» UP PSC Vacancy Details » UP PSC Educational Qualification » UP PSC Application Form

» UP PSC Important Instruction » UP PSC Plan of Examination » UP PSC Syllabus

Lower Mathematics (Subject Code - 06) Section-A

1. Algebra:

(A1) Set Theoretic Concepts: Bets, operations on sets (Union, Intersection, Complement Difference), Power sets, Cartesian products of sets, Relations, Compsite and inverse relations, Equivalence and order relations Examples, Maps (functions), injective, map, subjective map bijective maps, inverse maps, Examples,

(A2) Number System: Natural numbers, peano's axioms, Principle of Mathematical induction, integers, Divsion algorithm in integers, G.C.D. & L.C.M. Euclidean alogorthrh for G.C.D. Prime numbers, Prime factrozation of integers (Fundamental theorem of Arithmatics), Rational numbers.

(A3) Groups: Definition and example of groups, Subsgroups, Homomorphism, Cosel decomposition, Lagrenges theorem. Normal subgroups, Quotient groups. Premutations groups.

(A4) Rings: Definition and examples of rings, integral Domains & Fields, Homomorphisms of rings, Field of Quotiendts of integral Domains, Subrings, and ideals, Quotient Rings, Polynomiar over Fields, Division Algorithm, G.C.D. & L.C.M. of Polynomials, Remainder and Factor theorems, Roots of Polynomials and Newton's identities (about sum of products of roots, taken r at a time)

2. Linear Algebra: Vector space over a field, subspaces, Linear dependence and independence, Linear span of subsects, Bases, Dimention, Linear Maps, Image & Kernel, Rank and Nullity of Linear Maps, Matrices, Algebra of matrices.

Matrics Representation of Linear Maps, Row and Column rank of Matrices. Determinants and their properties, Cofactor & Adjoint of a Matrics, Inverse matrices, similar Matrices, Systems of Linear Equations, Crammer's Rule, Eigenvalues and elgen-vectors., Cayley Hamilton Theorem, Diagonalization of square matrics with distinct eigen values.

3. Analytical Geometry:

(G1) Two Dimensional: Straight lines, Circles, Parabolas, Ellipses, Hyperbolas, General equation of second degree, Polar equations of conies. Examples.

(G2) Three Dimensional: Planes and straight lines (Cartesian and Vector forms of the equations), Spheres.

Section - B

4. Calculus: (CD Axiomatic description of the set of Real numbers as a complete ordered field, Modulus and Greatest Integer functions, Archimedean property of Real numbers, Complex numbers, De Molver's Theorem N root of complex numbers.

Sequences, Limit of a Sequence Bounded Sequence. Algebra of limits. Convergence of series of positive terms (Comparison Test Radio Test, Root Test and Higher Ratio Tests). Limits and continuity of functions, Algebra of limits Discontinuities, properties of continuous functions.

Examples. Differentiability. Algebra of Derivatives, Leibniz Theorem for successive differentiation Chain rule of differentiation, Rolle's Theorem mean value theorem (Dagranges Form) Taylor's Theorem, Expansion of functions, Tangents and Normals, Maxima and Minima, Indeterminate Forms -(DeHospital's rule).

Limit and continuity of Real Valued Functions of two real variables.

Partial Derivatives, Simple examples (C2) Techniques of Integration: Integration of Standard Functions, Definite intergrals and their properties".

Integration by parts and substitutions, Integration of Rational Functions using Partial Fractions, Integration of Irrational Algebraic Functions, Integration of Transcendentral Function Reduction Formulas.

5. Differential Equations :

(D1) First Order Differential Equations: Separable Equations, Equations with Homogenous Coefficients, Exact Equations, Integrating Factors, Linear Equations, Bernouli's Equations, Equations of first order and Higher degree, Clairaul's Equations, Singular solutions.

(D2) Linear Differential Equations: Linear independence of Solutions, Wronskians, Statement of Existence and Uniquences Theorems, Fundamental system of Solutions, Method of variation of parameters for second order Equations, Reduction of order Method, Linear Differential Equations with Constant Coefficients, Auxiliary Equations, Complementary Functions and Particular solutions, Inverse Operator Method, Exponential Shift Enter-Cauchy equations, Linear system of First Order equations with constant Coefficients, Method Elimination.

Section-C

6. Statics: Forces and their vector representations, Moments of Forces, Couples, Equilibrium of a Rigid Body acted on by forces in one plane. Virtual Work, Centre of Gravity, Catenary, Examples.

7. Dynamics: Momentum, Angular, Momentum Rectilinear motion of a particly, Simple harmonic Motion, Elactic strings, Motion under inverse equare law. Radial and Transverse Velocities and Accelerations, Tangential and Normal Velocities and Accelerations, Examples. Projectiles, Central Orbits.

UPPSC - Combined Lower Subordinate Services Exam 2009 Mathematics Syllabus