, Electric Field Gauss’s Law, Electric potential. Possion and Laplace equations for homogenous dielectric, uncharged conducting sphere in a uniform field, point charge and infinite conducting plane. Current electricity; Kirchoff’s laws and its applications: Wheatstone bridge, Kelvin’s double bridge, Carey-foster’s bridge. Bio-Savart law and applications. Ampere’s circuital law and its applications, Magnetic induction and field strength, Magnetic shell Magnetic field on the axis of circular coil Helmboltz coil, Electromagnetic induction, Faraday’s and Lenz’s law, Self and Mutual inductances, Alternating currents L.C.R. circuits, series and parallel resonance circuits, quality factor. Maxwell’s equations and electromagnetic waves, Transverse nature of electromagnetic waves, Poynting vector, Magnetic fields in Matter: Dia, para, Ferro, Antiferro and Ferrimagnetism (Qualitative approach only). Hsteresis.

2. Modern Physics: Bohr’s theory of hydrogen atom Electron spin, Optical and X-ray Spectral Stern-Gerlach experiment and spatial quantkation, Vector model of the atom spectral terms, fine structure of spectral fines. J-J and L-S coupling Zeeman effect, Pauli’s exclusion principle, spectral terms of two equivalent and non-equivalent electrons. Gross and fine structure of electronic band spectra. Raman effect, Photoelectric effect, Compton effect De-Broglie waves. Wave- Particle duality, uncertainty principle, postulates of quantum mechanics. Schrodinger wave equation with application

(i)particle in a box,(ii) motion across a step potential, One dimensional harmonic oscillator eigen values and eigen functions. Radioactivity, Alpha, Beta and gamma radiations. Elementary theory of the alpha deca. Nuclear binding energy. Mass spectroscopy, semi empirical mass formula. Nuclear fission and fusion. Elementary Reactor Physics, Elementary particles and their classification, strong and weak Electromagnetic interactions. Particle accelerators, cyclotrol. Linear accelerators. Elementary ideas of superconductivity.

3. Electronics : Band theory of solids, conductors insulators and semiconductors. Intrinsic and extrinsic semiconductors, P.N. junction, Thermistor Zener diodes. Reverse and forward based P.N. Junction, solar cell. Use of diodes and transistors for rectification, amplification oscillation, modulation and detection r.f. waves. Transistor, receiver. Television, Logic Gates and their truth table, some applications.

Algebra : Groups, subgroups, normal subgroups, 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 Euciidean domains, Field Extensions, Finite fields.

Real Analysis : Metric spaces, their topology with special reference to ‘R’ sequence in metric space Cauchy sequence completeness. Completion, continuous functions. Uniform continuity. Properties of continuous function on Compact sets. Riemann Steltjes Integral. Improper integrals and their condition’s of existence. Differentiation of function of several variables. Implicit function theorem, maxima and minima. Absolute and conditional Convergence of series of real 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, Cachley’s Residue theorem and Contour integration.

Partial Differential Equations : Formation of partial differential equations. Types of integrals of partial differential equations of first order, Charphs method, Partial differential equation with constant coefficients.

Mechanics : Generalised Coordinates, constraints, holonomic and non-holonomic systems, D ‘Alemberts' Principle and Langrange’s equations, Moment of inertia. Motion of rigid bodies in two dimensions.

Hydrodynamics : Equation of continuity, momentum and energy, inviscid flow theory, Two dimensional motion, streaming motion sources and Sinks.

Numerical Analysis : Transcendental and polynomial Equations-Methods of tabulation, bisection, reaula-false secants and Newton-Renhsonand order of is converagence. Interpolation and Numerical differentiation Polynomial interpolation with equal or unequal step size. Numerical differentiation formulae with error terms.

Numerical integration of Ordinary differential Equations : Euler’s method, mulistepperdictors Corrector methods. Adam’s and Milne’s method convergence and stability, Runge Kutta Methods. Operational Research : Mathematical Programming, Definition and some elementary properties of convex sets, simplex methods, rectangular games and their solutions.