The UPSC Mathematics Optional Syllabus is a comprehensive and structured curriculum designed for candidates opting for Mathematics in the UPSC Civil Services Examination. Mathematics remains one of the most rewarding and logical optional subjects for UPSC aspirants, especially for candidates from engineering, physics, or mathematics backgrounds. The structured nature of the subject and its objective scoring potential make it an attractive choice.
This article provides a clear and updated overview of the UPSC Maths Optional Syllabus 2025, including both Paper 1 and Paper 2, along with preparation tips & recommended books. Also, download the UPSC Mathematics Optional Syllabus PDF!
The UPSC Mathematics Optional Syllabus is divided into two papers:
UPSC Maths Optional Syllabus |
|||
Paper |
Subject |
Marks |
Duration |
VI |
Mathematics Optional – Paper I |
250 |
3 Hours |
VII |
Mathematics Optional – Paper II |
250 |
3 Hours |
Total |
500 |
Each paper carries 250 marks, totaling 500 marks for the mathematics syllabus for UPSC.
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Paper 1 of UPSC Maths syllabus focuses on foundational concepts in mathematics that build analytical thinking and problem-solving skills.
UPSC Maths Optional Syllabus 2025– Paper I |
|
Topic |
Description |
Section A |
|
Linear Algebra |
Vector spaces over ℝ and ℂ, subspaces, basis, dimension, linear transformations, matrices, rank, nullity, eigenvalues, Cayley-Hamilton theorem, types of matrices (Hermitian, orthogonal, etc.). |
Calculus |
Limits, continuity, differentiability, Taylor’s theorem, maxima/minima, curve tracing, functions of several variables, Jacobians, Lagrange multipliers, Riemann integration, improper integrals, double/triple integrals. |
Analytic Geometry |
3D Cartesian and polar coordinates, conics and quadrics in 3D, canonical forms, planes, lines, spheres, ellipsoids, paraboloids, hyperboloids. |
Section B |
|
Ordinary Differential Equations (ODE) |
First and second-order equations, integrating factor, Clairaut’s equation, Euler-Cauchy equations, Laplace transforms, applications to initial value problems. |
Vector Analysis |
Gradient, divergence, curl, vector identities, line and surface integrals, Green’s, Gauss’ and Stokes’ theorems, curvature and torsion, Serret-Frenet formulas. |
Dynamics and Statics |
Rectilinear and projectile motion, conservation laws, virtual work, friction, equilibrium in 3D, Kepler’s laws, constrained motion. |
Paper 2 delves into more advanced topics including algebraic structures, real and complex analysis, programming, and fluid dynamics.
UPSC Maths Optional Syllabus 2025– Paper II |
|
Topic |
Description |
Section A |
|
Modern Algebra |
Groups, subgroups, cosets, Lagrange’s theorem, rings, fields, homomorphisms, isomorphisms, Euclidean domains, PID, UFDs. |
Real Analysis |
Real number system, sequences and series, convergence, continuity, Riemann integrals, uniform convergence, functions of several variables. |
Complex Analysis |
Analytic functions, Cauchy-Riemann equations, Cauchy’s theorem, Taylor/Laurent series, residues, contour integration. |
Linear Programming |
Graphical and simplex methods, duality, transportation and assignment problems. |
Section B |
|
Partial Differential Equations (PDE) |
Formulation of PDEs, first-order PDEs, Cauchy’s method, canonical forms, solutions of wave, heat, and Laplace equations. |
Numerical Analysis & Computer Programming |
Methods for solving equations, interpolation, numerical integration, Euler and Runge-Kutta methods, basic programming logic, binary/octal/hex systems, Boolean algebra, logic gates. |
Mechanics & Fluid Dynamics |
Generalized coordinates, Lagrange’s and Hamilton’s equations, moment of inertia, Euler’s equation, potential flow, streamlines, Navier-Stokes equations. |
To effectively prepare for the UPSC Maths Optional Syllabus, consider the following resources:
Click here to know more on the UPSC Maths Optional Books!
Opting for Mathematics as an optional subject in the UPSC examination offers several advantages:
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