UPSC CSE Mathematics Optional Syllabus 2026

Mathematics as an optional subject in UPSC Civil Services Exam(CSE) is considered a technical optional subject. Mathematics is often considered as one of the good scoring optional subjects, majorly for the candidates having a technical educational background in their graduation. Many selected candidates in the past have scored highest marks by choosing Mathematics as their optional subject.

Mathematics optional paper has a weightage of 500 marks for both Paper-I and Paper-II. Opting Mathematics as an optional subject can actually make a difference in the All India Rank(AIR) of the candidate in the UPSC final result list.

UPSC CSE Mathematics Optional Syllabus Paper-I

Topic	Detailed Syllabus
(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, congruences 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
(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 equation Application to geometry: Curves in space, curvature and torsion, Serret–Frenet's formulae Gauss and Stokes' theorems, Green's identities

UPSC CSE Mathematics Optional Syllabus Paper-II

Topic	Detailed Syllabus
(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–Jordan (direct), Gauss–Seidel (iterative) methods Newton's (forward and backward) interpolation, Lagrange's interpolation Numerical integration: Trapezoidal rule, Simpson's rule, 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	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 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

Download CSE Mathematics Optional Previous Year Papers

Follow the steps to download the papers:

Open the NEXT IAS website
Go to the ‘Free resources’ tab and click on the ‘Previous Year Papers’ in the menu.
You will be redirected to the page where you can find all the previous year exam papers of the UPSC CSE.
You can also navigate using the dropdown box on the side, for example, navigating to prelims, mains or optional papers.
Search through the list of available papers to locate the Mathematics Optional Paper.
You can both view and download the papers.
Download the Mathematics optional subject PDF by clicking the download icon or using the ‘Save As’ option from the browser.
Once finished up with the download, save the document in a folder for future reference.

Download Mathematics Optional Previous Year Papers

Mathematics Optional Past Year Toppers

Shoham Teberiwal (AIR 77, CSE 2023) has the recent highest Mathematics optional score with 332/500 (Paper-I: 164 + Paper-II: 168).
Shravan Kumar Reddy (AIR 62, CSE 2024) scored 320, with highest Maths marks in that year.
Kanishk Kataria (AIR 1, 2019) scored 361 (170 + 191).
Anubhav Singh (AIR 8, 2017) scored 361/500 in total across papers.
Siddharth K Misra (AIR 34) scored 311, and Manan Agarwal (AIR 17) scored 310.
Shubhankar Pratyush Pathak (AIR 11, 2021) secured 300+ consistently.
Their emphasis is always on timed practice, Previous Year Questions, standard texts such as Kreyszig and Herstein, and error-free solutions.
Maths assures 270–300+ with a success rate of 6–8%, ideal for technical backgrounds with no GS overlap.

FAQs on Mathematics Optional Syllabus

What is UPSC Maths Optional structure?

Two papers (250 marks each, 3 hours). Paper-I: Linear Algebra, Calculus, Geometry, ODEs, Dynamics, Vectors. Paper-II: Algebra, Real/Complex Analysis, LP, PDEs, Numerics, Programming, Mechanics.

What are the Paper-I topics of Mathematics optional?

Vector spaces, eigenvalues, multivariable calculus, conics/3D geometry, first/higher-order ODEs (Laplace transforms), Newtonian mechanics, grad/div/curl operations.

What are the key differences in Paper-II of mathematics optional?

Advanced: Groups/rings/fields, metric spaces/Lebesgue, residues/contours, simplex method, wave/heat equations, Runge-Kutta, binary logic, fluid dynamics.

Is Mathematics optional suitable for non-Maths candidates?

Mathematics optional is tough for beginners but best for engineers/Maths graduates. 6-8% success, 300+ marks possible with 4-5 months rigorous practice, zero error tolerance.