The Odisha Public Service Commission is the state agency authorized to conduct the Civil Services Examination for entry-level appointments to the various civil services of Odisha. Here is a Detailed Syllabus for Odisha Civil Services Main Exam Paper Mathematics.
Odisha Civil Services Examination
OPSC Optional Paper - Mathematics
Paper-I
Section – A
- Abstract Algebra :
- Integers, Congruences.
- Groups, Subgroup, Normal Subgroups, Permutation groups, Homomorphism, Isomorphism, Counting Principles, Sylow’s Theorem, Caley’s Group.
- Rings, Integral Domain, Field, Subring, Homomorphism, Ideal, Principal Ideal Ring, Maximal Ideal, Polynomial rings, Unique Factorization Theorem.
- Linear Algebra :
- Vector space, Linear dependence, Independence, Subspaces, Basis, Dimension, Finite Dimensional Vector space, Linear Transformation, Rank-nullity Theorem.
- Matrices, Determinants, Igenvalue, Igenvectors, Row-column reduction, Echelon form, Orthogonal, Symmetrical, Skew-symmetrical, Unitary, Hermitian Matrices.
- Analytic Geometry :
- 2-D Geometry : Straight lines, Pairs of lines, Circle, System of Circles, Conic sections.
- 3-D Geometry : Planes, Lines, Skew-lines, Sphere, Intersection of Plane and sphere, Cone, Cylinder, Conicoids, Tangent plane to conicoids .
Section – B
- Real and Complex Analysis :
- Real Analysis : Real number system, Order relation, Bounds, l.u.b. g.l.b., Cauchy sequence, Completeness, Compactness, Continuity, Uniform Continuity of functions, Riemann-Theory of Integration, Fundamental Theorem of calculus, Convergence of sequence and series, Uniform convergence.
- Complex Analysis : Analytic function, Cauchy Riemann Equation, Cauchy Integral Formula, Taylor, Laurent’s series, Singuralities, Poles, residues, Contour Integral.
- Calculus :
- Functions of one variable : Limit, Continuity, Differentiability, Meanvalue theorem, Maxima, Minima.
- Asymptotes and Curvatures : Rectification, Area , Volume and Surface area of revolution (Equations in Cartesian and Parametric forms only)
- Functions of several variables : Limit, Continuity, Differentiability, Jacobians, Euler’s theorem.
- Improper integrals : Convergence, Gamma and Beta functions./
- Multiple integrals : Double and Triple integrals and their Evaluations.
- Vector Analysis :
- Dot and Vector products, Vector and scalar Triple Products.
- Differentiation of Vector functions, Divergence, Gradient, Curl of Vectors
- (in Cartesian forms only).
- Green, Gauss and Stokes theorems and applications.
- Tangent, normal and binormal of curves in space, serret-frenet formulas.
Paper – II
Section – A
- Numerical Analysis :
- Interpolation : Lagrange, Newton divided difference forms, Forward and back ward interpolation polynomials.
- Approximations : Least squares approximations and curve fitting.
- Numerical solution of non-linear equations : Bisection, Secant, NewtonRaphson and fixed point iteration techniques.
- Numerical differentiation and integration: Differentiation formulas involving differences, Newton-Cotes rules, Compound rules, Gauss –
Legendre 2 and 3 point rules.
- Numerical solution. of I.V.P.: Euler method, Taylor’s method, RungeKutta Method of order two
- Graph Theory :
Simple graphs, Regular, Complete graphs, Bipartite graphs, Matrix representation of graphs, Connected graphs, Isomorphic graphs, Trees, Planar graph, Hamiltonian and Eulerian graphs, Vertex colouring of graphs and Chromatic number.
- Ordinary and Partial differential equations.
- Linear first order O.D.E.
- Higher order linear differential equations with constant and variable coefficients.
- Series solution of O.D.E.
- Solution of O.D.E. by Laplace transformation techniques.
- Solution of equations Pdx + Qdy + Rdz=O and dx/P = dy/Q = dz/R
- Char pits method for partial differential equations.
- Linear second order P.D.E. and solutions.
Section – B
- Computer programming :
- Flow charting and algorithms.
- Basics of Fortran language, arithmetic and logical operations, Arithmetc and Logical Statements.
- GO TO and Computed GO TO Statements, Arithmetic and Logical IF, IF… THEN….ELSE Statements, DO Loops.
- Arrays and subscripted variables.
- Functions, Subprograms and Subroutines.
- Programme writing in Fortran.
- Mechanics and Hydrodynamics.
- Statics : Law of parallelogram of forces, Equilibrium of forces, Couple and Moments, Frictions.
- Dynamics : Laws of motion, D’ Alemberts principle, Motion of a particle I in a plane, Projectiles, Motion of rigid bodies, Moment of inertia.
- Hydrodynamics : Equation of continuity, Euler equation of motion (in Cartesian forms) Stream lines, Path Line, Potential flow, Stream functions and Potential functions, Sources, Sinks and Image system with respect to
- Plane and Circle.
- Operations Research :
- Formulation of L.P.P., Graphical solution.
- Simplex method and Duality.
- Transportation and Assignment problems
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