UPSC Maths Optional Syllabus 2025 – Download PDF for Mains Paper 1 & 2
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!
UPSC Maths Optional Syllabus
The UPSC Mathematics Optional Syllabus is divided into two papers:
- Paper I: Focuses on foundational topics such as Linear Algebra, Calculus, Analytic Geometry, Ordinary Differential Equations, Dynamics & Statics, and Vector Analysis.
- Paper II: Delves into advanced topics including Algebra, Real Analysis, Complex Analysis, Linear Programming, Partial Differential Equations, Numerical Analysis & Computer Programming, and Mechanics & Fluid Dynamics.
|
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.
UPSC Beginners Program
Get UPSC Beginners Program - 60 Days Foundation Course SuperCoaching @ just
People also like
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. |
UPSC Maths Optional Syllabus 2025 – Paper II
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:
- Linear Algebra: “Linear Algebra” by K.C. Joshi
- Calculus: “Calculus” by M.D. Raisinghania
- Analytic Geometry: “Coordinate Geometry” by S.L. Loney
- Ordinary Differential Equations: “Ordinary and Partial Differential Equations” by M.D. Raisinghania
- Dynamics & Statics: “Engineering Mechanics” by R.K. Bansal
- Vector Analysis: “Vector Analysis” by Murray R. Spiegel
- Algebra: “Higher Algebra” by Hall and Knight
- Real Analysis: “Real Analysis” by S.C. Malik and Savita Arora
- Complex Analysis: “Complex Analysis” by S.C. Malik and Savita Arora
- Linear Programming: “Operations Research” by P.K. Gupta and D.S. Hira
- Partial Differential Equations: “Partial Differential Equations” by M.D. Raisinghania
- Numerical Analysis and Computer Programming: “Numerical Methods” by S.S. Sastry
Click here to know more on the UPSC Maths Optional Books!
How to Prepare for UPSC Mathematics Syllabus?
- Understand the Syllabus: Thoroughly go through the UPSC Mathematics Optional Syllabus to get acquainted with the topics and their weightage.
- Create a Study Plan: Develop a structured timetable that allocates sufficient time to each topic, ensuring comprehensive coverage.
- Practice Regularly: Consistent practice is key. Solve previous years' question papers and take mock tests to improve problem-solving speed and accuracy.
- Revise Frequently: Regular revision helps reinforce concepts and improves retention. Maintain a formula sheet for quick reference.
- Join a Test Series: Enrolling in a test series can provide valuable feedback and simulate the actual exam environment.
Why Choose Mathematics as an Optional Subject?
Opting for Mathematics as an optional subject in the UPSC examination offers several advantages:
- Scoring Potential: The objective nature of the subject allows for precise answers, leading to higher scores.
- Logical Structure: The well-defined syllabus and logical progression make it easier to grasp concepts.
- Overlap with General Studies: Topics like statistics and data interpretation have relevance in the General Studies papers.
- Less Subjectivity: Mathematics answers are less prone to subjective interpretation, ensuring fair evaluation.
Ace your UPSC preparation by downloading the Testbook App now!
More Articles for IAS Preparation
- Ordinance Making Power Of Executives In India
- Hriday Scheme
- Types of Vedas UPSC Notes: Rig, Yajur, Sama, & Atharva Veda
- Hunter Commission
- Natural Vegetation In India
- Chronology Of Ancient To Modern History Of India
- Philosophy Optional Syllabus
- Economic Survey
- UPSC Sociology Syllabus
- Public Administration Syllabus for UPSC
