Rajasthan Public Service Commission
RAS Syllabus (Main): Mathematics (Code No. 20)
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
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
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,
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
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
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.
- 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
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
- 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,
- 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
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