# UPSC Maths Optional Syllabus

UPSC Maths Optional Syllabus can be read or downloaded from here. A large number of candidates appear in UPSC Exams every year. UPSC conducts exams for different government jobs in India. To get good marks in the exam candidates need to study syllabus properly.

UPSC full form is **Union Public Service Commission.** This commission has responsibility to prepare candidates for government jobs in India.

### UPSC Mains Syllabus in Detail

**B.MAIN EXAMINATION:**

The written examination will consist of the following papers

Qualifying Papers:

Paper-A

(One of the Indian Language to be selected by the candidate from the Languages included in the Eighth Schedule to the Constitution). **300 Marks**

Paper-B

English **300 Marks**

**Papers to be counted for merit**

Paper-I – Essay – 250 Marks

Paper-II – General Studies I – 250 Marks

(Indian Heritage and Culture, History and Geography of the World and Society)

Paper-III – General Studies – II – 250 Marks

(Governance, Constitution, Polity, Social Justice and International relations)

Paper-IV – General Studies – III – 250 Marks

(Technology, Economic Development, Bio-diversity, Environment, Security and Disaster Management)

Paper-V – General Studies -IV – 250 Marks

(Ethics, Integrity and Aptitude)

Paper-VI – Optional Subject – Paper 1 – 250 Marks

Paper-VII – Optional Subject – Paper 2 – 250 Marks

**Sub Total (Written test) – 1750 Marks**

**Personality Test – 275 Marks**

**Grand Total – 2025 Marks**

Candidates may choose any one of the optional subjects from amongst the list of subjects given below

**List of optional subjects for Main Examination:**

(1) Agriculture

(ii) Animal Husbandry and Veterinary Science

(iii) Anthropology

(iv) Botany

(v) Chemistry

(vi) Civil Engineering

(vii) Commerce and Accountancy

(viii) Economics

(ix) Electrical Engineering

(x) Geography

(xi) Geology

(xii) History

(xiii) Law

(xiv) Management

(xv) Mathematics

(xvi) Mechanical Engineering

(xvii) Medical Science

(xviii) Philosophy

(xix) Physics

(xx) Political Science and International Relations

(xxi) Psychology

(xxii) Public Administration

(xxiii) Sociology

(xxiv) Statistics

(XXV) Zoology

**(xxvi) Literature of any one of the following languages:**

Assamese, Bengali, Bodo, Dogri, Gujarati, Hindi, Kannada, Kashmiri, Konkani, Maithili, Malayalam, Manipuri,

Marathi, Nepali, Oriya, Punjabi, Sanskrit, Santhali, Sindhi, Tamil, Telugu, Urdu and English.

NOTE:

(i) The question papers for the examination will be of conventional (essay) type.

(ii) Each paper will be of three hours duration.

(iii) Candidates will have the option to answer all the question papers, except the Qualifying Language Papers, Paper-A and Paper-B, in any of the languages included in the Eighth Schedule to the Constitution of India or in English.

(iv) Candidates exercising the option to answer Papers in any one of the languages mentioned above may, if they so desire, give English version within brackets of only the description of the technical terms, if any, in addition to the version in the language opted by them. Candidates should, however, note that if they misuse the above rule, a deduction will be made on this account from the total marks otherwise accruing to them and in extreme cases, their script(s) will not be valued for being in an unauthorized medium.

(vi) The question papers (other than the literature of language papers) will be set in Hindi and English only.

(vii) The details of the syllabi are set out in Part B of Section III.

### UPSC Maths Optional Syllabus in Detail

UPSC Mathematics syllabus is divided in two paper. Paper 1 is divided in 6 topics whereas paper 2 is divided in 7 topics.

UPSC Maths Optional Paper 1 Topics

**Linear Algebra****Calculus****Analytic Geometry****Ordinary Differential Equations****Dynamics and Statics****Vector Analysis**

UPSC Maths Optional Paper 2 Topics

**Algebra****Real Analysis****Complex Analysis****Linear Programming****Partial Differential Equations****Numerical Analysis and Computer Programming****Mechanics and Fluid Dynamics**

MATHEMATICS

PAPER – 1

**(1) Linear Algebra :**

Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions, Linear transformations, rank and nullity, matrix of a linear transformation.

