# UPSC Preliminary Examination Syllabus Mathematics

## 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.

## 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.

## 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.

## 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.

## Ordinary Differential Equations

Order and Degree of a Differential Equation, First order differential Equations, Singular solution, Geometrical interpretation, Second order equations with constant coefficients.

# 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.

# 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 Shit/Rotate operators, Algorithms and Flow Charts.

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