IAS Mathematics Question

1. Algebra : Elements of Set Theory; Algebra of Real and Complex numbers including Demovire's theorem; Polynomials and Polynomial equations, relation between Coefficients and Roots, symmetric functions of roots; Elements of Group Theory; Sub-Group, Cyclic groups, Permutation, Groups and their elementary properties.
Rings, Integral Domains and Fields and their elementary properties.
2. Vector Spaces and Matrices : Vector Space, Linear Dependence and Independence. Sub-spaces. Basis and Dimensions, Finite Dimensional Vector Spaces. Linear Transformation of a Finite Dimensional Vector Space, Matrix Representation. Singular and Nonsingular Transformations. Rank and Nullity.
Matrices : Addition, Multiplication, Determinants of a Matrix, Properties of Determinants of order, Inverse of a Matrix, Cramer's rule.
3. Geometry and Vectors : Analytic Geometry of straight lines and conics in Cartesian and Polar coordinates; Three Dimensional geometry for planes, straight lines, sphere, cone and cylinder. Addition, Subtraction and Products of Vectors and Simple applications to Geometry.
4. Calculus : Functions, Sequences, Series, Limits, Continuity, Derivatives.
Application of Derivatives : Rates of change, Tangents, Normals, Maxima, Minima, Rolle's Theorem, Mean Value Theorems of Lagrange and Cauchy, Asymptotes, Curvature. Methods of finding indefinite integrals, Definite Integrals, Fundamental Theorem of integrals Calculus. Application of definite integrals to area, Length of a plane curve, Volume and Surfaces of revolution.
5. Ordinary Differential Equations : Order and Degree of a Differential Equation, First order differential Equations, Singular solution, Geometrical interpretation, Second order equations with constant coefficients.
6. Mechanics : Concepts of particles-Lamina; Rigid Body; Displacements; force; Mass; weight; Motion; Velocity; Speed; Acceleration; Parallelogram of forces; Parallelogram of velocity, acceleration; resultant; equilibrium of coplanar forces; Moments; Couples; Friction; Centre of mass, Gravity; Laws of motion; Motion of a particle in a straight line; simple Harmonic Motion; Motion under conservative forces; Motion under gravity; Projectile; Escape velocity; Motion of artificial satellites.
7. Elements of Computer Programming : Binary system, Octal and Hexadecimal systems. Conversion to and from Decimal systems. Codes, Bits, Bytes and Words. Memory of a computer, Arithmetic and Logical operations on numbers. Precisions. AND, OR, XOR, NOT and Shift Not Shit operators, Algorithms and Flow Charts. 
Civil services( pre.) exam is not too far away now.for the student, who has taken mathematics as his optional paper, it is busy time.there are few tips ,to effectively utilize this time.
1..make notes for surface area, volume, asymptotes , length of curve , centre of gravity ,equation of evelope of the important and well known curve.
2..statics and dynamics are two topics, which student takes lightly.they are small in portion.Simple harmonic motion,rectilinear motion, projectile, central forces, moment of interia, pedal equation are important for examination.
3.calculas questions manily based on important curves.so prepare each and every aspect of the curves.
4..do not try questions in booklet sequence, first of all try easy one.it will encourage you and there are lesser chances of errors.
5…practice more on visulisation of curves and their properties.
6..78-82 is good score in math is a good score for math( pre.).
7.stress more on accuracy of the question rather than number of questions.
8..vector analysis is another topic, which is easy to handle and contribute 3-4 questions.
9..in recent years a lot of questions are coming from real analysis, abstract algebra and linear algebra. These section produce 15 -18 questions.questions are concept based, so try to understand deep meaning of theorems and results.
10..a lot of topics covered in math (pre.) paper.so distribute it in sections for profer planning.