Algebra of Matrices; Row and column reduction, Echelon form, congruences and similarity; Rankof a matrix; Inverse of a matrix; Solution of system of linear equations; Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, Symmetric, skew-syntheti, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues,

**(2) Calculus :**

Real numbers, functions of a real variable, limits, continuity, differentiability, mean-value theorem, Taylor’s theorem with remainders, indeterminate forms, maxima and minima asymptotes; Curve tracing: Functions of two or three variables: Limits, continuity, partial derivatives, maxima and minima, Lagrange’s method of multipliers, Jacobian.

Riemann’s definition of definite integrals; Indefinite integrals: Infinite and improper integral; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

**(3) Analytic Geometry:**

Cartesian and polar coordinates in three dimensions, second degree equations in three variables, reduction to Canonical forms; straight lines, shortest distance between two skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties

**(4) Ordinary Differential Equations :**

Formulation of differential equations; Equations of first order and first degree, integrating factor; Orthogonal trajectory: Equations of first order but not of first degree, Clairaut’s equation, singular solution

Second and higher order liner equations with constant coefficients, complementary function, particular integral and general solution.

Section order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using method of variation of parameters.

Laplace and Inverse Laplace transforms and their properties, Laplace transforms of elementary functions. Application to initial value problems for 2nd order linear equations with constant coefficients.

**(5) Dynamics and Statics :**

Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; Constrained

motion; Work and energy, conservation of energy; Kepler’s laws, orbits under central forces.

Equilibrium of a system of particles; Work and potential energy, friction, Common catenary;

Principle of virtual work; Stability of equilibrium, equilibrium of forces in three dimensions.

**(6) Vector Analysis :**

Scalar and vector fields, differentiation of vector field of a scalar variable; Gradient, divergence and curl in cartesian and cylindrical coordinates; Higher order derivatives; Vector identities and vector equation.

Application to geometry : Curves in space, curvature and torsion; Serret-Furenet’s formulae.

Gauss and Stokes’ theorems, Green’s indentities.

PAPER II

**(1) Algebra :**

Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s theorem.

Rings, subrings and ideals, homomorphisms of rings; Integral domains, principal ideal domains, Euclidean domains and unique factorization domains; Fields, quotient fields.

**(2) Real Analysis :**

Real number system as an ordered field with least upper bound property; Sequences, limit of a sequence, Cauchy sequence, completeness of real line; Series and its convergence, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Continuity and uniform continuity of functions, properties of continuous functions on compact sets.

Riemann integral, improper integrals; Fundamental theorems of integral calculus.

Uniform convergence, continuity, differentiability and integrability for sequences and series of

functions; Partial derivatives of functions of several (two or three) variables, maxima and minima.

**(3) Complex Analysis :**

Analytic function, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, power series, representation of an analytic function, Taylor’s series; Singularities; Laurent’s series; Cauchy’s residue theorem; Contour integration.

**(4) Linear Programming:**

Linear programming problems, basic solution, basic feasible solution and optimal solution; Graphical method and simplex method of solutions; Duality. Transportation and assignment problems.

**(5) Partial Differential Equations :**

Family of surfaces in three dimensions and formulation of partial differential equations; Solution of quasilinear partial differential equations of the first order, Cauchy’s method of characteristics; Linear partial differential equations of the second order with constant coefficients, canonical form; Equation of a vibrating string, heat equation, Laplace equation and their solutions.

**(6) Numerical Analysis and Computer Programming:**

Numerical methods: Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi and Newton-Raphson methods, solution of system of linear equations by Gaussian Elimination and Gauss-Jorden (direct), Gauss-Seidel iterative) methods. Newton’s (forward and backward) and interpolation, Lagrange’s interpolation.

Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula.

Numerical solution of ordinary differential equations : Eular and Runga Kutta methods.

Computer Programming : Binary system; Arithmetic and logical operations on numbers; Octal and Hexadecimal Systems; Conversion to and from decimal Systems; Algebra of binary numbers.

Elements of computer systems and concept of memory: Basic logic gates and truth tables, Boolean algebra, normal forms.

Representation of unsigned integers, signed integers and reals, double precision reals and long integers.

Algorithms and flow charts for solving numerical analysis problems.

**(7) Mechanics and Fluid Dynamics :**

Generalised coordinates; D’Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions.

Equation of continuity; Euler’s equation of rnotion for inviscid flow; Stream-lines, path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.

This was the syllabus of UPSC Maths Optional Syllabus. Candidates can also visit UPSC Official Website – Click Here

Read Also – UPSC Political Science Optional Syllabus