Civil Service Exam Mains – IAS Exam Mains 2017 Syllabus for Mathematics Paper II - Paper I
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.