IAS Maths Syllabus For UPSC Mains

Union Public Service Commission (UPSC) Civil Services Mains Exam comprises Mathematics as one of the Optional Subjects with 2 papers (Paper I and Paper II).The Maths Optional for CSE 2022 remains the same as was in 2021. Check UPSC Notification 2022 to stay updated with the current IAS Maths Syllabus. The IAS Mathematics Optional papers are 250 marks each with a total of 500 marks. Mains in IAS Exam has 9 papers.This article provides you with the IAS Mathematics Syllabus in detail. Candidates may refer to the pattern of UPSC Mains in the linked article.

UPSC Maths Optional Syllabus For Paper I:

Given below is the Maths Optional syllabus for Paper I of the civil services exam: PAPER – I

(1) Linear Algebra: Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension; linear transformations, rank and nullity, matrix of a linear transformation. Algebra of Matrices; row and column reduction, echelon form, congruence and similarity; rank of a matrix; inverse of a matrix; solution of a system of linear equations; eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, symmetric, skew-symmetric, 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 integrals; 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 linear equations with constant coefficients, complementary function, particular integral and general solution. Second 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 & 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 equations. Application to geometry: curves in space, curvature and torsion; Serret-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s identities.Candidates preparing for UPSC 2022 can refer to the linked article.

UPSC Maths Optional Syllabus For Paper II:

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 the least upper bound property; Sequences, the 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 functions, 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 a system of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel(iterative) methods. Newton’s (forward and backward) interpolation, Lagrange’s interpolation. Numerical integration: Trapezoidal rule, Simpson’s rules, Gaussian quadrature formula. Numerical solution of ordinary differential equations: Euler and Runge 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: Generalized 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 motion 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. 

Also, for a detailed UPSC Syllabus for other optional subjects, as well as the Preliminary examination, candidates can visit the linked article.

IAS Mathematics Syllabus PDF:- 

Booklist for UPSC Maths Optional:

UPSC Maths Optional is a preferred choice of optional for engineering students owing to its objective syllabus. Here’re a few topics that will help you in preparation for UPSC Maths optional:

• Develop Conceptual Understanding: It is important to develop a crystal-clear conceptual understanding of a subject as concept-focused as Maths. Hence, first and foremost, develop a good understanding of each of the topics that is a part of UPSC Maths Optional.
• Revise Optimally: Secondly, it is important to keep revising whatever you’re studying to retain the information. Hence, allocate fixed time for revision for maximum retention of the information. Practice through the previous year's full-length papers and mock tests.
• Be systematic: Presentation matters a lot in UPSC answer-writing. Hence, crack the art of answer-writing for your optional by glancing through toppers’ answer scripts or getting yours evaluated by the mentors.
• Do not cram mathematics: Do not try to cram mathematics, rather focus on building the logical flow of the questions. It will help you in solving all the types of questions that are asked in the exam.
• Prepare a formula sheet: Maths is a subject of formulas and theorems. It is essential to learn them to solve the questions. Hence, maintain a separate formula sheet or notebook that is handy. Keep revising it from time to time to ensure that you do not forget any important formula.
• Avoid Silly Mistakes: Practice enough to ensure that you’re not doing any silly mistakes while solving the questions.

Given below are a few valuable tips from IAS topper Utsav Kaushal:

• Firstly, for scoring good marks in the mathematics optional you need to put a lot of hard work. You should solve as many problems as you can. Practice alone can help you here. Don’t shy away from hard work!
• Don’t copy solutions. It is important that you try and solve the problems on your own. Do not jump to the solution as and when you encounter a problem. You will be able to learn only when you practice problems yourself.
• Abstract algebra is a topic that is anathema for many candidates. Most of the candidates leave this topic and many are scared of it. Skipping this topic would be a mistake since, in previous years, many questions have been asked from this section. There is no way around it since you cannot afford to skip a chunk of the questions. So, all you can do is practice problems on this topic really well. Adequate practice will definitely help you understand it better and also improve your ability to solve problems.
• Mechanics and Fluid Dynamics is another topic that you have to do. A lot of questions are asked in the compulsory part.
• Dynamics & Statics: This topic is relatively easy and many candidates defer studying it until they have but a few days left for the main exam, also because they would have studied it in their XI and XII standards. Kaushal’s advice to students is that they do this section initially. Leaving it totally is unwise since you need practice in mathematics no matter how easy a topic seems.
• There are some questions in maths which are in parts. You will be asked to prove a certain theorem in the first part and based on that, you have to solve the second part. Many candidates, on finding that they cannot solve the first part of the question, altogether leave the question. Kaushal advises candidates to answer whichever part of the question they know. It is possible to get some marks on that question and in the UPSC exam, every mark counts.
• Coordinate Geometry: Here, questions are in a set pattern. Kaushal recommends candidates practice ‘Analytic Geometry’ by PN Chatterjee.
• In calculus, the theoretical part takes a lot of time and Kaushal advises candidates to take it up only if time permits.

How many take Mathematics optional?

As per the latest available data, in 2015, 258 candidates had taken mathematics as their optional subject out of which 31 cleared the exam giving the subject a success rate of 12% that year. The following table shows the number of candidates who had appeared with the maths optional.Table showing maths optional success rate