UPSC Mathematics Syllabus 2025, Download Paper 1 and 2 PDF

The Union Public Service Commission (UPSC) Civil Services Mains Examination includes Mathematics as one of two optional subjects (Paper I and Paper II). The UPSC Mathematics Optional papers carry 250 marks each, and a total of 500 marks for both papers. The IAS Exam Mains consists of nine papers. You can find the UPSC Mathematics Syllabus in this article. This article provides you with full information regarding the UPSC Mathematics syllabus that’s required to prepare for the UPSC Syllabus 2025.

UPSC Mathematics Optional Syllabus

The UPSC Mathematics Syllabus is static in nature, and candidates who have studied Math during their undergraduate studies will easily cover the topics by brushing up on the basics. Because arithmetic is unrelated to current events, applicants just need to review the theory and formulas. Furthermore, the questions in the Mathematics Optional Syllabus have a definitive answer, thus applicants can easily obtain high scores. The topics included in the UPSC Mathematics Optional Syllabus are logic-based, providing candidates with a scoring opportunity. Other than formulas and theorems, aspirants do not need to memorise much information to cover the UPSC Mathematics Optional Syllabus. Applicants with great mathematical aptitude may take maths as an optional subject for their UPSC exam.

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UPSC Mathematics Syllabus 2025

In UPSC Mains, the UPSC Mathematics Optional Syllabus is divided into two papers, Paper 1 and Paper 2. Papers 1 and 2 each carry 250 marks, for a total of 500 marks. You must first qualify for the UPSC Prelims before proceeding to the Mains Examination.

UPSC Mathematics Paper 1 Syllabus 2025

UPSC Mathematics Syllabus Paper 1 includes the following topics Linear Algebra, Calculus, Analytic Geometry, Ordinary, Differential Equations, Dynamics and Statics, and Vector Analysis. The detailed syllabus is given below

(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’s 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, the 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 the first order and first degree, integrating factor; Orthogonal trajectory; Equations of first order but not of the first degree, Clairaut’s equation, singular solution. Second and higher-order linear equations with constant coefficients, complementary functions, particular integral and general solutions. Second-order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using a 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.

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UPSC Mathematics Paper 2 Syllabus

UPSC Mathematics Syllabus Paper 2 includes the following topics Algebra, Real Analysis, Complex Analysis, Linear Programming, Partial Differential Equations, Numerical Analysis, Computer Programming, Mechanics and Fluid, and Dynamics. The detailed syllabus is given below

(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, 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 backwards) interpolation, and Lagrange’s interpolation. Numerical integration: Trapezoidal rule, Simpson’s rules, Gaussian quadrature formula. Numerical solution of ordinary differential equations: Euler 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:

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, the path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.

This article discusses the complete UPSC Mathematics syllabus for UPSC/IAS. Candidates can download the PDF in this article. For more details related to UPSC Examination; students can visit the official website of StudyIQ UPSC Online Coaching.

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