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# (Syllabus) RAS Syllabus (Main): Mathematics (Code No. 20)

## Rajasthan Public Service Commission

## RAS Syllabus (Main): Mathematics (Code No. 20)

## Mathematics Paper-I

**1. Linear Algebra**

Vector spaces, subspaces, bases and dimension of a finitely generated space. Quotient spaces, Rank and nullity of a linear transformation, Matrix of a linear transformation, Row and Column reduction, Echelon form, Equivalence. Congruence and Similarity, Reduction to canonical forms. Cayley-Hamilton theorem, Eigenvalues and Eigenvectors, Application of matrices and determinants to solutions of simultaneous linear equations upto four unknowns; determinants upto fourth order.

**2. Abstract Algebra**

Groups, Subgroups, Normal subgroups, Permutation groups, Quotient groups, Homomorphism groups, Isomorphism theorems, Cayley and Lagrange's theorems. Automorphisms. Rings, Integral domains. Fields, ideals, Principal ideal domains. Simple rings, Prime ideals and Prime fields. Maximal ideals in Commutative rings.

**3. Calculus**

Limits, Continuity, Differentiability, Mean Value theorems, Taylor's theorem, Indeterminate forms, Partial derivatives, Maxima & Minima of functions of one and two variables, Curvature, Asymptotes, Curve Tracing, Envelopes, Definite integrals, Rectification and Quadrature, Volumes and Surfaces (standard curves) of solids of revolutions. Double & Triple integrals. Beta and Gamma functions. Changes of order of integration, Triple integrals and simple application, Dirichlet's integral.

**4. Real Analysis**

Real numbers as a complete ordered field, Dedekind's theory of Real numbers,
Linear sets, Lower and upper bounds, Limit points, Bolzanoweierstrass theorem,
closed and open sets, concept of compactness enumerable sets. Heine-Borel therem,
connected sets, Real sequences, Limit and convergence, Cauchy's general
principle of convergence.

Continuity and Differentiability, types of discontinuities, Properties of
derivable functions, Darboux's and Roll's theorem.

Riemann integration, Mean value theorems and fundamental theorem of Integral
Calculus. Riemann-stieltje's integral.

Convergence of series of real variables. Absolute convergence, conditionally
convergent series of real numbers, Uniform convergence of sequence and series of
functions.

**5. Analytical Geometry of Two and Three Dimensions**

Conic sections in two dimensions referred to Cartesian and polar coordinates, Plane, straight lines, sphere, cylinder and cone in standard forms and their properties.

**6. Complex Analysis**

Analytical functions, Cauchy's theorem, Cauchy's integral formula, power series, Taylor's and Laurent's series, Singularities, Cauchy's Residue theorem and Contour integration.

## Mathematics Paper-II

**Vector Analysis :**Vector algebra. Differentiation of a Vector function of scalar variable, Gradient, divergence and Curl (rectangular coordinates) and their physical interpretation, vector identities, Gauss's, Stokes' & Green's theorems.**Statics and Hydrostatics :**General conditions of equilibrium of a rigid body under coplanar forces, Friction, Common Catenary, Principle of Virtual Work.

Fluid pressure on plane surfaces, Thrust on curved surfaces, Centre of pressure. Equilibrium of floating bodies.**Dynamics :**Kinetics and Kinematics, Simple Harmonic motion, Hook's law, Motion of a particle attached to horizontal and vertical elastic strings, Motion in a plane under variable forces, Motion under resisting medium, Direct impact of smooth bodies, Projectiles, Motion on a smooth vertical circle and Cycloid, Orbits under central forces, Kepler's laws of motion.**Linear Programming :**Graphical method of solution of linear programming in tow variables. Convex sets and their properties, Simplex methods, degeneracy, duality and sensitivity analysis, Assignment Problems, Transportation Problems.**Numerical Analysis and Difference Equations :**Polynomial interpolation with equal or unequal step size. Lagrange's interpolation formula, Truncation error. Numerical differentiation, Numerical integration, Newton-Cotes quadrature formula, Gauss's quadrature formulae, Convergence, Estimation of errors.

Transcedental and polynomical equations, bisection method, Regulafalsi method, method of interaction, Newton-Raphson method, Convergence.

First and higher order aomogeneous linear difference equations, non-homogeneous linear difference equations:

Complementary function, particular integral.**Differential Equations :**Ordinary differential equations of first order and first degree, Differential equations of first order but not of first degree, Clairut's equation-its general and singular solution. Linear Differential equations with constant coefficients, Homogeneous linear differential equations with variable coefficients. Simultaneous linear differential equations of first order. Linear differential equations of second order-change of variables, normal form, method of variation of parameters.