| Section | Topics | Details |
| Section A | Linear Algebra | Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions; 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. |
| 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, surfaces, and volumes. | |
| Analytic Geometry | Cartesian and polar coordinates in 3D; 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 (one and two sheets) and their properties. | |
| Section B | Ordinary Differential Equations | Formulation of differential equations; First-order and first-degree equations, integrating factor; Orthogonal trajectories; Equations of first order but not first degree, Clairaut’s equation, singular solution; Second and higher-order linear equations with constant coefficients – complementary function, particular integral, general solution; Second-order equations with variable coefficients – Euler-Cauchy equation; Method of variation of parameters when one solution is known; Laplace and inverse Laplace transforms, properties, Laplace transforms of elementary functions; Application to initial value problems of second-order linear equations with constant coefficients. |
| Vector Analysis | Scalar and vector fields; Differentiation of vector field of a scalar variable; Gradient, divergence, curl in Cartesian and cylindrical coordinates; Higher-order derivatives; Vector identities and equations; Applications to geometry – curves in space, curvature, torsion, Serret-Frenet’s formulae; Green’s theorem, Gauss’ theorem, and Stokes’ theorem. | |
| Dynamics and 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. |
The UPSC Maths Optional remains one of the most structured and static optional subjects for the Civil Services Examination. Mathematics, chosen by aspirants with a strong numerical background and logical aptitude, offers clear-cut solutions and high-scoring potential. Unlike other subjects that change with evolving current affairs, the syllabus of mathematics remains stable, making it a reliable choice for many candidates.
In this article, we will explore the detailed UPSC Maths Optional Syllabus 2026 for both Mains Paper 1 and Paper 2, along with preparation strategies.
Why Choose Mathematics as an Optional?
Many aspirants often wonder why they should go for Mathematics in the UPSC examination. The answer lies in the unique quality of the UPSC Maths optional syllabus 2025, which makes it one of the most reliable and rewarding choices.
- First, you will get a clear and structured syllabus, as the UPSC Maths optional syllabus is straightforward and leaves no room for confusion.
- Second, you can enjoy the benefit of objective evaluation, since answers in the UPSC Maths optional syllabus are based on formulas, theorems, and logical steps.
- Third, you can score higher marks with accuracy, because the UPSC Maths optional syllabus provides step-by-step marking.
- Moreover, it will strengthen your analytical and problem-solving skills, as the UPSC Maths optional syllabus trains you to think logically and systematically.
- Lastly, you will face less subjectivity, as the UPSC Maths optional syllabus relies on definite answers rather than interpretations.
If you’re exploring other options, you can also check our detailed guide on UPSC Optional Subjects to compare subjects before finalizing your choice.
What Is There in the UPSC Maths Optional Syllabus?
The UPSC Maths optional syllabus stands out because of its clarity, precision, and scoring potential. Unlike many humanities subjects, it relies on formulas, theorems, and logical reasoning, making evaluation more objective. In essence, the syllabus is systematic, covers a wide range of mathematical concepts, and demands consistent practice. Many aspirants prefer Mathematics because it offers accuracy, a defined structure, and high-scoring opportunities. Below, we have highlighted the key reasons:
1. Clear and Well-Defined Syllabus
Firstly, the UPSC Maths optional syllabus is fixed and does not change frequently. Topics such as linear algebra, calculus, differential equations, dynamics, and statistics are straightforward. Therefore, aspirants can plan their preparation with clarity and avoid surprises during the exam.
2. High Scoring Potential with Objective Answers
Secondly, answers in the UPSC Maths optional syllabus are based on proofs, derivations, and calculations. With step-by-step solutions, candidates can secure marks for every correct step. This objectivity makes Mathematics one of the most rewarding optional subjects.
3. Strong Analytical and Logical Training
Furthermore, the UPSC Maths optional syllabus builds analytical thinking and logical problem-solving. Practicing mathematical problems sharpens accuracy and time management, which also helps in CSAT and other analytical portions of the exam.
4. Wide Availability of Resources
Moreover, there is a plus point that books, materials, and previous year papers for the UPSC Maths optional syllabus are easily available. Toppers’ notes and solved examples further support aspirants in developing a strong command over the subject.
Overview of UPSC Maths Optional Syllabus 2026
The UPSC Maths Optional Syllabus 2026 includes fundamental and advanced topics such as linear algebra, calculus, analytical geometry, differential equations, real analysis, complex analysis, vector analysis, and mechanics. Hence, aspirants aiming for this subject should possess a genuine passion for problem-solving and conceptual clarity.
With two papers carrying 250 marks each, with a total of 500 marks, the syllabus demands consistency and practice. Yet, its objective nature allows aspirants to score well with the right approach.
| Overview of UPSC Maths Syllabus 2025 | |||
| S. No. | UPSC IAS Mains Papers | Subject | Marks |
| 1 | Paper VI | Optional Subject Paper-I | 250 |
| 2 | Paper VII | Optional Subject Paper-II | 250 |
| TOTAL | 500 | ||
| Time Duration | 3 hours | ||
UPSC Maths Optional Syllabus 2025 – Paper 1
Paper 1 carries a total of 250 marks and is divided into Section A and Section B.
