RAS PRE Syllabus-Mathematics (Code No. 20)

Mathematics (Code No. 20)

Algebra : 

Sets, relations, equivalence relations, Natural numbers, Integers, Rational numbers, Real and Complex numbers, division algorithm, greatest common division polynomials, division algorithm, derivations, Integral, rational real and complex roots of a polynomial, Relation between roots and Coefficients, repeated roots, elementary symmetric functions, Groups, rings, fields and their elementary properties.

Matrices : 

Addition and multiplication, elementary row and column operation, rank determinants, inverse, solutions of systems of linear equations.

Calculus : 

Real numbers, order completeness property, standard functions, limits, continuity, properties of continuous functions in closed intervals, differentiability, Mean value Theorem, Taylors Theorem, Maxima and Minima, Application to curves tangent normal properties, Curvatute, asymptotes, double points, points of inflection and tracing.

Definition of a definite integral of continuous function as the limit of a sum, fundamental theorem of integral Calculus, methods of integration, rectification quadrature, volume and surfaces of solids of revolution.

Partial differentiation and its application.

Simple test of convergence of series of positive terms, alternating series and absolute convergence.

Differential Equations : First order differential equations, Singular solutions, geometrical interpretations, linear differential equations with constant coefficients.

Geometry : 

Analytic Geometry of straight lines and conics referred to Cartesian and polar Coordinates; three dimensional geometry for planes, straight lines, sphere, Cone and Cylinder.

Mechanics : 

Concept of particle, lamina, rigid body, displacement, force, mass, weight, concept of scalar and vector quantities, Vector Algebra, Combination and equilibrium of coplanar forces, Newton's Laws of motion, motion of a particle in a straight line; Simple Harmonic motion, projectile, circular motion, motion under central forces (inverse square law), escape velocity.