UPSC Maths Syllabus 2022 - Mathematics Optional Syllabus for Paper 1 & 2, Preparation Tips

UPSC Maths Syllabus 2022 - Mathematics Optional Syllabus for Paper 1 & 2, Preparation Tips

UPSC Maths Syllabus 2022: The Union Public Service Commission (UPSC) releases the notification at and mentions the UPSC 2022 syllabus in it. Candidates planning to appear in the UPSC IAS exam and opting for maths as their optional subject, must know the UPSC Maths syllabus before they start their IAS exam preparing.

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Maths is one of the IAS 2022 optional subjects with 2 papers (Paper 1 and Paper 2). The syllabus of UPSC Maths 2022 consists of topics and sections on which the question paper is based. The difficulty level of UPSC Maths syllabus 2022 is of graduation level. UPSC 2022 Maths syllabus comprises Algebra, Real Analysis, Complex Analysis, Mechanics of Rigid Bodies, Mechanics of Deformable Bodies and much more. In this article, we have shared the detailed syllabus of UPSC Maths 2022 to help IAS 2022 aspirants for preparation.

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UPSC Maths syllabus 2022

Both the papers of the IAS Mathematics Optional paper are of 250 marks each. The UPSC IAS 2022 mains has 9 papers - GS 1, GS 2, GS 3, GS 4, Ethics and Integrity, General English, Regional subject and two optional papers.

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Free download: Complete UPSC IAS Prelims Mock Test-1 (100 Questions, Ans. & Explanation

Free download: Complete UPSC IAS Prelims Mock Test-2 (100 Questions, Ans. & Explanation)

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UPSC Maths syllabus 2022 - For Paper 1

From the following table, candidates can check the UPSC maths optional syllabus for paper 1.

UPSC Maths Syllabus Paper 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 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


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.

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.

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.

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.

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.

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UPSC Maths Syllabus 2022 - Paper 2

From the following table, we have shared the UPSC 2022 Maths syllabus for paper 2. Candidates can check the IAS maths optional syllabus for paper 2 here in detail.

UPSC Maths syllabus - Paper 2


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.

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.

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.

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.

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.

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-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. Algorithms and flow charts for solving numerical analysis problems. 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.

Mechanics and Fluid Dynamics

Moment of inertia; Motion of rigid bodies in two dimensions., Generalized coordinates; D’ Alembert’s principle and Lagrange’s equations; Hamilton equations; Equation of continuity; Euler’s equation of motion for inviscid flow; Stream-lines, path of a particle; Potential flow; Sources and sinks, Two-dimensional and axisymmetric motion; vortex motion; Navier-Stokes equation for a viscous fluid.

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UPSC IAS Syllabus 2022

Along with knowing the syllabus of UPSC Maths 2022, candidates must also know about the overall syllabus of the exam. UPSC mentions the syllabus in its official IAS notification. Candidates preparing for the exam must go through the syllabus to know the sections, subjects and topics on which the question paper of IAS will be based. The IAS syllabus is divided into two parts - Prelims and Mains and optional subjects as well for the mains part. In this article we have shared the UPSC IAS maths optional syllabus. There are a total of 26 optional subjects in the Civil services exam 2022.

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UPSC IAS Exam Pattern 2022

Candidates must know the IAS exam pattern along with the UPSC Maths syllabus 2022, for a sorted preparation strategy of the IAS exam 2022. UPSC mentions the UPSC IAS exam pattern in the notification. The UPSC exam pattern consists of the number of questions, stages, time duration and marking scheme of the exam. As per the exam pattern, there are 3 stages - preliminary, mains and personality test/interview. The prelims consists of 2 papers which are MCQ-based, the mains consists of 9 papers in descriptive mode. There is negative marking in the preliminary stage of the UPSC IAS exam. Candidates qualifying each stage will be eligible to appear in the next stage.

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UPSC IAS Maths Preparation Tips 2022

Candidates preparing for the IAS exam can refer to the following pointers on how to approach their preparations for IAS maths exam.

  • Refer to the detailed syllabus of UPSC Maths 2022 and check the subjects.

  • Pick the subjects from IAS Maths syllabus which are tough and keep them aside.

  • Enroll for subject specific classes and try to get your concepts clear for the tougher subjects.

  • Start solving the UPSC IAS maths optional question papers. The question papers can be downloaded from the official website of UPSC.

  • Keep analysing your performance from time-to-time to check the progress of the preparations.

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UPSC IAS Maths Books 2022 - For Paper 1 and 2

Apart from the UPSC 2022 Maths syllabus, candidates must also get to know about the best IAS books through which they can prepare for the Maths optional subject efficiently.

UPSC Maths Books for Paper 1 (topic-wise)


Name of Books


Analytic Geometry

Analytical Solid Geometry

Shanti Narayan and P K Mittal

Solid Geometry

P N Chatterjee


Mathematical Analysis

S C Malik and Savita Arora

Elements Of Real Analysis

Shanti Narayan and M D Raisinghania

Linear Algebra

Schaum Series

Seymour Lipschutz

Linear Algebra

Hoffman and Kunze

Dynamics And Statics

Me Statics & Dynamics


Ordinary Differential Equations (Ode)

Ordinary And Partial Differential Equations

M D Raisinghania

Vector Analysis

Schaum Series

Murray R. Spiegel

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Topic-wise UPSC Maths Books for Paper 2


Name of the Books

Author/ Publisher

Complex Analysis

Complex Variable (Schaum's Outlines

Spiegel, Lipschitz, Schiller, Spellman

Real Analysis

Elements Of Real Analysis

Shanti Narayan and M D Raisinghania


Contemporary Abstract Algebra

Joseph Gallian

Partial Differential Equations

Advanced Differential Equations

M D Raisinghania

Linear Programming

Linear Programming And Game Theory

Lakshmishree Bandopadhyay

Mechanics And Fluid Dynamics

Fluid Dynamics

Krishna Prakashan

Numerical Analysis And Computer Programming

Computer-Based Numerical And Statistical Techniques


Numerical Methods

Jain, Iyengar, and Jain

Digital-logic And Computer Design

M. Morris Mano

Other optional subjects syllabus of IAS exam 2022

UPSC Ethics Syllabus

UPSC Geology Syllabus

UPSC History Syllabus

UPSC Chemistry Syllabus

UPSC Economics Syllabus

UPSC Sociology Syllabus

UPSC Agriculture Syllabus

UPSC Psychology Syllabus

UPSC Political Science Syllabus

UPSC Public Administration Syllabus

UPSC Geography Syllabus

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