UPSC Maths Optional Syllabus 2025 – Paper 2
Paper 2, like Paper 1, carries 250 marks and is also divided into Section A and Section B.
| Section | Topics | Details |
| Section A | Modern 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, 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 convergence – absolute and conditional, real and complex terms, rearrangement of series; Continuity and uniform continuity of functions, properties of continuous functions on compact sets; Riemann integral and improper integrals, fundamental theorems of integral calculus; Uniform convergence, continuity, differentiability, and integrability of sequences and series of functions; Partial derivatives of functions of several (two or three) variables, maxima and minima. | |
| 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. | |
| Linear Programming | Linear programming problems, basic solution, basic feasible solution, optimal solution; Graphical method and simplex method of solutions; Duality theory; Transportation and assignment problems. | |
| Section B | Partial Differential Equations (PDEs) | Family of surfaces in 3D and formulation of partial differential equations; Solution of quasilinear PDEs of first order – Cauchy’s method of characteristics; Linear PDEs of second order with constant coefficients, canonical form; Equation of vibrating string, heat equation, Laplace equation, and their solutions. |
| Numerical Analysis & Computer Programming | Numerical Methods: Solution of algebraic and transcendental equations of one variable – Bisection, Regula-Falsi, Newton-Raphson methods; Solution of system of linear equations – Gaussian elimination, Gauss-Jordan (direct), Gauss-Seidel (iterative); Newton’s interpolation (forward & backward), Lagrange’s interpolation; Numerical integration – Trapezoidal rule, Simpson’s rules, Gaussian quadrature; Numerical solution of ODEs – Euler and Runge-Kutta methods. Computer Programming: Binary system, arithmetic & logical operations on numbers; Octal & Hexadecimal systems, conversion to/from decimal; Algebra of binary numbers; Elements of computer systems, memory concepts; Basic logic gates & truth tables, Boolean algebra, normal forms; Representation of integers & reals (unsigned, signed, double precision, long integers); Algorithms & flowcharts for solving numerical analysis problems. | |
| Mechanics and Fluid Dynamics | Mechanics: Generalised coordinates, D’Alembert’s principle, Lagrange’s equations, Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions.Fluid Dynamics: Equation of continuity; Euler’s equation of motion for inviscid flow; Streamlines, particle path; Potential flow; 2D and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid. |
Benefits of Choosing Mathematics as an Optional Subject
Selecting mathematics as an optional comes with clear advantages:
- High Scoring Potential: Firstly, questions are objective with definite solutions.
- Logical Approach: Second, it builds analytical and problem-solving skills.
- Overlap with Other Subjects: Second, the concepts help in economics, science, and technology-related topics.
- Stable Syllabus: Also, it is independent from current events, ensuring predictable preparation.
- Structured Problem Solving: Lastly, it will help you in developing logical reasoning and clarity.
Preparation Strategy for UPSC Maths Optional Syllabus 2025
To excel in the UPSC Maths Optional Syllabus 2025, aspirants must follow a systematic strategy:
- Strengthen Fundamentals: Develop clarity in core topics like calculus, algebra, and differential equations.
- Practice Regularly: Solve previous year question papers and mock tests.
- Organise Topics: Study real analysis alongside calculus, and ODEs alongside PDEs for conceptual linkage.
- Maintain Formula Sheets: Revise important formulas and theorems frequently.
- Answer Presentation: Practice writing stepwise solutions to score maximum marks.
- Revise Consistently: Allocate a fixed time for revision before the exam.
Conclusion
The UPSC Maths Optional Syllabus 2026 provides a well-structured and predictable roadmap for aspirants with strong mathematical skills. With proper preparation, consistent practice, and strategic revision, mathematics can become one of the most rewarding optional subjects. Its static nature and objective solutions make it a popular choice among engineering graduates and candidates with a logical bent of mind.
Are you getting ready for the UPSC 2025? Enroll in our SPM IAS Academy’s UPSC foundation batches to improve your readiness. Enroll right away!
Previous Years’ Question Papers
| 2025 Maths Optional Paper | |
| 2025 Maths Optional Paper- 1 | 2025 Maths Optional Paper- 2 |
| 2024 Maths Optional Paper | |
| 2024 Maths Optional Paper- 1 | 2024 Maths Optional Paper- 2 |
| 2023 Maths Optional Paper | |
| 2023 Maths Optional Paper- 1 | 2023 Maths Optional Paper- 2 |
| 2022 Maths Optional Paper | |
| 2022 Maths Optional Paper- 1 | 2022 Maths Optional Paper- 2 |
| 2021 Maths Optional Paper | |
| 2021 Maths Optional Paper- 1 | 2021 Maths Optional Paper- 2 |
| 2020 Maths Optional Paper | |
| 2020 Maths Optional Paper- 1 | 2020 Maths Optional Paper- 2 |
| 2019 Maths Optional Paper | |
| 2019 Maths Optional Paper- 1 | 2019 Maths Optional Paper- 2 |
| 2018 Maths Optional Paper | |
| 2018 Maths Optional Paper- 1 | 2018 Maths Optional Paper- 2 |
| 2017 Maths Optional Paper | |
| 2017 Maths Optional Paper- 1 | 2017 Maths Optional Paper- 2 |
| 2016 Maths Optional Paper | |
| 2016 Maths Optional Paper- 1 | 2016 Maths Optional Paper- 2 |
FAQs on UPSC Maths Optional Syllabus 2025
The syllabus covers linear algebra, calculus, differential equations, vector analysis, real analysis, complex analysis, modern algebra, numerical methods, and mechanics.
There are two papers – Paper 1 and Paper 2 – each carrying 250 marks, making a total of 500 marks.
Yes, it is suitable if candidates have a strong mathematical foundation, though engineering graduates generally find it more comfortable.
Mathematics is considered scoring because answers are objective, less subjective to interpretation, and can yield full marks with correct solutions.
