# Exam Strategy for Mathematics

## Paper I

### Section A

- Linear Algebra: Vector, space, linear dependance and independance, subspaces, bases, dimensions. Finite dimensional vector spaces. Eigenvalues and eigenvectors, eqivalence, congruences and similarity, reduction to canonical form, rank, orthogonal, symmetrical, skew symmetrical, unitary, hermitian, skew-hermitian formstheir eigenvalues.
- Calculus: Lagrange's method of multipliers, Jacobian. Riemann's definition of definite integrals, indefinite integrals, infinite and improper integrals, beta and gamma functions. Double and triple integrals i.e.. Evaluation techniques only. Areas, surface and volumes and centre of gravity.
- Analytic Geometry: Sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

### Section B

- Ordinary Differential Equations: Clariaut's equation, singular solution. Higher order linear equations, with constant coefficients, complementary function and particular integral, general solution, Euler-Cauchy equation. Second order linear equations with variable coefficients, determination of complete solution when one solution is known, method of variation of parameters.
- Dynamics, Statics and Hydrostatics: However, ignore the whole section, if you have thoroughly prepared for other sections.
- Vector Analysis: Triple products, vector identities and vector equations. Application to Geometry: Curves in space, curvature and torision. Serret-Frenet's formulae, Gauss and Stokes'theorems, Green's identities.

## Paper II

### Section A

- Algebra: Normal subgroups, homomorphism of groups quotient groups basic isomorophism theorems, Sylow's group, principal ideal domains, unique factorisation domains and Euclidean domains. Field extensions, finite fields.
- Real Analysis: Riemann integral, improper integrals, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Uniform convergence, continuity, differentiability and integrability for sequences and series of functions. Differentiation of functions of several variables, alteration in the order of partial derivatives, implicit function theorem, maxima and minima. Multiple integrals.
- Complex Analysis: However, ignore the whole section, if you have thoroughly prepared for other sections.
- Linear Programming: Basic solution, basic feasible solution and optimal solution, Simplex method of solutions. Duality. Transportation and assignment problems. Travelling salesman problems.

### Section B

- Partial differential equations: Solutions of equations of type dx/p = dy/q = dz/r; orthogonal trajectories, pfaffian differential equations; partial differential equations of the first order, solution by Cauchy's method of characteristics; Char-pit's method of solutions, linear partial differential equations of the second order with constant coefficients, equations of vibrating string, heat equation, laplace equation.
- Numerical Analysis and Computer programming: Numerical methods, Regula-Falsi and Newton-Raphson methods Numerical integration: Simpson's one-third rule, tranpesodial rule, Gaussian quardrature formula. Numerical solution of ordinary differential equations: Euler and Runge Kutta-methods.
- Computer Programming: Binary system. Arithmetic and logical operations on numbers. Bitwise operations. Octal and Hexadecimal Systems. Convers-ion to and from decimal Systems.
- Mechanics and Fluid Dynamics: D ‘Alembert's principle and Lagrange’ equations, Hamilton equations, moment of intertia, motion of rigid bodies in two dimensions.

Sail through IAS Mains and Prelims: Fully-explained Prelims (Both GS & Aptitude) problems with detailed solutions. Notes & detailed answers for Mains GS, Essay, Complulsory (Hindi and English) papers and optionals.

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